Paper Title: The impact of the phase transition on Quark-Gluon Plasma with an extremely strong magnetic field in holographic QCD.


APS



caoxm@jnu.edu.cn
 Department of Physics and Siyuan Laboratory, Jinan University, Guangzhou 510632, China

tliuhui@jnu.edu.cn
 Department of Physics and Siyuan Laboratory, Jinan University, Guangzhou 510632, China

§ ABSTRACT 
We investigate the phase transition within an extremely strong magnetic background field, employing a holographic Quantum Chromodynamics (QCD) model with a focus on entropy and pressure properties. At relatively modest magnetic field strengths, our study discerns a crossover transition between the normal phase and the Quark-Gluon Plasma (QGP) phase as the temperature rises. In contrast, under the influence of an extremely strong magnetic field, a first-order phase transition is observed. A critical point is identified at (eB_c, T_c) ≈ (2.8623  GeV^2, 0.1191  GeV), which corresponds to a second-order phase transition. This phase structure is found to be in qualitative agreement with lattice simulation predictions reported in [Phys. Rev. D 105, 034511 (2022)].
Furthermore, we explore the impact of the magnetic field on the jet quenching parameter across various phases. At zero magnetic field (eB = 0 ), the normalized jet quenching parameter q̂ / T^3 exhibits a monotonic increase with temperature. However, in the presence of a magnetic background field, the normalized jet quenching parameters not only display directional anisotropy but also experience a universal enhancement, particularly in the vicinity of the critical temperature region. This observation suggests that the jet quenching parameter could potentially act as an indicator of phase transitions.


The impact of the phase transition on  Quark-Gluon Plasma with an extremely strong magnetic field in holographic QCD
    Hui Liu
    August 5, 2024
====================================================================================================================






§ INTRODUCTION

The investigation of the quantum chromodynamics (QCD) properties under a magnetic field is one of the most important fields in high energy physics <cit.>. The background magnetic field could be generated and widely exist in non-central heavy ion collisions <cit.>, in the early cosmology <cit.>, and in magnetar <cit.>. Many important and interesting phenomena are found in this area, such as the (inverse) magnetic catalyses <cit.> and the chiral magnetic effects <cit.>.

The chiral and confining properties of QCD are directly influenced by the presence of a magnetic field. An extremely strong magnetic field can even change the order of the phase transition, which could be important for our understanding of the early universe <cit.>. One of the most powerful methods for studying QCD in a finite magnetic background field is the lattice simulation.  Lattice simulations <cit.>, which employed standard staggered fermions, demonstrated a rise of transition temperature in chiral condensates with Polyakov loops. However, these findings are in contrast with the results of lattice simulations  <cit.> that incorporated improved staggered fermions.
 The discrepancy is primarily attributed to lattice cutoff effects <cit.>. Moreover, lattice simulations have demonstrated that a first-order transition exists when eB ≳ 11 m_π^2 with N_f=3 <cit.>. Extrapolation of the lattice results predicts a critical point beyond eB=3.25   GeV^2 with N_f=1+1+1 <cit.>. In more recent investigations, a lattice simulation for N_f = 2 + 1 <cit.>, which employs stout-improved staggered fermions and try to control the cut-off effect, has indicated that the critical temperature decreases as the magnetic field strengthens. Despite the absence of technical issues such as the sign problem at finite chemical density, lattice simulations remain constrained by limitations in computational power. They observed a crossover transition at eB=4  GeV^2 and a first order phase transition at eB=9  GeV^2, indicating the presence of a critical end point between these values. 

The phase structure of QCD under magnetic fields has also been extensively investigated using low energy effective theories and models, such as the bag model <cit.>, chiral perturbation theory (χPT) <cit.>, the Nambu-Jona- Lasinio (NJL) model <cit.> and the quark-meson (QM) model <cit.>. However, a unified consensus on the phase diagram at finite magnetic fields and temperatures has not yet been reached across these various studies. The phase diagram in an extremely strong magnetic field remains an open topic. Therefore, it is necessary to conduct further research in this domain with a variety of numerical and theoretical methods.

In addition to the traditional methods, holographic duality is another powerful approach for studying strongly coupled systems <cit.>. Given that the quark-gluon-plasma (QGP)  exists as a strongly coupled state <cit.>, holographic duality has become an important tool in the QCD investigations <cit.>. Regarding investigations involving magnetic backgrounds, various studies have employed different  constructive models, including bottom-up holographic models  in the Veneziano limit <cit.>, magnetized Einstein-Maxwell model <cit.> and Magnetized Einstein-Maxwell  model <cit.>. An abundance of research has been dedicated to studying the Quantum Chromodynamics (QCD) phase diagram within the finite temperature and chemical potential plane <cit.>. However, in this study, we employ the magnetized Einstein-Maxwell-Dilaton (EMD) model to explore the phase transition in the context of an extended magnetic field region, especially the extremely strong magnetic field. In the previous investigations, they utilized the magnetized EMD model <cit.> and have yielded some findings. They have predominantly concentrated on crossover transitions and have not yet provided an exhaustive examination of the domain characterized by extremely strong magnetic fields. Building upon this foundation, we have developed numerical code to extend the parameter space  of the magnetic field and temperature.
Then, we study the extremely strong magnetic field region and observe first-order phase transitions. 

Taking into account the first-order transitions, we discuss their influence on the properties of QGP.  Among various QGP signals in heavy-ion collisions, jet quenching is a crucial one <cit.>. The holographic calculation of the jet quenching parameter q̂ has been previously provided  <cit.> and extensively tested in some holographic models <cit.>. In this work, we focus on the behaviors of anisotropic jet quenching parameters under both crossover and first order phase transitions.
 


This paper is organized as follows. In section II, we introduce the construction of EMD model and  how it realizes duality to QCD. In section III, we numerically obtain the mapping between the input parameters (ϕ,ℬ) and the physical parameters (T,eB). We then study the entropy variation with temperature at fixed magnetic field, determine the phase boundary of the first-order transition and identify the critical end point in T-eB plane. In section IV, we investigate the effect of the first-order transition to the jet quenching parameters. Section V is a brief conclusion.



§ THE EINSTEIN-MAXWELL-DILATION MODEL

In this section, we provide a brief introduction to the EMD model with a finite magnetic field B≠ 0 and zero chemical potential μ_B=0, as constructed in Refs. <cit.>. This bottom-up approach aims to mimic the QCD system by fitting lattice results from the holographic perspective, a method originally proposed by Gubser and Nellore in Ref. <cit.>.


In the EMD model, a scalar field ϕ, i.e., a dilaton field, and an abelian gauge field A_μ are included.  The five dimension action is

    S=1/2 κ^2∫ d^5 x√(-g)[R-1/2(∂_μϕ)^2-V(ϕ)-f(ϕ)/4F_μν^2],

with

    V(ϕ)   =   -v_0  cosh(v'_0 ϕ)+ v_2 ϕ^2+v_4 ϕ^4+v_6 ϕ^6,
    κ^2   =   8π G_5,
    
        f(ϕ)   =   c_3  sech(c_0+c_1 ϕ+c_2ϕ^2),

and the field strength tensor F_μν=∂_μ A_ν-∂_ν A_μ. The dilaton potential V(ϕ)  and gravitational constant κ^2≡ 8π G_5 is determined by fitting the lattice data of equation of state at B=0 in Ref. <cit.> [For the fitting of dilaton potential, the Breitenlohner-Freedman bound should be satisfied <cit.>.]. The coupling between the gauge field and dilaton f(ϕ) is determined by fitting the lattice results of the magnetic susceptibility at B=0 in Ref. <cit.>. The values of these parameters are summarised in Table <ref>. For the asymptotically AdS_5 geometries, the ultraviolet asymptotically expansion should satisfy V(ϕ→ 0)≈ -12/L^2+m^2ϕ^2/2 with L the asymptotic AdS_5 radius, in which the mass m and the scaling dimension Δ of the dual operator in the gauge theory satisfy the relation

    m^2 L^2=-νΔ=(4-Δ)Δ.

Here,  the radius is always set to be unity, i.e., L=1. Thus, one can derive Δ= 2.73294.




In this EMD model, the magnetic field ℬ is introduced through the abelian gauge field at the boundary and set to be a constant in the z direction. Therefore, one can take the metric ansatz as

    ds^2   =   e^2a(r)[-h(r)dt^2+dz^2]+e^2c(r)(dx^2+dy^2)
          +e^2b(r)dr^2/h(r),
    ϕ   =   ϕ(r),
    
        A   =   A_μ dx^μ=ℬ xdy⇒ F=dA= ℬdx∧ dy,

with five dimension coordinates (t,x,y,z,r).  As a result of the magnetic field, the rotation symmetry is broken from SO(3) to SO(2). Along the holographic direction, the boundary of the asymptotically AdS_5 is at r→∞ and the horizon of the black hole at r_H, which defined by h(r_H)=0. Under this metric in (<ref>), the equations of motion (EOMs) can be derived as,



    ϕ”+ (2 a'+2 c'-b'+h'/h )ϕ'-e^2b/h(∂ V(ϕ)/∂ϕ +ℬ^2 e^-4c/2∂ f(ϕ)/∂ϕ)   =   0,
    
        a”+(14/3c'-b'+4h'/3h)a'+8/3a'^2+2/3c'^2+2h'/3hc'+2e^2b/3hV(ϕ)-1/6ϕ'^2   =   0,
    
        c”-(10/3a'+b'+h'/3h )c'+2/3c'^2-4/3a'^2-2h'/3ha'-e^2b/3hV(ϕ)+1/3ϕ'^2   =   0
    
        h”+(2a'+2c'-b')h'   =   0,

with a constraint equation,

    a'^2+c'^2-1/4ϕ'^2+(a'/2+c')h'/h+4a'c'+e^2b/2h(V(ϕ)+ℬ^2 e^-4c/2f(ϕ))   =   0,



where the prime denotes the derivative with respect to the fifth dimension r.

To investigate the properties of the EMD model, one should solve the EOMs in Eqs. (<ref>) and extract the thermaldynamic quantities. The first step involves obtaining the near-boundary and near-horizon asymptotic expansions for the bulk field ϕ(r) and a(r), c(r), h(r). We adopt the domain-wall gauge b(r)=0. Consequently, the ultraviolet asymptotic expansions can be derived as follows:

    
    a(r)   =   α(r)+⋯,
    
            c(r)   =   α(r)+c_0^ far-a_0^ far+⋯,
    
            h(r)   =   h_0^  far+h_4^  far e^-4α(r)+⋯,
    ϕ(r)   =   ϕ_Ae^-να(r)+ϕ_B e^-Δα(r)+⋯,

with 

    α(r)   =   a_0^ 
     far+r/√(h_0^ 
     far).


and ⋯ denote higher-order terms. The integration constants a_0^ far, c_0^ far, h_0^ far and ϕ_A are related to the thermodynamic quantities. Near the horizon r_H=0, the asymptotic expansions of the bulk fields are given by:

    Y(r)=∑_n=0^∞Y_n (r-r_H)^n,

where Y=a,c,h,ϕ. In the expansion coefficients, due to the freedom in rescaling properties, one can set r_H=a_0=c_0=0 and h_1=1. One can also assume h_0=0. After these settings and assumptions, only ϕ_0 and ℬ remain undetermined among the seven parameters in the near horizon expansions.  Once ϕ_0 and ℬ are given, the values X(ϵ) [Due to the singularity at horizon, a small deviation ϵ is used in the numerical strategy. We choose ϵ=10^-15 in this work.] and their first-order derivative values X'(ϵ) of the near horizon expansion are determined. One can numerically solve the EOMs in Eqs (<ref>)  by fitting the UV asymptotic expansions in Eq. (<ref>) with these numerical solutions to obtain the corresponding integration constants a_0^ far,c_0^ far, h_0^ far and ϕ_A. From our numerical tests, we find that choosing the maximum radial coordinate r_max=10 is sufficient for the near-boundary asymptotic expansions. Thus, the numerical solutions are all given in the interval r∈ [ϵ, r_max].

As previously shown, we can obtain the numerical results with ϕ_A≠ 0, which can be understood as renormalization group flows induced  by a slight deformation of a relevant operator. However, to accurately mimic QCD, ϕ_A must be maintained constant.  To accomplish this, it is necessary to perform a coordinate transformation from the numerical coordinates to the standard coordinates. In the standard coordinates, the metric and the UV asymptotic expansions are

    ds̃^2   =   e^2ã(r̃)[-h̃(r̃)dt̃^2+dz̃^2]+e^2c̃(r̃)(dx̃^2+dỹ^2)+dr̃^2/h̃(r̃),
    ϕ̃   =   ϕ̃(r̃),
    Ã   =   Ã_μ dx̃^μ=B̂x̃dỹ⇒F̃=dÃ=B̂dx̃∧ dỹ,

and 


    ã(r̃)   =   r̃+⋯,
    c̃(r̃)   =   r̃+⋯,
    h̃(r̃)   =   1+⋯,
    ϕ̃(r̃)   =   e^-νr̃+⋯.


with the boundary at r̃→∞ and the horizon at r̃=r̃_H. The numerical coordinates and the standard coordinates are equivalent and satisfy

    ds̃^2=ds^2,    ϕ̃(r̃)=ϕ(r),    B̂dx̃∧ dỹ=ℬ dx∧ dy.

Using the relation in Eq. (<ref>) and comparing the UV asymptotic expansions in numerical and standard coordinates in Eqs. (<ref>) and  (<ref>), one can derive


    r̃=r/√(h_0^ far)+a_0^ far-ln(ϕ_A^1/ν),
       t̃=ϕ_A^1/ν√(h_0^ far)t,
       x̃=ϕ_A^1/ν e^c_0^ far-a_0^ farx,
       ỹ=ϕ_A^1/ν e^c_0^ far-a_0^ farx,
       z̃=ϕ_A^1/νz,
    


    
    ã(r̃)=a(r)-ln (ϕ_A^1/ν),
       c̃(r̃)=c(r)-(c_0^ far-a_)^ far)-ln (ϕ_A^1/ν),
       h̃(r̃)=h(r)/h_0^ far,
       ϕ̃(r̃)=ϕ(r),


and

    B̂=e^2(a_0^ far-c_0^ far)/ϕ_A^2/νℬ.




Therefore, one can solve the EOMs in the  numerical coordinates and obtain the numerical solutions  of the QCD quantities in the standard coordinates through the relations in Eqs. (<ref>) and (<ref>). In the upper panel of Fig. <ref>, the curve represents the upper bound of the the input parameters (ϕ_0,ℬ).
From the relation in Eqs. (<ref>) and asymptotic expansions at horizon in Eqs. (<ref>), one can obtain the QCD temperature as

    T̂=.√(-g̃_t̃t̃^'g̃^'r̃r̃)/4π |_r̃=r̃_H=e^ã(r̃_H)/4π|h̃'(r̃_H)|=1/4πϕ_A^1/ν√(h_0^ far),




To mimic to the real QCD, the temperature corresponding to the minimum of the sound speed c_s^2 in the holographic results is compared to the lattice results. Based on this comparison, a scaling factor is determined,

    λ=T^ lattice_ min c_s^2/T̂_ min c_s^2≈ 1058.83  MeV.

Therefore, the QCD quantities X can be obtained by scaling X̂ with λ, following the relation X=X̂λ^p [MeV^p], where p is determined by dimensional analysis. For example T=T̂λ [ MeV] and B=B̂λ^2 [ MeV^2].
In the lower panel of Fig. <ref>, we show the numerical results for (T, eB), where each curve corresponds to a fixed ϕ_0 and a series of ℬ. Normally, small ϕ_0 maps to high T, large ϕ_0 maps to low T and small ℬ maps to small eB. As ℬ approaches its upper bound, a slight variation in ℬ induces a large change of eB. Noticing that the fixed ϕ_0 curves in  lower panel of Fig. <ref> are convex for large ϕ_0 ( i.e., low T) but concave for small ϕ_0 ( i.e., high T), these curves may exhibit overlaps at the moderate temperatures in the range of T=100∼ 150 MeV.







§ THE PHASE TRANSITION WITH FINITE MAGNETIC FIELD

In this section, we investigate the QCD properties in a magnetic background field. The strategy is as follows: first, we  draw a vertical line to intersect with the curves in the lower panel of Fig. <ref>, and then numerically map from the intersection points back to determine the corresponding input parameters (ϕ_0,ℬ). For example, as shown in Fig. <ref>, we plot three curves of the selected input parameters (ϕ_0,ℬ), which are corresponding to the magnetic background fields set at eB=0.1, 1 and 10 GeV^2, respectively. Finally, after determining several sets of the input parameters for fixed magnetic fields, we solve the EOMs in Eq. (<ref>) and obtain the corresponding values of the QCD quantities.




 §.§ The entropy and pressure

For the mapping between (ϕ_0,ℬ) and (T,B) shown in the lower panel of Fig. <ref>, there is a special region where a pair of (T,B) corresponds to different pairs of (ϕ_0,ℬ) [A similar result of the multiple correspondence, has been observed in Ref. <cit.>, in which the authors studied the critical point at finite temperature and chemical potential.]. Specifically, as eB increases from small to large, this special region begins at a particular point. This behavior could be a signal of the first-order phase transition. To confirm this conjecture, we  study the behaviors of the entropy density s and pressure P in this region in the following part.

Entropy is an extensive quantity, which can be extracted through the Bekenstein-Hawking relation as S=A/4G_N with A the area of the horizon. Thus, the entropy density can be obtained by dividing the volume V,

    s=S/V=2π/κ^2e^ã(r̃_H)+2c̃(r̃_H)=2π e^2(a_0^ far-c_0^ far)/κ^2ϕ_A^3/ν.


In  the upper panel of Fig. <ref>, we show the numerical results of the temperature dependence of the normalized entropy density s/T^3 for four different fixed magnetic fields, eB=0, 1, 2.8326, and 4 GeV^2.  In the absence of a magnetic background field, the normalized entropy density is small at low temperatures and exhibits a monotonic increase as the temperature rises.  At high temperatures, it behaves according to the relation s∼ T^3. This may indicate that the QCD system undergoes a crossover from the low-temperature normal phase to the high-temperature quark-gluon-plasma (QGP) phase. 
With a finite magnetic background field,  the normalized entropy density in both the low and high-temperature limits consistently approaches the values observed in the absence of a magnetic field. In the intermediate temperature region, there is a noticeable enhancement of the normalized entropy density, which becomes more pronounced with an increase in the magnetic field. When eB≳ 2.8326  GeV^2, as shown in the inset in the upper panel of Fig. <ref>, the normalized entropy density as a function of temperature exists multiple value region, signaling the onset of the first-order phase transition. When eB≤ 2.8326  GeV^2, we the specific heat C=T∂ s/∂ T with μ_B=0 shown in the lower panel of Fig. <ref> diverges at a certain temperature T_c≈ 0.1191   GeV. Below the critical magnetic field eB_c=2.8326  GeV^2, the susceptibility of entropy density with respect to the temperature is smooth and continuous, indicating a crossover transition. Therefore, the point (eB_c, T_c)=(2.8326  GeV^2, 0.1191   GeV) is a critical endpoint, corresponding to a second-order phase transition, where the first-order transition changes to a crossover.



To determine the phase boundary of the first-order transition, we study the behavior of the pressure P. At a fixed magnetic background field, the pressure can be approximately determined as

    P(T)≈∫_T_ low^T s(T)  dT,

where we select a low temperature T_ low as the lower boundary rather than absolute zero due to the impracticality of achieving zero temperature within this model. This selection introduces only a constant difference and does not affect the results of our analysis. For convenience, we choose T_ low=0.053  GeV in our calculations. In Fig. <ref>, we show the temperature dependence of the entropy density (S-shape) and pressure (Swallowtail shape) in the upper panels and lower panels, respectively. The corresponding turning points of the entropy and pressure are labeled with red cross symbols. The first-order phase transition temperature and the unstable region can be  determined  by employing the Maxwell construction, where the areas separated by the isotherm in the thermodynamic curves are equal.  In the lower left panel, the five-pointed star symbol determines the transition temperature T_t=0.114  GeV,  at which the low-temperature branch of the entropy density curve directly jumps to the high-temperature branch through the red vertical line of left upper panel, with the remaining part representing the unstable or nonphysical states. For any other magnetic background field eB>2.8326  GeV^2, we can also determine the transition temperature using this method. Finally, we show the phase diagram in the magnetic field-temperature plane in Fig. <ref>,  where the five-points star symbol represents the critical endpoint (CEP) and the black line represents the first-order boundary.  As the results demonstrate, the temperature at which the first-order phase transition occurs diminishes with an increase of the magnetic field.



§ JET QUENCHING PARAMETER

From the study of the phase transition in the temperature and magnetic field plane in the frame of holographic QCD, we find there are crossover, first-order and second-order phase transitions. It would be interesting to investigate the jet quenching parameter within these different phase transition backgrounds. This investigation of jet quenching parameter is essential for gaining insights into the QGP from both theoretical and experimental perspectives, as well as for understanding the phase transitions within the framework of QCD.  Although laboratory-generated magnetic fields may not reach extremely high values, the investigation for the extremely magnetic field region remains meaningful. On one hand, relevant phenomena can be compared to the high density case. On the other hand, extremely strong magnetic fields might have existed in the early universe making the study of first-order transition very important.









Within the framework of holographic QCD, the calculation of the jet quenching parameter was first obtained by Hong Liu, Krishna Rajagopal, and Urs Achim Wiedemann in Ref. <cit.>. Since then, many studies have been conducted under various conditions and using different holographic models. However, the jet quenching parameter in extremely strong magnetic fields remains unexplored. Our calculations suggest that the phase transition in this condition could be of the first-order,  thereby rendering the discussion of jet quenching parameters in this scenario both compelling and significant.

Due to the magnetic background field, the spatial SO(3) symmetry is broken down to SO(2). In the anisotropic medium, there are three different types of jet quenching parameters. First, the light parton moves along the direction of the magnetic field, with transverse momentum broadening perpendicular to the magnetic field direction. Second, the light parton moves perpendicular to the magnetic field, with transverse momentum broadening in the magnetic field direction. Third, the light parton moves perpendicular to the magnetic field direction, with transverse momentum broadening also perpendicular to the magnetic field direction.
Following the steps in Refs. <cit.>, one can derive the jet quenching parameters q̂ as [In the numerical process, the integration interval is chosen to be [ϵ,r_max].]


    q̂_p(k)=√(λ_t)/π(∫_r̃_H^∞ dr̃1/g̃_kk√(g̃_rr/g̃_tt+g̃_pp))^-1,



where p is the direction of the light parton moving and k is the direction of the transverse momentum broadening. λ_t is the 't Hooft coupling. Substituting the relation Eq. (<ref>) into the expression of the jet quenching parameter Eq. (<ref>), one can obtain the expressions for these normalized jet quenching parameters.


    
    q̂_||(⊥)/T^3= 64π^2 √(λ_t) h_0^ far/∫_r_H^r_maxd re^-2c(r)-a(r)+2(c_0^ far-a_0^ far)/√(h(r)[h_0^ far-h(r)])
    q̂_⊥(||)/T^3=32π^2 √(λ_t) h_0^ far/∫_r_H^r_maxd re^-2a(r)/√(h(r)[h_0^ fare^2c(r)-2(c_0^ far-a_0^ far)-h(r)e^2a(r)])
    q̂_⊥(⊥)/T^3=32π^2 √(λ_t) h_0^ far/∫_r_H^r_maxd re^-2c(r)+2(c_0^ far-a_0^ far)/√(h(r)[h_0^ fare^2c(r)-2(c_0^ far-a_0^ far)-h(r)e^2a(r)])




In Fig. <ref>, we show the results of q̂ without a magnetic field.
As shown by the red curve, we have the normalized jet quenching parameter q̂/T^3 as a function of the temperature. In our calculations, we choose the 't Hooft coupling λ_t=1/2.  We have found that the holographic outcomes are in agreement with the Monte Carlo simulations, which were conducted using an initial quark jet with an energy of  E=10 GeV, as well as the results from the deep inelastic scattering (DIS)  <cit.>. The holographic curve lies within the region of the Monte Carlo and the DIS results. The significant difference between the low and high temperatures indicates a crossover from the normal phase to the QGP phase.


In Fig. <ref>, we show the normalized jet quenching parameter q̂_||(⊥) as a function of temperature for different fixed magnetic background fields. Comparing the sub-figures, one can see that q̂_||(⊥) is generally enhanced by the magnetic field, favoring  the transverse momentum broadening. This enhancement is especially significant in the phase transition temperature region and tends to match the values of the zero magnetic field case in both the low and high-temperature limits. Furthermore, with relatively small magnetic fields, q̂_||(⊥)/T^3 remains a monotonic function of temperature.  However,  in the presence of an extremely strong magnetic field, for instance, eB=4 and 10  GeV^2, q̂_||(⊥)/T^3, there exists a region around the phase transition temperature where q̂_||(⊥)/T^3 takes on multiple values. This results in three different values of q̂_||(⊥)/T^3 for a single temperature, as indicated by the red dashed lines in Fig. <ref>, which is the typical characteristic of the first-order transition. At the critical value of the magnetic field, i.e., eB=2.8326  GeV^2, the first-order derivative of jet quenching parameter with respect to the temperature ∂q̂_||(⊥)/∂ T diverges at T_c.

For the other two types of jet quenching parameters, q̂_⊥(||) and q̂_⊥(⊥), the numerical results are shown in Fig. <ref>. The overall trends of these jet quenching parameters are almost the same as q̂_||(⊥) shown in Fig. <ref>. Specifically, q̂_⊥(||)+q̂_⊥(⊥) is larger than q̂_||(⊥); q̂_⊥(⊥) is larger than q̂_⊥(||) under the same temperature and magnetic field conditions [This is consistent with the results in a narrower region of magnetic field  <cit.>].





§ CONCLUSIONS

In this paper, we investigate the impact of a magnetic background field on QCD properties using the 
 EMD model. Our findings reveal that at extremely high magnetic fields, the QCD phase transition is of first-order. As the magnetic field decreases, the first-order phase transition boundary terminates at a critical endpoint, located at (eB,T_C)= (2.8326  GeV^2, 0.1191   GeV), before which the transition turns into a crossover.  The entropy density and pressure behavior confirmed this phase transition boundary, with typical S-shape and Swallowtail shape curves. The first-order transition temperature decreases with the increasing of magnetic field. These are qualitatively consistent with lattice results in Ref. <cit.>. Additionally, both the entropy and pressure are significantly enhanced by the magnetic fields. 

Furthermore, we also discuss the variation of the jet-quenching parameter during the phase transition process in the presence of a magnetic field. For comparison, in the absence of a magnetic field, our holographic results are primarily consistent with the Monte Carlo and the DIS experiment  <cit.>, provided that λ_t=1/2. In the presence of a magnetic field, it highlights the anisotropic nature of QGP  in heavy ion collisions. The magnetic field reduces the spatial SO(3) symmetry to SO(2), leading to distinct jet quenching parameters that vary with the direction of the parton’s motion and the orientation of the transverse momentum broadening. Under the phase transition, the jet quenching parameters are greatly affected by the magnetic field. The normalized jet quenching parameters are enhanced around the phase transition temperature, suggesting that magnetic fields promote the transverse momentum broadening as well as the jet energy loss in the QGP. The parameters converge to the zero magnetic field cases at both low and high-temperature extremes. Moreover, as a consequence of the first-order phase transition, they can exhibit multiple values which indicates that jet quenching could serve as a signal to probe the phase diagram of QCD phase transition.



§ ACKNOWLEDGEMENT

We thank Danning Li for his beneficial discussion. This work is supported by the Guangdong Basic and Applied Basic Research Foundation under Grant No. 2024A1515012931, the National Natural Science Foundation of China (NSFC) under Grant Nos. 12305142, 12275108  and the Science and Technology Planning Project of Guangzhou, China under Grant No. 2024A04J3243.





 Paper Title: Holographic thermodynamics of a five-dimensional neutral Gauss-Bonnet AdS black hole.


 Enabling Next-Generation V2X Perception: 
 Wireless Rigid Body Localization and Tracking

    Niclas Führling0000-0003-1942-8691, Hyeon Seok Rou0000-0003-3483-7629, Graduate Student Members, IEEE,

Giuseppe Thadeu Freitas de Abreu0000-0002-5018-8174, David González G.0000-0003-2090-8481, Senior Members, IEEE, and Osvaldo Gonsa0000-0001-5452-8159




N. Führling, H. S. Rou and G. T. F. Abreu are with the School of Computer Science and Engineering, Constructor University, Campus Ring 1, 28759, Bremen, Germany (emails: [nfuehrling, hrou, gabreu]@constructor.university).

D. González G. and O. Gonsa are with the Wireless Communications Technologies Group, Continental Automotive Technologies GmbH, Guerickestrasse 7, 60488, Frankfurt am Main, Germany (emails: david.gonzalez.g@ieee.org, osvaldo.gonsa@continental-corporation.com).

    August 5, 2024
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§ INTRODUCTION


Black hole thermodynamics in asymptotically anti-de Sitter (AdS) spacetime has advanced to the forefront of current research activities in theoretical high energy physics. One of the intriguing aspects of theoretical physics is duality which has revolutionized perspectives of physics at a fundamental level. Dualities relate two different facets of nature and thus they lead to an enhanced understanding of physics. An implication of these dualities is that, while the mathematical structures of the two theories may be identical, there may exist differences from a physical standpoint. One of the interesting dualities is the anti-de Sitter/conformal field theory (AdS/CFT) correspondence <cit.> which connects the CFT on a flat spacetime to string theory. This is interesting because string theory is a theoretical framework for quantum gravity, whereas quantum field theory is not. 

Now, even though these two theories are different, the AdS/CFT correspondence assumes the existence of a correspondence between the two theories and this is one of the most attractive features of this duality, which pushes research for its deeper understanding and possible applications. A renowned example of a phase transition in black hole spacetimes is the radiation/black hole first-order phase transition of Hawking and Page, observed for Schwarzschild-AdS black holes immersed in a bath of radiation <cit.>. Such a phenomenon has a dual interpretation for a boundary quantum field theory via the AdS/CFT correspondence and is related to a confinement/deconfinement phase transition in the dual quark gluon plasma <cit.>. For charged black hole in AdS spacetime one likewise observes a small/large black hole first-order phase transition reminiscent of the liquid/gas phase transition of the Van der Waals fluid <cit.>. But when the cosmological constant is interpreted as the thermodynamic pressure and the black hole mass is interpreted as the thermodynamic enthalpy <cit.>, the phase behavior of the charged AdS black hole is exactly the same with that of the Van der Waals fluid and they have the same critical exponents <cit.>. This version of the black hole thermodynamics in the presence of negative cosmological constant is interpreted as the extended phase space thermodynamics or generically speaking, the black hole chemistry <cit.>. The phase structures in the extended phase space is much richer, including reentrant  phase transition <cit.>, superfluid-like phase transition <cit.>, triple points <cit.> etc. Recent advancements have yielded numerous successful interpretations of the origins of the extended phase space thermodynamics. Notable among these are the extended Iyer-Wald formalism <cit.>, the coupling constant of gauge field perspective <cit.>. Additionally, a mechanism for the higher-dimensional origin of a dynamical cosmological constant was proposed <cit.>.

Despite significant advancements have been achieved in the field of thermodynamics for black holes in AdS spacetime, the holographic interpretation of black hole chemistry has remained an enigma <cit.>. From the viewpoint of the AdS/CFT correspondence, the negative cosmological constant Λ sets the asymptotic structure of the bulk spacetime and is related to the number of colours N in the dual gauge field theory <cit.>

    kL^d-2/16π G=N^2,

where L is the AdS radius and the numerical factor k depends on the details of the particular holographic system. Hence, variation of the negative cosmological constant Λ corresponds to changing the number of colors N and hence varying the dual gauge field theories in the boundary <cit.>.

Recently there have been considerable progress in developing an exact dictionary between the black hole chemistry and the thermodynamics of the dual gauge field theory. Considering the viewpoint that Newton's constant is treated as a variable <cit.>, and extended to the more general CFT metric with boundary radius different from the AdS radius, Visser argued that holographic thermodynamics requires a chemical potential for color and obtained the thermodynamic Euler equation for high energy states of large-N gauge theories <cit.>. Systematic works have been done based on this viewpoint <cit.>. 
Building on Visser’s viewpoint that the radius of the CFT metric need not coincide with the AdS radius, and allowing the conformal factor to vary while keeping Newton’s constant fixed, thereby resulting in the boundary volume 𝒱 and the central charge C being completely independent,  Ahmed et al. constructed an exact dictionary between the thermodynamics of the charged AdS black hole and the dual conformal field theory <cit.>. Based on this perspective, a lot of work has been done for the holographic thermodynamics for AdS black holes, including the charged AdS black hole <cit.>, Kerr AdS black hole <cit.>, Kerr-Newman AdS black hole <cit.>. Inspired by this viewpoint, we investigate the holographic CFT thermodynamics for the five-dimensional neutral Gauss-Bonnet AdS black hole. Different from the recent work of Cui et al. <cit.>, where the Newton's constant is regarded as a dynamical variable, we take the viewpoint of Ahmed et al. in which the Newton's constant is treated as a fixed quantity but the conformal factor is supposed to vary. 

From the perspective of the gauge/gravity correspondence, the thermodynamics of AdS black holes is completely equivalent to the thermodynamics of the dual gauge fields. Through the extended bulk thermodynamics for the five-dimensional Gauss-Bonnet AdS black hole, we derive the first law of the CFT thermodynamics which is obtained by directly translating the arbitrary conformal factors in the dual CFT and investigate the phase behavior and criticality of the CFT. We find that the free energy exhibits a characteristic swallowtail behavior and the critical points and critical exponents are the same as that of the five-dimensional Gauss-Bonnet AdS black hole. 


The outline of the paper is as follows. In Sec. <ref>, we derive the AdS/CFT dictionary and the laws for the CFT residing in the boundary dual to the five-dimensional Gauss-Bonnet AdS black hole. In Sec. <ref>, we investigate the phase behavior and criticality of the CFT. Finally, in the last section we discuss the results and conclude them. 




§ HOLOGRAPHIC THERMODYNAMICS OF FIVE-DIMENSIONAL GAUSS-BONNET ADS BLACK HOLE



In this section, we tentatively establish a connection between the extended thermodynamics of the five-dimensional neutral Gauss-Bonnet AdS black hole and the corresponding extended thermodynamics of dual CFT. By exploring the relationship between these two realms, we aim to shed light on the profound similarities and underlying principles that govern their thermodynamic properties.




 §.§ Extended bulk thermodynamics



In this subsection, we briefly review the main points of the extended black hole thermodynamics for the five-dimensional neutral Gauss-Bonnet AdS black hole. A d-dimensional neutral Gauss-Bonnet AdS black hole is a solution for the gravitational field equation derived by the action <cit.>

    S=∫ d^dx√(-g)1/16π G_N(ℛ-2Λ+α_GBℒ_𝒢ℬ)

with the Gauss-Bonnet term

    ℒ_𝒢ℬ=ℛ_μναβℛ^μναβ -4ℛ_μνℛ^μν+ℛ^2,

where G_N is the Newton constant, α_GB is the Gauss-Bonnet coupling constant, and the cosmological constant is given by

    Λ=-(d-1)(d-2)/2L^2

with L the AdS radius.






Within the context of the low energy effective action formalism in heterotic string theory, the Gauss–Bonnet term is the leading order quantum correction to gravity <cit.>. The Gauss–Bonnet coupling constant is related to the string scale and can be identified with the inverse of the string tension <cit.>. Hence, we only consider positive Gauss-Bonnet coupling constant in this paper.

The theory admits an exact neutral black hole solution in five dimensions. The metric for the five-dimensional neutral Gauss-Bonnet AdS black hole is <cit.>

    ds^2=-f(r)dt^2+f^-1(r)dr^2+r^2[dθ^2+sin^2θ (dϕ^2+sin^2ϕ dψ^2)]

with the metric function given by

    f(r)=1+r^2/2α(1-√(1+32α G_NM/3π r^4+2αΛ/3)),

where α=2α_GB.

The solution is a steady spherical solution with the event horizon r_h determined by f(r_h)=0. From this equation, we can express the mass of the black hole as

    M=π/8G_N(-1/2Λ r_h^4+3r_h^2+3α).

From the extended black hole thermodynamic perspective, the mass of the black hole is interpreted as the enthalpy H=M rather than the internal energy U of the gravitational system.

By requiring the absence of a conical singularity at the event horizon in the Euclidean action, we can get the Hawking temperature of the black hole, which is given by

    T_H=f'(r_h)/4π=3r_h-Λ r_h^3/6π r_h^2+12πα.

From the viewpoint of the extended black hole thermodynamics, the cosmological constant Λ is interpreted as the thermodynamic pressure and the black hole mass is regarded as the thermodynamic enthalpy H <cit.>. The thermodynamic pressure is

    P=-Λ/8π G_N=3/4π G_N1/L^2,

and its conjugate thermodynamic volume is

    V=π^2 r_h^4/2.

The Gauss-Bonnet coupling constant α is a thermodynamic variable, with the corresponding conjugate thermodynamic variable given by

    𝒜=(∂ M/∂α)_S,P=-π/8G_N-4Λ r_h^4+9r_h^2-6α/r_h^2+2α.

The entropy of the black hole is

    S=∫_0^r_h(∂ M/∂ r_h)_P,αdr =π^2 r_h/2G_N(r_h^2+6α).

Evidently, the presence of the Gauss-Bonnet correction to Einstein's theory of gravity leads to a notable revision of the Bekenstein-Hawking area law when considering the black hole's temperature as its surface gravity, as emphasized in Ref. <cit.>.

The first law of black hole thermodynamics and the Smarr relation for the five-dimensional Gauss-Bonnet AdS black hole are <cit.>

    dM    =    TdS+VdP+ 𝒜dα,
    
      2M    =    3TS-2PV+2𝒜α.


Using the first law of the bulk thermodynamics, we proceed to derive the boundary thermodynamics in the next subsection. This derivation establishes a tentative relationship between the thermodynamic properties of the bulk system and its boundary counterpart, allowing for a comprehensive analysis of their interconnected behavior. The insights gained from this derivation might enhance our understanding on the overall thermodynamic framework.





 §.§ Extended boundary thermodynamics



From the perspective of the AdS/CFT correspondence, the dual CFT lives on the conformal boundary of the asymptotically AdS spacetime. According to the recent development of the holographic chemistry <cit.>, the boundary metric of the CFT dual to the five-dimensional Gauss-Bonnet AdS spacetime is <cit.>:

    ds^2=ω^2(-dt^2+L^2dΩ_3^2 ),

where dΩ_3^2 is the metric on the three-dimensional unit sphere:

    dΩ_3^2=dθ^2+sin^2θ (dϕ^2+sin^2ϕ dψ^2),

and ω=R/L is the dimensionless conformal factor that is free to vary <cit.>. It reflects the conformal symmetry of the boundary theory. R is the radius of the boundary.

In the gauge/gravity duality, the central charge of the CFT is related to the AdS radius L and the gravitational coupling constant G_N in the bulk theory. For Einstein gravity the central charge usually reads <cit.>

    C=Ω_d-1 L^d-1/16π G_N,

where Ω_d-1 is the volume of a (d-1)-dimensional unit sphere. Since the precise coefficient of the central charge is irrelevant for our purpose and only the combination dC/C appears in the first law of the CFT, the dimensional dependent coefficient is normalized such that it agrees with the coefficient A of the Euler density in the trace anomaly <cit.>.

From the first law of black hole thermodynamics and the Smarr relation for the five-dimensional Gauss-Bonnet AdS black hole, we can express the first law in terms of the boundary thermodynamics

    δ(M/ω)=(T/ω)δ S+M-TS-𝒜α/ωδ(L^3/G_N)/(L^3/G_N)+√(G_N)𝒜/ω Lδ(α L/√(G_N))-(Ẽ/3𝒱)δ𝒱,

where 𝒱=2π^2 R^3 is the spatial volume of the boundary on which the CFT resides.
In the derivation of the first law in terms of the boundary thermodynamics, the Newtown constant G_N is regarded as a fixed constant, but the conformal factor ω is free to vary.
Employing the following holographic dictionary:

    Ẽ   =M/ω,                 T̃=T/ω,               S=S, 
    𝒜̃   =√(G_N)𝒜/ω L,
                    α̃=α L/√(G_N),        𝒫=Ẽ/3𝒱, 
    
          C   =π L^3/8G_N,              μ=1/C(Ẽ- T̃ S-𝒜̃α̃),

the extended first law of thermodynamics for the five-dimensional Gauss-Bonnet AdS black hole is dual to the following thermodynamic first law in the CFT:

    δẼ=T̃δ S+𝒜̃δα̃+μδ C-𝒫δ𝒱,

where 𝒫 is the field theory pressure, and μ is the chemical potential associated with the central charge C.  The generalized Smarr relation (<ref>) for the AdS black hole is dual to an Euler equation in a large N gauge field theory, which takes the following form:

    Ẽ=T̃ S+𝒜̃α̃+μ C.

One noticeable point is that the Euler equation does not include a 𝒫𝒱 term.












From the above AdS/CFT dictionary, the central charge C in the extended boundary thermodynamics is dual to the pressure P of the extended bulk thermodynamics. Hence it is natural to take the viewpoint that the conjugate thermodynamics of the central charge—the chemical potential μ in the extended boundary thermodynamic is dual to the thermodynamical volume V of the extended black hole thermodynamics. 

We define the dimensionless parameters x and y as

    x=/L,      y=α/L^2.

Then, in terms of the parameters x and y, the CFT thermodynamic quantities are given by

  
  * energy:
  
    Ẽ  =3C/R(x^4+x^2+y),

  
  * temperature:
  
    T̃=1/2π R2x^3+x/x^2+2y,

  
  * entropy:
  
    S  =4π Cx(x^2+6y),

  
  * pressure:
  
    𝒫 =C/2π^2R^4(x^4+x^2+y),

  
  * chemical potential:
  
    μ=1/Rx^2[-x^2(x^2-1)+6y(x^2+1)]/x^2+2y,

  
  * the Gauss coupling constant α̃ and its conjugate CFT thermodynamic quantity 𝒜̃:
  
    α̃=√(8CL^3/π)y,     𝒜̃ =-1/R√(π C/8L^3)24x^4+9x^2-6y/x^2+2y.








Having the thermodynamics for the CFT dual to the five-dimensional Gauss-Bonnet AdS black hole, we delve into the exploration of the phase transition  and critical phenomena for the CFT.





§ THERMODYNAMIC ENSEMBLES IN THE DUAL CFT



The extended thermodynamics and phase transition for CFT states that are dual to the singly-spinning asymptotically AdS black hole were investigated by Ahmed et al., and various phase transitions and critical behavior in the CFT were uncovered <cit.>. Gong et al. found that the critical exponents for the CFT states in the vicinity of the critical point are the same with that of the Kerr-Newman AdS black hole <cit.>. In this section, we investigate the phase behavior and criticality exhibited by the canonical ensemble within the CFT

framework.





 §.§ Canonical ensemble: swallowtail behavior and critical point



The canonical ensemble for five-dimensional Gauss-Bonnet AdS black hole has been well study in the framework of extended black hole thermodynamics, i.e. black hole chemistry <cit.>. In these studies it was found that the five-dimensional Gauss-Bonnet AdS black hole displays the swallowtail behavior, which indicates the small/large black hole phase transition reminiscent of the Van der Waals like phase transition. Besides, it was shown that the critical point for the five-dimensional Gauss-Bonnet AdS black hole phase transition is <cit.>

    r_hc=√(6α),      P_c=1/48πα,       T_c=1/2√(6α)π.

In this subsection, we investigate the phase behavior of the CFT states dual to the five-dimensional neutral Gauss-Bonnet AdS spacetime.

From the previous subsection for the study of the boundary thermodynamics, we can get the relevant Helmholtz free energy in the fixed (C, 𝒱, α̃) canonical ensemble:

    F=Ẽ-T̃S=-C/Rx^6+18x^4y-x^4+3x^2y-6y^2/x^2+2y.


To illustrate the phase behavior of the CFT states, we draw the diagram for the Helmholtz free energy F vs the temperature T in Fig. <ref>. 
Though we have chosen the boundary radius R=10^4 and the thermodynamic variable of the CFT α̃=1 in the figures, the scale of the boundary radius R and α̃ do not affect the qualitative thermodynamic behavior of the system. The figures show the characteristic swallowtail behavior which is similar to the liquid/gas phase transition for the Van der Waals fluid. The brown dashed and blue curves in Fig. <ref> resemble the shape of a swallowtail. The swallowtail behavior for the CFT states demonstrates the existence of a first-order phase transition. When the central charge C equals the critical value C_c, it becomes a second-order phase transition. However, the swallowtail disappears when the central charge C is smaller than the critical value C_c.

As depicted in Fig. <ref>, each swallowtail consists of three distinct piecewise smooth branches. The parameter x increases along the curve, as evidenced by the direction of the arrows traced along the curve in Fig. <ref>. Since the entropy of the CFT states is a monotonic increasing function of x, as indicated by eq. (<ref>), the entropy increases along the direction of the arrows depicted on the curve. The black line starting from zero temperature is the low entropy branch, since the value of x is the smallest on this branch. The entropy increases along the read dashed curve, which is the intermediate entropy branch. The branch that the Helmholtz free energy F extends to minus infinite is the largest entropy branch. For any swallowtail curve in the figure, starting at low temperature T, the branch that minimizes the Helmholtz free energy is the thermodynamically favored state. Hence the low entropy branch is thermodynamically stable for low temperature. As the self-intersection point for the low entropy branch and high entropy branch has the same Helmholtz free energy, it is the phase transition point. The low and high entropy phases can exist together. After crossing the self-intersection point, the higher entropy branch is more stable. 




The stability of the CFT states can also be captured by the heat capacity. For positive heat capacity, the system is locally thermodynamically stable; while negative heat capacity indicates thermodynamically unstable. For the canonical ensemble with fixed C, 𝒱 and α̃, the heat capacity is

    𝒞_C, 𝒱, α̃=T(∂ S /∂T)_C, 𝒱, α̃= 12 πC(2 x^3+x) (x^2+2 y)^2/x^2 (12 y-1)+2 x^4+2 y.



The heat capacities for the low and high entropy branches of the Helmholtz free energy in Fig. <ref> are both positive, hence they are thermodynamically stable. While the heat capacity for the intermediate entropy branch is negative, hence it is thermodynamically unstable. We also draw the heat capacity for the CFT states as in Fig. <ref>. For the brown dashed and blue line, the central charge C is larger than the critical value C_c=9πα̃/(2√(G_N)). For the red large dashing line, the central charge is equal to the critical value C = C_c. For the black dotted line, the central charge C is smaller than the critical value C_c. As we can see from the figure, the heat capacity is continues and finite for small central charge C< C_c. However, there are points where the heat capacity is divergent for large central charge C ≥  C_c. The divergent points of the heat capacity correspond to the intersection points between the small/large entropy branches and intermediate entropy branches in Fig. <ref>.





Besides the Helmholtz free energy, the oscillatory behavior of the temperature T against the entropy S can also depict the phase transition of the system. 
The oscillatory behavior for curves in the T̃-S diagram in Fig. <ref> shows the phase transition of the boundary CFT. For large central charge C>C_c, the curves (the brown dashed and blue lines) in the T̃-S plane display the oscillatory behavior. The oscillatory behavior of the T̃-S diagram indicates there exists a first-order phase transition. For small central charge C≤ C_c, the oscillatory behavior disappears and the temperature T̃ is a monotonic increasing function of the entropy S, as can be seen in the figure for the red dashing and black dotted lines.




From the T̃-S oscillatory curve in Fig. <ref>, we can get the critical point for the phase transition. The minimal and maximal points of the T̃-S oscillatory curve coincide and exhibit a horizontal tangent at the critical point. Mathematically, the critical point satisfies the following equations:

    (∂T̃/∂ S)_C, 𝒱, α̃=0,     (∂^2T̃/∂ S^2)_C, 𝒱, α̃=0.

From the above two equations, we can get the critical point, which is determined by the following two parameters

    x_c=1/√(6),      y_c=1/36.

Some straightforward calculations reveal that the critical point for the boundary CFT satisfies

    r_hc=√(6α).

Evidently, the critical point of the thermodynamics for the boundary CFT is the same as the critical point for the bulk phase transition as indicated in eq. (<ref>). 
At the critical point, the other thermodynamic quantities are

    T̃_c   =√(6)/2π R,       𝒜̃_c=-9 π/8√(G_N) R,
         𝒫_c= α̃/2 π√(G_N)R^4, 
     
    S_c   =√(6)/√(G_N)π^2α̃,       
    C_c=9π/2√(G_N)α̃,
        μ_c=1/4 R.


From the critical point, we can easily find an interesting relation among the critical pressure 𝒫_c, critical volume 𝒱_c, critical temperature T̃_c, and the critical entropy S_c:

    𝒫_c𝒱_c/T̃_c S_c=1/3.






 §.§ Maxwell's equal area law and coexistence curve




The physical processes of crossing the coexistence phase often unveil intriguing phenomena, such as latent heat <cit.>. As shown by previous work <cit.>, both the swallowtail behavior of the free energy and Maxwell's equal area law can reveal phase transition of a thermodynamic system and they are consistent with each other. In this subsection,  we derive Maxwell's equal area law from the change in the Helmholtz free energy during the phase transition between low and high entropy states. Then we employ Maxwell's equal area law to construct an analytical formula for the coexistence curve in the T̃-S plane.

To solve the coexistence curve for the low and high entropy states of the boundary CFT, we need to solve the phase transition points in the F-T̃ plane in Fig. <ref>. We focus on the self-intersection points in Fig. <ref>. These two points are low and high entropy phases. These two coexistence phases correspond to point A and point C in the T-S plane, respectively. A phase of the boundary CFT thermodynamic system in the fixed (C, 𝒱, α̃) canonical ensemble can be determined by the parameter x. We denote the low entropy phase A as (x_L,y,R,C) and the high entropy phase C as (x_H,y,R,C).

For the phase transition between the low entropy state A and high entropy state C in Fig. <ref>, the Helmholtz free energies of the two phases are the same and the difference is zero:

    Δ F=0.

 From the Helmholtz free energy

    F=Ẽ-T̃S,

and the first law (<ref>), we have

    dF=SdT̃-𝒫d𝒱+𝒜dα+μ dC.

Integrating it and referring to eq. (<ref>), we obtain

    ∫^C_A dF=∫^C_A SdT-∫^C_A Pd 𝒱+∫^C_A𝒜̃dα+∫^C_Aμ dC=0.

Evidently, the exactly result of each integral in eq. (<ref>) depends on the physical process.
For the fixed C, 𝒱, and α canonical ensemble, eq. (<ref>) becomes

    ∫^C_A SdT=0.

The result shows that the areas in the region 1 and region 2 in Fig. <ref> are the same. It suggests that Maxwell's equal area law holds for the phase transition of the boundary CFT in the canonical ensemble. As we can see from Fig. <ref>,

it is more convenient and direct to formulate Maxwell's equal area law (<ref>) in the following form

    ∫^C_ATdS=T(A)[S(C)-S(A)].

From the temperature in eq. (<ref>) and the entropy in eq. (<ref>) of the CFT, we can get the integration, which is

    ∫^C_ATdS=∫^x_H_x_LT(∂ S/∂ x)dx=3C/R(x_L^2+x_H^2+1)(x_L+x_H)(x_H-x_L).

Besides, the right-hand side of eq. (<ref>) is

    T̃(A)[S(C)-S(A)]
           =2C/R2x_L^3+x_L/x_L^2+2y(x_L^2+x_Lx_H+x_H^2+6y)(x_H-x_L).

Then, Maxwell's equal area law (<ref>) indicates the following equation

    3(x_L+x_H)(x_L^2+2y)(x_L^2+x_H^2+1)
            =2(2x_L^3+x_L)(x_L^2+x_Lx_H+x_H^2+6y).

On the other hand, the temperatures of the phase A and phase C are the same:

    T̃(A)= T̃(C).



From Maxwell's equal area law in eq. (<ref>) and eq. (<ref>), we can get the parameters characterizing the low entropy phase A and high entropy phase C:

    x_L   =√(1-24 y-√(48 y (9 y-1)+1))/√(2), 
    
            x_H   =6 √(2) y/√(1-24 y-√(48 y (9 y-1)+1)).

The isothermal temperature for phases A and C, representing the temperature of coexistence, can be determined. The temperature indicates the precise conditions at which coexistence takes place. The isothermal temperature is

    T̃=√(1-24 y-√(48 y (9 y-1)+1))(2-24 y-√(48 y (9 y-1)+1))/√(2)π  R (1-20 y-√(48 y (9 y-1)+1)).

Since we are considering the fixed (C, 𝒱, α̃) canonical ensemble, the function for the coexistence curve for the canonical ensemble is given by eq. (<ref>) and

    C=π/8√(G_N)α̃/y.

The above equations (<ref>) and (<ref>) are the parameter equations for the coexistence curve. The temperature for the coexistence of the two phases is a function of the parameter y and boundary volume 𝒱. To write the equation in a more concise manner, we define

    τ=T̃/T̃_c,      c=C/C_c.

The coexistence curve equation can be written as

    τ= √(-√(3)√(1/c^2-4/c+3)-2/c+3)(c (√(3)√(1/c^2-4/c+3)-6)+2)/c (3 √(3)√(1/c^2-4/c+3)-9)+5.

One of the most remarkable properties of the above equation for the coexistence curve is its independence of the CFT thermodynamic variable α̃, as well as the boundary volume 𝒱. From the coexistence curve in Fig. <ref>, the temperature for the low and high entropy coexistence phases decreases with the increasing of the inverse of the central charge 1/C.

In the coexistence curve, a first-order phase transition occurs between the low and high entropy states for C> C_c. At the critical point, a second-order phase transition occurs. However, there is no phase transition for C<C_c.





 §.§ Critical exponents



Physics near the critical point often exhibits fascinating phenomena. Critical exponents characterize the behavior of physical quantities in the vicinity of a critical point. These critical exponents are considered universal, as they remain independent of the specific characteristics of the physical system. In this subsection, we study the critical behavior of some physical quantities and calculate the corresponding critical exponents for the thermodynamics of the CFT residing on the boundary.

In the vicinity of the critical point, the critical behavior of the CFT dual to the five-dimensional neutral Gauss-Bonnet AdS black hole can be precisely characterized by the critical exponents denoted as α', β, γ, and δ. These critical exponents are explicitly defined as follows:


  * The critical exponent α' governs the behavior of the heat capacity of the CFT at constant chemical potential as compared to the heat capacity of an AdS black hole at constant volume

    C_μ=T̃(∂ S/∂T̃)_μ∼(-T-T_c/T_c)^-α'.


  * The critical exponent β determines the behavior of the order parameter η. Since the entropy increases along the arrows in the swallowtail curve in Fig. <ref>, and the difference of the entropies S_H-S_L changes discontinuously in the first-order phase transition but continuously at the critical point, we choose the entropy difference between high and low entropy phases as the order parameter. Then the critical exponent β is defined as

    η=  S_H-S_L/S_c∼(-T̃-T_c/T_c)^β.













  * The critical exponent δ determines the behavior of the pressure on the critical point for extended black hole thermodynamics. Through the dual relation, the critical exponent δ for the CFT is defined as

    C-C_c∼ (  μ-μ_c)^δ.



These critical exponents characterize the critical phenomena for the boundary CFT. To calculate the critical exponents for the CFT, we define

    t=T̃/T̃_c-1,     
        s=S/S_c-1,     χ=μ/μ_c-1.

At the critical point, the heat capacity for the fixed chemical potential is

    C_μ,𝒱,α̃=T̃(∂ S/∂T̃)_μ,𝒱,α̃=4 √(6)/√(GN)π^2 α̃,

which is not divergent.  Hence the critical exponent  governing the behavior of the heat capacity for the CFT is α'=0.

To calculate the critical exponent governing the behavior of the order parameter, we start at the entropy of the CFT at the phase transition point. Together with the entropy (<ref>), and the parameter labeling the low and high entropy states (<ref>), the entropy of the phases A and C in Fig. <ref> are

    S_L   =√(2)π  C √(1-24 y-√(48 y (9 y-1)+1))(1-12 y-√(48 y (9 y-1)+1)),
    
        S_H   =144 √(2)π  C y^2 (1-12 y-√(48 y (9 y-1)+1))/(1-24 y-√(48 y (9 y-1)+1))^3/2.

In terms of the parameter y, the order parameter can be expressed as

    η   =S_H-S_L/S_c
       =(12 y+√(48 y (9 y-1)+1)-1)^2 (36 y+√(48 y (9 y-1)+1)-1)/8 √(3) y (-24 y-√(48 y (9 y-1)+1)+1)^3/2.

From the temperature T̃ in eq. (<ref>),  we have

    t   =T̃/T̃_c-1
       =√(1-24 y-√(48 y (9 y-1)+1))(24 y+√(48 y (9 y-1)+1)-2)/√(3)(20 y+√(48 y (9 y-1)+1)-1)-1.

Solving eq. (<ref>), we can get a simple result for the parameter y in terms of t:

    y= 1/36(-2 t^2-4 t+1).

Together with eqs. (<ref>) and (<ref>), the order parameter can be formulated in terms of the temperature parameter t:

    η   =4 (t (t+2)-√(3)√(t (t+1)^2 (t+2))+1)^2 (√(3)√(t (t+1)^2 (t+2))-3 t (t+2))/(1-2 t (t+2)) (4 t (t+2)-2 √(3)√(t (t+1)^2 (t+2))+1)^3/2
       =4√(6)t^1/2+25√(6)t^3/2+𝒪(t^5/2).

Hence, the critical exponent is given by β=1/2.

To calculate the critical exponent δ, through the definition of t and the equation for the temperature (<ref>), we can solve the third-order equation for x in terms of the parameters t and y. Then the chemical potential χ can be formulated as a function of the parameters t and y. 
 







Taking y=y_c=1/36 and expand χ near the critical point, we obtain

    χ(1+3c)(c-1)^1/3/3c^4/3∼(c-1)^1/3,

where c is the reduced central charge as defined in eq. (<ref>). Then we have

    c-1 ∼χ^3=(μ/μ_c-1)^3.

Hence, the critical exponent is δ=3.

Thus, we have obtained the critical exponents α', β and δ associated with the T̃-S criticality of the CFT,  where the conformal factor ω is free to vary but with fixed Newton's constant. Our results suggest that these critical exponents associated with the T̃-S criticality of the CFT are the same as those of the five-dimensional Gauss-Bonnet AdS black hole <cit.>.




§ CONCLUSION AND DISCUSSION



In the AdS/CFT correspondence, black holes in AdS spacetime correspond to thermal states of the boundary field theories. Based on the recently proposed viewpoint of the holographic thermodynamics, which treats the conformal factor ω as a variable, we derived the first law and Euler equation for the thermodynamics of the CFT, and established a holographic dictionary between the thermodynamics of the five-dimensional  neutral Gauss-Bonnet AdS black hole and the dual boundary thermodynamics. Having the thermodynamics for the dual CFT, we investigated the phase transition and criticality of the CFT in the fixed (C, 𝒱, α̃)  canonical ensemble. Just like the free energy for the five-dimensional neutral Gauss-Bonnet AdS black hole, the free energy for the CFT displays a characteristic swallowtail behavior. This indicates a first-order phase transition below the critical point and a second-order phase transition at the critical point. Through the T̃-S oscillatory behavior, we obtained the critical point for the thermodynamics of the CFT in the  T̃-S plane. The critical point for the CFT is exact the same with that of the five-dimensional neutral Gauss-Bonnet AdS black hole. Using Maxwell's equal area law, we got the coexistence curve for the high and low entropy phases of the CFT. Besides, we got the critical exponents for the CFT, and found that these critical exponents associated with the T̃-S criticality are the same as those of the five-dimensional Gauss-Bonnet AdS black hole.

Critical exponents describe the singularity behavior of thermodynamic variables such as the heat capacity, order parameter etc. They are universal and satisfy the scaling hypothesis. The critical exponent γ governing the isothermal compressibility of black hole thermodynamics is defined as

    κ_T=-1/v(∂ v/∂ P)_T∼(-T-T_c/T_c)^-γ.

As we argued in Sec. <ref>, the black hole thermodynamic pressure P is dual to the CFT central charge C, and the black hole thermodynamic volume V is dual to the CFT chemical potential μ.
In our investigation of the critical exponents for the CFT thermodynamics in the T̃-S plane, we found that if we define the critical exponent γ just replacing the pressure P and the  specific volume v by the central charge C and the chemical potential μ respectively

    κ_T̃=-1/μ(∂μ/∂ C)_T̃,𝒱,α̃∼(-T-T_c/T_c)^-γ,

then, the critical exponent is given by γ=2/3, which is different from the prediction of the scaling hypothesis. Perhaps this is due to the fact that it is the black hole thermodynamic volume V instead of the specific volume v that is dual to the CFT chemical potential μ. This leads to the result that the isothermal compressibility is not defined just by replacing the pressure P and the  specific volume v by the central charge C and the chemical potential μ in the T̃-S plane.

Criticality is a very important phenomena for thermodynamic systems and critical exponents are universal. Our investigation of the thermodynamics and criticality from the gravity/gauge duality viewpoint might lead to new insight to the understanding of the Van der Waals like phase behavior of the five-dimensional Gauss-Bonnet AdS black hole.







We are grateful to Chuan Chen, Shan-Ping Wu, and Yu-Peng Zhang for useful discussions. This work was supported by National Key Research and Development Program of China (Grant No. 2021YFC2203003), the National Natural Science Foundation of China (Grants No. 12305065, No. 12247178, No. 12347177,  No. 12075103 and No. 12247101), the China Postdoctoral Science Foundation (Grant No. 2023M731468), the 111 Project (Grant No. B20063), and Lanzhou City's scientific research funding subsidy to Lanzhou University.


















JHEP




 Paper Title: Positive Mass in General Relativity Without Energy Conditions.













alves.nickolas@alumni.usp.br    
Center for Natural and Human Sciences, Federal University of ABC,
Av. dos Estados, 5001, 09210-580, Santo André, SP, Brazil

andre.landulfo@ufabc.edu.br
Center for Natural and Human Sciences, Federal University of ABC,
Av. dos Estados, 5001, 09210-580, Santo André, SP, Brazil


bcosta@troy.edu
Center for Relativity and Cosmology, Troy University,
 Troy, AL, 36082, USA







§ ABSTRACT 
A long-standing problem in physics is why observed masses are always positive. While energy conditions in quantum field theory can partly answer this problem, in this paper we find evidence that classical general relativity abhors negative masses, without the need for quantum theory or energy conditions. This is done by considering many different models of negative-mass “stars” and showing they are dynamically unstable. A fortiori, we show that any barotropic negative-mass star whose pressure is not a monotonically increasing function of the energy density must be dynamically unstable. Furthermore, we argue that all acceptable barotropic models of negative-mass stars must be in this class, and are thus unstable.




Positive Mass in General Relativity Without Energy Conditions
    Bruno Arderucio Costa 0000-0001-5182-2010
    July 29, 2024
=============================================================





§ INTRODUCTION

A simple, yet profound, question one can ask about gravity is: “why do things fall down?” This is a conspicuous experimental fact about gravitational physics, and it is certainly fundamental for the formation of structures in the Universe and the existence of life itself. The answer, however, may dive deep into the foundations of physics. 

From a purely quantum field theoretical point of view, one could answer that things fall down because gravity is mediated by a spin 2 field—the graviton. As is well-known <cit.>, even-spin mediators lead to attractive forces for particles with charges with the same sign. Hence, positive masses must attract each other. This is in contrast to the situation with electromagnetism, in which the spin 1 photon makes opposite charges attract each other, but similar charges repel each other. 

A follow-up question arises: why do we only observe “gravitational charges” (, masses) of the same sign? The absence of negative masses is a prominent feature of nature which can be deemed curious since one at first would expect the situation for positive and negative masses to be symmetric.

The fact is that assuming that the equivalence principle holds in such cases (i.e., inertial and gravitational masses are equal), the situation is not symmetric. The reason is the curious gravitational behavior of masses with different signs that can be understood in Newtonian gravity, for simplicity. Consider a pair of masses. If both the masses are positive, then the situation is standard and the masses will be attracted to each other. In particular, the gravitational force on each particle and their acceleration point in the same direction (see figure <ref>). If the masses have opposite signs, the forces are repulsive, but the net effect is that the negative mass pursues the positive mass, while the latter runs from the former. This is because the force and acceleration on the negative mass point to different directions (F = - ma). Finally, for a similar reason, the gravitational force between two negative masses is attractive, but the effect is repulsive. This is pictured in figure <ref>. These examples make it clear that the gravitational dynamics of negative masses is fairly different from that of positive masses. While negative and positive masses can lead to similar orbits in Newtonian mechanics <cit.>, we know of no experimental evidence supporting the existence of negative masses.



Within classical general relativity, this makes sense. One has a limited amount of matter types (, of stress-energy-momentum tensors) that can be considered in the theory. These correspond to the fields of the standard model of particle physics and their emergent consequences. It turns out that all known classical forms of matter satisfy certain energy conditions, which are impositions made on the stress tensor to restrict the allowed behavior of the matter in a certain spacetime. Within classical general relativity, it is well-known that the so-called dominant energy condition (DEC), together with other reasonable assumptions, is sufficient to ensure the positivity of the total mass in an asymptotically flat spacetime at a given time <cit.>. Hence, classical general relativity provides an answer to why we do not observe negative masses: as long as the matter content obeys the classical energy conditions, it follows that the total mass must be positive.

It happens, however, that quantum mechanical systems can easily violate the classical energy conditions, including the DEC. This means that quantum mechanical systems can have regions with negative energy density. For example, the Casimir effect <cit.> leads to a negative energy density between two conducting plates. It is then natural to wonder whether one could produce a macroscopic object with negative mass by exploiting quantum effects. 

With this in mind, Costa and Matsas recently analyzed whether quantum mechanical effects could lead to macroscopic negative masses by considering the Casimir system <cit.>. They noticed that to keep the Casimir plates separated, it is necessary to hold them apart with “struts.” Assuming these struts are classical, they must satisfy the DEC, the authors showed that the positive mass due to the struts is sufficient for the complete system to have a positive mass.

While Costa and Matsas analyzed the Casimir effect in Minkowski spacetime, their conclusions suggested the conjecture (already stated at the end of Ref. <cit.>) that a “cosmic-weight watcher must rule out from nature regular asymptotically flat stationary solutions of Einstein’s equations with M < 0”. This statement is vague regarding what could count as a “cosmic-weight watcher,” but some other conditions such as restrictions on the matter content or stability are known to be needed. For example, in Ref. <cit.>, Novikov, Bisnovatyi-Kogan, and Novikov considered a number of possible stellar solutions with negative energy density. Their interest was mostly in the fact that a star with negative energy density tends to expand due to gravitational effects and contract due to the star's pressure—which is the exact opposite of the roles pressure and gravity play in regular stars. With this goal, the authors studied a few different models that show how general relativity allows these solutions, but do not comment on whether the matter necessary to form these stars exists or whether these solutions are stable. 

If one decides to take input from the matter theory, then there is evidence that negative masses should be forbidden. While quantum fields do not obey the classical energy conditions, there is evidence that they obey weakened energy conditions, which hold only on average. In rough terms, while quantum field theory allows negative energy densities in a given region, these negative energy densities must be balanced by positive energy densities somewhere else. With this assumption, and assuming that the Einstein field equations hold semiclassically, one can conclude under some geometric assumptions on the underlying spacetime that mass must always be positive. This conclusion is mostly supported by a theorem due to Penrose, Sorkin, and Woolgar <cit.>, which must be complemented by a theorem due to Borde <cit.> to translate the result into a condition about the stress tensor.

It is interesting, however, to return attention to the original cosmic-weight watcher conjecture and ask: can general relativity discard negative mass solutions without the aid of quantum field theory? This would provide an alternative mechanism for the nonoccurrence of negative masses that does not rely on quantum theory at all. Hence, if true, it would show that gravity itself abhors negative masses, regardless of the matter content.

A natural candidate for the role of a cosmic-weight watcher is the stability of solutions. More specifically, we conjecture that there are no stable regular asymptotically flat stationary solutions of Einstein’s equations with M < 0. This time, we consider the possibility that equilibrium solutions are allowed but discard their relevance based on whether or not they are dynamically stable. Our findings provide strong evidence for this refined conjecture. Namely, all models of negative-mass stars we considered turned out to be unstable under linear perturbations or presented other problems. A fortiori, we establish that any stars with (*Pρ)_s < 0 at any point must be dynamically unstable, and all acceptable negative-mass stars must satisfy this condition somewhere (and therefore are unstable). This analysis does not require the use of energy conditions, and thus the conclusion is very different from the earlier results on positive mass theorems.

The structure of the paper is as follows. Section <ref> reviews the Tolman–Oppenheimer–Volkoff equation, which is the basic equation for the hydrostatic equilibrium of a static and spherically symmetric star in general relativity. Section <ref> reviews, for completeness, the role of energy conditions in ruling out negative masses, with particular emphasis on the semiclassical scenario, in which the averaged null energy condition (ANEC) is relevant. Section <ref> discusses how to study the stability of static, spherically symmetric stars in general relativity and how to adapt that formalism to stars with negative masses. Section <ref> then provides concrete examples of negative-mass stars. Section <ref> gives our main results on how negative-mass stars are hydrodynamically unstable. We conclude in section <ref>. Appendix <ref> exhibits how classical energy conditions can be violated in quantum field theory. Appendix <ref> expresses the ANEC integral (which is essential for verifying whether a given spacetime satisfies the ANEC) in a simpler form for a TOV-like star. Appendix <ref> computes a pair of integrals involving bump functions that are relevant for the stability analysis of a constant density star, and later for general stars.

We employ the same conventions used in Ref. <cit.>, which includes abstract index notation and has the sign convention +++ in the Misner–Thorne–Wheeler classification <cit.>. Latin indices stand for abstract indices. We also use geometric units with G=c=1. 



§ THE TOLMAN–OPPENHEIMER–VOLKOFF SOLUTION

To study negative-mass stars, we work on the framework of standard general relativity with some simplifying symmetry assumptions about the source T_a_b and metric. At this point we refrain from interpreting its origin.

The first symmetry we demand is that the spacetime is stationary, which is motivated by the fact we would like to describe a negative-mass star that can retain equilibrium. Then we ask for spherical symmetry to keep the calculations simple and manageable. Finally, we impose that the material composing the star is isotropic, meaning all of the three principal pressures of the stress tensor coincide. This simplifies the analysis, but this condition could likely be lifted at the expense of the calculations becoming more complex. 

Within these assumptions, there is a spherical coordinate system in which the line element can be written in the form

    s^2 = - e^2ϕ(r)t^2 + e^2ψ(r)r^2 + r^2 Ω^2,

where Ω^2 is the line element for the round metric in the unit two-sphere. r is such that the area of the spheres with constant coordinate radius r is 4 π r^2.

The symmetry assumptions also restrict the stress tensor to have the form

    T_a_b = ρ(r) u_au_b + P(r) (g_a_b + u_au_b),

where

    u^a = e^-ϕ(r)(t)^a.

Notice that u^a is a normalized timelike vector which is everywhere parallel to the stationary Killing vector field. 

The problem of solving the Einstein field equations with these ansätze was originally considered in these coordinates by Tolman, Oppenheimer, and Volkoff <cit.> and is reviewed in standard textbooks in general relativity <cit.> and stellar structure <cit.>. It can be shown that the solution inside the star is given by the line element

    s^2_< = - e^2ϕ(r)t^2 + (1 - 2m(r)/r)^-1r^2 + r^2 Ω^2,

where the subscript < in s^2_< indicates this is the interior solution. The functions ϕ and m are determined by means of the differential equations

    mr   = 4 πρ(r) r^2, 
    ϕr   = 4 π P(r) r^3 + m(r)/r[r - 2m(r)], 
    Pr   = -(ρ + P) 4 π P(r) r^3 + m(r)/r[r - 2m(r)].

Equation (<ref>) is known as the Tolman–Oppenheimer–Volkoff (TOV) equation.

The system of differential equations given by Eqs. (<ref>) to (<ref>) is underdetermined, since it involves four unknowns (ρ, P, m, and ϕ), but only three equations. In most applications, the remaining equation is supplied in the form of a thermodynamical equation of state relating P and ρ. The procedure to solve the TOV system is then to integrate Eqs. (<ref>) and (<ref>) from the center to the border of the star with the aid of the equation of state. This is often done numerically by providing the initial conditions m(0) = 0 and P(0) = P_0 [It is also possible to provide ρ(0) = ρ_0 instead of P(0) = P_0. The condition m(0) = 0 is used to avoid the presence of a physical singularity at the origin (this singularity would be analogous to the singularity at the origin of Schwarzschild spacetime). Notice also that in practical numerical computations, one does not give the initial condition precisely at r = 0, where the right-hand side of the TOV equation is indeterminate at best. Rather, one introduces a small cutoff r_0 > 0 and provides initial conditions at r_0. Here, we use this method with initial conditions of the form ρ(r_0) = ρ_0 and m(r_0) = 4/3πρ_0 r_0^3.]. One then proceeds with the integration until the boundary of the star is reached at r=R, defined by the condition that 

    P(R) = 0.

One then usually assumes that the star ends at R and ρ and P are understood to vanish for r > R. In this outer region, the metric is given by the standard Schwarzschild metric in vacuum with mass parameter

    M = m(R).


Once ρ, P, and m are known, one can solve Eq. (<ref>) to obtain ϕ. The boundary condition is then that the metric of the spacetime is continuous across the stellar boundary at r = R and translates into the condition [In numerical computations, it may be more efficient to solve for ϕ while solving for the remaining variables. This can be done by imposing any initial condition for ϕ at the center of the star, and then using the fact that Eq. (<ref>) is linear in ϕ to subtract the value obtained at the boundary and add the value desired at the boundary, hence fixing the appropriate boundary condition.]

    e^2ϕ(R) = (1 - 2M/R).


The boundary conditions at R, Eqs. (<ref>) to (<ref>), can be deduced from the Israel junction conditions <cit.> with the additional assumption that the stress-energy tensor is not singular at the stellar surface. Stars with discontinuous pressure can be described by allowing a thin matter shell at r=R, in which case the boundary conditions are relaxed as well.

In most practical applications in astrophysics, one is interested in a particular model for a star and wants to understand how this model gravitates. For example, one can consider an equation of state modeling a neutron star and use it to solve the TOV system. Nevertheless, this approach is restrictive in the study of negative-mass stars, whose equation of state is not predetermined. Instead of using only equations of state to fix a stellar solution, one can impose that the star should have a particular energy density profile. In other words, one imposes by hand that the function ρ(r) is given by an ansatz. With this function fixed, one then solves the TOV system with the condition that the star ends at some predefined radius r=R. This method was employed in Ref. <cit.> and seems more convenient for generating negative-mass stars since one can fine-tune the behavior of the stellar mass. 



 §.§ Example: The Schwarzschild Star

    A first example of a negative-mass star is based on a star of constant density profile ρ(r) = ρ_0. This is known as a Schwarzschild star <cit.>. Under this ansatz, the TOV system can be handled analytically. The mass parameter is given by
    
    m(r) = 4 πρ_0 r^3/3.

    The boundary condition m(R) = M fixes the value of M according to
    
    ρ_0 = 3 M/4 π R^3.

    We are interested in the case with M < 0 (and hence ρ_0 < 0).

    The solution for the pressure is the same as in the positive-mass case. It is given by 
    
    P(r) = ρ_0 [√(1 - 2M/R) - √(1 - 2M r^2/R^3)/√(1 - 2M r^2/R^3) - 3 √(1 - 2M/R)].

    The pressure is everywhere non-negative and finite if M/R < 4/9, which holds for a wide class of equations of state and is known as the Buchdahl limit <cit.>. In particular, notice the pressure is everywhere non-negative and finite when the star has negative mass. 

    Having found a star with a negative mass, there are two important questions we would like to address. 
    
        
  i. Is the matter composing this star allowed by quantum field theory?
        
  ii. Is this configuration stable?
    
    We consider them in the next two sections.



§ ENERGY CONDITIONS

It is not surprising that the Einstein equations allow for negative-mass stars. After all, any Lorentzian geometry is a solution to the Einstein equations as long as one chooses the right stress-energy-momentum tensor. 

To put constraints on the physical reasonableness of a given solution, one often imposes energy conditions <cit.>. These are restrictions on the stress tensor that enforce it to have some interesting properties, typically understood as energy being positive in a suitable sense. The weakest of the classical energy conditions is the null energy condition (NEC), reviewed for example in Refs. <cit.>. It states that for all null vectors k^a the stress tensor satisfies the bound

    T_a_bk^ak^b≥ 0.

In the particular case of interest, a perfect fluid, this condition states that ρ + P ≥ 0. 

The NEC is not the only energy condition of classical interest. It is important to point out the dominant energy condition (DEC), which states that for all future-directed causal vectors ξ^a it holds that

    -T^a_bξ^b

is causal and future-directed. This condition essentially imposes that all observers see causal, future-directed energy fluxes. For a perfect fluid, the DEC states that ρ≥P. Notice that the DEC implies the NEC. The interest in this condition is that it was used by Schoen, Yau, and Witten to obtain the first positive-mass theorems in general relativity <cit.>. Hence, within suitable assumptions, the dominant energy condition ensures that the spacetime has a positive mass.

All known forms of classical matter satisfy the NEC [It is known that some forms of classical matter, such as a non-minimally coupled scalar field, violate the NEC—see, , Refs. <cit.>. Nevertheless, we are restricting our focus to the classical fields associated with the standard model and we assume general relativity to be the theory describing gravitational interactions. In particular, once we assume gravity to be described by general relativity, we are also assuming that all standard model fields are minimally coupled. In any case, this does not affect the main points of our discussion.], and in fact even the DEC. Does our star?

The positive mass theorems due to Schoen, Yau, and Witten already imply that the star cannot satisfy the DEC. Using Eq. (<ref>), one can promptly show that

    ρ(r) + P(r) = ρ_0 [2 √(1 - 2M/R)/3 √(1 - 2M/R) - √(1 - 2Mr^2/R^3)].

By close inspection, one can tell that the term in brackets is always positive for 0 ≤ r ≤ R. Since ρ_0 < 0, we conclude that ρ(r) + P(r) < 0. Hence, the star cannot be built from any known form of classical matter.

This is not surprising. After all, we started this discussion by considering the Casimir effect, which is inherently quantum. There is no a priori reason to expect that classical configurations of matter could lead to negative energy solutions, but, in principle, one may question whether quantum configurations can lead to macroscopic negative masses since they are known to allow local violations of the positivity of energy—in fact, all classical energy conditions, including the NEC, can be easily violated by exploiting quantum effects, as shown in appendix <ref>. Is there a quantum analog of the classical energy conditions? 

There is. The averaged null energy condition (ANEC), in the conventions of Ref.  <cit.>, states that, for any inextendible null geodesic γ with affine parameter λ, it holds that

    ∫_-∞^+∞T_a_b(γ(λ)) γ̇^a(λ) γ̇^b(λ) λ≥ 0

whenever the integral is absolutely convergent. T_a_b is now understood as the renormalized expectation value of the stress tensor. Notice the idea is then that the NEC can be locally violated, but it still holds on average—an idea originally due to Tipler <cit.>. An even weaker condition is the achronal ANEC (AANEC), which states that the ANEC must hold only for achronal geodesics, , for null geodesics such that no pair of its points can be connected by a timelike curve. The restriction of the ANEC to achronal geodesics was originally considered by Wald and Yurtsever <cit.>, and it is interesting because there is physical reason to believe that the AANEC is true in all physically reasonable circumstances and that it could be implied by a fundamental principle of full quantum gravity—see Refs. <cit.>. Furthermore, the ANEC is equivalent to the quantum null energy condition <cit.>, which is interesting in its own right and is implied in turn by an interesting conjecture pertaining to the quantum focusing of congruences of geodesics <cit.>.

The fact that the ANEC demands that energy be positive on average is remarkably similar to how Costa and Matsas dealt with the Casimir system: while the energy density could be negative within the plates, it was positive somewhere else, and the net energy was positive. Hence, the ANEC appears to provide an interesting criterion to establish the net positivity of mass. Instead of focusing on forcing the energy density to be positive everywhere, we simply require that negative energy densities be compensated somewhere else. Furthermore, it should be mentioned that the ANEC does hold in the Casimir system <cit.>.

Assuming that every AANEC integral is absolutely convergent, this intuition is correct. This is a corollary of works by Borde <cit.> and Penrose, Sorkin, and Woolgar <cit.>, which we briefly review below. 

Borde's theorem is a statement about the occurrence of conjugate points in geodesics satisfying a weaker form of an averaged energy condition. His original result, focusing theorem 2 in Ref. <cit.>, applies to any causal geodesic, but we will focus on the case of null geodesics. Borde's theorem is the following. 


    Consider a complete null geodesic with tangent vector k^a and affine parameter λ. For each ϵ > 0, assume there is some b > 0 such that for any λ_1 < λ_2 there is a pair of intervals I^- < λ_1 and I^+ > λ_2—each with length larger than or equal to b—such that
    
    ∫_λ'^λ”R_a_bk^ak^bλ≥ - ϵ∀λ' ∈ I^-, ∀λ”∈ I^+.

    If k^ck^dk_[_aR_b_]_c_d_[_ek_f_]≠ 0 at some point on γ, then γ contains a pair of conjugate points.


This is a generalization of an earlier result due to Hawking and Penrose that establishes the occurrence of conjugate points based on the validity of the null convergence condition, which states that R_a_bk^ak^b≥ 0 for all null vectors k^a <cit.>. The basic idea of both proofs is to use the curvature condition to ensure that the geodesics do not defocus, while the condition k^ck^dk_[_aR_b_]_c_d_[_ek_f_]≠ 0 ensures that the geodesic interacts with curvature at some point for the focusing to start. The main technique in the proof consists in manipulating the Raychaudhuri equation <cit.>. 

Under the hypothesis of absolute convergence of the integral ∫_-∞^∞ R_abk^ak^bλ, we can show the following intuitive corollary.


    Consider a complete null geodesic with tangent vector k^a and affine parameter λ. Suppose that
    
    ∫_-∞^+∞R_a_bk^ak^bλ≥ 0

    and that the integral converges absolutely. If it holds that k^ck^dk_[_aR_b_]_c_d_[_ek_f_]≠ 0 at some point on γ, then γ must contain a pair of conjugate points.


The assumption of absolute convergence is used to ensure that the “tails” of the integrand as λ→ +∞ are negligible compared to the “bulk” contributions. Notice that if the Einstein field equations hold (as we assume they do), then the condition on the integral of R_a_bk^ak^b is equivalent to the ANEC integral being non-negative. Furthermore, it is known that achronal null geodesics cannot have conjugate points (Proposition 4.5.12 in Ref. <cit.>). Hence, we get the following result.


    Consider a complete null geodesic with tangent vector k^a and affine parameter λ. Suppose that the Einstein equations and the ANEC hold. If k^ck^dk_[_aR_b_]_c_d_[_ek_f_]≠ 0 at some point on γ, then γ is not achronal.


This was known to Penrose, Sorkin, and Woolgar at the time of writing of Ref. <cit.>. The result they established is then a result concerning the positivity of mass based on the existence of achronal geodesics in the spacetime. 

The basic idea behind the argument runs as follows. The Shapiro time delay <cit.> shows that null geodesics that pass closer to a positive mass are delayed (in coordinate time) relative to geodesics that pass farther away. Analogously, one concludes that geodesics passing close to a negative mass will be “faster” (in coordinate time) than geodesics that pass far from the negative mass. With this in mind, Penrose, Sorkin, and Woolgar constructed the fastest causal curve from the past null infinity to the future null infinity of the conformal completion of an asymptotically flat spacetime and showed this is an achronal geodesic. This geodesic can pass either through the interior of the spacetime or avoid it completely by travelling only through the conformal boundaries of the spacetime. Due to the Shapiro time delay, if the spacetime has negative mass, this geodesic must enter the spacetime. They then conclude that if the spacetime has no complete achronal null geodesics, it must follow that the spacetime mass must be non-negative. 

The precise statement is as follows <cit.>:


    Let M ∈ℝ and let B ⊆ℝ^3 be a ball with radius R > 2 M. Consider the metric *g^M_a_b on ℝ× (ℝ^3 ∖ B) with line element
    
    s_M^2 = - (1 - 2M/r) t^2 + (1 - 2M/r)^-1r^2 + r^2 Ω^2.

    A metric g_a_b on ℝ× (ℝ^3 ∖ B) is said to be uniformly Schwarzschildean if there is an M ∈ℝ such that, in the coordinates of Eq. (<ref>), it holds that
    
    g_μ_ν - *g^M_μ_ν = o(M r^-1), 
    ∂_μ(g_ν_ρ - *g^M_ν_ρ) = o(M r^-2).

    In the case M = 0, the above notation is understood to mean the metric is flat for r > R, where R ≥ 0 is some constant.



    Consider a spacetime (M,g) taken to be asymptotically flat at null and spatial infinity and uniformly Schwarzschildean. Let 𝒟 = I^-(ℐ^+) ∩ I^+(ℐ^-) be the domain of outer communications of the spacetime. Assume that 𝒟∪ℐ^+ ∪ℐ^- is globally hyperbolic as a subset of the conformal extension of M. If there are no achronal null geodesics connecting ℐ^- to ℐ^+ and passing through 𝒟, then it follows that the ADM four-momentum of M is future-causal.  


Recall that the ADM four-momentum (after Arnowitt, Deser, and Misner <cit.>) is a notion of four-momentum of the whole spacetime measured at a single instant of time (which is to be understood as a spacelike hypersurface of the spacetime). 

The Penrose–Sorkin–Woolgar theorem can then be combined with corollary <ref> to yield the following result. 


    Consider a spacetime (M,g) taken to be asymptotically flat at null and spatial infinity and uniformly Schwarzschildean. Let 𝒟 = I^-(ℐ^+) ∩ I^+(ℐ^-) be the domain of outer communications of the spacetime. Assume that 𝒟∪ℐ^+ ∪ℐ^- is globally hyperbolic as a subset of the conformal extension of M. Suppose the Einstein field equations hold, that the AANEC integrals are always absolutely convergent and non-negative, and that the null generic condition holds—, that for all null geodesics, there is some point at which k^ck^dk_[_aR_b_]_c_d_[_ek_f_]≠ 0. Then the ADM four-momentum of the spacetime is future-causal.


In the above result, the AANEC is sufficient (as opposed to the ANEC), because we only need to rule out the achronal geodesics. If a geodesic is already known to be chronal, it can be ignored. Notice then that the AANEC will typically be a vacuous truth. 

The exterior solution for a spherically symmetric star in general relativity is always the Schwarzschild solution due to Birkhoff's theorem. Hence, all spherically symmetric stellar spacetimes must be uniformly Schwarzschildean, and hence satisfy the conditions for the Penrose–Sorkin–Woolgar theorem. It follows that a star satisfying the AANEC and the null generic condition can never have negative mass in general relativity. Due to the strong evidence supporting the AANEC <cit.>, it seems unlikely that one can build negative-mass stars within the domain of the semiclassical Einstein equations. 



§ DYNAMICAL STABILITY OF STARS

From the perspective of averaged energy conditions, it seems unlikely that one can build negative-mass stars. However, this requires a lot of input from quantum theory, and it would be interesting to try to discard negative masses based purely on classical physics. With this in mind, in this section, we study this problem from the perspective of stability analysis. 

There are three obvious ways in which a star can be spontaneously destroyed: through hydrodynamic instabilities, thermodynamic instabilities, and the consumption of nuclear fuel. In regular stars, the time scales associated with hydrodynamic processes are much shorter than those associated with thermal processes, which, in turn, are much shorter than those associated with nuclear processes. Ref. <cit.>, for example, considers the Sun and estimates the time scales for hydrodynamic processes at around t_H ∼e3, for thermal processes at t_T ∼3e7, and t_N ∼e10 for the burning of nuclear fuel. Hence, hydrodynamic stability is the most important one in regular stars.

While the estimates for the Sun cannot be trusted when dealing with negative-mass stars, it still seems natural to focus on the hydrodynamic stability. While this is partially motivated by the time scale estimates we just mentioned, there is also the important reason that a thermodynamic or nuclear analysis would require a precise model of the matter constituting the star, for example in the form of an equation of state. This model, however, is not available, since we are precisely avoiding making too many assumptions on the composition of matter. Hence, we have insufficient information for discussing the thermal or nuclear stabilities of negative-mass stars. Furthermore, since they would necessarily be made of unconventional matter (or purely of quantum effects), one lacks a good justification for the use of traditional techniques developed for regular matter. Thus, we focus on the hydrodynamic stability.

The simplest way of discussing the hydrodynamic stability of a star is to consider linear perturbations of the stellar parameters and metric coefficients about the background spacetime provided by the TOV solution. One then writes the metric and the stellar parameters as the equilibrium solutions to the TOV equations plus small time-dependent perturbations. The Einstein equations for these perturbations are linearized. The end goal is to see whether the perturbations grow in time or remain bounded. If they grow in time, the star is deemed unstable. 

This problem was originally considered by Chandrasekhar <cit.> (see also the review in Chap. 26 of Ref. <cit.>), who simplified the problem to the study of solutions of the so-called Chandrasekhar pulsation equation. This is an equation that describes the Lagrangian displacement of the perturbation ξ. This is a measure of how much each fluid element in the star is dislocated by the perturbation. If the fluid element was originally at r before the perturbation, it is taken to r+ ξ(t,r) by the perturbation. The pulsation equation is given by

    r[p(r) χr] + q(r) χ(t,r) = w(r) [2]χt,

where χ is a “renormalized Lagrangian displacement”, from which all remaining perturbations can be promptly calculated. It is given in terms of the “true Lagrangian displacement” ξ by 

    χ(t,r) = r^2 e^-ϕ(r)ξ(t,r).

The coefficient functions p (not to be mistaken for the pressure P), q, and r are determined by the background stellar spacetime according to


    p(r)    = e^ψ + 3 ϕγ P/r^2, 
    
        q(r)    = e^ψ + 3 ϕ[1/r^2(P + ρ)(Pr)^2 - 4/r^3Pr - 8πP/r^2  e^2 ψ (P+ρ)], 
    
        w(r)    = (ρ + P)/r^2 e^3ψ + ϕ.


The coefficient p in the Chandrasekhar pulsation equation involves the “effective polytropic index”

    γ = 1/P(Pn)_ρ[n - (ρ + P) (nρ)_P],

where n is the baryon number density (with antibaryons counted negatively).

Typically, the boundary conditions for the problem are that <cit.> 

    lim_r → 0χ(t,r)/r^3 < ∞and[γ P e^ϕ/r^2χr]_R = 0.

The first one ensures χ and its derivative stay finite at the center of the star, while the second one enforces that fluid elements on the boundary of the star stay at the boundary of the star [This is done by imposing that the Lagrangian perturbation of the pressure vanishes at the boundary.]. Additional conditions may be imposed to keep ξ finite at the boundary <cit.>.

The solution of Eq. (<ref>) is carried out by the separation of variables χ(t,r) = τ(t)ζ(r). The resulting equations are 

    r[p(r) ζr] + q(r) ζ(r) + σ^2 w(r) ζ(r) = 0,

also known as the Chandrasekhar pulsation equation, and

    [2]τt + σ^2 τ = 0.


The boundary conditions take the form

    lim_r → 0ζ(r)/r^3 < ∞and[γ P e^ϕ/r^2ζr]_R = 0.


The solution to Eq. (<ref>) is 

    τ(t) = τ(0) cos(σ t) + τ̇(0)/σsin(σ t),

where the dot denotes a time derivative. Notice, in particular, that if σ^2 < 0 (, if σ is imaginary), then τ(t) grows exponentially with time for a generic initial condition. In this case, the perturbation is unstable, and so is the star. The main question for stability with the Chandrasekhar pulsation equation is then to study whether σ^2 > 0 always holds or whether there are exceptions. If there is a single admissible value of σ for which σ^2 < 0, then the star is unstable, because τ(t) will grow exponentially in time for this mode.

Eq. (<ref>), when combined with the boundary conditions of Eq. (<ref>), corresponds to a Sturm–Liouville problem <cit.> with eigenvalues σ^2. This determines the allowed values of σ. For example, in standard astrophysical applications, it holds that p(r) > 0 and w(r) > 0 in (0,R). This ensures the problem is a self-adjoint eigenvalue problem in a Hilbert space, and thus σ^2 is always real.

The standard methods for studying stability with the Chandrasekhar pulsation equation involve techniques from Sturm–Liouville theory. For example, method 2-C in Ref. <cit.> obtains the number of unstable modes by solving numerically the Chandrasekhar equation with σ^2 = 0 and counting the number of zeros in the solution. By the Sturm Comparison Theorem <cit.>, this number of zeros can be understood as the number of eigenvalues smaller than zero, , the number of eigenvalues with σ^2 < 0. This is only possible because, for usual stars, p(r) > 0 and w(r) > 0 for r ∈ (0,R), which means the problem is sufficiently regular to use standard results from Sturm–Liouville theory (although the endpoints r=0 and r=R are often singular). Alternatively, method 2-D in Ref. <cit.> uses a variational technique to obtain the sign of the smallest eigenvalue, which is a technique based on the so-called min-max principle of functional analysis <cit.>.

There are a few reasons, however, that keep the standard Chandrasekhar pulsation equation from being straightforward when dealing with negative-mass stars. The first is that the NEC violations which are required in negative-mass stars (otherwise the Penrose–Sorkin–Woolgar theorem would imply the star has positive mass) lead to sign flips in the coefficients p and w. This means we are apparently outside the domain of the regular Sturm–Liouville problem, and there is no a priori reason to believe that results such as the Sturm Comparison theorem still hold. Hence, at first glance, it seems challenging to extract the sign of σ^2 in a NEC-violating star.

The second reason is also practical in nature and involves the difficulty in knowing the equation of state for the star. Recall that the Chandrasekhar pulsation equation involves the effective polytropic index γ. For an adiabatic perturbation, one can rewrite this expression as <cit.>

    γ = ρ + P/P(Pρ)_s,

where s is the entropy per baryon in the star. In the case of a barotropic equation of state (, an equation of state of the form P = P(ρ)), we can write

    (Pρ)_s = Pr(ρr)^-1

and compute γ as a function of r without using further information about the equation of state. However, for an equation of state that is not barotropic, this ratio of radial derivatives could correspond to the partial derivative with different quantities being held constant because (*Pρ)_T, (*Pρ)_s, and (*Pρ)_n, for example, need not coincide. 

If the star in question was obtained from an equation of state, this is not a problem. However, stars obtained from energy density profiles provide us more freedom to exploit when building examples. For these stars, there is no a priori reason to believe that its constituent matter is barotropic. Hence, the need to know γ is particularly inconvenient.

Finally, a third objection to the Chandrasekhar pulsation equation is conceptual. One of the assumptions that enter Chandrasekhar's derivation is baryon number conservation. Some authors go as far as saying that “The most fundamental law of thermodynamics—even more fundamental than the `first' and `second' laws—is baryon conservation” <cit.>. Nevertheless, it is not clear whether this is applicable in the case of a negative-mass star. In the negative-mass case, it is natural to begin by generalizing “baryon number” to “particle number”, where particle refers to the particles of whichever field composes the star. Nevertheless, even with this generalization the conservation of particle number is still unclear. For example, the quantum state violating the NEC shown in appendix <ref> is not an eigenstate of the particle number operator, and as a consequence, the number of particles is not defined. Hence, the number of particles in a negative mass star is, in general, ill-defined. Hence, we prefer to avoid techniques which rely on baryon number conservation.

Luckily, within suitable assumptions, these difficulties can be overcome. The regularity issue can be solved by noticing that all coefficients flip signs precisely when P+ρ vanishes (p and q can undergo additional sign flips, but these are unimportant). Due to this, the problem can be multiplied by the sign of P+ρ to yield a sufficiently regular problem. This allows the use of some of the techniques in Ref. <cit.>. The main available technique is method 2-D, which is numerically costly, but manageable for our purposes. 

The difficulty with γ can be overcome in the particular case of a barotropic equation of state, and the validity of this hypothesis can be verified by making a parametric plot of P(r) against ρ(r) and checking whether the resulting graph yields a function. While this is not proof that the equation of state is barotropic, it gives evidence that the star can be well-approximated by a barotropic equation of state. In practice, however, it may not be possible to obtain this equation of state.

The difficulty concerning baryon number conservation can be solved in the particular case of an adiabatic perturbation. The hypothesis of baryon number conservation enters Chandrasekhar's derivation in order to express the Eulerian pressure perturbation P as a function of the Lagrangian displacement ξ. For a general perturbation, this method seems adequate. For an adiabatic perturbation, however, we can impose simply that the Lagrangian pressure perturbation Δ P satisfies

    Δ P = (Pρ)_s Δρ.

With the aid of Eq. (<ref>), one can show that this prescription yields the same result obtained by Chandrasekhar <cit.> using baryon number conservation. Recall that the Lagrangian perturbation Δα of a quantity α is the perturbative change in a quantity as one follows a fluid element, whereas the Eulerian perturbation α is the change as one considers a fixed point in space <cit.>. They are related, to linear order, by

    Δα(t,r) = α(t,r) + ξ(t,r) αr(r),

where *αr refers to the background (unperturbed) value of α.

We notice that an incorrect approach one could consider to avoid these difficulties is to impose that the perturbations satisfy a simple equation of state that is not necessarily the same as the one of the underlying star. For example, one could impose that the Lagrangian perturbation of the pressure of the star always vanishes. This would correspond to a dust-like perturbation. However, in this case, the equations describe the evolution of the dust particles on a background describing a static star, rather than the motion of the star itself. Hence, one cannot conclude anything about stability in such an approach, but rather only about the behavior of the new matter in the star. For example, one can conclude that grains of dust in the Sun move either toward the surface or the center, not that the Sun is unstable.

With all of these considerations, the remaining mathematical difficulty in the problem is that both the coefficients p and w in the Sturm–Liouville problem change signs. In particular, the fact that w changes sign means we are not considering an eigenvalue problem in a Hilbert space, but rather in a more general vector space with an indefinite inner product. This keeps us from using standard results from functional analysis on Hilbert spaces. However, we can manipulate the Chandrasekhar pulsation equation into a form corresponding to a self-adjoint problem in a Hilbert space. This is because all of the coefficient functions in the Sturm–Liouville problem change sign simultaneously, precisely at the point in which the NEC is violated. We can then remove the sign of the “NEC factor” P + ρ and get to a problem that is better behaved. In more detail,



 §.§ Alternative Form of the Chandrasekhar Pulsation Equation

    We assume the perturbations to be adiabatic, so that Eq. (<ref>) holds. Recall the Chandrasekhar equation in the Sturm–Liouville form of Eq. (<ref>), with the relevant functions being given in Eqs. (<ref>) to (<ref>) and (<ref>). Recalling that the TOV equations yield 
    
    Pr = - (P+ρ) ϕr,

    we find that the Sturm–Liouville coefficient functions become 
    
    
    p(r)    = e^ψ + 3 ϕP+ρ/r^2(Pρ)_s, 
    
            q(r)    = e^ψ + 3 ϕ[(P + ρ)/r^2(ϕr)^2 + 4(P+ρ)/r^3ϕr - 8πP/r^2  e^2 ψ (P+ρ)], 
    
            w(r)    = (ρ + P)/r^2 e^3ψ + ϕ,

    
    which used Eq. (<ref>). 
    
    We notice that all three functions above are proportional to P+ρ, which can flip its sign inside the star due to NEC violations and cause w to reverse its sign as well. Suppose we eliminate this sign term from the equation. In that case, we will get to a more standard Sturm–Liouville problem, which in the worst-case scenario will still correspond to a self-adjoint eigenvalue problem in a Hilbert space (rather than an indefinite inner product space).

    This can be easily done by defining the function 
    
    s(r) = sign(P(r)+ρ(r)).

    Where the sign function is defined by
    
    sign(x) = 
                +1,    if  x > 0, 
    
                0,    if  x = 0, 
    
                -1,    if  x < 0.


    We can now write the Chandrasekhar pulsation equation as
    
    r[s(r)p(r) ζr] + s(r)q(r) ζ(r) + σ^2 s(r)w(r) ζ(r) = 0,

    which is possible because p, q, and w all vanish at the point in which s(r) does. In particular, the derivative of s(r) does not lead to any problems because p(r) vanishes at the point in which the derivative does not. By inspecting Eq. (<ref>) one can tell that s(r)w(r) is manifestly non-negative, and this ensures the Sturm–Liouville problem takes place in a Hilbert space. 

    For future convenience, we introduce the shorthand notation
    
    p_⋆(r) = s(r)p(r)

    and similarly for q_⋆ and w_⋆.



 §.§ Variational Technique for Stability Analysis

    To see whether a Sturm–Liouville problem admits negative eigenvalues, we can use the so-called min-max principle (Theorem XIII.1 in Ref. <cit.>). For our purposes, it states that given a self-adjoint operator L with spectrum spec(L) on a Hilbert space, it holds that
    
    infspec(L) ≤⟨ψ|Lψ⟩/⟨ψ|⟩

    for all ψ∈Dom(L) with ψ≠ 0. Ref. <cit.> makes the additional assumption that L is bounded from below, but since we are only interested in knowing whether spec (L) extends to negative values, we do not need this additional assumption (having a spectrum which extends to minus infinity means there is a negative value in the spectrum). This is a costly albeit standard technique in the study of stellar stability used by Chandrasekhar in the same references in which the pulsation equation was introduced <cit.>. It corresponds to method 2-D in Ref. <cit.>. This method is also widely used in textbook quantum mechanics to find upper bounds on the ground state energy of a system <cit.>.
    
    In the Sturm–Liouville problem with coefficient functions p_⋆, q_⋆, and w_⋆ the inner product is given by 
    
    ⟨φ|ψ⟩ = ∫φ(r) ψ(r) w_⋆(r) r,

    where we assume the functions to be real. We can then cast the problem in the form
    
    L ζ = σ^2 ζ,

    with 
    
    L = - 1/w_⋆(r)[r[p_⋆(r) r] + q_⋆(r)].

    
    L is self-adjoint [The standard references <cit.> do not pay close attention to the difference between symmetric and self-adjoint operators, so neither will we. Details of this type within Sturm–Liouville theory can be found, for example, in Ref. <cit.>.] when we assume homogeneous boundary conditions (our case of interest). Since w_⋆ can only be zero at points in which p_⋆ and q_⋆ vanish (which turns out to be a point in which p_⋆' also vanishes) and all of these functions vanish in precisely the same way, the operator L is well-defined.
    
    Using integration by parts one can write
    
    ⟨ψ|Lψ⟩ = p_⋆(0)ψ'(0)ψ(0) - p_⋆(R)ψ'(R)ψ(R) 
     + ∫_0^R p_⋆(r) ψ'(r)^2 - q_⋆(r)ψ(r)^2 r.

    The boundary conditions we are interested in, given in Eq. (<ref>), are such that we can assume the boundary contributions to vanish. Hence, we can write that 
    
    infspec(L) ≤∫_0^R p_⋆(r) ψ'(r)^2 - q_⋆(r)ψ(r)^2 r/∫_0^R w_⋆(r) ψ(r)^2 r≡ S[ψ],

    where ψ is assumed to satisfy the boundary conditions in Eq. (<ref>). Since the denominator on the right-hand side is always positive, the question of stability is reduced to whether S[ψ] admits negative values for some ψ satisfying the boundary conditions.



 §.§ Bardeen's Technique for Stability Analysis

    An alternative method of studying the stability of a star using the Chandrasekhar pulsation equation is due to Bardeen <cit.> (see also Method 2-C in Ref. <cit.>). It uses the Sturm Comparison Theorem <cit.> and the fact that the n-th eigenfunction of a Sturm–Liouville problem has n roots to establish the number of unstable modes in the star. The basic idea is that increasing the eigenvalue in the Sturm–Liouville equation increases the number of roots a solution with fixed initial (not boundary) conditions has in the interval (this follows from the Sturm Comparison Theorem). As a consequence, one can count how many roots the numerical solution with σ^2=0 and this will correspond to the number of eigenvalues with σ^2 < 0. This number therefore corresponds to the number of unstable modes in the star.

    The basic process is the following. One numerically solves the pulsation equation with the ansatz σ^2 = 0. If the stellar surface is a regular point for the Sturm–Liouville problem, then one starts from there. Otherwise, one starts from the center. One counts how many roots the numerical solution has inside the star and this gives the number of unstable modes. If the numerical solution reaches the other end of the star (either the center or the boundary, depending on where integration started) and satisfies the boundary condition there, then σ^2 = 0 is one of the eigenvalues of the problem, meaning there is a mode with neutral stability.

    This method is much more efficient than the variational technique but has some drawbacks. The variational technique relies on a very general result from functional analysis (the min-max principle), while Bardeen's method depends on the Sturm Comparison Theorem, which is a result about a certain class of Sturm–Liouville problems. In particular, the theorem requires that p_⋆ be positive on (0,R), and hence Bardeen's technique cannot be applied to stars in which P+ρ flips signs or in which (*Pρ)_s < 0. This excludes all acceptable models of negative-mass stars (see section <ref>), but the method is still convenient to illustrate how the variational technique works.



 §.§ Example: Relativistic Polytropes

    As a warm-up, let us use these stability criteria against some simple examples. Namely, we consider relativistic polytropes in the sense originally considered by Tooper <cit.>. These models were also studied by Chandrasekhar in Ref. <cit.> as an example application of the original pulsation equation.

    The equation of state we will consider is
    
    P = ρ^Γ,

    where Γ is a constant. Since we will do the computations numerically, from this point onward we also set R_0 = 1, where R_0 is some arbitrarily chosen length scale. This length scale is to be understood as the typical length scale of the negative mass objects under consideration. Different choices for R_0 affect the quantitative results in the sense that the equations of state and density profiles considered are different depending on the value of R_0. For example, Eq. (<ref>) would typically involve a dimensional constant in front of ρ^Γ, which we are setting to 1. This constant depends on R_0. Due to the absence of a preferred length scale to be considered, we choose to keep R_0 unspecified. R_0 cannot be too small to the point of the classical description of spacetime to be untrustworthy, but very large values of R_0 would likely be uninteresting on experimental grounds.
    
    The mass-radius diagrams and mass per central density diagrams for some values of Γ are shown in figures <ref>, <ref>, and <ref>. In the Newtonian case, Γ = 4/3 and Γ = 5/3 are common stellar models <cit.>.

    

    

    

    The spiral structures in the mass-radius diagrams for these three different sequences of stars are a very generic feature of the TOV equation, as per Theorem 6.4 on Ref. <cit.>. The most important feature for stability analyses is the sign of 
    
    Mρ_0,

    where M is the equilibrium value of the mass, for it provides a back-of-the-envelope stability criterion (see, , Ref. <cit.>). Indeed, consider a sequence of stars and pick a star with mass M̅ > 0 and Mρ_0 > 0. Perturb this star by increasing the central density according to
    
    ρ_0 →ρ_0 + ρ_0

    (supposing ρ_0, ρ_0 > 0), assuming that M̅ is kept constant during this process. Since *Mρ_0 > 0, M̅ is smaller than the mass M of the equilibrium star with central density ρ_0 + ρ_0. This means the gravitational field of the perturbed star is weaker than it should be to retain equilibrium. Consequently, the increased pressure forces the star to expand, thereby decreasing the central pressure and density, and bringing the perturbed star back to the original equilibrium state. Notice that if we had *Mρ_0 < 0 instead, the perturbation would grow and the star would be unstable. It should be pointed out that this is a necessary, but not sufficient, criterion for stability. 

    To perform more detailed stability analyses, we need to fix the boundary conditions to be considered in the Chandrasekhar pulsation equation. Following Eq. (<ref>), we take 
    
    ζ(r) ∼ r^3  at  r ≈ 0

    and
    
    ζr = 0  at  r = R.

    
    We will first use the variational technique to obtain the sign of the lowest mode of oscillation. We pick a trial function and compute S[ψ] to check its sign. If it is negative, the star is unstable. To abide by the boundary conditions, we will use trial functions of the form 
    
    ψ_n(r) = r^3 - 3 R^-n/3+n r^3+n

    where the coefficient was adjusted so that ψ_n'(R) = 0. The constant n must be positive to obey the boundary conditions, but it is otherwise arbitrary. These types of trial functions were considered in Ref. <cit.>, where they were used with more elaborate techniques to estimate the first few eigenvalues of the Chandrasekhar pulsation equation. In scenarios in which the star is very inhomogeneous these trial functions can lead to misleading estimates. Yet, since we are only interested in the sign of the smallest eigenvalue we do not need to worry about precise approximations.

    Using these trial functions, we can compute S[ψ_n] and check its sign. The results are shown in table <ref>. We see that the variational criterion reproduces the back-of-the-envelope expectations we had for the sequences with Γ = 2 and Γ = 5/3, but not for Γ = 4/3. For the sequence with Γ = 4/3, the star with ρ_0 = 5.623e-1 passes the necessary condition for stability from the back-of-the-envelope criterion, but more careful calculations show it is not stable. In fact, it is well known <cit.> that in Newtonian theory a polytrope is only stable if Γ > 4/3, and the bound gets tighter in relativity. This is consistent with the fact that the back-of-the-envelope criterion is only a necessary, not sufficient, condition for stability.
    
    

    Of course, the positive values of S[ψ_n] in table <ref> do not imply stability, they simply fail to falsify it. In the absence of other methods, we could repeat the computation for other values of n and choices of test functions.

    In these examples, the pulsation equation is sufficiently well-behaved to employ Bardeen's technique. We report the results in table <ref>, all of which agree with the results obtained through variational methods. Notice that Bardeen's technique can establish stability and confirms that the positive values of table <ref> indeed correspond to stable stars.

    



§ EXAMPLES OF NEGATIVE-MASS STARS

We are now ready to show different stellar models with negative masses and study their stability. 

The two main methods for generating negative-mass stellar models are explored in Ref. <cit.>: proposing an equation of state, or giving an energy density profile. The latter is more adequate for generating unusual stars that partially satisfy the ANEC since it allows fine-tuning the stellar energy. We will focus on this method first. 



 §.§ Solutions with Density Profiles

    Generating solutions with a density profile involves different techniques than solving the TOV system with an equation of state. With an equation of state one integrates from the center outward to ensure the condition that m(0) = 0, which is necessary to avoid a Schwarzschild-like singularity at the origin. With a density profile, however, this condition is already ensured because one can define m directly as 
    
    m(r) = ∫_0^r 4 πρ(r') r'^2 r',

    where ρ(r) is a previously chosen function. Hence, it is secured that m(0) = 0. Hence, the initial condition we use instead is that P(R) = 0 for some previously chosen R. For simplicity, we take R=1. This immediately determines that the star's mass is M = m(1), which can be fine-tuned by carefully selecting ρ(r). We now solve the TOV system for the pressure, as the remaining variables ρ and m are already known. 
    
    It will also be interesting to consider the value of the quantity 
    
    F[ρ] = ∫_0^R (P+ρ) e^-ϕ(r)/√(1 - 2m(r)/r)r

    which, on account of Eq. (<ref>), is proportional to the value of the ANEC integral along a radial null geodesic. The sign of F[ρ] for a given density profile ρ decides whether the ANEC is satisfied for radial null geodesics. We know that, due to the Penrose–Sorkin–Woolgar theorem and Borde's theorem, the AANEC will fail in all of the negative-mass stellar spacetimes we are about to consider. However, F[ρ] allows us to understand this failure in more detail. 
    
    
    
    Table <ref> exhibits the masses and values of F[ρ] for different choices of profiles. All models have positive pressures in the interior of the star. Models <ref>, <ref>, and <ref> are particular cases of the ones considered in Ref. <cit.>. Models <ref>, <ref>, and <ref> to <ref> are natural generalizations of those models. All of these models have a large amount of negative energy density in the center of the star, and the absolute value of the energy density decreases from the center to the border. This is similar to a regular star, apart, of course, from the sign. Models <ref> to <ref> consistently violate the NEC in the sense that P(r) + ρ(r) < 0 for all r < R (some of them have P(R)+ρ(R) = 0). In this sense, they are completely made of negative energy.

    Models <ref> to <ref> are less usual. They start with a small amount of negative energy density at the center (often taken to be zero for simplicity) and the absolute value of the energy density grows as one gets farther away from the center. This is an uncommon situation and has the peculiar feature that the equation of state yields a non-vanishing pressure for a vanishing energy density, but we include this case for completeness because it is still easy enough to analyze. These models partially satisfy the NEC because in the deep interior of the star, the pressure is positive, but the energy density is very small. The NEC is still violated on the outer layers (as the Penrose–Sorkin–Woolgar theorem demands).

    Models <ref> to <ref> are the most interesting examples. Their density profiles are plotted in figure <ref>. These density profiles flip signs inside the star. In these two particular cases, the stars have a positive mass core, but their outer layers are made of negative mass densities. Since the pressure is everywhere positive inside the star, the core satisfies the NEC, but the outer layers do not. As one can tell from table <ref>, this can be fine-tuned in such a way that the ANEC is respected by radial geodesics (even though the Penrose–Sorkin–Woolgar theorem implies it must be violated for other geodesics). The reason is that radial null geodesics will cross a region with a substantial amount of positive energy, which suffices to balance out the negative amounts obtained in the outer shells. Nevertheless, since the outer shells have negative energy densities, a “glancing” null geodesic, , one that barely penetrates the stellar surface can cross a region of strictly negative energy and violate the ANEC. This is illustrated in figure <ref>. Incidentally, the Penrose–Sorkin–Woolgar theorem entails that at least one of these glancing geodesics must be inextendible and achronal.
    
    

    
    
    Once we consider the gravitational dynamics of particles with different mass signs as discussed in the introduction, configurations with sign flips appear to be unstable. The positive core is repelled by the outer layers but has nowhere to go. The perfect spherical symmetry of the star keeps the core in what seems to be an unstable equilibrium. Meanwhile, the outer layers fall toward the core. This argument suggests the impossibility of the reversed situation: a negative core would be attracted by the positive layers, which would be repelled by the core. It seems one would need, at the very least, negative pressures in the positive-energy region to keep the system together. Indeed, suppose the total mass M is negative and that ρ(R) > 0. Since P(R) = 0, we see that the TOV equation leads to
    
    Pr_R = - ρ(R) M/R(R-2M) > 0.

    Hence, P(r) must be negative for r < R sufficiently close to R. Negative pressures have been considered elsewhere in the study of ultra-compact objects, such as in the Mazur–Mottola gravastar <cit.>. We shall not focus on this direction in this work due to the reasons discussed at the end of section <ref>. 
    
    The above argument suggests that negative-mass stars with a sign flip should be unstable against perturbations that break spherical symmetry. It should be mentioned, however, that negative-mass stars without any sign flips also have curious equilibrium dynamics <cit.>. Since the gravitational interaction between negative masses is repulsive, the gravitational force tries to make a negative-mass star explode. Meanwhile, the pressure gradient applies an outward-pointing force on the fluid elements, which are then accelerated inward due to the negative mass sign. Hence, in a negative-mass star, the pressure pulls the star inward while gravity tries to make the star explode (rather than collapse). Thus, the roles of pressure and gravity in the stellar equilibrium dynamics are reversed.



 §.§ Solutions with Equations of State

    Next, we consider examples of stars with a negative mass arising from a (barotropic) equation of state. 

    First, we should mention it seems to be particularly difficult to use this method to generate finite stars with a sign flip, such as models <ref> and <ref> of table <ref>. The reason is as follows. Assuming the star has positive pressure, there is a point inside the star in which the NEC holds (because the pressure is always positive and the sign flip implies there is some point with positive energy density). If the NEC held everywhere, the star would have positive mass due to the Penrose–Sorkin–Woolgar theorem. Hence, the NEC must be violated somewhere. By continuity, this implies that there is a point with P+ρ = 0. Since the pressure obeys the TOV equation and we are assuming the equation of state to be barotropic, we find that
    
    ρr = - (P+ρ) 4π P r^3 + m/r[r-2m]ρP.

    We see then that, at the point with P+ρ = 0, both the derivatives of P and ρ typically vanish on account of Eqs. (<ref>) and (<ref>). Since the differential equations are of first order, this means P and ρ become constant from that point onward, which implies the star will be infinite.

    This behavior can be bypassed if the equation of state is fine-tuned so that *ρP diverges precisely when P+ρ vanishes. This makes P+ρ traverse the point where it is zero, and the TOV equation allows the pressure to continue evolving. 

    Such fine-tuning can be easily enforced when creating an equation of state, so it is not problematic as long as one is aware of it. There is, however, a second concern. Both the TOV equation and Eq. (<ref>) also have a factor of the form
    
    4 π P r^3 + m.

    Assuming the NEC holds at the center of the star and that the pressure is positive, one can conclude that the strong energy condition (SEC) holds at the center of the star: ρ(0) + 3P(0) ≥ 0 and ρ(0) + P(0) ≥ 0 <cit.>. In fact, we have ρ(0) + 3P(0) > 0, since P(0) > 0. This implies that, for sufficiently small r,
    
    4 π P r^3 + m = 4π/3∫_0^r (3P(0) + ρ(r')) r'^2 r' > 0,

    which follows from the fact that ρ(0) + 3P(0) > 0 implies ρ(r) + 3P(0) > 0 for sufficiently small r, by continuity. Hence, 4 π P r^3 + m is positive near the star's center. Meanwhile, it is negative at its boundary because there we have P(R) = 0 and m(R) < 0. Hence, it must flip sign at least once due to the intermediate value theorem. At the point in which 4 π P r^3 + m = 0, we run into a new problem as before: the evolution of P and ρ reaches an extremum. Since the derivative of m does not vanish, the solution manages to traverse this point and keep evolving. However, unless *ρP diverges to keep *ρr nonzero, ρ will “turn around”: the extremum point r_0 is reached and, if ρ decreased for r<r_0, it starts to grow for r>r_0. This means that if ρ approaches the value ρ^* in which P(ρ^*) = 0 for r<r_0 (signaling the end of the star), it departs from ρ^* for r>r_0.

    Counteracting this effect requires an even finer tuning of the equation of state to ensure that *ρP diverges exactly at the point in which 4 π P r^3 + m = 0. This difficulty is illustrated for model <ref> of table <ref> in figures <ref> and <ref>.

    

    

    With this in mind, we focus on stars with negative-semidefinite energy density. Novikov, Bisnovatyi-Kogan, and Novikov previously considered two examples of this kind <cit.>, so we consider one of their models (their other model, with a linear equation of state, does not lead to finite-radius stars) and a few generalizations. More specifically, we will consider stars with an equation of state with the form
    
    P = (-1)^Γρ^Γ,

    where Γ is a positive integer for simplicity[In astrophysics, it is more common to drop the signal in Eq. (<ref>) and consider Γ = 1 + 1/n. This would be a relativistic polytrope, such as the ones considered in section <ref>. However, since ρ is now negative, we keep Γ an integer so that the equation of state is differentiable.]. Ref. <cit.> considered this equation of state with Γ=1—which leads to an infinite star—and with Γ=2. Notice this model is essentially a relativistic polytrope with negative mass. 

    As noted in Ref. <cit.>, there is a bound on the parameters of the equation of state for the star to have finite size. If the (negative) central density ρ_0 is such that
    
    ρ_0 ≤ρ_c ≡ - 3^1/1-Γ,

    then the star is infinite. The reason is that for ρ_0 = ρ_c the star admits a solution with a constant density (and hence constant pressure). Notice that in the Γ=1 case, the bound diverges, and all central densities lead to infinite stars. 

    Figure <ref> exhibits the mass-radius diagrams and mass per central density diagrams for a few stars satisfying the equation of state (<ref>). It is easy to see how the masses (and thus the radii) diverge as ρ_0 approaches the critical central density ρ_c = -3^1/1-Γ. 

    

    It is interesting to point out that the models with the equation of state (<ref>) consistently violate the NEC.



§ STABILITY OF NEGATIVE-MASS STARS

The steps needed to perform the stability analysis of negative-mass stars are now the following:

    
  * choose a density profile or an equation of state (in which case a value for the central density will also be needed);
    
  * solve the TOV equation with this information and obtain the functions p_⋆, q_⋆, and w_⋆ of section <ref>;
    
  * if the model came from a density profile, verify that the parametric plot of (ρ(r),P(r)) is the graph of a function to ensure the model can be assumed to be barotropic;
    
  * employ the variational or the Bardeen technique to determine if the star is stable.


For some of the models we want to consider, the trial functions of Eq. (<ref>) will not be sufficient to establish instability. Hence, we define a new family of trial functions. To do so, we will first define the auxiliary function

    θ_0(x) = 
            0,    if  x ≤ 0, 
    exp-1/x,    if  x >0.

Here, θ_0 is a common example of a function that is everywhere smooth but fails to be analytic at x = 0. Using θ_0 we can construct the functions

    θ_a,b(x) = θ_0(a^2 - (x-b)^2)/θ_0(a^2).

These are standard examples of smooth bump functions. These functions are everywhere smooth but have compact support. Namely, θ_a,b(x) = 0 for any x such that x-b > a. For our purposes, we will consider bump functions with 0 < b-a < b+a < R, which ensures they are non-vanishing only in the interior of the star under consideration. Since the function and all of its derivatives vanish at the origin and the stellar surface, the boundary conditions for the pulsation equation are satisfied.



 §.§ Models with Density Profiles

Variational stability tests for the models arising from density profiles are shown in table <ref>. It turns out that, for many models considered in table <ref>, the trial function (<ref>) with n=2 was sufficient to establish instability. Recall that all of these models violate the NEC consistently inside the star: they are such that P(r) + ρ(r) < 0 for all r < R (but possibly not at r=R). Nevertheless, all of these models have *Pρ < 0, which renders p_⋆ negative. As a result, we cannot apply Bardeen's technique.



The trial function (<ref>) does not yield a good bound for the models that partially satisfy the NEC (models <ref> to <ref> in table <ref>). For them, a better result is obtained by using the bump function (<ref>). The results are shown in table <ref>. All models are unstable. Since p_⋆ changes its sign inside the star, we cannot apply Bardeen's technique.






 §.§ Schwarzschild Star

Tables <ref> and <ref> address almost all models of table <ref>, the exception being model <ref>—the Schwarzschild star. A naive calculation for model <ref> would lead to infinite *Pρ and to an ill-definition of p_⋆. This is because *ρr is zero since the density is constant. Nevertheless, one can treat (*Pρ)_s as finite by considering it corresponds to the value of the perturbations and that these perturbations do not have constant energy density. In Refs. <cit.>, Chandrasekhar does exactly this by treating γ (as defined on Eq. (<ref>)) as being constant. He then finds there is a critical value for γ which determines the onset of stability. We can establish a similar result. Define p̅_⋆ through p_⋆(r) = γp̅_⋆(r). For the Schwarzschild star, one can compute p̅_⋆, q_⋆, and w_⋆ exactly. The important properties about them is that p̅_⋆ and q_⋆ are negative throughout the interior of the star, p̅_⋆ only vanishes at the boundary, and q_⋆ never vanishes. The variational criterion establishes then that the star will be unstable if and only if,

    ∫_0^R p̅_⋆(r) ψ'(r)^2 rγ - ∫_0^R q_⋆(r)ψ(r)^2 r/∫_0^R w_⋆(r) ψ(r)^2 r < 0

for some ψ≠ 0 satisfying the boundary conditions. This means the necessary and sufficient condition for stability is 

    γ > ∫_0^R q_⋆(r)ψ(r)^2 r/∫_0^R p̅_⋆(r) ψ'(r)^2 r

for some ψ≠ 0 satisfying the boundary conditions. The expression on the right-hand side can never vanish. Hence, we can define a critical value for γ by

    γ_c = inf_ψ∫_0^R q_⋆(r)ψ(r)^2 r/∫_0^R p̅_⋆(r) ψ'(r)^2 r

where in the infimum it is understood that ψ≠ 0 and that the boundary conditions are satisfied. 

By considering the trial functions (<ref>), one can show that 

    γ_c ≤inf_n ∫_0^R q_⋆(r)ψ_n(r)^2 r/∫_0^R p̅_⋆(r) ψ_n'(r)^2 r≈0.777774

The actual infimum over n>0 can also be found analytically and occurs in the limit n → 0.

The bump functions (<ref>) generate much better estimates. Fix ϵ > 0 and denote I_ϵ = (ϵ,R-ϵ). Assume ϵ < b-a < b+a < 1 - ϵ, which means θ_a,b is compactly supported in I_ϵ. In this case, we have

    
    γ_c    ≤∫_0^R q_⋆(r)θ_a,b(r)^2 r/∫_0^R p̅_⋆(r) θ_a,b'(r)^2 r, 
       ≤max_I_ϵ q_⋆(r) ∫_0^R θ_a,b(r)^2 r/min_I_ϵp̅_⋆(r) ∫_0^R θ_a,b'(r)^2 r, 
       = c ∫_0^R θ_a,b(r)^2 r/∫_0^R θ_a,b'(r)^2 r,


where c is the finite and positive constant obtained from the ratio between the minimum and maximum. The minimum and maximum are finite and non-vanishing because we assumed only a compact region of the integral contributes. Notice that the value of b does not change the integral as long as we integrate over the whole support of θ_a,b. This leads to 

    γ_c ≤ c ∫_-a^+aθ_a,0(r)^2 r/∫_-a^+aθ_a,0'(r)^2 r≤ c I(a),

where I(a) is defined as

    I(a) = ∫_-a^+aθ_a,0(r)^2 r/∫_-a^+aθ_a,0'(r)^2 r.


These integrals can be evaluated analytically (see appendix <ref>). One finds that, for a ≪ 1, I(a) behaves as I(a) = a^4 + a^6 (Eq. (<ref>)). Hence, by taking a sequence of bump functions with smaller values of a we can bound γ_c by increasingly smaller values. In the limit when we take the infimum, we find that γ_c = 0.

Since γ_c vanishes, then the fact that the Schwarzschild star consistently violates the NEC together with Eq. (<ref>) implies that stability requires

    (Pρ)_s > 0.

We will comment on this later in this section.



 §.§ Models with an Equation of State

Stability tests for the models arising from the equation of state (<ref>) are shown in table <ref>. All tests indicated instability. For fixed Γ and n, the value of S[ψ_n] appeared to increase in absolute value as ρ_0/ρ_c increased. This suggests that more negative mass leads to less instability since σ^2 is bounded farther away from zero. Larger values of σ in absolute value are associated with faster exponential growth of the perturbations.



All of these models have negative values for p_⋆ due to the fact that (*Pρ)_s < 0. Hence, we cannot apply Bardeen's method.



 §.§ Missing Examples

When dealing with the Schwarzschild star, we found that it is possible to have stability if (*Pρ)_s > 0. If one pays close attention to the models we considered so far—both coming from profiles and from equations of state—one would notice that they all violate (*Pρ)_s > 0 at some point inside the star. Therefore, it is necessary to discuss this condition further.

Firstly, we notice that in a negative-mass star, it is expected that (*Pρ)_s < 0, which is the opposite of usual thermodynamics. This is motivated by the fact that regions with very large negative values of ρ will tend to induce stronger gravitational effects, and thus are more susceptible to “exploding” (recall that in a negative-mass star, the equilibrium dynamics are reversed). To counter this effect, large pressures are expected to be necessary to balance the forces and maintain equilibrium. One could argue, however, that this argument relies too heavily on intuition coming from positive energy physics, and therefore we will not depend solely on it.

Let us find a stronger argument then. Under the assumption of the absence of singularities and a barotropic equation of state, (*Pρ)_s > 0 implies negative energies are only possible if there are also negative pressures. The reason is as follows. First, notice this condition implies the pressure is a monotonically increasing function of the energy density. Next, notice that the TOV equation with vanishing energy density (, in the vacuum) leads to

    Pr = - 4 π P^2 r,

where we used the absence of singularities to ensure that m = 0. Since the equation of state is barotropic, we can write

    ρr = - 4 π P^2 r (ρP)_s = 0,

where the expression vanishes by the assumption that ρ = 0. Since (*ρP)_s ≠ 0, we conclude P = 0. Hence, the pressure must vanish when the energy density vanishes. Since P(ρ) is increasing, we conclude that (*Pρ)_s > 0 demands negative pressures for negative energy densities. Notice also that this means the NEC must be violated at all points with negative energy densities. In the slightly more general case with (*ρP)_s ≥ 0 we can conclude that the pressure is non-positive for negative energy densities (and thus the NEC is still violated).

Can a star with (*ρP)_s > 0 have negative mass and negative pressure? Consider a star with negative pressure throughout the interior: P(r) < 0 for all r ∈ [0,R). At the center, we have P(0) = P_0 < 0 and ρ(0) = ρ_0 < 0. Then the TOV equation tells us that, close to the center,

    Pr = - (P + ρ) 4 π P r^3 + m/r [r- 2m]≤ 0,

because P, ρ, and m are all negative there. We also have (*ρP)_s > 0, *ρr≤ 0 as well. Hence, P and ρ can only become more negative as one gets farther from the center. Hence, there is no point R with P(R) = 0 and the star is infinite. If we only assume (*Pρ)_s ≥ 0, we can adapt the argument by noticing that continuity impedes the positivity of *ρr at the points in which (*ρP)_s diverges.

This is expected. In a negative-mass star, gravity is responsible for expanding the star, while pressure is responsible for contracting it. For pressure to contract the star, the pressure gradient must point inward. Hence, if pressure is negative, the star cannot be finite because the pressure would need to grow from a negative value up to zero close to the border, and hence the pressure gradient would point in the wrong direction close to the border.

This can be made rigorous using the TOV equation. 


        Consider a spherically symmetric static star in general relativity with radius R > 0 defined by the condition that P(R) = 0, with P(r) < 0 for all R - ϵ < r < R, for some ϵ > 0. Assume that the star's pressure P is differentiable in [R-ϵ,R] and that the energy density ρ is continuous in this interval. If the star has a negative total mass, the NEC must be satisfied at a subinterval of [R-ϵ,R].
    

    Since the star has a negative total mass, we know that m(R) < 0. We also know that P(R) = 0 by definition. Notice then that
    
    4π P(R) R^3 + m(R) < 0

    and 
    
    R - 2m(R) > 0.


    Since P(r) < 0 for all R - ϵ < r < R, we know that *Pr≥ 0 at R. Hence, the TOV equation reads
    
    Pr_R = - (P(R)+ρ(R)) 4π P(R) R^3 + m(R)/R [R - 2m(R)]≥ 0.

    We see then that P(R) + ρ(R) ≥ 0.  

    If P(R) + ρ(R) > 0, the result follows from continuity. Assume next that P(R) + ρ(R) = 0. Since m(R) < 0, continuity implies there is some δ > 0 (δ < ϵ) such that m(r) < 0 for r ∈ (R-δ,R]. For r ∈ (R-δ,R) ⊊ (R-ϵ,R) we know that P(r) < 0. In particular,
    
    P(R - δ/2) < 0.

    Since P(R) = 0, the mean value theorem ensures that there is some point r^* ∈(R - δ/2,R) with P(r^*) > 0. At this point, we know that 4π P(r^*) (r^*)^3 + m(r^*) < 0. The TOV equation then enforces that P(r^*) + ρ(r^*) > 0. Continuity ensures positivity holds in an interval.


Hence, we learn that a negative mass star with negative pressure near the border must satisfy the NEC near the boundary. This contradicts our desires that (*Pρ)_s ≥ 0 with P(ρ=0) = 0. After all, there is at least one interval inside the star in which the pressure is negative, but the energy density is positive. We conclude that the scenarios with negative pressure and (*Pρ)_s ≥ 0 cannot lead to negative-mass stars. Since the star ends when the pressure changes sign, we conclude (*Pρ)_s < 0 somewhere in the star.

It should be noted that if (*Pρ)_s < 0 anywhere inside the star, then there is a region with p_⋆ < 0, as one can tell from Eq. (<ref>). This leads to two important observations.

    
  * Bardeen's technique is generally not applicable to negative mass stars, as it relies on the assumption that p_⋆ > 0 throughout the interior of the star (this is a condition for the Sturm Comparison Theorem).
    
  * One will often be able to use a small bump function with large enough derivatives to make S[θ_a,b] negative by restricting the support of θ_a,b to the region in which p_⋆ is negative.


This second observation corroborates that negative mass stars cannot be stable in general relativity. The variational criterion establishes that the star will be unstable if there is any ζ satisfying the boundary conditions (<ref>) such that

    ∫_0^R p_⋆(r) (ζ'(r))^2 r - ∫_0^R q_⋆(r) (ζ(r))^2 r < 0.

Suppose now that p_⋆ is negative and non-vanishing in some interval (c,d) ⊆ (0,R) (if p_⋆ is negative at any point, it is negative in an interval by continuity). Choose ζ to be a bump function of the form (<ref>) with support in (c,d). Then Eq. (<ref>) is equivalent to

    1 > ∫_c^d q_⋆(r) (ζ(r))^2 r/∫_c^d p_⋆(r) (ζ'(r))^2 r.

Notice that a sufficient condition for instability is that 

    1 > max q_⋆∫_c^d (ζ(r))^2 r/min p_⋆∫_c^d (ζ'(r))^2 r,

where the maximum and minimum are taken over the interval (c,d). We have now reduced the problem to something similar to the analysis of the Schwarzschild star. Namely, by taking ζ to be a sufficiently sharp bump function we can make the right-hand side arbitrarily small, which entails instability.

Hence, stars with (*Pρ)_s < 0 at any point inside the star are unstable. This does not require any assumption on the sign of the energy density [The argument fails in the particular scenario in which (*Pρ)_s < 0 only at points with P+ρ = 0, since then p_⋆ vanishes. Nevertheless, P+ρ = 0 means the pressure is constant by the TOV equation. Furthermore, since (*Pρ)_s < 0, applying the chain rule to the TOV equation also leads to the conclusion that *ρr = 0. Hence, if (*Pρ)_s < 0 at a point with P+ρ = 0, then the star will be infinite and the solution is not relevant for our purposes.]. 



 §.§ Instability Timescales

    We finally estimate the timescales associated with the hydrodynamical stabilities we considered. For this, we must recover the units.

    Let N(σ) denote the absolute value of the numerical value obtained for σ in the system of units we have been using so far, with G = c = R_0 = 1, as discussed at the end of section <ref>. Then the value of σ in an arbitrary system of units will be given by
    
    σ = N(σ) c/R_0.

    Accordingly, the timescale in which the instability happens is 
    
    T = R_0/N(σ) c.

    Notice then that the timescale is the time it takes for light to cross a length R_0 corrected by the numerical factor associated with the instability. 

    According to tables <ref>, <ref>, and <ref>, the smallest value we found for σ^2 was of order unity (Γ = 3, ρ_0 = 0.1ρ_c on table <ref>). Hence, the smallest value we found for N(σ) was at order N(σ) ∼ 1. Larger values would lead to smaller instability timescales, so this is the “most stable scenario”. 

    We thus find that the instability timescales for the models we considered are such that
    
    T ≲R_0/c.

    It would be incorrect to conclude from this expression that the instability is faster than light. One should recall that the instabilities we are considering are local in nature. They are derived from the pulsation equation, which describes how each fluid element moves around its equilibrium position. A way to picture this is to remember that Sturm–Liouville problems traditionally have an infinite and unbounded sequence of eigenvalues. Hence, even in a stable star, there are oscillation modes with arbitrarily large frequencies, which are thus associated with arbitrarily small oscillation periods. The phenomenon we are seeing here is the same. T does not measure the time it takes for the star to undergo gravitational collapse, for example, but rather the time it takes for the perturbations to grow significantly.

    For R_0 ∼7e8 (about the size of the solar radius), we get T ∼2e2, indicating that only larger structures with negative masses can live longer than a couple of minutes, posing the astrophysical question of their formation.



§ CONCLUSIONS

We have discussed the properties of negative-mass relativistic stars. In particular, we reviewed how quantum theory, grounded on Borde's and Penrose–Sorkin–Woolgar's theorems, hampers the existence of negative-mass stars. Furthermore, we found that classical general relativity abhors negative masses in the sense that all interesting models of negative mass stars considered in this work—arising both from an equation of state or an energy density profile—turned out to be unstable. A fortiori, we showed that any barotropic star with (*Pρ)_s < 0 at some point must be dynamically unstable, and also that all interesting barotropic negative-mass stars satisfy this condition. None of our stability analyses depend on the validity of energy conditions.

It is interesting to notice the implications of these results to the cosmic-weight watcher conjecture about negative masses. While general relativity cannot forbid an equilibrium solution with negative mass—after all any Lorentzian geometry is a solution to the Einstein field equations for the appropriate stress tensor—it may forbid stable solutions with negative mass. This happens in the cases considered in this work. While one can construct a negative-mass star, one cannot expect it to be stable under small hydrodynamical perturbations. 

Notice this gives a classical answer to a problem that may appear to be quantum in nature. One would typically expect that the energy conditions of quantum theory would be necessary, not only sufficient, to forbid negative-mass relativistic stars. The argument would be essentially that quantum field theory gives the input on the sorts of matter available to general relativity and thus forbids otherwise valid solutions. However, our results indicate that this is not the only mechanism used by the cosmic weight-watcher of Costa and Matsas to forbid negative masses. Stability is also an important mechanism that seems to rule out negative masses even at a classical level. 

We thank Daniel A. T. Vanzella, and George E. A. Matsas for illuminating discussions that initiated and fuelled this project. Part of the calculations in this work were carried out with the aid of  <cit.> (and in particular the  package <cit.>) in a license to the University of São Paulo (N. A. A.'s previous institution). N. A. A. was supported by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior—Brasil (CAPES)—Finance Code 001.




§ NEC VIOLATIONS IN QFT

    For completeness, in this appendix, we give a short, but standard, argument showing that one can violate the null energy condition in quantum field theory. The argument is similar to the one given in Refs. <cit.>. Shortly, we will find a state for which the expectation value of T̂_a_bk^ak^b— where T̂_a_b is the stress tensor operator, k^a is a null vector, and the colons denote normal ordering—becomes negative. This appendix uses Planck units G = c = ħ = 1.

    We consider a minimally coupled massive scalar field in Minkowski spacetime. The stress-energy tensor for such a field is known to be 
    
    T̂_a_b = ∇_aφ̂∇_bφ̂ - 1/2η_a_b[∇_cφ̂∇^cφ̂ + m^2 φ̂^2],

    where the hats remind us that we should consider these objects as operators. We will soon need to renormalize the stress tensor, which can be implemented by the normal-ordering prescription.

    For any null vector field k^a,
    
    T̂_a_bk^ak^b = k^a∇_aφ̂k^b∇_bφ̂≡ℋ̂,

    where we defined the “null Hamiltonian density” ℋ̂ with the sole intention of simplifying the notation. As a side note, if we were in the classical theory Eq. (<ref>) would be manifestly non-negative since we would have
    
    T_a_bk^ak^b = (k^a∇_aφ)^2 ≥ 0,

    showing that a minimally coupled classical scalar field always obeys the NEC.

    Our next step is to express the normal-ordered operator ℋ̂ in terms of creation and annihilation operators. For this, we decompose the quantum field φ̂ in creation and annihilation operators by writing
    
    φ̂(x) = 1/(2π)^d/2∫[â_p e^i p · x + â_p^† e^-i p · x] [d]p/√(2 ω_p),

    where p · x = p_μx^μ,
    
    p^ap_a = - (p^0)^2 + p^2 = -m^2

    is understood, and we denote ω_p = √(p^2 + m^2). For generality, we are assuming at this stage a d+1 dimensional spacetime.

    The decomposition in Eq. (<ref>) leads to the canonical commutation relations
    
    â_pâ_q^† = δ^(d)(p-q).


    From Eq. (<ref>) we extract
    
    k^b∇_bφ̂ = i/(2π)^d/2∫[â_p e^i p · x - â_p^† e^-i p · x] k^bp_b[d]p/√(2 ω_p).


    From Eqs. (<ref>) and (<ref>) we find that 
    
    
    ℋ̂ = -1/(2π)^d∫[â_pâ_q e^i (p + q) · x - â_p^†â_q e^-i (p - q) · x - â_q^†â_p e^-i (q-p) · x + â_p^†â_q^† e^-i (p + q) · x] (k^bp_b)(k^cq_c) [d]p[d]q/2 √(ω_pω_q).

    

    The next step is to find a state |ψ⟩ which yields ℋ̂(x)ψ < 0 in some region. We take
    
    |ψ⟩ = cosβ|0⟩ + sinβ|fg⟩,

    where β is a real parameter which we will fix later, |0⟩ is the (Minkowski) vacuum, and 
    
    |fg⟩ = (∫√(2 ω_p) f(p)/k^bp_bâ^†_p[d]p∫√(2 ω_q) g(q)/k^cq_câ^†_q[d]q) |0⟩

    is a smeared two-particle state. The smearing is necessary to later keep the expectation value of ℋ̂ well-defined. The factors of ω and k^ap_a are chosen for later convenience. The normalization of this state is given by
    
    ⟨f_2 g_2||f_1 g_1⟩ = (f_2,f_1) (g_2,g_1) + (f_2,g_1) (g_2,f_1),

    where 
    
    (f, g) = ∫ f^*(p) g(p) 2 ω_p[d]p/(k^bp_b)^2.

    For simplicity, we will take f = g. We obtain ⟨ff|=⟩ 1 imposing
    
    (f,f) = 1/√(2).

    
    We recall that the Minkowski vacuum is annihilated by all annihilation operators, and this implies
    
    ℋ̂(x)0 = 0.

    We also have the normalization condition ⟨0|=⟩ 1.

    We notice then that
    
    ℋ̂ψ = sin(2β) [0ℋ̂ff] 
     + sin^2βℋ̂ff.


    The matrix elements are given by
    
    0ℋ̂ff = - 2/(2π)^d∫ f(p) f(q) e^i (p+q) · x[d]p[d]q.

    and
    
    ℋ̂ff = 4/√(2) (2π)^d∫ f^*(p)f(q) e^- i (p-q)· x[d]p[d]q.


    In total, we find that
    
        
    ℋ̂(x)ψ = - 2 sin(2 β)/(2π)^d[∫ f(p) f(q) e^i (p+q) · x[d]p[d]q] + 4 sin^2β/√(2) (2π)^d∫ f^*(p)f(q) e^- i (p-q)· x[d]p[d]q.

    Once x is fixed, one can then choose β arbitrarily small so that the first term is negative. In the small β limit, the second term becomes negligible, exhibiting local violations of the NEC. 

    Let us consider now what happens when we integrate along a null geodesic parallel to k^a, which will allow us to consider the ANEC. For simplicity, we focus on a massive field m>0 in a two-dimensional Minkowski spacetime (d = 1). A curve parallel to k^a is described in coordinates by x^μ = λk^μ, where λ is an affine parameter. 

    With this in mind, we see that Eq. (<ref>) leads to the expression 
        
    ∫ℋ̂ψλ = - 2 sin(2 β)/(2π)^d-1[∫ f(p) f(q) δ((p+q) · k) [d]p[d]q] 
     + 4 sin^2β/√(2) (2π)^d-1∫ f^*(p)f(q) δ((p-q) · k) [d]p[d]q.

    
    The first integral vanishes because the delta's argument,
    
    (p+q)· k = (p + q) k - (ω_p + ω_q) k^0,

    is strictly positive since k^0 = k and ω_p > p. Notice the main reason this quantity cannot vanish is the fact that ω_p > 0, so we cannot have p^a = -q^a because ω_p = - ω_q is impossible.

    To deal with the second term, we use the simplifying assumption that d=1. Then the ANEC integral becomes
    
    ∫ℋ̂ψλ = 4 sin^2βω_p/√(2)∫f^*(p)f(p)/ω_p - pp,

    which is manifestly positive. Therefore, we see that the ANEC holds for the state |ψ⟩ of a massive scalar field in a two-dimensional Minkowski spacetime.



§ ANEC IN A STELLAR SPACETIME

In this appendix, we rewrite the ANEC integral in a way that is easier to compute in a spacetime representing a star. To do so, we begin by noticing that Eqs. (<ref>) and (<ref>) imply that 

    ∫T_a_bγ̇^aγ̇^bλ = ∫ (P+ρ) (u_aγ̇^a)^2 λ.


Notice that

    γ̇^a(λ) = tλ(t)^a + rλ(r)^a + φλ(φ)^a,

where we are considering an arbitrary null geodesic and will impose the radial condition only later. Notice that spherical symmetry allowed us to make the simplifying assumption that θ = π/2 without any loss of generality. 

The spacetime has a timelike Killing vector field and an axial Killing vector field (among others). This allows us to identify the conserved quantities 

    ε = - (t)^aγ̇^bg_a_b = e^2ϕ(r)tλ
    andℓ = (φ)^aγ̇^bg_a_b = r^2 φλ,

which are interpreted as energy and angular momentum.

It follows then that

    γ̇^a(λ) = ε e^-2ϕ(r)(t)^a + rλ(r)^a + ℓ/r^2(φ)^a,

and hence, using Eq. (<ref>),

    u_aγ̇^a = - e^-ϕ(r)ε.


The condition γ̇^aγ̇_a = 0 implies that 

    (rλ)^2 = (1 - 2m(r)/r)(ε e^-2ϕ(r) - ℓ^2/r^2).

Notice that (assuming 2m(r) < r for all r, which holds in our cases of interest) the point of closest approach, r_min, is characterized by

    r_min^2 e^-2ϕ(r_min) = ℓ^2/ε^2,

and if this equation admits multiple solutions, then r_min is the largest solution with r_min < R.

Let us focus on the time interval in which the coordinate r grows along the geodesic. We choose λ such that r(0) = r_min, so that this corresponds to computing the integral

    ∫_0^+∞T_a_bγ̇^aγ̇^bλ = ∫_0^+∞ (P+ρ) (u_aγ̇^a)^2 λ,

which we already know will yield 

    ∫_0^+∞T_a_bγ̇^aγ̇^bλ = ε^2 ∫_0^+∞ (P+ρ) e^-2ϕλ.

We can change the integration parameter from λ to r by using Eq. (<ref>) with the additional assumption that *rλ > 0. This yields, upon simplification,

    ∫_0^+∞T_a_bγ̇^aγ̇^bλ
     = ε∫_r_min^R(P(r)+ρ(r)) e^-ϕ(r)r/√((1 - 2 m(r)/r)(1 - ℓ^2/ε^2e^2ϕ(r)/r^2)).


To get the full ANEC integral, we notice that the expression is symmetric on whether r is increasing or decreasing. Hence, we find that

    ∫_-∞^+∞T_a_bγ̇^aγ̇^bλ
     = 2 ε∫_r_min^R(P(r)+ρ(r)) e^-ϕ(r)r/√((1 - 2 m(r)/r)(1 - ℓ^2/ε^2e^2ϕ(r)/r^2)).


In the particular case with ℓ = 0, corresponding to a radial geodesic, we get r_min = 0 and 

    ∫_-∞^+∞T_a_bγ̇^aγ̇^bλ = 2 ε∫_0^R (P+ρ) e^-ϕ(r)/√(1 - 2m(r)/r)r.




§ BUMP FUNCTION INTEGRALS

While dealing with the stability of the Schwarzschild star, we faced the integrals

    ∫_-a^aθ_a,0(r)^2 rand∫_-a^aθ'_a,0(r)^2 r.

In this appendix, we evaluate them exactly. We begin by defining

    I_1(a) = ∫_-a^aθ_a,0(r)^2 r

and

    I_2(a) = ∫_-a^aθ'_a,0(r)^2 r.


From the definition of θ_a,0 (Eq. <ref>), it is straightforward to show that

    I_1(a) = e^2/a^2∫_-a^a e^2/r^2-a^2r

and

    I_2(a) = 4 e^2/a^2∫_-a^ar^2 e^2/r^2-a^2/(r^2-a^2)^4r.


Using the fact that the integrands are even and performing the substitution r = ax we get that

    I_1(a) = 2 a e^2/a^2∫_0^1 e^2/a^2(x^2-1)x

and

    I_2(a) = 8 a^-5 e^2/a^2∫_0^1x^2 e^2/a^2(x^2-1)/(x^2-1)^4x.


Next, we make the substitutions 

    u = - 1/x^2-1,

which involve

    x = u/2 u √(u^2 - u).

This leads to 

    I_1(a) = a e^2/a^2∫_1^+∞e^-2u/a^2/u √(u^2 - u)u

and

    I_2(a) = 4 a^-5 e^2/a^2∫_1^+∞ e^-2u/a^2 u√(u^2 - u)u.


The problem of finding both I_1(a) and I_2(a) has now been reduced to computing integrals of the form

    J(z;α,β) ≡∫_1^+∞ e^- z u u^-α (u-1)^β-1u.

For I_1 we have z = 2/a^2, α = 3/2, and β = 1/2. For I_2, z = 2/a^2, α=-3/2, and β = 3/2. The trick to solve this integrals is the same one often used in symbolic integration: we express the integrals in terms of Meijer G-functions <cit.> and then apply an integration theorem. 

We thus begin by noticing that <cit.>

    e^-z u = G^1,0_0,1(z u| 0pt0),

which leads us to 

    J(z;α,β) = ∫_1^+∞ G^1,0_0,1(z u| 0pt0) u^β (u-1)^γu.

The result of this integral is known and given by <cit.>

    J(z;α,β) = Γ(β) G^2,0_1,2(z | 0ptαα-β,0),

where Γ is the gamma function. Therefore, we conclude that

    ∫_1^+∞e^-2u/a^2/u √(u^2 - u)u = √(π) G^2,0_1,2(2/a^2| 0pt3/20,1)

and

    ∫_1^+∞ e^-2u/a^2 u √(u^2 - u)u = √(π)/2 G^2,0_1,2(2/a^2| 0pt-3/2-3,0).


Bringing everything together, we find that

    I_1(a) = a √(π) e^2/a^2 G^2,0_1,2(2/a^2| 0pt3/20,1)

and

    I_2(a) = 2 √(π) a^-5 e^2/a^2 G^2,0_1,2(2/a^2| 0pt-3/2-3,0).


Ultimately, the quantity we are interested in is the function

    
    I(a)    = I_1(a)/I_2(a), 
       = a^6/2G^2,0_1,2(2/a^2| 0pt3/20,1)/G^2,0_1,2(2/a^2| 0pt-3/2-3,0).


More specifically, we are interested in the behavior of I(a) for small a. To that end, it is useful to use an asymptotic expansion for the Meijer G-functions. For large z ∈ℝ, z>0, it holds that <cit.>

    G^2,0_1,2(z| 0ptαα-β,0) ∼ e^-z z^-β(1 + ∑_k=1^+∞ M_k z^-k),

where M_k are constants determined by the specific values of α and β. We thus find that 

    G^2,0_1,2(z| 0pt3/20,1)/G^2,0_1,2(z| 0pt-3/2-3,0) = z + 1.


Using this asymptotic expression we conclude that, in the limit with a ≪ 1, it holds that

    I(a) = a^4 + a^6.

Hence, I(a) tends to zero as a → 0.






 Paper Title: Energy-time uncertainty relation from entropy measures.




An inhomogeneous p-laplacian equation with a Hardy potential
    Genival da Silva[email: gdasilva@tamusa.edu, website: <www.gdasilvajr.com>]
    August 5, 2024
===============================================================================





§ INTRODUCTION



Position (q) and momentum (p) are two complementary quantities of a quantum particle, satisfying an uncertainty relation first derived by H. Kennard <cit.> in 1927, and H. Weyl <cit.> in 1929, mathematically represented by

    Δ q Δ p ≥ħ/2,

where Δ q and Δ p represent the standard deviation of the position and momentum, respectively, and ħ is the reduced Planck constant. This equation expresses Heisenberg's Uncertainty Principle (HUP), which states that it is not possible to simultaneously determine the position and momentum of a quantum particle. The same expression is obtained by considering position and momentum as operators that satisfy the commutation relation

    [q̂, p̂] = iħ.

Substituting this into Robertson's uncertainty relation (1929) <cit.>,

    Δ A Δ B ≥1/2| ⟨ψ | [Â, B̂] | ψ⟩|,

where |ψ⟩ is the physical state of the system, we obtain (<ref>).

Heisenberg also postulated the energy-time uncertainty relation, expressed mathematically as

    Δ t Δ E ≥ħ/2,

where Δ E represents the energy dispersion and Δ t is defined as a specific time interval. The inequality (<ref>) cannot be interpreted in the same way as the well-known Heisenberg relation between position and momentum (<ref>). This distinction arises because in quantum mechanics, time is regarded as an evolution parameter rather than as an observable. Hence, it cannot be represented by a self-adjoint operator (conjugated to the Hamiltonian) as argued by Pauli due to the bounded nature of the continuous energy spectrum <cit.>.

The time-energy uncertainty relation was initially deduced in the work by Mandelstam and Tamm <cit.>, and subsequently discussed by various authors <cit.>.

There exist several methods to derive the energy-time uncertainty relation, (<ref>). Particularly noteworthy is the approach presented in <cit.>, which provides a derivation from first principles involving a canonical transformation in classical mechanics. This transformation yields a new canonical momentum (the energy E) and a new canonical coordinate T (referred to as tempus), which is conjugate to the energy. It is crucial to emphasize that tempus, as a new canonical coordinate, is conceptually distinct from the time t that denotes the system's evolution. The canonical invariance of the Poisson bracket ensures that energy and tempus follow the same algebraic relationships as momentum and position. Upon quantization of the system, the uncertainty relation (<ref>) emerges.



To introduce the quantum effects of gravity, the uncertainty relation for position and momentum, (<ref>), is extended to a Generalized Uncertainty Principle (GUP), which ensures a nonzero minimal uncertainty in position. In <cit.>, it is demonstrated that the GUP arises from the consideration of non-extensive entropies, denoted as S_±. The deformation parameters, which determine the magnitude of minimal uncertainty effects, are associated with S_+ and S_- entropies, and are negative and positive, respectively. This enables exploration of various possibilities in the quest for physical implications. Therefore, non-extensive statistics can be interpreted as a signature of quantum gravity.

In recent years some works have explored the relation between quantum gravity and non-extensivity as well from different perspectives<cit.>.

GUP is also derived from different proposals: in <cit.>, the scattering of strings at ultra-high energies is considered to analyze the divergences of quantum gravity at the Planck scale; in <cit.>, a gedanken experiment is proposed to measure the area of the apparent horizon of black holes in the context of quantum gravity; and <cit.> explores the idea that spacetime in the Planck region fluctuates, leading to the possibility of virtual micro-black holes affecting the measurement process.



The non-extensive entropies S_±, first introduced in <cit.>, were derived within the framework of Superstatistics <cit.>. These entropies depend only on the probabilities associated with the microscopic configuration of the system and enable exploration of non-equilibrium phenomena. These generalized entropies represent the entire family of entropies dependent solely on probability <cit.>. The statistics associated with S_+ and S_- exhibit distinct characteristics: the former gives rise to an effective potential related to an effective repulsive contribution term, while the latter contributes effectively attractively <cit.>.

In this work, we expand upon the results obtained in <cit.>. As observed, the generalized entropies ensuring quantum gravity effects lead to modifications in the uncertainty relation and, consequently, in the algebra between position and momentum. Similarly, these entropies are expected to modify the uncertainty relation between time and energy, (<ref>). 


The structure of the paper is as follows: In section <ref>, we present the derivation of the uncertainty relation between energy and time by introducing a canonical variable called tempus, which is conjugate to energy. In section <ref>, we discuss the origin of non-extensive entropies S_± in the domain of Superstatistics and how GUP emerges from them. In section <ref>, we derive the generalized energy-time uncertainty relation by examining the modified commutation relation between the tempus operator and the relativistic effective energy. Conclusions are given in section <ref>.




§ DERIVATION OF THE ENERGY-TIME UNCERTAINTY RELATION


A rigorous derivation of the uncertainty relation between time and energy was first provided by <cit.>. In this work, we will follow a different approach, as presented by <cit.>, where a new canonical variable, called tempus T, conjugate to energy E is defined through a canonical transformation. This approach ensures that they satisfy the classical Poisson algebra

    {T,E}=1,

similar to the position q and momentum p

    {q,p}=1.

According to <cit.>, tempus is related to time t in a conservative system as follows

    T=t-t_0,

here, t_0 is an arbitrary constant. Without loss of generality, t_0=0 can be set, making tempus equivalent to the time 
t in the classical regime.

































































































 
 
As we see from (<ref>) the tempus, and the energy are canonical conjugates. Therefore, when the system is quantized, the operators T̂ and Ê satisfy the commutation relation

    [T̂,Ê]=iħ,

and considering the Robertson relation (<ref>),



one obtains the uncertainty relation

    Δ TΔ E≥ħ/2.

The relationship between the standard deviation of the tempus operator, represented by Δ T, and the time interval Δ t is given by

    Δ T=|d⟨ T⟩/dt|Δ t,

and replacing in the tempus-energy uncertainty relation (<ref>), is obtained

    Δ tΔ E≥ħ/2/|d⟨ T⟩/dt|.

Now, with the help of Ehrenfest's Theorem, we can find the time derivative of the expected value of the operator T̂, considering that the operator itself is not explicitly time-dependent, i.e., ∂T̂/∂ t=0. Then,

    d⟨ T⟩/dt=1/iħ⟨[T̂,Ĥ]⟩,

and for a system with a time-independent Hamiltonian (conservative system) Ĥ=Ê, thus, from (<ref>) one arrives at energy-time uncertainty relation (<ref>). Reference <cit.> shows that the expectation value of the tempus operator is directly related to the time t of the system's evolution, which directly leads, from (<ref>), to (<ref>).






Another alternative way to obtain (<ref>) is from the uncertainty relationship for position and momentum (<ref>), considering Δ E=v_GΔ p and Δ q=v_GΔ t, where v_G is the group velocity of the wave packet. Considering these factors, an alternative derivation of the uncertainty relation between energy and time is obtained.




§ MODIFIED ENTROPIES AND THE GENERALIZED UNCERTAINTY PRINCIPLE


In the Heisenberg uncertainty relation, the gravitational interaction between particles was completely neglected, although this was somehow justified by the considerable weakness of gravity when compared with other fundamental interactions.

Incorporating gravity into quantum measurement processes led to the generalization of the uncertainty relation (<ref>). This modification is known as the Generalized Uncertainty Principle (GUP), which introduces additional terms that account for gravitational effects at very small scales, typically at the Planck scale.

GUP can alternatively be derived from modified entropies <cit.>, in particular of the ones that depend only on the probability <cit.>. These entropies are obtained in the Superstatistics formalism <cit.>, mathematically expressed as

    S_+   =   k_β∑_l^Ω(1-p_l^p_l),
    
        S_-   =   k_β∑_l^Ω(p_l^-p_l-1),

and in terms of the Boltzmann-Gibbs (BG) (Shannon) entropy S_BG=k_βlnΩ, for p_l=1/Ω, can be written as (considering the expansion)

    S_±=S_BG∓1/2!S_BG^2 e^-S_BG+⋯,

The associated probabilities and, consequently, the exponential and logarithms are also generalizations of the usual ones; the exponentials result in
   
    exp_±(-x)=exp(-x)∑_j=0^∞a^±_jx^j,

   where the dimensionless coefficients a^±_j are fixed <cit.>. One can then define in space of phase a generalized probability
   
    𝒫_±=exp(-H̅)∑_j=0^∞a̅^±_jH̅^j,

being H̅ the dimensionless Hamiltonian, and by these means generalized Hamiltonians <cit.>

    H_±=1/2m(p^2+α_±/3p^4+⋯),









in terms of low-energy momentum p. Then, it is possible to define a generalized momenta p_± in terms of p, as follows

    p_±=p+α_±/3 p^3+⋯,

where, α_±=α^0_±/M_x^2 (c=1 is assumed) is the deformed parameter, M_x is a certain energy scale, and α^0_± is a dimensionless constant

    α_±^0=3 ((a^±_1)^2-2a^±_2)/8 (1-a^±_1).

The commutation relation between p̂_± and q̂ can be calculated using equation (<ref>), and the usual commutation relation [q̂,p̂]=iħ

    [q̂,p̂_±]=iħ(1+α_±p̂^2+⋯),

taking into account that α_±p̂_±^2=α_±p̂^2+⋯, it allows us to express (<ref>) as

    [q̂,p̂_±]=iħ(1+α_± p̂_±^2).

A compact way to express the modified momentum p_± in (<ref>), which satisfies the commutation relation (<ref>), is

    p_±=1/√(α_±)tan(√(α_±)p),

where the domain of p is restricted to -π/2√(α_±)<p<π/2√(α_±). It is essential to mention that the coordinate q and the effective momentum p_± are no longer canonical conjugates <cit.>.

The generalized uncertainty relation associated with the modified algebra in (<ref>) is

    Δ qΔ p_±≥ħ/2(1+α_±(Δ p_±)^2),

which is a modification of the uncertainty relation in (<ref>), taking into account quantum corrections of gravity. The parameter α_±^0 can have positive and negative values, depending on the considered statistics, i.e.,  α^0_+=-0.560565 and α^0_-=0.361022 <cit.>.  When the positive value is set, (<ref>) introduce a minimal length in the position, (Δ q)_min=ħ√(α_-). If we choose the Planck scale M_x=M_Pl the parameter α^0_- is in order of unity and the minimal uncertainty in the position becomes the Planck length l_Pl.  On the other hand, when the negative value is chosen, a maximum uncertainty in the momentum is obtained, (Δ p)_max∼ 1/√(|α_+|).




§ GENERALIZED ENERGY-TIME UNCERTAINTY RELATION


It has been demonstrated that starting from non-extensive entropies, it is possible to obtain a generalization to the uncertainty relation (<ref>) to obtain (<ref>). Using the same arguments, we can find a modified uncertainty relation between time and energy, which should be a generalization of (<ref>) to include quantum gravity corrections.



Before finding the uncertainty relation between time and energy, we must express the relativistic version of the effective Hamiltonian H_± in (<ref>). In the relativistic regime, energy E and momentum p are written as the components of a four-momentum vector p_0=(E,p). Similarly, we should be able to write a modified four-momentum with components P_±=(E_±,p_±), where p_± is defined in (<ref>). On the other hand, E_± must be the relativistic energy for the Hamiltonian H_± in (<ref>). We define this as

    E_±=E+α_±/3 E^3+⋯.

It can be easily verified that this expression reduces to the Hamiltonian (<ref>) in the non-relativistic limit. With this purpose, we rewrite (<ref>) as follows

    E_±=E(1+α_±/3 E^2+⋯),

and considering the relativistic energy relation express as E= m√(1+p^2/m^2) in non-relativistic limit, we have

    E_±=m(1+p^2/2m^2)[1+α_±/3p^2+α_±/3m^2+⋯],

which reduces to

    E_±=m(1+α_±/3m^2)+(1+α_± m^2)p^2/2m+α_±/3p^4/2m+⋯.

As we have seen already, the deformation parameters α_± are expressed in terms of the Planck energy M_Pl through the relation α_±=α^0_±/M_Pl^2. This allows us to rewrite (<ref>) as

    E_±=m(1+α^0_±/3(m/M_Pl)^2)+(1+α^0_±(m/M_Pl)^2)p^2/2m+α_±/3p^4/2m+⋯,

where α^0_± is a dimensionless parameter defined in the previous section. On the other hand, m represents the energy at low scales, implying that M_Pl≫ m, and therefore m/M_Pl≪ 1.  Notice that the conditions m ≪ M_pl have to be stronger than the condition p ≪ m, i.e. m/M_Pl≪ p/m. This means that the mass of the particle has to be below the Planck scale, and therefore it can not be an excited state of string theory (for example). Considering this fact, we can write the effective energy (<ref>) in a familiar form

    E_±≃ m+1/2m(p^2+α_±/3p^4+⋯).

The first term represents the rest energy of the particle, and the second one represents the effective Hamiltonian (<ref>). With this, we demonstrate that (<ref>) is the relativistic expression of (<ref>). The same expression can be obtained from the effective momenta (<ref>) when a relatively high energy is applied to a particle with relatively low rest mass, it can be approximated as E≃ p in high-energy physics.

So, from (<ref>) we have

    E_±=p+α_±/3 p^3+⋯.

This expression is interpreted as a modified dispersion relation (MDR) derived from entropic principles. Building on other proposals in quantum gravity, similar MDRs to (<ref>) have been derived; for example, we cite <cit.>. In the non-relativistic approximation, the MDR in (<ref>) reduces, using (<ref>), to

    ω_±(k)=m/ħ+ħ k^2/2m(1+α_±ħ^2k^2/2m)+⋯,

where the de Broglie relations for energy and momentum for matter waves E = ħω and p = ħ k, respectively, have been utilized. Clearly, this expression reduces to the usual dispersion relation ω(k) ≈m/ħ + ħ k^2/2m when α_±→ 0. 

In <cit.>, it is argued that the emergence of MDR can be attributed to a fluctuating background dominated by quantum effects. This is because the dispersion relation of a particle propagating through a fluctuating spacetime differs from that of a particle moving in a smooth background. Furthermore, we can argue that the dispersion relation is modified due to non-extensivity, which is a signature of quantum gravity as it was discussed in <cit.>.









Starting from (<ref>), similar to what was done in (<ref>), we can calculate the commutation relation between the operator T̂ and the effective energy Ê_±, obtained from entropic principles. This yields

    [T̂,Ê_±]=iħ(1+α_±Ê^2+⋯),

which is a generalization of (<ref>), taking into account the quantum gravity effects associated with the non-extensivity of the entropies considered in (<ref>) and (<ref>). Unlike the standard case, tempus and energy aren't longer canonical conjugates. Replacing the deformed Heisenberg algebra (<ref>) in the general uncertainty relation (<ref>), we obtain

    Δ TΔ E_±≥ħ/2(1+α_± (Δ E)^2+⋯).

Similarly to the usual case, we can find the relationship between Δ T and Δ t by considering the fact that the operator T̂ has not been modified and satisfies the usual algebra with Ê. Therefore, for a conservative system, from (<ref>) we have that Δ T = Δ t. So, (<ref>) become in

    Δ tΔ E_±≥ħ/2(1+α_± (Δ E)^2+⋯).

This is the modified uncertainty relation for time and energy, derived from non-extensive entropies. Equation (<ref>) is a generalization of (<ref>) that incorporates high-energy corrections encoded in the non-extensive deformation parameter α_±. Additionally, if we consider the de Broglie relation for matter waves, then in (<ref>) we obtain the modified time-frequency uncertainty relation

    Δ tΔω_±≥1/2(1+α_±ħ^2 (Δω)^2+⋯),

for which we have considered Δ E_±=ħΔω_±. For example, in <cit.>, a Gaussian signal is considered, resulting in Δ t=1/√(2) and Δω=√((π-2)/2π), and therefore the usual uncertainty relation gives Δ tΔω=0.301. In this case, from (<ref>), we have Δ tΔω_+=0.301-2.38× 10^-54 and Δ tΔω_-=0.301+1.53× 10^-54. Clearly, in both cases, it can be seen that the first correction term is very small compared to 0.301. This leads us to interpret that the corrections introduced by the non-extensive entropies S_+ and S_- only play an important role at Planck scales (frequencies), which corroborates the results obtained in <cit.>.

Considering the usually relation Δ t Δ E ≥ħ/2, it can be concluded from (<ref>) that

    Δ E_±=Δ E(1+α_± (Δ E)^2+⋯).

This result can be interpreted in two ways. If the positive deformation parameter α_- is considered, the energy fluctuation increases in the region where the contribution of gravity is significant, that is, at Planck scales, which implies the appearance of an unobservable region Δ x that cannot be probed <cit.>. On the other hand, if the negative parameter α_+ is considered, the energy fluctuation decreases. Furthermore, as α_+ (Δ E)^2<1 must be satisfied in (<ref>), the energy fluctuation reaches its maximum in this scenario. This behavior has been observed in models that consider a crystal-like universe, whose lattice spacing is of the order of the Planck length <cit.>.

Now, we intend to calculate the quantum-corrected radiation temperature of a black hole. From (<ref>), we have

    Δ E_±ħ/2Δ t+α_±ħ/2Δ t(Δ E)^2+⋯.

In <cit.>, a heuristic derivation of the radiation temperature of a black hole is presented by considering particles as real gas at the horizon. Since these particles are moving in a space slice of thickness Δ x, each of them has an uncertainty in the kinetic energy equal to Δ E=3k_βΘ/2, where Θ is the temperature of this gas of real particles. This provides an estimate of the temperature. We will follow a similar procedure to calculate the contributions of quantum gravity effects to Hawking radiation. For a black hole, we have Δ t=2 R^2/GM and Δ E=ħ/16GM, where R=2GM is the Schwarzschild radius, G is the gravitational constant, and M is the mass of the black hole. Thus, in (<ref>), we have

    Δ E_±ħ/16 GM+α_±(ħ/16 GM)^3,

and the corresponding modified Hawking radiation temperature associated with the different statistics imposed by the entropies S_+ and S_- are

    Θ_+=   ħ/24k_β GM[1-9|α_+|k_β^2/4(ħ/24 k_β G M)^2],
    Θ_-=   ħ/24k_β GM[1+9α_-k_β^2/4(ħ/24 k_β G M)^2],

where k_β is the Boltzmann constant. The first term in (<ref>) and (<ref>) is the usual Hawking radiation temperature Θ=ħ/24k_β GM <cit.>, and the second term contains quantum gravity effects. In the classical limit, where the mass M of the black hole is sufficiently large, the correction terms in (<ref>) and (<ref>) are too small to be considered compared to the first term. For example, suppose M=M_sun=2× 10^30kg; the contribution of quantum gravity effects is on the order of 10^-96 compared to 1 for the first term. On the other hand, if we consider a micro black hole with a mass on the order of the Planck mass M∼ M_Pl, quantum gravity effects will be significant. This coincides with the observation in <cit.> that the modifications are relevant in the quantum gravity realm.  


From (<ref>), it can be seen that the radiation temperature for micro black holes, denoted as Θ_+, is lower than the Hawking radiation temperature Θ. This result implies that quantum gravity effects related to the non-extensivity associated with entropy S_+ decrease the black hole's radiation temperature. Conversely, quantum effects due to S_- increase the temperature of the quantum micro black hole, as shown in (<ref>), compared to the usual Hawking temperature. Several authors by other means have also obtained corrected black hole temperatures in different contexts <cit.>.



We can extend the commutation relation (<ref>) to a form analogous to (<ref>) considering all corrections terms in α_±, or equivalently, by setting α_±Ê_±^2=α_±Ê^2+⋯. Thus, we have

    [T̂,Ê_±]=iħ(1+α_±Ê_±^2),

and the corresponding uncertainty relation

    Δ TΔ E_±≥ħ/2(1+α_±(Δ E_±)^2).

Using the same argument as in (<ref>), we can express the modified uncertainty relation between time and energy from (<ref>), which is as follows

    Δ tΔ E_±≥ħ/2(1+α_±(Δ E_±)^2),

that is the generalized version of (<ref>), the energy-time Generalized Uncertainty Relation. As mentioned earlier, the deformation parameter α_± has two possible values, one positive, α_-, and the other negative, α_+, depending on the statistics considered. Choosing the positive parameter in (<ref>), associated with the entropy S_-, leads to a minimum time interval

    (Δ t)_min=t_min=ħ√(α_-).

On the contrary, if one chooses the negative value of the deformation parameter denoted by α_+, associated with the entropy S_+,  a maximum energy width is found

    (Δ E_+)_max=E_max=1/√(|α_+|).

Taking into account that in the Planck regimen α_±=α_±^0/M_Pl^2, then the minimum time in (<ref>) is on the order of the Planck time t_Pl, t_min=t_Pl√(α_-^0). In (<ref>), this implies that the maximum energy is on the order of the Planck energy M_Pl, i.e., E_max=M_Pl/√(|α_+^0|). Notice first that the extreme magnitudes are of the order of the Planck scale. Also, is important to see that from the different statistics we obtain minimum time or maximum energy. This is given by the signs of the α_±, meaning that one class of entropies corresponds to one case and the other entropy to the other one.


 Non-extensive statistics characterized by entropies S_± lead to distinct physical behaviors. For example, in (<ref>) and (<ref>), S_- leads to a minimum time and an increase in the radiation temperature of a mini black hole, while in (<ref>) and (<ref>), S_+ leads to maximum energy and a decrease in the Hawking temperature. This fact was already noted in <cit.> because the statistics associated with S_+ and S_- have different characteristics. The first entropy gives rise to an effective potential related to an effective repulsive contribution term, while the second one gives rise to an effective attractive contribution <cit.>.

 In <cit.>, the decay laws are discussed based on their relationship with the energy spectrum of the system <cit.>. Due to (<ref>), the representation of the energy eigenstates must change, which would result in a modified non-decay amplitude χ(t) <cit.>, and consequently, the decay law Q(t). In future work, we will analyze these and other related issues more carefully.




§ CONCLUSIONS


With the goal of introducing the quantum effects of gravity into measurement processes, in an effective manner, the uncertainty relation between position and momentum is modified (<ref>). This modification is named the Generalized Uncertainty Principle (GUP). Various approaches have demonstrated the emergence of GUP <cit.>. Notably, an intriguing connection has been established between non-extensive entropies and GUP, as S_± in (<ref>)-(<ref>), depending only on the probability, <cit.> (see (<ref>)). This leads us to the conclusion that non-extensivity can be interpreted as a consequence of quantum gravity. The statistics associated with S_+ and S_- yields distinct interpretations. For instance, those linked to S_+ result in a maximum uncertainty for the momentum, while those associated with S_- lead to a minimum uncertainty for the position. Furthermore, S_+ gives rise to an effective potential correlated with a repulsive interaction, whereas S_- gives rise to an effective attractive interaction <cit.>.

Using the probability distributions derived from the generalized entropies S_±, (<ref>), we obtain two effective Hamiltonians, labeled in (<ref>), which include correction terms at the Planck scale. At high-energy and high-velocity regimes, the energy and the momentum exhibit similar characteristics; therefore we express the effective Hamiltonian in its relativistic form (<ref>), in which the concepts of relativity and quantum theory are combined. As a result, from the generalized relativistic energy, the Modified Dispersion Relation (MDR) (<ref>) is derived. Unlike other proposals, such as <cit.>, this relation is obtained starting from entropic principles. This leads us to conclude that the dispersion relation is modified as a consequence of the non-extensivity, with the latter being a signature of quantum gravity <cit.>.

We employ the definition of the tempus operator <cit.>, computing the commutator with the relativistic effective energy. This yields a modified commutation relations (<ref>), leading to the generalized energy-time uncertainty relation (<ref>). Consistently with previous findings, the corrections arising from S_+ and S_- exhibit differences. The deformation parameter of the commutator α_+ derived from S_+ results in a decrease in energy fluctuation and the radiation temperature of a micro black hole, see (<ref>) and (<ref>), as well as in maximum uncertainty in energy (<ref>). Conversely, the deformation parameter α_- derived from S_- leads to an increase in the energy fluctuation and radiation temperature, as indicated in (<ref>) and (<ref>), as well as to the emergence of a minimum time interval (<ref>). In the Planck regime, the minimum time interval is on the order of the Planck time, and the maximum uncertainty in energy is of the order of the Planck energy.







The outcomes presented in this work strengthen the proposition that non-extensivity in entropy is a manifestation of quantum gravity <cit.>. The quantum corrections originating from the generalized entropies S_± prove to be substantial on both Planck scales (GUP). Grasping these findings unveils considerable potential for applying non-extensive entropies to understand phenomena entailing quantum gravity, a subject that will be explored further in forthcoming research.



N. C. B and W. Y thank the support of the University of Guanajuato grants:  CIIC 251/2024 ”Teorías efectivas de cuerdas y exploraciones de aprendizaje de máquina”,  CIIC 264/2022
"Geometría de dimensiones extras en teoría de cuerdas y sus aplicaciones físicas". NCB would like to thank the Grant CONAHCyT A-1-S-37752, “Teorías efectivas de cuerdas y sus aplicaciones
a la física de partículas y cosmología” and  University of Guanajuato Grant CIIC
224/2023 "Conjeturas de gravedad cuántica y el paisaje de la teoría de cuerdas". NCB would like to acknowledge 
support from the ICTP through the Associates Programme (2023-2029), and the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programme "Black holes: bridges between number theory and holographic quantum information" where work on this paper was undertaken. This work was supported by EPSRC grant no EP/R014604/1.  O. O and W. Y thanks to the grant by the University of Guanajuato CIIC 168/2023 "Non-extensive Entropies Independent of Parameters" and CIIC 156/2024 "Generalized Uncertainty Principle, Non-extensive Entropies, and General Relativity", as well as the CONAHCyT Grant CBF2023-2024-2923 "Implications of the Generalized Uncertainty Principle (GUP) in Quantum Cosmology, Gravitation, and its Connection with Non-extensive Entropies.
















JHEP




 Paper Title: Infinitely many new renormalization group flows between Virasoro minimal models from non-invertible symmetries.






YITP-24-92











 Infinitely many new renormalization group flows between Virasoro minimal models from  non-invertible symmetries



Takahiro Tanaka and Yu Nakayama









Yukawa Institute for Theoretical Physics,
Kyoto University, Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japan







§ ABSTRACT 
Based on the study of non-invertible symmetries, we propose there exist infinitely many new renormalization group flows between Virasoro minimal models  ℳ(kq + I, q) →ℳ(kq-I, q) induced by ϕ_(1,2k+1). They vastly generalize the previously proposed ones k=I=1 by Zamolodchikov, k=1, I>1 by Ahn and Lässig, and k=2  by Dorey et al. All the other ℤ_2 preserving renormalization group flows sporadically known in the literature (e.g. ℳ(10,3) →ℳ(8,3) studied by Klebanov et al) fall into our proposal (e.g. k=3, I=1). We claim our new flows give a complete understanding of the renormalization group flows between Virasoro minimal models that preserve a modular tensor category with the SU(2)_q-2 fusion ring.


empty 












§ INTRODUCTION

While we believe we know everything about the Virasoro minimal models in two-dimensional conformal field theories, the renormalization group flow between them, in particular when one of them is non-unitary has been largely unknown. The Virasoro minimal model ℳ(p,q) is characterized by two coprime integers p and q, and we can compute all the conformal data for an arbitrary choice of (p,q). It has been, however, a surprisingly difficult question to ask if they are connected by a renormalization group flow, given two randomly chosen integers (p,q) and (p',q'), Our goal is to answer this question by using the non-invertible symmetries to classify the flows. Indeed, we will show that non-invertible symmetries give rise to infinitely many new renormalization group flows between two Virasoro minimal models.

Our infinitely many new flows
 ℳ(kq + I, q) →ℳ(kq-I, q) induced by ϕ_(1, 2k + 1) vastly generalize the previously proposed ones k=I=1 by Zamolodchikov <cit.>, k=1, I>1 by Ahn <cit.> and Lässig <cit.>, and k=2 by Dorey et al <cit.>.[Dorey et al also studied the flows corresponding to k=1/2 with odd q <cit.>. Since the preserved non-invertible symmetries become smaller, we study the generalization of half-integer k separately in section <ref>.] It also encodes the other ℤ_2 preserving renormalization group flow sporadically known in the literature (e.g. ℳ(10,3) →ℳ(8,3) studied by Klebanov et al <cit.>). With a slight twist, the renormalization group flows between multi-critical Lee-Yang fixed points ℳ(2,q) studied in the literature <cit.> fall into our proposal.  We claim that our flows give a complete understanding of the renormalization group flows between minimal models that preserve a modular tensor category (or more precisely modular fusion category) with the SU(2)_q-2 fusion ring.[The fusion ring SU(2)_q-2 is given by the fusion rule of the primary operators in the SU(2) WZW model at level q-2. It is essentially angular momentum addition of J = 0, 1/2, ⋯ ,q-2/2 with a q dependent “cap": J_1+J_2 + J_3 ≤ q-2. Given the fusion ring, the modular tensor category is further classified by the solutions of the Pentagon identity. The relation between the solutions of the Pentagon identity and the renormalization group invariants we will discuss can be found in <cit.>.]  

In our discussions, the non-invertible symmetries or categorical symmetries,[Recent applications of non-invertible symmetries in two-dimensional conformal field theories include <cit.>. ] which are realized by topological defect lines in two-dimensional conformal field theories, will play an analog of the 't Hooft anomaly. Similarly to the 't Hooft anomaly, their properties are preserved under the renormalization group flows. When the renormalization group invariants obtained from the non-invertible symmetries are different, they cannot be connected by the renormalization group flow that preserves the symmetries. We will show that our proposed renormalization group flows are not only consistent with the constraint but also give the actual flows between two minimal models with the same preserved non-invertible symmetries.

Some of our results may appeal to more physical intuitions. For instance, the preserved non-invertible symmetries in ℳ(p,4) →ℳ(p',4)  flows are given by the ℤ_2 Tambara-Yamagami modular tensor category in math terms or the duality defect line in physics terms. We know from mathematics that there are two distinct quantum dimensions consistent with the ℤ_2 Tambara-Yamagami (or Ising) fusion rule. We also know from physics that the quantum dimensions are renormalization group invariants. Accordingly, there exist two intrinsically different duality defects in two-dimensional conformal field theories. Our new renormalization group flows know them and the flows are completely separated as long as the duality is preserved. One physical application of such a flow was to identify the fate of the non-supersymmetric Yukawa fixed point in two dimensions <cit.>. As a vast generalization, our new results should give us a new map to explore renormalization group flows in two-dimensional quantum field theories.


The organization of our paper is as follows. In section <ref>, we give a review of Virasoro minimal models and topological defect lines. In section <ref>, we present our main claim of the new renormalization group flows and give supporting evidence from the study of the renormalization group invariants associated with the non-invertible symmetries. In section <ref>, we study several physically interesting examples of our new renormalization group flows. In section <ref>, we conclude with some discussions for future directions.









§ VIRASORO MINIMAL MODELS

Let us first state our conventions of Virasoro minimal models. We specify the Virasoro minimal model ℳ(p, q) by two coprime integers p and q. While our convention is more or less the same as the one in the yellow book <cit.>, one notable exception is they always assume p > q in the yellow book, but we take q to be a fixed integer, and we investigate the renormalization group flow that changes p. For instance, the well-known renormalization group flow from the tricritical Ising model to the critical Ising model is ℳ(5,4) →ℳ(3,4) in our paper rather than ℳ(5,4) →ℳ(4,3) as in the yellow book.





The central charge of ℳ(p, q) is 

    c = c̅ = 1 - 6(p - q)^2pq

and it has (p - 1)(q - 1)/2 (chiral) primary operators ϕ_(r, s) where r and s are the Kac indices whose ranges are 1 ≤ r < q, 1 ≤ s < p. Two primary operators ϕ_(r, s) and ϕ_(q -  r, p - s) are identified: they have the same conformal weight 

    h_r, s = h_q - r, p - s = (pr - qs)^2 - (p - q)^24pq.

The fusion rule of the primary operators is given by

    ϕ_(r, s)×ϕ_(m, n) = ∑_k = 1 + | r - m |
     k + r + m = 12^min (r + m - 1, 2q - 1 - r - m)∑_l = 1 + | s - n |
     l + s + n = 12^min (s + n - 1, 2p - 1 - s - n)ϕ_(k, l).

For example, we can choose a fundamental domain Γ of the Kac table, which specifies (p - 1)(q - 1)/2 distinct Kac indices as

    Γ = { (r, s) | 1 ≤ r ≤ q - 1, 1 ≤ s ≤ p - 1, pr + qs < pq }.

See Figure <ref>.




The Virasoro minimal models are further classified by the modular invariant partition functions. In this paper, we focus on those with A-series modular invariant partition functions (a.k.a. A-series minimal models). Since most of the renormalization group flows that we will discuss preserve the ℤ_2 symmetry, which can be used to relate the A-series and D-series by orbifolding (when it can be gauged), most of the following discussions apply to the D-series minimal models. Exceptions are half-integer k flow discussed in section <ref> because the deformation may not exist in D-series. Furthermore, when we can gauge the ℤ_2 symmetry, we have fermionic minimal models <cit.> as well. Our discussions also apply to their cases. Again, exceptions are half-integer k flow discussed in section <ref>. Our discussions do not directly apply to the E-series minimal models, however, and we need separate considerations. Some studies of the topological defect lines in E-series minimal models in view of the renormalization group flows can be found in <cit.>.







 §.§ Non-invertible symmetries and topological defect lines

In two-dimensional quantum field theories, non-invertible symmetries are synonyms of topological defect lines. In Virasoro minimal models with A-series modular invariant partition functions, it is widely believed that all the topological defect lines are given by the Verlinde lines.[To avoid trivial counterexamples, we here exclude space-time symmetries such as Poincaré transformation or space-time parity.] Verlinde lines L_(r,s) have the same label (r,s) as the (chiral) primary operators ϕ_(r, s) of the same theory, so  (p-1)(q-1)/2 of them are independent.

 The action of L_(r, s) on a primary state |ϕ_(ρ, σ)⟩ = ϕ_(ρ, σ)|0⟩ is given by

    L_(r, s)|ϕ_(ρ, σ)⟩ = S_(r, s), (ρ, σ)S_(1, 1), (ρ, σ)|ϕ_(ρ, σ)⟩,

where S_(r, s), (ρ, σ) is modular S-matrix.  The explicit expression of the S-matrix is

    S_(r, s), (ρ, σ) = 2√(2pq)(-1)^1 + sρ + rσsin(πpqrρ)sin(πqpsσ),

which is real and symmetric.

As a special case, the action of L_(r, s) on the vacuum |0⟩ = |ϕ_(1, 1)⟩ is given by

    L_(r, s)|0⟩ = d_(r, s)|0⟩ = S_(r, s), (1, 1)S_(1, 1), (1, 1)|0⟩,

where the eigenvalue d_(r, s) is called quantum dimension of L_(r, s). The salient property of the quantum dimension is that it is a renormalization group invariant when L_(r,s) commutes with the deforming operators as we will see.

The Verlinde lines satisfy the same fusion rule as (chiral) primary operators:

    L_(r, s)× L_(m, n)   =  ∑ N^(k,l)_(r,s)(m,n) L_(k,l)   = ∑_k = 1 + | r - m |
     k + r + m = 12^min (r + m - 1, 2q - 1 - r - m)∑_l = 1 + | s - n |
     l + s + n = 12^min (s + n - 1, 2p - 1 - s - n) L_(k, l).

The fusion coefficients N^(k,l)_(r,s)(m,n) are related to the modular S-matrix by the Verlinde formula:

    N_ab^c = ∑_d S_ad S_bd S_dc/S_0d,

where we collectively denote (r,s) by a etc. This short-hand collective notation is sometimes used hereafter.



 §.§ Invertible TEZT symmetry

All the Virasoro minimal models with A-series modular invariance (except when p or q is 2) possess a special topological defect line L_(q - 1, 1) corresponding to the invertible ℤ_2 symmetry. It is invertible because the fusion rule (see e.g. (<ref>)) gives L_(q - 1, 1)× L_(q - 1, 1) = 1. 

 Under the action of L_(q - 1, 1), ℤ_2 charge of the primary operators ϕ_(r, s) can be summarized as  

    r-1      2   (when q is even) 
    
        s-1      2    (when p is even) 
    
        r+s-2      2    (when q and p are both odd)  .

See e.g. <cit.>.
The ℤ_2 symmetry is compatible with the fusion rule of the primary operators and operator identifications. If we gauge the ℤ_2 symmetry (or orbifold it in the conformal field theory language), A-series minimal models are exchanged with the D-series minimal models unless it is anomalous.

At this point, we note that the ℤ_2 symmetry is anomalous when p and q are both odd. One way to see this is to compute the quantum dimensions of L_(q - 1, 1), which is -1 rather than +1.  As we will review in the next subsection, the quantum dimension is a renormalization group invariant and it gives a selection rule of the renormalization group flow as the 't Hooft anomaly. We can also compute the spin contents of the defect Hilbert space, which will be again reviewed in the next subsection, and see that it is ±1/4 rather than 0,±1/2. As discussed in <cit.> this is a clear signal of the 't Hooft anomaly. Let us also recall that there is no D-series modular invariant partition function when p and q are both odd. This means that we cannot gauge the ℤ_2 invertible symmetry and it is consistent with the existence of the anomaly. 

In this paper, we focus on the renormalization group flow that preserves the ℤ_2 invertible symmetry. To be more precise, our renormalization group flow will preserve a larger modular tensor category with the SU(2)_q-2 fusion ring. This modular tensor category includes the ℤ_2 invertible symmetry but gives extra q-3 non-invertible symmetries.  In the next subsection, we will review how the preserved non-invertible symmetries will constrain the renormalization group flows.




 §.§ Renormalization group invariants

The salient feature of the topological defect lines regarded as non-invertible symmetries is that they give renormalization group invariants when they are preserved along the flow. Let us first discuss the condition that the topological defect lines are preserved along the renormalization group flow in the general setup.

Suppose we are at the ultraviolet conformal field theory with a topological defect line L_a. We deform the theory by a relevant operator ϕ_b. More generally we have to consider a collection of relevant operators {ϕ_b } to reach the desired infrared fixed point. In such a case, we assume that the following conditions are met with all the operators we need. Now, we say that L_a is preserved along the renormalization group flow when they commute with ϕ_b:  L_aϕ_b = ϕ_b L_a on the cylinder. More explicitly, they must satisfy

    L_a ϕ_b |Φ⟩ = ϕ_b L_a |Φ⟩

on any state |Φ⟩ on the cylinder. In unitary conformal field theories, it is sufficient to check it on the vacuum |0⟩ = |ϕ_(1, 1)⟩ <cit.>, but in non-unitary conformal field theories, we may have to study all the states. We will, however, explicitly see that in our discussions we do not gain any new constraint by studying all the states rather than the vacuum alone.

In the mathematics language, topological defect lines give a realization of the modular tensor category. As a modular tensor category, the physical properties of the topological defect lines are severely constrained. It is believed that the finite-rank modular tensor categories have no continuous parameters (so-called rigidity). Therefore, when the topological defect lines are preserved, we expect that the properties of the topological defect lines as a modular tensor category should be preserved by regarding the renormalization group flow as a continuous deformation.

Some properties of the topological defect lines are directly related to the 't Hooft anomaly i.e. an obstruction to gauge the symmetry, in which case it is not necessary to assume the non-existence of the continuous deformation of the modular tensor category to deduce the renormalization group invariance of the modular tensor category. It is simply the 't Hooft anomaly matching. 

In this paper, we focus on two such properties. One is the quantum dimensions of topological defect lines and the other is the spin contents of the defect Hilbert space. See e.g. <cit.> for studies of these properties in various examples.

We have already introduced the quantum dimensions of the topological defect lines L_a in (<ref>). 
 The renormalization group invariance of the quantum dimensions can be understood as follows. Recall that the fusion of the topological defect lines is

    L_a × L_b = ∑_c N_ab^cL_c,

where N_ab^c is the fusion coefficients. 

Let us now take the vacuum expectation value of this equality: ⟨ L_a ⟩ := ⟨ 0|L_a | 0 ⟩, where |0⟩ = |ϕ_(1,1)⟩ on the cylinder. The topological defect lines are topological, and we can place them far away. Then we may assume that the expectation value is factorized[Potentially, this argument can fail because we are applying the idea developed in unitary quantum field theories to non-unitary conformal field theories. It, however, turns out that the quantum dimensions that we can compute explicitly in non-unitary minimal models do satisfy the constraints obtained from this cluster decomposition ansatz.]

    ⟨ L_a ⟩×⟨ L_b ⟩ = ∑_c N_ab^c⟨ L_c ⟩.

The solutions of these quadratic equations turn out to be discrete, so it is invariant under the continuous renormalization group flows.

As the simplest example, let us consider the ℤ_2 invertible symmetry with the fusion rule L_(q - 1, 1)× L_(q - 1, 1) = L_(1, 1) (= 1). 
 By taking the vacuum expectation value, we can immediately see that ⟨ L_(q - 1, 1)⟩ = ± 1. The sign choice corresponds to the anomaly of the ℤ_2 invertible symmetry.[Whether one can choose the sign depends on the constraint from the other fusion rules. In Appendix C of <cit.>, we can find the discussion of why the minus sign is still consistent with the ℤ_2 selection rule in correlation functions.] In this case, the renormalization group invariance of the quantum dimension is equivalent to the 't Hooft anomaly matching. 


The other renormalization group invariants we will discuss are the spin contents of the defect Hilbert space associated with the preserved topological defect line L_a. To introduce the notion of the defect Hilbert space and the spin contents, we begin with the torus partition function with the insertion of L_a in the spatial direction.

    Z_L_a(τ, τ̅)    = Tr( L_a q^L_0-c/24q̅^L̅_0-c/24)    = ∑_b S_a b/S_0bχ_b(τ) χ̅_b(τ̅) ,

where q= e^2π i τ and χ_b is the Virasoro character of the primary operator ϕ_b.

Let us perform the S-modular transformation (τ̃ = -τ^-1) to regard the partition function as the sum over the defect Hilbert space ℋ_L_a:

    Z_L_a(τ, τ̅)    = ∑_b, c, dS_a b/S_0b S_bc S_bdχ_c(τ̃) χ̅_d(τ̅̃̅)     = ∑_c, d N_cd^aχ_c(τ̃) χ̅_d(τ̅̃̅)    = Tr_ℋ_L_a(  q̃^L_0-c/24q̅̃̅^L̅_0-c/24),

where we have used the Verlinde formula (<ref>). In the last line, the topological defect line is inserted in the time direction, and the quantization condition on the spatial circle is modified.
The spin contents of the defect Hilbert space ℋ_L_a are defined by the set of “Lorents spin" { h_c - h_d  (mod ℤ) } evaluated over non-zero N_cd^a. 
 

Let us now argue that the spin contents of preserved topological defect lines are renormalization group invariants (see e.g. <cit.>). The idea is that the topological defect lines are topological and the relevant deformation preserves the U(1) rotational symmetry of the quantum field theory. It implies that the twisted quantization condition on the defect Hilbert space cannot change in a discrete manner. More precisely, in <cit.> it is claimed that the spin content of the infrared theory must be a subset of the ultraviolet theory. The spin contents of the infrared theory can be a subset because the states might become heavy and decouple.





§ NEW RENORMALIZATION GROUP FLOWS

In this section, based on the study of non-invertible symmetries, we propose infinitely many new renormalization group flows in Virasoro minimal models. We give some formal checks of the agreement of renormalization group invariants under the flow. Examples will be given in the next section.



 §.§ Renormalization group flow TEXT


The main claim of our paper is there exist infinitely many renormalization group flows ℳ(kq + I, q) →ℳ(kq-I, q) induced by the ϕ_(1, 2k + 1) deformation, which satisfy the constraint from the non-invertible symmetries. We further claim that the renormalization group flow is complete when we preserve a modular tensor category with the  SU(2)_q-2 fusion rule that includes the ℤ_2 invertible symmetry. By complete, we mean that all the possible flows that preserve a modular tensor category with the SU(2)_q-2 fusion ring can be represented by a repeated use of our proposed flows ℳ(kq + I, q) →ℳ(kq-I, q).
For concreteness, we assume k and I are both integers in this subsection. The generalization for non-integer k (and I) will be discussed in the next subsection. The half-integer case does not preserve the modular tensor category with the SU(2)_q-2 fusion ring.

We first show that the ϕ_(1, 2k + 1) deformation preserves the SU(2)_q-2 subcategory L_(r, 1) (i= 1, ⋯, q-1) out of the non-invertible symmetries of the undeformed theory ℳ(kq + I, q). For this purpose, as reviewed in section <ref>, we should check L_(r,1) and ϕ_(1,2k+1) commute on the cylinder.

Let us start with the fusion rule (<ref>). We have

    ϕ_(1, 2k + 1)×ϕ_(ρ, σ) = ∑_l = 1 + | 2k + 1 - σ|
     l + σ = 02^min (2k + σ, 2(kq + I) - 2- 2k - σ)ϕ_(ρ, l) = ∑_l ∈Λϕ_(ρ, l)

in ℳ(kq + I, q), where we have defined Λ as 

    Λ = { l | 1 ≤ l ≤min (2k + σ, 2(kq + I) - 2- 2k - σ), l + σ = 0   mod 2  },

and then we can compare the action of L_(r, 1)ϕ_(1, 2k + 1) and ϕ_(1, 2k + 1) L_(r, 1)  on arbitrary primary states |ϕ_(ρ, σ)⟩:

    L_(r, 1)ϕ_(1, 2k + 1)|ϕ_(ρ, σ)⟩   = L_(r, 1)ϕ_(1, 2k + 1)ϕ_(ρ, σ)|0⟩ = ∑_l ∈Λ L_(r, 1)ϕ_(ρ, l)|0⟩
       = ∑_l ∈Λ L_(r, 1)|ϕ_(ρ, l)⟩ = ∑_l ∈ΛS_(r, 1), (ρ, l)S_(1, 1), (ρ, l)|ϕ_(ρ, l)⟩
    ϕ_(1, 2k + 1)L_(r, 1)|ϕ_(ρ, σ)⟩   = S_(r, 1), (ρ, σ)S_(1, 1), (ρ, σ)ϕ_(1, 2k + 1)|ϕ_(ρ, σ)⟩ = S_(r, 1), (ρ, σ)S_(1, 1), (ρ, σ)ϕ_(1, 2k + 1)ϕ_(ρ, σ)|0⟩
       = ∑_l ∈ΛS_(r, 1), (ρ, σ)S_(1, 1), (ρ, σ)ϕ_(ρ, l)|0⟩ = ∑_l ∈ΛS_(r, 1), (ρ, σ)S_(1, 1), (ρ, σ)|ϕ_(ρ, l)⟩.

Because l + σ is even in the above sum,

    .S_(r, 1), (ρ, l)S_(1, 1), (ρ, l)/ .S_(r, 1), (ρ, σ)S_(1, 1), (ρ, σ)   = .(-1)^(r - 1)lsin(πkp + I/prρ)sin(πkp + I/pρ)/ .(-1)^(r - 1)σsin(πkp + I/prρ)sin(πkp + I/pρ)
       = 1
 
and hence L_(1, 1), L_(2, 1), ⋯, L_(q - 2, 1), L_(q - 1, 1) commute with ϕ_(1,2k+1). This means that SU(2)_q-2 subcategory generated by L_(r,1) is preserved under the renormalization group flow induced by the ϕ_(1, 2k + 1) deformation. In particular, we stress that L_(q - 1, 1) generates the invertible ℤ_2 symmetry of the minimal models, so the renormalization group flow is ℤ_2 symmetric. 

In the above discussions, we realize that ϕ_(1,2k+1) can be replaced with ϕ_(1,2l+1) for any integer l such that  ϕ_(1,2l+1) is in the Kac table. The physical significance of choosing l=k is that then ϕ_(1,2k+1) will be the least relevant deformation within {ϕ_(1,2l+1)} if I<k.  When k>1, we typically have to introduce the other more relevant deformation ϕ_(1,2l+1) (with l<k) and fine-tune the deformation parameters to get to the critical point.[Generally we have to fine-tune k-1 parameters.] When I>k, we need fewer fine-tunings to reach the fixed point dictated by the proposed flow.



Let us now show that the quantum dimensions of all the preserved topological defect lines match under the proposed renormalization group flows ℳ(kq + I, q) →ℳ(kq-I, q).[The matching of the quantum dimensions when k=1 and k=2, corresponding to the previously known renormalization group flows, was discussed in <cit.>.] The proof follows by the direct calculation.
The quantum dimension d_(r, 1) of L_(r, 1) in ℳ(kq + I, q) is (- 1)^r + 1sin(πkq + I/qr)/sin(πkq + I/q) and that in ℳ(kq - I, q) is (- 1)^r + 1sin(πkq - I/qr)/sin(πkq - I/q). Its ratio is 

    sin(πkq + I/qr)sin(πkq - I/qr)sin(πkq - I/q)sin(πkq + I/q) = sin(πI/qr)sin(- πI/qr)sin(- πI/q)sin(πI/q) = 1.


We can further argue that the constraint from the quantum dimensions is sufficiently non-trivial. We observe that because kq + (± I + q) = (k + 1)q ± I, I takes a value in 1 ≤ I ≤ q - 1, and kq ± I and q are coprime, our proposed flows can be classified by φ(q) kinds of I, where φ(q) is Euler function. Now we want to see that each connected flows have different quantum dimensions. For this purpose, let us examine the quantum dimensions of L_(2,1) in ℳ(p,q) in particular; the constraint from the other preserved topological defect lines is typically not as strong. The explicit evaluation of the formula (<ref>) gives d_(2,1) = -2 cos(p/qπ) and we immediately observe d_(2,1) takes a distinct value for different p (mod q), given q. Thus, the quantum dimensions do distinguish our renormalization group flows.

From the general argument in section <ref>, under the new renormalization group flow, the spin contents of the preserved line  L_(r,1) (r = 1, ⋯, q - 1) should remain the same or become a subset.
While it may be non-trivial to check the agreement of all the spin contents in the most generic cases, one can systematically check the agreement in the following way. 

We first realize that N_(s, m)(t, n)^(r, 1)χ_(s, m)χ̅_(t, n) is non-zero only when m = n. Furthermore, the fusion coefficients necessary for us do not depend on m, so we can first determine the non-zero entry of N_(s, m)(t, n)^(r, 1)= δ_mn N_(s, 1)(t, 1)^(r, 1) from SU(2)_q - 2 fusion rule, and then we can compute the spin of the defect Hilbert space from (<ref>). 
In this way, it is possible to systematically compute the spin contents of L_(r,1) and see the agreement under the proposed renormalization group flow. We will show some examples in section <ref>.








Now let us show that the spin contents of ℋ_L_(2,1) in  ℳ(p,q) are preserved 
 under the proposed renormalization group flows. We first observe that the only non-zero fusion coefficients are N_(t, n) (t + 1, m)^(2, 1) and N_(t + 1, m) (t, m)^(2, 1). By computing ± (h_(t + 1, m) - h_(t, m)), we find the spin contents of ℋ_L_(2,1) are ±p/q1 + 2t/4 (mod 1/2ℤ) with t = 1, ⋯, q - 2. Thus, the spin contents of ℋ_L_(2,1) in ℳ(kq + I, q) contain those in ℳ(kq - I, q).

We further want to see that the spin contents of ℋ_L_(2,1) distinguish the renormalization group flows as the quantum dimensions. 
We observe that p/q1+2t/4 never equals to p/q1/4 mod ℤ/2, so p/q1/4  is always missing in the spin content. (Indeed, if this were true, we would have p/ql = ℤ, but since p and q are coprime, it should be a contradiction.) On the other hand, the missing ones (except 1/4q and 1/2q) always appear in p=2q-1 or p=2q-2 as long as q>3, so the spin contents of p=2q-1 and p=2q-2 are different from the other p's (mode q). We can repeat the analysis with the other p=2q-s with s=3, 4, ⋯, q-1 to realize that they all have different spin contents. The exceptional but easier q = 3 case can be treated separately (see section <ref>).

The renormalization group invariants cannot tell which of ℳ(kq+I,q) or  ℳ(kq-I,q) is the ultraviolet theory. To address this question, let us mention two evidence that  ℳ(kq+I,q) is the ultraviolet theory. The first evidence is the c_eff theorem <cit.>. While ℳ(p,q) can be non-unitary, it is believed that as long as the deformation preserves the 𝒫𝒯-symmetry (see e.g. <cit.> for a review), the renormalization group flow shows the monotonicity with respect to the effective central charge c_eff = 1 -6/pq along the flow. If we accept the c_eff theorem (with the assumption our flows preserve the 𝒫𝒯-symmetry), we may conclude ℳ(kq+I,q) is the ultraviolet theory.[One caveat here is that we have not verified our renormalization group flow preserves the 𝒫𝒯 symmetry. It is an interesting question to see if the 𝒫𝒯 symmetry can be realized as a topological defect line.] 

The other evidence comes from the number of (singlet) relevant deformations <cit.>. It was argued there that along the renormalization group flow, the number of (singlet) relevant deformations should decrease.
Intuitively, if we deform the theory by relevant perturbation, the deformed theory becomes less critical and we expect less fine-tuning is needed to reach the criticality. In <cit.>, it was put on firmer ground by studying the topology of the renormalization group flow. Again once we accept this conjecture, we realize ℳ(kq+I,q) is the ultraviolet theory because it has a larger number of relevant singlet operators. Note also that if our constraint from the non-invertible symmetries did not exist, we could have found counterexamples of the conjecture. For instance in the would-be flow ℳ(7,4) →ℳ(5,4), the number of relevant singlet operators did not decrease. We have, however, forbidden the flow by matching the non-invertible symmetries. In this discussion, it is important to note that by singlet, we mean the operators that commute with all the non-invertible symmetries {L_(r,1)} whose fusion ring is SU(2)_q-2 than just those that commute with the ℤ_2 invertible symmetry.







Here are some historical comments. In a classic paper <cit.>, Zamolodchikov studied the renormalization group flow k=I=1, which turns out to be integrable <cit.>. This case is most physically interesting because it describes the flow between unitary minimal models. Subsequently, Ahn <cit.> and Lässig <cit.> proposed the generalization to the non-unitary minimal models independently, corresponding to k=1, I>1. Later, Dorey, Dunning and Tateo <cit.> (see also <cit.>) studied the case with k=2 from the viewpoint of non-linear integral equations. Our approach, based on the non-invertible symmetries, also provides us with new insights into the previously known flows.





 §.§ Half-integer TEXT

Our proposed renormalization group flow
 ℳ(kq + I, q) →ℳ(kq - I, q) induced by ϕ_(1, 2k + 1)
 formally makes sense even if k is a half-integer, where I takes a half-integer or integer depending on q. The main difference is that 2k+1 is even so not all the topological defect lines L_(r, 1) commute with ϕ_(1, 2k + 1), but only the subcategory spanned by L_(2ℓ + 1, 1), where ℓ is an integer, is preserved. This restriction can be explicitly seen in (<ref>), where for half-integer k, l+σ is odd (rather than even for integer k case discussed there), so we have to assume r is an odd integer. This subcategory has the fusion ring of SO(3)_[q/2]-1.
 
If we restrict ourselves to these topological defect lines, all the discussions above can be repeated. In particular, the consistency of the proposed renormalization group flow with the preserved non-invertible symmetry holds. For instance, the invariance of the  quantum dimensions of L_(r,1) can be checked as in (<ref>), but with half-integer k, the ratio becomes (-1)^r+1. Thus only the preserved lines with r=2ℓ+1 have the same quantum dimensions as claimed. The strongest constraint from the spin contents comes from L_(3,1) (rather than L_(2,1) which does not exist). In ℳ(p,q), the spin contents of ℋ_L_(3,1) are {0,±p/q(ℓ+1) } (ℓ=1, ⋯, q-3), and we can check it is consistent with our flow ℳ(kq + I, q) →ℳ(kq - I, q) with half-integer k.


Some specific renormalization group flows have been studied in the literature.
The particular case with k=1/2 was discussed in Dorey et al <cit.>, but note (p,q) there was swapped compared with our unified notation so that their ϕ_(1,2) is our ϕ_(2,1). The non-invertible symmetries, in this previously known case, were studied in <cit.>. Again, we generalize their works by letting k be an arbitrary half-integer.



Let us comment on the fate of the ℤ_2  invertible symmetry under the half-integer k flow.
 When k is a half-integer and q is odd, ϕ_(1, 2k + 1) is odd under the ℤ_2 invertible symmetry, so the preserved modular tensor category does not include L_(q - 1, 1). Nevertheless, after the flow, we do recover the entire modular tensor category with the SU(2)_q-2 fusion ring, including the invertible ℤ_2 symmetry. They must be emergent symmetries. While this must be the case since this half-integer k flow allows us to change the (non-)anomalous nature of the ℤ_2 symmetry, the microscopic details of the renormalization group flow may be of interest and needs further study.


 




§ SOME EXAMPLES


In this section, we study explicit examples of ℳ(kq+I,q) →ℳ(kq-I,q) flows for q=3,4,5, emphasizing physical interpretations. As a potentially exceptional case, we then examine q=2 separately.






 §.§ TEXT and anomaly matching


In this case, our proposal with integer k boils down to the conclusion that there exists distinct renormalization group flows among  ℳ(2l+1,3) and among ℳ(2l,3). The only non-trivial topological defect line preserved by the ϕ_(1, 2k + 1) deformation is L_(2,1) that generates the invertible ℤ_2 symmetry. The constraint from the quantum dimensions as well as spin contents will be equivalent to the 't Hooft anomaly matching. We recall that the ℤ_2 invertible symmetry is anomalous in  ℳ(2l+1,3) and non-anomalous in ℳ(2l,3) so they cannot mix under the ℤ_2 preserving renormalization group flow.

While the quantum dimensions of L_(2,1) can be directly computable from (<ref>), the spin contents of ℋ_L_(2,1) can be computed by the algorithm mentioned in section <ref>.
As argued there, non-zero entry of N_(s, m)(t, n)^(2, 1) implies m = n and (s, t) = (2, 1) or (1, 2), which have the opposite spins. From the dimensions of primary operators (<ref>), we find the spin of the defect Hilbert space ℋ_L_(2,1) is given by  ±(h_(2, m)-h_(1, m)) = ±1/4(p - 2m) with 1≤ m ≤ p - 1. The spin contents are therefore {±1/4} when p is odd and {0, ±1/2} 
 when p is even. 

We have summarized the quantum dimensions and spin contents of ℳ(p,3) models in Table <ref> and Table <ref>. Note that for the anomalous ℤ_2 invertible symmetry, the quantum dimension is -1 and the spin content is {±1/4}, which is consistent with the general argument <cit.>.








We would like to note that in the literature <cit.>, one of the flows in this class ℳ(10,3) →ℳ(8,3) induced by ϕ_(1,7) (k=3, I=1) was used to study the Landau-Ginzburg-like Lagrangian constructions of non-unitary minimal models. The further flow ℳ(8,3) →ℳ(4,3) induced by ϕ_(1,5) (k=2, I=2) can be straightforwardly realized in their Landau-Ginzburg-like Lagrangian. Another flow ℳ(7,3) →ℳ(5,3) induced by ϕ_(1, 5) (k = 2, I=1) was studied 
in <cit.> from the truncated conformal space approach.

Let us briefly discuss the case with half-integer k. With the half-integer k, ϕ_(1,2k+1) deformation does not preserve any non-trivial topological defect lines in ℳ(p,3), so the constraint is none. We, however, observe that some of the predicted flow such as ℳ(10,3) ϕ_(1, 6)⟶ℳ(5,3) ϕ_(1, 4)⟶ℳ (4,3) may give an interesting Lagrangian description of ℳ(5,3), given the Lagrangian description of ℳ(10,3) is available. Indeed, the flow has some theoretical interest because of the (dis)appearance of the anomalous ℤ_2 symmetry of ℳ(5,3) 
 in the middle of the renormalization group flow.








 §.§ TEXT and duality defects


Here, under the ϕ_(1, 2k + 1) deformations with integer k, the preserved topological defect lines are L_(1,1), L_(2,1) (sometimes called 𝒩) and L_(3,1) (sometimes called η) and generate the ℤ_2 Tambara-Yamagami modular tensor category. 
There are φ(4) = 2 distinct renormalization group flows predicted from our proposal. 


Let us first study the quantum dimensions of L_(2, 1) and L_(3, 1), as shown in Table <ref>. As discussed in section <ref>, d_(2,1) distinguishes the two renormalization group flows inside ℳ(p,4). Since the ℤ_2 invertible symmetry is non-anomalous, d_(3,1) =1 for all p, and it is not useful as a constraint on the renormalization group flow.



We can also compute the spin contents of L_(2,1) and L_(3,1). Since the topological defect line L_(3,1) generates the non-anomalous ℤ_2 invertible symmetry, the spin contents must be {0, ±1/2} as can be checked easily, so we focus on L_(2,1). The result of the explicit computation is summarized in Table <ref>.



As derived in section <ref>, for ℳ(p, 4), the general formula for h_s, m - h_t, n and the spin content of ℋ_L_(2,1) can be derived:

    h_s, m - h_t, n : {±3p - 8m16, ±5p - 8m16 | 1 ≤ m ≤ p - 1},
    the spin contents of ℋ_L_(2,1) : ±116, ±716    (p = 3, 5 mod 8)
    ±316, ±516    (p = 1, 7 mod 8)

This is compatible with (6.5) in <cit.> on the spin content of the duality defect line. With this agreement in mind, let us argue that L_(2,1) is indeed the duality defect line.

As is well-known in the critical Ising model ℳ(3,4) and the tricritical Ising model ℳ(5,4), the ℤ_2 Tambara-Yamagami modular tensor category has a physical interpretation as the duality defect line. We would like to offer a similar interpretation to the general ℳ(p,4) Virasoro minimal models.

First, let us review the duality defect in more generality <cit.>. Consider a quantum field theory 𝒬 on a d-dimensional spacetime with a non-anomalous ℤ_N q-form global symmetry. We divide the spacetime into two regions and gauge the ℤ_N global symmetry in half of the spacetime. Then, we can construct a codimension-one topological interface between the original theory 𝒬 and its gauged theory 𝒬/ℤ_N. If the original theory 𝒬 is invariant under gauging the ℤ_N global symmetry

    𝒬≅𝒬/ℤ_N ,

the interface becomes the duality defect in the original theory 𝒬. 

When do we expect the duality defect in this so-called half-space gauging construction? Note that the gauged theory has ℤ_N≅ℤ_N (d - q - 2)-form dual (or quantum) symmetry, so the ungauged theory and the gauged theory may have the same symmetry  as in (<ref>) if we require

    q = d - 22.

Our cases discussed in this paper correspond to  (d, q) = (2, 0).

Given a duality defect in the ultraviolet conformal field theory, we want to understand its fate under the renormalization group flow. Let us assume that the duality procedure commutes with the deformation of the ultraviolet conformal field theory.[This assumption is stronger than just assuming the deformation preserves ℤ_2 invertible symmetry. For the difference, see the example of half-integer k flows below.] Then, we should have the duality defect with the same property in the infrared as well. Such an existence should provide us with a strong constraint on the renormalization group flow.




We want to argue that the topological defect line L_(2,1) in ℳ(p,4) is the duality defect line. Let us consider general ℳ(p,q) with the A-series modular invariant partition functions Z_A_q - 1, A_p - 1. Let us further assume the case when p or q is even (but not 2) so that the ℤ_2 symmetry is non-anomalous and can be gauged.
The partition function of the ℤ_2 gauged theory can be obtained by the orbifold technique:

    Z_ℳ(p, q)/ℤ_2 = 
            Z_D_q/2 + 1, A_p - 1   when q = 0 mod 2
    
            Z_A_q - 1, D_p/2 + 1   when p = 0 mod 2.
 
The right-hand side is generically a non-diagonal modular invariant partition function of D-series, but the exception is when q=4 (or p=4). 
Because A_3 ≅ D_3, A-series minimal models ℳ(p, q) are invariant under gauging ℤ_2 global symmetry if and only if p or q is 4. In other words,  ℳ(4, q) is self-dual under the ℤ_2 gauging and it has a duality defect line 𝒩. Moreover, we can show that the duality defect line 𝒩 satisfies the ℤ_2 Tambara-Yamagami fusion rule 
 such as η×η = 1, η×𝒩 = 𝒩×η = 𝒩 and  𝒩×𝒩 = 1+η.
This is nothing but the fusion rule of the topological defect lines L_(2,1) and L_(3,1) preserved under the renormalization group flow ℳ(4k+I,4) →ℳ(4k-I,4) induced by ϕ_(1,2k+1). 


After seeing how the duality line plays a significant role in the renormalization group flows of  ℳ(4k+I,4) →ℳ(4k-I,4) when k is an integer, let us briefly discuss the case when k is a half-integer. Here, ϕ_(1,2k+1) does not commute with L_(2,1) and the only preserved symmetry is L_(3,1). Accordingly, there are no obstructions for the renormalization group flows between any pair of p in ℳ(p,4). The simplest example would be ℳ(3,4)
 to “ℳ(1,4)" induced by ϕ_(1,2) (= ϵ), which is a massive flow. Physically, if we change the temperature of the Ising model, the Kramers–Wannier duality is broken and the criticality will be lost. See <cit.> for an interpretation of “ℳ(1,4)" from the viewpoint of the integral equations.






To conclude this subsection, we have summarized the renormalization group flows between ℳ(p,4) in Figure <ref>. As a physical application beyond the classic tri-critical Ising to critical Ising flow ℳ(5,4) →ℳ(3,4), the fermionic version of the renormalization group flow ℳ(11,4) to ℳ(5,4) was discussed in <cit.> to understand the fate of the non-supersymmetric Yukawa fixed point.














 §.§ TEXT 


We repeat our analysis on the renormalization group flows ℳ(5k+I,5) →ℳ(5k-I,5) induced by ϕ_(1,2k+1) that preserve 
 a modular tensor category with the SU(2)_3 fusion ring, assuming k is an integer. Here, we have φ(5)=4 distinct flows that cannot mix. The 't Hooft anomaly matching of the ℤ_2 invertible symmetry distinguishes ℳ(p,5) with even p and odd p, but the other non-invertible symmetries give a finer classification.

The quantum dimensions of preserved non-invertible symmetries are given in Table <ref>. Here, ϕ = 1+√(5)/2 is the golden ratio. As discussed in section <ref>, not only all the quantum dimensions are consistent with the proposed renormalization group flow, but L_(2,1) gives the strongest constraint such that it fully distinguishes our proposed renormalization group flows.


Let us also compute the spin contents of the preserved topological defect lines. On  ℋ_L_(2,1), the non-zero fusion coefficients are N_(1, m)(2, m)^(2, 1), N_(2, m)(1, m)^(2, 1),  N_(2, m)(3, m)^(2, 1), N_(3, m)(2, m)^(2, 1), N_(3, m)(4, m)^(2, 1), and N_(4, m)(3, m)^(2, 1), so we have

    {±(3p/20 -m/2), ±1/4(p-2m), ±(7p/20 -m/2)  } ,

where 1 ≤ m ≤ p - 1. Similarly on ℋ_L_(3,1), from the non-zero fusion coefficients N_(1, m)(3, m)^(3, 1), N_(3, m)(1, m)^(3, 1), N_(2, m)(4, m)^(3, 1), N_(4, m)(2, m)^(3, 1), N_(2, m)(2, m)^(3, 1), and N_(3, m)(3, m)^(3, 1), we have

    {±(2p/5-m), ±(3p/5 -m), 0 } .

Finally, on ℋ_L_(4,1), from the non-zero fusion coefficients N_(1, m)(4, m)^(4, 1), N_(4, m)(1, m)^(4, 1), N_(2, m)(3, m)^(4, 1), and N_(3, m)(2, m)^(4, 1),  we have 

    {±3/4(p-2m), ±1/4(p-2m) } .



The results are summarized in Table <ref>. We note that the constraints from the quantum dimensions and constraints from the spin contents are the same.






Since this is the first example of the appearance of p=2 as in the flow ℳ(8,5) →ℳ(2,5) induced by ϕ_(1, 3), let us comment on it. We will give more detailed discussions in section <ref>. Unlike in ℳ(p,5) with p>2, in ℳ(2,5), L_(4,1) is identified with L_(1,1) and L_(3,1) is identified with L_(2,1), so we have only half of the preserved non-invertible symmetries in the infrared compared with the ones in the ultraviolet. These symmetries may be spontaneously broken or disappear. Note, in particular, that The ℤ_2 invertible symmetry generated by L_(4,1) is non-anomalous, so it can simply disappear. Understanding the fate of these symmetries requires dynamical analysis which is beyond the scope of our paper, but further discussions follow in section <ref>.


Let us finally make a brief discussion on the half-integer k flow. The preserved topological defect lines are L_(1,1) and L_(3,1), which form the fusion rule of ℳ(2,5) (Lee-Yang fusion ring). It is also called the Fibonacci fusion ring (see the appearance of the golden ratio above). There exist two distinct renormalization group flows depending on d_(3,1), which is consistent with the classification of the modular tensor category with the Fibonacci fusion ring.
The ℤ_2 invertible symmetry is broken by the deformation, but it reappears as an emergent symmetry. This is how we can change the (non-)anomalous nature of the ℤ_2 invertible symmetry along the renormalization group flow.
 



 §.§ Flows to TEXT

As far as the non-invertible symmetry goes, there is nothing wrong with our flows ℳ(kq + I, q) →ℳ(kq-I, q) even when q=2 or kq-I = 2. A closer look tells us, however, the situation is more subtle essentially because there is no ℤ_2 invertible symmetry in ℳ(p,q) when either p or q is 2. From the viewpoint of the topological defect lines, the would-be  ℤ_2 topological defect line L_(q - 1, 1) is identified with the identity defect line L_(1, 1). In other words, there exist no charged primary operators under the ℤ_2 invertible symmetry.

By taking q = 2, our proposal suggests the existence of the renormalization group flow ℳ(2k + I, 2) →ℳ(2k - I, 2) induced by ϕ_(1, 2k + 1). The case with I=1, however, may sound problematic because  ϕ_(1,2k + 1) is outside of the Kac table. The most reasonable interpretation is that the flow exists but the primary operator that induces the renormalization group flow is a certain fine-tuned combination of ϕ_(1, l) with l=1,2, ⋯, k.[Under the identification, this is equivalent to taking l=1,3, ⋯, 2k-1.] Indeed, the numerical analysis based on the truncated conformal space approach shows the existence of such flows (for small k) <cit.>. Of course, we could be simply agnostic about these flows because the argument based on the non-invertible symmetries does not give any constraint at all here, but we are tempted to unify them within our proposal.

Another subtle case is q≠ 2 but kq - I = 2. In this case, the question is about the fate of the ℤ_2 invertible symmetry.
We start with the ultraviolet theory with a ℤ_2 invertible symmetry and we add the deformation that preserves it. In the infrared limit, we end up with the theory that does not possess any ℤ_2 invertible symmetry. What happened to the symmetry? One possibility is it is spontaneously broken. Another possibility is all the charged operators are decoupled in the infrared limit. 
 The latter is possible because the ℤ_2 symmetry here is non-anomalous.
 
 In the massive flow example of ℳ(4,3) →ℳ(2,3) induced by ϕ_(1,3) (formally k=1, I=1 flow), where a Lagrangian description is available, both can happen depending on the sign of the deformation. Regarding ℳ(4,3) as a critical ϕ^4 theory, adding m^2ϕ^2 with positive m^2 gives decoupling of the ℤ_2 charged field ϕ, and with negative m^2 it gives spontaneous breaking of the ℤ_2 symmetry.

Physically ℳ(2, q) can be realized in the multi-critical Lee-Yang fixed point in two dimensions.
So far, the canonical construction of the (multi-critical) Lee-Yang fixed point was to start with the ℤ_2 symmetric theory, say Ising model, and add the (imaginary) ℤ_2 breaking interaction, say pure imaginary magnetic field, to get to the fixed point. This new construction without explicitly breaking the ℤ_2 symmetry is novel and worth studying further.




§ DISCUSSIONS


In this paper, based on the study of non-invertible symmetries, we have proposed there exist infinitely many new renormalization group flows between Virasoro minimal models  ℳ(kq + I, q) →ℳ(kq-I, q) induced by ϕ_(1,2k+1). They vastly generalize the previously proposed renormalization group flows in the literature.

One should be able to check our new renormalization group flows by using the conformal perturbation theory or truncated conformal space approach <cit.>. Here, let us simply observe that the deformation operator ϕ_(1,2k+1) is more relevant when the separation of the renormalization group flow given by  I is larger. This agrees with our intuitive picture; the more relevant deformations, the longer the renormalization group trajectory becomes. It should also agree with the difference of the central charge at the fixed points. More recent approaches to the renormalization group flow based on the boundary states and interfaces can be found in <cit.>, and they may be useful to understand our new renormalization group flows better.
 

For a future direction, it is interesting to study the integrable structure of our new renormalization group flows  ℳ(kq + I, q) →ℳ(kq-I, q). As initiated in <cit.> the flows with I=1 (or I=1/2) can be described by the Thermodynamic Bethe Ansatz <cit.>. When I>1, there seem to be no known descriptions in terms of the Thermodynamic Bethe Ansatz, but the integral equations that describe the effective central charge along the flow were proposed 
 <cit.>. It should be interesting to generalize these works to our new renormalization group flows.


Another direction to be explored further is a concrete realization of our new renormalization group flows in statistical models or quantum field theories. It is known that any (non-unitary) minimal models have a statistical model realization based on the Restricted Solid-On-Solid (RSOS) model <cit.> (see also <cit.> for the related spin chain construction). It is interesting to understand the nature of the anomalous ℤ_2 symmetries, duality defects as well as general non-invertible symmetries preserved by our renormalization group flow.

As for the quantum field theory realization, we are still in our infancy even more than fifty years after the birth of conformal field theories. This is because the quantum field theory descriptions (e.g. Landau-Ginzburg theory <cit.> or gauge theory <cit.>) for non-unitary minimal models are largely unknown. We, nevertheless, note that 2d-4d correspondence <cit.> gives realizations of certain (but probably not all) non-unitary chiral algebra from supersymmetric conformal field theories in four dimensions. 
We hope our renormalization group flow may become a breakthrough in this direction. 

Concerning the explicit constructions of minimal models from known quantum field theories, while our discussions focus on the infrared flow, it seems interesting to look at the flow in the opposite way as an ultraviolet flow <cit.>. Since the ultraviolet flow is a non-renormalizable deformation, we may need some other input such as an integrability to pursue, but once done, it should give a strong hint toward the quantum field theory descriptions of the ultraviolet theory whose “Lagrangian description" were unknown.

Why do we crave Lagrangian descriptions of the Virasoro minimal models?
We wish to point out here that once the Lagrangian descriptions are available, we may generalize these models in higher dimensions. When we obtain the higher dimensional fixed points, we can ask whether the constraints on the renormalization flow from the non-invertible symmetries can also be uplifted. Since we have less explicit constructions of non-invertible symmetries in higher dimensions, this is an outstanding question to be pursued.



§ ACKNOWLEDGMENTS

TT is supported by JSPS KAKENHI Grant Number JP24KJ1500.
YN is in part supported by JSPS KAKENHI Grant Number 21K03581. YN would like to thank C. Ahn and K. Kikuchi for the correspondence and discussions.


h-physrev






intersections, calc, arrows












When we insert L_(r, s) along time direction, the torus partition function which is defined by a trace over defect Hilbert space ℋ_L_(r, s) is written as

Consider the torus partition function. Let us further insert  L_(r, s) along the time direction. The torus partition function with the defect inserted can be interpreted as the trace over the defect Hilbert space ℋ_L_(r, s):

    Z_L_(r, s)(τ, τ̅) = ∑_(r^', s^'), (ρ^', σ^') ∈Γ n_(r^', s^'), (ρ^', σ^')χ_(r^', s^')(τ)χ̅_(ρ^', σ^')(τ̅),

where n_(r^', s^'), (ρ^', σ^') is non-negative integer, τ is modular parameter and χ_(r^', s^')(τ) is (degenerate) Virasoro character.   When n_(r^', s^'), (ρ^', σ^')≠ 0, the spin content of ℋ_L_(r, s) contains h_r^', s^' - h_ρ^', σ^' mod ℤ where h_r^', s^' is the conformal dimension of the primary operator ϕ_(r^', s^').

To compute Z_L_(r, s)(τ, τ̅), we use the modular S-transformation. Because of modular S invariance, 

    Z_L_(r, s)(- 1τ, - 1τ̅) = Z^L_(r, s)(τ, τ̅) := Tr_ℋ(L_(r, s)q^L_0 - c/24q̅^L̅_0 - c/24)

where in the right hand side, the topological defect lines are inserted in the space direction. Once we know the action of L_(r,s) on primary states, the right hand side is explicitly computable. 

From (<ref>) and (<ref>), we get

    𝖫 = 𝖲𝖭𝖲

where 𝖫 = (𝖫_IJ), 𝖲 = (𝖲_IJ) and 𝖭 = (𝖭_IJ) are matrices defined by

    𝖫_IJ = diag(S_(r, s), (1, 1)S_(1, 1), (1, 1), ⋯), 𝖲_IJ = S_IJ, 𝖭_IJ = n_IJ,

where I, J = (r^', s^'), (ρ^', σ^') are multi indices. 





Potentially interesting new examples.

ℳ(5,3) →ℳ(4,3), 

ℳ(10,3) →ℳ(5,3) (this can give an interesting “Lagrangian" description of ℳ(5,3).)

ℳ(7,4) →ℳ(5,4) (this preserves only ℤ_2), 

ℳ(8,5) →ℳ(7,5) 


 


§ NOTATION

First, we check the notation of minimal models in this paper. Minimal models denoted by ℳ(p, q) are specified by two coprime integers p and q. p > q in the yellow book, but in this paper, we fix q to a fixed integer and investigate the renormalization group flow among ℳ(p, q) series where q is fixed. So the critical Ising model is denoted by ℳ(3, 4) rather than ℳ(4, 3) in the yellow book when we study ℳ(p, 4) series.



The central charge of ℳ(p, q) is 

    c = c̅ = 1 - 6(p - q)^2pq

and it has (p - 1)(q - 1)/2 primary fields ϕ_(r, s) where p and q are the Kac indices and meets 1 ≤ r < q, 1 ≤ s < p. Two primayy fields ϕ_(r, s) and ϕ_(q -  r, p - s) are identified which has the same conformal weight 

    h_r, s = h_q - r, p - s = (pr - qs)^2 - (p - q)^24pq.

The fusion rule of the primary fields is given by

    ϕ_(r, s)×ϕ_(m, n) = ∑_k = 1 + | r - m |
     k + r + m = 12^min (r + m - 1, 2q - 1 - r - m)∑_l = 1 + | s - n |
     l + s + n = 12^min (s + n - 1, 2p - 1 - s - n)ϕ_(k, l).

We choose the fundamental domain in the Kac table composed of (p - 1)(q - 1)/2 distinct Kac indices as below:

    Γ = { (r, s) | 1 ≤ r ≤ q - 1, 1 ≤ s ≤ p - 1, ps + qr < pq }



§ RENORMALIZATION GROUP FLOW OF   MINIMAL MODELS



 §.§ Non-invertible symmetry

Consider a QFT 𝒬 on a d-dimensional spacetime with a non-anomalous ℤ_N q-form global symmetry. We divide the spacetime into two regions and gauge ℤ_N global symmetry in half of the spacetime. Then, we can construct a codimension-one topological interface between the original theory 𝒬 and its gauged theory 𝒬/ℤ_N. If the original theory 𝒬 is invariant under gauging ℤ_N global symmetry

    𝒬≅𝒬/ℤ_N ,

the interface becomes the duality defect in the original theory 𝒬. The gauged theory has ℤ_N≅ℤ_N (d - q - 2)-form dual(quantum) symmetry. (<ref>) can be the case where

    q = d - 22.

In particular, q = 0 if d = 2. The operation of discrete gauging is topological and commutes with renormalization group flow. So the existence of the duality defect is renormalization group invariant. It gives us the strong constraint on renormalization group flow.



In this paper, we consider A-series minimal models ℳ(p, q) in two-dimensional conformal field theory, whose partition functions are diagonal modular invariant. They have ℤ_2 0-form global symmetry. When p and q are both odd, this symmetry is anomalous. We denote the partition function of A-series minimal models ℳ(p, q)(p > q) as Z_A_q - 1, A_p - 1, then the partition function of its gauged theory is

    Z_ℳ(p, q)/ℤ_2 = 
            Z_D_q/2 + 1, A_p - 1   when q = 0 mod 2
    
            Z_A_q - 1, D_p/2 + 1   when p = 0 mod 2.

Because A_3 ≅ D_3, A-series minimal models ℳ(p, q) is invariant under gauging ℤ_2 global symmetry if and only if p or q is 4. We fix q to 4 below. From the above, ℳ(p, 4) must flow to ℳ(p^', 4).




 §.§  -theorem

Zamolodchikov showed that the central charge decreases along renormalization group flow between two unitary CFTs. For non-unitary case, there is generalized c-theorem which states the effective central charenormalization groupe c_eff = c - 24h_min do not increase along renormalization group flow which connects two 𝒫𝒯 invariant conformal field theories. For A-series minimal models ℳ(p, q), c_eff = 1 - 6/pq. Therefore ℳ(p, 4) must flow to ℳ(p^', 4) where p > p^'.




 §.§ Spin content

First, we review some properties of topological defect lines of A-series minimal models ℳ(p, q). ℳ(p, q) has (p - 1)(q - 1)/2 topological defect lines (Verlinde lines) L_(r, s) for each of the primary fields ϕ_(r, s). The action of L_(r, s) on a primary state |ρ, σ⟩ = ϕ_(ρ, σ)|0⟩ is given by

    L_(r, s)|ϕ_(ρ, σ)⟩ = S_(r, s), (ρ, σ)S_(1, 1), (ρ, σ)|ϕ_(ρ, σ)⟩,

where S_(r, s), (ρ, σ) is modular S-matrix and given by

    S_(r, s), (ρ, σ) = 2√(2pq)(-1)^1 + sρ + rσsin(πpqrρ)sin(πqpsσ).

The action of L_(r, s) on the vacuum is given by

    L_(r, s)|0⟩ = d_(r, s)|0⟩ = S_(r, s), (1, 1)S_(1, 1), (1, 1)|0⟩,

where the eigenvalue d_(r, s) is called quantum dimension of L_(r, s). 




When we insert L_(r, s) along time direction, the torus partition function which is defined by a trace over defect Hilbert space ℋ_L_(r, s) is written as 

    Z_L_(r, s)(τ, τ̅) = ∑_(r^', s^'), (ρ^', σ^') ∈Γ n_(r^', s^'), (ρ^', σ^')χ_(r^', s^')(τ)χ̅_(ρ^', σ^')(τ̅),

where n_(r^', s^'), (ρ^', σ^') is non-negative integer, τ is modular parameter and χ_(r^', s^')(τ) is Virasoro character. The way to read the spin content of the defect Hilbert space ℋ_L_(r, s) is following. Because of modular S invariance, 

    Z_L_(r, s)(- 1τ, - 1τ̅) = Z^L_(r, s)(τ, τ̅) := Tr_ℋ(L_(r, s)q^L_0 - c/24q̅^L̅_0 - c/24)

From (<ref>) and (<ref>), we get

    𝖫 = 𝖲𝖭𝖲

where 𝖫 = (𝖫_IJ), 𝖲 = (𝖲_IJ) and 𝖭 = (𝖭_IJ) are matrices defined by

    𝖫_IJ = diag(S_rs, 11S_11, 11, ⋯), 𝖲_IJ = S_IJ, 𝖭_IJ = n_IJ,

where I, J = (r^', s^'), (ρ^', σ^') are multi indices. When 𝖭_IJ≠ 0, the spin content of ℋ_L_(r, s) contains h_I - h_J = h_r^', s^' - h_ρ^', σ^' mod ℤ where h_r^', s^' is the conformal dimension of the primary field ϕ_(r^', s^'). If L_(r, s) survives under renormalization group flow, the spin content of the defect Hilbert space ℋ_L_(r, s) in IR must be a subset of that in UV. 




We insert the duality defect line 𝒩 = L_(2, 1), generated by ℤ_2 0-form symmetry whose topological defect line is η = L_(3, 1), along time direction and compute the spin content of the defect Hilbert space ℋ_𝒩 for p = 3, 5, ⋯, 17. The result is in Table <ref>.



In general, we can expect that h_r^', s^' - h_ρ^', σ^' and the spin content of ℋ_𝒩 are

    h_r^', s^' - h_ρ^', σ^' : {±3p - 8i16, ±5p - 8i16 | 1 ≤ i ≤ m},
    the spin content of ℋ_𝒩 : ±116, ±716    (p = 3, 5 mod 8)
    ±316, ±516    (p = 1, 7 mod 8)

when we consider minimal model ℳ(p = 2m + 1, 4).
From the above, the following renormalization group flow can be inferred.


§ RG FLOW TEXT

Zamolodchikov states that there is a flow ℳ(q + 1, q) →ℳ(q -1, q) by ϕ_(1, 3) perturbation in and Ahn and Lässig state that there is a flow ℳ(q + I, q) →ℳ(q - I, q). Moreover, Dorey, Dunning and Tateo states that there is a flow ℳ(2q + I, q) →ℳ(2q - I, q). From these statements and the previous section, we can infer that there is a flow ℳ(kq + I, q) →ℳ(kq - I, q) by ϕ_(1, 2k + 1) perturbation. Because kq + (± I + q) = (k + 1)q ± I, I takes a value in 1 ≤ I ≤ q - 1. Moreover, kq ± I and q are coprime, so there are φ(p) kinds of I where φ(q) is Euler function. We show this conjectural flow below. Our strategy is to find the topological defect lines which commute with ϕ_(1, 2k + 1) and calculate these quantum dimensions which are renormalization group flow. This invariance can be understood as below. The fusion of the topological defect lines is

    L_i × L_j = ∑_k N_ij^kL_k

where N_ij^k is the fusion coefficients and take the vacuum expectaion value. The defect lines are topological, so we can place them far away and it is factorized

    ⟨ L_i ⟩×⟨ L_j ⟩ = ∑_k N_ij^k⟨ L_k ⟩.

The solution of these quadratic equations is discrete, hence it is invariant under continuous renormalization group flow.



From (<ref>), we get

    ϕ_(1, 2k + 1)×ϕ_(ρ, σ) = ∑_l = 1 + | 2k + 1 - σ|
     l + σ = 02^min (2k + σ, 2(kq + I) - 2- 2k - σ)ϕ_(ρ, l) =: ∑_some l'sϕ_(ρ, l).

in ℳ(kq + I, q) and then

    L_(r, 1)ϕ_(1, 2k + 1)|ϕ_(ρ, σ)⟩   = L_(r, 1)ϕ_(1, 2k + 1)ϕ_(ρ, σ)|0⟩ = ∑_some l's L_(r, 1)ϕ_(ρ, l)|0⟩
       = ∑_some l's L_(r, 1)|ϕ_(ρ, l)⟩ = ∑_some l'sS_(r, 1), (ρ, l)S_(1, 1), (ρ, l)|ϕ_(ρ, l)⟩
    ϕ_(1, 2k + 1)L_(r, 1)|ϕ_(ρ, σ)⟩   = S_(r, 1), (ρ, σ)S_(1, 1), (ρ, σ)ϕ_(1, 2k + 1)|ϕ_(ρ, σ)⟩ = S_(r, 1), (ρ, σ)S_(1, 1), (ρ, σ)ϕ_(1, 2k + 1)ϕ_(ρ, σ)|0⟩
       = ∑_some l'sS_(r, 1), (ρ, σ)S_(1, 1), (ρ, σ)ϕ_(ρ, l)|0⟩ = ∑_some l'sS_(r, 1), (ρ, σ)S_(1, 1), (ρ, σ)|ϕ_(ρ, l)⟩.

Because l + σ is even in the above sum,

    .S_(r, 1), (ρ, l)S_(1, 1), (ρ, l)/ .S_(r, 1), (ρ, σ)S_(1, 1), (ρ, σ)   = .(-1)^(r - 1)lsin(πkp + I/prρ)sin(πkp + I/pρ)/ .(-1)^(r - 1)σsin(πkp + I/prρ)sin(πkp + I/pρ)
       = 1

and hence L_(1, 1), L_(2, 1), ⋯, L_(q - 2, 1), L_(q - 1, 1) whose fusion is closed commute with ϕ_(1, 2k + 1) and they survive under renormalization group flow. The quantum dimension d_(r, 1) of L_(r, 1) in ℳ(kq + I, q) is (- 1)^r + 1sin(πkq + I/qr)/sin(πkq + I/q) and that in ℳ(kq - I, q) is (- 1)^r + 1sin(πkq - I/qr)/sin(πkq - I/q). Its ratio is 

    sin(πkq + I/qr)sin(πkq - I/qr)sin(πkq - I/q)sin(πkq + I/q) = sin(πI/qr)sin(- πI/qr)sin(- πI/q)sin(πI/q) = 1.



§ DISCUSSION



    
  * a puzzle when q=2. OK
    
  * Generalization of (2,1)  OK
    
  * deformations.Conformal perturbation theory.
    
  * conformal dimensions of ϕ_1,2k+1 and the length of RG flows.
    
  * RG flow from Cardy's ansatzs.
    
  * Gaiotto's RG defect




3/2= 3+0/2 = 2+1/2


2/2= 1+3/2 = 0+2/2 =1+1/2 = 2+2/2?


1/2= 1+0/2 = 3+2/2 = 1+2/2

 Paper Title: Spinning waveforms in cubic effective field theories of gravity.
































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        #1AB: #1#1GB: #1#1GT: #1#1PP: #1#1PVM: #1
        ^aCentre for Theoretical Physics, Department of Physics and Astronomy, 
        
        Queen Mary University of London, Mile End Road, London E1 4NS, United Kingdom^bNiels Bohr International Academy,
        Niels Bohr Institute, University of Copenhagen,
        
        Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmarka.brandhuber@qmul.ac.ukgraham.brown@qmul.ac.ukgang.chen@nbi.ku.dkg.travaglini@qmul.ac.ukp.vivesmatasan@qmul.ac.uk
        
        	QMUL-PH-24-11
        
        
        We derive analytic all-order-in-spin expressions  for  the leading-order time-domain waveforms  generated in the scattering of two Kerr black holes with arbitrary masses and spin vectors in the presence of all independent cubic deformations of Einstein-Hilbert gravity. These are  the  two parity-even interactions I_1 and G_3, and  the parity-odd ones 
        Ĩ_1 and G̃_3. 
        Our results are obtained using three independent methods: a particularly efficient  direct integration and tensor reduction approach; integration by parts combined with  the method of differential equations; and finally a residue computation. 
        For the case of the G_3 and G̃_3 deformations we can express the spinning waveform  in terms of the scalar waveform  with appropriately shifted impact parameters, which are reminiscent of Newman-Janis shifts. For I_1 and Ĩ_1 similar shifts occur, but  are   accompanied by additional contributions that cannot be captured by simply shifting the scalar I_1 and Ĩ_1 waveforms. 
        We also show the absence of leading-order corrections to gravitational memory.  
         Our analytic results are  notably compact,  and we compare the effectiveness of the three methods used to obtain them.  We also briefly comment on the magnitude of the corrections to observables due to cubic deformations.  
        
        
        
        
        
        Spinning  waveforms in cubic effective field theories of gravity 
        
            and Pablo Vives Matasan^a
            Submitted August 5, 2024
        ==================================================================
        
        
        
        § INTRODUCTION
        
        
        
        The high accuracy achieved in current observations of gravitational waves by the LIGO-Virgo-KAGRA collaboration, along with the potential of further  improvements  in upcoming experiments such as LISA, strongly motivates the pursuit of 
        increasing precision in theoretical calculations.
        One is then faced with the need to perform higher-loop calculations in Newton's constant 
        G, incorporating  spin effects, and also considering    potential modifications to the Einstein-Hilbert theory arising from as yet undiscovered higher-dimensional interactions. 
        
        
        
        Quadratic corrections in the curvatures are known to leave scattering amplitudes invariant <cit.>, hence the first  deformations to be studied appear at dimension six, that is they are cubic in the curvatures (see <cit.> for work on dimension-eight operators). 
        Specifically, we consider the deformations 
        
            I_1    R^αβ_μνR^μν_ρσR^ρσ_αβ ,      
            I_2 R^μνα_βR^βγ_νσR^σ_μγα .
        
        However,  instead of    I_2,  we  prefer working with  the combination 
        
            G_3  I_1 - 2 I_2  ,
        
        which is a topological term in six dimensions and has vanishing four-dimensional graviton amplitudes. 
        We will also study the effect of  the parity-odd couplings Ĩ_1 and G̃_̃3̃, which are obtained from the parity-even ones by replacing one of the Riemann curvatures R^μναβ by the dual R̃^μναβ=  
        (1/2)  ϵ^μνρσR_ρσ^αβ.
        Summarising, the effective action we will work with is 
        
            S = ∫d^4x  √(-g) (    -2/κ^2R      +  β_1 I_1 + β_2 G_3 + β̃_1 Ĩ_1 + β̃_2 G̃_3)  .
        
        Particular choices of the coefficients β_1, β_2, β̃_1, β̃_2  correspond to specific theories
        [For instance β_1 = -2/κ^2 α^'  2/48,  β_2 = -2/κ^2 α^'  2/24, β̃_1=β̃_2=0 reproduces the low-energy effective action of bosonic string theory.], but we will be agnostic and treat each deformation independently.  
        
        Cubic  deformations  have been studied  
        both using  general relativity approaches <cit.>, as well as 
        amplitude methods applied to the case of spinless binaries <cit.>, or systems where only one black hole is spinning and has a  much larger mass than the other black hole  <cit.>.
        Related work on other modified theories of gravity include <cit.>. We also note that in <cit.> a bound of ℓ_ EFT≤ 38.2 km was determined for the fundamental length scale of cubic theories, specifically for the  case where β_1 =β̃_1, β_2 =β̃_2=0. Furthermore,   <cit.> investigated the potential  detectability of cubic deformations of gravity in a gravitational-wave event from the merger of two stellar-mass black holes. 
        
        In a typical encounter, we expect both celestial bodies to have non-zero spin vectors, along with arbitrary masses, and this is the scenario we considered in <cit.>. 
        In that paper we focused   on  elastic processes, related to the two-to-two scattering amplitude of Kerr black holes,  and   using the KMOC formalism <cit.> we computed the leading-order  impulse (or momentum kick) and spin kick in a generic   hyperbolic encounter. These observables  vanish at tree level, and hence a one-loop calculation was required. 
        For the sake of gravitational-wave observations, the most relevant quantity to compute is the waveform, and this is what we  focus on in this paper. 
        
        
        Waveforms for hyperbolic spinning encounters in general relativity were derived to leading order using a worldline quantum field theory approach in <cit.> for the spinless case, reproducing the classic results of Kovacs and Thorne <cit.>. This was later extended to include spinning bodies in   <cit.>, with results valid  up to quadratic order in the spin. Results for the scattering waveform of spinless bodies at next-to-leading order were later derived using amplitude approaches in <cit.>,  while for spinning objects, 
        leading-order expressions 
        valid to high order in the spins were obtained in <cit.>, also using amplitude methods,   which could readily be extended to any spin order once the relevant Compton amplitudes are available. 
        
        
        In order to derive the waveforms  we will follow the adaptation of the KMOC approach presented in <cit.>. In that paper it was shown that, to leading order in the coupling, gravitational  waveforms can be  expressed as a Fourier transform to impact parameter space of the five-point amplitude describing the scattering of two spinning celestial objects with the emission of a graviton. In fact, only the physical singularities of this amplitude contribute to the waveform; as we approach the poles, the five-point amplitude factorises onto a four-point Compton amplitude and a three-point amplitude of two spinning objects with one graviton. 
        Compact expressions for  the Compton amplitude in the presence of cubic deformations were derived  in <cit.>, and we will use them to derive all-order in spin expressions for the tree-level waveforms of two Kerr black holes in a cubic background.
        
        
        After constructing the integrand from factorisation, we are left with the task of taking its Fourier transform. We will do this using three different approaches, which will give us the possibility of comparing their own merits. The first approach  is that of direct integration, building on  the seminal work of <cit.>.
        In this method, it is possible to write down ansätze for certain tensor (i.e. not necessarily scalar) integrals which are invariant under rescaling of the integration variable. Such integrals often contain poles on the integration contour and, for the rescaling argument to be valid, they  must  be regularised with a principal value prescription. This is possible  since  additional contributions arising from the   i ε prescriptions 
         are free of physical poles and hence cancel, as we have checked explicitly.  
         Higher-rank tensor integrals can then be produced efficiently with a generating function technique which is applicable thanks to the presence of a Fourier transform to impact parameter space. Effectively, integrals with  (loop-momentum dependent) numerators can be obtained by differentiating a master integral with a shifted impact parameter with respect to certain scalar auxiliary variables. This approach avoids  differentiation with respect to   four-momenta as well as lengthy Lorentz contractions, thereby leading to very compact results. 
        
         A second independent approach is based on integration by parts (IBP) reductions and the method of differential equations <cit.>. The waveforms corresponding to I_1 and its parity-odd version Ĩ_1
        contain spurious poles of the form (sinh x)/x, which we avoid by rewriting this function in an integral representation that is well suited for IBP reductions.  As a consequence, the Fourier transforms in the I_1 or Ĩ_1 cases are reduced to a single  simple master integral. We also note that this approach is systematic and  can be extended to one-loop waveforms with or without spin.  Finally, a third approach we  pursued employs Cauchy's residue theorem. This is straightforward to implement and highly  efficient, though it generally produces less compact  analytic results  compared to  the   other two methods, with  direct integration   yielding  the most compact results. 
        The three approaches give results in 
        perfect agreement. 
        
        Remarkably, we are able to write the spinning waveforms in terms of  Fourier transforms to impact parameter space where most (and, for the G_3 and G̃_3 cases, all)
        of the spin dependence is encoded in spin-dependent shifts of the impact parameter. These  shifts are reminiscent of the Newman-Janis shift,  which made an appearance  in amplitude contexts in <cit.>,  
        and more recently in the computation of spinning waveforms in <cit.>. 
        For the G_3 and G̃_3 waveforms, we are able to write the spinning waveforms entirely in terms of non-spinning ones (with shifted impact parameter). 
        
        The case of waveforms in the presence of parity-odd deformations is very easy to discuss and in fact does not require new calculations: parity-odd waveforms can be obtained from parity-even ones by simply swapping the “plus” and “cross” polarisations.  
        Investigating parity-odd deformations of gravity  is not merely an academic exercise, given the early 
         indications of parity violation in  the Cosmic Microwave Background and  the  
        large-scale structure of galaxies <cit.>.  
        
        A feature of all the leading waveforms in the presence of cubic deformations is that they do not modify the gravitational memory, which is easily proven using the connnection between the memory and soft theorems <cit.>. 
        
        
        Our  waveform results can be found in our   
        https://github.com/QMULAmplitudes/Cubic-Corrections-to-Spinning-Observables-from-Amplitudes Cubic Corrections to Spinning Observables from Amplitudes GitHub repository.  
        
        The rest of the paper is organised as follows. 
        In Section <ref> we summarise the expressions for the classical Compton amplitudes derived in <cit.>. 
        In Section <ref> we present the relevant formulae to derive the waveforms from  the factorisation diagrams of the five-point amplitudes. We also construct these factorisations using the classical Compton amplitudes and the classical three-point amplitudes of two spinning particles and one graviton. 
        In Section <ref> we present the direct integration method of the waveform, inspired by the work of <cit.>. The method is illustrated in great detail in the Appendices, while in this section  we use the results of the integrals to derive the  final expressions for the  waveforms with  G_3 and I_1 deformations. 
        In Section <ref> we present our second alternative derivation of the waveforms using a  systematic method proposed in <cit.> for the tensor integral generating functions <cit.>.    
        A third derivation of the waveforms is then briefly shown  in 
        Section <ref> using residues, much in line with our previous work <cit.>. Section <ref> illustrates the waveforms for various values of the relative velocities of the black holes and their spins, for the G_3 and I_1 deformations. These waveforms  clearly  show the absence of a  contribution to the gravitational memory, which we   demonstrate  in 
        Section <ref>. 
        Finally, in 
        Section <ref>  
        we conclude by comparing the size of cubic corrections to that of Post-Minkowskian (PM) corrections. 
        
        
        Two Appendices deal with the evaluation of  the  integrals needed to compute our waveforms. Specifically, in 
        Appendix <ref> we evaluate the necessary master integrals using a combination of explicit evaluations and educated guesses; and in 
        Appendix <ref> we present the generating function technique described earlier, which allows to compute with great economy all the necessary higher-tensor integrals. 
        
        
        
        
        § SUMMARY OF CLASSICAL SPINNING COMPTON AMPLITUDES WITH CUBIC DEFORMATIONS
        
        
        
        In this section we briefly review the Compton amplitudes in the presence of parity-even and parity-odd cubic deformations derived in <cit.>. 
        We denote by k_i and ε_k_i, i=1,2, the momenta and polarisations of the emitted gravitons, while m and a are the mass and ring radius of the black hole, with p being the classical momentum.  
        
        
        
         §.§ Parity-even deformations
        
        For the parity-even cubic deformations, the classical Compton amplitudes 
        in the gauge k_1 ε_k_2 = k_2 ε_k_1 =0 
        are found to be
        [From now on,   we will omit the coefficients of the cubic interactions β_1, β_2, β̃_1, β̃_̃2̃, which  can  be reintroduced at the end of the calculations. Furthermore, we  note that in our normalisations Newton's  constant is defined as   G κ^2 /(32π).]<cit.>
            M_I_1(p, k_1, k_2)    = i ( κ/2)^4   24 (k_1  k_2)^2 (ε_k_1ε_k_2) 
                {cosh ( a q ) [ 2 (pε_k_1) (pε_k_2) - m^2 (ε_k_1ε_k_2)  ]
                -i sinh (a q)/a  q[ 
            (pε_k_1 ) ϵ( ε_k_2 p a q) +(pε_k_2 ) ϵ( ε_k_1 p a q) 
            ] }  ,
        
            and
        
            M_I_2(p, k_1, k_2)    =
                i   ( κ/2)^4 6   (k_1  k_2)  (ε_k_1ε_k_2)^2  
                {cosh ( a q ) [ ( p (k_1 - k_2))^2 + m^2 (k_1  k_2)   ]
                -2i  sinh (a q)/a  q p (k_1 - k_2) ϵ( k_1 k_2  p a)  
            ] }  ,
        
            while 
            
            M_I_1(p, k_1^±±, k_2^∓∓)= M_I_2(p, k_1^±±, k_2^∓∓)=0 
                     ,
        
                where 
                q=k_1 + k_2.
                
        The amplitudes  (<ref>) and (<ref>) can also be rewritten in spinor-helicity variables:
        [Our spinor conventions are the same as in  <cit.>.]
            M_I_1(p, k_1^++, k_2^++)    = 
                i( κ/2)^4 3! [1  2]^4/q^2{ - 4 cosh(a q)   (p k_1)(p k_2) 
                +1/2p(k_1-k_2) sinh (a q)/a  q( [1|p|2⟩ [2|a|1⟩ - [2|p|1⟩ [1|a|2⟩) 
             }  ,
        
            and 
            
            M_I_2(p, k_1^++, k_2^++)    = i/2( κ/2)^4   3! [1  2]^4/q^2{cosh (a q) [ ( p (k_1 - k_2))^2 + m^2 (k_1  k_2)   ]
                        
               +1/2p(k_1-k_2) sinh (a q)/a  q( [1|p|2⟩ [2|a|1⟩ - [2|p|1⟩ [1|a|2⟩)  } .
        
        In the following we will focus on the G_3 deformations instead of I_2. The corresponding classical Compton amplitude in spinor-helicity variables is  
        
            M_G_3(p, k_1^++, k_2^++) = -3   i m^2  ( κ/2)^4 cosh (a q) [1  2]^4   
                .
        
        Note that this amplitude has no term proportional to (sinh a q) /(a q).
        
        
        
         §.§ Parity-odd deformations
        
        
        Finally, as discussed in <cit.>
        the amplitudes with parity-odd deformations can be obtained easily from the parity-even  ones:  for the case of positive (negative)  helicity, they are equal to those for the corresponding parity-even case multiplied by a factor of  +i (-i), and we recall that the only non-vanishing amplitudes are those where the two gravitons have the same helicity.  
        
        
        
        
        § CONSTRUCTING THE TIME-DOMAIN WAVEFORMS
        
        
         §.§ General results
        
        
        The derivation of the time-domain waveforms using the KMOC formalism was presented  in <cit.>.  
        We will work in the  far-field limit, that is at large observer distance r|x⃗|  and time t with fixed retarded time u t-r.  
        The quantity of interest  is then 
        
            ⟨ h_+± i h_×⟩   ⟨ h_μν^ out⟩^μν_(±±)1/4π r (h_+^∞± i h_×^∞)   ,
        
        for which one finds that
        [See for example Section 5 of <cit.> for a detailed derivation.]
            h^∞_+± i  h^∞_×  =    
              
              κ∫_-∞^+∞dω/2π e^-i ω u 
             [ 	 
            		 
               θ(ω) .W(b, k^±)  |_k= ω (1, 𝐱̂)
             +   
             θ(-ω) . W^∗(b,k^∓) |_k= -ω (1, 𝐱̂)]
            		,
        
        where  W=W(b⃗, k; h)  is the spectral waveform for the emission of a graviton of momentum k and helicity h, 
        which satisfies,  at tree level, 
            
            .    W^∗ (b, k^h)|_k=-ω(1, 𝐱̂) = 
            .    W(b, k^-h)|_k=ω(1, 𝐱̂)  .
        
        Using this result, one can easily combine  the two terms in (<ref>) into 
        
            h^∞_+± i  h^∞_× = 
            κ∫_-∞^+∞dω/2πe^- i ω u. 
            W(b, k^±)|_k=ω(1, 𝐱̂)  .
        
        At leading order (tree level) this can be neatly obtained from the  five-point classical bremsstrahlung amplitude ℳ_5(q_1, q_2, a_1, a_2; h) describing the scattering of two objects with ring radii a_1 and a_2 with the emission of a graviton with momentum k= q_1 + q_2 and helicity h: 
        
            W(b, k^h )   -i ∫dμ^(4)  e^i(q_1 b_1 + i q_2  b_2)  ℳ_5(q_1, q_2, a_1, a_2; h)   ,
        
        where  
        
            dμ^(D)d^Dq_1/(2π)^D-1d^Dq_2/(2π)^D-1  (2π)^D  δ^(D) (q_1 + q_2 - k) δ(2  _1 q_1 ) δ(2  _2 q_2 )  ,
        
        and   q_1,2= p_1,2- p_1,2^' are the momentum transfers. The barred variables 
        <cit.>  are defined as 
        
            p_1    = p̅_1 + q_1/2  ,      p_1^' = p̅_1 - q_1/2  , 
            
            		p_2    = p̅_2 +  q_2/2  ,      p_2^' = p̅_2 - q_2/2  ,
        
        and satisfy  
            p̅_1 q_1 =p̅_2 q_2 =0   .
        
        However, at leading order we can actually drop the distinction between p̅_i and p_i, hence we will henceforth drop bars. Furthermore, as in <cit.>,
        in order to simplify the notation  we will  work with the quantity 
        
            h^∞(u)    κ∫_-∞^+∞dω/2πe^- i ω u. 
            W(b, k)|_k=ω(1, 𝐱̂)
               = -iκ∫_-∞^+∞dω/2πe^- i ω u∫d^4q_1/(2π)^2δ(2  p_1 q_1 ) δ(2  p_2 (k-q_1) )  e^i(q_1 b_1 + q_2  b_2) ℳ_5   ,
        
        where we omitted the helicity dependence, 
        performed the q_2 integration using momentum conservation from (<ref>)
        and defined
        
            kωk̂ =ω(1, 𝐱̂)  .
        
        This is the main formula we will use. We also note that b b_1 - b_2 is the asymptotic impact parameter, which  satisfies 
        
            b v_1 = b v_2 = 0  .
        
        
         §.§ Kinematics of the five-point scattering
        
        
        The time-domain waveform (<ref>) requires knowledge of the classical part of the five-point bremsstrahlung amplitude, that is the amplitude of two spinning particles with  one emitted graviton. 
        The kinematics of the process is shown below: 
        
            [ [scale=12,baseline=([yshift=-1mm]centro.base)]
            			00 at (0+,0+) (centro) ;
            			 at (-3pt+,-3pt+) (uno) p_1;
            			 at (-3pt+,3pt+) (due) p_2;
            			 at (3pt+,3pt+) (tre) p_2^';
            			 at (3pt+,-3pt+) (quattro) p_1^';
            			 at (5.55pt+,) (cinque)
            			
            			k= q_1 + q_2;
            
            			 [thick,double,orange] (uno) – (centro);
            			 [thick,double,amber] (due) – (centro);
            			 [thick,double,amber] (tre) – (centro);
            			 [thick,double,orange] (quattro) – (centro);
            			 [vector,double] (cinque) – (centro);
            
            			 [->] (-2.8pt+,-2pt+) – (-1.8pt+,-1pt+);
            			 [<-] (2.8pt+,-2pt+) – (1.8pt+,-1pt+);
            			 [<-] (-1.8pt+,1pt+) – (-2.8pt+,2pt+);
            			 [->] (1.8pt+,1pt+) – (2.8pt+,2pt+);
            			 [->] (2.3pt+,-0.6pt+) – (3.75pt+,-0.6pt+);
            
            			 [shading=radial] (centro) circle (1.5pt);                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   ]
        
        The  two Kerr black holes  have  masses m_1 and m_2  with 
        
            p_1^2 = (p_1^')^2 = m_1^2  ,       p_2^2 = (p_2^')^2 = m_2^2  .
        
        It is also convenient to  introduce four-velocities v_1 and v_2 as
        
            p_1  m_1 v_1  ,       p_2  m_2 v_2  ,
        
        so that  v_1^2 = v_2^2 = 1. The two black holes have ring radii a_1 and a_2, where for a single black hole we recall that the ring radius a^μ is related to the spin vector as a^μ S^μ/ m <cit.>, where 
        
            S^μ = 1/2 mϵ^μναβ p_ν S_αβ ,
        
        with S^μν being the spin tensor. 
        The ring radius satisfies the covariant spin supplementary condition    p a(p) =0. 
         Finally, we also define the Lorentz factor
        
            σ v_1  v_2  ≥ 1 .
        
        Note that σ=1/√(1-ẋ⃗̇^2), with  ẋ⃗̇ being the relative velocity of one of the two black holes  in the rest frame of the other. For instance, in  the rest frame of particle 1 we can write  v_1^μ=(1,0,0,0) and   v_2^μ=σ(1, ẋ⃗̇). 
        
        
        
         §.§ Waveform integrands
        
        
        An important simplification  in the calculation of the waveforms  consists in the fact that only the residues on the physical factorisation channels  are needed in order to obtain  the waveform. These can  be computed from the two factorisation diagrams 
        
            [baseline=([yshift=-0.4ex]current bounding box.center)]every node=[font=]
            		 (p1) 1;
            			 [above=1.5cm of p1](p2)2;
            			 [right=1.5cm of p2] (u1) [HV]H;
            			 [right=1.2cm of u1] (p3)2^';
            			 [right=1.5cm of p1] (b1) [dot];
            			 [right=1.2cm of b1] (p4) []1^';
            			 [above=0.4cm of p1] (cutLt);
                         [right=0.6cm of cutLt] (cutL);
            			 [right=1.8cm of cutL] (cutR)k;
            			 [right=0.5cm of b1] (cut1);
            			 [above=0.2cm of cut1] (cut1u);
            			 [below=0.2cm of cut1] (cut1b);
                        [vector, double] (u1) – (b1); 
            			* 
            			(p2) – [thick, amber, double] (u1) – [thick, double, amber] (p3),
            			(b1)–[opacity=0,momentum=q_1](u1),  (u1)– [vector, double] (cutR), (p1) – [thick, orange, double] (b1)– [thick, orange, double] (p4), (cutL)–[dashed, red,thick] (cutR),
            			;
            		      [baseline=([yshift=-0.4ex]current bounding box.center)]every node=[font=]
            		 (p1) 1;
            			 [above=1.5cm of p1](p2)2;
            			 [right=1.5cm of p2] (u1) [dot];
            			 [right=1.2cm of u1] (p3)2^';
            			 [right=1.5cm of p1] (b1) [HV]H;
            			 [right=1.2cm of b1] (p4) []1^';
            			 [above=1.1cm of p1] (cutLt);
                         [right=0.6cm of cutLt] (cutL);
            			 [right=1.8cm of cutL] (cutR)k;
            			 [right=0.5cm of b1] (cut1);
            			 [above=0.2cm of cut1] (cut1u);
            			 [below=0.2cm of cut1] (cut1b);
                        [vector, double] (b1) – (u1); 
            			* 
            			(p2) – [thick, amber, double] (u1) – [thick, amber, double] (p3),
            			(u1)–[opacity=0,momentum'=q_2](b1), (b1)– [vector, double] (cutR), (p1) – [thick, orange, double] (b1)– [thick, double, orange] (p4), (cutL)–[dashed, red,thick] (cutR),
            			;
        
        which  correspond to the two possible factorisations as q_1^2→ 0 or q_2^2→ 0, with q_1 + q_2 = k. 
        In each diagram, the key ingredients are the Compton amplitudes for two spinning objects, which were computed for the parity-even and parity-odd cubic deformations  in <cit.> and are quoted in Section <ref>. Using these, we now construct the integrands for the  parity-even and parity-odd cubic deformations of gravity. 
        
        For simplicity, from now on we focus on the case of an emitted graviton with positive helicity. The two factorisation diagrams are then given by 
        
            _I, q_1^2    = i/q_1^2 M_3(-p_1, q_1^–) M_I(-p_2, - q_1^++, k^++)    , 
            _I, q_2^2    = i/q_2^2 M_3(-p_2, q_2^–) M_I(-p_1, - q_2^++, k^++)   ,
        
        with  
        
            _5, I  = _I, q_1^2+ _I, q_1^2 +  contact  terms  .
        
        Here  M_I denotes  the Compton amplitude in the presence of a deformation I,  with I∈ (I_1, G_3, Ĩ_1, G̃_3), as given in Section <ref>. 
        M_3 is  
        a classical three-point amplitude with one graviton of momentum q and two massive 
        spinning particles with momenta p and -(p+q) and ring radius a. For the two possible helicities of the emitted gravitons one has <cit.>
            M_3 (p, q^++) = -i ( κ/2) e^a q(
                ⟨ξ | p | q]/⟨ξ q⟩)^2  ,      
                M_3 (p, q^–) = -i ( κ/2) e^-a q(
                ⟨ q  | p | ξ ]/[ q ξ] )^2  ,
        
        where ξ and ξ̃ are reference spinors. 
        Finally, we observe that the second factorisation diagram in (<ref>) can be obtained from the first as 
        
            _I, q_2^2 = ._I, q_1^2|_1↔2  ,
        
        where 1↔2 means swapping  the masses, spin vectors and momenta, in addition to also exchanging q_1 and q_2.
        
        
        With these ingredients it is very easy to construct the integrand of the waveforms. For the case of the G_3 deformation,  and choosing as reference spinor of the three-point amplitude the spinor of the external graviton, we get 
        
            _G_3, q_1^2 = - i (κ/2)^53  m_1^2 m_2^2/q_1^2  [k| v_1 q_1 | k]^2  e^- a_1  q_1cosh( a_2 (q_1 - k) )  .
        
        For I_1, the result is
        
            _I_1, q_1^2   =  -i (κ/2)^53  m_1^2 m_2^2/q_1^2[k| v_1 q_1 | k]^2/ q_1 ke^- a_1  q_1×
               [ - 4 cosh [ a_2 (k-q_1)] (-v_2 q_1) (v_2  k) - 2 i  v_2  (k+q_1) sinh a_2 (k-q_1)/a_2 (k-q_1)ϵ ( k v_2 q_1 a_2)]  .
        
        Note that because of the first  δ-function in (<ref>)
        we can simplify the above expression using v_1 q_1=0 into 
        
            _I_1, q_1^2   = -  i (κ/2)^512    m_1^2 m_2^2/q_1^2  (q_1 k)  
              [k| v_1 q_1 | k]^2(v_2  k )  e^- a_1  q_1×
               [   cosh [ a_2 (k-q_1)]  (v_2  k) -  i sinh a_2 (k-q_1)/a_2 (k-q_1)ϵ ( k v_2 q_1 a_2)]  .
        
        
        § TIME-DOMAIN WAVEFORMS FROM DIRECT INTEGRATION 
        
        
        There are several approaches to explicitly perform the integrations in (<ref>). 
        In this section  we follow  the direct integration approach of <cit.>, while in Section <ref> we present  an alternative derivation  based on Cauchy's  residue theorem. 
        
        
        The first step in the direct integration method consists in 
        performing the ω integration
        in (<ref>) using
        the delta function δ (2 p_2  (k-q_1)). This localises the graviton energy ω to the value ω^∗ given by  
        
            ω^∗ = q_1 v_2/k̂ v_2  ,
        
        where we recall from (<ref>) that 
           kωk̂.
        One then quickly arrives at <cit.>
            h^∞(u) = - i κ/2 m_1 m_2 (k̂ v_2)∫d^4 q_1/(2π)^3δ(2 q_1  v_1) e^i q_1b̃._5|_k = ω^∗k̂  ,
        
        with the following definitions: 
        
            b̃   b̃_1 - b̃_2  ,  
                
            b̃_i     b_i + u_i v_i  ,      
               u_i  u - k̂ b_i/k̂ v_i  ,      i=1,2.
        
        We note that the modified (shifted) impact parameter b̃ has the useful  property that 
        
            k̂b̃=0  ,
        
        since k̂b̃_i =u. We are then left to perform integrations of the form (<ref>). For clarity we have derived and collected all required  master integrals in Appendix <ref> 
        and <ref>, where a detailed description of the direct integration method is included. 
        
        
        
        
        
        
        
         
        
        
        
         §.§ The G_3 waveform
        
        We now discuss the G_3 waveform. 
        The relevant integral is (<ref>), where we can replace the five-point amplitude with the contributions of the two poles,  _G_3, q_1^2, _G_3, q_2^2 and discarding contact terms (see (<ref>)).   
        To proceed, we first rewrite  
        
            |k] √(ω) |k̂]  ,
        
        and recall that ω is localised at the q_1-dependent value ω^∗ given in (<ref>), and where | k̂] is ω-independent. 
        We also rewrite the cosh term in M_G_3(-p_2, - q_1^++, k^++) in terms of exponentials, with  the net effect of shifting the impact parameters. Doing so, we obtain for  
        the contribution to the time-domain waveform  arising from the first diagram in (<ref>), 
        
            h^∞, (1)_G_3 (u) = - i κ/2 m_1 m_2 (k̂ v_2)∫d^4 q_1/(2π)^3δ(2 q_1  v_1) e^i q_1b̃._G_3, q_1^2|_k = ω^∗k̂
               =- 3/2(κ/2)^6
                m_1 m_2/(k̂ v_2)^3∫d^4q_1/(2π)^3δ(2 q_1 v_1) (q_1  X)^2 (q_1 v_2)^2/q_1^2[ e^i q_1  [  b̃ +  i (ã_1 - ã_2) ] + e^i q_1  [  b̃ + i (ã_1 + ã_2)]] 
                = - 3/2(κ/2)^6
                m_1 m_2
                /(k̂ v_2)^3 X_μ X_ν v_2 α v_2β[ 
                .^μναβ|_b̃→b̃ + i(ã_1 - ã_2) + .^μναβ|_b̃→b̃+ i(ã_1 + ã_2)] 
                = - 3/2(κ/2)^6
                m_1 m_2
                /(k̂ v_2)^3[ 
                .^(1)|_b̃→b̃ + i(ã_1 - ã_2) + .^(1)|_b̃→b̃+ i(ã_1 + ã_2)]  ,
        
        where we used the expression for _G_3, q_1^2 derived earlier in (<ref>). We also  defined
        
            X_μ = [k̂| v_1 σ_μ |k̂]   ,
        
        and
        
            ã_i   a_i - v_i a_ik̂/v_ik̂ ,
        
        which allows us, using (<ref>), to rewrite
        [In writing (<ref>) we have also used that q_1 v_1=0 and replaced a_1 q_1 with ã_1 q_1. This is useful in order to maintain the symmetry between the two factorisation diagrams in (<ref>) so that the second can easily be obtained from the first upon performing suitable replacements, see (<ref>). 
        ]
            a_2  (k-q_1) = -ã_2  q_1  .
        
        The definitions of the tensor integrals ^μ_1 …μ_n can be found in (<ref>). 
         
        The integrals in the second line of (<ref>) can be computed elegantly using the method of generating functions introduced in Appendix <ref>. In particular  (<ref>) and (<ref>) are relevant for this case, with the result
        
            ^(1)   =∫d^4q_1/(2π)^3δ(2 q_1 v_1) (q_1  X)^2 (q_1 v_2)^2/q_1^2 e^i q_1 b̃
               = -1/8 π(∂^4/∂^2 t_1 ∂^2 t_21/|b̃_(1)+t_1 v_2(1) + t_2 X_(1)|)_t_1=t_2=0 .
        
        Here the subscript (1)  indicates the projection imposed by the delta function and implemented by the projector 
        
            P_1^μν=η^μν-v_1^μ v_1^ν ,
        
        which allows us to define
        
            V^μ_(1) = P_1^μν V_ν ,
        
        and
        
            |V_(1)| = √(-V_(1) V_(1))
            = √(-V  P_1  V) .
        
        We also note that X^μ_(1) = X^μ.
        
        With these definitions and performing the derivatives in (<ref>) we find
        
            ^(1)     = -3/8 π|b̃_(1)|^5[ 
            2 (X  v_2 (1))^2
                
            + 5/|b̃_(1)|^2(v_2(1)^2 (X b̃_(1))^2 
            + 4 (X  v_2 (1)) 
            (X b̃_(1))(b̃_(1) v_2(1))
            +7 (b̃_(1) v_2)^2/|b̃_(1)|^2 (X b̃_(1))^2)
            ]  ,
        
        which can be simplified further into
        [We have also rewritten b̃_(1) v_2(1) as b̃_(1) v_2, since for any two vectors m^μ and n^μ we have m_(1) n_(1) =m n_(1) = m_(1) n, with m_(1)^μ P_1^μν m_ν, with the projector P_1 defined in (<ref>).]
            ^(1) = -3/8 π[ 
            2 (X  v_2)^2/|b̃_(1)|^5  
            + 5 (X b̃)/|b̃_(1)|^7(v_2(1)^2 (X b̃) 
            + 4 
            (b̃_(1) v_2) (X  v_2) 
            +7 (b̃_(1) v_2)^2/|b̃_(1)|^2 (X b̃))
            ]  .
        
        Alternatively, we can evaluate the third line of (<ref>) by twice differentiating (<ref>), which gives  
        
            ^μναβ(b̃)     = - 3/8 π|b̃_(1)|^5[ 
                ( P_1^μν P_1^αβ + P_1^μα P_1^νβ + P_1^να P_1^μβ) 
                
               + 5/|b̃_(1)|^2  (
                P_1^μνb̃_(1)^αb̃_(1)^β + P_1^μαb̃_(1)^νb̃_(1)^β + P_1^ναb̃_(1)^μb̃_(1)^β
                +
                P_1^αβb̃_(1)^μb̃_(1)^ν + P_1^μβb̃_(1)^αb̃_(1)^ν + P_1^νβb̃_(1)^αb̃_(1)^μ)
                
                + 35/|b̃_(1)|^4   b̃_(1)^μb̃_(1)^νb̃_(1)^αb̃_(1)^β]   ,
        
        where the projector P_1^μν is defined in (<ref>), 
        
        b̃_(1)^μ in (<ref>) and 
        |b̃_(1)| 
        in (<ref>). 
        
        Note that all the terms proportional to P_1^μν in 
        (<ref>) do not contribute to  the contractions in (<ref>).
        Pleasingly, we find complete agreement of the contractions of this tensor integral with the result obtained from the generating function,  
        
            X_μ X_ν v_2 α v_2β ^μναβ(b̃) = ^(1) .
        
        We then have 
        
            h^∞, (1)_G_3 (u)  
                    = - 3/2(κ/2)^6
                m_1 m_2
                /(k̂ v_2)^3[ 
                .^(1)|_b̃→b̃ + i(ã_1 - ã_2) + .^(1)|_b̃→b̃+ i(ã_1 + ã_2)] 
                  .
        
        We  note that effectively we have rewritten the spinning waveform in terms of the scalar waveform, but with shifted impact parameters: 
        
            h_G_3^∞, (1) (u) =1/2[ . h_G_3^∞, (1) (u) |_a_i=0,  b̃→b̃+ i (ã_1 + ã_2)  +  . h_G_3^∞, (1) (u) |_a_i=0,  b̃→b̃+ i (ã_1 - ã_2)]   .
         
        We call the two contributions of these two terms as arising from two “sectors”, with shifted impact parameter
        
            (1)               (2)        
            b̂ = b̃ + i (ã_1+ ã_2)        b̂ = b̃ + i(ã_1- ã_2)   ,
        
        where we recall that ã_i was defined in (<ref>). 
        The result for I_1 will not have such a simple form, but it can be expressed in terms of the same two sectors. 
        
        Finally, the  complete waveform, including contributions from both factorisation channels, is 
        
            h^∞_G_3 (u) = h^∞, (1)_G_3 (u) + h^∞, (2)_G_3 (u)  ,
        
        where
        
            h^∞, (2)_G_3 (u) = .h^∞, (1)_G_3 (u)|_(m_1, a_1,  v_1, b_1)↔ (m_2, a_2,  v_2, b_2)  .
        
        We also observe  that under this exchange the variable b̃ defined in (<ref>) changes sign, b̃↔  - b̃. 
        
        
        
        
        
        
        
        
        
        
        
         §.§ The I_1 waveform
        
        
        In the presence of the I_1 deformation, the time-domain waveform  is 
        
            h^∞, (1)_I_1 (u) = - i κ/2 m_1 m_2 (k̂ v_2)∫d^4 q_1/(2π)^3δ(2 q_1  v_1) e^i q_1b̃._I_1, q_1^2|_k = ω^∗k̂  ,
        
        where we have, from (<ref>),
        
            ._I_1, q_1^2|_k = ω^∗k̂   =   -  i (κ/2)^512    m_1^2 m_2^2  (ω^∗)^3/q_1^2  (q_1k̂)  
              (X  q_1)^2(v_2 k̂ )  e^- a_1  q_1×
               [   cosh [ a_2 (ω^∗k̂-q_1)]  (v_2 k̂) -  i sinh a_2 (ω^∗k̂-q_1)/a_2 (ω^∗k̂-q_1)ϵ ( k̂ v_2 q_1 a_2)]  ,
        
        with ω^∗  given in (<ref>). 
        Therefore we obtain  
        
            h^∞, (1)_I_1 (u)
                = - (κ/2)^6 
            6m_1 m_2/(k̂ v_2)^3∫d^4q_1/(2π)^3δ(2 q_1 v_1) (X  q_1)^2  (q_1 v_2)^3/q_1^2  (q_1k̂)×
               [ ( v_2 k̂ + i ϵ(k̂ v_2 q_1 a_2)/q_1 ã_2)  e^i q_1  [  b̃ +  i (ã_1 + ã_2) ] + ( v_2 k̂  - i ϵ(k̂ v_2 q_1 a_2)/q_1 ã_2)  e^i q_1  [  b̃ +  i (ã_1 -  ã_2) ]]
                  ,
        
        where the shifted spin vectors  ã_i were defined in (<ref>). 
        
        We now  derive using the  method of generating functions presented in Appendix <ref>
        the two master integrals  needed to evaluate the waveform integral  in  (<ref>), corresponding to the first factorisation diagram. 
        Before doing so, we  comment that 
        its integrand   contains the spurious pole 1/ (q_1k̂), which originates from the propagator i/(q^2 + i ε) =  i / ( -2 q_1k̂ + i ε) in the Compton amplitude (<ref>) onto which the five-point amplitude factorises. We have checked explicitly that this pole is cancelled by a corresponding spurious pole in the second factorisation diagram, and have double-checked this conclusion by a detailed comparison with the full five-point amplitude obtained from a diagrammatic computation. In practice it is convenient to perform the integrations of the two factorisation diagrams separately, which requires a choice of regularisation of the pole. The sum of the two diagrams should be independent of the  regularisation, since the pole is spurious; 
        we have confirmed this expectation by performing the integration with either the Feynman i ε prescription or the Principal Value (PV)  prescription,  finding complete agreement.
        [In fact, using the PV prescription the spurious contributions vanish separately in each diagram; this is related to the tracelessness of the integral 𝒦^μν
        introduced in (<ref>), see end of Appendix <ref> for a thorough discussion of this point.
        ]
        In the following we will choose the PV prescription, and for completeness    we  give in Appendix <ref> the master integral 𝒥^μ in (<ref>) with both prescriptions (while for 𝒦^μν we only present the result in the PV prescription).
        
        The first type of integral we need is
        
            ^(1)   ∫d^4q_1/(2π)^3δ(2 q_1 v_1) (q_1  X)^2 (q_1 v_2)^3/q_1^2 (q_1 k̂) e^i q_1 b̃ , 
               = [  ∂^4/∂^2 t_1 ∂^2 t_2(
             𝒥 v_2 )_b̃→b̃+t_1 v_2 + t_2 X]_t_1=t_2=0
                = 
            [  ∂^4/∂^2 t_1 ∂^2 t_2(v_2  K_1 k̂/8 π |b̃_(1)|  [(v_1 k̂)^2 |b̃_(1)|^2-(k̂b̃_(1))^2 ])_b̃→b̃+t_1 v_2 + t_2 X]_t_1=t_2=0  ,
        
        where we have used (<ref>) and 
        K_1^μν = |b̃_(1)|^2 P_1^μν + b̃_(1)^μb̃_(1)^ν from (<ref>).
        
        The second type is
        
            ℰ^(1)   ∫d^4q_1/(2π)^3δ(2 q_1 v_1) (q_1  X)^2 (q_1 v_2)^3 ϵ(k̂ v_2 q_1 a_2)/q_1^2 (q_1 k̂)(q_1 ã_2) 
            e^i q_1 b̃
               = -[  ∂^4/∂^2 t_1 ∂^2 t_2(Y 𝒦 v_2 )_b̃→b̃+t_1 v_2 + t_2 X]_t_1=t_2=0 ,
        
        with 
        
            Y 𝒦 v_2 = 
              - ( ã_2  K_1k̂) (Y  K_1 v_2) - (ã_2  K_1 Y) (k̂ K_1 v_2)-
            (ã_2  K_1 v_2) (k̂ K_1 Y)/8 π  |b̃_(1)| ( ã_2  K_1ã_2 ) ( k̂ K_1k̂) ,
        
        where we have used
        Y^μϵ(k̂ v_2μ a_2), as well as (<ref>) and (<ref>).
        
        Putting everything together, we find that (<ref>) evaluates to
        
            h^∞, (1)_I_1 (u)
                = - (κ/2)^6 
            6m_1 m_2/(k̂ v_2)^3×
               [ ( (k̂ v_2) ^(1)+ i ℰ^(1))_b̃→b̃ +  i (ã_1 + ã_2) + 
            ( (k̂ v_2) ^(1)- i ℰ^(1))_b̃→b̃ +  i (ã_1 - ã_2)]
                  .
        
        As for the G_3 case, the complete I_1 waveform is 
        
            h^∞_I_1 (u) = h^∞, (1)_I_1 (u) + h^∞, (2)_I_1 (u)  ,
        
        where
        
            h^∞, (2)_I_1 (u) = .h^∞, (1)_I_1 (u)|_(m_1, a_1,  v_1, b_1)↔ (m_2, a_2,  v_2, b_2)  .
        
        Finally, we  note the appearance in (<ref>) of shifted impact parameters b̃ +  i (ã_1 ±ã_2),  similarly to (<ref>) in the G_3 case. However there is an important difference with that case: while  the G_3 waveform can be obtained from the scalar  integrated waveform by performing shifts in the impact parameter as dictated by the sectors, in the I_1 case these shifts occur at the level of the  integrand. For the cosh part of the I_1 integrand they translate to shifts that can be performed on the integrated result; however the part of the I_1 proportional to the sinh function vanishes in the scalar case, and thus provides a new contribution that cannot be obtained from the spinless waveform.  
        
        
        
        
        
        
         §.§ Waveforms for parity-odd deformations
        
        
        To derive the waveforms for parity-odd deformations we only need to recall from Section <ref> that the Compton amplitudes for parity-odd deformations are obtained from the parity-even ones by multiplying them by a factor of ± i depending on whether the two gravitons have positive or negative helicity. The same property is inherited by the cut five-point amplitudes which we use to construct the waveforms, that is 
        
            -i _5^ P.O. (k^+) = i (-i _5^ P.E.(k^+))  ,     (-i _5^ P.O. (k^-))^∗ = i (-i _5^ P.E.(k^-))^∗  ,
        
        from which it follows that 
        
            W^ P.O. (k^+) = i W^ P.E.(k^+)  ,     (W^ P.O. (k^-))^∗ = i (W^ P.E.(k^-))^∗  ,
        
        where P.O. and P.E. stand for parity odd and even, respectively.
        Combining (<ref>) with (<ref>) 
        we see that 
        
            .    (W^ P.O. (k^-))^∗|_k = - ω (1,𝐱̂)  = .W^ P.O. (k^+)|_k =  ω (1,𝐱̂)  = 
            .i  W^ P.E. (k^+)|_k = ω (1,𝐱̂)   .
        
        The two terms in  
        (<ref>), can then be combined as in the parity-even case, 
        and hence we conclude that 
        
            (h^∞,  P.O._+ + i    h^∞,  P.O._×)    = 
                κ∫_-∞^+∞dω/2πe^- i ω u. 
            W^ P.O.(b, k)|_k=ω(1, 𝐱̂)
               =
                i   (h^∞, P.E._+ + i    h^∞,  P.E._×)  .
        
        This implies that  
        
            h^∞,  P.O._+ = -    h^∞,  P.E._×  ,       
                h^∞,  P.O._× = h^∞,  P.E._+  ,
        
        that is,  the “plus” and “cross” polarisations are then swapped  in the way prescribed by (<ref>).
        
        
        
        § TIME-DOMAIN WAVEFORMS FROM  TENSOR INTEGRAL GENERATING FUNCTIONS, RELOADED
        
        Alternatively, in the spirit of modern multiloop amplitude calculations, we can compute the integrals by a systematic method proposed in <cit.>.
        For the tree-level waveform, we have two D-dimensional master integrals,    
        
            _1[𝐲]∫d^Dq_1/(2π)^D-1δ(2 q_1 v_1) q_1^2  e^i q_1 b̂ ,      _2[𝐲]∫d^Dq_1/(2π)^D-1δ(2 q_1 v_1) q_1^2 q_1k̂  e^i q_1 b̂ ,
        
        where 𝐲 (y_1, y_2, y_3,ŷ_4,ŷ_5,ŷ_6,ŷ_7), with 
        
            y_1   σ=v_1 v_2,     y_2    v_1k̂ ,    y_3    v_2k̂ , 
            ŷ_4   b̂k̂√(-b̂b̂)=0,    ŷ_5   b̂ v_1 √(-b̂b̂),    ŷ_6   b̂ v_2√(-b̂b̂),    ŷ_7b̂b̂ .
        
        One can rewrite the two master integrals as 
        
            _j[𝐲]∫_-∞^∞dt   e^it _j[𝐲, t]   :=∫_-∞^∞dt   e^it∫d^D-4q_1/(2π)^3δ(2 q_1 v_1) q_1^2 (q_1 k)^j-1δ(q_1b̂-t)
               .
        
        The differential equation in t for _j[𝐲, t] is very simple:
        
            ∂_t  _1[𝐲, t]= D-4/t_1[𝐲, t],       ∂_t  _2[𝐲, t]= D-5/t_2[𝐲, t] .
        
        It is easy to see that  the t dependence is factorised from the other variables, and we can solve for it directly as 
        
            _1[𝐲, t]= t^D-4 D-4_1[𝐲] ,       _2[𝐲, t]= t^D-5_2[𝐲]  .
        
        Then the two integrals in (<ref>) are rewritten as  
        
            _1[𝐲]   =(∫_-∞^∞dt   e^it t^-4 + D D-4)_1[𝐲]=-π _1[𝐲]  ,
            _2[𝐲]   =(∫_-∞^∞dt   e^it t^-5 + D)_2[𝐲]=iπ _2[𝐲]  ,
        
        where we have already used D = 4 - 2ϵ and taken the → 0 limit.
        The differential equations in four  spacetime dimensions  of the  𝐲-dependent part _1,2[𝐲] are 
        
            ∂_ŷ_5_1[𝐲]= -ŷ_5/ŷ_5^2+1_1[𝐲],       ∂_y_2_2[𝐲]= -1  y_2_2[𝐲].
        
        The solutions  are 
        
            _1[𝐲]=-π/(2π)^3√(-b̂b̂)√(ŷ_5^2+1),      _2[𝐲]=c_2  y_2 ,
        
        where c_2=0 from the boundary conditions. 
        Hence we can omit _2[𝐲] in the calculation and we are thus left with a single master integral. 
        
        All other integrals can be reduced to this master integral. For example,  consider the case of the integral
        
            𝒱[𝐲]   ∫d^Dq_1/(2π)^D-1δ(2 q_1 v_1)q_1 v_2 q_1^2 q_1k̂  e^i q_1 b̂  .
        
        We can rewrite it as 
        
            𝒱[𝐲]=∫_-∞^∞dt  e^it 𝒱[𝐲, t]   =∫_-∞^∞dt   e^it∫d^Dq_1/(2π)^D-1δ(2 q_1 v_1)q_1 v_2 q_1^2 (q_1 k)δ(q_1b̂-t)  .
        
        By  IBP reduction, the q_1-integral is reduced to the single master integral  _1[𝐲], and finally we get 
        
        
        
        
        
            𝒱[𝐲]
            	   =(-π)y_1 (y_2+ŷ_4 ŷ_5)-y_3 (ŷ_5^2+1)-ŷ_4 ŷ_6+y_2 ŷ_5 ŷ_6/y_2^2+2 ŷ_4 ŷ_5 y_2-ŷ_4^2_1[𝐲]  .
        
        We have checked that this formula is  consistent with (<ref>).   
        
        
        
        
        
        
        
        
        
         The integral in (<ref>) contains a spurious pole in the entire function (sinh x)/x (see <cit.> for general discussions). However we can eliminate it  by introducing the integral representation   <cit.>
            sinh x  x= ∫_0^1dz  cosh(z  x)  .
        
        Then there are four sectors in the waveform, with exponential factors e^i q b̂. The particular  form of the new variable b̂ depends on the  sector as follows:
        [We note that in the previous approach we employed to compute the I_1 waveform there were  two sectors, corresponding to the cases   (1) and  (2) below. We now have two additional sectors because 
         the integral representation    (<ref>) introduces a   dependence on the integration variable z.]
            (1)              (2)              (3)              (4)       
            b̂= b̃+i(ã_1+ ã_2)        b̂=b̃+i(ã_1- ã_2)       b̂= b̃+i(ã_1+ z ã_2)       b̂= b̃+i (ã_1- z ã_2)  .
        
        Then the waveform can be represented as a combination of tensor integrals:
            
            h^∞, (1)_G_3 (u)
                = - (κ/2)^6
            6m_1 m_2/(k̂ v_2)^3(1 4X^μ_1X^μ_2v_2^μ_3v_2^μ_4∑_i=1^2 ^(i)_μ_1μ_2μ_3 μ_4(y))   ,
               h^∞, (1)_I_1 (u)
                = - (κ/2)^6
            6m_1 m_2/(k̂ v_2)^3(v_2k̂ X^μ_1X^μ_2v_2^μ_3v_2^μ_4∑_i=1^2 '^(i)_μ_1μ_2μ_3 μ_4(y)
                                                  +X^μ_1X^μ_2v_2^μ_3v_2^μ_4Y^μ_5∑_i=3^4 '^(i)_μ_1μ_2μ_3 μ_4μ_5(y)) 
              ,
        
        where Y^μϵ(k̂ v_2μ a_2),  and  y (y_1, y_2, y_3, y_4, …, y_18)  denotes all the independent scalar products among external kinematic vectors v_1, v_2, k̂, ã_1, ã_2, b̃, with y_1, y_2, y_3 defined in (<ref>) and 
        
            y_4   b̃k̂√(-b̃b̃)=0,      y_5   b̃ v_1 √(-b̃b̃),     y_6   b̃ v_2√(-b̃b̃),      y_7   b̃b̃ ,  
            
                  y_8   ã_1k̂√(-b̃b̃)=0,     y_9   ã_1 v_1√(-b̃b̃),     y_10   ã_1 v_2√(-b̃b̃),      y_11   ã_1 ã_1 -b̃b̃    
             
                   y_12   ã_2k̂√(-b̃b̃)=0,      y_13   ã_2 v_1√(-b̃b̃),      y_14   ã_2 v_2√(-b̃b̃),     y_15   ã_2 ã_2 -b̃b̃    
             
                   y_16   ã_1 b̃ -b̃b̃,    y_17   ã_2 b̃ -b̃b̃,    y_18   ã_1 ã_2 -b̃b̃  .
        
        The tensor integrals are generated as  
        
            ^(i)_μ_1…μ_r(y)   = ∂_b̃^μ_1⋯∂_b̃^μ_r _1^(i)[𝐲]= ∂_b̃^μ_1⋯∂_b̃^μ_r∫_0^1dz  _1^(i)[𝐲]  ,  
            '^(i)_μ_1…μ_r(y)   = ∂_b̃^μ_1⋯∂_b̃^μ_r 𝒱^(i)[𝐲]=∂_b̃^μ_1⋯∂_b̃^μ_r∫_0^1dz 𝒱^(i)[𝐲]  .
        
        We note that one cannot use _2 to generate the tensor integral as its value is trivial. Furthermore,  it is not possible to   set y_4=0 in 𝒱^(i) and ^(i)_1 directly,  as ∂_b̃^μy_4 is nonvanishing.  The superscripts in 𝒱^(i) and ^(i)_1 denote the tensor generating function with b̂ in different sectors. Then we only need to perform the z integration for the simple cases of _1^(3,4)[𝐲] and 𝒱^(3,4)[𝐲]. By using IBP and partial fractioning, the z integral can  then  be reduced  to a basis of  three integrals:
        
            ∫_0^1 dz/𝒴,       ∫_0^1 dz/b̂b̂ 𝒴,       ∫_0^1  dz  z/b̂b̂ 𝒴  ,
        
        where  𝒴^2  -b̂b̂ ŷ_5^2-b̂b̂, which 
         are straightforward to evaluate directly.
        
        
        
        
        
        
        
        
        
        
        
        
        
        
            
            
            
            
            
        
        
        
        
        
        § TIME-DOMAIN WAVEFORMS FROM RESIDUES 
        
        A final approach, followed in <cit.>, 
        makes use of Cauchy's residue theorem. We start by rewriting the original expression for the waveform in (<ref>) as follows,
        
            h_I^∞(u) 
               = -iκ∫_-∞^+∞dω/2πe^- i ω (u-k̂_2)∫d^4q_1/(2π)^2δ(2  p_1 q_1 ) δ(2  p_2 (k-q_1) )  e^iq_1 ℳ_I   ,
        
        where k=ωk̂. Further, we  split up the waveform into contributions coming from the two cuts (<ref>), and rescale the momentum transfers as q_i=ωq̂_i,  
        
            h_I^∞, (i) (u) 
            = -iκ∫d^4q̂_1/(2π)^2∫_-∞^+∞dω/2π    ω^2 e^- i ω (u-k̂_2 -q̂_1)δ(2  p_1q̂_1 ) δ(2  p_2 (k̂-q̂_1) ) 
               ℳ_I,q_i^2(ωk̂,ωq̂_1,ωq̂_2) .
        
        The basic idea now is to rewrite the integral over q_1 as a contour integral encircling the physical poles. To do this, it is worth introducing a parameterisation which is best suited to the various sectors of the problem. 
        Explicitly, we can factor out the exponential dependence on the spin for each cut, in a similar manner to (<ref>),
        
            _I,q_1^2=e^-_1  q_1∑_l=±e^l _2 q_2_I,q_1^(l) ,    _I,q_2^2=e^-_2  q_2∑_l=±e^l _1 q_1_I,q_2^(l) .
        
        Focusing on the q_1 cut, this allows us to rewrite the waveform as 
        
            h_I^∞, (1) (u) 
               = -iκ∫d^4q̂_1/(2π)^2∫_-∞^+∞dω/2πω^2  
                δ(2  p_1q̂_1 ) δ(2  p_2 (k̂-q̂_1) ) 
                 ∑_l=±e^- i ω (u-k̂_2 -q̂_1-i_1q̂_1+ i l_2(k̂-q̂_1))ℳ_I,q_1^2^(l)(ωk̂,ωq̂_1,ωq̂_2)
               = -iκ∫d^4q̂_1/(2π)^2∫_-∞^+∞dω/2πω^4  
                δ(2  p_1q̂_1 ) δ(2  p_2 (k̂-q̂_1) ) 
                 ∑_l=±e^- i ω (_l-_l q_1)ℳ_I,q_1^2^(l)(k̂,q̂_1,q̂_2)  ,
        
        where we have extracted the ω dependence of ℳ_I,q_i^2^(l), which is simply a factor of ω^2, and we have defined
        
            _l u-k̂_2+ i   l _2k̂  ,   _l^μ^μ + i _1^μ +i   l _2^μ .
        
        To evaluate (<ref>), we parameterise the q_1 integral in each sector using <cit.>
            z_1 v_1 q_1,    z_2    v_2 q_1,     z__l_l q_1,    z_o_l o_l q_1 ,
        
            with 
            
            o_l^μϵ(v_1 v_2 _lμ ) .
        
        The Jacobian for this transformation is simply 
        
            | ∂ q_1^μ/∂ z_j |= 1/|o_l o_l| .
        
        With this, the q_1 cut contribution to the waveform becomes
        
            h_I^∞, (1) (u)   = -iκ/(4π)^2 m_1 m_2(∂/∂ u)^4 ∑_l=±∫d^4 z_j/|o_l o_l|δ(z_1) δ(z_2-v_2k̂)δ(z__l-_l) _I,q_1^2^(l)
               =-iκ/(4π)^2 m_1 m_2(∂/∂ u)^4 ∑_l=±∫_-∞^∞d z_o_l/|o_l o_l|_I,q_1^2^(l)|_z_1→0, z_2→ v_2k̂, z__l→_l .
        
        The final integral in z_o_l can be computed using Cauchy's residue theorem. There are (at most) three poles present in _I,q_1^2^(l):
        
            Physical Pole:    1/q_1^2∼1/(z_o_l-A)(z_o_l-A^⋆) ,  
               Spurious Poles:  1/q_1 k∼1/(z_o_l-B) ,  1/(k-q_1) a_2∼1/(z_o_l-C) ,
        
        where A,B,C are functions of the external kinematics. The residues in 1/q_1 k and 1/(k-q_1) a_2 must cancel upon summing over the sectors l and cuts since the poles are spurious.[ More precisely the residue in  1/(k-q_1) a_2 must cancel when summing over the sectors l in each cut. The residue in 1/q_1 k will only cancel after summing both cut contributions.] Thus, in practice, we only need to compute the residue on the physical pole 1/q_1^2. This way of computing the integrals also makes it clear why terms 
        with only spurious poles do not contribute to the waveform: they have vanishing residues. 
        
        One may check that the integral (<ref>) has no pole at infinity (after applying the ∂/∂ u derivatives) and that the integrand falls off sufficiently fast at infinity to close the contour above or below the axis. This contrasts with the case of  Einstein-Hilbert gravity <cit.> where the pole at infinity exists and must be evaluated using a principal value prescription. In fact, we could have predicted this from the start. In  <cit.>, it was shown that the pole at infinity is given by the leading soft theorem of the five-point amplitude. Our five-point amplitude is built from higher-derivative corrections to general relativity and cannot change this universal leading soft theorem. Indeed, we can check that our amplitude vanishes in the soft limit ω→ 0, which manifests in the time-domain waveform as vanishing linear memory, as discussed in Section <ref>. 
        
        To compute the contribution of the q_2 cut, we follow an identical procedure to the above but using the sectors for _q_2^2 in (<ref>). Upon summing both contributions, we obtain the full waveform for I_1 and G_3:
        
            h_I^∞ (u)= h_I^∞, (1) (u)+h_I^∞, (2) (u) .
        
        The expressions obtained using this method are lengthier than those found in Section <ref>. Reassuringly, they are in complete agreement.
        
        
        
        
        § SHOWCASE OF WAVEFORMS
        
        
         
        In this section we show several  plots of the waveforms for the  G_3 and I_1 deformations, with and without spin. We recall that the waveforms for the parity-odd interactions G̃_3 and Ĩ_1 are related to the parity-even ones by (<ref>), that is a simple swap of their real and imaginary parts; thus  separate plots for the case of parity-odd deformations are not needed. 
        We will   parameterise our external kinematics as follows:
        [Our waveforms are valid for arbitrary spin configurations; however in the figures shown in this section we will focus on the case where the spins of the two bodies are aligned.]
            v_1^μ=(σ,0,0,√(σ^2-1)) ,      v_2^μ   =(1,0,0,0) ,
               k̂^μ=(1,sinθcosϕ ,sinθsinϕ ,cosθ ) ,     b^μ   =b(0, 1, 0, 0) , 
               a_1^μ=a_1(0,0,1,0) ,     a_2^μ   =a_2(0,0,1,0) , 
               
               [k|_α̇=√(2)[         sinθ2; -cosθ2  e^-iϕ                             ]  .
        
        For the graphs below, we will choose θ=ϕ=π/3 and fix the spins to be either aligned or anti-aligned with the orbital angular momentum. We will also set b = 1 which means in practice 
        that the spins a_i and the retarded time u are measured in units of b. Finally we will ignore a prefactor so that the waveforms are plotted in units of κ^6 m_1 m_2 β_i.
        
        
        
        
        
         §.§ Spinless case
        
        
        We begin by showing the G_3 and I_1 waveforms for the scalar case, for various values of σ=v_1 v_2. 
        
        
        
        
         §.§ Spinning case
        
        We now show a few  G_3 and I_1 waveforms for  spinning objects. In Figures <ref> and <ref> we consider the cases where only the spin of one of the two bodies is nonvanishing, say a_1≠ 0 and a_2=0, and  do so for increasing values of a_1.
        In Figures <ref> and <ref> we show the G_3 and I_1  waveforms when both objects are spinning,  with their spins aligned, for various choices of the ratio a_2/a_1.
        Note that the  gravitational memory is absent  also in the spinning case. 
        We also recall that in setting b=1 we are measuring the spins a_i in units of b. Thus, the values we are plotting here, e.g. a_i=± 0.5, are rather large. Finally,  the waveforms also have poles at b=± a_1± a_2,
        [These poles arise  from treating  the  spin dependence   exactly and are  also present  in the scattering  angle <cit.>.] although  these points are not problematic since we assume |a_i|≤  G m_i< b.
        
        
        
        
        § (NO) GRAVITATIONAL MEMORY
        
        
        
        As it is well known, gravitational memory is related to soft limits of the five-point amplitude <cit.>, and a discussion of the memory in the General Relativity waveforms to all orders in spin was presented in <cit.>. We now wish to study if   cubic deformations alter in any way the gravitational memory. 
        Defining the memory  as 
        
            Δ (h^∞_+± i  h^∞_×) . (h^∞_+± i  h^∞_×)|_u→ + ∞ - . (h^∞_+± i  h^∞_×)|_u→ -  ∞  ,
        
        one finds that (see e.g. <cit.>)
        
            Δ (h^∞_+± i  h^∞_×)    = 
              -i  κ S_ W^ HEFT(k̂, -i ∂/∂ b; ±) δ_ HEFT  ,
        
        where
        
            δ_ HEFT∫dμ^(D) e^iq_ b ( -i _4^ HEFT)(q)  ,
        
        is the Fourier transform to impact parameter space of the four-point classical amplitude _4^ HEFT<cit.>, which now has to be evaluated in the presence of cubic deformations; while 
        and S_ W^ HEFT is the classical (or HEFT) Weinberg soft factor   <cit.>
            S_ W^ HEFT   = 
               - κ/2 1/ωε_μν^(h)(k) [
            		p_1^μ q^ν +p_1^ν q^μ/p_1k̂ - p_1^μ p_1^νqk̂/(p_1k̂)^2  -   1↔ 2]
                 .
        
         However, at leading order (tree level) in the presence of cubic deformations the classical four-point amplitude vanishes,  _4^ HEFT, (0)=0; indeed,  this quantity  starts receiving corrections at one loop, or 2PM, which were computed  in <cit.>. We conclude that the leading-order gravitational memory in the presence of cubic deformations vanishes. This is also confirmed by the explicit plots shown in 
         the 
         
         figures  in Section <ref>. 
        
        
        § DISCUSSION
        
        
        
        
         
        
        In this paper we have considered  parity-even and -odd cubic deformations to general relativity, and computed analytically the corresponding corrections to the waveforms to leading order in the deformations and in Newton's constant. We have done so following three independent approaches, yielding  results in complete agreement. The direct integration method provides the most compact expressions, though the IBP reduction and residue methods are possibly more easily extendable beyond tree level. The residue method is especially   rewarding in that it easily shows  the absence of contributions from spurious poles. 
        
        To conclude, we  briefly muse on the size  of the corrections due to the cubic deformations studied here and in <cit.>. Specifically, we   focus  on an observable quantity such as the impulse (or momentum kick) .Δℙ|_R^3, and   ask when the corrections arising from cubic deformations become comparable to those  in general relativity at a certain PM order. 
        
        Our cubic vertices have the form (schematically) ∫d^4x βκ^3 (∂^2h)^3. We  redefine the coupling constant in such a way that the three-point vertex scales as κ, thus introducing β = β̂ / κ^2; in terms of β̂ the vertex has the form κβ̂∫d^4x  (∂^2h)^3, and we  note that the dimension of β̂ is ( length)^4, so that  we will set β̂ = ℓ_ EFT^4. 
        We found in <cit.> that, 
        to leading order in the deformation and in G, the corrections to the impulse arising from cubic deformations have the form (see (6.61)–(6.64) of that paper)
        
            .Δℙ|_R^3∼βκ^6 m^3/b^6∼β̂/κ^2 G^3 m^3/b^6∼ m ×(ℓ_ EFT/b)^4 (Gm/b)^2   .
        
        On the other hand, to nPM order in general relativity the impulse  receives corrections    of the type 
        
            .Δℙ|_ EH,  n∼ m ×(Gm/b)^n
                  ,
        
        so that 
        
            .Δℙ|_R^3/.Δℙ|_ EH,  n∼(ℓ_ EFTb)^4/(Gmb)^n-2  .
        
        For hyperbolic encounters of interest we can choose for illustration  G m/b ∼  1/5, which ensures that  perturbation theory is still valid while also giving rise to a sizeable effect. By further requiring that, for observability,  .Δℙ|_R^3/ .Δℙ|_ EH,  n∼ 1, we arrive at  the estimate 
        
            ℓ_ EFT∼ b   (1/5)^n-2/4  .
        
        As an   example, we can ask when   the corrections  from cubic deformations  become comparable to those arising at 5PM (or four loops)  in general relativity.
        [Parts of  the 5PM corrections to the impulse and the scattering angle for non-spinning objects were recently computed in <cit.>.]
        From  (<ref>), and further choosing b∼ 100   km we arrive at the estimate  ℓ_ EFT∼ 30   km, which is within the regime of validity of the effective field theory,
        and also happens to be in agreement with the estimate   provided 
        in <cit.>. We also note that in <cit.>, bounds on the fundamental length scale  ℓ_R^4 in a quartic effective field theory considered earlier in <cit.> were established, resulting in ℓ_R^4≲ 150  km.  
        
        
        
        Similarly, we find that the waveform 
        h^∞ in general relativity at order n in the PM expansion scales as
        
            h^∞|_ EH,  n∼ b ×( G m/b)^n  ,
        
        where now  n=2 corresponds to tree level and n=3  to one loop. 
        For the leading cubic correction computed in this paper we found (see for example (<ref>) and (<ref>))
        
            h^∞|_R^3∼ b ×( G m/b)^2 ( ℓ_ EFT/b)^4  ,
        
        leading to similar conclusions as above for the impulse (<ref>). Furthermore, if we assume that ℓ_ EFT∼ G m we can say that the effects of cubic interaction are in practice four orders higher in the PM expansion. This may be challenging given the currently available state of the art results in the PM expansion, but results in the PN framework might already give access to higher orders in G.
        
        A final comment on the spin radii a_i. In the expressions for the impulse and waveform,  they appear through the shifted impact parameters (schematically) b̃ = b + a (omitting Lorentz indices and numerical factors). Recalling that for a physical Kerr black hole
        a/(G m) ≤ 1,  we can express this as
        
            b̃ = b ( 1 + a/G mG m/b)  .
        
        This implies that effectively every factor of a increases the  PM order by one,  since GM/b is the effective loop counting parameter. For example,  the linear in spin part of our results  should be accompanied  by the one-loop correction of the scalar waveform in the cubic theory.
        We hope to return to these questions in the  near future. 
        
        
        
        
        
        
        
        
        
        
        
        § ACKNOWLEDGEMENTS
        
        We would like to thank  
        Joshua Gowdy and Paolo Pichini 
        for stimulating conversations.
        GT thanks the Physics Department at the University of Rome “Tor Vergata” for their warm hospitality.   
        This work was supported by the Science and Technology Facilities Council (STFC) Consolidated Grants   ST/T000686/1 and  ST/X00063X/1 “Amplitudes, Strings  & Duality”.
        The work of GRB and PVM is supported by STFC quota studentships.   GC has received funding from the European Union's Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement No. 847523 “INTERACTIONS”. 
        GT is also supported by a Leverhulme research fellowship RF-2023-279\ 9.
        No new data were generated or analysed during this study.
        
        
          
        
        
        § ALL TENSOR INTEGRALS WE NEED
        
        
        We summarise here relevant integrals needed to compute the time-domain waveforms in our paper. We focus here on integrals involving the 1/q_1^2 propagator, while integrals involving the 1/q_2^2 propagator can be  obtained by appropriate relabelling of the results quoted below. In the following we define q^μ q_1^μ and introduce the projector
        
            P_1^μνη^μν - v_1^μ v_1^ν .
        
        Henceforth, we denote projected four-vectors  and their modulus as
        
            V^μ_(1) P_1^μν V_ν  ,
        
            and 
            
            |V_(1)| √(-V_(1) V_(1)) = √(-V  P_1  V).
        
        The first and simplest integral we encounter is
        
            ℐ    = ∫d^4q/(2 π)^3δ(2 q  v_1) e^i q ·b̃/q^2
                = ∫d^4q/(2 π)^3δ(2 q  v_1) e^i q ·b̃_(1)/q^2
                = -1/8 π |b̃_(1)| ,
        
        where in the second line we exploited the fact that due to the δ-function we can insert for free the projector P_1 between q and b̃ in the exponent. We actually need tensor versions of this integral, 
        
            ℐ^μ_1 μ_2 …μ_n = 
                ℐ[q^μ_1 q^μ_2⋯ q^μ_n] = 
                ∫d^4q/(2 π)^3δ(2 q v_1 )   e^i q b̃ q^μ_1 q^μ_2⋯ q^μ_n/q^2 ,
        
        which can be simply obtained by acting  n times on ℐ  with the derivative operator
        
            -i ∂/∂b̃_(1) μ_i ,
        
        and using the identity 
        <cit.>
            ∂b̃_(1)^μ/∂b̃_(1)ν = P_1^μν .
        
        Two examples are 
        
            ^μ = -i ∂ℐ/∂b̃_(1)μ = i/8 πb̃_(1)^μ/|b̃_(1)|^3  ,
        
            and 
            
            ^μν = - ∂^2 ℐ/∂b̃_(1)μ∂b̃_(1)ν = |b̃_(1)|^2 P_1^μν+3 b̃^μ_(1)b̃^ν_(1)/8 π |b̃_(1)|^5 .
        
        In a similar fashion, we can obtain the other two more complicated tensor integral families. 
        The first one is 
        
            𝒥^μ∫d^4q/(2 π)^3δ(2 q  v_1) e^i q ·b̃ q^μ/q^2  k̂ q ,
        
        which, as already mentioned at the end of Section <ref>, requires a regulator because of the 1/k̂ q pole.
        This is a spurious pole that cancels in the sum of the two diagrams contributing to  the waveform and, of course, the final result is independent of the regulator. For  illustration we will present here  the evaluation of 𝒥^μ using both the principal value (PV) and iε prescriptions, while in the main text we  only make use the PV prescription.
        
        We begin discussing  the PV prescription.
        We  denote the PV regulated integral by 𝒥^μ,  which can  be expressed in terms of the scalar integral 
        
            𝒥∫d^4q/(2 π)^3δ(2 q  v_1) e^i q ·b̃/q^2  k̂ q ,
        
        as
        
            𝒥^μ = -i ∂𝒥/∂b̃_(1)μ .
        
        Following <cit.>,  we now notice that  𝒥 is invariant under rescaling of b̃. Indeed a rescaling b̃→α b̃ can be undone by the rescaling q → q/α which leaves the integrand unchanged. Hence, we infer that 
        
            b̃_(1)μ𝒥^μ = -i  b̃_(1)μ∂𝒥/∂b̃_(1)μ = 0  ,
        
        and we further notice that v_1 μ𝒥^μ = 0 since v_1  q = 0 due to the δ-function in the integral. Hence 𝒥^μ must live in a two-dimensional subspace orthogonal to v_1^μ and b̃_(1)^μ. In order to implement this,  it is convenient  to introduce the symmetric tensor <cit.>
            K_1^μν = |b̃_(1)|^2 P_1^μν + b̃_(1)^μb̃_(1)^ν ,
        
        obeying K_1^μν v_1ν = K_1^μνb̃_(1)ν = 0. This allows us to write an ansatz for 𝒥^μ as
        
            𝒥^μ = c   K_1^μνk̂_ν .
        
        We can now solve for c by observing that 
        
            𝒥^μk̂_μ = ℐ = -1/8 π |b̃_(1)| = c  k̂ K_1 k̂  ,
        
            from which it follows that 
            
            c = -1/8 π |b̃_(1)| k̂ K_1 k̂ .
        
        Therefore, we find that 
        
            𝒥^μ = K_1^μνk̂_ν/8 π |b̃_(1)|  [(v_1 k̂)^2 |b̃_(1)|^2-(k̂b̃_(1))^2 ] ,
        
        and from this result we can obtain 𝒥^μμ_1 …μ_n with any number of extra q^μ_i insertions by taking derivatives with respect to b̃_(1) μ_i as explained above.
        
        In order to obtain tensor integrals,  the form above is preferred as it makes the b̃_(1)-dependence fully transparent. On the other hand, one can obtain a more compact expression  in terms of the shifted, unprojected impact parameter b̃ as follows. 
        First, it is important to note that b̃k̂=0 as assumed throughout the paper. Note that any shift of b̃ by ã_i
        preserves this property
        since ã_i k̂=0.
        Under this assumption we can then write 
        
            b̃^μ = b̃_(1)^μ - k̂b̃_(1)/k̂ v_1 v_1^μ  ,
        
        from which it follows that 
        
            |b̃  |^2 -b̃b̃ =  |b̃_(1)|^2 - 
            (k̂b̃_(1))^2/(k̂ v_1)^2 .
        
        We  can then  rewrite  𝒥^μ in the more compact form (if b̃k̂=0)
        
            𝒥^μ =  K_1^μνk̂_ν/8 π  (v_1 k̂)^2 |b̃_(1)|   |b̃  |^2 .
        
        Next, we examine  the same integral,  this time  regulated with an iε prescription:
        
            𝒥_ε^μ   ∫d^4q/(2 π)^3δ(2 q  v_1) q^μ e^i q ·b̃/q^2  (k̂ q - i ε)
                =   i ∫_-∞^0 dτ∫d^4q/(2 π)^3δ(2 q  v_1) q^μ e^i q · (b̃ + τk̂)  + τε/q^2
                = -1/8 π∫_-∞^0 dτb̃^μ_(1) + τk̂^μ_(1)/[ -(b̃+τk̂)  P_1  (b̃+τk̂)]^3/2 ,
        
        where in the second line we have used the Schwinger trick and in the last line we have used (<ref>). Performing the τ integration gives
        
            𝒥_ε^μ =
            |b̃_(1)| k̂^μ_(1) - v_1 k̂ b̃^μ_(1)/8 π (v_1 k̂) |b̃_(1)| (|b̃_(1)| v_1 k̂ + b̃_(1)k̂ ) ,
        
        and similarly, replacing i ε→ -i ε we find
        
            𝒥_-ε^μ =
            |b̃_(1)| k̂^μ_(1) + v_1 k̂ b̃^μ_(1)/8 π (v_1 k̂) |b̃_(1)| (|b̃_(1)| v_1 k̂ - b̃_(1)k̂ ) .
        
        It can then be checked that these integrals are related to the PV integral (<ref>) as
        
            𝒥^μ = 1/2( 𝒥^μ_ε + 𝒥^μ_-ε)  .
        
        Finally, we tackle (with PV regularisation) the most complicated integral,
        
            𝒦^μν = ∫d^4q/(2 π)^3δ(2 v_1  q)
            e^i q b̃ q^μ q^ν/q^2  k̂ q  a_2  q_2 
            = ∫d^4q/(2 π)^3δ(2 v_1  q)
            e^i q b̃ q^μ q^ν/q^2  k̂ q  (- ã_2  q) ,
        
        where 
        we have introduced the shifted spin radii 
        
            ã_i^μ = a_i^μ - a_i k̂/v_i k̂ v_i^μ  ,      ã_i k̂ = 0   ,
        
        for i=1,2.
        Also note that q_2^μ = k^μ - q_1^μ = ω^* k̂^μ - q^μ = v_2  q/v_2 k̂k̂^μ - q^μ, and therefore we can rewrite a_2  q_2 = - ã_2  q.
        𝒦^μν has the following important properties:
        
            𝒦^μν v_1ν = 𝒦^μνb̃_ν
                = 𝒦^μνb̃_(1)ν =𝒦^μνη_μν = 0  .
        
        The first is due to the δ-function, the second and third property are consequence of the rescaling invariance of 𝒦^μ, and the fourth property (tracelessness) can be seen as follows: taking the trace and
        choosing to work in the rest frame v_1=(1,0,0,0), we find
        
            𝒦^μνη_μν    = ∫d^4q/(2 π)^3 δ(2 q v_1 )
            e^i q b̃/k̂ q   (- ã_2  q)
                = -∫_-∞^0dt_1 dt_2 ∫d^3q⃗/2 (2 π)^3  e^-i q⃗ (b⃗̃⃗+t_1 k⃗̂⃗ + t_2 ã⃗_2)
                = -1/2∫_-∞^0 dt_1 dt_2  δ^(3)(b⃗̃⃗+t_1 k⃗̂⃗ + t_2 ã⃗_2)  ,
        
        which vanishes for generic choices of the vectors.
        
        Due to the properties mentioned above, the integral (<ref>) must be a symmetric tensor living in the two-dimensional subspace orthogonal to v_1 and b̃, 
        
            𝒦^μν = c_1 K_1^μν + 2 c_2 (ã_2  K_1)^(μ (k̂ K_1)^ν) ,
        
        where 
            K_1^μν |b̃_(1)|^2 P_1^μν + b̃_(1)^μb̃_(1)^ν  .
        
            Notice that no other structures are allowed in this ansatz, since the result must be symmetric under swapping k̂↔ -ã_2 and be rescaled by 1/(α_1 α_2) if we perform the replacement  {k̂, ã_2}→{α_1 k̂, α_2 ã_2}. 
        Requiring also tracelessness, we instantaneously find
        c_1 = -c_2 ( ã_2  K_1k̂).
        On the other hand
        
            𝒦^μνk̂_μã_2ν = - ℐ = 1/8 π |b̃_(1)| = c_2 ( ã_2  K_1ã_2 )( k̂ K_1k̂) = - c_2 ( ã_2  K_1ã_2 ) (v_1 k̂)^2 |b̃|^2  ,
        
        and hence we find
        
            𝒦^μν = -( ã_2  K_1k̂) K_1^μν -2 (ã_2  K_1)^(μ (k̂ K_1)^ν)/8 π  |b̃_(1)| ( ã_2  K_1ã_2 ) ( k̂ K_1k̂) .
        
        If we further assume b̃k̂=0, which is
        true for all possible shifted versions of the impact parameter b̃± i ã_1 ± i ã_2,
        we find
        
            𝒦^μν = ( ã_2  K_1k̂) K_1^μν -2 (ã_2  K_1)^(μ (k̂ K_1)^ν)/8 π (v_1 k̂)^2 |b̃|^2 |b̃_(1)| ( ã_2  K_1ã_2 )  .
        
        Finally, we comment on potential terms in the waveform integrand that do not have physical poles 1/q_i^2 with i=1,2, but instead have a pole of the form 1/(w  q) (or a product of such factors). In such cases the same manipulation as in (<ref>) guarantees that such terms make vanishing contributions to the waveform. In  other words,  the tracelessness of integrals like 𝒦^μν in (<ref>) guarantees that terms with only spurious poles do not contribute to the waveform. This fact was also noticed in the Einstein-Hilbert case in <cit.>. For this reason, it is not necessary to use the full five-point amplitude,  but the sum of the two factorisation channels (<ref>) suffices. Similarly, all polynomial terms that only give rise to (derivatives of) δ-functions can be ignored.
        
        
        
        
        
        § TENSOR INTEGRALS FROM GENERATING FUNCTIONS
        
        We now describe an alternative (and faster) way to compute tensor integrals. All of the integrals we are interested in  have the form
        
            _m_1… m_n^ [] (b̃)     ∫d^4q/(2 π)^3 δ(2 q v_1 )  
            e^i q b̃ (q_1 c_1)^m_1⋯  (q_1 c_n)^m_n/ ,
        
        for some denominator , and where c_i, i=1, … , n are four-vectors. These can be obtained very easily from the integral 
        
            ^[] (b̃; t_1, …, t_n)    ∫d^4q/(2 π)^3 δ(2 q v_1 ) e^i q ( b̃ + ∑_i=1^n t_i c_i)/
                = ^[] (b̃ + ∑_i=1^n t_i c_i; 0, …, 0)
               ,
        
        as
        
            _m_1… m_n^[]   = .(-i)^∑_i=1^n m_i∂^∑_i=1^n m_i/∂^m_1t_1 ⋯∂^m_nt_n^[] (b̃; t_1, …, t_n)|_t_1 = t_2 = ⋯ = t_n = 0  .
        
        For instance, for the case of the G_3 waveform we need an integral of the form 
        
            ∫d^4q/(2 π)^3 δ(2 q v_1 )  
            e^i q b̃ (q  v_2)^2(q  X)^2/q^2  ,
        
        where X^μ[k̂| v_1 
        σ^μ | k̂]. This can be evaluated as 
        
            . ∂^4/∂^2t_1  ∂^2t_2∫d^4q/(2 π)^3 δ(2 q v_1 ) e^i q ( b̃ + t_1v_2 + t_2 X)/q^2|_t_1 = t_2  
            = 0
                
            = -1/8 π(∂^4/∂^2t_1  ∂^2t_21/
            |b̃_(1) + t_1 v_2(1) + t_2 X_(1)|)_t_1 = t_2 = 0 ,
        
        where we used (<ref>), and as usual we adopt the notation defined in (<ref>) for projected vectors (but note that X_(1) = X). 
        
          
        Summarising, in this approach we can easily relate tensor integrals to simpler ones (not necessarily scalar integrals, as seen in the previous section), without the need to perform Lorentz contractions and with the advantage of having to perform differentiations with respect to scalar parameters rather than vectors. 
        
        JHEP


 Paper Title: Bulk-induced D-brane deformations and the string coupling constant.




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    ⊗ℕℤℂℝ𝕋 tr I D K Hå A B M O P N E C S U†∂Å𝔸ℚ𝕏𝕃𝕍𝕎𝔹ℂℍȷımℜ𝒱EOMωω_cω_oω̂ϵ_cϵ_omln∂U
         Bulk-induced D-brane deformations 
         and the string coupling constant
        
         Carlo Maccaferri^(a)[Email: maccafer at gmail.com], Alberto Ruffino^(a)[Email: ruffinoalb at gmail.com]  and  Jakub Vošmera^(b)[Email: jakub.vosmera at ipht.fr] 1 cm
        ^(a) Dipartimento di Fisica, Università di Torino, 
        INFN  Sezione di Torino 
        
        Via Pietro Giuria 1, I-10125 Torino, Italy .5 cm
         .5 cm
        ^(b) Institut de Physique Théorique
        
        	CNRS, CEA, Université Paris-Saclay
        
         Orme des Merisiers, Gif-sur-Yvette, 91191 CEDEX, France
        
         Abstract
        We consider computing the on-shell disk action of open-closed string field theory as a gauge-invariant way of capturing the shift in D-brane tension that is induced by a deformation of the bulk CFT. We study the effect of bulk matter deformations (both marginal and relevant) on a wide range of boundary conditions in a number of CFTs up to subleading (two-loop) order in perturbation theory. In all analyzed examples, we find that the shift in the g-function of the matter boundary state is always accompanied by a boundary-independent shift in the string coupling constant, whose leading behaviour is universally proportional to the sphere two-point function of the deforming bulk operator.
        
        =16pt
        
        
        § INTRODUCTION
        
        
        It is known since early times that string field theory (SFT)  provides a way to investigate the space of 2d (B)CFT's <cit.> by connecting different conformal vacua  via classical solutions. This has recently received a renewed attention <cit.> as an alternative convenient approach to the standard conformal perturbation theory (CPT) <cit.>. 
        The advantage of SFT versus CPT is that it provides an automatically consistent way of regulating  singularities from collisions of perturbing fields by recognizing such collisions as originating from degenerations of punctured Riemann surfaces (in this particular application spheres and disks). In this picture the collisions of the perturbing  operators happen  at a corner of moduli space that is always associated to a SFT Feynman diagram, with the SFT propagator taking care of the degenerating region in a way which is analogous to the iϵ prescription of quantum field theory <cit.>. In this way all contact divergences are systematically regulated.
        Along this direction, following the  pioneering analysis of <cit.>, there has been progress <cit.> in the study of particular examples of bulk RG flows triggered by a slightly relevant operator 𝕍 of conformal dimensions (1-y,1-y)   (with y very small and positive) whose OPE closes on itself
        
            𝕍(z,z̅) 𝕍(0,0)=1/|z|^4(1-y)+C_𝕍𝕍𝕍 𝕍(0,0)/|z|^2(1-y)+ ( regular) .
        
        Calling t the coefficient  (coupling) of the deforming field  T=t(y) 𝕍, in this case the leading order beta function has the structure
        
            β_t∼ 2y t(y)-C_𝕍𝕍𝕍 t(y)^2+… ,
        
        and therefore we can search for a perturbative solution in the small parameter y so that a new non-trivial conformal point (β=0) is found at
        
            t(y)∼2y/C_𝕍𝕍𝕍+𝒪(y^2) .
        
        This represents an IR fixed point which is parametrically close to the initial perturbative vacuum t=0 (UV fixed point) and we refer to this  as a  short RG-flow.   The paradigm example of short RG flows are the Zamolodchikov minimal model flows triggered by the relevant field 𝕍=ϕ_(1,3), which connect the m-th (UV) to the (m-1)-th  (IR) minimal models, in the large m∼2/y limit <cit.>.
        
        Going to higher order in y  in CPT is notoriously hard and the details of the calculation depend on the chosen regularization scheme, see for example <cit.>.  On the other hand, the analogous analysis in SFT requires to solve classical (from the target space perspective) equations of motion which are concretely defined, once the SFT is constructed. This is carried out by embedding the CFT of interest in a critical c=26 matter CFT, together with the c=-26bc system and searching for a solution 
        
            Φ(y)∼ y t_1 cc̅𝕍+𝒪(y^2) ,
         
        solving order by order in y the classical equation of motion.  The SFT solution Φ(y) is a state in the full Hilbert space of the combined matter-ghost  c=26-26=0 CFT and it should be thought of as a collection of vacuum expectation values for the states that are responsible for the RG flow plus an infinite dressing of irrelevant and auxiliary fields which are needed to solve the full SFT equation of motion. In particular, after having integrated out these extra fields, the equations of motion for the relevant fields should be thought of as the SFT incarnations of the β functions of CPT, where the ambiguities related to the definition of the regularization scheme on the CPT side are reflected in the freedom of choosing the SFT data and gauge fixing. The conformal data of the IR fixed point are expected to be encoded in gauge invariant quantities involving the (non gauge invariant) classical solution Φ(y). For example the physical fluctuations around the solution are expected to encode the conformal weights of primaries at the IR fixed point. Another natural gauge invariant quantity is the classical SFT action. In closed string field theory however the value of the action on a classical solution is expected to formally vanish <cit.> because of the existence of the ghost-dilaton D=c∂^2 c-c̅∂̅^2 c̅ which can be used to arbitrarily rescale the action <cit.>. Nevertheless an explicit computation <cit.> recently showed that the action evaluated on the Φ(y) classical solution appears to be universally proportional to the shift in the central charge
        
            S[Φ(y)]=-1/12Δ c=𝒪(y^3) .
        
        This has been non-trivially verified up to 𝒪(y^4) by comparing with the known expectation from Zamolodchikov flows of minimal models.
        This seems to be at odds with the expected vanishing action from <cit.> and indeed a closer inspection shows that Φ(y) is not really a solution in that it violates the equation of motion in the direction of the dual ghost dilaton state D̃=c_0^+D
            EOM(Φ(y))∼ y^3 D̃+… .
        
        This problem was realized already in <cit.> and attributed to the fact that the solution is trying to change the central charge of the mater CFT while string theory requires a critical fixed central charge. This basic tension is manifestated in the ghost dilaton obstruction (<ref>).  However, as diccussed in <cit.>, this obstruction can be trivialized by assuming the existence of a linear dilaton factor in the matter CFT. If this factor is present, then D̃ can be trivialized by an allowed ghost number two state Θ (see (<ref>))
        
            D̃=1/β Q_ cΘ ,
        
        where β is the background charge of the linear dilaton and Q_ c the closed string BRST operator.
        This then allows to correct the Φ(y) solution by adding a y^3Θ term which cancels the ghost-dilaton obstruction. Moreover Θ has the effect of shifting the linear dilaton background charge and thus the central charge so that the full solution now describes a flow of central charge from the initial perturbed CFT to the linear dilaton factor in such a way that the total central charge stays constant and the full string theory background remains critical. It remains to be seen what are the consequences of this on the closed SFT action, but this is not the direction we want to take in this paper.
        
        In general, a closed SFT solution will not just change the bulk CFT background but also the string coupling constant g_s. Detecting a change of the string coupling via a gauge invariant observable in closed string field theory does not  seem to be straightforward, though. In this regard it was pointed out in <cit.> that one can use probe D-branes for this task. This can be done by adding an open string sector to the closed SFT and studying how the open string vacuum state  Ψ=0 is shifted to a new solution Ψ^* as a response to the change in the closed string background induced by a given closed string solution Φ^*. The natural observable that is associated to this process is the combined disk vacuum energy generated by the closed SFT solution Φ^*, together with the open string vacuum shift Ψ^*. This quantity Λ(Φ^*,Ψ^*) was conjectured in <cit.> to compute the shift in the string theory disk partition function 
        
            Λ(Φ^∗,Ψ^∗)=  -1/2π^2(g^∗/g_s^∗-g/g_s) ,
        
        where g and g^∗ are the g-functions of the initial and final boundary states and g_s and g_s^∗ are the initial and final string coupling constants.  We call this the   generalised  Sen's conjecture. 
         By checking (<ref>) with a sufficient number of boundary states subject to the same closed string field theory solution one can  isolate a universal part in Λ(Φ^∗,Ψ^∗) (independent of the boundary conditions) which represents the change in g_s. In this paper we perform this analysis in two rather different cases and observe that the change in the string coupling constant is universally the same, in the sense to be described below. 
         
         To start with, extending the analysis of <cit.>, we compute the on-shell disk action for various boundary states subject to a marginal deformation of a Narain lattice  of d free bosons. By comparing with the expected answer of the deformed g-functions from the exact (B)CFT sigma model, we are able to detect that the bulk solution is changing the string coupling constant as described in (<ref>).
         
          In the second, and more important, application we study how a short bulk  RG flow affects the initial boundary state on which we define the open-closed SFT. Under the assumption that the induced boundary flow is also short, we are able to find a closed-form expression for the gauge invariant quantity Λ(Φ^∗,Ψ^∗) that is exact up to 𝒪(y^2) (thus one order beyond the CPT analysis of <cit.>). This gives
          
            g^∗/g(g_s^∗/g_s)^-1=1+ B_𝕍1/g(y/C_𝕍𝕍𝕍)+
              (B_𝕍1/g2 A^ f.p._TTTT/3 C_𝕍𝕍𝕍+ A^ f.p._TT)(y/C_𝕍𝕍𝕍)^2+𝒪(y^3) ,
        
        where A^ f.p._TTTT is the regularized zero-momentum bulk four-point amplitude computed in <cit.>, B_𝕍1 is the bulk-boundary OPE between the bulk deforming field and the boundary identity and finally A^ f.p._TT is the regularized disk 2-point amplitude defined in (<ref>). This is the first main result of this paper.
        
        To extract the change in the string coupling constant triggered by the bulk relevant solution, we apply this formula to particular Cardy boundary states in large m Virasoro minimal models MM_m subject to the Zamolodchikov short RG flow triggered by the fieldϕ_(1,3) and compare with the known exact changes in the g-functions. This can be done independently for two different classes of boundary conditions and we unambiguosly find that the string coupling constant changes as (<ref>).
          
           In both the marginal and relevant cases the closed string solution is perturbative in the sense that it admits a power series expansion, either in the marginal parameter (the Narain modulus) or in the small parameter y associated to the small RG-flow MM_m→MM_m-1. In both cases we can thus write
         
            Φ^*=cc̅ T+( subleading) ,
        
         where T is a pure-matter marginal or (slightly) relevant field with normalization which is free (but small) for marginal deformations or fixed by the equation of motion to a (small) value, in the case of relevant deformations. In both cases when we compute the on-shell disk action for a set of different boundary conditions as described above we observe that the change in the string coupling constant is universally given by
         
            g_s^∗/g_s=1+1/8⟨ T,T⟩/⟨0|0⟩+( subleading) ,
        
        where ⟨·,·⟩ is the BPZ inner product in the matter bulk CFT and |0⟩ is its SL(2,ℂ) vacuum. This is the other central observation of this paper. Notice that this is eventually a statement about the purely closed string sector without any reference to the open string sector which, from this perspective,  has only been used as an auxiliary ingredient.
        
        The paper is organized as follows. In section <ref> we review the main results of <cit.> about the properties of the on-shell disk action of open-closed SFT. We then specialize to the case of bulk marginal deformations and, in particular, to deformations of a Narain lattice of d free bosons. By computing the disk action of a solution describing a deformation of the Narain modulus in the presence of generic boundary states, we are then able to identify how the string coupling constant is changed by the closed-string part of the SFT solution. In section <ref> we apply our framework to the case of short bulk RG-flows, consistently setting up a bulk-boundary perturbative scheme in order to associate an open-string vacuum-shift solution to a given closed SFT solution. We discuss under which conditions the open-string vacuum-shift solution is perturbative (as the seeding closed SFT solution is) and we write down the closed- and the open-string solutions to compute the disk action up to subleading order. This would correspond to a two-loop calculation in CPT. Section <ref> contains the main technical results of this paper and it is fully devoted to the computation of the disk action for small RG flows. We perform the computation in the limit of large closed-string stubs and, as a non-trivial consistency check of our construction, we show that our disk action (<ref>) turns out to be fully independent of the SFT data used to define the interaction vertices of the open-closed SFT. In section <ref>, we test our general result for the disk action against the expected change in the boundary entropy for a set of Cardy boundary states in large m minimal models subject to Zamolodchikov short RG flows triggered by the fieldϕ_(1,3). After matching the g-function part of the disk action, we unambiguously identify a second order change in the string coupling constant, which we find to be fully parallel to the change that was detected in the case of the marginally deformed Narain lattice. In the concluding section <ref> we further discuss our main results and we outline various research directions suggested by our work. Finally, in appendix <ref>, we compute (for the first time, to our knowledge) the bulk two-point function of the fieldϕ_(1,3) on a disk in the large m limit by explicitly solving the third order differential equation implied by the existence of a level-three null state and by fixing the integration constants using the known bulk-boundary BCFT data. 
        
        
        
        
        
        
        § THE ON-SHELL DISK ACTION
        
        
        
        In this section we will review the main results and observations of <cit.> relating to the disk part of the open-closed SFT action evaluated on a classical closed string-field configuration Φ^∗, as well as on an open-string field configuration Ψ^∗ which can be found as a classical solution of open-string equations of motion on the closed-string background given by Φ^∗.
        
        
        
         §.§ Generalised Sen's conjecture
        
        
        In the context of pure open-string field theory, it has been conjectured by Ashoke Sen that the on-shell action gives a direct measure of the difference between the g-functions (tensions) of the D-brane associated with the perturbative vacuum Ψ=0 and the D-brane which is represented by a classical open-string field configuration Ψ^∗. We will now discuss a generalisation of this statement to the situations where one is allowed to also change the closed-string background by turning on a non-trivial classical closed-string field configuration Φ^∗≠ 0.
        
        
        
          §.§.§ Changing the closed-string background
        
        
        Let us start by writing down the pure-closed classical string field theory action <cit.>
            S_sphere[Φ] = Z_sphere-1/g_s^2∑_k=1^∞1/(k+1)!ω_c(Φ,l_k(Φ^∧ k)) .
        
        Here, the degree-even dynamical string field Φ∈ℋ_c is a linear combination of states in the combined matter and ghost CFT which determine the worldsheet theory of a closed string propagating on a particular (classical) background. Furthermore, ω_c: ℋ_c⊗ℋ_c⟶ℂ denotes a symplectic form given by the BPZ product on ℋ_c, while l_k: (ℋ_c)^⊗ k⟶ℋ_c are the (cyclic, symmetric, degree-odd) closed-string products on a sphere, starting with the closed-string BRST operator l_1 = Q_c. Finally, g_s denotes the string coupling constant and Z_sphere the sphere partition function, which is characteristic to the chosen closed-string background. Requiring that the action (<ref>) solves the classical BV master equation 
        
            1/2(S_sphere,S_sphere)=0
        
        is equivalent to demanding that the products l_k satisfy the L_∞ relations
        
            ∑_k=1^r l_k l_r+1-k=0 .
        
        Following the Feynman rules dictated by the action (<ref>), one can compute tree-level on-shell closed-string amplitudes around the chosen background. On the other hand, varying this action with respect to Φ yields the equation of motion
        
            ∑_k=1^∞1/k! l_k(Φ^∧ k)=0 .
        
        A solution Φ^∗ of (<ref>) should then be thought of as representing a new consistent classical closed-string background described in terms of a new worldsheet theory CFT^∗ and, in general, a new value g_s^∗ of the string coupling constant. By the token of the background-independence conjecture <cit.>, 
        on-shell closed-string amplitudes on this new background can be computed by expanding the action (<ref>) in small fluctuations around Φ^∗ and applying a suitable field redefinition. 
        
        
        
          §.§.§ Sphere-disk open-closed vertices
        
        
        Enlarging our scope to include worldsheets with boundaries and arbitrary genus, one can formulate a consistent field theory of a degree-even open-string field Ψ∈ℋ_o (which is a linear combination of states in the boundary spectrum of a combined matter and ghost BCFT given by a boundary state B⟩⟩) and a degree-even closed-string field Φ∈ℋ_c, whose action S_oc[Φ,Ψ] satisfies the quantum BV master equation <cit.>
            1/2(S_oc,S_oc)+Δ S_oc=0 .
        
        The action S_oc is therefore well-suited for a complete (quantum) treatment of a backreacting system of D-branes, encoded in the boundary state B⟩⟩, on a given closed-string background specified by a worldsheet CFT and string coupling constant g_s<cit.>.
        Expressing S_oc as a sum over worldsheet topologies (parametrized by the genus g and boundary number b), it contains the sphere part (<ref>), as well as the disk part
        
            S_disk[Φ,Ψ]    = Z_disk-1/g_s∑_k=0^∞1/(k+1)!ω_c(Φ,l_k,0(Φ^∧ k))+
               -1/g_s∑_k=0^∞∑_l=0^∞1/k!1/l+1ω_o(Ψ,m_k,l(Φ^∧ k;Ψ^⊗ l)) ,
        
        which are then followed by other (higher-g and higher-b) contributions. The new ingredients introduced in (<ref>) include the open-string symplectic form ω_o:ℋ_o⊗ℋ_o⟶ℂ (defined in terms of the BPZ product on ℋ_o), the (cyclic, degree-odd, symmetric) closed-string valued disk products l_k,0:(ℋ_c)^⊗ k⟶ℋ_c, the open-string valued degree-odd products m_k,l: (ℋ_c)^⊗ k⊗ (ℋ_o)^⊗ l⟶ℋ_o and the disk partition function 
        
        
        
            Z_disk = -1/2π^2g/g_s ,
        
        where g=⟨ 0 B⟩⟩ denotes the g-function of the consistent conformal boundary state B⟩⟩.
        
        Furthermore, as a consequence of the (classical) criticality of the vacuum Φ=Ψ=0, we have m_0,0=0. The remaining products m_k,l are symmetric in their closed-string slots and cyclic in their open-string slots. For k>0, the corresponding disk string vertices can be equivalently parametrized in terms of the closed-string valued disk products l_k,l:(ℋ_c)^⊗ k⊗ (ℋ_o)^⊗ l⟶ℋ_c. The transition between the two descriptions is facilitated by the relation
        
            ω_o(Ψ,m_k,l(Φ^∧ k;Ψ^⊗ l)) = ω_c(Φ,l_k-1,l+1(Φ^∧ (k-1);Ψ^⊗ (l+1))) .
        
        Altogether, the BV master equation (<ref>) implies the Sphere-Disk Homotopy Algebra (SDHA) which, on top of the L_∞ relations (<ref>), includes the homotopy relations
        
            ∑_k=1^r [l_k,l_r-k,0] +∑_k=1^r-1 l_k-1,1m_r-k,0   =0 ,
            ∑_k=1^r m_k,s-1 l_r+1-k+ ∑_k=0^r∑_n=1^s m_k,nm_r-k,s-n   =0 .
        
        Notice that in (<ref>), one can recognize the Kajiura-Stasheff Open-Closed Homotopy Algebra (OCHA) <cit.>.  See <cit.> for an analysis of the full algebraic structure of quantum open-closed SFT action vertices.
        
        
        
        
          §.§.§ Open-string field theory on a classical closed-string background
        
        
        Restricting the closed string field Φ to satisfy the classical closed-string equation of motion (<ref>), one can show that the action
        
            S_Φ^∗[Ψ] = S_disk[Φ^∗,Ψ]
        
        satisfies the classical open-string BV master equation
        
            (S_Φ^∗,S_Φ^∗)_open=0 .
        
        This means that (<ref>) represents a consistent classical action for the open-string field Ψ on a closed-string background configuration given by Ψ^∗. In particular, one should emphasize that the action (<ref>) is therefore well-suited for the description of classical dynamics of D-branes on a changing closed-string background in the probe approximation: while S_Φ^∗ will be able to encompass the response of a given D-brane system to the change in the closed-string background given by the classical closed-string field configuration Φ^∗, it will not provide any information about the backreaction of the D-branes on the bulk. The vertices of the action S_Φ^∗[Ψ] can be conveniently parametrized in terms of open-string valued products m̃_l:(ℋ_o)^⊗ l⟶ℋ_o defined as
        
            m̃_l(Ψ^⊗ l)  = ∑_k=0^∞1/k!m_k,l((Φ^∗)^∧ k;Ψ^⊗ l) .
        
        One can then write the action (<ref>) in the form
        
            S_Φ^∗[Ψ] = Z_disk+Λ_c(Φ^∗)- 1/g_s∑_l=0^∞ω_o(Ψ,m̃_l(Ψ^⊗ l)) ,
        
        where we have isolated the contribution
        
            Λ_c(Φ^∗)=   -1/g_s∑_k=0^∞1/(k+1)!ω_c(Φ^∗,l_k,0((Φ^∗)^∧ k))
        
        of purely-closed disk vertices into the constant part of S_Φ^∗.
        The BV relation (<ref>) is then equivalent to stating that the products m̃_l satisfy an A_∞ algebra. Crucially, this algebra is weak, because the product m̃_0 is generally non-zero and gives rise to a tadpole in (<ref>). In particular, this means that Ψ=0 no longer represents a critical open-string background. So as to remove the tadpole and hence restore the full open-closed criticality, one has to re-expand the action (<ref>) in small fluctuations around an open-string field configuration Ψ^∗, which solves the equation of motion
        
            ∑_l=0^∞m̃_l(Ψ^⊗ l)  = 0 .
        
        The pair (Φ^∗,Ψ^∗) of string fields should then be thought of as interpolating between the original perturbative vacuum Φ=Ψ=0 and a new consistent classical open-closed background described in terms of a new bulk worldsheet theory CFT^∗, new string coupling g_s^∗ and the corresponding response B^∗⟩⟩ of the D-brane system to the change of the bulk. In order to gain some mileage on what exactly the new background is, one has to formulate (and then, ideally, calculate) some observable quantities. To this end, we will be interested in the constant part of the action S_Φ^∗[Ψ] expanded around Ψ^∗, which is given simply by the on-shell disk action
            S_Φ^∗[Ψ^∗]=S_disk[Φ^∗,Ψ^∗]=Z_disk+Λ(Φ^∗,Ψ^∗) ,
        
        where we have denoted
        
            Λ(Φ^∗,Ψ^∗)    =  Λ_c(Φ^∗)- 1/g_s∑_l=0^∞ω_o(Ψ^∗,m̃_l((Ψ^∗)^⊗ l))
               =-1/g_s∑_k=0^∞1/(k+1)!ω_c(Φ^∗,l_k,0((Φ^∗)^∧ k))+
               -1/g_s∑_k=0^∞∑_l=0^∞1/k!1/l+1ω_o(Ψ^∗,m_k,l((Φ^∗)^∧ k;(Ψ^∗)^⊗ l)) .
        
        Strong evidence for the observability of the on-shell disk action (<ref>) is provided by the fact that the quantity Λ(Φ^∗,Ψ^∗) can be shown <cit.> to be invariant not only with respect to the (weak) A_∞ gauge-variation of the open-string solution Ψ^∗, but also with respect to the L_∞ gauge-variation of the closed string solution Φ^∗.
        Following the spirit of background independence, one is then led to conjecture that the disk action S_disk[Φ^∗,Ψ^∗]
        should be identified as the disk partition function Z_disk^∗ of the new open-closed background (CFT^∗,g_s^∗,B^∗⟩⟩). Expressing the original, as well as the new disk partition function in terms of the g-function and the string coupling constant, we finally arrive at the generalised Sen's conjecture
            -1/2π^2(g^∗/g_s^∗-g/g_s) = Λ(Φ^∗,Ψ^∗) .
        
        In the special class of cases when the closed-string background does not undergo any change (that is Φ^∗=0), one has to put g_s^∗ =g_s while, so that the l.h.s. of (<ref>) is directly proportional to the change in the g-function of the D-brane system. Since, at the same time, the r.h.s. of (<ref>) then reduces to the on-shell OSFT action, one recovers the classical Sen's conjecture. In the more general case when Φ^∗≠ 0, the conjecture (<ref>) not only provides a way of tracking the change in the g-function of probe D-branes as the bulk is deformed, but it also represents an indirect method of measuring any potential changes in the string coupling constant: as the new value g_s^∗ should depend neither on the choice of the initial boundary state B⟩⟩, nor on the open-string solution Ψ^∗, one should be able to isolate the change in the string coupling constant by analyzing the r.h.s. of (<ref>) for a number of distinct open-string backgrounds and assuming that one knows the expected change in the g-function for some of these. This demonstrates how coupling the bulk closed-string theory to probe D-branes may be beneficial for providing observables which help with keeping track of pure closed-string dynamics.
        
        
        
        
         §.§ Bulk marginal deformations
        
        
        The formal statement of the generalised Sen's conjecture can be illustrated by taking the closed-string solution Φ^∗ to represent a marginal deformation.
        
        
        
          §.§.§ Perturbative solutions TEXT and TEXT
        
        
        We will consider an exactly marginal deformation of the closed-string background which, at the leading order of the (continuous) deformation parameter μ, is given by an on-shell ghost-number 2 operator V. We will mostly focus on the cases when V = cc̅𝕍, where 𝕍 is a matter primary with dimensions (h,h̅)=(1,1). The exact marginality of the deformation needs to be guaranteed order by order in μ<cit.>. For instance, at second order, this entails requiring that
        
            P_0^+ l_2(μ V,μ V) =0
        
        up to Q_c-exact terms. The full unobstructed solution Φ^∗(μ) can then be expanded as
        
            Φ^∗(μ)    = μ V - 1/2!b_0^+/L_0^+P̅_0^+ l_2(μ V, μ V) +𝒪(μ^3) .
        
        Furthermore, let us assume that given the exactly marginal deformation Φ^∗(μ), one can find a corresponding solution Ψ^∗(μ) to the open-string equation of motion (<ref>) which remains perturbatively close to the original background at μ=0. This again necessitates clearing possible cohomological obstructions order by order in μ, starting with
        
            P_0 m_1,0(μ V)=0
        
        up to Q_o-exact terms. Failure of these would signify an inability of the D-brane system at hand to adapt to the marginal deformation which we are turning on in the bulk. Instead, one would then have to search for a non-perturbative solution to (<ref>) in order to restore criticality of the background. In the absence of any obstructions, the open-string solution Ψ^∗(μ) would be expanded in μ as
        
            Ψ^∗(μ)   = -b_0/L_0P̅_0 m_1,0(μ V)+𝒪(μ^2) .
        
        
          §.§.§ The deformed disk partition function
        
        
        Substituting the solutions (<ref>) and (<ref>) into (<ref>) yields a μ-expansion of the change in the disk partition function. Combining this with the statement (<ref>) of the generalised Sen's conjecture, one obtains 
        
            1/2π^2[g^∗(μ)/g(g_s^∗(μ)/g_s)^-1-1] =
               =1/gω_c(μ V,l_0,0)+1/2[1/gω_c(l_0,0,b_0^+/L_0^+P̅_0^+ l_2(μ V ,μ V))+
               +1/gω_c(μ V, l_1,0(μ V))+1/gω_o(m_1,0(μ V),b_0/L_0P̅_0 m_1,0(μ V))]+𝒪(μ^3) .
        
        To evaluate the r.h.s. of (<ref>), it is practical to make concrete assumptions about the form of the SFT vertices. See <cit.> for a particular choice of vertices which are based on SL(2,ℂ) maps and which will be also put to use in later sections of this paper when discussing relevant deformations. One should nonetheless keep in mind that as the disk action is an observable quantity, the endpoint of calculating the r.h.s. of (<ref>) should be manifestly independent of any off-shell SFT data (such as the local coordinate maps around punctures and their parameters). This serves as a useful consistency check. Following the steps outlined in <cit.>, one can then arrive at the expression
        
            g^∗(μ)/g(g_s^∗(μ)/g_s)^-1 = 1+ 2 1/g⟨μ𝕍(i,i̅)⟩_UHP+ ∫_0
            reg.^1 ds (s^2-1) 1/g⟨μ𝕍(i,i̅) μ𝕍(is,i̅s̅)⟩_UHP+𝒪(μ^3)
        
        for the evolution of the worldhseet disk partition function with μ. The integral in the second term runs from s=0 (open-string degeneration) to s=1 (closed-string degeneration) and needs to be regulated at both endpoints so as to avoid divergences coming from the propagation of the open- and closed-string tachyons. In particular, we define <cit.>
            ∫_0
            reg.^1  ds  = lim_ϵ_c→ 0
            ϵ_o→ 0 [ ∫_0^a ds  s^ϵ_o+∫_a^1 ds (1-√(s)/1+√(s))^ϵ_c] ,
        
        where the parameter a∈ (0,1) serves to separate the two degeneration regions (and can be related to the open-string stub parameter λ_o as a = 1/λ_o^2). The result of the integration is however always independent of a, as can be readily checked by differentiating with respect to it. Also note that the function appearing in the second term of (<ref>) regulating the closed-string collision can be naturally interpreted in terms of the radial coordinate y on the disk, so that near the closed-string degeneration (y=0), the tachyon-divergence is regulated with y^ϵ_c, as appropriate. Finally, note that the integral is not endangered by any divergences due to propagation of massless modes as these are absent precisely by virtue of the conditions (<ref>) (closed-string channel) and (<ref>) (open-string channel).
        
        
        
        
         §.§ Example: navigating the Narain moduli space
        
        
        As an illustration, we will now consider computing the on-shell disk action in the case when the on-shell closed-string field Φ^∗ represents a finite deformation of the Narain modulus E_μν = G_μν+B_μν of a toroidal compactification of d free bosons. We will assume that our probe D-brane is represented by an elementary matter boundary state which satisfies the Ishibashi conditions
        
            [(α_n^L)_μ+Ω_μ^ν(α_-n^R)_ν]B⟩⟩ =0 ,
        
        where the gluing automorphism matrix Ω is constrained to satisfy Ω G Ω^T=G. Such boundary state is built up from U(1)^d Ishibashi states, which are labelled by the left- and right-moving momenta
        
            k^L   = k +E w ,
            
                k^R   = k -E^T w ,
        
        where the momentum k_μ and winding w^ν are both ℤ^d-valued. Specializing the relation (<ref>) for the zero-modes n=0, one learns that k and w need to be restricted so that
        
            (1+Ω)k+(E-Ω E^T)w=0 .
        
        Requiring that (<ref>) is solved by a d-dimensional sublattice of ℤ^d⊕ℤ^d imposes further restrictions on Ω, which in fact make the set of allowed gluing matrices Ω discrete. 
        
        
        
          §.§.§ BCFT expectation
        
        
        Let us first discuss what change in the g-function of a Narain boundary state one should expect (based on pure BCFT considerations) as we deform the Narain modulus by sending E→ E+ϵ_σ. In conformal perturbation theory, this change can be implemented by the marginal deformation
        
            S[X;E]⟶ S[X;E]+ 1/4π∫ d^2 z 𝕍^ϵ_σ(z,z̅) ,
        
        where S[X;E] is the action of d free bosons X^μ compactified on lattice with Narain modulus E_μν and 
        
            𝕍^ϵ_σ(z,z̅)  =(ϵ_σ)_μν  X^μ(z)  X^ν(z̅)
        
        is the deforming operator.
        To first order in ϵ_σ, a one-loop calculation in conformal perturbation theory reveals that the g function changes as
        
            δ^(1)g/g = 1/4[Ω^T G^-1ϵ_σ] .
        
        Furthermore, varying the relation (<ref>) with respect to E, it is straightforward to realise that
        
            δ^(1)Ω  = 1/2(ϵ_σ - Ω ϵ_σ^T)G^-1(1+Ω) .
        
        Given the results (<ref>) and (<ref>), one can straightforwardly find the second variation δ^(2)g of g with respect to E. Combining this with the first variation (<ref>), we learn that the finite change Δ g of the g-function can be expanded to second order in ϵ_σ as
        
            1+ Δ g/g   = 1+δ^(1)g/g+1/2δ^(2)g/g+𝒪(ϵ_σ^3) 
               = 1+ 1/4[Ω^T G^-1ϵ_σ]+1/32[Ω^T G^-1ϵ_σ]^2 - 1/16[(Ω^T G^-1ϵ_σ)^2]+
               -1/8[Ω^T G^-1ϵ_σ  G^-1ϵ_σ]+1/16[ G^-1ϵ_σ^T  G^-1ϵ_σ]+𝒪(ϵ_σ^3) .
        
        Notice that all terms in (<ref>) depend on the gluing matrix Ω (and thus on the boundary condition) except for the last one, which is proportional to the two-point function coefficient C^ϵ_σ_𝕍𝕍1 of the deforming operator 𝕍^ϵ_σ(z,z̅) on the sphere. Indeed, substituting from (<ref>), one has
        
            ⟨𝕍^ϵ_σ(z,z̅) 𝕍^ϵ_σ(w,w̅)⟩_ℂ = C_𝕍𝕍1^ϵ_σ/|z-w|^4 ,
        
        where
        
            C_𝕍𝕍1^ϵ_σ = 1/4[ G^-1ϵ_σ^T  G^-1ϵ_σ] .
        
        
          §.§.§ SFT calculation
        
        
        We will now implement the marginal deformation which sends E to E+ϵ_σ as a classical solution (of the form (<ref>)) in closed string field theory. To this end, we put
        
            μ𝕍(z,z̅)≡𝕍^ϵ(z,z̅) ≡ϵ_μν  X^μ (z)  X^ν(z̅) .
        
        It is not difficult to check that for such a marginal state, the condition (<ref>) holds, which in turn guarantees exact marginality of the deformation to second order in ϵ.
        First, one can compute the change in conformal weights of the bulk momentum plane waves induced by the solution (<ref>). This calculation proceeds by analyzing the cohomology of the BRST operator shifted around the classical solution. Comparing the result with the shift in conformal weights which is expected upon deforming the Narain modulus E by sending E→ E+ϵ_σ, we can fix the relation between the SFT deformation parameter ϵ and the CFT deformation ϵ^σ to be <cit.>
            ϵ = ϵ_σ - 1/2ϵ_σ G^-1ϵ_σ + 𝒪(ϵ_σ^3) .
        
        Second, considering any Dp-brane described by a gluing matrix Ω, we would like to evaluate the disk action (<ref>) of the open-string solution (<ref>) computed on a classical closed-string background given by the solution (<ref>). This is a healthy solution, as one can readily verify that the condition (<ref>) holds for any Ω. It describes how the D-brane adapts as we perturb the Narain modulus of the bulk CFT.
        Evaluating the relevant UHP correlators, we obtain
        
            1/g⟨𝕍^ϵ(i,i̅)⟩_UHP   =1/8[ϵΩ^T G^-1] ,
            1/g⟨𝕍^ϵ(i,i̅) 𝕍^ϵ(is,i̅s̅)⟩_UHP   =1/4([ϵΩ^T G^-1]^2/16s^2+ [ϵ G^-1ϵ^T G^-1]/(1-s)^4+[ϵΩ^T G^-1ϵΩ^T G^-1]/(1+s)^4) .
        
        Using the prescription (<ref>), we can also evaluate
        
            ∫_0
            reg.^1  ds s^2-1/16s^2   =+1/8 ,
            ∫_0
            reg.^1  ds s+1/(s-1)^3   =+1/8 ,
            ∫_0
            reg.^1  ds s-1/(s+1)^3   =-1/4 .
        
        Finally, substituting the results (<ref>), (<ref>) and (<ref>) into the expression (<ref>) for the disk action, we obtain
        
            (1+ Δ g/g)(1+ Δ g_s/g_s)^-1 =
               =(1+1/4[ϵ_σΩ^T G^-1]+1/32[ϵ_σΩ^T G^-1]^2-1/16[(ϵ_σΩ^T G^-1)^2]+
               -1/8[  ϵ_σ G^-1ϵ_σΩ^T G^-1]+1/16 [ϵ_σ G^-1ϵ_σ^T G^-1])(1+1/32 [ϵ_σ G^-1ϵ_σ^T G^-1])^-1+𝒪(ϵ_σ^3) .
        
        Quite non-trivially, on the r.h.s. of (<ref>) we recover the expected response (<ref>) of the g-function of the Dp-brane boundary state to the bulk deformation E→ E+ε_σ. This is multiplied by a universal factor which is independent of Ω. Attributing this factor to a change Δ g_s in the string coupling constant g_s which is induced by the closed-string solution (<ref>), we can write
        
            1+ Δ g_s/g_s= 1+1/8C_𝕍𝕍1^ϵ_σ+𝒪(ϵ_σ^3) ,
        
        where the two-point function coefficient C_𝕍𝕍1^ϵ_σ was computed in (<ref>).
        
        
        
        
        
        
        
            
        
        
        
        
        
        
        § SHORT BULK RG FLOWS WITH BOUNDARIES
        
        In this section, we will apply the open-closed SFT framework to investigate how relevant bulk CFT perturbations affect boundary states. In particular, we will construct the classical open SFT solutions that describe D-branes after the closed-string background has undergone a `short' RG flow.The possibility of studying such closed-string perturbations via SFT was considered long ago by Mukherji and Sen <cit.> and more recently in <cit.> and <cit.>. Here we would like to extend this analysis to the presence of world-sheet boundaries representing D-branes.
        
        We will begin our exposition by describing the pure 2d CFT setup and briefly reviewing the results of <cit.>. We will then proceed with an explicit construction of the open SFT solutions.
        
        
        
        
        
         §.§ Conformal perturbation theory description
        
        Let us start with an overview of conformal perturbation theory (CPT) for a theory CFT_0 with boundary described by a boundary state B_0⟩⟩. In general, bulk deformations of 2d CFTs with boundaries can induce boundary RG flows. To write down the RG equations, we consider a generic bulk and boundary deformation given in terms of bulk primary fields 𝕆_k with scaling dimensions Δ_k and boundary primary fields o_l with scaling dimensions h_l as
            S_ CFT_0→  S_ CFT_0+∑_iλ_iϵ^Δ_i-2∫ d^2 z  𝕆_i (z,z̅)+∑_jμ_jϵ^h_j-1∫ dx  o_i (x) ,
        
        where λ_i and μ_j are dimensionless coupling constants, and ϵ is the length scale used as a natural cutoff to treat the contact divergences arising in correlators of the perturbed theory. Following the usual CPT approach (see, for example,  <cit.>), we obtain the following RG equations
        
            dλ_k/dlog(ϵ)=(2-Δ_k)λ_k+∑_ijπ C_ijkλ_iλ_j+𝒪(λ^3) ,
               dμ_l/dlog(ϵ)=(1-h_l)μ_l+∑_i1/2B_il/gλ_i+𝒪(λμ)+𝒪(λ^2)+𝒪(μ^2)  ,
        
        where g is the g-function of the initial BCFT and C_ijk and B_il/g are the OPE coefficients
        
            𝕆_i(z,z̅) 𝕆_j(w,w̅)=∑_kC_ijk/| z-w|^Δ_i+Δ_j-Δ_k𝕆_k(w,w̅)+… ,
               𝕆_i(x+is,x-is)=∑_l1/(2s)^Δ_i-h_lB_il/go_l(x)+… .
        
        In particular, we will consider a perturbation triggered by a nearly marginal primary bulk field 𝕍(z,z̅)which has conformal dimensions (1-y,1-y) with y small and positive. We will also assume that the corresponding Virasoro representation satisfies the fusion rule
            𝕍×𝕍=1+𝕍+… ,
        
        where the dots represent possible irrelevant contributions. The bulk and the bulk-boundary OPEs of 𝕍 can then be written as
        
            𝕍(z,z̅)𝕍(w,w̅)   z→ w∼1/| z-w|^4(1-y)+1/| z-w|^2(1-y)C_𝕍𝕍𝕍𝕍(w,w̅)+ reg ,
            𝕍(x+is,x-is)   s→ 0∼1/(2s)^2(1-y)B_𝕍1/g+1/(2s)^1-yB_𝕍v/gv(x)+ reg ,
        
        where we normalised the nearly marginal field as ⟨𝕍|𝕍⟩ =C_𝕍𝕍1= 1. We have also introduced the boundary field v of dimension h = 1 - y transforming in the same Virasoro representation as the deforming bulk field 𝕍. Note that sometimes, it may happen that this representation is absent from the spectrum of the boundary fields as determined by the Cardy condition (annulus crossing equation). In such cases, the OPE coefficient B_𝕍v/g has to vanish. 
        
        
        Under these assumptions the RG  equations (<ref>) and (<ref>) can be rewritten as 
        
            dt/dlog(ϵ)   =2yt -C_𝕍𝕍𝕍t^2+𝒪(t^3) ,
            dτ/dlog(ϵ)   =y τ +B_𝕍v/gt/2+𝒪(t τ)+𝒪(t^2)+𝒪(τ^2) ,
        
        where τ -μ_vπ. In turn, from (<ref>) and (<ref>) we can generally read off the fixed points at leading order in y as
        
            t^∗(y)   =2y/C_𝕍𝕍𝕍+𝒪(y^2) ,
            τ^∗(y)   =-1/C_𝕍𝕍𝕍B_𝕍v/g+𝒪(y) .
        
        Looking at the first line, we can notice that y adjusts the length of the bulk RG flow by making it short in the y→ 0 limit, in the sense that we have a perturbative bulk RG fixed point t^∗(y). On the other hand, in the case with boundaries, we have a perturbative fixed point (with τ^∗(0)=0) only if 
        
            B_𝕍v/g|_y=0=0 .
        
        Therefore, to have a short bulk RG flow with boundaries, we must work under the assumption B_𝕍v/g y → 0∼𝒪(y). This includes the cases when the bulk-broundary OPE coefficient B_𝕍v/g vanishes exactly. In such situations, one has a choice of not triggering the boundary RG flow at all. Finally, in the cases when the field v is not present in the boundary spectrum, the boundary RG equation (<ref>) becomes vacuous and there is no boundary RG flow to be induced by the bulk deformation triggered by 𝕍.As we are now going to discuss, from the SFT perspective, the bulk-boundary perturbation triggered by 𝕍 can be described by 1. starting with a consistent open-closed world-sheet background incorporating the initial matter theory CFT_0 and boundary state B_0⟩⟩, 2. solving perturbatively the closed-string equation of motion for a classical solution Φ^∗(y), 3. shifting the open-closed background by this solution and then, finally, 4. solving the tadpole-sourced open-string equation of motion to find the classical solution Ψ^∗(y) which captures the changes in the D-brane system along the RG flow.
        
         §.§ Short bulk RG flows from SFT
        
        
        
        Starting with the bulk RG flows, we embed the matter theoryCFT_0 with central charge c, into a worldsheet CFT given by the tensor product of three sectors
        
            CFT_ tot^(0)= CFT_0⊗ CFT_ aux⊗ CFT_gh ,
        
        where CFT_gh is the usual bc-ghost system with central charge c_ gh = -26 and CFT_aux is an auxiliary CFT with c_ aux = 26 - c, which ensures that the theory is critical,
        
            c_ tot=c+c_ aux+c_ gh=0 .
        
        Now, our aim is to perturbatively solve the classical closed-string equation of motion (<ref>). Doing so, one should find an explicit expression for the solution Φ^∗(y) which describes the critical closed-string background corresponding to fixed point (<ref>) of the short RG flow. Since our eventual goal  will be to evaluate the the on-shell disk action (<ref>) up to quadratic order, we will only be interested in finding Φ^∗(y) up to second order in y. In CPT, this would correspond to a rather hard two-loop computation. 
        Also recall that as we are neglecting backreaction of the D-branes on the bulk, the closed-string equation of motion is unaffected by the presence of the boundaries, exactly in the spirit of the bulk RG equations (<ref>).
        
          §.§.§ Closed-string equation of motion and obstructions
        
        
        Following the strategy outlined in <cit.> and <cit.>, we introduce a projector P that projects onto the space of eigenstates of the operators L_0 and L̅_0 which have eigenvalues (0,0) and (-y,-y).It is not hard to see that P commutes with the BRST charge, that is[P,Q_ c]=0. Afterwards, we split the closed string field through the action of P and P̅=1-Pby defining
        
            Φ=PΦ+P̅Φ W + R .
        For the “tachyon” component W of the string field, we then consider the ansatz
            W = t(y) T ,
        
        where T=cc̅𝕍 is the nearly on-shell tachyon and t(y) is a function that will be fixed by imposing the closed-string equation of motion. In particular, this function can be expanded as
        
            t(y)=t_1 y+ t_2 y^2+𝒪(y^3) .
        
        Our aim will be to explicitly find t_1 and t_2 in terms of CFT data.
        
        First, let us project the full equation of motion
        
            EOM_ c[Φ]=Q_ cΦ+1/2l_2(Φ,Φ)+1/3!l_3(Φ,Φ,Φ)+𝒪(y^4)
        into the two (orthogonal) subspaces given as the image of P and P̅, namely
            P  EOM_ c[Φ]=Q_ cW+1/2Pl_2(Φ,Φ)+1/3!Pl_3(Φ,Φ,Φ)+𝒪(y^4) ,
               P̅  EOM_ c[Φ]=Q_ cR+1/2P̅l_2(Φ,Φ)+1/3!P̅l_3(Φ,Φ,Φ)+𝒪(y^4) .
        Since it is possible to invert Q_c on the image of P̅, this splitting enables us to write down a recursive solution for R in terms of W. At leading order, this reads (in the Siegel gauge b_0^+=0)
            R(W)=-1/2b_0^+/L_0^+P̅l_2(W,W)+𝒪(y^3) .
        Plugging (<ref>) back into (<ref>), the component of the full equation of motion which lies in the image of P can be recast purely in terms of W as
            P  EOM_ c[Φ]=Q_ cW+1/2Pl_2(W,W)-1/2Pl_2(W,b_0^+/L_0^+P̅l_2(W,W))+1/3!P l_3(W,W,W)+𝒪(y^4) .
        
        Looking in detail at the r.h.s. of (<ref>), we can notice that there is an obstruction in solving the equation of motion. Indeed, the last term turns out to give rise to terms proportional to the anti-ghost dilaton, c_0^+(c^2 c-c̅^2c̅) at cubic order in y, which cannot be balanced by any of the remaining terms on the r.h.s. of (<ref>). The meaning of this obstruction is related to the fact that the deformation by 𝕍 gives rise to an 𝒪(y^3) change in the central charge of the matter CFT.With no further provisions, this would have resulted into a non-critical total world-sheet CFT. 
        
        A solution to this problem was suggested already in <cit.> and very recently concretised in <cit.>. The key point is to modify the auxiliary sector of the initial worldsheet CFT by adding a linear dilaton sector CFT_ D_β. This has the central charge c_β=1+3β^2 where β denotes the background charge. Hence, the total worldsheet theory CFT_ tot^(0) now factorizes as
            CFT_ tot^(0)= CFT_0⊗ CFT_ D_β⊗ CFT_ aux⊗ CFT_ gh .
        Denoting by Y the scalar field of the linear dilaton CFT, the anti-ghost dilaton can then be BRST-trivialized by the field
            Θ = 1/2(c^2 c-c̅^2c̅)Y+c_0^+(c Y- c̅Y) .
        Indeed, we have
            Q_ c Θ = √(α^'/2)β c_0^+(c^2 c-c̅^2c̅) .
        This makes it possible to compensate for the obstruction by introducing a term proportional to Θ in the SFT solution, namely 
        
            W=t(y)T+θ(y)Θ ,
        
        where θ(y) is a function  which starts at the order y^3.
        From the worldsheet CFT viewpoint, this corresponds to implementing the RG flow both in the matter and in the linear dilaton sector. The latter will induce a change in the background charge and, therefore, in c_β. This, in turn, exactly compensates for the variation of the matter central charge, resulting in a total worldsheet CFT that remains critical. We therefore recover
            c_ tot^∗=c^∗+c_β^∗+c_ aux+c_ gh=0 ,
        
        where ∗ indicates that the corresponding quantities refer to the perturbed theory. 
        
        Finally, having discussed the subtleties of searching for a closed-string solution which changes the matter central charge, we recall that our primary focus in this paper is on computing the disk action on the classical solution up to second order in y. Hence, the third-order term θ(y)Θ will not be of any practical interest in the remainder of our analysis.
        
          §.§.§ The classical closed-string solution
        
        Let us now impose the equation of motion (<ref>) and substitute the ansatz (<ref>). We obtain 
        
            Q_ cT+t(y)/2Pl_2(T,T)+t(y)^2/3!(P l_3(T,T,T)-3Pl_2(T,b_0^+/L_0^+P̅l_2(T,T)))+𝒪(y^3)=0 .
        By calculating the symplectic form of T against the l.h.s. of (<ref>), we can write
            (T,Q_ cT)+t(y)/2(T,l_2(T,T))+t(y)^2/3! A_TTTT+𝒪(y^3)=0 ,
        
        where we defined
        
            A_TTTT(T, l_3(T,T,T))+3 (l_2(T,T),b_0^+/L_0^+P̅l_2(T,T)) .
        
        The quantity A_TTTT is proportional to the zero-momentum amplitude of four slightly off-shell tachyons on the sphere.
        
        Let us now separately describe the three terms on the l.h.s. of (<ref>). Starting with the first one, we can write
            (T,Q_ cT)=⟨ cc𝕍(0,0)| c_0^-(c_0L_0+c̅_0L̅_0)| cc𝕍(0,0)⟩ =y ,
        
        in which we used the fact that we are working in the Siegel gauge and with operators normalized so that⟨ 0| c_-1c̅_-1 c_0c̅_0 c_1c̅_1|0⟩=-1 and ⟨𝕍|𝕍⟩=1. To evaluate the second term in (<ref>), we must introduce the cubic string productl_2. In particular, we will use the SL(2,ℂ) vertices defined in <cit.>. We will describe these in more completeness in the following section. For now, we will just provide the definition of l_2, namely
            | l_2(Φ_1,Φ_2)⟩ = b_0^-δ(L_0^-)f_1∘Φ_1(0,0)f_2∘Φ_2(0,0)| 0⟩_ SL(2,ℂ) ,
        
        with 
        
            f_1(w)=1/λ_ cw-λ_ c/3λ_ c+w=-f_2(w) ,
        
        where λ_ c>1 is a tunable stub parameter. Using this definition to compute the second term in(<ref>), we get
        
            (T,l_2(T,T))   =⟨ cc𝕍(0,0)| c_0^-b_0^-δ(L_0^-)| f_1∘  cc𝕍(0,0)f_2∘  cc𝕍(0,0)⟩
               =⟨ I_ c∘  cc𝕍(0,0) f_1∘  cc𝕍(0,0)f_2∘  cc𝕍(0,0)⟩_ℂ
               =-C_𝕍𝕍𝕍(3/2λ_ c)^6y ,
        
        where, in the first line, we have reabsorbed the projector onto the level-matched Hilbert space in the bra, and, in the second line, we introduced the closed-string BPZ inverse map defined as I_ c(w)=1/w. 
        
        
        Summarizing our progress up to this point, plugging back (<ref>) and (<ref>) into (<ref>) and expressing t(y) as a power-series expansion, we obtain
        
            y-1/2(t_1 y+t_2 y^2)(1+6ylog(3/2λ_ c))C_𝕍𝕍𝕍+1/3!t_1^2y^2   A_TTTT+𝒪(y^3)=0 .
        This enables one to determine the coefficients t_1 and t_2 as
            t_1   =2/C_𝕍𝕍𝕍 ,
            
                t_2   =-(2/C_𝕍𝕍𝕍)^2(3C_𝕍𝕍𝕍log(3/2λ_ c))+1/3!(2/C_𝕍𝕍𝕍)^3 A_TTTT|_y=0 .
        It remains to evaluate the amplitude A_TTTT of four slightly relevant operators. This is one of the main results of <cit.>. It turns out that in the large stub limit λ_ c→∞, one can write it as
            A_TTTT=12(C_𝕍𝕍𝕍)^2log(3/2λ_ c)+ A_TTTT^ f.p. ,
        where A_TTTT^ f.p. is a finite number. Indeed, in computing this amplitude, there are contact divergences which can be regularized through a cutoff naturally provided by the closed stub parameter in the limit λ_ c→∞. The idea is then to expand the four-tachyon fundamental vertex in conformal blocks and then add and subtract divergent terms to obtain a finite part A_TTTT^ f.p. in which we can truly perform the limit λ_ c→∞.
        The remaining divergent terms explicitly depend on the closed-string stub parameter λ_c. 
        Note that by considering the limit λ_c→∞, we do not lose generality because any observable which we compute must be independent of the off-shell SFT data. We will explicitly see this at work in the following sections, where we will prove that the classical on-shell disk action does not depend on λ_ c. 
        
        
          
         
        Finally, by inserting (<ref>) into (<ref>), we obtain
         
            t_2=( A^ f.p._TTTT/3 C_𝕍𝕍𝕍+ C_𝕍𝕍𝕍log(3/2λ_ c))(2/C_𝕍𝕍𝕍)^2 .
        Hence, using (<ref>), (<ref>), (<ref>), (<ref>) and (<ref>), we find that the classical closed-string solution expanded up to quadratic order in y reads 
         
            Φ^∗(y)   =[2y/C_𝕍𝕍𝕍+( A^ f.p._TTTT/3 C_𝕍𝕍𝕍+ C_𝕍𝕍𝕍log(3/2λ_ c))(2y/C_𝕍𝕍𝕍)^2]cc𝕍+
               -1/2(2y/C_𝕍𝕍𝕍)^2b_0^+/L_0^+P̅l_2(cc𝕍,cc𝕍)+𝒪(y^3) .
        When one substitutes the solution (<ref>) into the sphere part of the string field theory action, all dependence on the stub parameter λ_c indeed cancels, as it should, since the on-shell value of the action is a gauge-invariant (observable) quantity. In <cit.>, this quantity has been conjectured to be related to the change in the central charge of the matter CFT under the short bulk RG flow triggered by 𝕍, and this has been non-trivially verified up to 𝒪(y^4) in the case of Zamolodchikov flows of large m minimal models.
        
         §.§ Fate of probe D-branes under short bulk RG flows
        In this section, we will make a key intermediate step towards computing the disk part of the on-shell action in the situation when the closed-string sector (which is coupled to a probe D-brane system) is undergoing a short RG flow in the matter sector: we will compute the solution Ψ^∗(y) to the tadpole-sourced open-string equation of motion (<ref>).We will see that since we are interested in expanding the disk action up to second order in y, we can truncate the open-string solution at first order in y.This is because for a critical open-string background, the disk open-string tadpole vanishes, so that second order terms in the open-string solution Ψ^∗(y) only start contributing into the on-shell disk action at third order in y.
        
        The initial open-string background can be defined by a consistent boundary CFT. This in turn can characterized by a boundary state which can be written as a tensor product of boundary states for each sector of the bulk theory, that is
        
            B_ tot⟩⟩=B_0⟩⟩⊗B_ aux⟩⟩⊗B_ gh⟩⟩ .
        In order to find Ψ^∗(y), we will follow a strategy similar to the above-described case of the pure closed-string background. Namely, we will split the open-string field into two orthogonal components through the application of the projector P, which projects onto the sub-space of the open-string Hilbert space which is spanned by the L_0 eigenvectors with eigenvalues 0 and -y. In other words, we write
            Ψ=PΨ + P̅Ψ w+r .
        For the component w (which lies in the image of P), we consider an ansatz proportional to the nearly marginal boundary field
        
            w = τ(y)  c v .
        To obtain an equation fixing the function τ(y), we consider the tadpole-sourced open-string equation of motion evaluated on the classical closed-string backgroundΦ^∗(y) which is given by (<ref>). Assuming that the open-string solution Ψ^∗ will start at order y, this can be truncated as
            EOM_Φ^∗(y)[Ψ]   =Q_ oΨ+m_0,2(Ψ,Ψ)+
               +m_1,0(Φ^∗(y))+m_1,1(Φ^∗(y), Ψ)+m_2,0(Φ^∗(y),Φ^∗(y))+𝒪(y^3) .
        Solving this equation can be facilitated by considering its projections into the image of P and its orthogonal complement, namely
            P  EOM_Φ^∗(y)[Ψ]   =Q_ ow+Pm_0,2(Ψ,Ψ)+
               +Pm_1,0(Φ^∗(y))+Pm_1,1(Φ^∗(y), Ψ)+Pm_2,0(Φ^∗(y),Φ^∗(y))+𝒪(y^3) ,
            P̅  EOM_Φ^∗(y)[Ψ]   =Q_ or+P̅m_0,2(Ψ,Ψ)+
               +P̅m_1,0(Φ^∗(y))+P̅m_1,1(Φ^∗(y), Ψ)+P̅m_2,0(Φ^∗(y),Φ^∗(y))+𝒪(y^3) .
        Since the open-string BRST operator Q_o is invertible on the image of P̅, the second equation can now be recursively solved in the Siegel gauge b_0=0 as
            r^∗(y)=-b_0/L_0P̅m_1,0(Φ^∗(y))+𝒪(y^2)=-2y/C_𝕍𝕍𝕍b_0/L_0P̅m_1,0(cc̅𝕍)+𝒪(y^2) .
        Notice that, in contrast to the closed-string case, the component of the open-string solution lying in the image of P̅ now starts at order y and not y^2. This is due to the presence of a closed-string sourced open-string tadpole in the disk action.It remains to solve the equation of motion (<ref>) for w (and therefore for τ(y)). First, 
        note that when the slightly relevant boundary field v is not part of the spectrum of the matter BCFT, the image of the projector P can be taken as empty.[This is provided that we choose to work with vertices which do not generate the pure-ghost boundary field c.] In such cases, the equation of motion (<ref>) is simply absent and the open-string solution Ψ^∗ (y) lies purely in the image of P̅.On the other hand, when  the boundary field vis part of the matter BCFT, then, under our assumption (<ref>), we can see that the P-component w of Ψ^∗(y) will start at 𝒪(y).Indeed, the contribution
            Q_ ow=τ(y) Q_ o c v=y τ(y)   c c  v = 𝒪(y^2) .
        to the r.h.s. of (<ref>) then generally comes at the same order as  Pm_0,2(Ψ^∗ (y),Ψ^∗ (y)), as well as Pm_1,0(Φ^∗ (y)) which acts as a source. Hence, if we put w(y)=τ_1 y  c  v+𝒪(y^2), the equation of motion (<ref>) can be balanced and, at least in principle, solved for the coefficients τ_i. This is how our assumption (<ref>) guarantees perturbativity of the bulk-induced boundary deformation.However, it turns out that in order to achieve our goal of finding the on-shell disk action up to second order in y, we do not have to be interested in finding the explicit expression for the solution w^∗(y)=PΨ^∗(y) because it only would have started to contribute at 𝒪(y^3). Indeed, the leading contribution to (<ref>) involving the component of the open-string solution lying in the image of P would go as
            [w^∗(y),m_1,0(Φ^∗(y))]∼ y^2 B_𝕍v/g+…∼𝒪(y^3)
        where two powers of y come from expanding the string fields w^∗(y) and Φ^∗(y) in y, while another power of y comes from the bulk-boundary OPE coefficient B_𝕍v/g∼𝒪(y) as per our assumption (<ref>).On the other hand, the P̅ component of the solution contributes to (<ref>) at order y^2
            [r^∗(y),m_1,0(Φ^∗(y))]∼ y^2 (B_𝕍1)^2/g+…∼𝒪(y^2) ,
        
        because generally B_𝕍1(y)/g(y)∼𝒪(y^0).
        
        
        To summarize, for the purposes of evaluating the on-shell disk action up to second order in y, we may take the open string solution to read
        
            Ψ^∗(y)=-2y/C_𝕍𝕍𝕍b_0/L_0P̅m_1,0(cc̅𝕍)+… ,
        
        where the dots … represent not only 𝒪(y^2) terms, but also terms at order y which, however, would have contributed at cubic order into the on-shell disk action.
        
        § DISK ACTION OF SHORT RG FLOWS
        
        In this section, we will compute (up to quadratic order in y) the disk action of open-closed SFT evaluated on the classical solutions Φ^∗(y) and Ψ^∗(y)which were constructed in section <ref> to describe short RG-flows in the matter (B)CFT. According to the generalised Sen's conjecture (<ref>), this will allow us to obtain the induced change in the ratio of the worldsheet boundary state g-function and the string coupling constant in terms of CFT data.
        
         §.§ Choice of vertices and initial setup
        
        In order to evaluate the on-shell disk action, the first step is to properly define the vertices that appear in (<ref>), (<ref>), as well as on the r.h.s. of (<ref>). Specifically, we will use the definition of multi-string products given in <cit.>, which is based on the construction of fundamental vertices using SL(2,ℂ) maps. The reason why this construction is particularly useful is that it enables us to set the multi-string product l_1,0 to zero, as demonstrated in <cit.>. Indeed, in such a setup, the interior of the moduli space, associated with the amplitude with two closed-string punctures on the disk, is fully covered through Feynman diagrams given by fundamental vertices arising at lower order in string perturbation theory. In other words, the SL(2,ℂ) products turn out to satisfy the corresponding homotopy relation without l_1,0, namely
        
            l_2l_0,0+l_0,1m_1,0=0 .
        
        Let us then explicitly list the string products which appear in (<ref>), (<ref>) and (<ref>). These were defined in <cit.> as
        
            | l_2(Φ_1,Φ_2)⟩   = b_0^-δ(L_0^-)f_1∘Φ_1(0,0)f_2∘Φ_2(0,0)| 0⟩_ SL(2,ℂ) ,
            | l_0,0⟩   =1/(2π i)^2λ_ b^-L_0^+ B_0⟩⟩ ,
            
                (-)^d(Φ)| m_1,0(Φ)⟩   =1/2π i[m∘Φ(0,0) ]| 0⟩_ SL(2,ℝ) ,
        
        where λ_ b>1 is a tunable stub parameter and (⋯) means that the bulk-boundary OPE is understood to have been performed. Finally, theSL(2,ℂ) maps f_1(w), f_2(w) and m(w) are defined as
            f_1(w)   =1/λ_ cw-λ_ c/3λ_ c+w=-f_2(w) ,
            
                m(w)   =i/λ_ o1+w/β_2/1+w/β_1 ,
        
        in which λ_ c>1 is a free stub parameter whereas λ_ o, β_1 and β_2 are fixed by the homotopy relation (<ref>)  as 
        
            λ_ o   =3λ_ bλ_ c+1/3λ_ bλ_ c-1 ,
            β_1   =3λ_ bλ_ c+1/3λ_ bλ_ c-1λ_ c ,
            β_2   =3λ_ bλ_ c-1/3λ_ bλ_ c+1λ_ c .
        
        Notice that (<ref>) gives a relation between the open-string and closed-string stub parameters, which is explicitly showing how the open-closed SFT naturally interpolates between the two limits where the dynamics is mostly dominated by the open and closed strings, respectively <cit.>. Indeed, the large closed-string stub limit λ_ c→∞ implies the short open-string stub limit λ_ o→𝒪(1), which means that the moduli space will be mostly covered by open-string propagators and closed-string fundamental vertex regions. Conversely, if λ_ o→∞, then λ_ c→𝒪(1), so that the moduli space is mostly covered by closed-string propagators and open-string fundamental vertices. As we have advertised in the previous sections, we will focus on the former limit.
        
        Given this particular choice of SFT vertices, we can further simplify the expression (<ref>) by setting l_1,0=0, namely
        
            1/2π^2[g^∗/g(g_s^∗/g_s)^-1-1]= 1/g (Φ^∗(y),l_0,0)+1/21/g (Ψ^∗(y),m_1,0(Φ^∗(y)))+𝒪(y^3) .
        Substituting the explicit expressions (<ref>) and (<ref>) for the classical solutions Φ^∗(y) and Ψ^∗(y) into the r.h.s. of (<ref>) and carefully collecting all contributions, we obtain
            1/2π^2[g^∗/g(g_s^∗/g_s)^-1-1]=
               =[2y/C_𝕍𝕍𝕍+( A^ f.p._TTTT/3 C_𝕍𝕍𝕍+ C_𝕍𝕍𝕍log(3/2λ_ c))(2y/C_𝕍𝕍𝕍)^2]  A^ disk_T-1/2(2y/C_𝕍𝕍𝕍)^2  A^ disk_TT+O(y^3) ,
        
        where A^ disk_T and A^ disk_TT are the zero-momentum one- and two-point amplitudes of the nearly on-shell tachyon cc̅𝕍 on the disk, which are rescaled by the ratio of the initial g-function and the string coupling constant. That is to say, we define
            A^ disk_T   1/g (cc𝕍, l_0,0) ,
             A^ disk_TT   1/g (b_0^+/L_0^+Pl_2( cc𝕍, cc𝕍),l_0,0)+1/g (b_0/L_0Pm_1,0( cc𝕍),m_1,0( cc𝕍)) .
        
        Looking at (<ref>), it would naively appear that the on-shell disk action could depend on the stub parameter λ_ c. This would stand in contradiction with the interpretation of the on-shell disk action as an observable quantity because as such, it should not depend on the choice of the local coordinate maps.
        Therefore, it will be crucial to prove that, after careful evaluation of the two amplitudes A^ disk_T and A^ disk_TT, all dependence on λ_c drops out, so that one ends up with an expression that is independent of the off-shell SFT data. In the following subsection, we will explicitly compute these two amplitudes in the large closed-string stub limit λ_c→∞.
        
         §.§ Computing  TEXT and  TEXT
        
        Let us begin by calculating the amplitude A^ disk_T of one tachyon on the disk. By substituting into (<ref>) the explicit expression (<ref>) for the closed-string tadpole and rewriting the symplectic form as the BPZ inner product, we obtain
        
            A^ disk_T=-1/4π^2 g⟨  cc𝕍(0,0)| c_0^-λ_ b^-L_0^+ B_0⟩⟩ 
                 =1/4π^2B_𝕍1/gλ_ b^2y,
        
        where in the second equality, we acted with the BPZ-even operator λ_ b^L_0^+ on the bra and we also computed the ghost correlator
        
            ⟨ cc̅(0,0̅)| c_0^- B_ghost⟩⟩=1/8⟨ c(i)c(-i)( c(i)- c(-i)) ⟩_ UHP=-1 .
        
        As far as the matter contribution is concerned, we only have to deal with the one-point function on the disk ⟨𝕍(0,0) B_0⟩⟩, which is identically equal to B_𝕍1. Finally, expanding the stub contribution up to first order in y, we end up with the expression
            A^ disk_T=1/4 π^2[B_𝕍1/g+(1/2B_𝕍1/gC_𝕍𝕍𝕍log(λ_ b))(2y/C_𝕍𝕍𝕍)]+𝒪(y^2) ,
        which is exact up to linear order in y. This precision is sufficient as we recall that 𝒜_T^disk is already multiplied by y in the expression (<ref>) for the on-shell disk action.Second, let us deal with the amplitudeA^ disk_TTwhich was defined in(<ref>). In this case, we will only be interested in the leading order 𝒪(y^0) because in  (<ref>), A^ disk_TT appears already multiplied by y^2. Focusing first on the closed-string exchange, we can write
            1/g (b_0^+/L_0^+Pl_2( cc𝕍, cc𝕍),l_0,0)= 
                =1/g⟨ l_0,0| c_0^-b_0^+/L_0^+P| l_2( cc𝕍(0,0), cc𝕍(0,0))⟩
                =-1/4 π^2 g⟨⟨ B_0 c_0^-λ_ b^-L_0^+b_0^+/L_0^+Pb_0^-c_0^-| l_2( cc𝕍(0,0), cc𝕍(0,0))⟩
                =-1/4 π^2 g⟨⟨ B_0 c_0^-λ_ b^-L_0^+|ξ_i⟩⟨ξ_i^c|b_0^+/L_0^+Pb_0^-|ξ_j^c⟩⟨ξ_j| c_0^-| l_2( cc𝕍(0,0), cc𝕍(0,0))⟩ .
        
        Here, in the second equality, we wrote out the closed-string tadpole in its explicit form (<ref>) and we also made manifest the projection onto the b_0^- exact states in the intermediate channel. Finally, in the last equality, we made two insertions of the identity operator 1_ H_ c=|ξ_i⟩⟨ξ_i^c|=|ξ_i^c⟩⟨ξ_i| on the Hilbert space H_ c of the level-matched closed-string states (where sums over repeated indices are understood). The basis vectors |ξ_i⟩ and their BPZ duals |ξ_i^c⟩ are normalized so that ⟨ξ_i^c|ξ_j⟩=⟨ξ_i|ξ_j^c⟩=δ_ij. 
        
        Following the argument described in <cit.>, thanks to the projector P̅ in front of the closed-string propagator, the only contribution received by (<ref>) in the large stub limit comes from the propagation of the universal tachyon|ξ_j⟩=|ξ_i⟩=| c_1c_1⟩.Hence, the part (<ref>) of the amplitude A^ disk_TT associated with the closed-string exchange can be rewritten as
            1/g (b_0^+/L_0^+Pl_2( cc𝕍, cc𝕍),l_0,0)=
               =-1/4 π^2 g⟨⟨ B_0 c_0^-λ_ b^-L_0^+| c_1c_1⟩⟨ c_0c_0c_1c_1|b_0^+/L_0^+Pb_0^-| c_0c_0c_1c_1⟩⟨ c_1c_1| c_0^-| l_2( cc𝕍(0,0), cc𝕍(0,0))⟩ .
        Let us now separately evaluate the three factors on the r.h.s. of (<ref>). The computation of the first one is entirely analogous to what was done in (<ref>)provided that we replace 𝕍 with the identity in the matter sector. We then obtain
            -1/4 π^2 g ⟨⟨ B_0 c_0^-λ_ b^-L_0^+| c_1c_1⟩=1/4 π^2  λ_ b^2 .
        
        Regarding the second factor, we trivially obtain
        
            ⟨ c_0c_0c_1c_1|b_0^+/L_0^+Pb_0^-| c_0c_0c_1c_1⟩=1 ,
        while the third factor can be computed using (<ref>) and (<ref>) as
        
            ⟨ c_1c_1| c_0^-| l_2( cc𝕍(0,0), cc𝕍(0,0))⟩   =⟨ c_1c_1| c_0^-b_0^-δ(L_0^-)| f_1∘  cc𝕍(0,0)f_2∘  cc𝕍(0,0)⟩
               =⟨ I_ c∘  cc(0,0)  f_1∘  cc𝕍(0,0)  f_2∘  cc𝕍(0,0)⟩_ℂ
               =-(3/2λ_ c)^2+𝒪(y) ,
        
        where I_ c(w)=1/wdenotes the closed-string BPZ inversion. Putting all these results together, we get
        
            1/g (b_0^+/L_0^+Pl_2( cc𝕍, cc𝕍),l_0,0)=-1/4π^2 (3/2λ_ cλ_ b)^2+𝒪(y)
        for the closed-string channel.Let us now focus on computing the channel of A^ disk_TT which is associated with the open-string exchange (second term in (<ref>)). Substituting the explicit form (<ref>) of the open-closed product m_1,0, we can first express
            1/g (b_0/L_0Pm_1,0( cc𝕍),m_1,0( cc𝕍))=
               = -1/4 π^2 g∫_0^1dt/tt^ϵ_o⟨ I_ o∘ m ∘  cc𝕍(0,0)  b_0 t^L_0  m ∘  cc𝕍(0,0)⟩_ UHP|_ϵ_o→ 0 ,
        
        where we wrote the Siegel propagator in the Schwinger representation and introduced the open-string BPZ inversion I_o(w)=-1/w. Furthermore, we introduced the factort^ϵ_o to regulate the divergences which generally arise near the open-string degeneration t→ 0 due to propagation of open-string tachyons. In our particular case, the only divergence will come from the identity channel in the bulk-boundary OPE of the matter field 𝕍. Also notice that on the r.h.s. of (<ref>), we have dropped the projector P̅. This is possible by realizing that
        our assumption (<ref>) guarantees that at the order 𝒪(y^0), no states in the image of P propagate in the open-string channel of the amplitude A^ disk_TT so that the P̅ insertion in front of the open-string propagator on the l.h.s. of (<ref>) becomes indistinguishable from the identity. If (<ref>) were not assumed, the states in the image of P propagating in the open-string channel would have to be removed by hand.
        With these provisions in mind, we can implement the action of the conformal maps (<ref>) on the r.h.s. of (<ref>) and compute the ghost-part of the correlator. Keeping only the contributions at the leading order in y, we eventually get
        
            1/g (b_0/L_0Pm_1,0( cc𝕍),m_1,0( cc𝕍)) 
                   =-1/4 π^2 g∫_0^1/λ_ o^2ds s^ϵ_o  4(s^2-1)⟨𝕍(i,i)  𝕍(is,is)⟩_ UHP|_ϵ_o→ 0 ,
        where the details can be checked by following an analogous computation which was presented in <cit.>.
        
          §.§.§ Open and closed degenerations in TEXT
        At this point, we will focus on making explicit the terms on the r.h.s. of (<ref>) which diverge as we remove the regulators λ_o and ϵ_o, putting particular emphasis on the regime of closed-string degeneration. This regime can be achieved as the Schwinger parameter s approaches 1 and therefore becomes available as we send λ_o→ 1. As showcased by the relation (<ref>), this is equivalent to the limit λ_bλ_c→∞ in which the closed-string stub parameters become large. By subtracting and adding the divergent contributions, we will be able to isolate the explicit dependence of (<ref>) on the stub parameters λ_b, λ_c. This will eventually facilitate the proof that the value of the on-shell disk action does not depend on the SFT data.
        Let us start by discussing the open-string degeneration s→ 0. In this limit, the operator insertion 𝕍(is,-is) in (<ref>)
        can be replaced by the corresponding bulk-boundary OPE, namely
        
            𝕍(is,-is)s→ 0∼1/(2s)^2B_𝕍1/g+1/2sB_𝕍v/gv(0)+ reg .
        Notice that the second term of the above relation would have been annihilated by the P̅ appearing in front of the Siegel-gauge propagator on the l.h.s. of (<ref>) and as such, it would have to be removed by hand when evaluating the open-string channel of the amplitude. However, as per our assumption (<ref>), we have B_𝕍v/g∼𝒪(y), so that at the order 𝒪(y^0), this is taken care of automatically. As a result, the only divergence arising is the one due to the identity channel, which can be regulated through the parameter ϵ_o in (<ref>). Consequently, in the limit s→ 0, one can isolate the divergent contribution to the correlator as
            ⟨𝕍(i,i̅) 𝕍(is,i̅s̅)⟩_ UHPs→ 0∼1/(2s)^2B_𝕍1/g⟨𝕍(i,i) ⟩_ UHP=1/(4s)^2(B_𝕍1)^2/g .
        For the sake of later convenience, let us also introduce an explicit notation
            div_ o(s)1/(4s)^2(B_𝕍1)^2/g
        for the open-string divergence in the 2-point correlator of 𝕍. 
        
        
        
        
        
        
        
        The closed string degeneration occurs for s → 1, and it is therefore regulated through the stub parameter λ_ o^-2. This regulator can be lifted by taking the limit λ_ c→∞ which, according to (<ref>), corresponds to λ_ o→ 1. As the Schwinger parameter s approaches 1, we observe a collision between the two bulk fields and, therefore, we can read off the divergent contribution of the correlator to the amplitude A^ disk_TT by using the bulk OPE
        
            ⟨𝕍(i,i̅) 𝕍(is,i̅s̅)⟩_ UHPs→ 1∼1/(1-s)^4 g+ 1/4(1-s)^2 C_𝕍𝕍𝕍B_𝕍1 .
        Note that here we are ignoring the simple pole in 1-s because thanks to the c-ghost measure 4(s^2-1), this gives rise to a finite contribution to the amplitude.In an exact analogy with the open-string channel, we also introduce the notation
        
            div_ c(s)1/(1-s)^4g+ 1/4(1-s)^2C_𝕍𝕍𝕍B_𝕍1
        
        for the closed-string divergence.
        
        Let us then subtract and add (<ref>) and (<ref>) to the 2-point correlator of 𝕍 when evaluating the integral over s in (<ref>). This gives
        
            1/g (b_0/L_0Pm_1,0( cc𝕍),m_1,0( cc𝕍)) =
               =-1/4 π^2  A^ f.p._TT -1/4 π^2 g∫_0^1/λ_ o^2ds s^ϵ_o 4(s^2-1)( div_ o(s)+ div_ c(s))|_ϵ_o→ 0 ,
        
        where we define  a finite contribution
            A^ f.p._TT1/g∫_0^1ds  4(s^2-1)[⟨𝕍(i,i) 𝕍(is,is)⟩_ UHP- div_ o(s)- div_ c(s)] ,
        to the amplitude A^ disk_TT, in which all regulators can be removed as we are subtracting all sources of divergent behaviour. The value of A^ f.p._TT will of course depend on the details of the particular 2d CFT in which the short RG flow is considered.
        On the other hand, the second term in
        (<ref>) can be evaluated as
            -1/4 π^2 g∫_0^1/λ_ o^2ds s^ϵ_o 4(s^2-1)[ div_ o(s)+ div_ c(s)]|_ϵ_o→ 0=
               =-1/4 π^2 g∫_0^1/λ_ o^2ds s^ϵ_o 4(s^2-1)[1/(4s)^2(B_𝕍1)^2/g+1/(1-s)^4g+ 1/4(1-s)^2C_𝕍𝕍𝕍B_𝕍1]|_ϵ_o→ 0
               =-1/4π^2[1/2(B_𝕍1/g)^2+1/2+C_𝕍𝕍𝕍B_𝕍1/g(1+2log(2)-2log(3/2λ_ bλ_ c))-(3/2λ_ bλ_ c)^2] .
        
        Notice that the last term in the square brackets of (<ref>) exactly cancels out the closed-string channel contribution (<ref>). 
        
        In total, the amplitude A^ disk_TT of two nearly on-shell tachyons on the disk at the order 𝒪(y^0) becomes
        
            A^ disk_TT=-1/4 π^2  A^ f.p._TT-1/4π^2[1/2(B_𝕍1/g)^2+1/2+C_𝕍𝕍𝕍B_𝕍1/g(1+2log(2)-2log(3/2λ_ bλ_ c))] .
        
        We observe that all remaining dependence of A^ disk_TT on the off-shell SFT data is encapsulated in the logarithmic term
        
            1/2π^2 C_𝕍𝕍𝕍B_𝕍1/glog(3/2λ_ bλ_ c) .
        
        
          §.§.§ The final formula for the on-shell disk action
        At this point, we have in place all the quantities needed for writing down the on-shell disk action in terms of CFT data. By substituting the results (<ref>) and (<ref>) for the amplitudes A^ disk_T and A^ disk_TT
        into the expression (<ref>) for the disk action, 
        we can explicitly see that the stub-dependent part (<ref>) of the amplitude A^ disk_TT exactly cancels with the log(3/2λ_ c) term in the prefactor of A^ disk_T in (<ref>) coming from the closed-string solution (<ref>), as well as with the log(λ_ b) term in the result (<ref>) for A^ disk_T.We therefore obtain the following expression for the on-shell open-closed SFT disk action
            g^∗/g(g_s^∗/g_s)^-1=1+ B_𝕍1/g(y/C_𝕍𝕍𝕍)+
              (B_𝕍1/g2 A^ f.p._TTTT/3 C_𝕍𝕍𝕍+ A^ f.p._TT)(y/C_𝕍𝕍𝕍)^2+𝒪(y^3) ,
        where the shifted finite part A^ f.p._TT is defined as
            A^ f.p._TT A^ f.p._TT+1/2(B_𝕍1/g)^2+1/2+C_𝕍𝕍𝕍B_𝕍1/g(1+2log(2)) .
        This result is manifestly independent of any off-shell SFT data and is expressed purely in terms of 2d CFT quantities.
        
        § EXAMPLE: VIRASORO MINIMAL MODELS
        In this section, we will present an application of the above-described construction of open-closed SFT solutions
        
        to the explicit example of Virasoro minimal models at c close to 1. From the pure 2d CFT point of view, this setup has already been famously analyzed in the case of pure-bulk RG flows by Zamolodchikov in <cit.> and, in the case of pure-boundary perturbations, by Recknagel, Roggenkamp and Schomerus in <cit.>. Here, our aim will be to showcase how open-closed string field theory can be conveniently used to evaluate the response of conformal boundaries to the Zamolodchikov perturbation in the bulk, to the order which would correspond to a two-loop calculation in conformal perturbation theory. Apart from confirming the conjectured flows of <cit.> at subleading order in perturbation theory, we will also show that the classical SFT solution describing the Zamolodchikov bulk RG flow changes the string coupling constant at a rate which is given by the same formula (in terms of the sphere 2-point function of the deforming operator) as in the case of the Narain marginal deformations discussed in section <ref>.
        
         §.§ Review of bulk and boundary data
        As we have advertised in the introductory paragraph, in this section we will assume that the matter sector of the worldhseet CFT contains a factor given by a unitary A-series Virasoro minimal model. These are 2d CFTs arising at a central charge 0<c<1 which can be discretely parametrized as
            c_m=1-6/m(m+1) ,
        where m=3,4,5,… For such values of c, the allowed highest weight representations of the Virasoro algebra can, in turn, be parametrized by the Kac labels(r,s) where 1≤ r ≤ m-1 and 1≤ s ≤ m, with the identification (r,s)∼ (m-r,m+1-s). The corresponding weights h_(r,s) are given as
        
            h_(r,s)=[(m+1)r-ms]^2-1/4m(m+1) .
        As we take m→∞, the leading contribution to the weights is determined purely by |r-s|, as we can write h_(r,s) = 1/4(r-s)^2+𝒪(1/m).In the bulk, the highest-weight representations (r,s) give rise to the (diagonal) primary fields ϕ_(r,s) with scaling dimensions Δ_(r,s)=2h_(r,s)= 1/2(r-s)^2+𝒪(1/m). 
        
        
        
        
        
        In particular, to implement a short RG flow in the bulk, one can study the relevant perturbation induced by the bulk field𝕍=ϕ_(1,3). Indeed, since its scaling dimension Δ_(1,3) reads 
        
            Δ_(1,3)=2-4/m+1 2(1-y) ,
        we can see that ϕ_(1,3) approaches marginality as m→∞ (or, equivalently, as c_m→ 1). The expansion parameter y of the corresponding short RG flow (which measures the failure of the relevant perturbing operator 𝕍 to be marginal) can thus be identified as
            y=2/m+1 .
        Furthermore, it can be shown that the representation (1,3) satisfies the fusion algebra
            (1,3)× (1,3) = (1,1)+(1,3)+(1,5) .
        Since the representation (1,5) gives rise to an irrelevant bulk field (with scaling dimension Δ_(1,5)=8+𝒪(y)), this implies that 
        the nearly marginal bulk field ϕ_(1,3) satisfies a fusion rule of the type (<ref>), which is a necessary condition for the existence of a short RG flow.Additionally, normalizing the two-point function coefficients of all bulk primaries to unity, the three-point function coefficient C_(1,3)(1,3)(1,3)(m) of three ϕ_(1,3) bulk fields 
        turns out to be <cit.>
            C_(1,3)(1,3)(1,3)(m) ≡C_𝕍𝕍𝕍 = 4/√(3)(1 - 3/2y + 𝒪(y^2))
        
        in the small y limit (or, equivalently, in the large m limit by using (<ref>)). Since this is non-zero, all conditions for the existence of a nearby fixed point of the RG flow triggered by ϕ_(1,3) are met.As we have mentioned earlier, in the pure-bulk case, this RG flow was described already by Zamolodchikov in <cit.>. His analysis showed that flowing from the m^ th minimal model in the UV, one ends up with the (m-1)^ th minimal model in the IR. This results in a change in the central charge
            c^∗-c=-3/2y^3+𝒪(y^4) ,
        which starts at cubic order in y.
        
        Regarding the open-string background, we will assume that the worldsheet matter boundary state factorizes into a minimal-model boundary state and some boundary state of the auxiliary CFT. The most general elementary boundary state in a diagonal Virasoro minimal model can be labeled by two integers a(a_1, a_2), which run over the same range as the Kac labels (r, s) of highest-weight representations and are subject to the same identifications. As argued by Cardy <cit.>, these boundary states can be expanded as
        
            a⟩⟩ _m= ∑_(r,s)S_(a_1,a_2)^(r,s)/√(S_(1,1)^(r,s)) |r,s ⟩⟩ ,
        where the Ishibashi states |r, s ⟩⟩<cit.> are normalized as lim_τ→∞⟨⟨ r, s |e^-2πτ(L_0+L̅_0-Δ_(r,s)) | r', s' ⟩⟩ = δ_r,r'δ_s,s', and S_(a_1,a_2)^(r,s) are the components of the modular S-matrix 
        
            S_(a_1,a_2)^(r,s)(m)=(-1)^1+a_1 s+a_2 r√(8/m(m+1))sin(m+1/mπ a_1 r)sin(m/m+1π a_2 s) .
        To diagnose the effect of a short bulk RG flow on a boundary state, it will be crucial to investigate the structure of the bulk-boundary OPE of the perturbing bulk operator. In particular, for the ϕ_(1,3) bulk field in the presence of a general Cardy boundary state a, we can write
            ϕ_(1,3)(x+is,x-is)s→ 0∼1/(2s)^2(1-2/m+1)B^(a)_(1,3)(1,1)(m)/g^(a)(m)+1/(2s)^1-2/m+1B^(a)_(1,3)(1,3)(m)/g^(a)(m) ψ_(1,3)(x)+ reg .
        where ψ_(1,3)v is the boundary field transforming in the (1,3) representation of Virasoro algebra.Here g^(a) denotes the g-function (boundary entropy) of the boundary state, which can be identified with the coefficient in the expansion (<ref>) in front of the (1,1) (identity) Ishibashi state. This means that we can write
            g^(a)(m)⟨ 0 a⟩⟩_m=S_(a_1,a_2)^(1,1)(m)/√(S_(1,1)^(1,1)(m)) .
        Second, the coefficient B^(a)_(1,3)(1,1) of the identity channel in the bulk-boundary OPE (<ref>) can be equated with the coefficient of the (1,3) Ishibashi states in (<ref>), meaning that in terms of the modular S-matrix, we can write
            B^(a)_(1,3)(1,1)(m)/g^(a)(m)⟨ϕ_(1,3)a⟩⟩_m/g^(a)(m)=√(S_(1,1)^(1,1)(m)/S_(1,1)^(1,3)(m))S_(a_1,a_2)^(1,3)(m)/S_(a_1,a_2)^(1,1)(m) .
        Finally, the OPE coefficient B^(a)_(1,3)(1,3)(m) in front of ψ_(1,3) can generally be computed using the 2-point function bootstrap on the upper-half plane. For arbitrary m, this was achieved in <cit.>, where all structure constants of the A-series of Virasoro minimal models were expressed in terms of the fusing matrix 𝖥 and the modular S-matrix. In appendix <ref>, we derive the corresponding crossing equation in the strict m→∞ limit (see (<ref>)), which suffices for our needs in this paper. 
        In particular, note that for the boundary states of the type a=(1,a_2), the representation (1,3) is actually not present in the spectrum of boundary fields, as can be explicitly checked using the Cardy condition. In such cases, the coefficient B^(a)_(1,3)(1,3)(m) identically vanishes to all orders in y.
        
        More generally, we will be interested in such boundary states a for which the ratio B^(a)_(1,3)(1,3)/g^(a) vanishes at the leading order 𝒪(y^0). Recalling our discussion in section <ref>, this will guarantee perturbativity of the boundary deformation induced by the bulk RG flow triggered by ϕ_(1,3). As analyzed in more detail in appendix <ref>, setting B^(a)_(1,3)(1,3) to zero in the crossing relation (<ref>) yields two distinct solutions for the coefficient B^(a)_(1,3)(1,1) which, in turn, can be associated with two distinct classes of boundary states for which the bulk-induced boundary deformation remains perturbative. These classes will be described in more detail below.
        
        
         §.§ Short RG flows in Virasoro minimal models with boundaries
        
        In this subsection, we will briefly review the exhaustive analysis of <cit.>, where the authors study the RG flows in the Virasoro minimal models with boundaries, which are induced by the nearly-marginal bulk field ϕ_(1,3), as well as by the nearly-marginal boundary field ψ_(1,3) in the limit of large m. 
        One of the main results of their work is a diagram which we reproduce in figure <ref> and which summarizes the possible RG flows in this setup. In particular, the upper horizontal line in this diagram refers to the minimal model m, while the lower horizontal line refers to the minimal model m-1. This means that the horizontal arrows indicate pure-boundary RG flows while the vertical ones denote RG flows induced by the bulk field ϕ_(1,3).
        We can notice that the RG flow chain naturally organizes itself horizontally so that the value of a_2 increases from right to left. On the other hand, by changing the value of a_1, one would generally produce a separate chain.In summary, starting from a boundary state with labels (a_1,a_2) in minimal model m, one can identify essentially two distinct sequences of deformations. 
        
        The first one connects the initial boundary state (a_1,a_2) with the fixed points (I), (II) and (III) (see figure <ref>). This proceeds 1. via a pure-boundary RG flow triggered by ψ_(1,3) in the minimal model m which takes us from (a_1,a_2) to the fixed point (I), then 2. via a pure-bulk flow induced by ϕ_(1,3) which is interpolating between the fixed points (I) and (II) and then finally, 3. via a pure-boundary flow, again triggered by ψ_(1,3) but this time in the minimal model m-1, mapping between the fixed points (II) and (III). Identifying the fixed points with specific boundary states in the two respective minimal models, we can write this sequence of deformations as
            (a_1, a_2)_m→⊕_l=1^ min(a_1,a_2) (a_1+a_2+1-2l,1)_m→⊕_l=1^ min(a_1,a_2) (1,a_1+a_2+1-2l)_m-1→(a_2, a_1)_m-1 .
        Crucially, for values a_1 and a_2 which are kept fixed as we take the limit m→∞, this sequence of deformations is perturbative. Correspondingly, in figure <ref> it is highlighted in red color. In more detail, to explicitly confirm perturbativity of the sequence (<ref>), we can use (<ref>) to compute the corresponding change in the g-function. This gives
            g^(a_2,a_1)(m-1)/g^(a_1,a_2)(m)=√(S_(1,1)^(1,1)(m)/S_(1,1)^(1,1)(m-1))S_(a_2,a_1)^(1,1)(m-1)/S_(a_1,a_2)^(1,1)(m)=1+3y/4+21y^2/32+𝒪(y^3) ,
        which goes to 1 as y approaches 0, consistently with our statement. Also notice that since, we are interested in computing the variation of the g-function only up to second order in y, our analysis will not be sensitive to those steps in the sequence (<ref>) which consist of pure-boundary flows, namely steps 1 and 3, which gives contribution at 𝒪(y^3). From the point of view of constructing the corresponding SFT solutions, these two steps would be reflected by turning on a perturbative open-string solution corresponding to a short boundary RG flow both at the level of the initial perturbative background around the boundary state (a_1,a_2)_m, as well as at the level of the small fluctuations around the vacuum-shift solution (<ref>) which describes the fixed point (II).Hence, to the order in y up to which we are working, the on-shell disk action of the open-closed SFT solution computed in section <ref> should encode the difference of g-functions associated with the deformation
            (a_1, a_2)_m→ (a_2, a_1)_m-1 ,
        which can be viewed as a streamlined version of the step-by-step deformation (<ref>). As we have advertised in the previous section, this class of perturbative deformations is associated with one of the two solutions (<ref>) for B^(a)_(1,3)(1,1)/g^(a) of the crossing equation (<ref>) where we put B^(a)_(1,3)(1,3) to be 𝒪(y): indeed, for arbitrary labels a_1 and a_2, as long as we keep them fixed as we send m→∞, it follows from (<ref>) that B^(a)_(1,3)(1,1)/g^(a)=√(3)+𝒪(y^2).
        The second type of deformation in this setup brings us from the boundary state (a_1,a_2)_m to the fixed point (IV). 
        As we can see on figure <ref>, this can proceed either via the fixed points (I) and (II) (from where a pure-boundary flow triggered by ψ_(1,3) with the opposite sign must be followed), or, via the fixed points (I') and (II') by first triggering a pure-boundary flow by -ψ_(1,3) in the minimal model m, then following a pure-bulk flow induced by ϕ_(1,3) and then finally, riding down yet another pure-boundary flow in the minimal model m-1. One can also follow an intermediate path by simultaneously turning a combination of bulk and boundary deformations starting from (a_1,a_2)_m. Altogether, this gives us the flow
            (a_1,a_2)_m→(a_2-1,a_1)_m-1 ,
        which, if we keep the labels a_1 and a_2 fixed as we take m→∞, is non-perturbative. Indeed, in this case we have
         
            g^(a_2-1,a_1)(m-1)/g^(a_1,a_2)(m)=√(S_(1,1)^(1,1)(m)/S_(1,1)^(1,1)(m-1))S_(a_2-1,a_1)^(1,1)(m-1)/S_(a_1,a_2)^(1,1)(m)=1-1/a_2+3(a_2-1)y/4a_2+𝒪(y^2) ,
        which does not approach 1 in the small y limit. On the other hand, near the middle of the chain displayed in figure <ref>, specifically for a_2=m+1/2+α, where we are fixing α as we take m→∞, perturbativity of the flow (<ref>) is achieved, as can be seen by analyzing the g-functions
            g^((m-1)/2+α, a_1)(m-1)/g^(a_1, (m+1)/2+α)(m)=√(S_(1,1)^(1,1)(m)/S_(1,1)^(1,1)(m-1))S_((m-1)/2+α,a_1)^(1,1)(m-1)/S_(a_1,(m+1)/2+α)^(1,1)(m)=1-y/4-3y^2/32+𝒪(y^3) .
        Therefore, near the middle of the chain, a second independent short RG flow
            (a_1,m+12+α)_m→(m-12+α,a_1)_m-1
        becomes available. It can be readily checked using (<ref>) that for such values of the labels a_1 and a_2, the bulk-boundary OPE coefficient of the bulk field ϕ_(1,3) in front of the boundary identity channel becomes B^(a)_(1,3)(1,1)/g^(a)=-1/√(3)+𝒪(y^2), which can be identified with the second solution of the crossing equation (<ref>).
        
        
         §.§ SFT analysis
        Our final goal in this paper will be to confirm the two independent short RG flows (<ref>) and (<ref>) up to second order in y by computing the change in the g-functions using the SFT framework which we described in great detail in sections <ref> and <ref>. In conformal perturbation theory, such a calculation would require going to the two-loop level. On the other hand, in our SFT scheme, one only needs to compute manifestly finite one-dimensional integrals, as it is apparent from the r.h.s. of the final formula (<ref>) for the on-shell disk action of the SFT solution describing short bulk RG flows in the presence of conformal boundaries. Also recall that the quantity which the on-shell SFT disk action computes is really the difference in the worldsheet disk partition functions between the corresponding two string backgrounds. As such, it is generally sensitive not only to a change in the boundary-state g-function, but also to a possible change in the value of the string coupling constant g_s. This is clearly manifested on the l.h.s. of (<ref>). Thus, we need to devise a method of separating these two pieces of information. As in the case of the Narain deformations of compact free bosons which we discussed in section <ref>, our strategy will be to show that for both of the aforementioned classes (<ref>) and (<ref>) of short RG flows in Virasoro minimal models, the SFT on-shell disk action (<ref>) can indeed be identified with the ratio g^∗/g of the g-functions corresponding to the endpoints of the flows times a universal factor, which is the same for both classes of flows (thus independent of the open-string solution Ψ^∗(y) and only sensitive to the closed-string solution Φ^∗(y)). This universal factor will then be identified with the ratio (g^∗_s/g_s)^-1. Such a strategy is equivalent to assuming that the change in the g-function is known for one of the flows and using this knowledge to fix the change in the string coupling constant. Then the SFT on-shell disk action predicts the change in the g-function for the second flow and thus serves as an indepedent check of the (conjectured) RG flow.
        Let us start by quantifying various ingredients needed for evaluation of the on-shell disk action, as given by the final result (<ref>) of section <ref>. Recall that the main non-trivial ingredient entering the subleading order of this calculation was the finite part A^ f.p._TT of the two-point disk amplitude of two slightly off-shell tachyons, introduced in (<ref>).
        
        
        
        
        Given the analysis of appendix <ref>, we can now substitute the explicit expression for the regulated  correlator (<ref>) of two bulk fields ϕ_(1,3) on the UHP, obtaining
            A^ f.p._TT=-1- B_𝕍1/g C_𝕍𝕍𝕍(1+2log 2).
        At the same time, we recall the result A^ f.p._TTTT=-8 of <cit.> for the finite part of the sphere amplitude of four slightly off-shell tachyons. Finally, we also remember the y-expansion (<ref>) of the three-point structure constant C_(1,3)(1,3)(1,3), as well as the the observation that in the case of both flows (<ref>) and (<ref>), the m→∞ limit of the bulk-boundary coefficient B^(a)_(1,3)(1,1)/g^(a) only starts receiving finite m corrections at the order 𝒪(y^2).
         In view of these results, the general formula  (<ref>) becomes
        
            g^∗/g(g_s^∗/g_s)^-1=
             
             
             
             
             
             
             
             
             
             
              (1+√(3)/4B_𝕍1/g y
              +3/8B_𝕍1/g(1/√(3)+1/4B_𝕍1/g) y^2 +𝒪(y^3))( 1+ 3y^2/32+𝒪(y^3))^-1,
        where, at this point, the ratio B_𝕍1/g can be replaced with its strict m→∞ limit. Also, we have rearranged the expression so that the 𝒪(y^2) term which does not depend on the boundary conditions is explicitly factored away from the rest, in a hope that we will be able to attribute this universal factor to the ratio (g_s^∗/g_s)^-1 of the old and the new string coupling constant. We will now provide an unambiguous check for this expectation by showing that the first factor on the r.h.s. of (<ref>) precisely reproduces the ratio g^∗/g of the g-functions of the endpoints of the conjectured flows (<ref>) and (<ref>).
        Considering first the flow (<ref>), which starts from the boundary state (a_1,a_2)_m with both a_1,a_2 fixed as we take m→∞, we can substitute
            B_𝕍1/g→B^(a_1,1)_(1,3)(1,1)(m)/g^(a_1,1)(m)=√(S_(1,1)^(1,1)(m)/S_(1,1)^(1,3)(m))S_(a_1,1)^(1,3)(m)/S_(a_1,1)^(1,1)(m)=√(3)+𝒪(y^2)
        to show that the first factor in (<ref>) evaluates to
            1+3y/4+21y^2/32+𝒪(y^3) ,
        which stands in precise agreement with the large-m expansion (<ref>) of the ratio g^(a_2,a_1)(m-1)/g^(a_1,a_2)(m) of g-functions for the expected endpoints of the flow (<ref>). Similarly, we can consider the remaining class of flows (<ref>), for which the UV BCFT is given by the boundary state (a_1,a_2)_m with a_2 = m+1/2+α with both a_1 and α kept fixed in the limit m→∞. In this case, substituting
            B_𝕍1/g→B^(a_1,(m+1)/2+α)_(1,3)(1,1)(m)/g^(a_1,(m+1)/2+α)(m)=√(S_(1,1)^(1,1)(m)/S_(1,1)^(1,3)(m))S_(a_1,(m+1)/2+α)^(1,3)(m)/S_(a_1,(m+1)/2+α)^(1,1)(m)=-1/√(3)+𝒪(y^2) ,
        into the first factor in (<ref>) yields
            1-y/4-3y^2/32+𝒪(y^3) ,
        which exactly matches the large-m expansion (<ref>) of the ratio g^((m-1)/2+α, a_1)(m-1)/g^(a_1, (m+1)/2+α)(m).
        The results (<ref>) and (<ref>) confirm that the value of the SFT on-shell disk action factorizes 1. into the ratio g^∗/g of the g-functions of the initial BCFT and the BCFT describing the classical solution and 2. into the ratio (g_s^∗/g_s)^-1 of the corresponding string coupling constants. At the same time, the latter can be isolated as
        
            g_s^∗/g_s=1+3y^2/32+𝒪(y^3)=1+1/8( 2y/C_𝕍𝕍𝕍)^2+𝒪(y^3)=1+1/8t(y)^2+𝒪(y^3) .
        Finally, we note that the quantity t(y)^2 can be identified with the sphere two-point function of the deforming matter CFT operator t(y) 𝕍. Thus, comparing (<ref>) with (<ref>), we conclude that, at least to the leading order in y, the change in the string coupling constant is described by the same expansion in terms of the sphere two-point function of the deforming operator as in the case of Narain deformations.
        
        § DISCUSSION
        
        
        In this paper we have extended  our understanding of how D-branes adapt to a perturbative change in the closed string background, in the context of (bosonic) open-closed SFT. 
        
        As a first main result we have computed exactly at next-to-leading order the on-shell disk action that is generated by a short bulk RG-flow in the initial matter CFT. As already discussed in <cit.>, this quantity receives contributions both from the closed SFT solution describing the change of the closed string background, as well as from the corresponding open-string vacuum-shift solution.  In quite a full generality, this observable is given by eq. (<ref>). This formula is valid under the assumption that the barely relevant bulk field, which is deforming the closed string background, has a sub-leading bulk-boundary OPE with the corresponding barely-relevant boundary field. This ensures that the main driving force triggering the open-string background deformation is provided by the boundary-identity contribution to the bulk-boundary OPE of the relevant deforming field in the bulk. 
        
        
        Broadening our perspective, the analysis presented in this paper has confirmed and reinforced the observation already made  in <cit.> that, although the bulk deformation happens exclusively at the level of matter CFT, the solution nevertheless changes the string coupling constant, even if the ghost dilaton is not switched on  (at least up to the perturbative order where our analysis is performed). We have now evidence that both in the case of exactly marginal deformations and of short RG flows triggered by a deforming matter field T, there is always a 𝒪(T^2) change in the string coupling constant, which is universally proportional to the sphere two-point function of the deforming bulk field as shown in eq. (<ref>). 
        
        There are several questions which naturally arise.
        
          *  The change in the string coupling constant is a closed-string effect. While we have used D-branes as probes to detect this change, it should be possible to capture it within the framework of the closed SFT alone. Any alternative method to access this quantity in purely closed SFT would thus be welcome. A standard way for accessing it would be the computation of a 3-point amplitude in the deformed background which would however effectively require a 5-point amplitude (two deforming fields and 3 external legs) in the original theory. Perhaps simpler and more accessible quantities exist.
        
          *  The ghost-dilaton D and related excitations are expected to play an important role. To start with, one can envisage correcting the observed change in the string coupling constant induced by a matter deformation by addingto the solution a second order correction proportional to the ghost-dilaton. In addition, in case of relevant bulk RG-flows, the (BPZ-dual) ghost-dilaton fieldD̃ and its trivializing field Θ are needed to cancel the third order obstruction to the solution induced by the change in the central charge. It would be interesting to better understand the role of these fundamental ghost fields, which do not seem to have their counterparts in the pure 2d CFT description.
        
          *  The status of the on-shell closed-string field theory action in presence of non-trivial ghost excitations (as the ones mentioned in the previous point) remains largely unexplored. Ideally, the on-shell closed SFT action of a short RG flow solution could furnish an independent way of measuring a change in the string coupling constant, possibly together with a shift in the matter central charge <cit.>.[This is provided that the on-shell action is not identically vanishing <cit.> thanks to some subtle boundary-term contribution.] It would be exciting to make progress in this direction.
        
          *  At the moment, (open-closed) SFT is the only fully consistent framework where to address these questions. However, recent progress from 
        the point of view of sigma-model deformations has been reported in <cit.>. It would be interesting to see how some of our results and questions get translated into this other approach.
        
        We hope that our paper furnishes a direction which may prove useful for future explorations of these basic aspects of string theory.
        
        
        
        
        
        
        
        
        
        
        § ACKNOWLEDGMENTS
        
        
        
        We thank Marco Meineri, Jaroslav Scheinpflug and Martin Schnabl for useful discussions. We are also grateful to the organizers and participants of the Workshop on Matrix Models and String Field Theory held from May 5 to May 16, 2024 in Benasque, who provided a stimulating environment which enabled completion of the bulk of this work. JV also thanks INFN Turin and CEICO, Institute of Physics, Czech Academy of Sciences for their hospitality during the initial and final stages of this work, respectively. The work of CM  and AR  is partially supported by the MIUR PRIN contract 2020KR4KN2 “String Theory as a bridge between Gauge Theories and Quantum Gravity” and by the INFN project ST&FI “String Theory and Fundamental Interactions”. The work of JV is supported by the ERC Starting Grant 853507.
        
        
        
        § 2-POINT FUNCTION OF 
        TEXT ON THE UHP
        
        
        
        In this appendix, we will compute the large-m limit of the 2-point correlation function
        
            1/g^(a) ⟨ϕ_(1,3)(i,i̅) ϕ_(1,3)(is,i̅s̅)⟩_UHP^(a) ,
        
        where ϕ_(1,3) is the primary with Kac label (1,3) in the unitary Virasoro minimal model m=3,4,… and the parameter s runs from 0 to 1. The conformal boundary conditions along the real line will be specified by a Cardy boundary state with Kac labels a=(a_1,a_2) and boundary entropy g^(a) = ⟨ 1⟩_UHP^(a). To evaluate the correlation function (<ref>), our strategy will be to exploit the existence of a null state at level 3 in the degenerate module (1,3) in order to show that (<ref>) satisfies a third-order differential equation. Using the known values of bulk and boundary OPE coefficients for the A-series of Virasoro minimal models, we will then fix a particular solution to this equation which has the correct factorization properties in the bulk and boundary channels.
        
        Let us start by noting that one can use the doubling trick to rewrite (<ref>) in terms of a chiral 4-point function on the full complex plane
        
            ⟨ϕ_(1,3)(z_1) ϕ_(1,3)(z_2) ϕ_(1,3)(z_3) ϕ_(1,3)(z_4)⟩_ℂ = (z_12z_34)^-2h 𝒢(η) ,
        
        subject to assigning z_1 = i, z_2 = -i, z_3=is, z_4=-is and h=h_(1,3) = m-1/m+1=1-y.
        Note that on the right-hand side of (<ref>), we have made use of global conformal invariance to recast the correlator in terms of a function 
        𝒢(η) of the cross-ratio
        
            η = z_12z_34/z_13z_24 = -4s/(1-s)^2 .
        
        We will see below that sometimes, it may be more convenient to work with a related function 𝒢̃(η̃)≡𝒢(η(η̃)) which is expressed in terms of the redefined parameter
        
            η̃(η) = η^2/1-η .
        
        One can then exploit the existence of the null-state 
        
            [(h+2)L_-3-2L_-1L_-2+1/h+1(L_-1)^3]| ϕ_(1,3)⟩
        
        to infer that the function 𝒢̃(η̃) satisfies the third-order differential equation
        
            0   =(η̃^3+4η̃^2)d^3𝒢̃/dη̃^3+[  (4-2 h) η̃^2+(10-8 h) η̃]d^2𝒢̃/dη̃^2+
               +[(2 - 5 h  - h^2 )η̃+2 - 7 h + 3 h^2  ]d𝒢̃/dη̃+2 h^2 (1 + h) 𝒢̃ .
        
        Equation (<ref>) admits a solution for 𝒢̃(η̃) which, for general h, can be explicitly written as a linear combination of hypergeometric functions of the type _3 F_2 in the variable -η̃/4. However, as we are really interested only in the leading contribution to the correlator (<ref>) as we take the limit m→∞, we can set h=1 and instead consider solving the simpler differential equation
        
            0   =η (1-η)^3 d^3𝒢/dη^3-2 (1-η)^2(1+η) d^2𝒢/dη^2+2(1-η)(1-2η)d𝒢/dη+4(2-η)𝒢
        
        for 𝒢(η). Doing so, we learn that the 2-point correlator (<ref>) takes the form of a rational function in s
            1/g^(a) ⟨ϕ_(1,3)(i,i̅) ϕ_(1,3)(is,i̅s̅)⟩_UHP^(a) =
               =1/s^2(1-s^2)^4[K_1 (s^8+28 s^6-314 s^4+28 s^2+1)+K_2(s^8+28 s^6-826 s^4+28 s^2+1)+
               +K_3(s^7+7 s^5+7 s^3+s)] ,
        
        where K_1, K_2 and K_3 are some integration constants.
        
        
        
        
        
        
        These can be fixed by identifying the states propagating in the bulk channel and the boundary channel of the correlator (taking the limits s→ 1 and s→ 0, respectively, on the r.h.s. of (<ref>)) and then matching the coefficients in front of poles with the corresponding OPE structure constants. 
        
        First, in the bulk channel, one can expand the r.h.s. of (<ref>) as
        
            -16 K_1-48 K_2+K_3/(1-s)^4+32 K_1+64 K_2+K_3/2 (1-s)^2+𝒪[(1-s)^-1] .
        
        Here, the quartic pole is to be associated with the propagation of the bulk identity ϕ_(1,1) in the  bulk channel, while the quadratic pole arises due to propagation of ϕ_(1,3).
        Comparing (<ref>) with the result of taking the bulk OPE of the two bulk operators ϕ_(1,3) inside the correlator (<ref>), we obtain the constraints 
        
            -1   =16 K_1+48 K_2-K_3 ,
            1/4 C_(1,3)(1,3)(1,3)B_(1,3)(1,1)^(a)/g^(a)   =16 K_1+32 K_2+K_3/2 .
        
        We recall that in the strict limit m→∞, the bulk 3-point structure constant C_(1,3)(1,3)(1,3) is equal to 4/√(3), while the value of the disk one-point function coefficient B_(1,3)(1,1)^(a) will generally depend on the particular choice a of the Cardy boundary condition. 
        
        Second, expanding the r.h.s. of (<ref>) in the limit s→ 0 (boundary channel), one gets
        
            K_1+K_2/s^2+K_3/s+𝒪(s^0) ,
        
        where the quadratic and simple poles are due to propagation of the identity and the (1,3) module in the boundary channel, respectively. This prompts us to identify
        
            1/16[B_(1,3)(1,1)^(a)/g^(a)]^2    =K_1+K_2 ,
            1/4[B_(1,3)(1,3)^(a)/g^(a)]^2   = K_3 ,
        
        where B_(1,3)(1,3)^(a) denotes the coefficient which enters the two-point function between a (1,3) field in the bulk and a (1,3) field on the boundary.
        
        By solving the constraints (<ref>), (<ref>) and (<ref>), one can fix the integration constants K_1, K_2 and K_3 (and hence the correlator (<ref>)) in terms of C_(1,3)(1,3)(1,3), B_(1,3)(1,1)^(a) and g^(a). The remaining constraint (<ref>) then provides a non-trivial bulk-boundary crossing relation for the structure constant B_(1,3)(1,3)^(a), namely
        
            [B_(1,3)(1,3)^(a)/g^(a)]^2 = 2-2[B_(1,3)(1,1)^(a)/g^(a)]^2+C_(1,3)(1,3)(1,3) B_(1,3)(1,1)^(a)/g^(a) .
        
        Validity of this relation can, in principle,  be checked against the known values of the structure constants in minimal models <cit.> in the limit m→∞.
        
        To facilitate applications to the SFT computation of the change in the g-function of Cardy boundary conditions upon introducing a bulk perturbation (section <ref>), we should keep in mind our assumption (<ref>), which ensures that the deformation triggered by the bulk field ϕ_(1,3) on the boundary remains perturbative. From now on, we will therefore fix the Cardy boundary condition a so that the l.h.s. of (<ref>) vanishes. This implies two distinct solutions  
        
            B_(1,3)(1,1)^(a)/g^(a) = {[     √(3)+𝒪(y); -1/√(3) +𝒪(y) ].
        
        of the crossing equation (<ref>) for the structure constant B_(1,3)(1,1)^(a).
        
        The first solution √(3) can be realized by taking the Kac labels (a_1,a_2) of the Cardy boundary state arbitrary but fixed when taking the limit m→∞. Correspondingly, one can establish that for such boundary states, the bulk field ϕ_(1,3) can indeed be used to trigger a sequence of perturbative bulk and boundary deformations. These turn out to relate the initial boundary condition (a_1,a_2) in the minimal model m to the boundary condition (a_2,a_1) in the minimal model m-1. 
        
        This is true in particular for the boundary conditions of the type (a_1,1) for which the module (1,3) is absent from the boundary spectrum so that the structure constant B_(1,3)(1,3)^(a) vanishes identically to all orders in y. In these cases, there is a direct RG flow taking the boundary condition (a_1,1)_m to (1,a_1)_m-1. See <cit.>, as well as section <ref> for details.
        
        On the other hand, the second solution -1/√(3) arises for the boundary states with Kac labels (a_1,a_2), where a_1 is kept fixed while we put a_2 = m+1/2+α for m odd (and for a fixed integer α). This is in precise agreement with the existence of an RG flow triggered by the bulk field ϕ_(1,3)  taking the boundary condition (a_1, a_2)_m directly to the boundary condition (a_2-1,a_1)_m-1. In the aformentioned regime of the Kac labels (a_1,a_2), such a flow is indeed perturbative.
        
        Focusing our attention on those structure constants B_(1,3)(1,1)^(a) which solve the crossing equation (<ref>) with zero l.h.s. (consistent with our assumption (<ref>)), then, in the strict m→∞ limit,
        
        
        
        
        
        
        
        
        
        the desired two-point correlator can finally be fixed as
        
            1/g^(a) ⟨ϕ_(1,3)(i,i̅) ϕ_(1,3)(is,i̅s̅)⟩_UHP^(a)=
               =C_(1,3)(1,3)(1,3) B_(1,3)(1,1)^(a)/g^(a)s^4+30 s^2+1 /32 s^2 (s^2-1)^2+ (s^4+14 s^2+1)^2/16 s^2 (s^2-1)^4 .
        
        Furthermore, subtracting the bulk and the boundary channel divergences in a way described in section <ref>, we obtain
        
            1/g^(a) [⟨ϕ_(1,3)(i,i̅) ϕ_(1,3)(is,i̅s̅)⟩_UHP^(a)-div_c(s)-div_o(s)]=
               =1/(1+s)^4+C_(1,3)(1,3)(1,3) B_(1,3)(1,1)^(a)/g^(a) s+3/4 (1-s) (1+s)^2 .
        
        This is the function which is to be integrated against the c-ghost measure 4(s^2-1) in (<ref>) when calculating the contribution 𝒜_TT^f.p. to the on-shell disk action at subleading order in y in section <ref>.
        
        
        
        
        
        99Mukherji:1991tb
        S. Mukherji and A. Sen,
        “Some all order classical solutions in nonpolynomial closed string field theory,”
        Nucl. Phys. B 363 (1991), 639-664
        doi:10.1016/0550-3213(91)80037-M
        
        
        
        Mukherji:1991kz
        S. Mukherji, S. Mukhi and A. Sen,
        “Black hole solution and its infinite parameter generalisations in c = 1 string field theory,”
        Phys. Lett. B 275 (1992), 39-46
        doi:10.1016/0370-2693(92)90848-X
        
        
        
        Freedman:2005wx
        D. Z. Freedman, M. Headrick and A. Lawrence,
        “On closed string tachyon dynamics,”
        Phys. Rev. D 73 (2006), 066015
        doi:10.1103/PhysRevD.73.066015
        [arXiv:hep-th/0510126 [hep-th]].
        
        
        
        Kudrna:2012re
        M. Kudrna, C. Maccaferri and M. Schnabl,
        “Boundary State from Ellwood Invariants,”
        JHEP 07 (2013), 033
        doi:10.1007/JHEP07(2013)033
        [arXiv:1207.4785 [hep-th]].
        
        
        
        
        
        
        
        
        
        
        Scheinpflug:2023osi
        J. Scheinpflug and M. Schnabl,
        “Conformal perturbation theory from open string field theory,”
        [arXiv:2301.05216 [hep-th]].
        
        
        
        
        
        Scheinpflug:2023lfn
        J. Scheinpflug and M. Schnabl,
        “Closed string tachyon condensation revisited,”
        [arXiv:2308.16142 [hep-th]].
        
        
        
        Mazel:2024alu
        B. Mazel, J. Sandor, C. Wang and X. Yin,
        “Conformal Perturbation Theory and Tachyon-Dilaton Eschatology via String Fields,”
        [arXiv:2403.14544 [hep-th]].
        
        
        
        
        Zamolodchikov:1986gt
        A. B. Zamolodchikov,
        “Irreversibility of the Flux of the Renormalization Group in a 2D Field Theory,”
        JETP Lett. 43 (1986), 730-732
        
        
        
        Zamolodchikov:1987ti
        A. B. Zamolodchikov,
        “Renormalization Group and Perturbation Theory Near Fixed Points in Two-Dimensional Field Theory,”
        Sov. J. Nucl. Phys. 46 (1987), 1090
        
        
        
        Zamolodchikov:1989hfa
        A. B. Zamolodchikov,
        “Integrable field theory from conformal field theory,”
        Adv. Stud. Pure Math. 19 (1989), 641-674
        
        
        
        Affleck:1991tk
        I. Affleck and A. W. W. Ludwig,
        “Universal noninteger 'ground state degeneracy' in critical quantum systems,”
        Phys. Rev. Lett. 67 (1991), 161-164
        doi:10.1103/PhysRevLett.67.161
        
        
        
        Friedan:2003yc
        D. Friedan and A. Konechny,
        “On the boundary entropy of one-dimensional quantum systems at low temperature,”
        Phys. Rev. Lett. 93 (2004), 030402
        doi:10.1103/PhysRevLett.93.030402
        [arXiv:hep-th/0312197 [hep-th]].
        
        
        Sen:2019jpm
        A. Sen,
        “String Field Theory as World-sheet UV Regulator,”
        JHEP 10 (2019), 119
        doi:10.1007/JHEP10(2019)119
        [arXiv:1902.00263 [hep-th]].
        
        
        
        Larocca:2017pbo
        P. V. Larocca and C. Maccaferri,
        “BCFT and OSFT moduli: an exact perturbative comparison,”
        Eur. Phys. J. C 77 (2017) no.11, 806
        doi:10.1140/epjc/s10052-017-5379-3
        [arXiv:1702.06489 [hep-th]].
        
        
        
        
        Witten:2013pra
        E. Witten,
        “The Feynman i ϵ in String Theory,”
        JHEP 04 (2015), 055
        doi:10.1007/JHEP04(2015)055
        [arXiv:1307.5124 [hep-th]].
        
        
        
        
        Gaberdiel:2008fn
        M. R. Gaberdiel, A. Konechny and C. Schmidt-Colinet,
        “Conformal perturbation theory beyond the leading order,”
        J. Phys. A 42 (2009), 105402
        doi:10.1088/1751-8113/42/10/105402
        [arXiv:0811.3149 [hep-th]].
        
        
                  Erler:2022agw
            T. Erler,
             “The closed string field theory action vanishes,”
            [arXiv:2204.12863 [hep-th]].
        
             Bergman:1994qq
               O. Bergman and B. Zwiebach,
               “The Dilaton theorem and closed string backgrounds,”
               Nucl. Phys. B  441 (1995) 76
               doi:10.1016/0550-3213(95)00022-K
               [hep-th/9411047].
              
        
        
        
        
        cosmo
        C. Maccaferri and J. Vošmera,
        “The classical cosmological constant of open-closed string field theory,”
        JHEP 10 (2022), 173
        doi:10.1007/JHEP10(2022)173
        [arXiv:2208.00410 [hep-th]].
        
        
        Fredenhagen:2006dn
        S. Fredenhagen, M. R. Gaberdiel and C. A. Keller,
        “Bulk induced boundary perturbations,”
        J. Phys. A 40 (2007), F17
        doi:10.1088/1751-8113/40/1/F03
        [arXiv:hep-th/0609034 [hep-th]].
        
        
         Zwiebach:1992ie
            B. Zwiebach,
           “Closed string field theory: Quantum action and the B-V master equation,”   Nucl. Phys. B  390 (1993) 33      doi:10.1016/0550-3213(93)90388-6      [hep-th/9206084].
             
        
        
        
        
        Sen:1993mh
        A. Sen and B. Zwiebach,
        
        Nucl. Phys. B 414 (1994), 649-714
        doi:10.1016/0550-3213(94)90258-5
        [arXiv:hep-th/9307088 [hep-th]].
        
        
        
        Sen:1993kb
        A. Sen and B. Zwiebach,
        
        Nucl. Phys. B 423 (1994), 580-630
        doi:10.1016/0550-3213(94)90145-7
        [arXiv:hep-th/9311009 [hep-th]].
        
        
        
        
        Zwiebach:1990qj
        B. Zwiebach,
        “Quantum open string theory with manifest closed string factorization,”
        Phys. Lett. B 256 (1991), 22-29
        doi:10.1016/0370-2693(91)90212-9
        
        
        
        Zwiebach:1997fe
        B. Zwiebach,
        “Oriented open - closed string theory revisited,”
        Annals Phys. 267 (1998), 193-248
        doi:10.1006/aphy.1998.5803
        [arXiv:hep-th/9705241 [hep-th]].
        
        
        
              
              
        
        Maccaferri:2023gcg
        C. Maccaferri, A. Ruffino and J. Vošmera,
        “The nilpotent structure of open-closed string field theory,”
        JHEP 08 (2023), 145
        doi:10.1007/JHEP08(2023)145
        [arXiv:2305.02843 [hep-th]].
        
        
        
        
        Maccaferri:2023gof
        C. Maccaferri, A. Ruffino and J. Vošmera,
        “Open-closed string field theory in the large N limit,”
        JHEP 09 (2023), 119
        doi:10.1007/JHEP09(2023)119
        [arXiv:2305.02844 [hep-th]].
        
        
        
        
            
        
        Kajiura:2004xu
           H. Kajiura and J. Stasheff,
           “Homotopy algebras inspired by classical open-closed string field theory,”
           Commun. Math. Phys. 263 (2006), 553-581
           doi:10.1007/s00220-006-1539-2
           [arXiv:math/0410291 [math.QA]].
           
         Kajiura:2005sn
           H. Kajiura and J. Stasheff,
           “Open-closed homotopy algebra in mathematical physics,”
           J. Math. Phys. 47 (2006), 023506
           doi:10.1063/1.2171524
           [arXiv:hep-th/0510118 [hep-th]].
        
        
        
        Munster:2011ij
        K. Munster and I. Sachs,
        
        Commun. Math. Phys. 321 (2013), 769-801
        doi:10.1007/s00220-012-1654-1
        [arXiv:1109.4101 [hep-th]].
        
        
        
        
        
        Maccaferri:2021ksp
        C. Maccaferri and J. Vošmera,
        
        JHEP 09 (2021), 047
        doi:10.1007/JHEP09(2021)047
        [arXiv:2103.04919 [hep-th]].
        
        
        
        Fredenhagen:2009tn
        S. Fredenhagen, M. R. Gaberdiel and C. Schmidt-Colinet,
        “Bulk flows in Virasoro minimal models with boundaries,”
        J. Phys. A 42 (2009) no.49, 495403
        doi:10.1088/1751-8113/42/49/495403
        [arXiv:0907.2560 [hep-th]].
        
        
        
        Runkel:1998he
        I. Runkel,
        “Boundary structure constants for the A series Virasoro minimal models,”
        Nucl. Phys. B 549 (1999), 563-578
        doi:10.1016/S0550-3213(99)00125-X
        [arXiv:hep-th/9811178 [hep-th]].
        
        
        
        Recknagel:2000ri
        A. Recknagel, D. Roggenkamp and V. Schomerus,
        “On relevant boundary perturbations of unitary minimal models,”
        Nucl. Phys. B 588 (2000), 552-564
        doi:10.1016/S0550-3213(00)00519-8
        [arXiv:hep-th/0003110 [hep-th]].
        
        
        Cardy:1989ir
        J. L. Cardy,
        “Boundary Conditions, Fusion Rules and the Verlinde Formula,”
        Nucl. Phys. B 324 (1989), 581-596
        doi:10.1016/0550-3213(89)90521-X
        
        
        Ishibashi:1988kg
        N. Ishibashi,
        “The Boundary and Crosscap States in Conformal Field Theories,”
        Mod. Phys. Lett. A 4 (1989), 251
        doi:10.1142/S0217732389000320
        
        
        
        Ahmadain:2024hgd
        A. Ahmadain and R. Khan,
        “A Worldsheet Derivation of the Classical Off-shell Boundary Action for the Dilaton in Half-Space,”
        [arXiv:2406.00712 [hep-th]].
        
        
        
        
        Ahmadain:2022eso
        A. Ahmadain and A. C. Wall,
        “Off-Shell Strings II: Black Hole Entropy,”
        [arXiv:2211.16448 [hep-th]].
        
        
        
        
        Ahmadain:2022tew
        A. Ahmadain and A. C. Wall,
        “Off-Shell Strings I: S-matrix and Action,”
        [arXiv:2211.08607 [hep-th]].
        
        
        
        Firat:2023gfn
        A. H. Fırat,
        “String vertices for the large N limit,”
        Nucl. Phys. B 1000 (2024), 116485
        doi:10.1016/j.nuclphysb.2024.116485
        [arXiv:2311.00747 [hep-th]].
        
        
        
        Zwiebach:1992bw
        B. Zwiebach,
        “Interpolating string field theories,”
        Mod. Phys. Lett. A 7 (1992), 1079-1090
        doi:10.1142/S0217732392000951
        [arXiv:hep-th/9202015 [hep-th]].


