Paper Title: Weak rates in strongly coupled cold quark matter.

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             Weak rates in strongly coupled cold quark matter
                Carlos Hoyos^1[hoyoscarlos@uniovi.es], Andrea Olzi^2[andrea.olzi@unifi.it],
            David Rodriguez-Fernandez^3[david.rfernandez@upm.es]
                August 5, 2024
            ====================================================================================================================================
            
            
            
            0.75[][s]^1Department of Physics and Instituto de Ciencias and Tecnologías Espaciales de Asturias (ICTEA)[][l] Universidad de Oviedo, c/ Leopoldo Calvo Sotelo 18, ES-33007 Oviedo, Spain[][s]^2INFN, Sezione di Firenze and Dipartimento di Fisica e Astronomia, Università di Firenze,[][l] Via G. Sansone 1,I-50019 Sesto Fiorentino (Firenze), Italy.[][s]^3Departamento de matemática aplicada a las TIC, Universidad Politécnica de Madrid[][l] Nikola Tesla, s/n, ES-28031 Madrid, Spain
            
            § ABSTRACT 
            The rates of flavor-changing weak processes are crucial in determining the conditions of beta equilibrium in neutron stars and mergers, influencing the damping of oscillations, the stability of rotating pulsars, and the emission of gravitational waves. We derive a formula for these rates at nonzero temperature, to leading order in the Fermi coupling and exact in the QCD coupling. Utilizing a simple phenomenological holographic model dual to QCD, we study massless unpaired quark matter at high densities. We numerically compute the rate for small deviations from beta equilibrium and derive an analytic approximation for small temperatures. Our findings reveal that, compared to the perturbative result, the rate is suppressed by logarithmic factors of the temperature.
            
            
            
            
            
            
            
            
            § INTRODUCTION
            
            
            
            Weak interactions play a crucial role in determining the properties of neutron stars and their binary mergers. The equation of state, which dictates the mass and radius of the star, is constrained by the condition of beta equilibrium. This condition establishes relationships among the relative amounts of different quark (or baryon) and lepton species. In mergers, the beta equilibrium condition may be modified due to the temperature dependence of the rates <cit.>. When a perturbation in the density drives the matter of the star out of beta equilibrium, weak processes work to restore it and dampen the perturbation. This results in an effective bulk viscosity that is significantly enhanced when the characteristic time of the perturbation resonates with the timescale determined by the rate of weak processes and the density in the star <cit.> (see <cit.> for a review of transport in neutron stars).
            
            Oscillations of a rotating neutron star produce gravitational waves that cause the star to lose angular momentum and spin down, a process that is more pronounced for rapidly rotating stars. Despite this, fast-rotating pulsars with periods on the order of milliseconds are observed within a limited range of temperatures. The effective bulk viscosity induced by beta equilibration is likely necessary to explain the existence and location of this stability window <cit.>. This viscosity also affects the continuous emission of gravitational waves by a quiescent star <cit.> and could impact the gravitational wave signal produced during the inspiral <cit.> and post-merger evolution of a neutron star collision <cit.>.
            
            Although most matter in a neutron star and the remnant of a merger is expected to be primarily composed of hadrons, there are indirect hints of quark matter in the core of heavy neutron stars <cit.>. Simulations of binary mergers suggest that pockets of quark matter may be produced just after the collision <cit.>. Therefore, it is interesting to evaluate the bulk viscosity of quark matter induced by weak processes. At the densities involved, quantum chromodynamics (QCD) cannot be treated perturbatively, and lattice QCD suffers from the sign problem, necessitating other non-perturbative approaches. Holographic models can circumvent these issues and have been applied to the study of dense matter in neutron stars (see <cit.> for comprehensive reviews). Specifically, the bulk viscosity induced by weak processes was computed using holographic models in <cit.>[The holographic estimate of the QCD contribution to the bulk viscosity was computed previously and found to be much smaller than the expected weak contribution <cit.>.]. However, the bulk viscosity formula involves weak reaction rates that have been computed only perturbatively, so it is not a completely self-contained approach. Furthermore, the existing formulas for weak rates are obtained through tree-level scattering matrix elements of quarks and leptons <cit.>. Since the observables obtained from gauge/gravity duality are correlators of gauge-invariant operators, there is no straightforward way to obtain the rates by extrapolating the tree-level formulas. Unfortunately, a main feature of the bulk viscosity, namely its temperature dependence, is highly sensitive to the behavior of the rate. Thus, the present estimation of the bulk viscosity is hindered by significant uncertainty in this respect.
            
            As a step towards improving the bulk viscosity estimates, we will provide a non-perturbative formula for the rates based on correlators of gauge-invariant operators. We will then use it to obtain the temperature dependence of the rates in a simple holographic model of unpaired quark matter. The model, a phenomenological holographic dual description of QCD <cit.>, has been used extensively, with recent related works including the description of nuclear matter in neutron stars and mergers <cit.> and the calculation of neutrino emissivities from flavor current correlators <cit.>. Our goal is not to obtain completely realistic values for the rates, for which more refined models would be needed, but to ascertain possible qualitative changes in the temperature dependence at strong coupling.
            
            Holographic models, despite their utility, come with inherent limitations. They are typically constructed as approximations that capture certain features of QCD, but they do not encompass all the complexities of the theory. One significant limitation is the lack of a first-principles derivation from QCD, leading to uncertainties in their applicability to specific phenomena. Furthermore, these models often rely on large-N_c (number of colors) approximations, which may not accurately reflect the behavior of QCD at the finite N_c relevant to real-world scenarios. The models also face challenges in accurately representing the dynamics at low temperatures and densities where confinement and chiral symmetry breaking are essential. Therefore, while holographic models provide valuable insights and a non-perturbative framework, their predictions should be interpreted with caution and validated against other approaches and experimental data.
            
            Keeping the limitations above in mind, for direct application to the description of neutron stars and their binary mergers, the rate should be computed in the model used to describe matter in the star. The most employed holographic models so far have been those inspired by top-down brane intersections, namely the D3-D7 <cit.> and Witten-Sakai-Sugimoto models <cit.>, and a bottom-up model designed to fit lattice QCD data and other known observables, the V-QCD model <cit.>. The D3-D7 model is the simplest but is limited to describing quark matter in the star <cit.> and must be combined with other nuclear matter models to construct hybrid equations of state for the matter inside the neutron star. The V-QCD model can describe confined matter but is used primarily at large densities, requiring a hybrid approach to describe matter at lower densities outside the core of the star <cit.>. Hybrid equations of state have also been constructed with the Sakai-Sugimoto model <cit.>, but there are proposals to describe nuclear matter in neutron stars at all densities purely holographically <cit.>. The Sakai-Sugimoto model has also been used to describe more exotic quark stars <cit.>. Other proposals to describe dense matter in neutron stars include Einstein-Maxwell-Dilaton theories and other holographic phenomenological models <cit.>. Computing the weak rate and bulk viscosity to estimate the location of the stability window of rotating pulsars would serve as an additional constraint on these models, as well as on any other phenomenological model for which such a calculation is possible.
            
            The outline of the paper is as follows. In Section <ref>, we introduce the weak reaction rates for beta equilibrium of quark matter and derive a non-perturbative formula in terms of correlators of flavor currents. Next, in Section <ref>, we use a simple holographic model to compute flavor current correlators in a high-density state and study their dependence on frequency and spatial momentum. Finally, in Section <ref>, we combine the results of the previous sections to produce analytical and numerical estimates of the weak rates at low temperatures. We compare these to the perturbative results and discuss potential improvements in Section <ref>. Technical details regarding the calculation of correlators in the holographic model are provided in the appendices.
            
            
            
            
            
            
            § WEAK REACTION RATES
            
            
            
            In this section, we derive non-perturbative formulas in the coupling of the strong interactions for the weak reaction rates of quark matter. The formulas correspond to the leading contribution of the electroweak interactions, whose corrections could in principle be computed perturbatively. We also assume that the times and length scales involved are much larger than the scale set by the inverse of the W boson mass.
            
            For the densities expected in the interior of neutron stars and mergers, one should be able to restrict to three quark flavors (q=u,d,s) and two lepton flavors (ℓ=e,μ). The relevant weak processes, or reactions, are produced by a W-boson exchange or emission:
            
                u+d⇌ u+s,   u+ℓ^- →  d,s+ν_ℓ,    d,s → u+ℓ^- +ν̅_ℓ,  ℓ_1^- →ℓ_2^-+ν_ℓ_1+ν̅_ℓ_2 .
            
            Assuming neutrinos are not trapped, the beta-equilibrium conditions in quark matter implied by these reactions are
            
                μ_u+μ_ℓ=μ_d=μ_s,    μ_ℓ=μ_e=μ_μ .
            
            Even if quarks are confined to hadrons the same reactions will change their type among protons, neutrons, and hyperons. In this situation, the beta equilibrium conditions should be given in terms of baryon chemical potentials. Note that in order for any of the species of fermions to be present it is necessary that the chemical potential is larger than the mass of the corresponding particle.
            
            
            If a perturbation takes the chemical potentials out of their equilibrium values, the weak reactions will not be in balance, and the densities will change according to the weak rates Γ:[We have omitted the label on the neutrinos when there is just one lepton involved and put the same numerical sub-index as the corresponding lepton when there are two.]
                d n_u/dt   =   ∑_ℓ=e,μ( Γ_d→ uℓν̅-Γ_uℓ→ dν+ Γ_s→ u ℓν̅-Γ_uℓ→ sν) ,
                d n_d/dt   =   Γ_su→ ud-Γ_ud→ su+∑_ℓ=e,μ( Γ_uℓ→ dν-Γ_d→ uℓν̅) ,
                d n_s/dt   =   Γ_ud→ su-Γ_su→ ud+ ∑_ℓ=e,μ(Γ_uℓ→ sν-Γ_s→ u ℓν̅) ,
                d n_ℓ_1/dt   =   Γ_d→ uℓ_1ν̅-Γ_uℓ_1→ dν+Γ_s→ u ℓ_1ν̅-Γ_uℓ_1→ sν+Γ_ℓ_2→ℓ_1ν_2ν̅_1-Γ_ℓ_1→ℓ_2ν_1ν̅_2 .
            
            It is easy to confirm that the total baryon number and electric charge are conserved by these processes
            
                n_B=1/3(n_u+n_d+n_s),   n_Q=2/3n_u-1/3n_d-1/3n_s-n_e-n_μ,    dn_B/dt=dn_Q/dt=0 .
            
            
            Assuming the deviation of the chemical potentials from their beta-equilibrium values δμ_f^a is small compared to the baryon chemical potential μ_B=μ_u+μ_d+μ_s, we can expand the rates to linear order in the differences
            
                Γ_ud→ su-Γ_su→ ud   ≈   λ_ds (δμ_s-δμ_d),
                Γ_uℓ→ dν-Γ_d→ uℓν̅   ≈   λ_ud^ℓ (δμ_d-δμ_u-δμ_ℓ),
                Γ_uℓ→ sν-Γ_s→ uℓν̅   ≈   λ_us^ℓ (δμ_s-δμ_u-δμ_ℓ),
                Γ_e→μν_2ν̅_1-Γ_μ→ eν_1ν̅_2   ≈   λ_eμ (δμ_μ-δμ_e).
            
            
            In some cases, for instance when quark matter is close to being symmetric, the lepton chemical potential could be below the muon mass μ_ℓ<m_μ. When this happens, muons quickly decay and the only leptons present are electrons. The equilibration equations simplify somewhat
            
                d n_u/dt   =   Γ_d→ u eν̅-Γ_ue→ dν+ Γ_s→ u eν̅-Γ_u e→ sν ,
                d n_d/dt   =   Γ_su→ ud-Γ_ud→ su+Γ_u e→ dν-Γ_d→ u eν̅ ,
                d n_s/dt   =   Γ_ud→ su-Γ_su→ ud+ Γ_u e→ sν-Γ_s→ u eν̅ ,
                d n_e/dt   =   Γ_d→ ueν̅-Γ_ue→ dν+Γ_s→ u eν̅-Γ_ue→ sν .
            
            For small deviations from beta-equilibrium, the rates are expanded as (<ref>), but taking into account only the electron contribution. Moreover, the coefficient λ_eμ does not involve strong coupling physics, so we will ignore it in the following, as it can be computed using the standard methods. Our strategy to derive the other coefficients is to arrive at an expression like (<ref>) from Ward identities capturing the non-conservation of flavor currents due to the coupling to the weak sector. 
            
            
            
            
             §.§ Flavor Ward identities
            
            
            The theory of strong interactions, QCD, respects flavor symmetries so that there are no flavor-changing processes induced by them. Weak interactions on the other hand are not flavor symmetric, and processes such as those in (<ref>) are possible. At energy scales much below the W mass, the flavor-changing processes are captured by adding Fermi's interaction term to the effective action
            
                L_ Fermi=-2√(2) G_F (J_ch^μ)^† J_ch μ,
            
            where the J_ch^μ is the left-charged current
            
                J_ch^μ=ν_e Lγ^μ e_L+ν_μ Lγ^μμ_L+cosθ_C u_Lγ^μ d_L+sinθ_C u_Lγ^μ s_L,
            
            with G_F Fermi's constant and θ_C the Cabibbo angle giving the change of basis between quark mass and flavor eigenstates. In addition to Fermi's term, flavor symmetry can also be broken by flavor-preserving mass terms and electromagnetic and weak neutral interactions.
            
            In the absence of any flavor-breaking terms, there would be a SU(3)_L× SU(3)_R global symmetry for flavor rotations among quarks and a SU(4)_L× SU(4)_R flavor global symmetry for leptons. To each of these symmetries, we can associate a conserved current J^μ_f χ, where f=l,q for leptons or quarks and χ=L,R for left or right chirality. Since the symmetry is non-Abelian the current components transform under flavor rotations. Denoting by T^A_f χ the generators of the flavor symmetry, the transformation of the current is
            
                δ_θ_f χ J^μ_f χ=iθ_f χ^A[J^μ_f χ,T^A_f χ] .
            
            We will take the usual normalization for the generators tr(T_f χ^A T_f χ^B)=1/2δ^AB. 
            
            A vector flavor transformation corresponds to a transformation with equal angles for both chiralities θ_f L^A=θ_f R^A=θ_f^A. In the following vector currents and generators will be denoted as J^μ_f and T^A_f. If there are flavor-symmetry breaking terms in the action L_ br, the Ward identity associated with a vector transformation becomes
            
                2θ_f^A tr(∂_μ J^μ_f T^A_f)=-δ_θ^A_f L L_ br-δ_θ^A_f R L_ br|_θ_f L^A=θ_f R^A=θ_f^A.
            
            In the following, it will be convenient to introduce the notation for the currents
            
                (J_f χ^μ)^a_ b=f̅_χ bγ^μ f_χ^a ,     f^a=q^a,l^a;   q^a=u,d,s;  l^a=ν_e,e,ν_μ,μ.
            
            
            We are interested in the flavor-changing contributions introduced by Fermi's interaction. We can simplify the analysis by noticing that since Fermi's interaction is symmetric under a rotation between the two lepton generations (ν_e,e)↔ (ν_μ,μ), we can actually restrict lepton transformations to SU(2)× SU(2)⊂ SU(4), with each SU(2) rotating a charged lepton and its corresponding neutrino within each generation. We leave details of the calculation, which is straightforward but tedious, to Appendix <ref>. The Ward identities for the diagonal components of the vector flavor currents are
            
                ∂_μ(J_q^μ)^u_ u   =     i√(2)G_F η_μν∑_ℓ=e,μ[cosθ_C(J_q L^μ)^u_ d(J_l L^ν)^d_ u-(J_q L^μ)^d_ u(J_l L^ν)^ν_ℓ_ ℓ.
                
                        . +sinθ_C(J_q L^μ)^u_ s(J_l L^ν)^ℓ_ ν_ℓ-(J_q L^μ)^s_ u(J_l L^ν)^ν_ℓ_ ℓ],
                
                ∂_μ(J_q^μ)^d_ d    =    i√(2) G_Fη_μν[cosθ_C sinθ_Cη_μν(J_q  L^μ)^u_ s (J_q L^ν)^d_ u-(J_q L^μ)^s_ u(J_q L^ν)^u_ d.
                
                        .-cosθ_C∑_ℓ=e,μ(J_q L^μ)^u_ d(J_l L^ν)^ℓ_ ν_ℓ-(J_q L^μ)^d_ u(J_l L^ν)^ν_ℓ_ ℓ],
                
                ∂_μ(J_q^μ)^s_ s   =    i√(2) G_Fη_μν[-cosθ_C sinθ_Cη_μν(J_q  L^μ)^u_ s (J_q L^ν)^d_ u-(J_q L^μ)^s_ u(J_q L^ν)^u_ d.
                
                        .-sinθ_C∑_ℓ=e,μ(J_q L^μ)^u_ s(J_l L^ν)^ℓ_ ν_ℓ-(J_q L^μ)^s_ u(J_l L^ν)^ν_ℓ_ ℓ],
                
                ∂_μ(J_l^μ)^ν_ℓ_ ν_ℓ   =    i√(2)G_F η_μν[-cosθ_C(J_q L^μ)^u_ d(J_l L^ν)^ℓ_ ν_ℓ-(J_q L^μ)^d_ u(J_l L^ν)^ℓ_ ν_ℓ.
                
                        . -sinθ_C(J_q L^μ)^u_ s(J_l L^ν)^ℓ_ ν_ℓ-(J_q L^μ)^s_ u(J_l L^ν)^ν_ℓ_ ℓ.
                
                        .-(J_l L^μ)^ν_ℓ'_ ℓ'(J_l L^ν)^ℓ_ ν_ℓ-(J_l L^μ)^ℓ'_ ν_ℓ'(J_l L^ν)^ℓ_ ν_ℓ],
                
                ∂_μ(J_l^μ)^ℓ_ ℓ   =    i√(2)G_F η_μν[cosθ_C(J_q L^μ)^u_ d(J_l L^ν)^ℓ_ ν_ℓ-(J_q L^μ)^d_ u(J_l L^ν)^ν_ℓ_ ℓ.
                
                        . +sinθ_C(J_q L^μ)^u_ s(J_l L^ν)^ℓ_ ν_ℓ-(J_q L^μ)^s_ u(J_l L^ν)^ν_ℓ_ ℓ.
                
                        .+(J_l L^μ)^ν_ℓ'_ ℓ'(J_l L^ν)^ℓ_ ν_ℓ-(J_l L^μ)^ℓ'_ ν_ℓ'(J_l L^ν)^ν_ℓ_ ℓ].
            
            Where (ℓ,ℓ')=(e,μ) or (μ,e). Note that other terms that break the non-Abelian flavor symmetry, but are diagonal in flavor, do not contribute to the Ward identities of the diagonal components. One can easily confirm that the expressions above are consistent with the conservation of baryon and lepton numbers as well as the electric charge:
            
                ∂_μ(J_q^μ)^u_ u+(J_q^μ)^d_ d+(J_q^μ)^s_ s=0,  ∂_μ(J_l^μ)^ν_ℓ_ ν_ℓ+(J_l^μ)^ℓ_ ℓ=0, 
                   ∂_μ2/3(J_q^μ)^u_ u-1/3(J_q^μ)^d_ d-1/3(J_q^μ)^s_ s-∑_ℓ=e,μ (J_l^μ)^ℓ_ ℓ=0.
            
            The time components of the diagonal components of the flavor vector currents equal the corresponding quark or lepton densities
            
                n_u=(J_q^0)^u_ u, n_d=(J_q^0)^d_ d, n_s=(J_q^0)^s_ s, n_ℓ=(J_l^0)^ℓ_ ℓ.
            
            Compared with (<ref>) for homogeneous configurations, the rates are determined by the expectation value of a non-diagonal operator quadratic in the flavor currents appearing on the right-hand side of the Ward identities. We can think of such a quadratic operator as obtained from a two-point correlator in the (properly regularized) limit of coincident points. In the absence of flavor-breaking terms, the most general form of the flavor current correlators would be
            
                (J_f  L^μ)^a_ b (J_f L^ν)^c_ d= G_1 δ^a_bδ^c_d+G_2 δ^a_d δ^c_b .
            
            If there are mass terms in the action of the form L_m=f̅_a M^a_f b f^b, or chemical potentials introduced through a background flavor gauge field A^a_ f b, there could also be terms where we replace some or all of the deltas in (<ref>) by M_f or A_f, or contractions like M^a_f c M^c_f b,  M^a_f c A^c_f b, etc. This potentially can produce many new terms, however, for diagonal mass matrices and chemical potentials, the non-diagonal current correlators appearing on the right-hand side of the Ward identities would still vanish, and similarly for contributions originating from interactions that are diagonal in flavor. Therefore, the non-diagonal current correlators contributing to the Ward identities must be proportional to the flavor-changing coupling G_F. We will derive in the following a closed expression for the O(G_F^2) contribution to the rates in thermal field theory.
            
            
            
            
            
             §.§ Leading contribution in the Fermi coupling
            
            
            We will assume the theory is at finite temperature and use the real-time formalism to compute the current correlators. In general, thermal correlation functions are computed by insertions in a path integral defined along a contour in complex time C. The two-point function of two operators O_1 and O_2 is
            
                T_C[ O_1(x_1) O_2(x_2)].
            
            The complex time path, or Schwinger-Keldysh path, extends along (t_i,t_f)∪ (t_f,t_f-iσ)∪ (t_f-iσ,t_i -iσ)∪ (t_i-iσ,t_i-iβ), with σ>0 and  t_i→-∞, t_f→+∞. We take the point x_1 to be at the first leg of the complex time path on the real axis. Then we have two possible contributions, if x_2 is on the same leg, then the thermal correlator coincides with the usual time-ordered Feynman correlator
            
                T_C[ O_1(x_1) O_2(x_2)]=T[ O_1(x_1) O_2(x_2)]=G^F_ O_1 O_2 (t_1-t_2,x_1-x_2).
            
            If, on the other hand, x_2 is on the third leg (t'=-σ), the correlator is the Wightman correlator
            
                T_C[ O_1(x_1) O_2(x_2)]= O_2(x_2) O_1(x_1)=G^<_ O_1 O_2 (t_1-t_2+iσ,x_1-x_2).
            
            With this in mind, let us find the effect of Fermi's interaction (<ref>) when we treat it perturbatively. The thermal expectation value of an operator O will be, to leading order in the Fermi coupling,
            
                O(x)_G_F=T_C[ O(x) e^i∫_C   L_ Fermi]_0≈ O(x)_0+i∫_C d^4 x'T_C[ O(x)   L_ Fermi(x')]_0.
            
            In the expression above X_0 indicates that the expectation value of X is computed in the absence of Fermi's interaction. Since x is in the first leg of the complex time contour and  x' runs over its full extent, we have contributions from both the first and third leg. 
            
                ∫_C d^4 x'T_C[ O(x)   L_ Fermi(x')]_0=∫ d^4x' [G_ O L_ Fermi^F(t-t',x-x')+G_ O L_ Fermi^<(t-t'+iσ,x-x')].
            
            In the evaluation of the right-hand side of the Ward identities (<ref>)O is quadratic in the flavor currents, as well as L_ Fermi. Therefore, the leading O(G_F^2) contribution is proportional to correlators of the form
            
                [(J_f^μ)^a_ b(x) (J_f^ν)^c_ d(x)][(J_f'^α)^a'_ b'(x') (J_f'^β)^c'_ d'(x')]_0≈
                   δ_ff'[(J_f^μ)^a_ b(x)(J_f^α)^a'_ b'(x')_0(J_f^ν)^c_ d(x) (J_f^β)^c'_ d'(x')_0+(J_f^μ)^a_ b(x) (J_f^β)^c'_ d'(x')_0(J_f^ν)^c_ d(x)(J_f^α)^a'_ b'(x')_0].
            
            The factorization above is an approximation. For holographic models dual to a large-N_c theory it will happen naturally as part of the large-N_c expansion. More generally, the non-factorizable contributions would originate from effective eight-fermion interactions, so it is perhaps not unreasonable to assume that they would introduce relatively small corrections even in the finite-N_c strongly coupled theory. 
            
            As we discussed before, in the absence of the Fermi interaction, non-vanishing current correlators should be diagonal in flavor, thus
            
                G^μα_f;ab,a'b'(x-x')≡(J_f^μ)^a_ b(x)(J_f^α)^a'_ b'(x')_0≠ 0,  ⇒  (a=b   a'=b')   (a=b'   a'=b).
            
            This limits the number of non-zero contributions to the four-current thermal correlators. We will now write the leading contributions in the Fermi coupling to the Ward identities (<ref>) in terms of the Fourier transforms of the Feynman and Wightmann correlators of the flavor currents
            
                G^K μα_f;ab,a'b'(x-x')=∫d^4 q/(2π)^4e^iq· (x-x') G^K μα_f;ab,a'b'(q),     K=F,<  .
            
            It will be convenient to define
            
                γ__f_1^a f_2^b→ f_1^cf_2^d=γ^F__f_1^af_2^b→ f_1^c f_2^d+γ^<__f_1^af_2^b→ f_1^c f_2^ d,
            
            with (K=F,<)
            
                γ_f_1^a f_2^b→ f_1^c f_2^d^K=η_μνη_αβ∫d^4 q/(2π)^4[G^K μα_f_1;ac,ca(q)G^K νβ_f_2;bd,db(-q)-G^K μα_f_1;ca,ac(q)G^K νβ_f_2;db,bd(-q)] .
            
            In terms of these quantities, the leading order contributions in the Fermi coupling to the flavor Ward identities are
            
                ∂_μ(J_q^μ)^u_ u   ≈    -4 G_F^2∑_ℓ=e,μ(cos^2θ_C γ_uℓ→ dν+sin^2θ_C γ_uℓ→ sν),
                ∂_μ(J_q^μ)^d_ d   ≈    4 G_F^2 (-sinθ_C^2cos^2θ_C γ_ud→ su+cos^2θ_C ∑_ℓ=e,μγ_uℓ→ dν),
                ∂_μ(J_q^μ)^s_ s   ≈    4 G_F^2 (sinθ_C^2cos^2θ_C γ_ud→ su+sin^2θ_C ∑_ℓ=e,μγ_uℓ→ sν),
                ∂_μ(J_l^μ)^ν_ℓ_ ν_ℓ   ≈    4 G_F^2 (cos^2θ_C γ_uℓ→ dν+sin^2θ_C γ_uℓ→ sν+γ_ℓν'→νℓ'),
                ∂_μ(J_l^μ)^ℓ_ ℓ   ≈    -4 G_F^2 (cos^2θ_C γ_uℓ→ dν+sin^2θ_C γ_uℓ→ sν+γ_ℓν'→νℓ').
            
            Where in the last two rows (ℓ,ℓ')=(e,μ) or (μ,ℓ) and ν,ν' are the neutrinos corresponding to ℓ,ℓ' respectively.
            
            
            
            
             §.§ Formulas for the rates
            
            
            A direct comparison of the equilibration equations  (<ref>) and the Ward identities (<ref>) results in the identification of the leading order contribution in the Fermi coupling to the rates as
            
                Γ_uℓ→ dν-Γ_d→ u ℓν̅   ≈    4 G_F^2cos^2θ_C γ_uℓ→ dν  ,
                Γ_u ℓ→ sν-Γ_s→ u ℓν̅   ≈     4 G_F^2sin^2θ_C γ_uℓ→ sν  ,
                Γ_ud → su-Γ_su → ud   ≈     4 G_F^2sinθ_C^2cos^2θ_C γ_ud→ su  ,
                Γ_ℓ_1→ℓ_2ν_1ν̅_2-Γ_ℓ_2→ℓ_1ν_2ν̅_1   ≈    4 G_F^2γ_ℓ_1ν_2→ν_1ℓ_2 .
            
            With the factors γ defined in (<ref>) and (<ref>) in terms of thermal correlation functions. We can further simplify the expressions by first noting that, if the system has parity and time reversal invariance, the Feynman correlator satisfies
            
                G_f;ab,ba^F μν(-q_0,-q)=G_f;ba,ab^F νμ(q_0,q).
            
            A straightforward calculation then shows that γ^F__f_1^af_2^b→ f_1^c f_2^d=0 in this case. The Wightmann correlator on the other hand is proportional to the spectral function, defined as the imaginary part of the retarded correlator:
            
                G^< μν_f;ab,ba(q_0,q)=n_f;ab(q_0)ρ_f;ab^μν(q_0,q),  ρ_f;ab^μν=-2 (G^R μν_f;ab,ba),
             
            where the factor multiplying the spectral function is the thermal distribution
            
                n_f;ab(q_0)=1/e^β(q_0+μ_f^a-μ_f^b)-1 ,
            
            with β=1/T the inverse of the temperature and μ_f^a the chemical potential for the fermion species f^a. We now introduce (<ref>) in (<ref>), and use that
            
                n_f;ab(-q_0)=-(1+n_f;ba(q_0)),     ρ_f;ab^μα(-q_0,-q)=-ρ_f;ba^αμ(q_0,q) ,
            
            where the last identity follows from time-reversal invariance and parity. After this, one arrives at
            
                γ__f_1^a f_2^b→ f_1^cf_2^d=γ^<__f_1^af_2^b→ f_1^c f_2^ d=η_μνη_αβ∫d^4 q/(2π)^4[(n_f_1;ac(q_0)-n_f_2;db(q_0))ρ^μα_f_1;ac(q_0,q)ρ^βν_f_2;db(q_0,q)] .
            
            This quantity will vanish when the chemical potentials satisfy the relations
            
                μ_f_1^a-μ_f_1^c=μ_f_2^d-μ_f_2^b.
            
            Then, demanding that each of the rates in (<ref>) vanish gives the conditions
            
                μ_u-μ_d    =    μ_ν_ℓ-μ_ℓ,
                μ_u-μ_s    =   μ_ν_ℓ-μ_ℓ,
                μ_u-μ_s    =   μ_u-μ_d,
                μ_ν_ℓ_1-μ_ℓ_1   =   μ_ν_ℓ_2-μ_ℓ_2.
            
            Setting μ_ν_ℓ=0, it is easy to check that this coincides with the beta equilibrium conditions in (<ref>). We want to expand (<ref>) to the lowest order around the equilibrium values, μ_f^a=μ_f^a^eq+δμ_f^a. The thermal distributions have a pole at small frequencies, that we need to deal with. Let us introduce an arbitrary frequency ω_0 such that δμ_f_1^a-δμ_f_1^c∼δμ_f_2^d-δμ_f_2^b≪ω_0 ≪ T. Then, there is a contribution that we can expand to linear order in the chemical potentials plus a correction from the poles
            
                γ__f_1^a f_2^b→ f_1^cf_2^d≈(δμ_f_2^d-δμ_f_2^b-(δμ_f_1^a-δμ_f_1^c))Λ_f_1^a f_2^b→ f_1^cf_2^d^0+δγ__f_1^a f_2^b→ f_1^cf_2^d ,
            
            where, after defining Δμ^eq=μ_f_1^a^eq- μ_f_1^c^eq=μ_f_2^d^eq- μ_f_2^v^eq, the coefficient Λ is
            
                Λ__f_1^a f_2^b→ f_1^cf_2^d^0=   η_μνη_αβ∫d^4 q/(2π)^4ρ^μα_f_1;ac(q_0,q)ρ^βν_f_2;db(q_0,q) 
                   ×[1/4Tsinh^2(q_0+Δμ^eq/2T)-T/(q_0+Δμ^eq)^2Θ(ω_0^2-(q^0+Δμ^eq)^2)].
            
            We remind the reader that in this formula the fermion species are f_1,f_2=q (quarks) or l (leptons) and the possible flavors are q^a=u,d,s, l^a=ν_e,e,ν_μ,μ. For q_0+Δμ^eq≳ T the function decays exponentially ∼ e^-(q_0+Δμ^eq)/T, thus most of the contributions in the frequency integral will come from an interval of width ∼ T around q^0+Δμ^eq. 
            
            The correction to the rate originating from the poles is approximately
            
                δγ__f_1^a f_2^b→ f_1^cf_2^d=   η_μνη_αβ∫_|q^0+Δμ^eq|<ω_0d^4 q/(2π)^4ρ^eq μα_f_1;ac(q_0,q)ρ^eq βν_f_2;db(q_0,q) 
                   ×[T/q^0+Δμ^eq+δμ_f_1^a-δμ_f_1^c -T/q^0+Δμ^eq+δμ_f_2^d-δμ_f_2^b] .
            
            We will use the principal value integral
            
                ∫_-ω_0-Δμ^eq^ω_0-Δμ^eqd q^0/2πT/q^0+Δμ^eq+δμ S(q^0) ≈T/2πS(-Δμ^eq)logω_0+δμ/ω_0-δμ≈T/πω_0S(-Δμ^eq) δμ.
            
            Thus, we can write
            
                γ__f_1^a f_2^b→ f_1^cf_2^d≈(δμ_f_2^d-δμ_f_2^b-(δμ_f_1^a-δμ_f_1^c))Λ_f_1^a f_2^b→ f_1^cf_2^d ,
            
            with
            
                Λ__f_1^a f_2^b→ f_1^cf_2^d≈Λ__f_1^a f_2^b→ f_1^cf_2^d^0+η_μνη_αβT/πω_0∫d^3 q/(2π)^3ρ^eq μα_f_1;ac(-Δμ^eq,q)ρ^eq βν_f_2;db(-Δμ^eq,q).
            
            If the spectral function vanishes at q^0=-Δμ^eq the last term in (<ref>) is zero and one can take the ω_0→ 0 limit, in which case the expression for the rate simplifies
            
                Λ__f_1^a f_2^b→ f_1^cf_2^d≈η_μνη_αβ∫d^4 q/(2π)^4ρ^eq μα_f_1;ac(-Δμ^eq,q)ρ^eq μα_f_2;db(-Δμ^eq,q)/4Tsinh^2(q_0+Δμ^eq/2T).
            
            
            Combining the expression (<ref>) with (<ref>) and (<ref>), we arrive at the following formulas for the coefficients of the rates
            
                λ_ds   ≈    4 G_F^2 sin^2θ_Ccos^2θ_C Λ_ud→ su,
                λ_ud^ℓ   ≈    4 G_F^2 cos^2θ_C Λ_uℓ→ dν,
                λ_us^ℓ   ≈    4 G_F^2 sin^2θ_C Λ_uℓ→ sν,
                λ_eμ   ≈    4 G_F^2 Λ_μν_e→ν_μ e.
            
            When leptonic contributions can be neglected, as it is usually the case in the evaluation of the bulk viscosity in neutron stars, the only relevant coefficient is λ_1=λ_ds, following the conventions in <cit.>. In the remaining of the paper, we will give an estimate of λ_1 at strong coupling using a holographic model.
            
            
            
            
            
            
            § HOLOGRAPHIC MODEL
            
            
            
            We will use a holographic model dual to a SU(N_c) gauge theory with N_f flavors to compute the rate. We use a bottom-up approach, where we construct a gravitational action with the necessary ingredients to capture the physics of interest, but that does not correspond to a string theory construction where both sides of the duality are known.
            
            The glue sector is given by an action consisting of just the Einstein-Hilbert action plus a negative cosmological constant
             
            S_c = 1/16π G_5∫ d^5 x  √(-g)[ R - 2Λ],  Λ=-6/L^2 . 
             
            In the absence of matter, the maximally symmetric solution is an AdS_5 space with radius L. Including the flavor sector will modify the geometry, but it will still asymptote to the same AdS_5, so we can apply the usual holographic dictionary between the fields in the gravity theory and sources and operators in the gauge theory dual.
            
            The flavor sector consists of U(N_f)_L× U(N_f)_R gauge fields, L_M and R_M, dual to the flavor currents, and a complex scalar field X in the bifundamental representation dual to a quark bilinear. The action for the flavor fields is the one commonly used for holographic QCD models, originating with <cit.>
            S_f = 1/16π G_5N_f/N_c∫ d^5 x √(-g)  Tr[- | D X|^2 + 3 | X|^2 -L^2/4( F_(R)^2 + F_(L)^2) ] .
             
            The field strength is defined as F_(A)MN = ∂_M A_N - ∂_N A_M - i[ A_M,A_N]. The covariant derivative of the scalar field is defined as follows 
             
            D_N X = ∂_N X - i L_N X + i X R_N ,      D_N X^† = ∂_N X^† + i X^† L_N - i  R_N X^† .
             
            The mass of the scalar is fixed in such a way that the field is dual to a scalar operator of scaling dimensions Δ=3, as corresponds to a quark bilinear.
            
            The action is fixed up to an overall normalization of the scalar and gauge fields. In <cit.> (see also <cit.>) the normalization was fixed by matching the large Euclidean momentum form of current correlators derived from the holographic model to the OPE of QCD. The gauge fields and scalars are rescaled to put the flavor sector of the action in the form  
            
            S_f =∫ d^5 x √(-g)  Tr[g_X^2(- | D X|^2 + 3 | X|^2) -1/4 g_5^2( F_(R)^2 + F_(L)^2) ] .
             
            Then, the matching to QCD in the UV fixes 
            
                1/g_5^2=N_c/12π^2,   g_X^2=N_c/4π^2 .
            
            
            The equations of motion for the metric are Einstein's equations with a cosmological constant and the energy-momentum tensor sourced by the gauge fields and the scalars
            
                R_MN-1/2g_MNR+ g_MNΛ=N_f/N_cT_MN.
            
            The equations of motion for the gauge and scalar fields read
            1/√(-g) D_M [√(-g) F^MN_(R)]    =     j^N_(R) ,
            1/√(-g) D_M [√(-g) F^MN_(L)]    =    j^N_(L) ,
            1/√(-g) D_M [√(-g)  D^M X  ]    =      -3 X .
            
            
            where the bulk currents appearing in the equations of the gauge fields are
            
            j^M_(R)    =    i [( D^M X)^† X - X^† D^M X] , 
            
            j^M_(L)    =    i [ ( D^M X) X^† - X (D^M X)^†] .
            
            We will look for solutions dual to a deconfined state of the gauge theory at nonzero temperature and baryon density, which are charged black brane geometries. In order to simplify the analysis we will turn off the scalar field, which amounts to having zero quark mass and condensate for all the flavors. In this case, imposing the condition of beta equilibrium fixes all quark chemical potentials to be equal, since deviations from symmetric quark matter due to nonzero quark masses are absent. Since we are ultimately interested in comparing with QCD results, we will fix the rank of the flavor fields N_f=3.
            
            
            
            
             §.§ Background solution
            
            
            The metric of the charged black brane takes the general form, in Poincaré coordinates,
            
                ds^2 = L^2z^2(-f(z)dt^2+dz^2f(z)+dx ^2),      f(z)=1-M z^4/z_h^4+Q^2 z^6/z_h^6 .
            
            The AdS_5 asymptotic boundary is at z=0, and the horizon is at the value where the black body factor vanishes f(z_h)=0. The values of M and Q depend on the background scalar and gauge fields. The leading term in the asymptotic expansion of the background scalar at the boundary determines the quark masses 
            
                X ∼([ m_u        ;     m_d    ;         m_s ])z+O(z^3) .
            
            We will set all quark masses to zero m_u=m_d=m_s=0, which allows to have a trivial solution of the scalar X^ bkg=0. The background gauge fields dual to the quark densities are
            
                (L_0^ bkg)^a_ b=(R_0^ bkg)^a_ b=μ_q/2(1-z^2/z_h^2)δ^a_ b .
            
            Where μ_q=μ_u=μ_d=μ_s is the quark chemical potential, which is the same for all flavors at beta equilibrium. The backreaction over the metric is limited to the gauge fields, so the geometry is the AdS Reissner-Nordstrom charged black brane (<ref>), with 
            
                M=1+Q^2 ,      Q^2=z_h^2μ_q^2/2 .
            
            The temperature in the dual field theory is equal to the Hawking temperature, given by
            
                T = 1π z_h(1-Q^22) .
            
            One can use these formulas to obtain the position of the horizon as a function of the temperature and quark chemical potential
            
                z_h=2/μ_q(√(1+(π T/μ)^2)-π T/μ) .
            
            Later we will often use
            
                T̅= z_h T ≈2 T/μ_q .
            
            
            
            
             §.§ Two-point functions of flavor currents
            
            
            
            The spectral function entering the formulas for the rates can be obtained from the imaginary part of the retarded correlator of the left flavor currents. Following the usual holographic dictionary, we can extract the correlator from the solutions to the linearized equations of fluctuations. We will impose the boundary conditions 
            
                lim_z→ 0 L_μ(x,z)=l_μ(x),  lim_z→ 0R_μ(x,z)=0,  lim_z→ 01/zX(x,z)=0.
            
            This turns off the sources for the right flavor currents and the quark bilinears and sets l_μ(x) as the source for the left flavor currents. The boundary asymptotic expansion of the Fourier transform of the left gauge fields is
            
                (L_μ)^a_ b(ω,k,z)∼( δ^ ν_μ+(c_μ^ ν)_ab z^2 logz/L+z^2 (g_μ^ ν)_ab+⋯)(l_ν)^a_ b(ω,k).
            
            Where, keeping the dependence on the quark masses,
            
                (c_μ^ ν)_ab = 12((k_α k^α+m_a^2+m_b^2)δ_μ^ν - k_μ k^ν).
            
            The two-point function of the left flavor current can be obtained from the one-point function identifying the coefficient of the source l_μ. The one-point function is extracted from the boundary value of the regularized canonical momentum
            
                J_q L^μ=-lim_z→ 0L^2/16π G_5N_f/N_c[√(-g) F_L^zμ+(logz/L+log(LΛ))(L D_α(√(-γ) F_L^αμ)-1/L√(-γ) j_(L)^α)],
            
            where γ_μν=g_μν is the induced metric on a radial slice and Λ is an energy scale of the theory. The additional terms are obtained from the variation of the counter-term action
            
                S_c.t=L/16π G_5N_f/N_c(logz/L+log(LΛ))∫ d^4 x√(-γ)[ L^2/4 F_L^μνF_L μν+(D_μ X)^† D^μ X] .
            
            
            Using the expansion (<ref>), the one-point function to linear order in the fluctuation is
            
                (J_q L^μ)^a_ b= L^3/8π G_5N_f/N_c G^μν_ab(l_ν)^a_ b,     G^μν_ab=-η^μα((g_α^ ν)_ab-(c_α^ ν)_ablogΛ L/e^1/2)   .
            
            Taking the variation with respect to the source gives the two-point function
            
                (J_q L^μ)^a_ b (J_q L^ν)^a'_ b'= L^3/8π G_5N_f/N_c G_ab^μνδ^a_ b'δ^a'_b  .
            
            In cases where the holographic dual is a known gauge theory L^3/8π G_5∼ N_c^2, so the overall factor appearing on the correlator is expected to be proportional to N_c N_f. In the following, we will denote the normalization constant simply as
            
                N=L^3/8π G_5N_f/N_c .
            
            The fit of the model to QCD at large Euclidean momentum (<ref>) and (<ref>) fixes
            
                N= N_c/6π^2 .
            
            The spectral function is obtained from the imaginary part of the retarded correlator, which can only receive contributions from the coefficient of the O(z^2) term in the boundary expansion (<ref>)
                ρ^μν_ab,ba=2 Nη^μα Im (g_α^ ν)_ab .
            
            In order to compute the correlators of flavor currents involved in the formulas for the weak reaction rates, we must consider linearized fluctuations of the gauge fields around the background solution. We can restrict to fluctuations that are off-diagonal in flavor, and take into account that the background is diagonal. When the quark masses are zero and the scalar profile is trivial, the energy-momentum tensor of the gauge fields is at least quadratic in the fluctuations. Therefore, metric fluctuations can be turned off. Furthermore, the fluctuations of the left gauge fields decouple from the right gauge field and the scalar, so it is enough to restrict the analysis to the left gauge fields.
            
            
            
            
             §.§ Linearized fluctuations
            
            
            We will work in the radial gauge L_z=R_z=0 and expand the fluctuations in Fourier modes
            
                Y(x,z)=∫dω/2π∫d^3 k/(2π)^3 e^-iω x^0+ik·x Y(ω,k,z) ,  
                Y=L_μ,R_μ,X,X^† .
            
            Our fluctuations correspond to the off-diagonal flavor components Y≡ Y^a_ b, but we will omit flavor indices unless there is some ambiguity.
            
            It will be convenient to rescale coordinates by the horizon position 
            
                z=z_h Z,    x^μ=z_h X^μ ,
            
            and work with the dimensionless frequency and  momentum  
            
                ω̅=z_h ω,  k̅=z_h k ,  (k̅=|k̅|) .
            
            
            We are interested in temperatures much smaller than the chemical potential T/μ_q ≪ 1. Since the integral over frequencies that determines the rates (<ref>) is concentrated on frequencies smaller or of the order of the temperature, we can use a small frequency approximation ω/μ≪ 1. We will use the approach of <cit.> to compute low-frequency correlators. When the temperature is zero, the background solution is an extremal black brane that develops an AdS_2 throat at the horizon. One can then divide the geometry into an inner and outer region relative to this throat. When the temperature is non-zero, but very small compared to the chemical potential, there is still a throat but the inner geometry is a non-extremal AdS_2 with a horizon. From (<ref>), we see that the near-extremal solutions are for Q^2≲ 2. Let us parametrize
            
                Q^2=2/Z_*^6,     Z_*-1=|ω̅| Z_*^2/12 ζ_0,     Z_*-Z=|ω̅| Z_*^2/12 ζ .
            
            The temperature in units of the horizon position is πT̅=1-Z_*^-6. This gives the approximate relation
            
                ζ_0≈|ω̅|/2πT̅=|ω|/2π T .
            
            Expanding the metric (<ref>) for |ω̅|→ 0 and Z_*≈ 1 (since T̅≪ 1), one gets
            
                ds^2≈L^2/12 ζ^2(-F(ζ)dτ̅^2+dζ^2/F(ζ))+L^2dX ^2 ,      F(ζ)=1-ζ^2/ζ_0^2,  τ̅=|ω̅| X^0=|ω| x^0  .
            
            This is AdS_2×ℝ^3 Schwarzschild geometry with an AdS_2 radius L^2__AdS_2=L^2/12 and a horizon at ζ=ζ_0. In the zero temperature limit Z_*→ 1, ζ_0→∞ the throat extends to values ζ→∞.
            
            The equations for linearized fluctuations can be solved in the inner region imposing an ingoing boundary condition at the horizon ζ→ζ_0. In the region towards the end of the AdS_2 throat, the solutions have an expansion
            
                L_μ∼ A_μ^-ζ^Δ_-+A_μ^+ ζ^Δ_+ ,    Δ_+>Δ_-.
             
            This matches with the behaviour of solutions in the outer region close to the horizon Z→ 1
                L_μ = C^-_μη_-(Z)+C^+_μη_+(Z),  η_±(Z)∼ (1-Z)^-Δ_± ,
            
            which allows the identification
            
                C^±_μ=A_μ^±(|ω̅|/12)^Δ_± .
            
            The ingoing condition fixes a relation between A_μ^+ and A_μ^- that translates into a relation between C_μ^+ and C_μ^-:
            
                C_μ^+=𝒢(ω,k) C_ν^- ,
            
            with 𝒢(ω,k) usually interpreted as an infrared (IR) Green's function associated with a 0+1 CFT dual to the AdS_2 gravitational theory. This IR function determines the Green's function of the full theory. In order to see this, we need to translate the above relation between coefficients to the outer normalizable and non-normalizable solutions
            
                L_μ = l_μϕ_NN(Z)+g_μϕ_N(Z),
            
            with expansions close to the boundary Z→ 0
                ϕ_NN(Z)∼ 1+k̅^2/2Z^2log Z+O(Z^4),  ϕ_N(Z)=Z^2+O(Z^4) .
            
            The two sets of outer solutions are related by the connection coefficients
            
                η_±(Z)=α_±ϕ_NN(Z)+β_±ϕ_N(Z) .
            
            Thus, the full solution is
            
                L_μ =C_μ^-( η_-(Z)+𝒢η_+(Z))=C_μ^-(α_- +α_+𝒢)ϕ_NN(Z)+C_μ^-(β_- +β_+𝒢)ϕ_N(Z) .
            
            Fixing the coefficient of the non-normalizable solution to be the source l_μ, we get that the coefficient of the normalizable term is
            
                g_μ=β_- +β_+𝒢/α_- +α_+𝒢 l_μ .
             
            More precisely, the connection coefficients and IR Green's function depend on whether the fluctuation is transverse or longitudinal. The calculation of the solutions and connection coefficients up to O(ω^2/μ_q^2) corrections is detailed in Appendix <ref>. It turns out that at leading order in ω/T the longitudinal components do not contribute.
            
            
            
            
             §.§ Spectral function
            
            
            Since only the transverse components of the left gauge fields are relevant at low frequencies, the spectral function determined by (<ref>) is (ρ^00≈ 0, ρ^0i≈ 0),
            
                ρ^ij_ab,ba≈ Nμ_q^2/2(δ^ij-k^i k^j/k^2)ρ_L(ω/2π T,k̅,T̅) ,  ρ_L(ω/2π T,k̅,T̅)=β_+α_- - β_-α_+(α_-+ α_+Re 𝒢_L)^2 + (α_+)^2(Im 𝒢_L)^2Im 𝒢_L .
            
            The IR Green's function of the transverse modes of the left gauge fields is
            
                𝒢_L(ω̅,k̅,T̅)≈Γ(-ν_k) Γ(1/2-i  ω/2π T+ν_k)/Γ(ν_k) Γ(1/2-i  ω/2π T-ν_k)(πT̅/12)^2ν_k.
            
            Up to corrections suppressed by factors of T/μ_q. At large spatial momentum the IR Greens' function approaches
            
                𝒢_L(ω̅,k̅,T̅)∼ -cos(πk̅/√(12)+iω/2T)/sin(πk̅/√(12))(πT̅/12)^k̅/√(3) .
            
            At low temperatures, there is a strong suppression due to the T̅-dependent factor. When T/μ_q→ 0 (T̅→ 0), there is a strong suppression of the correlator already at small momenta
            
                𝒢_L∼(πT̅/12)e^-k̅^2/12log12/πT̅ .
            
            The support of the Green's function is concentrated on values k̅≲ |log(T/μ_q)|^-1≪ 1. In this case, we can approximate the Green's function by
            
                𝒢_L(ω̅,k̅,T̅)≈(πT̅/12)e^-(2ν_k-1)log12/πT̅(iω/π T+ (2ν_k-1))+O((2ν_k-1)^2,iω̅(2ν_k-1)) .
            
            The connection coefficients are real and only depend on the momentum. We have computed them numerically and determined their asymptotic behaviour at large spatial momentum (c.f. Appendix <ref>)
            
                α_±∼∓ 6^∓k̅/√(12)k̅^-1/2 ,
                   β_±∼∓ 6^∓k̅/√(12) k̅^3/2 logk̅.
            
            Considering the large momentum asymptotic behaviour of the IR Green's function (<ref>) and connection coefficients (<ref>), we see that the spectral function vanishes exponentially fast at large spatial momentum. A special point is k̅=3 (ν_k=1), where 𝒢_L diverges and the spectral function has its first zero. For a numerical estimate, one can thus restrict the momentum integral to smaller values.
            
            When T/μ_q→ 0 we can approximate the connection coefficients by their k̅→ 0 values, that we can compute analytically
            
                α_- ≈ 1+O(ν_k-1/2),   β_-≈ -3/2(2ν_k-1)+O[(ν_k-1/2)^2],
                   α_+≈4/3(2ν_k-1)+O(1),  β_+≈ 4+O(ν_k-1/2).
            
            Thus β_+α_–β_-α_+≈ 6 and
            
                ρ_L(ω/2π T,k̅,T̅)≈6(2ν_k-1)^2(2ν_k-1)^2 + (4 A_k(T̅) ω/3π T)^2  A_k(T̅)ω/π T≈ 6A_k(T̅)ω/π T .
            
            Where we have defined
            
                A_k(T̅)=(πT̅/12)e^-(2ν_k-1)log12/πT̅ .
            
            
            Numerically, at low frequencies the IR Green's function is well approximated by
            
                𝒢_L(ω̅,k̅,T̅)≈   Γ(-ν_k) Γ(1/2+ν_k)/Γ(ν_k) Γ(1/2-ν_k)(πT̅/12)^2ν_k(1-i/2tan(πν_k)ω/T +
                   + π ^2 tan ^2(πν_k )-ψ ^(1)(1/2-ν_k )+ψ ^(1)(1/2+ν_k)8π^2ω^2/T^2)+O(ω^3/T^3),
            
            where ψ^(1) is a polygamma function. In order to see how good is the approximation we plot the real and the imaginary part of the full IR Green function and its approximated expression. We define a reduced IR Green function
            
                g_L(ω̅,k̅) = Γ(ν_k)/Γ(-ν_k) (12/πT̅)^2ν_k𝒢_L(ω̅,k̅,T̅),
            
            so that the plots will be independent of T̅. In Figure <ref> we plot the full imaginary part of g_L (dashed lines) with the approximated formula (solid lines) for some values of ν_k, as a function of ω/T ∈ (0,2). Then, we plot the full real part of g_L (dashed lines) with the approximated formula (solid lines), as a function of ω/T ∈ (0,2), keeping just the first term in the expansion, Figure <ref>, and then adding also the quadratic correction, Figure <ref>. The values of ν_k are 0.6,0.7,0.75,0.8,0.9 which correspond to colors from blue to light purple. We observe that as k becomes smaller, the real part becomes better approximated by the leading term.
            
            
            
            
            
            § ESTIMATE OF WEAK REACTION RATES AT STRONG COUPLING
            
            
            
            We will proceed to estimate the weak rate. Since all the quark chemical potentials are equal at beta equilibrium and the spectral function (<ref>) vanishes at zero frequency, we can use the simplified formula (<ref>) to compute the rate. Taking into account that only the transverse components contribute to leading order, and rescaling the variables in the momentum integral q^0=2 π T s, k=μ_q k̅/2, 
            
                Λ_ud→ us≈ N^2μ_q^7/2^7π^2∫_-∞^∞ ds ∫_0^∞ dk̅ k̅^2 ρ_L^2(s,k̅,T̅)/sinh^2(π s) .
            
            Introducing (<ref>), and changing variables to k̅^2=3(4ν^2-1),
            
                Λ_ud→ us≈ N^26^2μ_q^7/2^7π^22(πT̅/12)^2
                  ∫_0^∞ ds (2s)^2/sinh^2(π s)∫_1/2^∞ dν 12 ν√(3(4ν^2-1))  e^-2(2ν-1)log12/πT̅ .
            
                Λ_ud→ us≈ N^23√(3)μ_q^5 T^2/4∫_0^∞ ds s^2/sinh^2(π s)∫_1/2^∞ dν ν√((4ν^2-1))  e^-2(2ν-1)log12/πT̅ .
            
            We have
            
                ∫_0^∞ ds s^2/sinh^2(π s)=1/6π
            
            Changing variables ν=1/2+u,
            
                Λ_ud→ us≈ N^2√(3)μ_q^5 T^2/8π∫_0^∞ du  (1+2u) √(u(u+1))  e^-4ulog12/πT̅ .
            
            We use that the main contributions to the integral come from u≪ 1:
            
                ∫_0^∞ du  (1+2u) √(u(u+1))  e^-4ulog12/πT̅≈∫_0^∞ du  u^1/2  e^-4ulog12/πT̅=Γ(3/2)/(4log12/πT̅)^3/2=√(π)/2(4log12/πT̅)^3/2 .
            
            Then,
            
                Λ_ud→ us≈ N^2/128√(3/π)μ_q^5 T^2(log6μ_q/π T)^-3/2 .
            
            We have computed numerically the rate for values of the temperature T/μ_q=10^-1,10^-2,10^-3,10^-4, considering the full dependence of the connection coefficients, and several approximations for the IR Green's function, namely the leading order (<ref>), the second order approximation (<ref>), and the full result (<ref>). In the numerical integration over momentum, we cut the integral at ν_k=1, where the spectral function has its first zero. The exponential suppression already present at this value of the spatial momentum makes higher momentum contributions to be exceedingly small, so we believe this is a good approximation. We show our numerical results together with the analytic approximation at low temperatures (<ref>) in Figure <ref>. We see that indeed the analytic formula approximates well the numerical result at low temperatures, even up to T/μ_q ≲ 10^-2, but for T/μ_q∼ 10^-1 the deviations are already sizeable, although the analytic formula still serves as an estimate of the order of magnitude.
            
            
            
            
            
            
            § DISCUSSION AND OUTLOOK
            
            
            
            In this work, we have derived formulas (<ref>) and (<ref>) for the weak equilibration rate of quark flavor densities n_a (a=u,d,s) to leading order in the Fermi coupling and exact in the QCD coupling. Although we have focused on quark matter, the formulas are derived in an explicitly gauge-invariant way and rely solely on symmetry arguments. Thus, they should be valid for any state of QCD at nonzero temperature, including hadronic (confined) phases. In a confined phase, the natural degrees of freedom are baryons rather than quarks. Baryons are composed of three quarks and can be grouped according to their flavor composition. Therefore, the chemical potential for a baryon B^abc∼ q^a q^b q^c would be μ_B^abc = μ_a + μ_b + μ_c, with μ_a,b,c being the quark chemical potentials corresponding to each flavor. The associated baryon densities should satisfy the conditions
            
                n_a = 3 n_B^aaa + 2 ∑_b ≠ a n_B^aab + ∑_b,c ≠ a n_B^abc,
            
            where the order of the flavor labels is immaterial, and n_B^abc would, in principle, include all contributions from baryon resonances with the same flavor composition and with a mass below the value of the baryon chemical potential, assuming the decay channels to lower resonances become closed. With these identifications, one can combine the expressions in (<ref>) to obtain the rates for the evolution of the density of different baryon species, or rather for the evolution of some combinations of them.
            
            Returning to our analysis, we have obtained the leading order low-temperature behavior of the weak reaction rate in a large-density state of strongly coupled quark matter at beta equilibrium. For simplicity, we have only considered massless quarks, from which we were able to extract an analytic expression (<ref>) for the rate coefficient as defined in (<ref>) and (<ref>), resulting in
            
                λ_ds≈ G_F^2 sin^2 θ_C cos^2 θ_C  N^2/32√(3/π)μ_q^5 T^2 ( log6 μ_q/π T)^-3/2.
            
            A fit to the large momentum behavior of QCD correlators fixes the normalization constant to N = N_c / (6 π^2), Eq. (<ref>). We can compare this with the perturbative result <cit.> including the leading QCD correction <cit.>
                λ_ds≈ G_F^2 sin^2 θ_C cos^2 θ_C 64/4 π^2μ_q^5 T^2 (1 + 4 α_s/9 πlogκ α_s μ_q/T)^4,
            
            where κ≈  0.28(2/3π)^1/2≈ 0.13. The logarithmic enhancement of the rate at low temperatures originates from a non-Fermi-liquid behavior of quark matter due to long-range interactions (see also <cit.>). Although the logarithm should not be too large in order for the perturbative expansion to be valid, the fourth power with which it appears can produce a significant effect on the rate. Considering the strong coupling result, the holographic dual geometry has also been conjectured to describe a non-Fermi (“marginal”) liquid <cit.>. However, the logarithmic factor operates in the opposite way, suppressing the rate. Furthermore, compared to the perturbative result, the logarithm will be much larger, as there are no coupling-dependent factors in the argument suppressing the ratio μ_q/T. As shown in Figure <ref>, the analytic formula (<ref>) provides a good approximation for temperatures T/μ_q below 10^-2, but at higher temperatures, corrections to this formula start to become significant. However, their effect is to further suppress the rate.
            
            Several improvements could be applied to the model, with perhaps the most significant being the inclusion of non-zero quark masses. In the perturbative calculation of the rate, the mass enters the dispersion relation of the fermions in the scattering amplitude and removes the low-energy contributions from the massive quarks. In the holographic model, the scalar field dual to the quark bilinears would be non-zero, and its backreaction can change the near-horizon geometry, thus affecting the rate behavior. A generic expectation is that the AdS_2 throat in the near-horizon geometry would be replaced by a hyperscaling Lifshitz geometry (see <cit.> and references therein), which would probably result in the rate becoming proportional to a different power of the temperature depending on the dynamical exponent and the hyperscaling parameters.
            
            Another extension of the model would be to study phases of paired quark matter, as quark pairing can affect transport properties <cit.>. A paired state would have a quark condensate with a non-trivial flavor structure, which could be captured by turning on a profile of the scalar field reproducing the same structure.
            
            In conclusion, our study provides a framework for understanding weak interaction rates in strongly coupled quark matter and highlights the importance of non-perturbative methods in studying dense QCD phases. Future work should focus on refining these models, incorporating more realistic features such as non-zero quark masses and quark pairing, to enhance our understanding of neutron star interiors and their observational signatures.
            
            
            
            
            
            § ACKNOWLEDGMENTS
            
            
            We thank Niko Jokela for useful discussions. This work is partially supported by the AEI and the MCIU through the
            Spanish grant PID2021-123021NB-I00. A.O. thanks the University of Oviedo for the kind hospitality during the preparation of this work. 
            
            
            
            
            
            
            
            
            § NON-CONSERVATION EQUATIONS OF FLAVOR CURRENTS
            
            
            
            With three massless flavors of quarks q^i=(u,d,s) and one flavor of massless leptons l^a=(ν_e,e) (with only left-handed component for the neutrino), the flavor symmetry group is SU(3)_q_L× SU(3)_q_R× SU(2)_l_L× U(1)_e_R. Unequal mass terms for the quarks and the electron break explicitly the flavor group to U(1)^5, where the quarks and electron U(1)s are vector-like.
            
            The EW sector, in particular, W-boson exchange introduces a term in the effective action of the form (Fermi model)
            
                Δ L_EW=-2√(2) G_F (J_ch^μ)^† J_ch μ,
            
            where the J_ch^μ is the left-charged current
            
                J_ch^μ=ν_e Lγ^μ e_L+cosθ_C u_Lγ^μ d_L+sinθ_C u_Lγ^μ s_L,
            
            with θ_C the Cabbibo angle, which gives the change of basis between mass and flavor eigenstates. Let us introduce the notation
            
                (J_q^μ)^i_ j=q̅_j γ^μ q^i,  (J_l^μ)^a_ b=l̅_b γ^μ l^a .
            
            Then, in terms of current components,
            
                J_ch^μ=(J_l L^μ)^2_ 1+cosθ_C (J_q L^μ)^2_ 1+sinθ_C  (J_q L^μ)^3_ 1,
            
            Where J_l L^μ is the current for the SU(2)_L lepton symmetry and J_q L^μ is the current for the SU(3)_L flavor symmetry. The EW term becomes
            
                Δ L_EW=   -2√(2) G_F [ cos^2θ_C (J_q L^μ)^ 2_ 1 (J_q L μ)^1_ 2+sin^2 θ_C  (J_q L^μ)^ 1_ 3 (J_q L μ)^3_ 1.
                   +cosθ_C sinθ_C( (J_q L^μ)^ 2_ 1 (J_q L^μ)^ 1_ 3+(J_q L^μ)^3_ 1(J_q L^μ)^ 1_ 2)
                   +cosθ_C ((J_l L^μ)^2_ 1(J_q L μ)^1_ 2+(J_l L^μ)^1_ 2(J_q L μ)^2_ 1)
                   .+sinθ_C ((J_l L^μ)^2_ 1(J_q L μ)^ 1_ 3+(J_l L^μ)^ 1_ 2(J_q L μ)^3_ 1)] .
            
            Let us neglect the terms coupling quarks and leptons. We can rewrite the remaining terms in the following way
            
                Δ L_EW=   -2√(2) G_F [ cos^2θ_C  tr(P_11 J_q L^μ P_22 J_q L μ) +sin^2 θ_C  tr(P_11 J_q L^μ P_33 J_q L μ).
                   .+cosθ_C sinθ_C( tr(P_11 J_q L^μ P_32 J_q L μ)  + tr(P_11 J_q L^μ P_23 J_q L μ) )] .
            
            Where we have defined the projectors (P_kl)^i_ j=δ^i_kδ_j^l.
            
            By doing SU(3)_L× SU(3)_R rotations of equal angle we can derive Ward identities for the vector SU(3) flavor currents. The currents transform as
            
                δ_θ J_q^μ=iθ_A [T^A,J^μ_q]
            
            
            Defining J_q^μ=J_q L^μ+J_q R^μ and J_l^μ=J_l L^μ+J_l R^μ. The covariant derivative under a background flavor gauge potential A_μ^f is
            
                D_μ J_q^μ=∂_μ J_q^μ-i[A_μ^f,J_q^μ] .
            
            We will take the normalization for the gauge group generators to be tr(T^A T^B)=1/2δ^AB. Then, the Ward identities are
            
                2tr(D_μ J_q^μ T^A)=
                   2i√(2) G_F[ cos^2θ_C  (tr(P_11 [T^A, J_q L^μ] P_22 J_q L μ)+tr(P_11 J_q L^μ P_22 [T^A,J_q L μ]  )).
                    +sin^2 θ_C  ( tr(P_11 [T^A,J_q L^μ] P_33 J_q L μ)+tr(P_11J_q L^μ P_33 [T^A,J_q L μ]  ))
                   +cosθ_C sinθ_C( tr(P_11 [T^A,J_q L^μ] P_23 J_q L μ)+tr(P_11 J_q L^μ P_23 [T^A,J_q L μ]  ). 
                   .. + tr(P_11 [T^A,J_q L^μ] P_32 J_q L μ)+ tr(P_11 J_q L^μ P_32[T^A, J_q L μ]  ) )] .
            
            Combining the traces and using the cyclic property we have that
            
                D_μ J_q^μ =
                   i√(2) G_F( cos^2θ_C  [ J_q L μ , P_11 J_q L^μ P_22 +P_22 J_q L^μ P_11 ].
                    +sin^2 θ_C  [ J_q L μ , P_11 J_q L^μ P_33 +P_33 J_q L^μ P_11 ]
                   .+cosθ_C sinθ_C(  [ J_q L μ , P_11 J_q L^μ P_23 +P_23 J_q L^μ P_11 ]+  [ J_q L μ , P_11 J_q L^μ P_32 +P_32 J_q L^μ P_11 ])) .
            
            In general, there will also be contributions on the right-hand side depending on the quark masses, but they should vanish for diagonal components.
            
            We can do a similar derivation taking into account the leptons if we introduce projectors
            
                (P̃_ai)^b_ j=δ_a^bδ^i_j,  (P̃^T_ia)^j_ b=δ^a_bδ^j_i,
            
            Then
            
                P̃J^μ_qP̃^T∼ J_l^μ,  P̃^T J^μ_lP̃∼ J_q^μ.
            
            And introduce the SU(2)_L× SU(2)_R rotations of the leptons, such that the diagonal rotations transform the vector currents as
            
                δ_ϕ J_l^μ=iϕ_α[τ^α,J^μ_l].
            
            With similar normalization for the generators tr(τ^ατ^β)=1/2δ^αβ.
            
            
            
            
            
            
            § LINEARIZED EQUATIONS OF MOTION OF FLUCTUATIONS
            
            
            
            The transverse spatial components of the gauge fields k^i L_i=k^i R_i=0 decouple from the other fluctuations. Their equations are
             
            0    =   L_i”+ (f'/f-1/Z)L_i'+
            ( ω̅^2/f^2-k̅^2/f)L_i  ,
            
            
             0    =   R_i”+ (f'/f-1/Z)R_i'+
            ( ω̅^2/f^2-k̅^2/f)R_i  , 
            
            When the quark masses are nonzero, the time L_0, R_0 and longitudinal L_∥ =k^i/k L_i, R_∥=k^i/k R_i components of the gauge fields are coupled to the scalar fluctuations, but they decouple otherwise. The left gauge field equations are
            
                0   =   L_∥”+(f'/f-1/Z)L_∥'+ω̅^2/f^2L_∥+k̅ω̅/f^2L_0, 
                
                     0   =   L_0”-1/ZL_0'-k̅^2/fL_0-k̅ω̅/fL_∥ ,
                
                     0   =   ω̅L_0'+k̅ f L_∥' .
            
            The right gauge field equations are
            
                0   =   R_∥”+(f'/f-1/Z)R_∥'+ω̅^2/f^2R_∥+k̅ω̅/f^2R_0,
                
                     0   =   R_0”-1/ZR_0'-k̅^2/fR_0-k̅ω̅/fR_∥,
                
                     0   =   ω̅R_0'+k̅ f R_∥' .
            
            The scalar equations are     
            
                0    =   X”+(f'/f-3/Z)X'+(ω̅^2/f^2-k̅^2/f+3/Z^2 f)X  .
            
            We can simplify the equations in the longitudinal sector by introducing the electric fields
            
                E_L=k̅ L_0+ω̅L_∥,     E_R=k̅ R_0+ω̅R_∥ .
            
            The constraints (<ref>) and (<ref>) fix the time and longitudinal components of the gauge fields in terms of the electric fields and the scalars
            
                L_∥'=ω̅E_L'/ω̅^2-k̅^2 f ,     L_0'=- k̅ fE_L'/ω̅^2-k̅^2 f ,
                   R_∥'=ω̅E_R'/ω̅^2-k̅^2 f ,     R_0'=- k̅ fE_R'/ω̅^2-k̅^2 f ,
            
            The combinations ω̅f/Z(<ref>)+k̅/Z(<ref>), and  ω̅f/Z(<ref>)+k̅/Z(<ref>) produce the equations for the left and right electric fields respectively
            
                0   =   ∂_Z (f/ZE_L'/ω̅^2-k̅^2 f)+1/f ZE_L, 
                
                0   =   ∂_Z (f/ZE_R'/ω̅^2-k̅^2 f)+1/f ZE_R .
            
            
            
            
            § SOLUTIONS IN ADS REISSNER-NORSDTROM
            
            
            
            We want to solve (<ref>) and (<ref>) without the terms depending on the masses, in the charged black brane geometry (<ref>) with parameters (<ref>). We can solve the equations of motion in the near-extremal AdS_2 throat (<ref>) and in the asymptotically AdS_5 metric (<ref>). When |ω̅|→ 0 the two sets of solutions have an overlapping region ζ→ 0 in the throat and Z→ 1 in the outer region. To leading order in the small |ω̅| expansion, the equations of motion of the left gauge fields (<ref>) and (<ref>) in the inner region are
            
                0    =   ∂_ζ^2 L_i +F'F∂_ζ L_i +1/ζ^2 F^2(ζ^2 -k̅^2/12F)L_i ,
                
                 0    =   ∂_ζ^2 E_L + ( F'/F-2/ζk̅^2/121/ζ^2-k̅^2/12F)∂_ζ E_L +1/ζ^2 F^2(ζ^2-k̅^2/12F)E_L .
            
            Outside the AdS_2 throat, we can approximate the geometry by the extremal (zero temperature) Reissner-Norstrom solution, with f(Z)=1-3Z^4+2Z^6. The equations in the outer region to leading order in the small frequency expansion are
            
                0    =   ∂_Z(f(Z)Z∂_Z L_i ) -k̅^2ZL_i+O(ω̅^2) ,
                
                 0    =   ∂_Z(∂_Z E_LZ) -k̅^2Z f(Z)E_L +O(ω̅^2) .
            
            
            
             §.§ Inner solutions of the transverse components
            
            
            Let us first find the solutions to the equations in the inner region (<ref>). At T=0 (F=1), there is a simple solution to the equation of the transverse components
            
                L_i,I(ζ) =√(ζ) ( C_i^(1)Θ(ω) H_ν_k^(1)(ζ)+C_i^(2)Θ(-ω)  H_ν_k^(2)(ζ)) ,
            
            where the coefficients have been chosen so that the solution satisfies ingoing boundary conditions and we have defined
            
                ν_k=√(1/4+k̅^2/12) .
            
            These solutions should match with the solutions of the outer region for ζ→ 0. The leading terms in the expansion are
            
                L_i,I(ζ) ∼  A_i ζ^1/2-ν_k+B_i ζ^1/2+ν_k ,
            
            where
            
                A_i    =      -i2^ν_kΓ(ν_k)/π(C_i^(1)Θ(ω)-C_i^(2)Θ(-ω)) ,
                
                B_i    =   1/2^ν_kΓ(ν_k+1)(C_i^(1)Θ(ω)+C_i^(2)Θ(-ω)+i(πν_k)(C_i^(1)Θ(ω)-C_i^(2)Θ(-ω))  ) .
            
            In terms of the original coordinate Z, related to ζ through (<ref>), the expansion (<ref>) is
            
                L_i,I(ζ) ∼  A_i(|ω̅|/12)^1/2-ν_k[(1-Z)^-1/2+ν_k+𝒢_L(ω̅,k̅) (1-Z)^-1/2-ν_k) ,
            
            where the IR Green's function is 
            
                𝒢_L(ω̅,k̅)=π2^2ν_kΓ(ν_k)Γ(1+ν_k)(-(πν_k)+i sign(ω))(|ω̅|/12)^2ν_k.
            
            At non-zero temperatures, it is still possible to find analytic solutions
            
                L_i,I(ζ) = A_i φ_ν_k^-(ζ)+B_iφ_ν_k^+(ζ) ,
            
            where
            
                φ_ν_k^±(ζ)=(1-ζ^2/ζ_0^2)^-i/2sign(ω)ζ_0(ζ/ζ_0)^1/2±ν_k_2 F_1(1/4-i/2sign(ω)ζ_0±ν_k/2 , 3/4-i/2sign(ω)ζ_0±ν_k/2 ,  1±ν_k ; ζ^2/ζ_0^2) .
            
            The condition that the solution is ingoing fixes
            
                B_i=Γ(-ν_k) Γ(1/2-i  sign(ω)ζ_0+ν_k)/2^2ν_kΓ(ν_k) Γ(1/2-i  sign(ω) ζ_0-ν_k)  A_i .
            
            Using that Z_*≈ 1, we arrive at 
            
                L_i,I(ζ) ∼  A_i(|ω̅|/12 ζ_0)^1/2-ν_k[(1-Z)^-1/2+ν_k+𝒢_L(ω̅,k̅,ζ_0) (1-Z)^-1/2-ν_k) ,
            
            where now
            
                𝒢_L(ω̅,k̅,ζ_0)=Γ(-ν_k) Γ(1/2-i  sign(ω)ζ_0+ν_k)/2^2ν_kΓ(ν_k) Γ(1/2-i  sign(ω) ζ_0-ν_k)(|ω̅|/12 ζ_0)^2ν_k.
            
            In terms of temperature the IR Green's function is approximately
            
                𝒢_L(ω̅,k̅,T̅)≈Γ(-ν_k) Γ(1/2-i  ω/2π T+ν_k)/2^2ν_kΓ(ν_k) Γ(1/2-i  ω/2π T-ν_k)(πT̅/6)^2ν_k.
            
            For frequencies smaller than the temperature, we can expand the IR Green's function as
            
                𝒢_L(ω̅,k̅,T̅)≈Γ(-ν_k) Γ(1/2+ν_k)/2^2ν_kΓ(ν_k) Γ(1/2-ν_k)(πT̅/6)^2ν_k(1-i/2tan(πν_k)ω/T)+O(ω^2/T^2).
            
            The imaginary term produces the leading contribution to the spectral function. We can obtain these first terms in the ω/T by first extracting the ingoing factor from the solutions
            
                L_i,I(ζ)=(1-ζ^2/ζ_0^2)^-i/2sign(ω)ζ_0ψ_i(ζ),
            
            and then expanding the equations in powers of ζ_0 and solving order by order. Changing variables to u=ζ^2/ζ_0^2, the equations for the first two orders ψ_i=ψ_i^0+ζ_0ψ_i^1+⋯ are
            
                0   =   ∂_u(√(u)(1-u)∂_u ψ_i^0)+1-4ν_k^2/16 u^3/2ψ_i^0 ,
                
                 0   =   ∂_u(√(u)(1-u)∂_u ψ_i^1)+1-4ν_k^2/16 u^3/2ψ_i^1+i sign(ω)√(u)(∂_uψ_i^0+1/4 uψ_i^0 ) .
            
            The solutions to zeroth order are
            
                ψ_i^0(u)= A_i^0 φ_ν_k^0,-(u)+B_iφ_ν_k^0,+(u),
            
            with
            
                φ_ν_k^0,±(u)= u^1/4(1± 2ν_k)_2 F_1(1/4±ν_k/2 , 3/4±ν_k/2 ,  1±ν_k ; u ) .
            
            Imposing regularity of this solution when u→ 1 (ζ→ζ_0), i.e. removing logarithmic terms, fixes the relation between the coefficients
            
                B_i^0=Γ(-ν_k) Γ(1/2+ν_k)/2^2ν_kΓ(ν_k) Γ(1/2-ν_k)  A_i^0 .
            
            The next-order solution can be obtained using a Green's function. First we identify the subleading solution when u→ 0 (ζ→ 0), φ^<, and the ingoing solution when u→ 1, φ^>,
            
                φ^<(u)=φ_ν_k^0,+(u) ,  φ^>(ζ)=φ_ν_k^0,-(u)+Γ(-ν_k) Γ(1/2+ν_k)/2^2ν_kΓ(ν_k) Γ(1/2-ν_k)φ_ν_k^0,+(u) .
            
            With these and the Wronskian of the two subleading and ingoing functions we construct a Green's function that maintains the boundary conditions of the leading order solution 
            
                G(u,u')=-1/ν_k{[ φ^<(u)φ^>(u')             ,          u<u'; φ^>(u)φ^<(u')             ,          u>u' ].
            
            Then, taking into account that ψ_i^0=A_i^0φ^>(u), the subleading solution is
            
                ψ_i^1(u)=-i sign(ω)A_i^0∫_0^1 du' G(u,u')√(u')(∂_u'φ^>(u')+1/4 u'φ^>(u') ) .
            
            We can now expand the leading and subleading solutions for u→ 0. One can check that all contributions from ψ_i^1 are subleading compared to the leading term ∼ u^1/4(1-2 ν_k). Also, there is just one contribution to the term ∼ u^1/4(1+2 ν_k), from a total derivative 
            
                ∫_0^1 du' G(u,u')√(u')(∂_u'φ^>(u')+1/4 u'φ^>(u') )∼φ^<(u)lim_u→ 0∫_u^1 du' ∂_u'(√(u')(φ^>(u'))^2/2)=φ^<(u)  (φ^>(1))^2/2 .
            
            Taking this into account, the relevant terms in the expansion are
            
                ψ_i^0    ∼   ( u^1/4(1- 2ν_k)+Γ(-ν_k) Γ(1/2+ν_k)/2^2ν_kΓ(ν_k) Γ(1/2-ν_k)  u^1/4(1+2 ν_k))A_i^0  ,
                ψ_i^1    ∼    -i sign(ω)πtan(πν_k) Γ(-ν_k) Γ(1/2+ν_k)/2^2ν_kΓ(ν_k) Γ(1/2-ν_k)  u^1/4(1+2 ν_k)  A_i^0 .
            
            It is straightforward to check that this reproduces the first two terms in the ω/T expansion of the  IR Greens' function (<ref>).
            
            
            
            
            
             §.§ Inner solutions of the longitudinal components
            
            
            The equation for the electric field in (<ref>) does not have a simple analytic solution, but since we are interested in frequencies ω/T≲ 1, we can apply the same expansion as for the transverse components to find the leading contribution to the spectral function.
            
            We first extract the ingoing factor from the solutions
            
                E_L,I(ζ)=(1-ζ^2/ζ_0^2)^-i/2sign(ω)ζ_0ψ_i(ζ),
            
            and then expand the equations in powers of ζ_0 and solve them order by order. Changing variables to u=ζ^2/ζ_0^2, the equations for the first two orders ψ_i=ψ_i^0+ζ_0ψ_i^1+⋯ are
            
                0   =   ∂_u(u^3/2∂_u ψ_i^0)+1-4ν_k^2/16 u^1/2(1-u)ψ_i^0 ,
                
                 0   =   ∂_u(u^3/2∂_u ψ_i^1)+1-4ν_k^2/16 u^1/2(1-u)ψ_i^1+i sign(ω)u^3/2/1-u(∂_uψ_i^0+3-u/4 u(1-u)ψ_i^0 ) .
            
            The solutions to zeroth order are
            
                ψ_i^0(u)= A_i^0 φ_ν_k^0,-(u)+B_iφ_ν_k^0,+(u),
            
            with
            
                φ_ν_k^0,±(u)= u^-1/4(1∓ 2 ν_k)_2 F_1(-1/4±ν_k/2 , 1/4±ν_k/2 ,  1±ν_k ; u ) .
            
            Imposing regularity of this solution when u→ 1 (ζ→ζ_0), i.e. removing logarithmic terms, fixes the relation between the coefficients
            
                B_i^0=Γ(-ν_k) Γ(3/2+ν_k)/2^2ν_kΓ(ν_k) Γ(3/2-ν_k)  A_i^0 .
            
            It turns out that this makes the solution vanish at the horizon, something that will be relevant below.
            
            The next-order solution can be obtained in terms of a Green's function. First, we identify the subleading solution when u→ 0 (ζ→ 0), φ^<, and the ingoing solution when u→ 1, φ^>,
            
                φ^<(u)=φ_ν_k^0,+(u) ,  φ^>(ζ)=φ_ν_k^0,-(u)+Γ(-ν_k) Γ(3/2+ν_k)/2^2ν_kΓ(ν_k) Γ(3/2-ν_k)φ_ν_k^0,+(u) .
            
            With these and the Wronskian of the two subleading and ingoing functions we construct a Green's function that maintains the boundary conditions of the leading order solution 
            
                G(u,u')=-1/ν_k{[ φ^<(u)φ^>(u')             ,          u<u'; φ^>(u)φ^<(u')             ,          u>u' ].
            
            Then, taking into account that ψ_i^0=A_i^0φ^>(u), the subleading solution is
            
                ψ_i^1(u)=-i sign(ω)A_i^0∫_0^1 du' G(u,u')u'^3/2/1-u'(∂_u'φ^>(u')+3-u'/4 u'(1-u')φ^>(u') ) .
            
            We can now expand the leading and subleading solutions for u→ 0. One can check that all contributions from ψ_i^1 are subleading compared to the leading term ∼ u^-1/4(1+2 ν_k). Also, there is just one potential contribution to the term ∼ u^-1/4(1-2 ν_k), from a total derivative 
            
                ∫_0^1 du' G(u,u')u'^3/2/1-u'(∂_u'φ^>(u')+3-u'/4 u'(1-u')φ^>(u') )
                       ∼φ^<(u)lim_u→ 0∫_u^1 du' ∂_u'(u'^3/2/1-u'(φ^>(u'))^2/2)=φ^<(u)  lim_u'→ 1(φ^>(u'))^2/2(1-u')=0 .
            
            Taking this into account, the relevant terms in the expansion are
            
                ψ_i^0    ∼   ( u^-1/4(1+2 ν_k)+Γ(-ν_k) Γ(3/2+ν_k)/2^2ν_kΓ(ν_k) Γ(3/2-ν_k)  u^-1/4(1-2 ν_k))A_i^0  ,
                ψ_i^1    ∼    0 ×  A_i^0  u^-1/4(1-2 ν_k) .
            
            Thus, the leading ω/T correction actually vanishes for the longitudinal components. This implies that for the calculation of the spectral function, it is enough to compute the transverse contribution.
            
            
            
            
            
             §.§ Outer solutions of the longitudinal components
            
            
            Although we will not need these solutions to compute the leading contributions to the rates, it will be useful to study them as they can be found analytically.
            
            We are looking for solutions of the electric field in (<ref>). First we will change coordinates u=Z^2, so the equations takes the form of a Schroedinger equation
            
                E_L,O”(u)-k̅^24u(2u+1)(u-1)^2E_L,O=0 .
            
            We can transform the above equation into a hypergeometric equation by the following change of coordinate and field redefinition
            
                E_L,O(u) = (1+2u)^c_±(1-u)^b_±t_±(u),      c_± = 12±ν_k,     b_± = 1-c_± = 12∓ν_k,
                   x=3u1+2u,           u =x3-2x.
            
             Then, we get the following hypergeometric equation
                
                x(1-x)t_±”(x)+(± 2ν_k -1)x t_±'(x) -k^212 t_±(x)=0.
            
                We can compare the above equation to the standard form x(1-x)t_±”(x) + (c_±-(a_±+b_±+1)x)t_±'(x) -a_± b_± t(x) =0, to get
            
                c_± = 0,         a_± = {1/2∓ν_k,-1/2∓ν_k},     b_± = {-1/2∓ν_k,1/2∓ν_k}.
            
            A general solution can be written as a linear combination of a normalizable and a non-normalizable mode at the boundary. In terms of the original radial variable, they have the form
            
                E_O^(0) = α ϕ_NN(Z) + β ϕ_N(Z),
                   ϕ_NN(Z) = (1-Z^2)^1/2+ν_k(1+2Z^2)^1/2-ν_k[((12-2ν_k^2)H_-1/2+ν_k-ν_k^2(-1+log 3)+
                             + 112(1+3log 3))3Z^21 + 2Z^2 _2F_1(12+ν_k,32+ν_k,2,3Z^21 + 2Z^2)+
                             +1Γ(-1/2+ν_k)Γ(1/2+ν_k)G_2,2^2,2(
                [ 3Z^21+2Z^2 ]|[ 1/2-ν_k,32-ν_k;            0,1;                ])],
                   ϕ_N(Z) = (1-Z^2)^1/2+ν_k(1+2Z^2)^1/2-ν_k3Z^21+2Z^2 _2F_1(12+ν_k,32+ν_k,2,3Z^21 + 2Z^2),
            
            where H_-1/2+ν_k is the (-1/2+ν_k)^th harmonic number and G is the Meijer function.
            
            The non-normalizable and normalizable solutions have been chosen so that their near-boundary expansion is
            
                ϕ_NN(Z) = 1 + 32(-1+4ν_k^2)Z^2log Z + O(Z^4log Z),               ϕ_N(Z) = Z^2 + O(Z^4) .
            
            We want to determine the coefficients α and β for solutions with a selected behaviour close to the horizon Z→ 1. The expansion close to the horizon is 
            
                E_L,O(Z) ∼ A(1-Z)^1/2+ν_k + B(1-Z)^1/2-ν_k +...         Z→ 1,
            
            These match with the expansions of the φ_ν_k^0,± inner solutions (<ref>) in the ζ→ 0 limit.
            By requiring that A = 1 and B = 0 we then determine the value of α and β for the minus mode while, if we impose A = 0 and B = 1 we find the coefficients for the plus mode. They turn out to be
            
                α_∓ =6^±ν_k -1/2Γ (±ν_k +1)√(π) Γ(±ν_k +3/2),
                   β_- = -2^ν_k -5/2 3^ν_k -1/2ν_k  Γ (ν_k ) ((6-24 ν_k ^2) H_ν -1/2-12 ν_k ^2 (1+log (3))+1+log (27))√(π) Γ(ν_k +3/2),
                   β_+ = -πν_k  ( (2 πν_k ) ((6-24 ν_k ^2) H_ν_k -1/2-12 ν_k ^2 (log (3)-1)+1+log (27))+3 π(4 ν_k ^2-1)  ^2(πν_k ))2^-ν_k +3/2 3^ν_k +1/2Γ(1/2-ν_k ) Γ(3/2-ν_k ) Γ (2 ν_k +1) .
            
            
            Since in order to compute the rates we have to perform an integral over all momentum, it is interesting to ascertain the behaviour of the spectral function at large spatial momentum, which will be in part determined by the behaviour of the connection coefficients. From the analytic formulas above one can see that, for ν_k→∞, their asymptotic form is
            
                α_- ∼ 6^ν_k-1/2(πν_k)^-1/2,  α_+ ∼ -6^-ν_k-1/2(πν_k)^-1/2(πν_k) ,
                   β_-∼ 6^ν_k+1/2ν_k^3/2log(3^1/2e^-1/2e^γ_Eν_k),
                   β_+ ∼  6^-ν_k+1/2ν_k^3/2√(π)(1-(πν_k)/πlog(3^1/2e^-1/2e^γ_Eν_k)) .
            
            We can recover this behaviour of the connection coefficients by using a WKB approximation at large spatial momentum. The equation of motion (<ref>) is already in Schroedinger form with a potential
            
                U_E(u)=k̅^24u(2u+1)(u-1)^2 .
            
            We expand the solutions as
            
                E_L,O(u) = Cexp(k̅ S_0(u) + S_1(u) + O(1/k̅)).
            
            The equation for the leading term in the 1/k̅ expansion is
            
                S_0'(u)^2-18 u^4-12 u^3+4 u = 0,
            
            which has an analytical solution
            
                S_0(u) = ±1√(3)tanh ^-1( √(3u2u+1)) + c_0.
            
            Then, the subleading term is given by
            
                S_1(u) = log(√(2) √(1- u) √(u (1+2u))) + c_1.
            
            Then the solution up to this order reads
            
                E_L,O(u)≈ψ(u)= C_± √(1- u) √(u (1+2u))exp(±k̅√(3)tanh ^-1( √(3u1+2u))).
            
            The WKB solution near the boundary has the following expansion
            
                ψ(u)∼ C_± u^1/4e^± k√(u)          u→ 0.
            
            Note that the two terms in the solution directly correspond to the "plus" and "minus" modes at the horizon
            
                ψ(u) ∼C_±(11-u)^±k̅/√(12)         u→ 1.
            
            We have to fix the overall coefficient to properly normalize the WKB solutions in the original coordinate z which are
            
                E(z) = C_± √(1- z^2) √(z) (1+2z^2)^1/4exp(±k̅√(3)tanh ^-1( √(3z^21+2z^2))).
            
            Then, the overall coefficients are found to be
            
                C_± = 6^∓k̅/√(12).
            
            Now we have to match these results with the solutions close to the boundary and read the boundary coefficients. Close to the boundary, the potential is to leading order
            
                U_E(u) ∼k̅^24u.
             
            A solution for this approximated potential can be given in terms of modified Bessel functions
            
                ψ(u)≈ c_1 √(u) K_1(k̅ √(u))+ c_2√(u) I_1(k̅ √(u)).
            
            The expansion for large k̅ is
            
                √(u) K_1(k̅ √(u))∼√(π2k̅)u^1/4e^-k̅√(u),                √(u) I_1(k̅ √(u))∼u^1/4√(2πk̅)e^k̅√(u),
            
            which is consistent with the WKB solution. We fix the coefficient c_1,2 with the formulas for C_± so that the WKB solution for z→ 0 and the solution close to the boundary for large k̅ coincide. Finally, we can read off from the z→ 0 expansion the coefficients α_± and β_± which are respectively the coefficients of the non-normalizable and normalizable mode. In a compact form, they are
            
                α_±∼∓ 6^∓k̅/√(12)k̅^-1/2 ,
                   β_±∼∓ 6^∓k̅/√(12) k̅^3/2 logk̅.
            
            Comparing to (<ref>), and taking into account ν_k∼k̅/√(12) in this limit, the WKB approximation captures the leading k̅ dependence up to the oscillatory factors.
            
            
            
            
             §.§ Outer solutions of the transverse components
            
            
            We are looking for solutions of the transverse components in (<ref>). We do not have a close analytic solution in this case, so we must resort to numerics. The general solution is a linear combination of two modes η_L,∓(Z) which when Z→ 1 match with the inner solutions (<ref>) for ζ→ 0,
            
                η_L,±(Z) ∼(1-Z)^-1/2∓ν_k + ...          Z→ 1.
            
            There is another basis of solutions we can use to write a general solution, consisting of the normalizable and non-normalizable modes, with a boundary expansion at Z→ 0
                ϕ_NN(Z) = 1 + 32(-1+4ν_k^2)Z^2log Z + O(Z^4log Z),               ϕ_N(Z) = Z^2 + O(Z^4)
            
            The change of basis is determined by the connection coefficients α_±,β_±:
            
                η_L,±(Z) = α_±ϕ_NN(Z) +β_±ϕ_N(Z).
            
            The connection coefficients can be determined from the solutions by matching the η_L ± solutions and their first derivative with a linear combination of the normalizable and non-normalizable solutions at any intermediate point 0<Ẑ<1:
            
                [ α_±; β_± ]
                =
                1W(Z)[ -∂_Zϕ_N(Z)     ϕ_N(Z); ∂_Zϕ_NN(Z)   -ϕ_NN(Z) ][    η_L,±(Z); ∂_Zη_L,±(Z) ]|_Z = Ẑ ,
            
            where W(Z) = ϕ_N(Z)∂_Zϕ_NN(Z)-∂_Zϕ_N(Z)ϕ_NN(Z).  We will obtain the connection coefficients using this formula, by finding numerically the η_L ± solutions using shooting from the horizon, and the ϕ_N,ϕ_NN solutions using shooting from the boundary. We use Ẑ=0.2 as the matching point, but we have checked that changing this value through the interval of numerical integration does not affect the result within our numerical precision.
            
            For the shooting from the boundary, we introduce a cutoff at Z=10^-5 and use a series expansion to give initial values for the solution and its first derivative at the cutoff. The expansion is 
            
                ϕ_N(Z) = ∑_n = 1^N_a a_n Z^2n,
                   ϕ_NN(Z) = b_0 + ∑_n = 2^N_b b_n Z^2n +log(Z)∑_n = 1^N_c c_n Z^2n.
            
            Where we take N_a=8, N_b=6 and N_c=7. The first coefficients of the expansion given that a_1 = b_0 = 1 are
            
                a_2 = k̅^28,    a_3 = 1 + k̅^4192,    a_4 = -12 + 5k̅^224 + k̅^69216, ...
                   b_2 = -3k̅^464,    b_3 = k̅^26 - 7k̅^62304  ...      c_1 =k̅^22,    c_2 = k̅^416, …
            
            
            For the shooting from the horizon we introduce a cutoff at Z=1-10^-5. We extract the leading power in the horizon expansion from the solutions
            
                η_L,∓(Z) = (1-Z)^-1/2±ν_kp_∓(Z) .
            
            The equation of motion we solve is
            
                ∂_Z^2 p_∓(Z) +(1+2Z(-1 ±ν_k)Z(-1+Z) + f'(Z)f(Z))∂_Z p_∓(Z)+
                   +(±ν_k-1)[(2-5Z ± 2Zν_k)f(Z) + 2(1-Z)Z(6(-1+Z)(1± 2ν_k) -f'(Z))]4 (Z-1)^2 Z f(Z) p_∓(Z).
            
            To give initial values at the cutoff we use the series expansion 
            
                p_∓ = ∑_n=0^N_j j_∓,n(1-Z)^n,
            
            and shooting it from the horizon towards the boundary. We expand up to N_j=8 terms, with the first coefficients in the expansion, given that j_∓,0=1, are
            
                j_∓,1 = 8√(3)√(3+k̅^2)∓ 24 ± 7k̅^212(√(3)√(3+k̅^2)± 3),    j_∓,2 = k̅^2 (49 √(3)√(k̅^2+3)±534)-156 (√(3)√(k̅^2+3)∓3)864 (√(3)√(k̅^2+3)±6), …
            
            
            Although we do not have an analytic formula for the connection coefficients, we can obtain analytic expressions for their asymptotic behaviour at large spatial momentum, by applying the same WKB approximation we used for the coefficients of the longitudinal components. First, we should put the equation of motion for L_i in (<ref>) in Schroedinger form. This is achieved by the change of variables u=Z^2 followed by the redefinition
            
                L_i(u)= (1-u) (1+2 u)^-1/2ψ_L_i(u) .
            
            The equation of motion is now
            
                ψ_L_i”(u)-U_L_i(u)ψ_L_i(u) = 0 ,      U_L_i(u) = k^2/4 (1-u)^2 u (2 u+1)-3 (u+1)/(1-u) (2 u+1)^2 .
            
            At large k̅ the potential is to leading order the same as for the longitudinal modes (<ref>)
                U_L_i(u)∼k^2/4 (1-u)^2 u (2 u+1)=U_E(u) .
            
            Therefore, the large k̅ WKB analysis proceeds in the same way as for the longitudinal components, and the connection coefficients have the save asymptotic behaviour given in (<ref>).
            
            
            We have plotted the connection coefficients in Figures <ref>- <ref>. In Figures <ref> and <ref> we plot α_+ and α_-. The coefficient of the non-normalizable mode, α_+, is multiplied by the factor 2(1-ν_k)(2ν_k-1) in order to remove the divergences in k̅ =0, 3. We will examine the origin of this divergence in more detail below. 
            In Figures <ref> and <ref> are shown the plots of 2(1-ν_k)β_+ and (2ν_k-1)^-1β_-. 
            
            
            
            
              §.§.§ Low momentum behavior of connection coefficients
            
            
            When k̅=0 (ν_k=1/2) the solutions in the outer region are
            
                η_L,-^0(Z)=1,    η_L,+^0(Z)=2/1-Z^2+4/3log[1+2Z^2/1-Z^2]-1/2-4/3log3/2 .
            
            The expansions at the boundary and the horizon of the η_L,+^0 solution are
            
                η_L,+^0(Z)    ∼   3/2-4/3log3/2+6Z^2+⋯,
                η_L,+^0(Z)    ∼    (1-Z)^-1-4/3log(1-Z)+⋯ .
            
            Therefore, the connection coefficients of these solutions are
            
                α_-^0=1, β_-^0=0,  α_+^0=3/2-4/3log3/2, β_+^0=6.
            
            We can relate these to the k̅→ 0 limit of the connection coefficients of the non-zero momentum solutions. First, note that the coefficient j_-,n in (<ref>) go to zero in the k̅→ 0 limit, so η_L,-→η_L,-^0. On the other hand j_+,1 in (<ref>) is singular in this limit, but the singularity does not propagate to other coefficients in the expansion, one can already see that it is absent in j_+,2. In order to remove the singular term we can take the following combination of the solutions
            
                η̃_L,+(Z)=η_L,+(Z)-j_1,+η_L,-(Z) .
            
            Expanding this solution at the horizon and for k̅→ 0 one can check that it coincides with the expansion in (<ref>). We then conclude that  η̃_L,→η_L,+^0 in the k̅→ 0 limit. This determines the k̅→ 0 behaviour of the connection coefficients
            
                α_-∼α_-^0,     β_-∼β_-^0,    α_+∼α_+^0+j_+,1α_-^0,     β_+∼β_+^0+j_+,1β_-^0 .
            
            Since j_+,1→∞ and β_-→ 0, in order to completely determine β_+ we need to find the non-zero subleading contribution to β_-. We expand
            
                η_L,-(Z)= (1-Z)^-1/2(1-2ν_k)(1+(ν_k-1/2)η_-^1(Z)+O[(ν_k-1/2)^2]) .
            
            The solution for η_-^1(Z) can be found explicitly
            
                η_-^1(Z)=   1/9(log (1-Z) (9-12 log (Z))+9 log (1+Z)+π ^2+9-9 log (2)-6Li_2(-2).
                   .+6 log (Z) (2 log(1+2
                   Z^2/1+Z)+3 Z^2/1-Z^2) + 6Li_2(-Z^2)-6Li_2(Z^2)) .
            
            The boundary and horizon expansions are
            
                η_L,-^1(Z)    ∼   1/9(9 + π^2 - 9log(2)- 6Li_2(-2)) + 3 Z^2 (-1 +2 log Z)+⋯,
                η_L,-^1(Z)    ∼   log(1-Z)+ 11/6(1-Z)+⋯ .
            
            Therefore
            
                β_-∼ -3/2(2ν_k-1) .
            
            Using now that to leading order
            
                j_+,1∼4/3(2ν_k-1),
            
            We find that when k̅→ 0
                α_-∼ 1+O(ν_k-12),    β_-∼ -3/2(2ν_k-1)+O[(ν_k-1/2)^2] ,
                   α_+∼43(2ν_k-1)+O(1),    β_+∼ 4+O(ν_k-1/2) .
            utphys ] ]


 Paper Title: Homotopy representations of extended holomorphic symmetry in holomorphic twists.


Point-supervised Brain Tumor Segmentation with Box-prompted MedSAM
    Xiaofeng Liu, Jonghye Woo, Chao Ma, Jinsong Ouyang, and Georges El Fakhri
X. Liu, C. Ma, J. Ouyang and G. El Fakhri are with the Department of Radiology and Biomedical Imaging, Yale University, New Haven, CT 06519.
J. Woo is with the Massachusetts General Hospital and Harvard Medical School, Boston, MA 02114.

    August 5, 2024
===========================================================================================================================================================================================================================================================================================================================

§ ABSTRACT 
We argue that holomorphic twists of supersymmetric field theories naturally come with a symmetry L_∞-algebra that nontrivially extends holomorphic symmetry. This symmetry acts on spacetime fields only up to homotopy, and the extension is only visible at the level of higher components of the action. We explicitly compute this for the holomorphic twist of ten-dimensional supersymmetric Yang–Mills theory, which produces a nontrivial action of a higher L_∞-algebra on (a graded version) of five-dimensional affine space.





§ INTRODUCTION AND SUMMARY

Twisting of physical theories <cit.> has attracted great interest in the physics literature.
In particular, the pure spinor formalism <cit.> (see reviews in <cit.>), which naturally describes such theories as supergravity <cit.>, supersymmetric Yang–Mills theory <cit.> and M2-brane models <cit.>,
accommodates twisting naturally <cit.>.

Physical theories come with representations of spacetime symmetry algebras, such as (super-)Poincaré algebras and (super-)conformal algebras. It has been long known that for theories with more than four supercharges it is often difficult to manifest this symmetry `off shell', that is, without using equations of motion. The pure spinor formalism provides a means of producing off-shell supermultiplets by introducing appropriate infinite towers of auxiliary fields and furthermore shows that the on-shell supermultiplets in fact carry a homotopy representation of the spacetime symmetries; the higher components of the action then correspond to the equations of motion needed to make the symmetry algebra close.

In this paper, we argue that holomorphic twists of supersymmetric field theories naturally come with more than just the holomorphic symmetry but rather a certain L_∞-extension of holomorphic symmetry. The extension is not visible at the level of strict representations, but spacetime fields naturally form a homotopy representation of this extended symmetry. We shall treat in detail the example of the holomorphic twist of ten-dimensional supersymmetric Yang–Mills theory. 
This twisted theory is holomorphic Chern–Simons theory on ^5 <cit.>, which enjoys a manifest isl(5) = sl(5) ⋉5 symmetry. 
As we shall see, it naturally comes with the extended holomorphic symmetry L_∞-algebra

    isl(5)(𝔰𝔩(5)⋉1005)

equipped with a certain higher bracket μ_4; and this L_∞-algebra acts on [z_1,z_2,z_3,z_4,z_5] (with appropriate grading) in the L_∞-algebraic sense.
This may be seen as a non-strict L_∞-algebra action of isl(5) on the (graded version of) five-dimensional complex affine space 𝔸^5.

We work with minimal models (of both the symmetry algebra and the field content), which canonically separates the physical information and makes clear the presence of higher-order structures (L_∞-algebras and their representations), rather than a larger strict model, which is not canonical and mixes in the physical degrees of freedom together with the unphysical auxiliary fields; this ensures that all information that we recover is physical and independent of the choice of auxiliary fields.

One way to think about this is to recall that twisting is akin to dimensional reduction <cit.> in which, rather than eliminating dependence on bosonic coordinates, we eliminate dependence on fermionic coordinates (restrict to Q-closed fields for a supersymmetry Q), which results in the `pair annihilation' of bosonic and fermionic coordinates. From this perspective, we have an `as above, so below' heuristic: the actions of twisted theories resemble those of their twistings, just like dimensional reduction preserves the forms of actions.
Using the pure spinor formalism, ten-dimensional supersymmetric Yang–Mills theory may be formulated as a holomorphic Chern–Simons theory on a complex (21|16)-dimensional pure spinor superspace (with 10 complexified ordinary spacetime coordinates, 16 ordinary superspace fermonic coordinates, and 11 bosonic pure spinor coordinates). The twisted theory has the same form of a Chern–Simons theory, but this time on 5|0 dimensions, where we have killed 16|16 coordinates. Under this `dimensional reduction', the ten-dimensional 𝒩=1 super-Poincaré symmetry, which is (55|16)-dimensional, reduces to a (39|0)-dimensional extended holomorphic symmetry. This dimensional reduction corresponds to twisting the supersymmetry algebra and taking the minimal models of the symmetry algebra and its homotopy representation on the field content. The additional factor 10 in (<ref>) and the concomitant μ_4 are the `dimensionally reduced' remnants of ten-dimensional super-Poincaré symmetry.

The discussion of the present paper is limited to the kinematics, that is, ignoring interactions and considering the linearized theory. This is not an essential restriction. A discussion of the interaction terms should make use of the L_∞-algebra formalism <cit.> for scattering amplitudes;
after colour-stripping, we should get a C_∞-algebra <cit.>, on which the extended holomorphic symmetry should act, forming an example of an open–closed homotopy algebra <cit.>. This, however, we leave to a future work.

While we focus on ten-dimensional supersymmetric Yang–Mills theory as a special case, the discussion is generic and applies, in principle, to the twists of any supersymmetric field theory. However, the twists in other dimensions often produce either a strict representation (with the 𝔦𝔰𝔩(d)-representation factoring through an 𝔦𝔰𝔩(d)-representation)
or a higher representation of an L_∞-superalgebra on affine superspace (with odd coordinates); 𝔸^5 is one of the few nontrivial purely bosonic examples that carry a higher symmetry. (For more discussion, see <ref>.)

All of our discussion is classical; there may be obstructions to quantization in the form of anomalies. For our main example of the holomorphic twist of ten-dimensional supersymmetric Yang–Mills theory, the twist (five-dimensional holomorphic Chern–Simons theory) is known to have anomalies unless it is coupled in a consistent fashion to Kodaira–Spencer gravity <cit.>.



 §.§ Organization of this paper

This paper is organized as follows. <Ref> reviews the generalities of twisting L_∞-algebras and modules over them and the appearance of higher components of the spacetime symmetry algebras and higher components of nonstrict representations of L_∞-algebras, both in the untwisted and twisted cases. <Ref> then computes the higher components of the representation of supersymmetry for ten-dimensional supersymmetric Yang–Mills theory, the higher products of the corresponding twisted extended holomorphic algebra, and the higher components of its representation on the twisted supermultiplet. <Ref> briefly surveys phenomena that appear in dimensions other than ten.

In the body of the paper, we will usually refer to irreducible representations of 𝔰𝔩(5) using their Dynkin labels, supplemented by Young tableaux where they are helpful.



§ MATHEMATICAL BACKGROUND

Here we briefly review the relevant concepts of twisting of L_∞-algebras and their modules. For more detailed reviews, see <cit.>.



 §.§ L∞-algebras

An L_∞-algebra is a homotopy generalization of the concept of a Lie algebra.

An L_∞-algebra (𝔤,{μ_k}_k≥ 1) consists of a graded vector space 𝔤=⊕_i∈ℤ𝔤^i together with skew-symmetric, multilinear maps μ_k𝔤^∧ k→𝔤 of degree 2-k for k∈{1,2,3,…} that satisfy the identity

    0=∑_i+j=n
    σ∈Sh(i,j)(-1)^jχ(σ,x)μ_j+1(μ_i(x_σ(1),…,x_σ(i)),…,x_σ(i+j))=0.

In the above, Sh(j_1,…,j_k) denotes the collection of shuffles, which are permutations σ of {1,…,j_1+…+j_k} such that σ(1)<…<σ(j_1) and σ(j_1+1)<…<σ(j_1+j_2) and so on up to σ(j_1+…+j_k-1+1)<…<σ(j_1+…+j_k).
The symbol χ(σ,x) denotes the skew-symmetric Koszul sign

    x_1∧…∧ x_k
        =χ(σ,x) x_σ(1)∧…∧ x_σ(k),

  defined for homogeneous elements x_1,…,x_k∈ g inside the exterior algebra ⋀^∙ g.

In what follows, we will often leave the products {μ_k^𝔤}_k≥ 1 implicit, and simply refer to an L_∞-algebra through its underlying graded vector space. The identities (<ref>) imply that μ_1∘μ_1=0 so that g is in particular a cochain complex.


  A morphism of L_∞-algebras ϕ(,{μ_k^𝔤}_k≥ 1)⇝(h,{μ_k^𝔥}_k≥ 1) consists of skew-symmetric, multilinear component maps
  
    ϕ^(n)^∧ n→ h

of degree 1-n for n∈{1,2,…}, satisfying the following coherence relations:
  
    ∑_j∈{1,…,i}
    k_1+…+k_j=i
    σ∈Sh(k_1,…,k_j)ζ(σ,k,x)/j!μ_j^𝔥(ϕ^(k_1)( x_σ(1), …, x_σ(k_1)),
          …, ϕ^(k_j)( x_σ(k_1+…+k_j-1+1), …,  x_σ(i)))
    
          =
          ∑_j+k=i
    σ∈Sh(j,k)(-1)^k χ(σ,x) ϕ^(k+1)(μ_j^𝔤( x_σ(1), …,  x_σ(j)),  x_σ(j+1), …,  x_σ(i)),

where

    ζ(σ,k,x)χ(σ,x)(-1)^∑_1 ≤ m<n ≤ j k_m k_n+∑_m=1^j-1 k_m(j-m)+∑_m=2^j(1-k_m)∑_k=1^k_1+…+k_m-1|x_σ(k)| .

We shall sometimes omit the μ_k and just write ⇝ h. 
L_∞-morphisms compose associatively, so that one has the category whose objects are L_∞-algebras and whose morphisms are L_∞-morphisms between them.

An L_∞-morphism is an  L_∞-(quasi-)isomorphism if the first component map is a (quasi-)isomorphism of the underlying cochain complexes.



  
Homotopy transfer of L_∞-algebras. 
L_∞-algebras admit a good homotopy theory in the sense that minimal models exist and can be computed by homotopy transfer using a strong deformation retract. Let us sketch how this works. Concretely, given an L_∞-algebra ( g,{μ_k^𝔤}_k≥ 1) one can always choose a strong deformation retract, denoted by a triple (i,p,h), from the underlying cochain complex ( g,μ_1) to its cohomology H( g):

    (𝔤,μ_1) [shift left, "p"] [loop left, "h"]     ( H(𝔤),0) [shift left, "i"]

(i.e. pi=id_ H( g) and ip=id_ g-[d,h]).
Then there exists an L_∞-algebra structure on the cohomology H(𝔤) together with an L_∞-quasi-isomorphism

    e H(𝔤)⇝𝔤,

whose first component is e^(1)=i; furthermore, there exist explicit formulae for the L_∞-algebra structure of H(𝔤) and the quasi-isomorphism e in terms of (i,p,h) <cit.>, e.g. using the tensor trick <cit.>, which can be interpreted as a sum over Feynman diagrams <cit.>. For example the ternary bracket μ_3^ H( g), is (modulo relative signs) the sum


    [scale=0.5,baseline=([yshift=-1ex]current bounding box.center)]
         [thick] (-2,0) – (-0.6,-1.4);
         [thick] (-0.35,-1.65) – (0,-2);
         [thick] (2,0) – (0,-2);
         [thick] (0,-2) – (0,-3);
         [thick] (-1,-1)–(0,0);
         at (-.5,-1.5) h;
         at (0,-3.5) p;
         at (0,0.5) i;
         at (-2,0.5) i;
         at (2,0.5) i;
        
         [gray!50] (-1,-1) circle (0.35cm);
         [thick] (-1,-1) circle (0.35cm);
         at (-0.98,-1) μ_2;
        [gray!50] (0,-2) circle (0.35cm);
         [thick] (0,-2) circle (0.35cm);
         at (0.02,-2) μ_2;
      
      +
      [scale=0.5,baseline=([yshift=-1ex]current bounding box.center)]
         [thick] (-2,0) – (0,-2);
         [thick] (2,0) – (0.65,-1.35);
         [thick] (0,-2) – (0.4,-1.6);
         [thick] (0,-2) – (0,-3);
         [thick] (1,-1)–(0,0);
         at (0.55,-1.45) h;
         [gray!50] (1,-1) circle (0.35cm);
         [thick] (1,-1) circle (0.35cm);
         at (1.02,-1) μ_2;
        [gray!50] (0,-2) circle (0.35cm);
         [thick] (0,-2) circle (0.35cm);
         at (0.02,-2) μ_2; at (0,-3.5) p;
         at (0,0.5) i;
         at (-2,0.5) i;
         at (2,0.5) i;
      
      +
      [scale=0.5,baseline=([yshift=-1ex]current bounding box.center)]
         [thick] (-2,0) – (-0.6,-1.4);
         [thick] (-0.35,-1.65) – (0,-2);
        
         [thick] (1,-1) – (0,-2);
         [thick] (0,-3) – (0,-2);
         [thick] (1,-1)–(0,0);
         [thick] (-1,-1) – (0.3,-0.57);
         [thick] (2,0) – (0.7,-.45);
         at (-.5,-1.5) h;
      
         at (0,-3.5) p;
         at (0,0.5) i;
         at (-2,0.5) i;
         at (2,0.5) i;
    
         [gray!50] (-1,-1) circle (0.35cm);
         [thick] (-1,-1) circle (0.35cm);
         at (-0.98,-1) μ_2;
        [gray!50] (0,-2) circle (0.35cm);
         [thick] (0,-2) circle (0.35cm);
         at (0.02,-2) μ_2;
      
      +
      [scale=0.5,baseline=([yshift=-1ex]current bounding box.center)]
         [thick] (-2,0) – (0,-2);
         [thick] (2,0) – (0,-2);
         [thick] (0,0) – (0,-2);
         [thick] (0,-2) – (0,-3);
         at (0,-3.5) p;
         at (0,0.5) i;
         at (-2,0.5) i;
         at (2,0.5) i;
        [gray!50] (0,-2) circle (0.35cm);
         [thick] (0,-2) circle (0.35cm);
         at (0.02,-2) μ_3;
      .

More generally, μ_k^ H( g) is computed by a sum[The explicit relative signs between the trees can be worked out by using the aforementioned tensor trick, for example.] over all rooted trees with k leaves, where one decorates the leaves with i, the n+1-ary vertices with μ_n, the internal edges with h, and the root with p.
The L_∞-algebra structure on H( g) is called the minimal model of 𝔤; minimal models are unique up to L_∞-isomorphisms.



  
Twisting L_∞-algebras
L_∞-algebras also admit a notion of twist with respect to a Maurer–Cartan element; for reviews, see <cit.>. In the definitions below, for a L_∞-algebra ( g,{μ_k}_k≥ 1), we assume for simplicity that μ_i=0 for sufficiently large i; this can be relaxed <cit.>.

Let (𝔤,{μ_k^𝔤}_k≥ 1) be an L_∞-algebra such that μ_i=0 for sufficiently large i.
A Maurer–Cartan element Q∈𝔤^1 of 𝔤 is an element of degree 1 such that

    ∑_i=1^∞1/i!μ_i(Q,…,Q)=0.



Let (𝔤,{μ_k^𝔤}_k≥ 1) be an L_∞-algebra such that μ_i=0 for sufficiently large i.
Let Q∈𝔤^1 be a Maurer–Cartan element.
The twist of 𝔤 with respect to Q is the L_∞-algebra 𝔤_Q whose underlying graded vector space is that of 𝔤 but whose brackets μ_k^Q are

    μ_k^Q      𝔤_Q^∧ k   →𝔤_Q
          (x_1,…,x_k)   ↦∑_i≥ 01/i!μ_i+k(Q,…,Q,x_1,…,x_k).





 §.§ L∞-representations

The notion of a representation of (or module over) a Lie algebra generalizes to the setting of homotopy algebras as follows.

  
    An L_∞-representation
of an L_∞-algebra (𝔤,{μ_k^𝔤}_k≥ 1) on a graded vector space M is an L_∞-algebra structure {μ_k^𝔤⋉ M}_k≥ 1 on the direct sum 𝔤⊕ M 
such that
    
    μ_k^𝔤⋉ M(x_1⊕0,…,x_k⊕0)=μ_k(x_1,…,x_k)

     and  μ_k+1^𝔤⋉ M(x_1,… x_k,m)∈ 0⊕ M for x_1,…,x_k∈𝔤 and m∈ M, and such that μ_k^𝔤⋉ M vanishes whenever at least two of its arguments belong to 0⊕ M⊂𝔤⊕ M. We will write g ⋉ M to refer to L_∞-algebras of this form.
We write
    
    ρ^(k)(x_1,…,x_k)μ_k+1^𝔤⋉ M(x_1,…,x_k,-) M→ M.

      Observe that, in particular, ρ^(k) carries degree 1-k.
    Note that ρ^(0) defines a differential on M, making it a cochain complex. We call an L_∞-representation strict whenever ρ^(k)=0 for k>1.

The L_∞-algebra homotopy Jacobi identities (<ref>) then can be written as a series of coherence relations amongst the ρ^(k)'s and μ_j^ g's.

As with L_∞-algebras themselves, L_∞-representations admit a good homotopy theory in that minimal models exist and homotopy transfer is possible. That is, given an L_∞-algebra 𝔤 and a g-representation M,
we can always choose a strong deformation retract

    [loop left,"(h,h')"] (𝔤⊕ M,μ_1+ρ^(0) )[shift left, "(p,p')"]    [shift left, "(i,i')"]  ( H(𝔤)⊕ H(M),0)

and perform homotopy transfer of L_∞-algebra structures along this retract to obtain an L_∞-algebra on H( g)⊕ H(M), which then defines the L_∞-representation of H(𝔤) on H(M).[The induced brackets on H( g)⊕ H(M) automatically satisfy the conditions given in <ref>. Indeed, as there are no brackets in g ⊕ M that reduce the number of factors of M, no such brackets can arise through composition.][This is the minimal model for the two-coloured operad of pairs of L_∞-algebras and their L_∞-representations, rather than the minimal model for the uncoloured operad of L_∞-representations over a fixed L_∞-algebra 𝔤.]

Given an L_∞-algebra 𝔤 and a g-representation M with structure maps ρ^(k),
then it is clear by inspection that a Maurer–Cartan element Q∈𝔤 is also Maurer–Cartan element of 𝔤⋉ M and that the twist (𝔤⋉ M)_Q factorizes as (𝔤⋉ M)_Q=𝔤_Q ⋉ M_Q <cit.>, where
M_Q comes with the structure maps

    ρ_Q^(k)(x_1,…,x_k) ∑_i=0^∞1/i!ρ^(i+k)(Q,…,Q,x_1,…,x_k).




§ HIGHER SYMMETRY OF TWISTED TEN-DIMENSIONAL SUPERSYMMETRIC YANG–MILLS THEORY

In the Batalin–Vilkovisky formalism <cit.>,
the field content of a perturbative gauge theory is a graded vector space (ℱ,d_ℱ) that comes equipped with a differential.
Field theories often respect symmetry algebras such as the super-Poincaré algebra, the (super-)conformal algebra, the (super-)(anti-)de Sitter algebra, the (super-)Galilean algebra, etc. 
The action of such a symmetry algebra may be off shell (i.e. on ℱ itself) or merely on shell (i.e. only on the space of solutions to the equations of motion[Since we ignore interactions, for us this is the linearized equations of motion, but in general one should consider the interacting case.]). The symmetry algebras are usually ungraded or ℤ/2ℤ-graded (i.e. superalgebras), and correspondingly the field space is ℤ/2ℤ-graded into bosons and fermions (in addition to the ℤ-grading corresponding to ghost number).
This ℤ/2ℤ grading may be often lifted to a ℤ grading; correspondingly, the ℤ/2ℤ grading of the field space may also be lifted to a ℤ-grading.[In order to lift the /2 grading of field space to ℤ, we work with polynomials over spacetime rather than smooth functions. This simplification can be avoided; see <cit.>.] 
The ℤ-grading enables a good homotopy theory of L_∞-algebras and L_∞-representations, and in particular off-shell realizations of symmetries can be in most cases lifted to a non-strict L_∞-representation of the corresponding symmetry <cit.>.

We turn now to our main example of interest, which is the twist of ten-dimensional super-Yang–Mills theory. (For simplicity and convenience with twisting, we assume all symmetries and fields to be complexified.)

We first discuss the twisted super-Poincaré algebra itself.
Let 

    V ≅^10
 
be a ten-dimensional complex vector space equipped with a nondegenerate symmetric bilinear form.
The ten-dimensional 𝒩=(1,0) super-Poincaré superalgebra is the Lie superalgebra

    𝔬(V)⋉( S_+⊕ V)

where S_± are the two 16-dimensional Weyl spinor representations of 𝔬(V) and  denotes parity reversal.
The ℤ/2ℤ grading of the super-Poincaré superalgebra can be lifted to a ℤ-grading as the graded Lie algebra

    𝔭𝔬(V)⋉(S_+[-1]⊕ V[-2])

in which the elements are graded as twice the conformal dimension (i.e. rotations in degree 0, supertranslations in degree 1, translations in degree 2).
For convenience, we can pick a basis _μν, _α, _μ of 𝔭^0, 𝔭^1, 𝔭^2 respectively. Then the structure constants for [𝔭^1,𝔭^1]⊂𝔭^2 are

    [_α,_β]=2γ^μ_αβ_μ,

where γ^μ_αβ are the chiral gamma (or Pauli) matrices in ten dimensions, i.e. the branching for the 𝔬(V)-representation S_+⊗ S_+→ V.



 §.§ The minimal model of the holomorphic twist algebra

Suppose we pick a nonzero Maurer–Cartan element of 𝔭, i.e. a nonzero Q=Q^α_α∈ S_+ such that [Q,Q]=0, that is,

    γ^μ_αβQ^α Q^β=0.

This picks out a subspace

    L = [Q,S_+]⊂ V.

This subspace L is a maximal isotropic subspace with respect to the bilinear form on V.
Indeed, using the Fierz identity 

    2γ^μ_α(βγ^0_μ|γ)δ=-γ^μ_βγγ^0_μαδ,

we have

    Q^β Q^γγ^μ_αβγ^0_μγδ∝(Q^β Q^γγ^μ_βγ)γ^0_μαδ=0;

given now any elements ψ,χ∈ S_+, consider the elements [Q,ψ],[Q,χ]∈ L. We have

    [Q,ψ]^μ[Q,χ]_μ =(Q^β Q^γγ_αβ^μγ_μγδ)ψ^αχ^δ=0.

Thus L is indeed contained in its own orthogonal complement, i.e. it is an isotropic subspace.
Isotropy implies L≤5. Furthermore, the set of Maurer–Cartan elements of 𝔭 consists of two 𝔬(V)-orbits, namely nonzero ones and {0}; and it can be shown that, when Q0, then L=5 <cit.>. That is, L is indeed a maximal isotropic subspace.

Thus, we have the short exact sequence of vector spaces

    0→ L→ V L^*→ 0

where the quotient q is via the composition VV^*↠ L^* in which VV^* is given by the bilinear form on V. Let us choose a splitting of (<ref>) to write

    V=L⊕ L^*.

This decomposition then fixes Lie subalgebras 𝔰𝔩(L)⊂𝔤𝔩(L)⊂𝔬(V), under which the ten-dimensional representation V canonically decomposes into 𝔰𝔩(L) irreducible representations as

    V≅_sl(L) L⊕ L^*≅_sl(L) (1000)_sl(L)⊕ (0001)_sl(L)

(Here and elsewhere we write (ijkl)_sl(L) for the irreducible representation with these sl(L) Dynkin labels, i.e. for the irreducible representation with highest weight iω_1 + j ω_2 + k ω_3 + lω_4 where ω_1,…,ω_4 are the fundamental weights.)

Similarly, the adjoint representation of o(V) decomposes into irreducible sl(L)-representations as

    𝔬(V)    ≅_sl(L)⊕𝔰𝔩(L)⊕ L^∧2⊕(L^*)^∧2
       ≅_sl(L) (0000)_sl(L)⊕ (1001)_sl(L)⊕ (0100)_sl(L)⊕ (0010)_sl(L),

and the spinor representations S_± decompose as

    S_+    ≅_sl(L)⊕_i=0^2(L^*)^∧(2i)≅_sl(L)(0000)_sl(L)⊕(0010)_sl(L)⊕(1000)_sl(L),
    
        S_-    ≅_sl(L)⊕_i=0^2(L^*)^∧(2i+1)≅_sl(L) S_+^*≅(0001)_sl(L)⊕(0100)_sl(L)⊕(0000)_sl(L).

Our chosen element Q∈ S_+ spans the one-dimensional sl(L)-submodule ≅ (L^*)^∧0.
The twist of p by Q is <cit.>

    𝔭_Q = 
    ([row sep=0]
    (0000)_sl(L)["id"]    (0000)_sl(L)(L^*)^∧0[-1]
    (0010)_sl(L)(L^*)^∧2["id"]    (0010)_sl(L)(L^*)^∧2[-1]
       (1000)_sl(L)(L^*)^∧4[-1]["id"]    (1000)_sl(L)L[-2]
    (1001)_sl(L)𝔰𝔩(L)  ⋉  (0100)_sl(L)L^∧2      (0001)_sl(L)L^*[-2]).


We work in an explicit basis (^i_j,^ij,_ij,,_ij,^i,^i,_i) of 𝔭 given in <ref>. In particular, the basis elements ^i_j span gl(L), and we shall write 

    ^i_j^i_j - 1/5δ^i_j^k_k

for the basis elements of sl(L). 
 
For a reason to become apparent in the theorem below, let us note that the representation L^* ⊗ (L^∧ 2)^∧ 3 of sl(L) decomposes into irreducibles as

    L^* ⊗ (L^∧ 2)^∧ 3≅ (0021)_sl(L)⊕ (0110)_sl(L)⊕ (1001)_sl(L)⊕ (2010)_sl(L)⊕ (2002)_sl(L).

In particular, the adjoint representation sl(L) ≅ (1001)_sl(L) occurs with multiplicity one. We shall write 

    P_L^*⊗(L^∧2)^∧3→𝔰𝔩(L)

for the projector onto this irreducible component. 

The minimal model of the ten-dimensional twisted ℤ-graded 𝒩=(1,0) super-Poincaré algebra 𝔭_Q is the L_∞-algebra whose underlying graded Lie algebra is

    H(𝔭_Q) = 𝔰𝔩(L)⋉( L^∧2⊕ L^*[-2]),

and whose higher brackets μ_i (i≥3)
are all zero except for μ_4, whose only nonvanishing component is given by

    μ_4(_i,^jk,^lm,^np)
           =-δ_i^[jε^k]lmqrδ^[n_q^p]_r
        +δ_i^[jε^k]npqrδ^[l_q^m]_r
           -δ_i^[lε^m]npqrδ^[j_q^k]_r
        +δ_i^[lε^m]jkqrδ^[n_q^p]_r
           -δ_i^[nε^p]jkqrδ^[l_q^m]_r
        +δ_i^[nε^p]lmqrδ^[j_q^k]_r
        P^jklmnp;_i^r_q^q_r,

where P^jklmnp;_i^r_q is the projector

    L^*⊗(L^∧2)^∧3→𝔰𝔩(L),

and where the skew-symmetrizations are unnormalized.


From <ref>, we see that there is an evident 𝔰𝔩(L)-equivariant strong deformation retract (i,p,h) of cochain complexes 

    [loop left,"h"] ( p_Q,ad_Q) [shift left, "p"]    [shift left, "i"]  ( H( p_Q),0)

from p_Q to its cohomology

    H(𝔭_Q) = (𝔰𝔩(L)⋉ L^∧2 0 L^*).

(The remaining sl(L) irreducible representations present in <ref> participate in trivial pairs; one defines the homotopy h to act as the inverse to the differential on these.)
The Lie algebra structure μ_2 on this cohomology H( p_Q) is given by restriction.

It remains to check what higher brackets μ_i are induced by homotopy transfer. 
We are to sum over rooted binary trees in which each vertex corresponds to the binary bracket μ_2^ p(-,-) = [-,-] of p (and thus of p_Q), each internal edge to the homotopy h, each leaf to i and the root to p <cit.>.

We will use Feynman-diagrammatic terminology, referring to elements as `states' (see <cit.>). 
Recall our notation (^i_j,^ij,_ij,,_ij,^i,^i,_i) for the basis elements of 𝔭 (whose underlying graded vector space we identify with that of 𝔭_Q) as given in <ref>. We will refer to a state as intermediate if it lies in the image of h.

Using the strong deformation retract (i,p,h), we try to construct the possible intermediate states, keeping track of the representations under 𝔰𝔩(L). Using the embedding i H(p_Q)↪p_Q, we will identify the basis elements ^i_j,^ij,_i in p_Q with those of H(p_Q). The products μ_i for i>2 can then be computed in a top-down recursive fashion by starting with two elements x, y∈H(p_Q), and then compute which intermediate states are allowed by considering

    h[i(x),i(y)].

The next intermediate states are then computed by plugging (<ref>), and one element a∈ i(H(p_Q))⊕(h[i,i]), into [-,-]. If the result lies in H(p_Q), we apply p, and we are done. If not, we apply h to obtain new intermediate states and then continue the procedure of pairing (using [-,-]) the newly obtained intermediate states with each other or with previously obtained intermediate states or states in the cohomology.

Starting with two elements of H(p_Q), applying h[-,-] can only yield the intermediate states

    h[_k,^ij]=δ^i_k^j-δ^j_k^i.

Using ^i together with ^i_j,^ij,_i, we can only further create 


    [^i,^j]    = 0, 
    
    [^i,_j]    = 0, 
    
    h[^ij,^k]    = -1/2^ijklm_lm,
    
    [^i_j,^k]   =-1/5δ^i_j^k+δ^k_j^i.


Among these, [^i_j,^k] does not belong to the cohomology, i.e. it is not Q-closed, nor can it produce a new intermediate state since h([^i_j,^k])=0.
Thus the only intermediate state we can create is given by h[^ij,^k]∝_lm. Applied to (<ref>), we obtain

    h[h[_k,^ij],^lm]=h[δ^i_k^j-δ^j_k^i,^lm]
        =1/2(
        δ^i_kε^jlmnp
        -δ^j_k
        ε^ilmnp)_np.

Using the new intermediate state _ij together with the previously created intermediate state ^i and the cohomology (^i_j,^ij,_i), we can create the following new states:


    [_ij,_k]    = 0, 
    
    [_ij,^k]    = 0, 
    
    [^i_j,_kl]   =
    -δ^i_k_jl-δ^i_l_kj
    +2/5δ^i_j_kl, 
    
    [^ij,_kl]    = δ^i_k^j_l-δ^j_k^i_l-δ^i_l^j_k+δ^j_l^i_k, 
    
    [_ij,_kl]    = 0.


Now, all these states sit in degree 0 and are thus killed by h, so that none of them can create further intermediate states. The only nontrivial thing we can now do is to project to the cohomology: [^i_j,_kl] never lies in the cohomology, whereas the traceless part of [^ij,_kl] does. Applied to (<ref>), we obtain

    p[h[h[_k,^ij],^lm],^qr]
           =-1/2(δ^j_k
        ε^ilmnp
        -
        δ^i_kε^jlmnp)[_np,^qr]
       =(
        δ^j_k
        ε^ilmnp
        -δ^i_kε^jlmnp)
        (
            δ^q_n^r_p
            -δ^r_n^q_p
        )
       =δ^[j_k
        ε^i]lmnpδ^[q_n^r]_p.

After graded-skew-symmetrization among the three arguments of the form ^ij, this yields the only nonvanishing component of μ_4.

There are no other μ_i since we have systematically constructed all possible nonzero tree Feynman diagrams (by constructing all possible intermediate states that occur in them).




 §.§ Action on 𝔸⁵

The pure spinor formalism <cit.> associates certain sheaves on (a derived replacement of) the variety of Maurer–Cartan elements to off-shell representations of the super-Poincaré algebra.
In particular, for the ten-dimensional 𝒩=(1,0) super-Poincaré algebra, it associates to the structure sheaf of the Maurer–Cartan variety Specℂ[λ^α]/(γ^μ_αβλ^αλ^β) the pure spinor supermultiplet <cit.>

    M   ℂ[x^μ,θ^α,λ^α]/(γ^μ_αβλ^αλ^β)
       ≅((10000)_𝔬(V))⊗(⋀(00010)_𝔬(V))⊗(⊕_i=0^∞(000i0)_𝔬(V))

where the 𝔬(V)-representation has been specified by Dynkin labels; we use the index notation where V indices are _μ and S_+ indices are _α (hence S_-≅ S_+^* indices are ^α). The formal variables x,θ,λ transform as V,S_-,S_- respectively under 𝔬(V), which in turn determines the 𝔬(V)-representation on M.
The generators carry the degrees[In fact, this grading can be refined into a bigrading <cit.>, and the 𝔭-representation respects this bigrading. But we do not need this fact.]

    |x|   =-2     |θ|    = -1     |λ|   = 0.

 The differential is <cit.>

    d λ^α(∂/∂θ^α-γ^μ_αβθ^β∂/∂ x^μ).

The 𝔬(V)-representation of M extends to a strict representation of 𝔭 as

  

    ρ^(1)_0(_μ)   = ∂/∂ x^μ, 
    ρ^(1)_0(_α)    = ∂/∂θ^α+γ^μ_αβθ^β∂/∂ x^μ.



We can twist this 𝔭-representation to obtain a (strict) representation  of 𝔭_Q on M_Q; the action of p_Q is through (<ref>), but the differential on M_Q is now d+ρ_0^(1)(Q). The cohomology of M_Q is the ring of regular functions on 𝔸^5.

The cohomology of M_Q is

    H(M_Q)=ℂ[z^i],

where z^i=(z^1,…,z^5) is a formal variable of degree -2 transforming under 𝔰𝔩(L) as the defining representation L, such that the L_∞-representation of the subalgebra 𝔦𝔰𝔩(L) of H(𝔭_Q) is

    ρ^(1)(^i_j)   =z^i∂/∂ z^j-1/5δ^i_jz^k∂/∂ z^k,   ρ^(1)(_i)   =∂/∂ z^i,

with ρ^(k)=0 for k≥2.

This corresponds to the fact that the holomorphic twist of ten-dimensional supersymmetric Yang–Mills theory is holomorphic Chern–Simons theory <cit.>, whose space of fields is[up to issues such as holomorphic versus algebraic functions, which we ignore] the algebraic Dolbeault complex of 𝔸^5, namely ℂ[z^i,z̅_i,dz̅^i], and whose cohomology is therefore ℂ[z^i].

Furthermore, the cohomology ℂ[z^i]= H(M_Q) is included into M_Q as a 𝔰𝔩(L)-subrepresentation <cit.>. Since 𝔰𝔩(L) is simple, there exists an sl(L)-equivariant strong deformation retract of cochain complexes
  
    [loop left,"(h,h')"]  p_Q⊕ M_Q [shift left, "(p,p')"]    [shift left, "(i,i')"]  H(𝔭_Q)⊕ H(M_Q),

whose restriction to 𝔭_Q↔ H(𝔭_Q) is the strong deformation-retract (i,p,h) given in (<ref>).
Thus, by taking the minimal model of (𝔭_Q,M_Q) along the strong deformation retract (<ref>), the above 𝔦𝔰𝔩(L)-representation extends into an L_∞-representation of the entirety of H(𝔭_Q), and such minimal models are unique up to quasi-isomorphisms of L_∞-algebra representations.
This minimal model is an L_∞-representation of the L_∞-algebra H( p_Q) on H(M_Q)=ℂ[z^i].
The following theorem computes this minimal model explicitly.

The minimal model of the L_∞-representation of 𝔭_Q on M_Q obtained using the strong deformation retract (i,i';p,p';h,h') is the H(𝔭_Q)-representation on H(M_Q)=ℂ[z^i] given by

    ρ^(1)(^i_j)    = z^i ∂/∂ z^j-1/5^i_jz^k∂/∂ z^k, 
    ρ^(1)(_i)    = ∂/∂ z^i, 
    ρ^(3)(^ij,^kl,^mn)    =
    1/2(z^[iε^j]klp[mz^n]
    -
    z^[iε^j]mnp[kz^l]
           +z^[kε^l]mnp[iz^j]
    -
    z^[kε^l]ijp[mz^n]
           +z^[mε^n]ijp[kz^l]
    -
    z^[mε^n]klp[iz^j])∂/∂ z^p
        P^ijklmn;_pq^rz^pz^q∂/∂ z^r,

with all other components vanishing (in particular, ρ^(1)(^ij)=0),
where P^ijklmn;_pq^r is the projection

    (0100)_𝔰𝔩(L)^∧3=(0020)_𝔰𝔩(L)⊕(2001)_𝔰𝔩(L)→(2001)_𝔰𝔩(L)

in terms of 𝔰𝔩(L) Dynkin labels
or, in Young tableau notation,

    1,1^∧3
        =
        2,2,2⊕3,1,1,1→3,1,1,1.


First, note that for degree reasons, we can only have nonzero ρ^(k) for odd k since ℂ[z^i] and H(𝔭_Q) are all concentrated in even degree and ρ^(k) carries degree 1-k. The leading component ρ^(1) is fixed simply by restriction of the 𝔭_Q-representation ρ_0 on M_Q to i( H(𝔭_Q))⊂𝔭_Q and i'( H(M_Q))⊂ M_Q as 

    ρ^(1)(x) = p'∘ρ_0(i(x))∘ i'

for x∈ H(𝔭_Q) (so that i(x)∈𝔭_Q);
in particular, ρ^(1)(^ij)=0. Furthermore, the ρ^(k) vanish whenever one of the arguments is ^i_j except when k=1 (<ref>).

Hence, it suffices to determine ρ^(3),ρ^(5),ρ^(7),… where all arguments are either ^ij or _i. Now, the possibilities of ρ^(k) are constrained by the fact that all operations μ_k, ρ^(k), and the strong deformation retract (i,i';p,p';h,h') are 𝔰𝔩(L)-equivariant. Suppose that ρ^(k) does not vanish when fed p arguments of the form ^ij and q arguments of the form _i with p+q=k≡12. Then, representation-theoretically, it must yield a nontrivial 𝔰𝔩(L)-representation that is a direct summand of

    ( L^∧2)^∧ p⊗(L^*)^∧ q.

On the other hand, it must carry the degree (1-p-q)+2q=1-p+q, and hence be a sum of terms of the form

    z^n(∂/∂ z)^n+(1-p+q)/2   if 1-p+q≥0
    
            z^n-(1-p+q)/2(∂/∂ z)^n   if 1-p+q≤0,

where z^n refers to a product z^i_1⋯ z^i_n, and similarly for (z)^n. Since z^i transforms as L and ∂/∂ z^i as L^*, this must transform under 𝔰𝔩(L) as a direct summand of

    L^⊙ n⊗(L^*)^⊙(n+(1-p+q)/2)   if 1-p+q≥0
    
            L^⊙(n-(1-p+q)/2)⊗(L^*)^⊙ n   if 1-p+q≤0.

Thus, the two 𝔰𝔩(L)-representations (<ref>) and (<ref>) must share some nontrivial subrepresentations if the corresponding ρ^(k) is to not vanish. <Ref> shows that this is only possible for (p,q)=(3,0) and (p,q)=(4,3), corresponding to

    ρ^(3)(^ij,^kl,^mn)= P^ijklmn;_pq^r(α_0z^pz^q∂/∂ z^r+α_1z^pz^qz^s∂^2/∂ z^r∂ z^s+…)

and

    ρ^(7)(^ij,^kl,^mn,^pq,_r,_s,_t)
    =P^ijklmnpq;_rst^u_v(β_0z^v∂/∂ z^u
        +
        β_1z^vz^w∂^2/∂ z^u∂ z^w+…),

respectively, where P^ijklmn;_pq^r is the projector (0100)^∧3_𝔰𝔩(L)→(2001)_𝔰𝔩(L) as in (<ref>) and P^ijklmnpq;_rst^u_v is the projector (0100)^∧4_𝔰𝔩(L)⊗(0001)^∧3_𝔰𝔩(L)→(1001)_𝔰𝔩(L).

Now, we must solve the coherence relations. One L_∞-representation coherence relation states[In this coherence relation, in our case, terms of the form [ρ^(0),ρ^(4)], [ρ^(2),ρ^(2)], ρ^(4)(μ_1), ρ^(3)(μ_2), and ρ^(2)(μ_3) vanish.]

    [ρ^(1)(_i),ρ^(3)(^jk,^lm,^np)]=ρ^(1)(μ_4(_i,^jk,^lm,^np)).

  Substituting ρ^(1)(_i)=∂/∂ z^i and (<ref>) into (<ref>) yields
  
    [
            ∂/∂ z^i,
            P^jklmnp;_qr^s(α_0z^qz^r∂/∂ z^s+α_1z^qz^rz^t∂^2/∂ z^s∂ z^t+…)
        ]
    
        =
        P^jklmnp;_i^r_qz^q∂/∂ z^r,

  where P^jklmnp;_i^r_q is the projector (<ref>).
  Solving this yields α_0=1 and α_n=0 for n>0.
  
  Next, we have the L_∞-module coherence relation[In this coherence relation, terms of the form [ρ^(0),ρ^(8)], [ρ^(2),ρ^(6)], [ρ^(3),ρ^(5)], [ρ^(4),ρ^(3)], ρ^(1)(μ_8), ρ^(2)(μ_7), …, ρ^(8)(μ_1) vanish.]

    0=[ρ^(7)(^ij,^kl,^mn,^pq,_[r,_s,_t),ρ^(1)(_u])].

Plugging in the ansatz (<ref>) into (<ref>) yields

    0=[P^ijklmnpq;_[rst|^v_w(β_0z^w∂/∂ z^v
        +
        β_1z^wz^x∂^2/∂ z^v∂ z^x+…),∂/∂ z^|u]].

Solving this shows that the coefficients β_0,β_1,… must all vanish since non-constant-coefficient differential operators do not commute with ∂/∂ z^l.



In the L_∞-representation of the L_∞-algebra H(𝔭_Q) on ℂ[z^i] obtained by homotopy transfer from M_Q using the 𝔰𝔩(L)-equivariant strong deformation retract (i,i';p,p';h,h'), we have

    ρ^(k)(^i_j,…) = 0

if k≥2.

We are to perform a homotopy transfer of L_∞-algebras along

    [loop left,"(h,h')"]  p_Q⊕ M_Q [shift left, "(p,p')"]    [shift left, "(i,i')"]  H(𝔭_Q)⊕ H(M_Q).

Let us again use Feynman-diagrammatic terminology to refer to elements as `states'.
If ρ^(k)(^i_j,…)0, this would mean that there is at least one tree Feynman diagram with at least one external leg corresponding to ^i_j. Assuming that k≥2, we have the following possibilities.


  *  The vertex connected to this leg may be directly connected to p' as

    p'(ρ_0(^i_j)X)
        =
      [scale=0.5,baseline=([yshift=-1ex]current bounding box.center)]
         [thick] (-2,0) – (0,-2);
         [thick, dashed] (2,0) – (0,-2);
         [thick, dashed] (0,-2) – (0,-3);
         at (0,-3.5) p';
         at (2.5,0.5) X;
         at (-2,0.5) ^i_j;
      ,

where X∈ M_Q. In this case, we may assume X to be an intermediate state X=h'(X̃). (The alternative, that X lies in the cohomology, only yields ρ^(1).)

  *  The vertex connected to this leg may feed into h and connect to the rest of the tree as

    p'(… h'(ρ_0(^i_j)X)…)
        =
      [scale=0.5,baseline=([yshift=-1ex]current bounding box.center)]
         [thick] (-2,0) – (0,-2);
         [thick, dashed] (2,0) – (0,-2);
         [thick,dashed] (0,-2) – (0,-5);
        [color=white, fill=white] (0,-3.5) circle (0.5);
         at (0,-3.5) h';
         at (0,-5.5) ⋮;
         at (2.5,0.5) X;
         at (-2,0.5) ^i_j;
      ,

where X∈ M_Q may be either an intermediate state X=h'(X̃) or belong to the cohomology (X∈ i'( H(M_Q))). In either case, we have h(X)=0.

  *  The vertex connected to this leg may feed into h and connect to the rest of the tree as

    p'(… h[^i_j,x]…)
        =
      [scale=0.5,baseline=([yshift=-1ex]current bounding box.center)]
         [thick] (-2,0) – (0,-2);
         [thick] (2,0) – (0,-2);
         [thick] (0,-2) – (0,-5);
        [color=white, fill=white] (0,-3.5) circle (0.5);
         at (0,-3.5) h;
         at (0,-5.5) ⋮;
         at (2.5,0.5) x;
         at (-2,0.5) ^i_j;
      ,

where x∈𝔭_Q is either an intermediate state x=h(x̃) or belongs to the cohomology (x∈ i( H(𝔭_Q))). In either case, h(x)=0.


In the first case (<ref>), since the strong deformation retract (i,i';p,p';h,h') is 𝔰𝔩(L)-equivariant, p'(ρ_0(^i_j)h'(X̃)) can be nonzero only if p'(h'(X̃)) is already nonzero. But this cannot be the case since (i',p',h') forms a strong deformation retract, whose definition requires p'∘ h'=0.

Similarly, in the latter case (<ref>),
since the strong deformation retract (i,i';p,p';h,h') is 𝔰𝔩(L)-equivariant, h'(ρ_0(^i_j)X) can be nonzero only if h'(X)) is already nonzero, but this is not possible.

Finally, in the last case (<ref>), again, since the strong deformation retract (i,i';p,p';h,h') is 𝔰𝔩(L)-equivariant, h[^i_j,x] can be nonzero only if h(x) is already nonzero, which is not possible.



For p+q odd and p+q≥3, the 𝔰𝔩(L)-representation

    R_p,q(L^∧2)^∧ p⊗(L^*)^∧ q

has no irreducible components in common with

    R̃_p,q⊕_n=0^∞ L^⊙ n⊗(L^*)^⊙(n+(1+q-p)/2)   if  1+q-p≥0 
    ⊕_n=0^∞ L^⊙(n-(1+q-p)/2)⊗(L^*)^⊙ n   if  1+q-p≤0

except when (p,q)=(3,0) or (4,3), in which case the irreducible components in common are
(2001)_𝔰𝔩(L) and (1001)_𝔰𝔩(L)
respectively.

We must compute the tensor product appearing in R̃_p,q. For 1+q-p≥0 and any nonnegative integer n≥0, it is easy to see that

    (n000)⊗(000(n+(1+q-p)/2))
        =⊕_i=0^n(i00(i+(1+q-p)/2)).

  Similarly, for 1+q-p≤0 we have
  
    ( (n-(1+q-p)/2))000)⊗(000n)
        =⊕_i=0^n((i-(1+q-p)/2))00i).

Thus, R̃_p,q only contains irreducible representations of the form

    (i00(i+(1+q-p)/2))   if 1+q-p≥0
    ( (i-(1+q-p)/2)00i)   if 1+q-p≤0    (i∈{0,1,2,…}).

Given this, iterating over[Recall that (L^∧ 2)^∧ k and (L^*)^∧ l are zero for k>10 and l>5.] p∈{0,1,…,10} and q∈{0,1,…,5} and verifying whether an irreducible representation of the above form appears in R_p,q (using e.g. a computer algebra system), one can see that (p,q)∈{(3,0),(4,3)} are the only possible solutions. 




§ OTHER DIMENSIONS AND AMOUNTS OF SUPERSYMMETRY

The above construction works for general supersymmetry algebras and general supermultiplets, but ten-dimensional 𝒩=(1,0) super-Poincaré algebra seems to be one of the very few in having nontrivial and purely bosonic higher products, at least if one is to start from the vector supermultiplet; a glance at <cit.> shows that this is the only case in which the cohomology is simply a polynomial ring in bosonic variables.

In general, for sufficiently large dimension n, the number of spinorial components in a super-Poincaré algebra increases as 𝒪(2^n) whereas bosonic components increase as 𝒪(n^2). Indeed, already at 14 dimensions, the minimal spinor has 128 components while 𝔦𝔬(14) has 105 components. So we are restricted to 13 or fewer dimensions (unless we twist again to eliminate more supertranslations) even if one did not take into consideration no-go theorems about higher-spin theories (since we ignore dynamics here). Similarly, the 11-dimensional case (starting with the supergravity multiplet) is discussed in <cit.>. There are two possible cases. In one case <cit.>, H(𝔭_Q) contains fermionic elements. Then we expect the action of H(𝔭_Q) to contain a ρ^(2) involving the remaining fermionic elements.
In the other case <cit.>, however, we expect to see a higher action of

    H(𝔭_Q) = (𝔤_2⊕𝔰𝔩(L)⊕ V_7⊗ L⊕ℂ)⋉ L

(which should carry nontrivial μ_2 and μ_4)
on

    𝔸(L^*)=Specℂ[z_1,z_2],

where L is a two-dimensional vector space and V_7 is a seven-dimensional vector space; here, ℂ[z_1,z_2]= H(Ω^0,∙(L)⊗^∙(V)) is the cohomology of the Dolbeault–de Rham complex on two complex and seven real dimensions.

On the other hand, if there are too few dimensions, higher products may vanish. For example, for the four-dimensional 𝒩=1 super-Poincaré algebra, the twist gives a decomposition of four-dimensional complexified spacetime V as V=L⊕ L^*, where L is a two-dimensional vector space, with the twisted super-Poincaré algebra being <cit.>

    𝔭_Q=(
             (L^*)^∧2    L^∧0
    
            (𝔰𝔩(L)⋉ L^∧2)⊕𝔤𝔩(1)_R    L^∧1    L 
    𝔤𝔩(1)_tr     L^∧2    L^*
        ),

where 𝔤𝔩(1)_R is the R-symmetry and 𝔤𝔩(1)_tr is the trace part of 𝔤𝔩(L).
Following the proof of <ref>, we see that there are no higher brackets for H(𝔭_Q) by constructing all possible intermediate states: apply μ_2( L^∧2,-) to L^*[-2] to get L[-2]; applying the homotopy h yields L^∧1; but now applying another μ_2( L^∧2,-) simply kills everything.



§ ACKNOWLEDGEMENTS

The authors (DSHJ, HK, and CASY) were supported in part by the Leverhulme Research Project Grant RPG–2021–092. The authors thank Fridrich Valach III0000-0003-0020-1999 and Charles Strickland-Constable0000-0003-0294-1253 for helpful discussions; casy thanks Brian R. Williams0000-0001-9641-861X, and dshj thanks Ingmar Akira Saberi0000-0002-2005-938X.


§ CONVENTIONS
 
In a basis adapted to the choice of pure spinor Q∈ S_+, the super-Poincaré algebra p =  p^0 ⊕ p^1 ⊕ p^2 in ten dimensions has the basis elements

    ^i_j, ^ij, _ij   ∈ p^0,   , _ij,^i   ∈ p^1,   ^i,_i   ∈ p^2,

with ^ij=-^ji, _ij=-_ji, and _ij=-_ji.
The graded-skew-symmetric Lie brackets among these basis elements are

    [ ^i_j, ^k_l ]    = δ^k_j ^i_l -δ^i_l ^k_j    [ ^i_j, ^kl]    = δ^k_j ^il + δ^l_j ^ik
    [ ^i_j, _kl]    = - δ^i_k _jl - δ^i_l _kj   [ ^ij, _kl]    = δ^i_k^j_l - δ^j_k^i_l - δ^i_l^j_k + δ^j_l^i_k
    [ ^i_j, ^k ]    =  δ^k_j ^i    [ ^i_j, _k ]    = -δ^i_k _j 
    [ ^ij, _k ]    =  δ^j_k ^i - δ^i_k ^j   [ _ij, ^k ]    =  δ_i^k _j - δ_j^k _i
    [ ^ij , ]    = 0    [ _ij, ]    = -_ij
    [ ^ij , _kl]    = (δ^i_l δ^j_k-δ^i_k δ^j_l)    [ _ij,  _kl]    = -ε_ijklm^m 
    [ ^ij , ^k ]    = -1/2ε^ijklm_lm   [ _ij,  ^k ]    = 0 
    [ ^i_j, ]    =  + 1/2δ^i_j    [ ^i_j, _kl]    =  + 1/2δ^i_j _kl - δ^i_k _jl - δ^i_l _kj
    [ ^i_j, ^k ]    = -1/2δ^i_j ^k + δ^k_j ^i    [  , ^i ]    = ^i 
    [ ^i , _jk]    = δ^i_k _j -δ^i_j _k     [ _ij , _kl]    = - ε_ijklm^m

with all remaining brackets of basis elements vanishing. Here the indices i,j,… run over {1,2,3,4,5}, and we employ the Einstein summation convention.
Here ^i_j span gl(5). The basis elements of the subalgebras sl(5) are

    ^i_j^i_j - 1/5δ^i_j^k_k.


unsrturl




 Paper Title: Confining modeling of quark propagator.





[]aradzh@icc.ru
Matrosov Institute for System Dynamics and Control Theory SB RAS, 664033, Irkutsk, Russia

[]shangxinle@impcas.ac.cn
Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China


§ ABSTRACT 

A confining extension of the quark model with nonlocal currents is proposed.
The quark propagator is modified by introducing a cut in α-space, which in momentum space corresponds to the subtraction of pole singularities.
A two-phase phase structure is proposed for modeling the confinement-deconfinement phase transition.
In the confined phase, the quark propagator does not have any pole singularities, while in the deconfined phase, there is a single quark pole.


Confining modeling of quark propagator
    X.L. Shang
    August 5, 2024
======================================



§ INTRODUCTION


Understanding of strong interactions is a complex task.
The fundamental theory of strong interactions, quantum chromodynamics (QCD), is formulated in terms of quarks and gluons.
Experimentally observed hadrons are interpreted as bound states of these fundamental constituents. This is due to the nonperturbative nature of QCD. The strong coupling constant α_s is a small parameter only at large Q^2.



Free quarks and gluons have not been observed in experiments, they 
are confined inside hadrons.

It is expected that under certain physical conditions, such as in heavy ion collisions or within neutron stars, a different state of matter may form, which differs from the hadronic world we know. In this state, fundamental particles become deconfined and can freely propagate as quarks and gluons in a quark-gluon plasma (QGP).

The only ab-initio, nonperturbative method for investigating strong interactions is lattice QCD simulation. In recent years, lattice QCD has made impressive progress.




Despite this, the theoretical understanding of the underlying degrees of freedom still requires modeling since lattice simulations only provide numerical results without deeper insight.

The symmetries of strong interactions can serve as a guiding principle, such as chiral perturbation theory <cit.>, which deals with Goldstone bosons or the quark version of Nambu-Jona-Lasinio (NJL) model <cit.>, where chiral symmetry is spontaneously broken due to quark condensate and massive constituent quarks are formed instead of current ones.  

More sophisticated approaches are based on coupled systems of Dyson–Schwinger equations (DSE) for quark and gluon
propagators and their vertices, along with Bethe–Salpeter equations for hadronic bound states <cit.>. 

These approaches describe quarks with a momentum-dependent mass  m(k^2) and wave-function renormalization  Z(k^2). 


At large momentum transfer, the mass function tends to the current mass. 

The nonlocal version of the Nambu-Jona-Lasinio (NJL) model <cit.> is somewhere between these two approaches. The structure of the interaction is similar to that of the NJL model, but the nonlocal quark current leads to a quark propagator with momentum-dependent mass
 
However, the absence of confinement can make calculations problematic. 


If the denominator of the quark propagator has zeroes[This expression is given in the Euclidean metric,  which is mainly used in paper. Only equations with gamma matrices (<ref>) and (<ref>) are given in Minkowski momentum space.]

    k^2+m^2(k^2)=0,
 
at real values, k^2=-m^2_pole,
this will result in an imaginary part in the polarization operators of mesons.
This imaginary part corresponds to the appearance of free quarks.






One way to avoid the occurrence of an imaginary component in polarization operators 
is 
to have a mass function m(k^2) that only results in complex-valued solutions of this equation.


This case is similar to the 
scheme <cit.> and could be interpreted as modeling confinement <cit.>.


Apart from technical issues in calculating Feynman diagrams, the physical meaning of complex singularities remains unclear.



In a medium with varying temperature or baryon density, the positions of the singularities change, further complicating their physical interpretation.


In this paper, we aim to demonstrate that by making relatively simple modifications to the model, it is possible to eliminate these undesirable singularities.


The general idea is that confinement should somehow change the model, which is based on chiral symmetry.


Therefore, physical observables related to free quarks should not exist in the hadron phase.

In addition, a method for realizing the confinement-deconfinement phase transition is required.

It should be noted that the Polyakov loop extension of the NJL model with an effective potential of gauge degrees of freedom is intended to address one aspect of quark confinement at finite temperature – the suppression of quark pressure in the confining phase <cit.>. 

We propose an additional extension to the quark model.

The inverse Laplace transform of the denominator of the quark propagator in the confined phase is modified to ensure that the transformed function is valid for arbitrary momentum.


The resulting momentum space-transformed function has no pole singularities.
The new momentum scale parameter associated with this modification can be interpreted as the confinement scale Λ_c.


In the deconfined phase, a similar procedure is applied when the quark pole has been separated.






§ NONLOCAL MODEL



The Lagrangian of the SU(2) nonlocal chiral quark model with pseudoscalar–scalar sectors is:  

    ℒ= ℒ_free+ℒ_P,S
     ,  
    
    ℒ_free = q̅(x)(i ∂̂-M_c)q(x), 
       
    ℒ_P,S = G/2((J_S^a(x))^2+(J^a_P(x))^2) ,

where M_c=diag(m_c,m_c) is the current quark mass matrix with diagonal elements m_c, G is 
the four-quark coupling constant. 





The nonlocal quark current can be taken in one-gluon-exchange-like (OGE) <cit.> or instanton liquid model (ILM) <cit.> forms. The structure of OGE-type currents is:




    J_M^a{,μ}(x)
    = ∫ d^4x_1 d^4x_2  δ(x-x_1+x_2/2) ×
    × g((x_1-x_2)^2) q̅(x_1)  Γ_M^a{,μ}  q(x_2),

with M=S,P and  
Γ_S^a=λ^a, Γ_P^a=iγ^5λ^a. 
For the SU(2) model,  
the flavour matrices are Pauli matrices: λ^a≡τ^a, a=0,..,3 with τ^0=1.


g(x) is the form factor encoding the nonlocality of the QCD vacuum. 



The bosonized Lagrangian after the Hubbard-Stratonovich transformation is: 

    ℒ_eff   =
    q̅(x)(i ∂̂_x -M_c)q(x)+σ_0 J_S^0(x) +
        - 1/2 G((P^a(x))^2+ (S̃^a(x)+ σ_0 δ^a_0)^2)+
       
    +
    P^a(x)J_P^a(x)+
    S̃^a(x)J_S^a(x)

Spontaneous chiral symmetry breaking leads to non-zero vacuum expectation value of the scalar isoscalar field  
⟨ S^0 ⟩_0=σ_0≠0.
After shifting the scalar isoscalar field S^0=S̃^0+σ^0,


the momentum-dependent
quark mass appears[The same notation is used for the Fourier transform of functions.]:

    m(p^2)=m_c+m_dg(p^2)

where scalar coefficient m_d=-σ^0
can be found from self-consistent equation
 




    m_d= G 8  N_c /(2 π)^4∫ d^4_Ek
    g(k^2)m(k^2)/k^2+m^2(k^2),

where N_c is the number of quark colors.

The quark propagator in Minkowski metric is 

    S(p)=(p̂-m(p))^-1.



The pion polarization loop is

    Π_π(p^2)=8 N_c/(2π)^4∫ d_E^4 k f^2(q^2) D(k_-^2)D(k_+^2) ×
       ×[ 
    (k_- · k_+)+m(k_+^2)m(k_-^2)]
    
    ,

where 
D(k^2)=(k^2+m^2(k^2))^-1 is scalar propagator 
and momenta are p=k_+-k_- and q=(k_++k_-)/2. 
In the general case, the flow of momentum in the diagram is arbitrary, with  k_+=k+ζ p and k_-=k-(1-ζ) p, where 0<ζ<1.
The result should be independent of this choice. 




We utilize this feature to verify calculations.

The pion mass M_π can be found from equation

    -G^-1+Π_π(-M_π^2)=0.















For numerical estimation the set of model parameters with q̅q=-(240MeV)^3 <cit.>(Scheme II) is used[m_c=5.8 MeV, m_d=424 MeV, Λ=752.2 MeV.]. 


 §.§ Singularities of quark propagator


Analytical structure of quark propagator in nonlocal model strongly depends on the form-factor. For a Gaussian form-factor

    g(k^2)=exp(-k^2/Λ^2),
 

the denominator of the quark propagator (<ref>) has an infinite number of solutions  for complex values of k^2.





The inverse function, which is a 
scalar propagator D(k^2), has corresponding pole singularities.

singularities are also found in DSE studies <cit.>.
For some model parameters, the first two poles could be for real negative-valued k^2, that is, k^2 = -m^2_pole. After analytically continuing the meson polarization loop (<ref>) to Minkowski space, p^2 becomes negative. The imaginary part appears when  -p^2 exceeds the threshold mass squared, M_thr^2 = 4m_pole^2.









For complex-valued poles the situation is 
more complicated. 
Imaginary parts of different poles cancel each other <cit.> but the real part of the polarization loop has a cusp[
Here M_thr^2=2Re(m^2_pole) + 2 √(Re(m^2_pole)^2+Im(m^2_pole)^2).]. Such a feature would not seem to be a physical one, as it would be observable through experiment. 





In order to perform calculations in a medium, the G and Λ are treated as constants and m_d is the only parameter that changes in the quark sector. Therefore, it is interesting to investigate the position of the poles of the quark propagator for an arbitrary value of 
m_d (see Fig.<ref>), since this is exactly the scenario that would occur in a medium.


At the vacuum value of m_d, the first two poles are complex-valued (circles in Fig.<ref>). 

As m_d decreases, the two poles start to move towards the real axis along the dashed lines.
At m_d^crit, these complex-conjugate poles become real-valued (square).
At finite temperature, 


m_d^crit corresponds to chiral phase transition point.


If the presence of real-valued poles is considered an indicator of the confinement-deconfinement phase transition, these two phase transitions synchronize.


However, 
the problems are:


  * How to perform analytical continuation of the meson polarization loops integral beyond point M_thr? 
The prescription for complex-valued poles <cit.> leads to the suppression of the imaginary part of polarization loops. However, there is a cusp in the real part of the loop at the pinch point,  
which seems unphysical. 

  * What is the physical meaning of the second real-valued solution? 

It has a wrong sign of residue, which make it similar to Lee-Wick QED model with Pauli-Villars regulator <cit.>.
The first real solution with decreasing of m_d goes to zero, meaning the quark becomes a current one, while the second mass in this limit approaches infinity. At m_d^crit they have the same value.







 §.§ Suggestions for remove quark thresholds


There are various methods for removing thresholds:


  * Represent the quark propagator as an entire function in virton model <cit.> (in Minkowski metric)

    S(p)=-Mexp(-lp̂-p^2 L^2/4),

where M, l and L are some model parameters.
Similarly, in the quark confinement model of hadrons
<cit.>, the confinement is realized through ansätze for quark propagator averaged in gluon background. 















  * IR-cutoff in NJL model with proper time regularization <cit.> 
with function

    1/s+M^2=∫_0^∞ dτe^-τ(s+M^2)  →
       →∫_τ_UV^2^τ_IR^2 dτe^-τ(s+M^2) = e^-τ_UV(s+M^2) -e^-τ_IR(s+M^2)/s+M^2

or with three dimensional IR cut-off <cit.>.
The function (<ref>) is quite often used to model confined propagators <cit.>.






  * Scalar (or vector) part of quark propagator as entire function <cit.>

    D(k^2)=1-exp(-(k^2+m_c^2)/Λ^2)/k^2+m_c^2.
 

  * Cut in α-space <cit.> of the whole expression for polarization loops. 





The expression for the polarization loop with 
quark propagators is represented by the usual prescriptions in α-space (Schwinger parameterization). In the integral over the sum of α, the upper limit is changed from infinity to 1/λ. 
 

 
It should be noted that there are two groups of approaches: modifying each propagator <cit.> or the whole loop integral <cit.>. 





In a nonlocal model, the problem with a momentum-dependent mass m(p^2) is that for gauge invariance, it is necessary to modify the vertices of interaction with external currents, along with the quark propagator. As a result, it is unclear how to implement the modification of the loop integral.



 §.§ Confining α-prescription for quark propagator










If the 
scalar propagator has only pole singularities, it can be represented as the sum of its poles

    D(k^2)= ∑_i=1^NR_z_i/k^2+z_i
 
where N is number of poles (for Gaussian form-factor N=∞
). 



Poles are sorted by the size of the real part of  z_i with  z_1  having the smallest real part. 
   
Another expansion of the D(k^2) 
is based on its high-energy k^2
behavior, i.e., expansion over m_d



    D(k^2)= 1/k^2+m_c^2-2 m_c m_d g(k^2) /(k^2 + m_c^2)^2
    +O(m_d^2).
 

The Laplace transform is defined as:

    D(k^2)= L {𝐃(α)},
      D(k^2)=
    ∫_0^∞ dαe^-α k^2𝐃(α).
 


Series corresponding to the above equations (<ref>) and (<ref>) can be obtained with inverse Laplace transforms  𝐃(α)= L^-1{D(k^2)} for each term:

    𝐃(α)   =∑_i=1^∞R_z_ie^-α z_i  ,  
    
    
    𝐃(α)   = e^-α m_c^2 -2m_c m_d e^-α_1 m_c^2α_1 θ(α_1) +.. ,
 
where α_i=α-1/Λ^2i. The first series is valid for arbitrary form-factors. 
In the second one, only the first term is model-independent, and the actual analytical form of the form factor (<ref>) should be used for the other terms


[
The inverse Laplace transform of the terms with g(k^2) (<ref>) is


    L^-1{g^n(k^2) /(k^2 + m_c^2)^m}=e^-α_n m_c^2(α_n)^m-1/(m-1)!θ(α_n).
].
These expansions complement each other, because in order to obtain 𝐃(0)=1, an infinite number of terms in series (<ref>) must be summed. In contrast, the first term in series (<ref>) immediately gives the correct answer. On the other hand, in order to obtain the behavior of 𝐃(α) for large α, the first terms are sufficient in (<ref>).







The region of convergence for the Laplace transform (<ref>) is k^2 > -Re(z_1).


In order to make this transformation applicable to arbitrary k^2, it is necessary to change the behavior of the function 𝐃(α) for large values of α.





The simplest modification is following[
At first glance, the only physical restriction seems to be  Λ_c < Λ. 
Otherwise, the only first term in the series (<ref>) would give a nonzero contribution, and 

𝐃_R(α)
becomes  m_d-independent.
] 




    𝐃_R(α)=𝐃(α)θ(1/Λ_c^2-α)
 
In this case, the Laplace transformation (<ref>) becomes valid for arbitrary k^2. 




Performing Laplace transform, one can obtain the scalar 
propagator in momentum space 






    D_R(k^2)   =1/k^2+m_R^2(k^2)=
    
       =D(k^2)
    
    - ∑_i=1^∞R_z_i Q(k^2+z_i)/k^2+z_i,
 
where Q(k^2)=exp(-k^2/Λ_c^2). D_R(k^2) is entire function, since Q(0)=1 and all the singularities have been subtracted. On the other hand, as k^2 becomes large, D_R(k^2) tends to  D(k^2), since Λ_c< Λ.

From the equation (<ref>), one can obtain the mass function[
Here, one may encounter a problem that for a real k^2, the expression under the square root becomes negative, i.e., the quark mass m_R(k^2) becomes imaginary. This situation appears unphysical. 




] m_R(k^2) = √(D_R^-1(k^2) - k^2). 








The behavior of 𝐃_R(α) is shown in Fig.<ref>. 
The asymptotic behavior of the function 𝐃(α) is determined by its two lowest complex conjugate poles, i.e. i=1,2 in (<ref>).

As illustrated in the Fig.<ref> that this mode occurs at α of the order of 3/Λ^2.

To calculate the polarization loop in (<ref>), the prescription for propagators with complex masses can be used <cit.>. The main difference is that, instead of pole singularities, branch cut points will appear in the complex plane.













The result of the calculation of the pion polarization loop is presented in Fig. <ref>. One can observe that the cusp-like behavior of the meson polarization loop around -1 GeV^2 is absent for the  D_R(k^2).
The possible choice of Λ_c is as follows: as mentioned in footnote <ref>, Λ_c < Λ.
On the other hand, if Λ_c is too small, the polarization operator of the pion will grow too much for -p^2 > 1 GeV^2. 
A value of Λ_c = Λ / √(6) appears to be a reasonable compromise which leads to a reasonable behavior of m_R(k^2), see footnote  <ref>. This corresponds to a numerical value of Λ_c =307  MeV.




Since quarks can freely propagate in the deconfined phase, a pole for real negative k^2 should be present.


As the poles on the right-hand side of Eq. (<ref>) occur simultaneously in D(k^2) and in the second term with sum, it is possible to start the summation at i=2.
In this case, 
one can rewrite the expression Eqs.(<ref>) for D_R(k^2) in the form 

    D_R(k^2)=
    
    D(k^2) - ∑_i=2^∞R_z_i
    Q(k^2+z_i)
     /k^2+z_i=
      =R_z_1/k^2+z_1-iϵ+[
    D(k^2)-R_z_1/k^2+z_1- 
    
    
    -∑_i=2^∞R_z_i
    Q(k^2+z_i)
    /k^2+z_i],
 
where the only pole is isolated and the expression in square brackets is an entire function. 

In the limit of m_d approaching zero, the first pole approaches z_1→ m_c^2, with a residue of R_z_1→ 1 , and the expression within the brackets approaches zero.



The two-phase confinement-deconfinement model is:


  * Confined phase: the subtraction sum in Eq.
(<ref>) starts from i=1. D_R(k^2) is an entire function.

  * Deconfined phase: the subtraction sum starts from i=2. D_R(k^2) has one pole which corresponds to the pole quark mass. 










An important question is about symmetries - what will happen to the gauge symmetry and chiral symmetry?

The situation is clearer with gauge symmetry: one simply needs to include m_R(k^2) in the expressions for effective vertices <cit.> instead of m(k^2), and the Ward identity is automatically satisfied in this case. 

For chiral symmetry, the situation is more complicated. By construction there is a fine-tuning between the gap equation (<ref>) and the pion polarization loop (<ref>) <cit.> and changing of only the mass due to confinement prescription breaks this fine-tuning.

One possible solution is to determine the form factor based on the mass using Eq. (<ref>).

Alternatively, one can adjust the normalization of g(p) at the pion vertex. Numerically, these corrections are around two percent for the model parameters used and Λ_c^2 = Λ^2 / 6.



§ FINITE T BEHAVIOR



The mean field thermodynamic potential is 
<cit.>

    Ω^MF =m_d^2/2 G -4 ∑_i=0,±∫_k,nln[k_n,i^2+m^2(k_n,i^2)] + 
    
    + U (Φ,Φ̅) + Ω_0,

where k_n,i=(ω_n^i-i μ)^2+𝐤^2 and ∫_k,n≡ T∑_nd^3p/(2π)^3. Fermionic Matsubara frequencies ω_n^i are partially shifted due to the presence of a Polyakov loop ω_n^±=ω_n±ϕ_3, ω_n^0=ω_n and ω_n=(2n+1)π T.
The thermodynamic potential is divergent due to the current quark contribution.

The infinite normalization constant, Ω_0, is chosen such that the pressure should be zero under vacuum conditions, which means at zero temperature, zero chemical potential, and for the vacuum m_d value.



In the presence of the Polyakov loop, the equations of motion for the mean fields are

    ∂Ω^MF/∂ m_d = 0 ,   ∂Ω^MF/∂ϕ_3 = 0.




For comparison, the three models are considered:


  * 
Nonlocal quark model

  * Confined quark model (the sum in eq. (<ref>) starts from i=1)

  * Confinement-deconfinement 
model (the sum  in eq. (<ref>) starts from i=1 in confined phase and i=2 in deconfined phase.)


The boundary between phases in model (III) is determined by the properties of quark propagator. 
To a first approximation, in the deconfined phase, there are real solutions to Eq. (<ref>), whereas in the confined phase, only complex-valued solutions are possible.


However, it seems that the region where two real poles almost coincide should also be included in the confined phase. In practice, we use a criterion that the difference between the two real poles is of the order of the first one, i.e., in the deconfined phase, z_2 - z_1 > z_1. The corresponding points of m_d are approximately 319 MeV, which is the border between real and complex pole solutions, and 310 MeV when z_2 - z_1 > z_1 (the position of the poles is shown in Fig.<ref>).






Since the modification (<ref>) and (<ref>) is m_d-dependent one should include this behavior when taking the derivatives from the thermodynamic potential (<ref>) in (<ref>).
Therefore if one uses the analytical form of thermodynamic potential eq. (<ref>), the expression for the m_d gap equation should be changed. Alternatively, one could keep the analytical form of the gap equation 

    ∂Ω_R/∂ m_d =m_d/G-8 ∑_i=0,±∫_k,n
    g(k_n,i^2)m(k_n,i^2)D_R(k_n,i^2),

and calculate the thermodynamic potential numerically from gap equation

    Ω_R = ∫_0^m_d d m_d ∂Ω_R/∂ m_d + C(T,μ),

using  C(T, μ) as the integration constant. C(T, μ) is equal to the value of Ω^MF, taken at zero m_d,

for a given values of T, μ, and ϕ_3 with normalization Ω_R=0 in vacuum.

It is very instructive to study the m_d dependence of the thermodynamic potential in almost vacuum, since there is no influence from the Polyakov loop in this case.
In Fig. <ref> ∂Ω_R/∂ m_d and Ω_R are shown for models (I)-(III).










The finite temperature behavior of the quark condensate and the Polyakov loop are shown in Fig. <ref>. 
One can see here that even for the confining model (II), the quark condensate (and similarly m_d) decreases as the temperature increases.


The reason for this is the gap equation (<ref>). In the numerator of the integrand, there is a form factor g(k^2). The Matsubara frequency is proportional to T, and the first fermionic frequency is nonzero (ω_0 = π T). Therefore, as the temperature increases, the second term on the right-hand side of (<ref>) becomes strongly suppressed due to the form factor. Therefore, in order to fulfill the gap equation, a corresponding decrease of m_d is needed.



The finite T phase transition in model (III) is of first order. This is due to the fact that the gap equations in the two phases are not completely synchronized. Fig. <ref> shows that the gap equation (upper part) is not smooth, and there is a jump in the behavior of ∂Ω_R/∂ m_d between the phases. In some conditions, both solutions are possible.
However, as can be seen in Fig. <ref>, the finite T behavior of model (III) is not significantly different from that of the model (I).


At finite chemical potential, the second minimum at low m_d appears in models (I) and (III), i.e., in the deconfined phase. 
At a certain critical value of m_d, the system undergoes a transition from a confined to a deconfined state.
In model (II), this is not the case by design.


§ CONCLUSIONS









The paper discusses a simple prescription based on modifying the Laplace transform of the quark propagator in order to phenomenologically model confinement. The main goal of the work is to improve the analytical behaviour of the quark propagator.



At finite T/μ, two phases are considered: in the confined phase, the quark propagator does not have pole singularities, and in the deconfined phase, it has a single physical pole. As a result, mean field calculations for the critical temperature and chemical potential in this scheme are almost unchanged. 


Pole singularities in the complex plane are absent, and therefore there are no physical consequences that can be caused by poles. Instead, the mass function has cuts in the complex plane, and instabilities discussed in <cit.> are present, although they are less significant.

One can expect that a more interesting situation will arise with 1/N_c corrections. 
Namely, the mesonic polarization loops at leading order in the nonlocal model exhibit a cusp-like behavior when poles in the complex plane are located inside the integration contour. For finite temperature and zero chemical potential, one can calculate the mesonic correction to the quark condensate using only the Euclidean properties of the polarization loops. At finite chemical potential, the cusp-like behavior of meson polarization loops can make calculations challenging.

However, our prescription is not free from problems. Some of these problems are technical and can be easily solved. In order for the pion to be massless in the chiral symmetric case, when m_c→ 0, it is necessary to modify 
the pion-quark vertex. The value of g(0), which represents the coupling between the pion and quark, should slightly deviate from 1. 


The first order phase transition at finite T for model (III) with a confinement-deconfinement phase transition occurs because the gap is not a smooth function of m_d.
 

The most important conceptual problem is the exponential growing of meson polarization loops in Minkowski space.
This issue arises because if the quark propagator has no singularities at finite p^2, it will have an essential singularity at infinity.


How to solve this problem remains unclear, but sub-leading 1/N_c corrections, such as pion dressing of the
quark propagator, may introduce physically motivated singularities.
The pion-quark dressing is discussed by V. N. Gribov in his confinement theory <cit.>. 




This work is supported by supported by the CAS President’s international fellowship initiative (Grant No. 2023VMA0015); the project of the Ministry of Education and Science of the Russian Federation (”Analytical and numerical methods of mathematical physics in problems of tomography,quantum field theory, fluid and gas mechanics”no. 121041300058-1); the National Natural Science Foundation of China under Grants No. 12375117; and the Youth Innovation Promotion Association of Chinese Academy of Sciences (Grant No. Y2021414).

The authors thank the I.V. Anikin, Yu.M. Bystritskiy, V.P. Lomov and B. Zhang for fruitful comments.  













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 Paper Title: Curvature-induced moduli stabilization.







flavio.tonioni@kuleuven.be
Instituut voor Theoretische Fysica, KU Leuven, Celestijnenlaan 200D, B-3001 Leuven, Belgium



§ ABSTRACT 
We discuss FLRW-cosmologies with negatively-curved spatial slices that induce a late-time stabilization of all the moduli of an effective field theory through Hubble friction, independently of the moduli-space curvature. This holds for pure moduli, which appear ubiquitously in string compactifications. Crucially, the cosmological solutions approach a Milne universe. Hence, the mechanism we describe in fact corresponds to late-time moduli stabilization in Minkowski spacetime.




Curvature-induced moduli stabilization
    Flavio Tonioni
    August 5, 2024
======================================






§ INTRODUCTION

In string compactifications, moduli stabilization typically requires extremely hard calculations, and it can face deep control issues.
Among others, the Dine-Seiberg problem represents a general argument as to why this is the case <cit.>, relating to the fact that all string-theoretic potentials are generated dynamically, with perturbative expansions that also control the validity of the effective field theory (EFT) itself.
In fact, proposed mechanisms that are argued to allow for lower-dimensional vacua with Minkowski or de Sitter (dS) geometries require a delicate interplay of (non)perturbative effects such that the moduli are stabilized somewhere in between the bulk and the asymptotic regions <cit.>; for recent reviews, see refs. <cit.>.
Working in intermediate regions, however, confronts one with the issue of controlling all corrections: see ref. <cit.> for a recent account.
Arguments based on the completion of an EFT into a full quantum-gravity framework, together with circumstantial evidence, actually led some to hypothesize the absence of potentials able to stabilize all moduli in asymptotic regions of moduli space of string compactifications as a general fact, at least if the potential is positive <cit.>; for a review of the criticisms to dS constructions in string compactifications, see ref. <cit.>.

In this note, we describe a general dynamical mechanism by which Friedmann-Lemaître-Robertson-Walker (FLRW) cosmologies with negatively-curved spatial slices automatically stabilize all moduli, at late-enough times.
Here, by “moduli”, we truly mean (pseudo)scalar fields with no potential.
The intuition behind their stabilization is extremely simple.
On the one hand, the curvature term is a fluid that saturates the strong energy condition (SEC).
On the other hand, pure moduli just represent a kinating fluid, whose energy density decays more quickly than a SEC-saturating fluid.
Therefore, given enough time, only the energy density of the curvature fluid survives.
Relative to the latter, the moduli kinetic energy must vanish.
It is in this sense that in the present work we intend the “stabilization”.
The dynamical mechanism is independent of the moduli-space curvature and of the details of the underlying string compactification.
A universe whose only component is its own negative spatial curvature, known as the Milne universe, is actually the interior of a future lightcone in Minkowski geometry, in suitable coordinates.
Hence, what we achieve is a dynamical method to stabilize all moduli in Minkowski spacetime.
This mechanism works exactly in the absence of a potential, thus circumventing all arguments related to the large slope and/or convexity of scalar potentials in string compactifications.
It also evades the claim that 10d supergravity solutions compactified to 4d Minkowski always admit a 4d massless scalar <cit.>.

The rest of this note is organized as follows.
In sec. <ref>, we present the details of the curvature-induced stabilization mechanism.
In sec. <ref>, we discuss a simple example.
Finally, a general discussion is in sec. <ref>.
Additional mathematical details are in app. <ref>.




§ DYNAMICAL MODULI STABILIZATION
 
Throughout this note, we study a set of n moduli φ^a that are associated to a moduli-space metric

    s_n^2 = G_ab (φ)  φ^a φ^b,

with a corresponding Christoffel connection Γ^a_bc; Einstein summation is always implied on the field-space indices a=1,…,n.
Curved moduli spaces emerge ubiquitously in string compactifications and in supergravity constructions.
Let the non-compact d-dimensional spacetime be described by the FLRW-metric

    d s̃_1,d-1^2 = -  t^2 + a^2(t)  [  r^21 - k r^2 + r^2 Ω_d-2^2 ].

Here, a is the scale factor, which defines the Hubble parameter H = ȧ/a, and k is the constant curvature of the spatial slice.
Finally, we consider the presence of a set of homogeneous and isotropic fluids of energy densities ρ^α and pressures p^α = (q^α-1) ρ^α, for α=1,…,b.
A barotropic fluid has an equation of state p = (q-1) ρ, where q is a constant in a multitude of cases including curvature anisotropies, radiation, matter, and cosmological constants, with q=2, 1/(d-1), 1, 0, respectively.

In this setup, the complete cosmological equations can be written as


    φ̈^a + Γ^a_bcφ̇^b φ̇^c + (d-1) H φ̇^a = 0, 
       ρ̇^α + (d-1) H (p^α + ρ^α) = 0, 
        H^2 = 2 κ_d^2(d-1) (d-2)[ 12  G_abφ̇^a φ̇^b + ∑_α=1^b ρ^α].


A combination of eqs. (<ref>, <ref>, <ref>) also gives the useful relationship

    Ḣ = - κ_d^2d-2 [ G_abφ̇^a φ̇^b + ∑_α=1^b (p^α + ρ^α) ].

The continuity eqs. (<ref>) for constant values of q^α can be solved as ρ^α = ρ^α_0   (a_0/a)^(d-1) q^α, where a_0 = a(t_0) is the scale factor at an initial time t_0.
Hence, the curvature term can be formally treated as an on-shell fluid with energy density ρ_k = - [(d-1)(d-2)/(2 κ_d^2)]   k/a^2 and equation of state given by the parameter

    q_k = 1 - d-3d-1.

Noticeably, this means that the negative-curvature fluid, i.e. the one with k=-1/ℓ^2, for an arbitrary length (or time) scale ℓ, saturates the SEC.




 §.§ Milne universe

In the presence of a potential V=V(φ) for the fields, eqs. (<ref>, <ref>) do not allow one to express the kinetic energy T = G_ab φ̇^a φ̇^b/2 in terms of the Hubble parameter and of the fluid energy densities.
In the Friedmann eqs. (<ref>, <ref>), one generally finds the linear combinations T+V and T-V, respectively.
However, if V=0, then T is a variable that drops out of the equation governing the evolution of the scale factor.

In fact, a simple manipulation leads to the equation

    (d-1)(d-2)2 κ_d^2 H^2 + (d-2)2 κ_d^2Ḣ = ∑_α2 - q^α2ρ^α.

The a-dependence of the fluid energy densities is known. Hence, the above is a second-order ordinary differential equation (ODE).
If the only fluid is negative FLRW-curvature, i.e. k=-1/ℓ^2, the equation takes the form

    H^2 + 1d-1Ḣ = d-2d-11ℓ^2 a^2.

One can find the analytic solution to this equation: details are in app. <ref>; for the case where the only fluid is the negative FLRW-metric spatial curvature, a previous derivation of such solution appeared in ref. <cit.>.

Here, we can shortly illustrate a way in which such a solution can be found.
Let us reparameterize the time t in terms of a time coordinate σ defined via <cit.>

    t^2 = σ^2σ^2σ_p^2 (ℓσ)^2(d-1) + 1,

for a constant σ_p ∈ℝ.
Then, eq. (<ref>) is solved by

    a(σ) = σℓ.

The fact that this is the only solution is discussed in more detail in app. <ref>. In particular, we observe that at late-enough times it is possible to approximate t(σ∼∞) ≃σ.

The main observation of the present note consists in the fact that, in terms of the cosmological time t, the late-time behavior of the scale factor is such that

    a_∞(t) = tℓ,

which corresponds to a Milne universe.
Therefore, by plugging this solution into either eq. (<ref>) or eq. (<ref>), one finds the requirement that

    φ_∞^a(t) = φ^a_∞,

where φ^a_∞∈ℝ are constants.
One can immediately see that eq. (<ref>) is a solution of eq. (<ref>).
For generic initial conditions, this solution is technically correct in the true asymptotic regime t ∼∞, where the scale factor takes exactly the form in eq. (<ref>).
One might fear that, before their kinetic energy dies off, the scalars might travel a long distance.
Actually, by an asymptotic expansion of the exact solution, one can verify that the scalar-field displacement falls off in a power-law form, with a time scale set by ℓ.
Hence, the field displacement is finite.

We can interpret the vanishing of the moduli kinetic energy as being due to Hubble friction.
To see this, we can observe that pure moduli represent a perfectly stiff fluid, with a parameter

    q_φ = 2.

It is indeed trivial to verify that, in the absence of a potential, the scalar field equation implies the continuity equation for a fluid with ρ_φ = p_φ = G_ab φ̇^a φ̇^b/2.
Comparing with eq. (<ref>), the energy densities associated to the negative spatial FLRW-curvature and the moduli fall off with the scale factor as

    
    ρ_k     = ρ_k 0 ( a_0a)^2, 
    ρ_φ    = ρ_φ 0 ( a_0a)^2(d-1).


Hence, by a reasoning that is typical in cosmology, eventually the fluid with with the lowest q-parameter dominates over the others.
One can also check that the scale factor in eq. (<ref>) determines a late-time Hubble parameter H_∞(t) = 1/t.
Hence, eq. (<ref>) requires φ̇_∞^a = 0.
We stress that none of these conclusions depends on the curvature of the moduli space.

If there are multiple fluids, the individual energy densities fall off more quickly for larger q-parameters.
Hence, after long enough, only the fluid with the smallest value of q survives.
Therefore, the mechanism we presented keeps working even if we include any number of SEC-fulfilling fluids, such as radiation, matter and curvature anisotropies.




 §.§ Minkowski spacetime

As is well known, the Milne universe coincides with the interior of a future lightcone in Minkowski spacetime <cit.>.
To see this, we can define the radial coordinate

    χ(r) = artanh [ r√(r^2 + ℓ^2)],

which is such that χ^2 =  r^2 / (r^2 + ℓ^2).
With this coordinate choice, if the scale factor is a(t)=t/ℓ, the metric in eq. (<ref>) reads

    d s̃_1,d-1^2 = -  t^2 + t^2 [ χ^2 + sinh^2  χ Ω_d-2^2 ].

Now, we can write the Minkowski metric as

    d s_𝕄^1,d-1^2 = - τ^2 + ρ^2 + ρ^2  Ω_d-2^2.

By the coordinate transformation

    
    τ    = t  cosh χ, 
    ρ    = t  sinh χ,


we recover exactly the negatively-curved FLRW-metric with a(t)=t/ℓ.
Crucially, the late-time regime t ∼∞ in the Milne universe is a truly late-time regime in Minkowski time τ∼∞, at any radial distance χ from the coordinate origin.

Solutions where the late-time scale factor is linear in cosmological time may be expected in FLRW-universes with negative spatial curvature, as the latter saturates the SEC.
However, such solutions are not necessarily Milne universes, and hence not Minkowski spacetimes after a coordinate redefinition.
To see this, let

    a(t) = α tℓ,

where k=-1/ℓ^2. Then, the Ricci scalar for the metric in eq. (<ref>) reads

    R̃ = (d-1)(d-2)t^2 ( 1 - 1α^2),

which is clearly non-zero for any value except α=1.
It is this specific value that corresponds to a Milne universe.
Scalar fields tend to have a “tracker behavior”: their q-parameter is not constant and in many known examples it tracks down the q-value of other fluids in the theory, at late times <cit.>.
For instance, a single scalar ϕ with a potential V = Λ ^- κ_d γϕ, for a constant γ, admits a solution a(t) = (1/√(1 - 4/[(d-2) γ^2]))   t / ℓ <cit.>.
It is only in the limit γ→∞ that one recovers a Milne universe, which indeed corresponds to V → 0.
This can be understood by noticing that it is the scalar potential that induces the tracker behavior: in the absence of a potential, a set of scalars is instead a purely kinating fluid, with constant parameter q_φ=2.




 §.§ Cosmic deceleration

One might wonder whether the late-time Minkowski geometry can be attained along a trajectory undergoing accelerated expansion, in the FLRW-formulation. Unfortunately, here we show that this is not the case.

Given an FLRW-metric as in eq. (<ref>), cosmic acceleration is defined as the set of conditions ȧ>0 and ä>0.
In particular, the condition on the second time derivative is equivalent to the condition that ϵ = -Ḣ/H^2<1.
In terms of the time coordinate σ defined in eq. (<ref>), we can express the ϵ-parameter as

    ϵ(t) = 1 + d-21 + ( σ(t)ℓ)^2(d-2).

This means that we have ϵ(t)>1 at all times t, and in particular that lim_t →∞ ϵ(t) = 1^+.
Hence, in the FLRW-formulation, the expansion is always decelerated.
This does not necessarily represent a problem for exploiting the stabilization mechanism into string embeddings of cosmic inflation, since the moduli might only need to be stabilized after the inflationary phase.
On the other hand, it does not provide a viable way for getting a transient quasi-dS expansion at late times.




§ EXAMPLE: GKP COMPACTIFICATIONS
 
A protypical example of a string compactifcation down to a Minkowski geometry with a set of unstabilized moduli is the class of Giddings-Kachru-Polchinski (GKP) backgrounds <cit.>.
These are type-IIB compactifications on a Calabi-Yau orientifold CY'_3 with RR-3- and NSNS-3-form fluxes in which the Kähler sector happens to be flat in all directions.
Here we illustrate how the dynamical stabilization of the latter can work.

As a compactification Ansatz, the 10d Einstein-frame line element ds_10^2 = ĝ_MN  x^M  x^N is taken to be

    ds_1,9^2 = ^2 A (y) d̃s̃_1,3^2 (x) + ^-2 A (y) d̆s̆_6^2 (y),

where the function ^2A is the so-called warp factor. The blocks g̃_μν (x) and ğ_mn(y) are a 4d metric depending only on the 4d coordinates x^μ and the metric of a 6d Calabi-Yau orientifold depending on the 6d internal coordinates y^m, respectively. Due to Lorentz invariance, the background axio-dilaton τ varies only over the internal manifold, i.e. τ = τ (y). Similarly, the complex 3-form flux G_3 = F_3 - τ H_3 only has internal indices, i.e. G_3 = G_mnl  y^m ∧ y^n ∧ y^l / 3!, and the self-dual 5-form flux F̃_5 is of the form F̃_5 = (1+*̂_10)  α∧ṽõl̃_1,3, for a real scalar α = α(y).
Finally, the theory includes a set of localized sources such as D- and anti-D-branes as well as O- and anti-O-planes, compatibly with the Ansatz metric.

The 4d components of Einstein's equations and the 5-form flux field equation can be combined in an enlightening way, giving <cit.>

    ∇̆^2 Φ_- = R̃_4     + (Φ_+ + Φ_-)^24  Im τ  G_3^-  ·̆ G_3^- 
        + 2Φ_+ + Φ_- Φ_- ·̆ Φ_- + T_4,6,

where G_3^- = [(*̆_6 G_3) -  G_3]/2, Φ_± = ^4 A±α, and T_4,6 is a combination of the energy-momentum tensor of the localized sources, which vanishes e.g. for appropriate D3-brane/O3-plane configurations; we use the notation
(A_q ·̆ B_q)  v̆ŏl̆_n = A_q ∧*̆_n B_q, on an n-dimensional space.
The left handside integrates to zero over a compact manifold, so the same must happen with the right-handside.
In a 4d Minkowski spacetime, with R̃_4=0, each term in the right handside must integrate to zero, too, which implies the conditions


    *̆_6 G_3     =  G_3, 
    ^4 A    = α.


Such conditions can also solve the remaining field equations.
In terms of a 4d effective theory, such fluxes stabilize the axio-dilaton and the complex-structure moduli.
However, the Kähler moduli are purely flat directions.
For instance, the overall volume is controlled by the imaginary part of the chiral supermultiplet ρ = θ +  ξ, whose field-space metric block can be read off the Kähler potential κ_4^2 K = -3  ln  [- (ρ - ρ)].
The mechanism we presented above stabilize both fields θ and ξ to finite values, while solving the 10d equations of motion.




§ DISCUSSION
 
In this note, we have described a generic framework in which all the moduli of a given EFT are dynamically stabilized into constant finite values in a Minkowski geometry.
The process takes place independently of the moduli-space curvature and of any possible compactification details, such as the internal geometry.
Strictly speaking, the stabilization mechanism takes place in the asymptotic future.
However, throughout the evolution, the kinetic energy of the fields becomes parametrically smaller and smaller compared to the energy set by the Hubble scale, in an associated formulation in terms of a FLRW-cosmology with negative spatial curvature.
The Hubble-sized energy is stored in the spatial FLRW-curvature.

It is of course not surprising that fluids (help) freeze the scalar kinetic energy, relative to the Hubble scale.
For instance, this is an integral element in the recent studies of kination phases in early moduli-driven stringy cosmologies, where radiation terminates the kination epoch <cit.>.
Hubble damping generally helps overcome the overshoot problem <cit.>.
In fact, such mechanisms have been exploited extensively in the literature: see e.g. refs. <cit.>.
What is special about negative spatial FLRW-curvature, however, is the fact that asymptotically it gives a perfect Milne universe, which is actually the Minkowski spacetime.
Hence, not only does the scalar kinetic energy vanish, compared to the overall energy scale of the problem (the Hubble scale, in a cosmological setting), but it does so into a Minkowski vacuum (and not necessarily in a cosmological setting).

The setup for the result is that of classical field theory.
The final state that one finds is a Minkowski vacuum, with all the scalars fixed at a finite value.
However, such scalars have no Lagrangian mass.
Hence, one should expect that their fluctuations mediate long-range forces, and possibly the time dependence of
fundamental constants; see e.g. ref. <cit.> for an account.
Looking at the problem from a different angle, the stabilization mechanism that we have presented is generally unstable under the presence of scalar potentials.
At some point, as the Hubble energy asymptotes to zero, any scalar potentials (that perhaps one may have consistently neglected at higher energies) might become relevant, thus changing the dynamics of the problem.
Nonetheless, if such potentials are irrelevant at least initially, the dynamics that we described might at least serve as an explanation as to why the field kinetic energy is small, and thus also circumvent the discussion on long-range forces.
This might be of use in constructions that require some degree of fine tuning of the initial conditions to accommodate for realizations of dark energy, such as in refs. <cit.>.
We leave studies of other phenomenological questions such as the cosmological moduli problem <cit.> for future work.

Although with a number of caveats, the instance of a dynamical moduli stabilization into a Minkowski vacuum presented in this note is applicable to string compactifications in generality.
Recently, Minkowski vacua with no running moduli have also been obtained in non-geometric backgrounds that are mirror duals of rigid Calabi-Yau manifolds <cit.>.





 §.§ Acknowledgments

I am grateful to Thomas Van Riet and Timm Wrase for inspirational conversations and for their precious feedback on the manuscript.
My work is supported by the FWO Odysseus grant GCD-D0989-G0F9516N.







§ EXACT SOLUTION
 

A solution a=a(t) to eq. (<ref>) can be found analytically; the overall sign is arbitrary, but we focus on solutions with a positive scale factor a>0.
It is convenient to express it as the inverse of a function b, i.e.

    a(t) = b^-1(t),

which reads

    b(t) = ℓ t  _2 F_1[ 12, - 121d-2, 122d-5d-2, pt^2(d-2)],

where _2 F_1=_2 F_1(a,b,c;z) is the ordinary hypergeometric function and p ∈ℝ is an integration constant.
An initial time gauge has been chosen for brevity.
Although this expression is complicated, it admits an extremely simple late-time behavior.
For simplicity, let p = s   t_p^2(d-2), with s = ± 1 and t_p>0.
An asymptotic expansion then reveals that

    b(t) = ℓ t [ 1 - 12s2d - 5 ( t_pt)^2(d-1) + O( t_pt)^4(d-1)].

At leading order in t_p/t, we may approximate the function b with the function b_∞(t) = ℓ t, which leads immediately to the late-time scale factor in eq. (<ref>).
Working at next-to-leading order is harder, but the expression of b(t) motivates the expansion

    a(t) = tℓ[ 1 - q  (t_pt)^m + O(t_pt)^m_1],

for a constant q ∈ℝ that is yet undetermined and two positive constants m>m_1>0.
Plugging this back into eq. (<ref>) indicates that one must fix m=2(d-2).
Plugging this solution into eq. (<ref>), one finds the late-time behavior

    T(t) = (d-2) q_dκ_d^2 t_p^2 (t_pt)^2(d-1) [ 1 + O(t_pt)^m_2],

for some positive constant m_2>0, and where we defined q_d = (d-1)(2d-5)   q for convenience.
This means that one must have q>0; the same expression is confirmed by plugging the scale factor into eq. (<ref>).
One can also check that we may write T = T_0   (a_0/a)^2 (d-1) [ 1 + O(a_0/a)^m_0], for some constant m_0>0 and some reference values a_0 and T_0, in agreement with the fact that the moduli represent a stiff fluid (see eq. (<ref>)).
For instance, in the case of a single canonical field, where T = φ̇^2/2, one finds

    φ(t) = φ_∞ - √(2 q_d)κ_d √(d-2) (t_pt)^d-2 [ 1 + O(t_pt)^m_3],

up to a sign, where φ_∞ is an integration constant and m_3>0 is another positive constant. The integrated scalar field excursion from an initial time t_0 until infinity, defined as δ = κ_d ∫_t_0^∞ t  φ(t), is

    δ = t_0d-3 √(q_d)√(d-2) (t_pt_0)^d-2[ 1 + O(t_pt_0)^m_3],

if d>3. One has δ=∞ if d=3.

The solution in eqs. (<ref>, <ref>) was first derived in ref. <cit.>.
Here we revisit this derivation to provide further understanding of the results in the main text.
Let the spacetime metric be parameterized as

    d s̃_1,d-1^2 = - f^2(σ)  σ^2 + a^2(σ)   h_i j(x)   x^i  x^j,

where h_ij (x) is the metric over a maximally-symmetric (d-1)-dimensional space 𝕏^d-1 with constant curvature k, for i,j = 1, …, d-1.
Let φ^a = φ^a(σ) be a set of scalars.
In terms of the time variable ω, defined through

    ωσ (σ) = f(σ)a^d-1(σ),

the action for the scalars takes the form

    S[φ] = 12 ∫ω∫_𝕏^d-1^d-1 x   G_ab(φ)  φ^aωφ^bω,

which is formally independent of the metric.
Hence, the scalars have trivial dynamics, as they just trace out geodesics in moduli space.
The Einstein equations for the function a(σ) then read

    (  aσ)^2 = [ κ_d^2   p^2(d-1)(d-2) 1a^2(d-2) - k ]   f^2,

where p^2 = G_ab(φ)   (φ^a / ω)   (φ^b / ω) is a constant.
This explains the disappearing of the scalar dynamics from the equations governing the metric evolution that we observe in eq. (<ref>).
A general solution takes the form

    a^2(σ) = ( σℓ)^2,

where

    f^2(σ) = 1ℓ^2 1κ_d^2   p^2(d-1)(d-2) (ℓσ)^2(d-2) - k.

Let us reparameterize time as in eq. (<ref>), namely through the definition t = f(σ)  σ.
In case k=-1/ℓ^2, we have a(σ(t)) = a(t), with a=a(t) in eqs. (<ref>, <ref>), confirming the result.
In terms of the time coordinate σ, it is already clear that the late-time behavior of the solution is a Milne universe.
Indeed, we see the behavior f^2(σ∼∞) ≃ 1, which trivially reduces the metric to the form in eq. (<ref>).
The late-time behavior is independent of the initial conditions.


apsrev4-1






 Paper Title: Effective Potential in Finite Formulation of QFT.













    â_+â_-ϕ̃α̃x̃m̃l̃k̃k_ renT̃q̃κ̃ρ̃Ø𝒪ϵŁℒγ^hhγ^θθξ_Gξ_hĥθ̂ϕ̂χ̂Â∂̣μ̃m_ p(ħ)∓m_ physλ_ physϕ_0^⋆𝒪(ħ)𝒪(ħ^2)h̅θ̅g_∂ h θ Ag_∂θ h Aarrows,shapestreesmatrix,arrows
        											
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         1.5cm
        
        
        Effective Potential in Finite Formulation of QFT 1.5cm
        
         Sander Mooij and Mikhail Shaposhnikov 1cm
        
        Institute of Physics,  École Polytechnique Fédérale de Lausanne, 
        CH-1015 Lausanne, Switzerland
        	
        	
         1cm
        
        sander.mooij@epfl.ch, mikhail.shaposhnikov@epfl.ch
        1cm
        
        
         Abstract
         0.5cm
        
        In recent works <cit.>, we have shown how n-point correlation functions in perturbative QFT can be computed without running into intermediate divergences. Here we want to illustrate explicitly that one can calculate the quantum effective potential by the same method. As a main example, we consider a theory with two fields having large and small vacuum expectation values (vev). We show that no fine-tuning between the  physical quantities is needed to keep the hierarchy between the vevs of different fields.
        
        emptyarabic
        
        § INTRODUCTION
        
        In recent works <cit.>, we have shown how to compute n-point Green's functions in scalar field theories without running into large or infinite quantum loop corrections that need to be carefully adjusted in theories with several distinct mass scales. We have argued that the existence of such a finite QFT formalism indicates that the so-called “hierarchy problem" is method-dependent and that no fine-tunings are needed between  physical parameters even if the theory contains light and superheavy particles.  In this sense, the physics at the electroweak scale does not, in general, feel the physics of a UV completion living at much higher energy scales <cit.>, with notable exceptions related to experimentally challenged attempts to describe the Higgs boson in terms of a composite bound state, or within the theories with dynamical supersymmetry breaking, or in theories with large extra dimensions and low fundamental Planck scale (the updated reviews can be found in <cit.>). 
        
        Our work <cit.> finds its place in a long debate about the meaning of infinite and “finite but large" contributions to correlation functions. As for the infinities, the discussion goes back to the days of Dirac <cit.> up to the present day, see for example the recent work <cit.>. Quadratically divergent quantum corrections in theories with fundamental scalar fields have been the major driving force of overwhelming literature on naturalness issues. A contemporary overview can be found, for example in <cit.>, plus references therein (and references in our paper <cit.>).
        
        In <cit.>, we mainly focussed on the computation of n-point correlation functions. This work aims to extend the formalism of finite QFT to the computation of the effective action of the theory, an instrument to uncover the ground state of the theories with spontaneous symmetry breaking <cit.>. 
        
        Historically, the gauge hierarchy problem has one of its roots in Grand Unified Theories (GUTs) of strong, weak and electromagnetic interactions <cit.>. It is required that one of the GUT multiplets (say, 24 of SU(5)) has a very large vacuum expectation value (vev) of the order of the GUT scale 𝒪(10^14) GeV to break the symmetry down to that of the Standard Model, and another (say, 5 of SU(5)) has a vev defining the Fermi scale. 
        
        In <cit.> it was argued that the radiative corrections set up an upper a bound on a ratio of the vector boson masses of the order of O(1/√(α)), where α is the fine structure constant, making GUTs impossible. It was soon realised that this was not the case, and there is no limit to gauge hierarchies coming from perturbation theory <cit.>. Still, it was reasoned that the hierarchy is not  “natural”, as getting it would require fine-tunings between the different orders of the perturbation theory. “Fine-tuning” means in general that a specific combination of two large numbers (say, mass parameters of  5 - and  24 - plets of SU(5)) should be adjusted in such a way that it is very small. An equivalent reformulation of this problem is known as the “doublet-triplet” splitting problem (for a review see <cit.> and for a recent reanalysis <cit.>).  The SU(5) GUT symmetry breaking splits a fundamental 5-plet  into a light doublet (the Higgs field) and a scalar leptoquark - SU(3) triplet whose masses remain of the order of the GUT breaking scale. To keep this hierarchy the same type of fine-tuning is needed. 
        
        Note that the gauge hierarchy (or doublet-triplet splitting) problem contains two layers. First, one needs an initial choice of parameters at the tree level to establish the hierarchy between the electroweak scale and the GUT scale. Second, one seemingly needs additional fine-tunings to keep this dramatic doublet-triplet splitting stable against loop corrections. 
        
        The problem has been settled a long time ago.  First, the necessity of the initial fine-tuning depends on the choice of the “fundamental parameters” of the action and thus cannot be expressed in measurable physical terms. For instance, if the “fundamental parameters” are the mass terms of different GUT scalar multiplets, fine-tuning is needed. If the “fundamental parameters” are the vevs of the multiplets, no fine-tuning is necessary (see, e.g.  <cit.>). As for the radiative corrections, they are not uniquely defined and depend on the renormalisation procedure. The latter can be constructed in such a way that no fine-tunings are needed in any order of perturbation theory <cit.>. In other words, there is the physics problem “why the GUT scale is so much larger than the Fermi scale”, but the problem about the stability of the hierarchy is artificial and related to the choice of formalism, but not the physics.
        
        In this paper, we show how to arrive at the same conclusions within the finite QFT method that we used in <cit.>.  We reframe this formalism in terms of the effective action. For definitiveness, we focus on the case of two interacting quartically coupled scalar fields with a large mass hierarchy in between them. We show how their effective action can be computed without the need of “artificially" invoking large counter-terms that need to be balanced against large bare loop corrections to shield the physics of the light field from the influence of the heavy field. Moreover, we show how the choice of parameters at the tree level suffices to avoid those large quantum corrections blowing up the vev and the masses of the light field. In other words, loop corrections do not spoil the hierarchy imposed at the tree level. Consequently, there is no need for additional fine-tuning. Once again, the delicate, fine-tuned interplay between the physics of widely separated mass scales that  keeps the Higgs vev small and the Higgs field light is just an artefact of the standard computational method. Precisely as we advocated in the previous paper, low-scale physics is insensitive to the way it is embedded in possible large-scale extensions.
        
        The paper is organised as follows. In Section <ref> we set our notation and quickly review the standard approach to compute the effective potential. Section <ref> is devoted to the generalisation of the finite Callan-Symanzik approach to computing correlation functions for the case of effective action. In Section <ref>, we show how formalism can be used in a theory of two interacting scalar fields. Implications for the quantum stability of the electroweak scale are reserved for Section <ref>. We conclude in Section <ref>.
        
        As for our notational convenience, we choose to reinstall ħ in all equations, unless explicitly stated. We will only consider a one-loop approximation, as it is sufficient for demonstrating the main point of our study.
        
        
        
        § STANDARD APPROACH
        
        In this section, we quickly review the standard approach <cit.> to computing the renormalised effective action Γ_ eff and effective potential Γ. We will restrict ourselves to the simple λϕ^4 theory.  We begin from the textbook definition of the effective action and specify renormalisation conditions. The final result is presented in its general form in  eq. <ref>. Its coefficients are given in (depending on the chosen renormalisation conditions) eqs. <ref>, <ref> and <ref>.
        
        We begin from the classical action[One could add a “cosmological constant" Λ, which defines a zero-point function in eq. <ref>. However, since in QFT, there is no physical meaning in a constant term we do not discuss it here. We leave the quartically divergent vacuum bubble diagram (the fourth diagram in Fig. <ref>) outside the discussion. Appendix <ref> shows how such a cosmological constant could be included in the framework that we are about to present.] (in -+++ metric):
        Γ_ cl =-i·∫ d^4 x [1/2(_̣μϕ_0)^2+m^2/2ϕ_0^2 +λ/4!ϕ_0^4 ] .
         
        The corresponding tree correlation functions are given by:
        Γ_ cl^(2) = i(k^2+m^2),         Γ^(4)_ cl = -iλ .
        
        The quantum effective action  Γ is the generating functional for the strongly connected Green's functions Γ^(n)(x_1… x_n) and reads 
        Γ_ eff =∑_n 1/n!∫ d^4x_1… d^4x_n  Γ^(n)(x_1… x_n)ϕ_0(x_1)…ϕ_0(x_n) .
        
        Equivalently, the effective action can be written as an expansion in powers of derivatives:
        Γ_ eff = -i∫ d^4 x [ Γ(ϕ_0)+ 1/2(_̣μϕ_0)^2 Z(ϕ_0)+…] .
         
        with the dots representing terms of higher order in _̣μ. The function Γ(ϕ_0) is the effective potential to be used for the study of spontaneous symmetry breaking beyond the classical approximation. Note there are no one loop corrections proportional to (_̣μϕ_0)^2, making Z(ϕ_0)=1 in this approximation.
        
        
        
        
        The tree and one-loop diagrams for the effective potential are shown in Fig. <ref>. For two and four external legs the one-loop contributions are divergent (and all further loop corrections as well) and can be regulated in several ways. We will use dimensional regularisation for definiteness.
        
        
        Next, one chooses a renormalisation scheme to instruct how these divergences should be removed. As always, different renormalisation schemes will lead to different results for the effective potential, but the resulting physical predictions are equal, at least up to next order in perturbation theory. Our choice is to impose that at a given renormalisation point, all quantum corrections to the effective potential vanish[Yet another common choice is to define the condition on d^4 Γ / d ϕ_0^4 at an arbitrary “renormalisation scale" μ. This approach connects closely to the renormalisation group. For our purposes we find the scheme in eqs. <ref> more instructive.]. To that end, we need to demand
        d^2 Γ/d ϕ_0^2|_ϕ_0=ϕ_0^⋆ = m^2+λϕ_0^⋆ 2/2 ,     d^4 Γ/d ϕ_0^4|_ϕ_0=ϕ_0^⋆ = λ,
        
        where Γ denotes the renormalised effective potential, and ϕ_0^⋆ denotes the renormalisation point: the value for ϕ_0 at which the renormalisation conditions are imposed. Note that since we want quantum corrections to vanish at the renormalisation point ϕ_0^⋆, the right hand sides of the conditions in eqs. <ref> are simply obtained from taking two or four derivatives of the classical potential m^2 ϕ_0^2/2 +λϕ_0^4/4!. 
        
        
        
        To execute this renormalisation program, one introduces counter-terms δ Z, δΛδ m and δλ, and finds their corresponding values that lead to the renormalised effective action that meets the conditions above. At any rate, one ends up with
        
        i·Γ_ eff   =   ∫ d^4 x [1/2(_̣μϕ_0)^2 +m^2/2ϕ_0^2 +λ/4!ϕ_0^4 
                      +ħ·m^4/32π^2·∑_n=3^∞(-1)^n+1·1/n· (n-1)·(n-2)·(λϕ_0^2/2m^2)^n +…]
         
        and ultimately[One could add an extra term α_0· m^4 to cancel a constant, ϕ_0-independent contribution from the first term, but until Appendix <ref> we will not bother about unphysical constant contributions to the effective action.]
        i·Γ_ eff   =   ∫ d^4 x [1/2(_̣μϕ_0)^2 +m^2/2ϕ_0^2 +λ/4!ϕ_0^4
                       +ħ/64π^2((m^2+λϕ_0^2/2)^2ln[ 1+λϕ_0^2/2/m^2] - α_1 · m^2 ϕ_0^2 - α_2 ·ϕ_0^4
        )], 
        with the coefficients α_1 and α_2 to be fixed by the renormalisation procedure (more precisely, by the choice for the classical field value ϕ_0^⋆ at which the renormalisation conditions in eqs. <ref> are imposed).
        
        
        
        
         §.§ Symmetric phase
        
        
        In the symmetric phase (m^2>0) a common choice is to impose the renormalisation conditions at ϕ_0=0:
        d^2 Γ/d ϕ_0^2|_ϕ_0=0 = m^2 ,  d^4 Γ/d ϕ_0^4|_ϕ_0=0 = λ. 
        In this case, one finds that the coefficients α_i in the general result of eq. <ref> are given by
        α_1= λ/2 ,     α_2 = 3λ^2/8.
        
         §.§ Broken phase
        
        When m^2 <0, spontaneous symmetry breaking occurs and the classical potential finds a minimum at a nonzero field value
        ϕ_0 = ±√(-6m^2/λ)≡± v .
         
        In this situation, one may decide to compute the effective potential around this classical vacuum expectation value ϕ_0=v. Now the conditions in eqs. <ref> read
        d^2 Γ/d ϕ_0^2|_ϕ_0=v = -2m^2 ,  d^4 Γ/d ϕ_0^4|_ϕ_0=v = λ .
        
        As noted before, the expressions at the right-hand side are simply given by the corresponding number of derivatives acting on the classical potential, evaluated at the vev v. 
        In this case the coefficients α_1 and α_2 are given by
        
        α_1 = λ(61/2+iπ+ln2) ,      α_2 = λ^2/4(9/2+iπ+ln2) . 
        Here the factors of iπ indicate that in a region ϕ_0^2 <-2m^2/λ around ϕ_0=0 the effective potential develops an imaginary part. At ϕ_0^2>-2m^2/λ the imaginary part disappears.
        
        Alternatively, one may replace the first condition in eqs. <ref> by a condition on the first derivative of the effective action:[Note that in the symmetric case, this condition makes no sense. Since the first derivative vanishes identically at ϕ_0=0, the condition is void. Consequently, we are one condition short to renormalise the theory.]d Γ/d ϕ_0|_ϕ_0=v = 0 . 
        
        This condition ensures that quantum corrections do not push away the vev from its classical location in field space.  Again, we obtain the general result of eq. <ref>, this time with coefficients
        α_1= λ(25/2+iπ+ln2)         α_2 = λ^2/4(9/2+iπ+ln2).
        
        
        § FINITE APPROACH
        The goal of this Section is to show that there exists an alternative way to compute the effective potential. Contrary to the traditional approach that we have just reviewed, this method never runs into any intermediate divergent quantity. In our opinion, the existence of such a method is of great conceptual interest. It counters the usual intuition that infinities are a fact of life in particle physics.
        
         §.§ Setup
        This first Subsection introduces the finite CS method that this paper employs. In Subsubsection <ref>, it is introduced in the original context: in terms of correlation functions. That was the formulation that we used in our papers <cit.>. Next, in Subsubsection <ref>, we formulate the method in terms of the effective potential. In <ref> we give the equations that we will use later on to compute the effective potential in a manifestly finite way.
        
          §.§.§ Correlation functions
        
        
        The finite QFT method to compute n-point correlation functions was presented in lecture notes for the Les Houches summer school in 1975 by Callan <cit.>. It was applied to quantum electrodynamics in <cit.>. Its central equations are similar (but not equal to) the Callan-Symanzik equation <cit.>. Therefore we refer to it as the “Callan-Symanzik (CS) method". It rests on a so-called “θ-operation". Graphically it can be thought of as cutting a propagator in two and pasting them together by a “θ-vertex", which comes with Feynman rule (-1). Algebraically, it acts on an n-point correlation function through
        Γ̃^(n)_θ≡ -i ×d/d m_0^2Γ̃^(n) ,    Γ̃^(n)_θθ≡ -i ×d/d m_0^2Γ̃_θ^(n) .
        
        Here Γ̃ denotes the bare Green's functions  , while m_0 denotes the bare mass. Bare quantities show up only in the derivation of the CS method, not in its actual use[As we are only concerned with executing the CS program rather than deriving it, there is no need to consider bare quantities at all. The point of the CS operation is precisely that once established, it circumvents the entire notion of bare quantities.]. The θ-operation lowers the degree of divergence by two units. The finite correlation functions Γ^(n), Γ^(n)_θ and Γ^(n)_θθ are connected through <cit.>
        2i m^2  (1+γ) Γ^(n)_θ   =   (2m^2  /ṃ^2 +β /λ̣    +n·γ) Γ^(n) ,
        
        2im^2 (1+γ)·Γ^(n)_θθ   =   (2m^2 /ṃ^2 +β /λ̣ +n·γ +γ_θ)Γ^(n)_θ. 
        Here β, γ and γ_θ are additional parameters that find their values in the course of computing the correlation functions[Note that a “fermionic version" of the CS approach would involve cutting fermion propagators in two by taking derivatives with respect to one power of the fermion mass instead of two. Such an operation would lower the degree of divergence by one unit <cit.>.].  β and γ are very reminiscent of the usual beta-functions and anomalous dimension, but we stress that in this context they are just parameters. The same goes for the CS equations themselves: they are reminiscent of the famous Callan-Symanzik equation used to compute the high energy behaviour of the Green's functions, but different. In partucular, they do not contain the derivatives with respect to momenta. Furthermore, the parameter “n" is the same as the “n" appearing in the superscripts of the correlation functions.
        
        
        
        
        Of course, one could continue and define correlation functions with any number of θ-subscripts, and the corresponding equations. However, in the λϕ^4 theory, two θ-operations are enough to render all n-point correlation functions with n>0 convergent. The set of equations in eqs. <ref> then allows for a computation, up to any desired loop order, of the remaining correlation functions with positive or zero formal degree of divergence (i.e., Γ^(2), Γ^(4) and Γ^(2)_θ)  without ever running into a divergent quantity. Therefore, at no point it is required to perform “divergent subtractions" to render the correlation functions finite.
        
        
        
          §.§.§ Effective potential
        
        
        
        From eq. <ref> it is direct to see that the set of equations in <ref> can be straightforwardly generalised from n-point correlation functions Γ^(n) to the effective potential Γ(ϕ_0)<cit.>:
        
        2 im^2  (1+γ)  Γ_θ   =   (2m^2  /ṃ^2 +β /λ̣    + γ·ϕ_0/ϕ̣_0)  Γ,
        
        2im^2 (1+γ)  Γ_θθ   =   (2m^2 /ṃ^2 +β /λ̣ +γ·ϕ_0/ϕ̣_0+γ_θ) Γ_θ, 
        where we have rewritten the factor of n in the first set of equations through a field derivative. In terms of bare quantities, one can define Γ_θ and Γ_θθ following eq. <ref>Γ̃_θ≡ -i ×d/d m_0^2Γ̃ ,    Γ̃_θθ≡ -i ×d/d m_0^2Γ̃_θ .
        
        
        As mentioned above, even if the θ-operation is defined in eqs. <ref> on bare correlation functions, all computations rely on the system of two equations above, which relate only the finite quantities. 
        
        Note that the equations above are in terms of the effective potential Γ(ϕ_0), rather than in terms of the effective action Γ_ eff. Of course, the CS equations can be rewritten in terms of (functional derivatives of) the effective action, making our discussion more complete.
        
        
        
        
        
        
        
        
        In spirit, nothing changes in the CS framework, but computations (and notations) become much more intricated. For the purposes of our paper (in particular, for the proof of principle that we want to provide), we prefer to present the simplified discussion in terms of just {Γ,Γ_θ,Γ_θθ}.
        
        Armed with the standard boundary conditions of eqs. <ref>, we are now ready to derive the effective potential. We omit the derivation (as it is identical to the one presented in <cit.>) that shows that β must begin at order λ^2, while the γ and γ_θ can have support at order λ as well.
        
        
        
        
         §.§ Computation
        We now get to the most technical part of the paper. We want to show explicitly that the equations derived in <ref> allow for a manifestly finite derivation of the effective potential. The endpoint of the computation is in eq. <ref>. From there on we are back on the “traditional path" (i.e., the standard approach described in Section <ref>) towards the standard result for the effective potential, eq. <ref>. Again, the central arguments of this paper are of conceptual nature. The physics ideas that we want to convey do not hinge on the details of the computation that we are about to present. In spirit, this Subsection can be summarised as just “The effective potential can indeed be derived by the finite CS method."  
        
        
        We begin from defining
        Γ = Γ_0+ħ·ϕ_0^4 ·Γ_1(ϕ_0^2/m^2)+.
        
        Here Γ_0 denotes the classical potential
        Γ_0 = m^2 ϕ_0^2/2 + λϕ_0^4/4!,
        
        while the dimensionless function Γ_1 is the one-loop correction to the effective potential that we are after.
        
        Similarly, we define
        Γ_θ   =   Γ_θ,0 +ħ·Γ_θ,1+ ,
        Γ_θθ   =   Γ_θθ,0 +ħ·Γ_θθ,1+ ,
        
        and also
        β = ħ·β_1 + ,         γ = ħ·γ_1 + ,        γ_θ = ħ·γ_θ,1 + .
        
        Evaluating the first CS equation in <ref> at tree level order (i.e., at zeroth order in ħ) gives
        Γ_θ,0 = -i·ϕ_0^2/2 .
        With that, we can evaluate the second CS equation in <ref> at tree level in order to find
        Γ_θθ,0 = 0 .
        
        Actually, now that we have decided to leave out the vacuum bubble diagram, all contributions to Γ_θθ are finite. Therefore, it can be computed without encountering intermediate divergences. See Fig. <ref>. This observation forms the core of the CS mechanism. At one loop (i.e., order ħ) we get
        
        
        
        
        
        
         
          
        
        
        
        
        Γ_θθ,1 = -1/32π^2log[  1+λϕ_0^2/2m^2] .
        
        Next we evaluate the first CS equation in <ref> at order ħ. In terms of a dimensionless variable x
        x≡ϕ_0^2/m^2
        we get
        Γ_θ,1/m^2 = i· x^2 [x Γ_1'(x)-β_1+4γ_1λ/48].
        
        
        
        
        
        
        
        The following step is to evaluate the second CS equation in eqs. <ref> at first order in ħ. Inserting the previous result, we end up with a second order differential equation for Γ_1:
        
        x^4 Γ_1”(x)+2x^3 Γ_1'(x) -x^2(β_1+4γ_1λ)/48 + x(2γ_1+γ_θ,1)/4= log[1+λ x/2]/32π^2.
        
        We get
        Γ_1    =   1/64π^2[ log[1+λ x/2]/x^2 +λ log[ 2+λ x](1/x+λ/4)  ] 
                       +log[x]·[ 1/4x(2γ_1+γ_θ,1-λ/16π^2) +1/48(β_1+4γ_1λ-3λ^2/16π^2)   ]
                       +1/4x(2γ_1+γ_θ-λ/32π^2+c_1) + c_2, 
        where c_1 and c_2 are two integration constants.
        
        Now we use that at one loop, there is no field renormalisation, so we can safely set the order ħ contribution γ_1→ 0[Strictly speaking, at this point we do not know that this CS parameter γ is just the usual field renormalisation parameter γ. However, we can take that fact from our paper <cit.> devoted to the CS computation of n-point functions Γ^(n). With that, we understand that at one loop, γ indeed disappears. See also the discussion below eq. <ref>]. Next, we use that imposing analyticity (i.e. imposing that the solution is regular at ϕ_0=0) forbids the terms proportional to log(x).  The requirement of analyticity of the effective potential at small ϕ is nothing but the requirement of existence of Green's functions (Taylor coefficients in the expanding of the effective action) in the  λϕ^4 theory with non-zero scalar mass. Though this is not guaranteed beyond perturbation theory (“triviality” of the model related to the Landau pole in the scalar coupling constant), these Green's functions do exist perturbatively.   With γ already out of the way, we can then solve for the remaining two CS parameters:
        β_1 →3λ^2/16π^2,        γ_θ,1→λ/16π^2. 
        With that, we are left with a well-behaving solution for Γ_1 (with newly defined integration constants c_1 and c_2):
        ϕ_0^4 ·Γ_1 =1/64π^2[ (m^2+λϕ_0^2/2)^2 log( 1+λϕ_0^2/2m^2)] + c_1· m^2ϕ_0^2 + c_2·ϕ_0^4 .  
        We have arrived on familiar grounds by now. By imposing two boundary conditions at a chosen field value we can solve for the integration constants c_1 and c_2. Everything works as in the standard computation: the boundary conditions of eq. <ref> lead to the result in eqs. <ref> with the coefficients in eq. <ref>, the boundary conditions of eq. <ref> lead to the same result in eq. <ref> but with the coefficients in eq. <ref>, etcetera.
        
        For completeness, we mention the result for Γ_θ,1:
        Γ_θ,1 = -i/32π^2(m^2+λϕ_0^2/2) log( 1+λϕ_0^2/2m^2) + c_1·ϕ_0^2,
        
        where c_1 is an integration constant.
        
        
        As a “divergent double check", one can of course compute Γ_θ through the conventional approach of regularising and renormalising divergent Feynman diagrams. This is illustrated in Fig. <ref> (for the particular case =0).
        
        
        
         
          
        
        
        
        
        
        It is worth noting that compared to the original CS papers, and to our works <cit.> as well, eq. <ref> is the first time that analyticity is used in the CS mechanism. Actually, that brings a conceptual advantage. In the computations of this paper (be it the conventional one, or the CS one), two renormalisation (boundary) conditions suffice to fix the two integration constants in eq. <ref>, or, equivalently, the two Lagrangian parameters m and λ that appear in the effective potential. Extending our discussion to include the effective action as well, a third condition sets the field parameter. Everything is in order. 
        
        
        
        
        Finally, it is straightforward to see how the finite CS method generalises to higher loop order. The function Γ_θθ is a convergent object and can therefore be computed up to all orders in ħ without ever running into divergences[Of course, one might run into subdivergences, but these can be safely dealt with through the standard skeleton expansion.]. For Γ_θ and Γ itself the situation is precisely as for the n-point correlation functions themselves, treated already in <cit.>. Once we have obtained their 𝒪(ħ^n) contributions, these can be inserted at the rightmost term in the CS equations. Evaluating the appropriate number of derivatives on the resulting equation and using the boundary conditions at the left-hand side of the CS equations results in an equation for the 𝒪(ħ^n+1) contributions. Meanwhile, imposing the boundary conditions yields the 𝒪(ħ^n+1) contributions to the CS parameters {β,γ,γ_θ}. This process can be repeated up to any desired order in ħ.
        
        
        
        
        § TWO FIELDS
        In this Section, we want to generalise the analysis of the previous Section to the case of two in­ter­acting scalar fields. The motivation for that is clear: it is the simplest context in which we can present the hierarchy arguments that this paper is meant to address. We spend some time on describing the vacuum configurations of the theory. After all, in the most elementary BSM scenarios we have our Higgs field sitting in its electroweak vacuum, while new heavy scalar fields are residing in their own vacuum.
        We study the theory described by the Lagrangian
        Ł = -1/2(_̣μϕ)(^̣μϕ) -1/2(_̣μΦ)(^̣μΦ)-m^2/2ϕ^2 - M^2/2Φ^2 - λ_ϕ/4!ϕ^4 - λ_Φ/4!Φ^4 - λ_ϕΦ/4ϕ^2 Φ^2 . 
        
        At the classical level, there are four different kinds of vacua now, as shown in Table <ref>. Depending on the sign of their mass parameter, both fields can be either in the symmetric phase or in the broken phase. We will be mostly interested in the vacua of type D, where both fields have negative mass squared parameters and reside at nonzero field values.
        
        
              
        At such type D vacua, the mass matrix can be easily depicted in terms of the vevs v and V of the fields ϕ and Φ respectively:
        ℳ^2 = ([  λ_ϕ v^2/3  λ_ϕΦV · v; λ_ϕΦ V · v  λ_Φ V^2/3 ])
        
        with v and V the vevs appearing in the first columns in Table <ref>.
        
        It is now clear that one single tuning of the Lagrangian parameters
        
        m^2 λ_Φ - 3M^2 λ_ϕΦ    →     0 
        can keep the vev v of the field ϕ_0 as small as we want.  This condition implies that m^2 is large, m^2∼ M^2  (we take λ_ϕ∼λ_ϕΦ∼λ_Φ for power counting). Eq. <ref> requires cancellations (fine-tunings) between different Lagrangian parameters m^2, M^2 and coupling constants. If the “fundamental parameters” of the theory are chosen to be the vevs v and V, the hierarchy condition v ≪ V does not require any cancellations between large numbers. Meanwhile, the same tuning sends the smallest eigenvalue of the matrix (<ref>) - the physical mass m_ phys of the lightest scalar to zero.  In the limit v ≪ V the ϕ and Φ are still the approximate true mass eigenstates of the theory.  (Of course, tuning the quantity M^2λ_ϕ-3m^2λ_ϕΦ leads to the same conclusions, but with the fields ϕ and Φ interchanged.)
        
        Moving on to the quantum level, the definition of the two-field effective action is a straightforward generalisation of eq. <ref>:
        Γ_ eff = ∑_n,N1/(n+N)!∫ d^4x_1… d^4 x_n+N Γ^(n,N)(x_1,…,x_n+N) ϕ_0(x_1)…,ϕ_0(x_n) Φ_0(x_n+1)…,Φ_0(x_n+N), 
        where n denotes the number of ϕ fields and N the number of Φ fields. In our case, our goal is again to compute the effective potential Γ which (following the notation of eq. <ref>) is related to the effective action Γ_ eff through 
        Γ_ eff = -i∫ d^4 x [ Γ(ϕ_0,Φ_0)+ 1/2(_̣μϕ_0)^2 Z_ϕ(ϕ_0,Φ_0) + 1/2(_̣μΦ_0)^2 Z_Φ(ϕ_0,Φ_0)+…].
        
        
         §.§ Standard computation
        This Subsection contains a review of the standard approach to derive the two-field effective potential. It is much less explicit than its one-field counterpart, Subsection <ref>. We write down renormalisation conditions and give the final result in eq. <ref>.
        To not obscure our reasoning, we will avoid explicit (intermediate) expressions as much as we can. The effective action follows from summing all Feynman diagrams with external classical lines sticking out of purely quantum loops. Diagrams with n light fields and N heavy fields on the external lines give contributions of order ϕ_0^nΦ_0^N. Divergences are regulated, for example by dimensional regularisation. The instructions for their subtraction are encoded in renormalisation conditions. Omitting the constant term, we have 7 renormalisation conditions for fitting 7 Lagrangian parameters (two kinetic terms for the fields, two masses, and three four-point couplings). Restricting ourselves to the effective potential as we did before, the number of Lagrangian parameters to fit and the conditions to do so reduces to 5. In the spirit of eqs. <ref>, we choose
        d^2 Γ/dϕ_0^2|_ϕ_0=v,Φ_0=V= d^2 Γ_ cl/dϕ_0^2|_ϕ_0=v,Φ_0=V ,    d^2 Γ/dΦ_0^2|_ϕ_0=v,Φ_0=V= d^2 Γ_ cl/dΦ_0^2|_ϕ_0=v,Φ_0=V ,
        
        i·d^4 Γ/dϕ_0^4|_ϕ_0=v,Φ_0=V= λ_ϕ ,     
        i·d^4 Γ/dϕ_0^2 dΦ_0^2|_ϕ_0=v,Φ_0=V= λ_ϕΦ ,    
        i·d^4Γ/dΦ_0^4|_ϕ_0=v,Φ_0=V= λ_Φ . 
        
        The right-hand sides of the first and second condition can be read off directly from Table <ref>. Again the philosophy is clear: we impose that at the vev, all quantum corrections to the second and fourth derivatives of the effective potential vanish.
        
        Alternatively, one can choose to replace the conditions on the second derivatives with conditions on the first derivatives:
        d Γ/dϕ_0|_ϕ_0=v,Φ_0=V   =    0,
        d Γ/dΦ_0|_ϕ_0=v,Φ_0=V   =    0 .
        
        Note that this approach only works for vacua of type D where both fields are in the broken phase. As we have seen in the one field case, in the symmetric phase the first derivative of the potential vanishes anyway. Only in case D the third and fourth conditions are valid (“not-void") and all divergences can be removed.
        
        Explicit expressions for the counter-terms required to meet these conditions are tedious and will be omitted. What matters for our argument is the observation that the counterterm for ϕ's mass (squared) parameter picks up contributions from diagrams with Φ running through the loops[Of course, there are also contributions δ M^2 ∝ m^2, but these are not relevant for the usual naturalness arguments.].
        δ m^2 ∝ M^2.  
        This observation is generic for all choices of renormalisation conditions and renormalisation points at which these conditions are to be evaluated. In the next section, where we will assume that m ≪ M, it leads to the naturalness worries that this paper is meant to address. 
        
        Going through all usual steps, one ends up with
        Γ   =   m^2/2ϕ_0^2 + M^2/2Φ_0^2 + λ_ϕ/4!ϕ_0^4 + λ_Φ/4!Φ_0^4 + λ_ϕΦ/4ϕ_0^2 Φ_0^2
                   +ħ/64π^2(m_1^4(ϕ_0,Φ_0)
        ln[m_1^2(ϕ_0,Φ_0)/m^2]
        +m_2^4(ϕ_0,Φ_0)ln[m_2^2(ϕ_0,Φ_0)/M^2]
        
        
                       -α_1· m^2 ϕ_0^2 - α_2· M^2 Φ_0^2 - α_3 ·Φ_0^4 - α_4 ·ϕ_0^2Φ_0^2- α_5 ·Φ_0^4
        ).  
        Here the coefficients {α_1,…,α_5} are (long) functions of the Lagrangian parameters and m_1,2^2(ϕ_0,Φ_0) are the eigenvalues of the mass matrix in the arbitrary background fields ϕ_0,Φ_0 that we will not bother to write out.  We stress that these coefficients are all finite. They are the result of subtracting infinite and large counterterms from infinite and large counterterms. At the end of the day, however, the result in eq. <ref> gives only small corrections to the tree effective potential in the vicinity of the classical minimum (ϕ_0,Φ_0) thanks to the imposed boundary conditions (<ref>) or (<ref>). Corrections coming from the heavy sector are suppressed by powers of v/V. Our goal in the next subsection will be to arrive at the same result without running into infinite and large loop corrections that need to be eliminated by carefully picking infinite and large counterterms.
        
         §.§ Convergent computation
        
        In our paper <cit.> we have given a detailed analysis of the two-field generalisation of the CS method for correlation functions. Here our task is to rewrite that to a two-field generalisation of the CS method for the effective action. We will go rather fast in this section since all details follow straightforwardly from the corresponding details described in <cit.>.
        
        What is more, we stress that the goal of this section (and of the whole paper) is not to convince the reader that from now on actual computations should be done by the manifestly convergent CS approach. The expressions appearing in this Subsection make clear that for actual computations, the standard textbook “divergent" framework is much more user-friendly. Our goal is only to show the  existence of a valid framework for an alternative computation of the (two-field) effective potential.  The central equations in this Subsection are <ref>, <ref> and <ref>. Their relevance lies in their mere existence, not in their actual shape and structure.
        
        With two fields and two masses on the scene, we have two kinds of θ-operations. The definitions in eq. <ref> are generalised to:
        Γ̃_θ,m≡ -i ×d/d m_0^2Γ̃ ,     Γ̃_θ,M≡ -i ×d/d M_0^2Γ̃
        and then
        Γ̃_θθ,mm≡ -i ×d/d m_0^2Γ̃_θ,m ,     Γ̃_θθ,Mm≡ -i ×d/d M_0^2Γ̃_θ,m , 
        Γ̃_θθ,mM≡ -i ×d/d m_0^2Γ̃_θ,M ,     Γ̃_θθ,MM≡ -i ×d/d M_0^2Γ̃_θ,M ,
        
        where of course Γ̃_θ,Mm = Γ̃_θ,mM. Again the tildes indicate bare quantities. As in the one field case the equations derived from these definitions (to be presented below) do not contain bare objects. 
        
        The corresponding CS equations (two-field generalisations of eqs. <ref>) show how the light and heavy sectors are intertwined. The two Γ_θs defined in eqs. <ref> can be united in a two-dimensional vector, while the four Γ_θθs form a two-by-two matrix. Written out in components, the first CS equation generalises to
        
        2i[m^2(1+γ_ϕ,m)Γ_θ,m + M^2 γ_Φ,mΓ_θ,M]    =   [2m^2/ṃ^2+∑_iβ_λ_i,m/λ̣_i
                      + γ_ϕ,m·ϕ/ϕ̣+γ_Φ,m·Φ/Φ̣]Γ
        and
        
        2i[M^2(1+γ_Φ,M)Γ_θ,M + m^2 γ_ϕ,MΓ_θ,m]    =   [2M^2/Ṃ^2+∑_iβ_λ_i,M/λ̣_i
                      + γ_ϕ,M·ϕ/ϕ̣+γ_Φ,M·Φ/Φ̣]Γ .
        Here the sum over λ_i runs over the three four-point couplings {λ_ϕ,λ_Φ, λ_ϕΦ}. 
        
        The second CS equation reads in full two-by-two glory
        
         i·      (
        [      1+γ_ϕ,m M^2/m^2γ_Φ,m; m^2/M^2γ_ϕ,M      1+γ_Φ,M ]) ×(
        [  Γ_θθ,mm  Γ_θθ,mM; Γ_θθ, Mm  Γ_θθ,MM ])
                      =
        [
        ([ /̣ṃ^2; /̣Ṃ^2 ])
        +∑_i ([ β_λ_i,m/2m^2; β_λ_i,M/2M^2 ])
        /λ̣_i
                              
        +1/2([ γ_ϕ,m/m^2; γ_ϕ,M/M^2 ])ϕ/ϕ̣
        +1/2([ γ_Φ,m/m^2; γ_Φ,M/M^2 ]) Φ/Φ̣]
        ([ Γ_θ,m Γ_θ,M ])
                          +
        1/2m^2×([ γ_θ,mmm γ_θ,Mmm; γ_θ,mmM γ_θ,MmM ]) 
          Γ_θ,m
                          
        +
        1/2M^2×([ γ_θ,mMm γ_θ,MMm; γ_θ,mMM γ_θ,MMM ])
          Γ_θ,M . 
        Note that the auxiliary CS parameters {β,γ,γ_θ} in the one field case have now generalised to 6 βs (related to two different derivatives on three couplings), 4 γs (related to two different derivatives on two fields) and 8 γ_θs (indirectly related to two different derivatives on two mass parameters related to two fields).
        
        In Fig. <ref> the structure of the two-field CS approach is outlined. For notational simplicity, we have only graphically depicted the θ-operations on one of the contributing diagrams, but the discussion is completely generic. (And again, given the Γ_θθs and the boundary conditions there is strictly speaking no need to think about Feynman diagrams whatsoever.)
        
        At the bottom line, we have three kinds of Γ_θθs. These can all be obtained by a divergence-free computation. (That is, if we decide again to leave out the constant contribution to the effective potential.) 
        
        Inserting these Γ_θθs in the second CS equation we obtain the two Γ_θs. Next, the first CS equation brings us to Γ.  This process yields both mass derivatives (with respect to m^2 and M^2) of the five objects in eq. <ref>, plus the two mass derivatives of the two Γ_θs. In the process of deriving them, one uses the CS parameters. Six of them (the βs) are involved in the computation of the two mass derivatives of the three quartic terms in Γ. Out of the other eight ones (the γ_θs), four of them are involved in the computation of the mass derivatives of the two mass terms in Γ. The last four are involved in the computation of the two mass derivatives of the quadratic terms in Γ_θ (using the second CS equation). All in all, to derive the full effective action we have 14 objects to fix, for which we use 14 renormalisation conditions and which yield the 14 CS parameters[Generalising the discussion to the effective action, we have four more objects (both mass derivatives on both kinetic terms) that can be fixed by the four γ parameters {γ_ϕ,m,γ_Φ,m, γ_ϕ,M, γ_Φ,M}.] We have all the ingredients to arrive at the result in eq. <ref> without meeting any intermediate divergence.
        
        
        
        
        § QUANTUM STABILITY OF THE TWO-FIELD EFFECTIVE POTENTIAL
        
        The theory of eq. <ref> treats ϕ and Φ on equal footing. However, for our purposes, we will now assume that  v≪ V. We also assume that the three four-point couplings {λ_ϕ, λ_ϕΦ, λ_Φ} are of the same order (or that at least the hierarchy between them is much smaller than the hierarchy between the two mass parameters).
        
        First of all, the interaction between ϕ and Φ gives off-diagonal elements to their mass matrix. Therefore, even before taking quantum corrections into account, the fields ϕ and Φ are not canonically normalised. However, in the situation  v ≪ V there will be a “light" canonical field ϕ_ light which almost coincides with ϕ, and a “heavy" canonical field ϕ_ heavy which almost coincides with Φ. (Here “almost" means: up to corrections proportional to  v/V.) To not lose ourselves in non-essential details, we will omit these redefinitions. Sloppily but safely we will refer to ϕ as the light field and to Φ as the heavy field.
        
        To make our point, we consider two-field vacua of kind D. We have seen that at tree level, one single fine-tuning of the factor (m^2 λ_Φ-3M^2 λ_ϕΦ) keeps both the vev of the light field and its mass small (i.e., free of contributions of order  V^2).
        
        Next, we turn to quantum corrections. In the standard approach, these follow from an expansion in Feynman diagrams that contains both infinite and “large" (order M^2) terms[Most notably, the bare effective potential contains terms of type M^4 ln[1+ϕ_0^2/M^2] which around ϕ's vev produce contributions proportional to M^2ϕ_0^2.]. In particular, we have diagrams of mass dimension two with two external classical lines which correct the second derivatives (the mass terms) of the effective action. To eliminate these divergences, we choose to work with the renormalisation scheme defined in eq. <ref>. By construction, at the vevs the masses of ϕ and Φ are equal to the tree-level masses, just because we impose that at the vev the quantum corrections to the second derivatives of the effective action disappear. However, it is clear that to achieve this, the standard approach requires a delicate cancellation between both infinite and “large" (order  V^2) loop corrections and counter-terms. 
        
        For a typical GUT theory with  v≃ 10^2  GeV and  V≃ 10^14  GeV, the necessary tuning between the mass (squared) parameters m^2 and M^2 would be of order 10^24, at least. It seems that to fulfil the renormalisation conditions, a new tuning is needed to restore the small (order  v) values for the vev of the field ϕ. Even worse, at every new order in perturbation theory, a new tuning between the Lagrangian parameters is required. This is how the technical hierarchy problem appears in the computation of the effective action. The physics of the light field seems to be unstable against quantum corrections. Without careful fine-tuning that needs to be “refreshed" at every next loop order, the coupling between ϕ and Φ shifts the mass of ϕ to values of order  V. 
        
        On the other hand, applying the finite CS formalism results in the same effective potential given in eq. <ref> that respects the boundary conditions of eqs. <ref>. Now quantum corrections follow from a diagrammatic expansion of the manifestly convergent object Γ_θθ. Since Γ_θθ has mass dimension minus two, by construction the finite computation cannot contain any contributions and cancellations between divergent or large (order  V^2) quantities.
        
        A last point that one might worry about is what quantum corrections do to the vev of the light field. After fixing Γ's second and fourth derivatives we have no more handles left to tune, we can only compute the resulting vevs and hope that v has not shifted to large values of order M. In other words, even if heavy exotic physics does not shift the mass of the light field, it might have shifted its vev, and the technical hierarchy problem reappears. This problem applies both to the standard method and to the CS method: at the end of the day, both have produced the same result. 
        
        In the particular case of the theory described by eq. <ref>, it is straightforward to see that the classical vev does not suffer enormous shifts when quantum corrections are taken into account. Since the resulting effective action in eq. <ref> is quadratic in ϕ_0 and Φ_0, its first derivative with respect to ϕ_0 must be proportional to ϕ_0. At the classical vev, it then becomes proportional to that same classical vev. With that, we understand that the same tree level tuning in eq. <ref> keeps the “quantum corrected" vev at the classical vev. Moreover, three of the four elements in the mass matrix of eq. <ref> are still proportional to that same factor. That ensures that the renormalisation scheme given in eqs. <ref> produces an effective action in which one single tuning shields the physics of the light field ϕ from the new physics of the heavy field Φ. Around the classical vevs, the renormalised theory does not show any imprint of the possible existence of new exotic physics living at a hypothetical scale  V ≫ v.
        
        However, what if we wish to go beyond the toy model of eq. <ref> and consider more realistic (GUT) settings? The reason that the small vev does not shift in our toy model was a discrete symmetry ϕ↔ -ϕ, forbidding the terms proportional to ϕΦ and ϕΦ^3 in the effective potential[If the ϕΦ, ϕΦ^3, ϕ^3Φ terms are introduced, the arguments about the hierarchy problem remain in force: in the CS approach with appropriate boundary conditions no tuning is necessary to keep the light scalar mass small in comparison with the heavy mass. However, the relationship between the vevs of the fields and the physical masses of the particles is lost - small vev of the light field does not automatically imply its small mass and vice-versa.]. In GUTs like SU(5), similar types of terms are forbidden by the fact that the two breaking stages of the unifying group are carried out by different multiplets. When we choose the boundary (renormalisation) conditions that forbid corrections to the vev (to the mass), we get for free that the mass (the vev) does not move to much larger values. 
        
        In summary, we have shown how all possible fine-tunings in the computation of the effective potential can be circumvented by applying the finite CS method. Also, we have made clear that in general the masses and vevs of the light field can be protected. Therefore, we can only conclude that the “hierarchy worries" suggested by the traditional approach are nothing but an unphysical artefact of that approach. In other words: there is no reason to worry about alleged cancellations between unphysical Lagrangian parameters.
        
        
        
        § CONCLUSIONS
        
        In this paper, we have presented a manifestly finite approach to the computation of the effective action. All intermediate divergences can be circumvented. We have explicitly shown how the method works for the simple case of λϕ^4 theory. We have then shown, by counting the number of parameters, that there is no fundamental obstruction in the two-field (or, n-field) generalisation of this computation. By no means do we suggest that one should actually  use this system for explicit computations. In practice, the standard “divergent" computation of the effective action (and potential) is much more direct and user-friendly. We just want to point out that to compute the effective action it is not necessary to resort to large (formally infinite) cancellations between bare loop corrections and ad hoc introduced artificial counter-terms.
        
        Our motivation for this exercise lies in the so-called “technical hierarchy problem." It is often assumed (see for example <cit.>) that in theories with a large hierarchy between (mass) scales careful tunings are necessary to shield low-energy physics from the influence of new exotic physics going on at much higher scales. In the particular case of the effective action/potential, it seems that at every order in perturbation theory, one needs a new tuning to prevent the vevs and masses of the light theory from running away to (much higher) energy scales associated with new physics. We have shown that the CS approach that we present in this paper does not need delicate “re-fine-tunings" to “keep light physics light". From there, we argue that the “hierarchy worries" about the alleged quantum-mechanical instability of the effective action/potential are method-dependent and therefore unphysical. In perfect agreement with the decoupling theorem, “accessible physics" at the electroweak scale never feels possible new physics at GUT scales or beyond.
        
        
        
        
        § ACKNOWLEDGEMENTS
        
        This work was supported by the Generalitat Valenciana grant PROMETEO/2021/083. SM wishes to thank the Nikhef Theory Group for their everlasting hospitality; part of this work was done during such a stay in Amsterdam.
        
        
        
        § INCLUSION OF COSMOLOGICAL CONSTANT
        
        
        In QFT without gravity, there is no physical meaning in the absolute value (that is, the constant, ϕ_0-independent part) of the effective potential. However, a priori there seems to be no obstruction in applying the CS program to the computation of that first, constant term that shows up in the effective potential. Our paper <cit.> contains a short section on this constant term. In this Appendix, we extend and clarify the discussion there. For simplicity, we restrict ourselves to the case of a single scalar field. Also, we will mostly work “top-down": we merely want to show that the CS approach to computing the effective potential including its constant term is completely well-defined and consistent. With that, it is straightforward to do actual computations as we did in Section <ref>.
        
        The constant part of the effective action follows from the first diagram appearing in the diagrammatic expansion of the effective action: the vacuum bubble diagram. As it has mass dimension four, it is clear that we will need three θ-operations to compute the full effective action. Consequently, we need one more CS equation. That equation should instruct us how to get Γ_θθ out of the manifestly convergent object Γ_θθθ. See Figure <ref> for a sketch of this extended CS program.
        
        Going through the derivation of the first two CS equations in Callan's original notes <cit.>, it seems that the third equation should be <cit.>
        2im^2 (1+γ) ·Γ_θθθ =(2m^2 /ṃ^2 +β /λ̣ +γ·ϕ_0/ϕ̣_0+2γ_θ)Γ_θθ. 
        On the left-hand side, we find the object Γ_θθθ, which has mass dimension minus two. As such, it can be computed from a series of convergent diagrams, just as we want. See again Figure <ref>. One readily finds
        Γ_θθθ=iħ/32π^2 m^2[1-λϕ_0^2/2m^2+λ^2 ϕ_0^4/4m^4+…] = iħ/32π^2·1/m^2+λϕ_0^2/2 .
        To show the consistency of this extended CS mechanism, we can take the standard result (computed by the traditional approach), throw it through the three CS equations beginning from the right top of Fig. <ref>, and see whether we end up with the above result for Γ_θθθ. 
        
        However, in doing so we stumble on a problem. We begin with the symmetric case. Throwing the result in eqs. <ref> and <ref> through the three CS equations given in eqs. <ref> and <ref>, we find
        Γ_θθθ=-iħ/32π^2 m^2·λϕ_0^2/2m^2[1-λϕ_0^2/2m^2+λ^2 ϕ_0^4/4m^4+…].
        Clearly, we have lost the first term in the expansion in eq. <ref>. 
        
        The solution to this paradox is to introduce a constant term (say, Λ) to our Lagrangian, and treat that as a proper parameter in the CS formalism. That is, we begin from the classical action
        Γ_ cl= -i·∫ d^4x [Λ +1/2(_̣μϕ_0)^2+m^2/2ϕ_0^2 +λ/4!ϕ_0^4 ].
        
        We renormalise the effective potential by imposing
        Γ|_ϕ_0=0   =   Λ
        d^2 Γ/d ϕ_0^2|_ϕ_0=0   =    m^2
        d^4 Γ/d ϕ_0^4|_ϕ_0=0   =   λ.
        
        That is going to give the result given in eqs. <ref> and <ref>, augmented by a cosmological constant Λ:
        
        i·Γ_ eff   =   ∫ d^4 x [Λ+1/2(_̣μϕ_0)^2 +m^2/2ϕ_0^2 +λ/4!ϕ_0^4
                       +ħ/64π^2((m^2+λϕ_0^2/2)^2ln[ 1+λϕ_0^2/2/m^2]-α_0· m^4 - α_1 · m^2 ϕ_0^2 - α_2 ·ϕ_0^4
        )],
        
        with
        α_0 = 0 ,    α_1= λ/2 ,     α_2 = 3λ^2/8.
        
        Meanwhile, the CS equations are generalised to
        
        2 im^2  (1+γ)  Γ_θ   =   (2m^2  /ṃ^2 +β /λ̣    + γ·ϕ_0/ϕ̣_0+γ_Λ /Λ̣) Γ,
        
        2im^2 (1+γ)  Γ_θθ   =   (2m^2 /ṃ^2 +β /λ̣ +γ·ϕ_0/ϕ̣_0+γ_θ+γ_Λ /Λ̣)Γ_θ,
        
        2im^2 (1+γ) ·Γ_θθθ   =   (2m^2 /ṃ^2 +β /λ̣ +γ·ϕ_0/ϕ̣_0+2γ_θ+γ_Λ /Λ̣)Γ_θθ. 
        In line with the CS logic so far, the appearance of the new parameter Λ in the theory results in a new CS parameter γ_Λ. As opposed to the other CS parameters, it is dimensionful. As a general Ansatz, we take for its one-loop description
        γ_Λ = Λ·(c_1 + ħ/32π^2·α_1 + Ø(ħ^2)) + m^4 ·(c_2 + ħ/32π^2·α_2 + Ø(ħ^2)).
        
        By a bottom-up computation, we find that taking
        
        c_1    =    4+Ø(λ),     α_1=-4λ + Ø(λ^2)
        
        c_2    =   Ø(λ^2),     α_2 = 1+Ø(λ^2)
        
        does the job. We conclude that with the inclusion of the constant term Λ and its associated CS parameter γ_Λ, the constant contribution that was missing in eq. <ref> reappears. With that, we have shown that indeed the constant contribution to the effective action can be accounted for by a slight generalisation of the Callan-Symanzik method.
        
        Finally, we mention that in the SSB case treated in the body of the paper, the story is precisely the same. In the traditional way one finds that the renormalisation conditions imposed at the theory's vev
        Γ|_ϕ_0=v   =   Λ-3m^4/2λ ,
        d^2 Γ/d ϕ_0^2|_ϕ_0=v   =    -2m^2 ,
        d^4 Γ/d ϕ_0^4|_ϕ_0=v   =   λ,
        
        again lead to the result of eq. <ref>, but now with (a direct generalisation of the result in eq. <ref>)
        α_0=285/2+iπ+ln2 ,         α_1 = λ(61/2+iπ+ln2) ,      α_2 = λ^2/4(9/2+iπ+ln2) . 
        
        Similarly, imposing
        Γ|_ϕ_0=v   =   Λ-3m^4/2λ ,
        d Γ/d ϕ_0|_ϕ_0=v   =    0 ,
        d^4 Γ/d ϕ_0^4|_ϕ_0=v   =   λ,
        
        leads to eq. <ref> with
        α_0=69/2+iπ+ln2 ,         α_1 = λ(25/2+iπ+ln2) ,      α_2 = λ^2/4(9/2+iπ+ln2) . 
        
        In both cases we find that same three CS equations in eqs. <ref> consistently connect these two results to the result for Γ_θθθ given in eq. <ref>.
        
        
        
        
        
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 Paper Title: Free energy from forward scattering in 1+1d.


Potential Mean-Field Games and Gradient Flows[Authors would like to thank Professors Dylan Possamaï and Ludovic Tangpi for insightful
comments.  Research
partially supported by the National Science Foundation grant
 DMS 2406762. ]
    Felix Höfer[
Department of Operations Research and Financial
Engineering, Princeton University, Princeton, NJ, 08540, USA, email: 
 fhoefer@princeton.edu. ] H. Mete Soner[
Department of Operations Research and Financial
Engineering, Princeton University, Princeton, NJ, 08540, USA, email: 
 soner@princeton.edu. ]
    August 5, 2024
=============================================================================================================================================================================================================================================================================================================================



	The free energy, or equivalently the ground state energy in finite volume, may be calculated from forward scattering amplitudes using a formula due to Dashen, Ma, and Bernstein <cit.>. However a naive treatment leads to singularities when considering the scattering of three or more particles. It is shown in detail how the approach can be applied to multi-particle scattering in various massive scalar theories in 1+1d, with or without integrability. The results for the sinh-Gordon, Lieb-Liniger, and O(N) non-linear sigma models are compared to exact results. It is shown how bound states can be considered in this approach by considering the attractive Lieb-Liniger model.
	
	

§ INTRODUCTION

			One of the most basic physical quantities that may be calculated from a quantum field theory is the free energy as a function of temperature. It is well-understood how to calculate this perturbatively using compactified Euclidean time, an approach referred to here as thermal field theory. The free energy should in principle encode much information about the zero-temperature dynamics of the theory, but since the size β of the thermal circle is involved in each propagator of a diagram, the thermal field theory approach mixes statistics with dynamics in an essential way.

Much better in this respect is the thermodynamic Bethe ansatz (TBA) <cit.>. The zero-temperature dynamics are encoded in kernels K which involve derivatives of the logarithm of the two-body S-matrix (see e.g. equation (<ref>)). These kernels then appear in an integral equation to determine the free energy. However, this approach only works for integrable systems.
	
	There is a less widely known third approach to calculate the free energy introduced by Dashen, Ma, and Bernstein (DMB) <cit.> which extends a formula by Beth and Uhlenbeck <cit.>. This approach explicitly separates the zero temperature dynamics from the temperature by writing the partition function in terms of the density of states ρ(E),
	
    Z=∫ dE e^-β Eρ(E).

	The density of states may be calculated in terms of forward scattering amplitudes at zero-temperature, even for non-integrable systems. In fact the expression for ρ involves a derivative of a logarithm of an S-matrix operator, and it is suggestive of the kernel K in the TBA approach,
	
    ρ(E)=ρ^(0)(E)+1/2π iETrlogŜ(E).

	Here ρ^(0) refers to the non-interacting density of states, and the operator Ŝ will be defined carefully in Sec <ref> where the derivation of the DMB formula is reviewed.
	
	The calculation of ρ in the DMB approach has certain subtleties, and the lack of explicit examples in the literature has perhaps hindered its adoption.  In this paper, in order to clearly illustrate the DMB approach we will consider certain massive scalar field theories in two spacetime dimensions for which we can compare with calculations in the thermal field theory and TBA approaches.



 §.§ Extending the Lüscher formula via Beth-Uhlenbeck

A reconsideration of the DMB approach <cit.>, as well as related approaches to calculating the self-energy and higher correlation functions <cit.>, may ultimately be useful for extracting information on multi-particle scattering  amplitudes from lattice simulations. Note that in 1+1d the free energy may alternatively be understood as a calculation of the ground state energy as a function of finite volume L. A better understanding of how to relate non-integrable scattering data to the energy levels as a function of L is relevant for the effective theory of Yang-Mills flux tubes <cit.>. There has been great success in using the TBA to relate the two-body S-matrix on the worldsheet of the flux tube to lattice calculations of the energy levels of the flux tube as a function of L <cit.>. Even so, it is understood that the worldsheet theory for both 3D and 4D Yang-Mills is not actually integrable <cit.>.
	
	There is an alternate approach due to Lüscher <cit.> which is well-known in this context, and which can be applied to the two-body energy levels in non-integrable systems. As will be explained below, the present investigation is an extension of the infinite volume limit of the Lüscher formula to multi-particle scattering states (see also <cit.>). This is not yet directly applicable to the effective string theory program since we are dealing with massive particles, but it is clearly of value to better understand how multi-particle scattering affects the finite size dependence of the energy levels in a simple case.
	
	The Lüscher formula in two dimensions is essentially the same thing as the two-particle asymptotic Bethe ansatz (ABA)
	
    2π i n = i k_n L + 2log S(k_n),

where n is an integer specifying the energy level[Since the equation is written only in terms of the relative momentum, n varies in steps of 2 for fixed center of mass momentum.], k_n is the relative momenta between the particles, and S(k_n) is the two-body S-matrix. Strictly speaking this formula is valid for massive particles in the limit of large mL, and any additional terms which vanish in this limit are suppressed in this section. Since the energy levels are closely spaced in this limit, it makes sense to instead consider the density of two-particle states ρ_(2),

    ρ_(2)(E)= (dE/dk)^-11/k_n+2-k_n= L/4π(dE/dk)^-1 +1/2π id/dElog S(E).

Here we are taking fixed center of mass momentum, so that the total energy E is a single-variable function of the relative momentum k, and the two body S-matrix is rewritten in terms of E. The first term is just the density of two-particle states in the non-interacting theory, and the remaining term δρ_(2) is
		
    δρ_(2)(E)  = 1/2π iElog S(E).

This formula was originally derived in ordinary quantum mechanics by Beth and Uhlenbeck <cit.>.

Of course, the density of states also has contributions from states with more than two particles. For integrable systems we may continue to use the ABA in order to calculate the density of states <cit.>. But such an approach is not directly applicable to non-integrable systems.

Instead we can extend the Beth-Uhlenbeck formula following DMB <cit.>. The DMB formula (<ref>) is superficially identical to the Beth-Uhlenbeck formula, but as will be discussed further in Sec <ref> the operator Ŝ(E) is an S-matrix operator that may be applied to arbitrary multiparticle states, and the E argument need not be equal to the energy of the states involved in the trace.
		
	

 §.§ Forward scattering singularities

	
		To see roughly how (<ref>) is to be interpreted, write the non-trivial part of the S-matrix operator as usual in terms of a T-matrix operator with an energy delta function factored out,
	
    Ŝ(E)=1-2π i δ(E-H_0)T̂(E).

	The factor logŜ(E) may be expanded to first order in powers of T̂, and after integrating by parts in E the partition function may be written as
	
    δ Z    = -β∫ dα e^-β E_αT_αα+T^2.

	Here the trace was taken over eigenstates α of H_0 with energy E_α, and dα schematically represents the appropriate measure for the trace. T_αα is the forward scattering amplitude with both an incoming and outgoing state of α.	As usual, we may consider instead the connected part of the T-matrix, T^c, and the free energy density[We use L for spatial volume, although the following formulas apply to arbitrary dimension.] f=-(β L)^-1log Z,
	
    δ f= L^-1∫ dα e^-β E_αT^c_αα+T^2.

	
	Here already we see a potential issue. A connected T-matrix amplitude T^c_βα may involve various delta functions, and in the forward scattering limit β=α the argument of these delta functions may go to zero. One delta function will just be due to momentum conservation and it will lead to an overall factor of L to cancel the L^-1 in (<ref>). But additional delta functions may arise from internal propagators in scattering amplitudes which can go on-shell. 
	
	For instance the lowest-order three-particle scattering amplitude in a relativistic scalar field theory in Fig <ref> is just proportional to the internal propagator
	
    1/(E_1+E_2-E_1')^2-(k_1+k_2-k_1')^2-m^2+iϵ,

	where the E are on-shell energies E=√(k^2+m^2) associated to the spatial momentum k of a given external leg. This propagator goes on-shell when k_1' equals k_1 or k_2. The propagator can be broken into a real principal part, and an imaginary delta function. These delta functions can not be ignored and in fact for integrable models they are necessary to ensure that the set of final momenta are equal to the initial ones (see the discussion in <cit.>). So the forward scattering amplitude T^c_αα will certainly involve delta functions with a vanishing argument, and these forward scattering singularities need to be interpreted for the DMB formula to make sense.
	
	As we will show perturbatively in Sec <ref>, this issue is resolved by including the higher order powers of T in the expansion in (<ref>). The delta function parts of the internal propagators in diagrams like Fig <ref> are canceled, and only the principal part remains. This introduces some new questions since for three-particle scattering in integrable theories the principal part is taken to vanish. Nevertheless we will show that there are two subtle types of contributions to the free energy due to the principal part: on-shell contributions and  collinear contributions.
	
	
	
	
	
	
	On-shell contributions are due to forward scattering diagrams that would have had an on-shell divergence, such as Fig <ref> with k_1' taken to be k_1 or k_2. The principal part of the internal propagator vanishes exactly on-shell, but it is large slightly off-shell, and the slightly off-shell region will contribute since the energy delta function involved in the S-matrix operator (<ref>) actually has a width that depends on iϵ.
	
	The calculation of the on-shell contributions has previously been discussed to a certain extent for the scattering of three or four particles <cit.>, as well as more generally using a different approach <cit.>. In section <ref> we give an arguably simpler calculation of the on-shell contributions for an arbitrary number of particles in relativistic theories from both the thermal field theory and DMB perspectives.
	
	Besides the on-shell contributions, there are also collinear contributions which appear at lowest order due to forward scattering diagrams such as Fig <ref> with k_1' = k_3, so that the internal propagator need not be on-shell. If k_1, k_2, k_3 are all distinct then the principal part vanishes as per the usual argument. But since we are integrating over all momentum in the DMB formula, we need to consider the cases where the particle momenta become collinear. It turns out that the principal part doesn't vanish in a small iϵ dependent region around the collinear point k_1=k_2=k_3, and this is actually a significant contribution to the free energy.
	
	 In the ϕ^4 theory there are two free energy diagrams at second order in the thermal field theory approach,  which may be called the bubble and melon diagrams (see Fig <ref>). We will show that the melon diagram in 1+1d is entirely the result of collinear contributions. Note that the massive <cit.> and massless <cit.> ϕ^4 theory have been investigated before in 3+1d via the DMB approach and the related approach of Barton <cit.>, but the essential features of the 1+1d case were not seen.  The 1+1d melon diagram is calculated in Sec <ref> and compared to a perturbative expansion of the TBA in Sec <ref>.  In Sec <ref> it is shown how these collinear contributions are necessary in perturbation theory since there are bound state contributions to the free energy upon flipping the sign of the coupling constant.
	


 §.§ Outline
 A brief outline of this paper is as follows. In Sec <ref> we will review the general derivation of the DMB formula and apply it to integrable models involving a single massive scalar such as sinh-Gordon. In Sec <ref> we consider sinh-Gordon and the non-integrable ϕ^4 theory at second order in perturbation theory, and show how free energy diagrams in the thermal field theory approach are connected to two- or three-body forward scattering diagrams in the DMB approach.
		
		In Sec <ref> diagrams involving only on-shell contributions are calculated at arbitrary order in perturbation theory, and it is shown how these give the leading contribution to the free energy for the large N extension of the ϕ^4 model. In Sec <ref> we investigate the treatment of bound states in the DMB approach by considering the non-relativistic Lieb-Liniger model with attractive coupling. We show how the collinear contributions at repulsive coupling are related to the contributions due to bound states for attractive coupling. In Sec <ref> we discuss future investigations into the DMB approach.
		


§ THE DMB FORMULA AND INTEGRABLE MODELS





 §.§ Derivation of the DMB formula

First let us review some material from scattering theory. The Lippmann-Schwinger equation expresses the in and out eigenstates Ψ^± of the Hamiltonian H=H_0+V in terms of the eigenstates Φ of H_0 with the same eigenvalue E,

    Ψ^± = Φ+1/E-H_0± iϵVΨ^±.


This can be written in terms of a quantity G_0 that is usually referred to as the propagator in this context,

    G_0(E)≡1/E-H_0+ iϵ.

An interacting propagator G may also be defined,

    G(E)≡1/E-H+ iϵ=1/1-G_0(E)VG_0(E).


Now we can define a T-matrix operator so that the Lippmann-Schwinger equation can be written as VΨ^+=T̂(E)Φ,

    T̂(E)≡ V1/1-G_0(E) V.

An S-matrix operator may also be defined in terms of T,

    Ŝ(E)   ≡ 1+(G_0(E)-G_0(E)^†)T̂(E)
       =(1-G_0(E)^† V) (1-G_0(E) V)^-1

The first line is just the usual relation between S and T since G_0-G_0^†→ -2π i δ(E-H_0) in the ϵ→ 0 limit.

The Ŝ(E) operator is defined even outside the usual limits in which scattering theory applies, but in order to interpret it as the usual S-matrix it must be acting on an H_0 eigenstate Φ with the same eigenvalue E as its argument, and the infinite volume and  ϵ→ 0 limits must be taken. Keeping these caveats in mind, we will suppress the E argument except when necessary.

Now the DMB formula follows from some simple manipulations using the formulas above and the fact that EG_0(E)= -G_0(E)^2.

    1/2π iETrlogŜ   = 		1/2π iETr[log(1-G_0^† V)-log(1-G_0 V)]   =	1/2π iTr[(G_0^†)^2V/1-G_0^† V-G_0^2V/1-G_0 V]   =	1/2π iTr[(1/1-G_0^† V-1)G_0^†-(1/1-G_0 V-1)G_0]   =-	1/2π iTr[(G-G^†)-(G_0-G_0^†)]   = Tr[δ(E-H)-δ(E-H_0)],

where the last line holds in the ϵ→ 0 limit. This is now in the form of the difference between the density of states ρ(E)-ρ^(0)(E) as required in the DMB formula (<ref>).

In situations where there is a conserved particle number N in both the free and interacting theories it further makes sense to introduce a chemical potential μ and corresponding fugacity y≡ e^βμ, in terms of which the grand canonical partition function is

    δ Z=∑_N=2^∞ y^N ∫ dE e^-β E1/2π iETr_(N)logŜ.


In what follows fugacity dependence will be used as a counting device to single out the trace over N-particle states Tr_(N), and we will set y=1 at the end. In the theories of massive particles considered here the term in the partition function proportional to y^N will have an asymptotic temperature dependence on the order of ∼ e^-N β m. So the fugacity expansion may equivalently be understood as a low temperature expansion in powers of e^-β m.

We will use a notation where a subscript in parenthesis denotes the order in the fugacity expansion. On the other hand, a superscript in parentheses denotes the order in a coupling constant (c or g^2) expansion, and a δ prefix denotes the difference of a quantity from the free theory. Later on we will introduce an n-body expansion that takes into account particle statistics, and this will be indexed by a subscript without parentheses.


 §.§ Integrable models: Sinh-Gordon and Lieb-Liniger

It should be clear that the DMB formula above is quite general since it simply follows from the definitions of the operators T̂ and Ŝ, but to illustrate it in practice we consider the sinh-Gordon model (SG) <cit.>, with Lagrangian

    _SG = 1/2(∂ϕ)^2+(m^2+δ m^2)/g^2(cosh( gϕ) - 1).

Here δ m^2 is a counterterm set to ensure that m is the physical mass.

In many respects the non-relativistic limit of SG may be understood to be the Lieb-Liniger model (LL) <cit.>, which involves bosons in ordinary quantum mechanics interacting through a repulsive two-body delta function potential 
    V_LL(x)=c/mδ(x).

The parameter g in the SG Lagrangian is related to the parameter c by

    c=msinπ g^2/8π+g^2  = m g^2/8-m g^4/64π+g^6.


Both SG and LL are integrable and the exact two-body S-matrix is given by

    S(p)=p-i c/p+ic.

In LL, p is taken to be the momentum difference between the particles, whereas in SG, p = m sinhθ with θ being the rapidity difference. 

Since both models involve a single particle type with no internal indices and no bound states, the TBA takes the simplest possible form (see <cit.> for a review). The free energy density f is calculated as

    f= -T∫dp/2πlog(1+e^-ϵ(p)/T)

where the pseudo-energy ϵ is determined by solving the TBA equation

    ϵ=E-μ-Tlog(1+e^-ϵ/T)⋆ K.

Here E is the single particle dispersion relation, and the ⋆ denotes convolution[The convolution is taken over the rapidity variable and the momentum variable in SG and LL, respectively.]

    f⋆ g (θ)≡∫ dθ' f(θ')g(θ-θ').

K is a kernel that is derived from the exact S-matrix. The kernels for LL and SG are respectively,

    K_LL = 1/2π id/dplog S(p)=1/πc/p^2+c^2,
    
    	K_SG = 1/2π id/dθlog S(msinhθ)=mcoshθ K_LL.

The formal similarity of the kernels to the DMB formula for the density of states is not an accident, since both approaches will reduce to the Beth-Uhlenbeck formula (<ref>) in the two-particle limit, as shown in the section below.


 §.§ Applying DMB to integrable models



Since SG and LL are integrable and only involve one species of particle, this implies that arbitrary multiparticle states α in the momentum basis are eigenstates of the S-matrix operator

    Ŝ(E_α)|α⟩ = S(α)|α⟩.

Here Ŝ refers to the operator appearing in the DMB formula, and S(α) is a complex eigenvalue which ends up being factorizable into two-body S-matrices.

The normalization of the eigenstates is fixed to be compatible with the measure in the sense that ⟨α|β⟩ acts like a delta function,

    ∫ dα⟨α|β⟩ f(α) =f(β).


Now let us use these eigenstates as the basis for the trace in the DMB formula,

    δ Z   =β/2π i∫ dE e^-β ETrlogŜ(E)   =-β/2π i∑_k=11/k∫ dE e^-β E∫ dα⟨α|(2π i δ(E-H_0)T̂(E))^k|α⟩   =-β∑_k=11/k∫ dα  e^-β E_α⟨α|T̂(E_α)(2π i δ(E_α-H_0)T̂(E_α))^k-1|α⟩.

Here the last step of using one delta function to evaluate the E integral is somewhat questionable since the delta function is really G_0-G_0^† for finite iϵ, and there may be problems with the vanishing iϵ limit in the T-matrix amplitudes. But for now let us proceed. This may be evaluated further by inserting an integration over a complete set of states between each δ(E-H_0)T, and using the idea that

    ⟨β | 2π i δ(E_α-E_β)T̂(E_α) |α⟩ = (1-S(α))⟨β |α⟩.

The result is

    δ Z		   =-β∑_k=11/k∫ dα dβ_1… dβ_k-1 e^-β E_α⟨α|T̂(E_α)|β_1⟩…⟨β_k-1 | 2π i δ(E_α-E_k-1)T̂(E_α) |α⟩   =β∫ dα  e^-β E_α(T(α)/1-S(α))log S(α),

where T(α)≡⟨α | T̂(E_α)|α⟩ is the on-shell forward scattering T-matrix. Note that considering (<ref>), T(α) appears to be proportional to 1-S(α) and the factor in parenthesis in (<ref>) appears to just be a kinematic factor.

This formula is able to reproduce the correct free energy for two-particle states. A quick way to see this is to use a normalization of two-particle states such that dα = dE dP with E and P being the total energy and momentum. If it is valid to cancel the energy delta function from both sides of (<ref>) then we can calculate T,

    2π i T(α)=(1-S(α))L/2π.
 
Then after integrating by parts, and rewriting in terms of the two-particle free energy density f_(2), (<ref>) becomes


    δ f_(2)=-T/2π∫ dP   dE  e^-β E1/2π iE.log S(α)|_P.

This is just the Beth-Uhlenbeck formula (<ref>) with the center of mass motion taken into account.

To write it more explicitly in SG, transform the integration measure to rapidity variables and use the kinematic relation E^2=P^2+4m^2cosh^2θ to rewrite the E derivative as a θ derivative,

    δ f_(2)=-T/2π1/2!∫ dθ_1 dθ_2  E  e^-β E1/2π iθlog S(θ)= -T∫dp_1/2πe^-β E_1∫ dθ_2 e^-β E_2K_SG(θ).

In the last equality this has been written in terms of a convolution integral over the SG kernel (<ref>), and it is a short step to see agreement with the TBA.

According to the formula for the grand canonical partition function (<ref>), the above expression should match the y^2 term in a fugacity expansion of free energy density (<ref>). This expansion may be carried out by expanding the pseudo-energy ϵ in powers of y and solving the TBA equation (<ref>) at each order. The result is

    δ f_(2) = (f-f^(0))_(2)   = -T∫dp/2π e^-β E(e^-β E⋆δ K).

Here we have subtracted off the the free energy density of a free boson gas,

    f^(0)=+T∫dp/2πlog(1- e^-β E),

and δ K is the interacting part of the kernel,

    δ K_LL(p)≡ K_LL(p)-δ(p),     δ K_SG= mcoshθ δ K_LL.


Clearly the TBA result (<ref>) matches the corresponding DMB result for SG (<ref>), and a similar agreement may be seen for the LL model. Agreement of the DMB formula with the TBA at y^2 has also been noted for the O(N) model <cit.>.



 §.§ Extending the argument to multiple particles

Now we would like to apply the seemingly general formula (<ref>) to the scattering of three or more particles, but it is difficult to see how this could succeed. If T/(1-S) is still just a kinematic factor, then since S is factorizable into two-body S-matrices the log S factor just breaks up into a disconnected sum. In fact we were too cavalier in the treatment of the iϵ terms in the derivation of (<ref>), and the contribution due to multi-particle scattering depends sensitively on this.

The contribution to the free energy due to three-particle scattering may be investigated by expanding the TBA to the third order in fugacity,

    δ f_(3)   = -T∫dp/2πe^-β E[e^-β E(e^-β E⋆δ K)+1/2(e^-2β E⋆δ K)]        -T∫dp/2π  e^-β E[(e^-β E(e^-β E⋆δ K))⋆δ K+1/2(e^-β E⋆δ K)^2].

This expression has also been derived by Kato and Wadati using an approach based on the asymptotic Bethe ansatz (ABA) <cit.>.

As will be discussed more at the end of Sec <ref>, the first line of (<ref>) involving a single appearance of δ K is better understood as a correction to the two-body contribution (<ref>) due to particle exchange. The two-body contribution δ f_2 to all orders in fugacity is

    δ f_2=+T∫dp/2π1/e^β E-1[log(1-e^-β E)⋆δ K].


The second line of (<ref>) involving two factors of δ K represents genuine three-particle scattering. Contributions due to connected scattering of even more particles may be derived by continuing the fugacity expansion of the TBA, or using the ABA approach of <cit.>. These expressions become increasingly complicated, and there is seemingly no way to derive them from the naive formula (<ref>).

To better understand how multi-particle scattering is treated in the DMB formula, we consider the second order in perturbation theory, which is the lowest order for which three-particle scattering is relevant. The operator logŜ may be expanded in powers of V,

    log(1+(G_0-G̅_0)T̂)^(2)   =(G_0-G̅_0)VG_0+G̅_0/2V   =(G_0-G̅_0)𝒫T̂^(2).

Note that the internal propagator of T̂^(2) is 

    G_0 = 1/E-H_0+iϵ = 𝒫 G_0 - iπδ(E-H_0).

For integrable theories, as long as all momenta of the particles are distinct (as is assumed in the ABA), the net contribution from the 𝒫G_0 terms vanishes when summed over all diagrams. Rather it is the delta function term which leads to an Ŝ operator which is diagonal in the momentum basis, which is assumed in the derivation of (<ref>). However (<ref>) asserts that only 𝒫G_0 contributes in the DMB formula, and not the delta function part.

One way in which 𝒫T^(2) can be non-zero is if E is not strictly set to be equal to the energy of the external particles in the scattering amplitude. This is possible since the “delta function” (G_0-Ĝ_0) has some width in terms of ϵ. These type of contributions will be referred to as on-shell contributions, since they involve diagrams for which the denominator of G_0 vanishes. On-shell contributions will be treated in DMB approach to arbitrary order in perturbation theory in Sec <ref>.

The other way in which 𝒫T^(2) can be non-zero is if the particle momenta are collinear. This is not taken to be the case in the ABA, but it is encountered in the trace over free particle states in the DMB formula. We will treat the collinear contributions at second order in perturbation theory in Sec <ref> and the associated Appendix <ref>. As discussed in Sec <ref>, the collinear contributions are closely connected to bound state contributions, and they are expected to be important at higher order in perturbation theory as well.

	

§ CUTTING THERMAL FIELD THEORY DIAGRAMS

The DMB formula gives contributions to the free energy in terms of forward scattering amplitudes. In this section we will show how the forward scattering diagrams arise by cutting lines of vacuum bubble diagrams in thermal field theory. As a concrete example we will consider the second-order diagrams in sinh-Gordon (<ref>) and its non-integrable truncation, the ϕ^4 theory,
		
    = 1/2(∂ϕ)^2+m^2+δ m^2 /2ϕ^2+m^2g^2/4!ϕ^4.

	

 §.§ Single vertex corrections

	
	
	
	
	The simplest free energy corrections to consider involve a single vertex proportional to ϕ^2n, which is contracted into a vacuum bubble diagram involving n tadpoles. See Fig <ref> for the n=2 and n=3 cases, which are relevant in sinh-Gordon up to second-order. Besides the diagram involving the ϕ^2n vertex, there will also be diagrams at the same order involving counterterm vertices proportional to δ m^2. For instance the total free energy density f^(1) at first order is
	
    f^(1)=m^2g^2/4!3(I^2-I_0^2)+δ m^2/2(I-I_0).
 
	This is written in terms of the tadpole integral I,
	
    I ≡∫_βd^2 p/(2π)^21/p^2+m^2=1/β∑_n∫dk/2π1/ω_n^2+k^2+m^2,

	where the β subscript shorthand indicates that Euclidean time is compactified, and the integral may be written explicitly in terms of Matsubara frequencies ω_n =2π n/β. This integral may be written as a sum of a zero-temperature part I_0 and a thermal part that involves the bosonic occupation number n_k, which is defined as
	
    n_k(β)≡1/e^β E_k-1,      	E_k ≡√(k^2+m^2).

	To make this decomposition, the sum over ω_n is transformed to an integral over ω by using the Poisson resummation formula
	
    I   =∑_l∫dk/2πdω/2πe^-i lβω/ω^2+k^2+m^2   = I_0+∫dk/2πn_k(β)/E_k.

	The l=0 term is the same as the zero-temperature tadpole integral, and this is referred to as I_0. In the l≠ 0 terms, the ω integral has been performed, and the geometric series summed.
	
	Now noting that the mass counterterm at this order is δ m^2 = -λ/2I_0, it is seen that all appearances of I_0 cancel in (<ref>)	
    f^(1)   =m^2 g^2/8(∫dk/2πn_k(β)/E_k)^2   = 1/L 1/2!∫d k_1/2π1/2E_1d k_2/2π1/2E_2e^-β(E_1+E_2)/(1-e^-β E_1)(1-e^-β E_2) m^2g^2 L.

If we momentarily ignore the two factors of (1-e^-β E)^-1, this is clearly in the form of the equation (<ref>),
	
    f^(1)= 1/L∫ dα e^-β E_αT^(1)_αα,

	where α represents a state of two identical relativistic particles with momenta k_1 and k_2, and dα is the appropriate measure for a trace over these states. The first-order forward scattering T matrix element is just the vertex factor m^2 g^2 times the volume L.
	
		Diagrams involving a single higher-order vertex, such as the second-order diagram involving the ϕ^6 vertex in Fig <ref>, follow similarly. The counterterm diagrams will again end up canceling the I_0 terms in the tadpole integrals, and the net contribution to the free energy density is
		 	
    f_ϕ^2n= 1/L 1/n!∫(∏_i=1^nd k_i/2π1/2E_ie^-β E_i/1-e^-β E_i) m^2g^2n-2 L.

	 		
	This has the appropriate measure for n relativistic particles, and involves the n-body forward scattering amplitude T_αα=m^2g^2n-2 L, but once again there are additional factors of  (1-e^-β E_i)^-1. These factors have to do with bosonic statistics, and they arise by considering auxiliary particles in the states α which do not participate in the two-body scattering. Due to identical particle exchange there will be parts of the forward scattering T matrix which are still connected in the sense that they involve a single factor of L and can not be written as a product of two or more free energy terms. An example is given in Fig <ref>. The auxiliary particle line leads to a delta function 2E_3 2πδ(k_3-k_1) that cancels the trace over the third particle momentum and leads to a correction which agrees with the first non-trivial term in the expansion of (1-e^-β E_1)^-1.
	
These (1-e^-β E_i)^-1 factors involve arbitrarily high particle number N as far as the fugacity expansion (<ref>) is concerned, but each term in the expansion involves only some finite number of n particles interacting through non-trivial scattering. It is useful to consider an expansion in the number of non-trivially interacting particles, which will be referred to as the n-body expansion. An “n-body correction” like (<ref>) will generically involve n factors of bosonic occupation numbers n_i.

For later reference we will write the three-body correction due to the ϕ^6 vertex explicitly in terms of occupation numbers as a function of rapidity

    f_ϕ^6   =1/3m^2 g^4/16(∫dθ/2πn(θ))^3,

	where n(θ)=(e^β mcoshθ-1)^-1.

	

	
	
	
	
	
	
	
	
	
	
	

 §.§ Cutting a general diagram

	
	
	The previous example involved using the Poisson resummation formula to write finite temperature propagators in terms of an infinite sum over an index l. The l=0 term which is present at zero-temperature was split from the rest. Now we will generalize this procedure.
	
	A general r-loop free energy diagram δ f may be written in terms of r independent frequencies, each with its own summation over exponentials due to the Poisson resummation formula,
	
    δ f =  ∫∏_j=1^rdk_j/2πdω_j/2π∑_l_je^-il_jβω_j/ω_j^2 + E_j^2ℳ(ω, k).
 
	The factor ℳ collects all of the combinatorial and vertex factors, as well as any additional propagators which are fixed by the delta functions at the vertices to be functions of the r independent ω_j, k_j.
	
	The summation over l_j may be broken up into the l_j=0 term and the l_j≠ 0 terms,
	
    ∑_l_j e^-i l_j βω_j= 1+ ∑_σ_j=± 1∑_l_j>0 e^-i σ_j l_j βω_j.

	After decomposing the r propagators into two terms in this manner, there are a total of 2^r terms in δ f. We refer to the propagators that keep the l≠ 0 terms as cut, and the those that keep the l=0 term as uncut.  We may index the 2^r terms in δ f by a set π which contains the indices of the cut propagators. The other uncut propagators are integrated over and absorbed into the ℳ factor, which is now denoted ℳ_π.
	
    δ f =  ∑_π∫∏_j∈πdk_j/2πdω_j/2π∑_σ_j=± 1∑_l_j>0e^-iσ_j l_jβω_j/ω_j^2 + E_j^2ℳ_π(ω, k).
 
	The quantity ℳ_π is easy to understand diagramatically. It is just given by cutting the cut propagators of the original free energy diagram and taking them to be external legs. The remaining uncut propagators of the diagram are taken to be at zero-temperature. ℳ_π is an off-shell forward scattering amplitude depending on the frequency and spatial momenta of the cut lines.
	
	If we momentarily assume that ℳ_π(ω) is analytic then only the pole at ω_j = -iσ_j E_j due to the cut propagator will contribute to the ω_j integral. Then after integration over ω and summation of the geometric series in l we get the formula,
	
    δ f =  ∑_π∫∏_j∈πdk_j/2πn(E_j)/2E_j∑_σℳ_π(-iσ E, k).
 
	
	This is now roughly in the form of the DMB formula (<ref>). ℳ_π has been put on-shell, and the signs σ determine whether the momentum of a cut line is considered to be incoming or outgoing. If there are n cut lines belonging to π then this is an n-body contribution to the free energy diagram δ f, and the factors of n(E) automatically contain the higher-order corrections due to particle exchange.
	
	Generically ℳ_π(ω) will not be analytic. There will be an additional contributions to the frequency integral from poles or branch cuts in ℳ_π. This will be easy enough to deal with in Sec <ref> for the case of diagrams with only on-shell contributions. The result will agree with the full DMB formula that includes the higher order in T̂ corrections which are neglected in (<ref>).
	

 §.§ The bubble and melon diagrams

	
	For now let us consider the free energy diagrams at second-order in the sinh-Gordon and ϕ^4 theories. As discussed in Sec <ref>, perturbation theory at this order will clarify how three-particle scattering amplitudes are treated in the DMB approach. The diagram involving a single sextic vertex in sinh-Gordon has already been considered in (<ref>). The remaining two diagrams —the bubble and melon— are shown in Fig <ref>.
	
	The previous section has made clear which forward scattering amplitudes are associated to a given thermal field theory diagram. Lines of the thermal field theory diagram are cut to create incoming and outgoing external lines with the same momenta. Conversely, we may `glue together' incoming and outgoing lines with the same momentum in a forward scattering amplitude (the lines labeled by the same number in Fig <ref>) to recover a free energy diagram.
	
	

  §.§.§ The bubble diagram

	The bubble diagram is shown on the left side of Fig <ref>. As in the single vertex case, free energy diagrams involving counterterms will cancel the uncut I_0 part of any tadpoles, so the two tadpoles on the sides of the diagram are always cut.
	
	The two-body part of the diagram f_bubble,  2 involves an uncut central bubble. Since in this case the amplitude ℳ_π does not depend on the cut momenta, it can be readily calculated

    f_bubble, 2   =-m^4 g^4/16(∫dθ/2πn(θ))^21/4π m^2.

	
	The scattering amplitude that would correspond to the three-body part of the bubble diagram f_bubble, 3 is given on the left of the bottom row of Fig <ref>. This diagram has a divergence due to the internal propagator being fixed by momentum-conservation to have the on-shell momentum labeled 3. This is an example of an `on-shell contribution.'
	
	Going back to the general cutting argument, this divergence just indicates that ℳ_π also has a pole as a function of ω_3, and this needs to be taken into account in the integration over ω_3. So dealing with the on-shell contribution is no major difficulty, it just involves integrating over a double pole
	
    f_bubble, 3   =-m^4 g^4/16(∫dθ/2πn(θ))^2	∫dk_3/2π∑_σ_3,l_3∫dω_3/2π e^-iσ_3 l_3 βω_3/(ω_3^2+E_3^2)^2   =-m^2 g^4/16(∫dθ/2πn(θ))^2	∫dk_3/2πm^2/2E_3^2(-d/dE_3+1/E_3)n(E_3).


	This expression will be derived from the DMB formula in Sec <ref>. By using similar manipulations to those in Appendix <ref> it can be brought to a simplified form
		
    f_bubble, 3   =1/3!m^2 g^4/16ββ(∫dθ/2πn(θ))^3.

	
	
	
	

  §.§.§ The melon diagram: two-body part

	
	
	The forward scattering diagrams corresponding to cutting the melon diagram are shown on the right of Fig <ref>. In the following sections we will use these diagrams in the DMB approach to evaluate the melon diagram and show it involves only contributions with collinear momenta. Of course the melon diagram may also be evaluated directly from thermal field theory, and this is shown using methods similar to <cit.> in Appendix <ref>.
	
	Note that the two-body amplitudes can be written in terms of the one-loop integral
		
    J(p)   ≡∫d^2 q/(2π)^21/q^2+m^21/(p-q)^2+m^2.

The bubble diagram was associated to the u-channel of two-body scattering, with no net momentum flowing through the loop, and (<ref>) indeed involves the factor J(0)=(4π m^2)^-1. The melon diagram instead involves the t-channel and s-channel. By the DMB approach, the two-body part of the melon should be
	
    f_melon, 2=-m^4g^4/21/2!∫(∏_j=1,2dk_j/2πn_j/2E_j) [J(-i(E_1-E_2))+𝒫J(-i(E_1+E_2))].

	The argument of the s-channel term J(-i(E_1+E_2)) sits on a branch cut, and some sort of iϵ prescription is needed to define it. The principal value prescription arising from the second-order DMB formula (<ref>) is the correct one.
	
This can be calculated further, using the expression

    J(p)=1/2π p^2 ξlog(ξ+1/ξ-1),    ξ≡√(1+4m^2/p^2).

The sum of terms in (<ref>) is proportional to a delta function,

    J(-i(E_1-E_2))+1/2[J(-i(E_1+E_2)+ϵ)+J(-i(E_1+E_2)-ϵ)]       =	log e^θ/4π m^2 sinhθ+1/2[-log(-e^θ+iϵ)/4π m^2 √(sinh^2 θ-iϵ)-log(-e^θ-iϵ)/4π m^2 √(sinh^2 θ+iϵ)]       =2π/16 m^2[i/π√(sinh^2 θ+iϵ)-i/π√(sinh^2 θ-iϵ)]       =2π/16 m^2δ(θ).

So in total

    f_melon, 2=-1/16m^2g^4/16∫dθ/2πn(θ)^2.

	
	
	
	
	

  §.§.§ The melon diagram: three-body part

	
	The three-body part of the melon diagram is similarly calculated from the three-body amplitudes in Fig <ref>,
	
    f_melon, 3=-m^4 g^41/3!∫(∏_j=1,2,3dk_j/2πn_j/2E_j)   [𝒫3/-(E_1+E_2-E_3)^2+(k_1+k_2-k_3)^2+m^2.       .+1/-(E_1+E_2+E_3)^2+(k_1+k_2+k_3)^2+m^2].

As long as the momenta are distinct the propagators are not on-shell, and the principal value regularization may be dropped.

The first propagator in (<ref>) may be written in terms of rapidity as,
	
    1/-(E_1+E_2-E_3)^2+(k_1+k_2-k_3)^2+m^2=e^θ_1+θ_2+θ_3/m^2(e^θ_1-e^θ_3)(e^θ_2-e^θ_3)(e^θ_1+e^θ_2).

There are two additional diagrams where index 3 is swapped with index 1 or 2 (hence the factor of three in (<ref>)). Adding these terms with permuted indices leads to

    e^θ_1+θ_2+θ_3/m^2(e^θ_1-e^θ_3)(e^θ_2-e^θ_3)(e^θ_1+e^θ_2)+(3↔ 1)+(2↔ 3) = e^θ_1+θ_2+θ_3/m^2(e^θ_1+e^θ_3)(e^θ_2+e^θ_3)(e^θ_1+e^θ_2).

	But this simply cancels with the second propagator in (<ref>),
	
    1/-(E_1+E_2+E_3)^2+(k_1+k_2+k_3)^2+m^2=-e^θ_1+θ_2+θ_3/m^2(e^θ_1+e^θ_3)(e^θ_2+e^θ_3)(e^θ_1+e^θ_2).

	
	The cancelation of the three-body forward scattering diagrams associated to the melon diagram is not surprising given the cancelations which must take place in the integrable sinh-Gordon model.[It can also be shown that the three-body diagrams associated with the bubble, displaced slightly from the forward direction, are canceled entirely by the three-body diagram given by the ϕ^6 vertex.] But actually the three-body part of the melon diagram does not vanish. The principal value regulation may not be ignored when the momentum become collinear, and a calculation in Appendix <ref> shows that the integrand of (<ref>) is proportional to a delta function, much as in the two-body case. The final result is
	
    f_melon, 3
       	=-1/24m^2 g^4/16∫dθ/2πn(θ)^3.

	
	
	 In Sec <ref> these same results will be derived from the TBA. It will be discussed why similar collinear contributions are expected at higher order in perturbation theory, and how they agree with the appearance of bound states upon switching the sign of the coupling in the theory.
	
	

§ ON-SHELL CONTRIBUTIONS AND LARGE N THEORIES

	The bubble diagram is part of a broader class of diagrams where the integrals over distinct loop momenta factorize. These are the same diagrams which would be singled out at leading order in a large N expansion. To study the net effect of these diagrams we consider extending the field in the ϕ^4 theory (<ref>) to an N component field ϕ with Lagrangian
	
    =1/2(∂ϕ·∂ϕ + (m^2+δ m^2) ϕ^2)+λ/4N(ϕ^2)^2.

	The counterterm δ m^2 = -λ I_0 is chosen to ensure that m^2 is the physical mass at large N and zero temperature. The quartic term may be simplified by introducing a Hubbard-Stratonovich field α
	
    =1/2(∂ϕ·∂ϕ + m^2 ϕ^2)-N/λα^2+α(ϕ^2-N I_0).

Here α has a quadratic term that vanishes in the strong coupling limit λ→∞. In this limit α can be understood as a Lagrange multiplier field, and the model reduces to the integrable O(N) non-linear sigma model.

Note that the counterterm was set to ensure that α has no expectation value at leading order for zero-temperature, but at finite temperature it does pick up an expectation value. We may absorb the expectation value into a temperature-dependent mass M(β), and a redefined field α_β≡α-1/2(M^2-m^2),

    =1/2(∂ϕ·∂ϕ + M(β)^2 ϕ^2)+α_β(ϕ^2-N I_0)-1/2(M(β)^2-m^2)N I_0.


The leading-order free energy of this form of the O(N) non-linear sigma model was considered by Balog and Hegedus in <cit.>. They used an exact expression for the free energy density which follows directly from (<ref>), 

    f=Nm^2/2π[-M^2/m^2∑_j=1^∞ K_2(jmβM/m) +1/4(M^2/m^2-1)].

This may be expanded in orders of fugacity by solving the gap equation <cit.> for M/m order-by-order,

    logM/m=2∑_j=1K_0(jβ m M/m).


Using this method, Balog and Hegedus were able to find the second-order in fugacity contribution f_(2) and show it agrees with the DMB formula for two-particle scattering. In principle there is no obstacle to using this method to calculate arbitrary orders f_(j), but matching to the DMB even for f_(3) encounters the problem of the forward scattering singularities. In the following sections we will give an alternate method to calculate the free energy density, and show how it may be matched to forward scattering amplitudes at arbitrary order.

However, note in passing that there is a direct connection between the thermal mass M(β) and forward scattering amplitudes along the same lines as the DMB relation. In <cit.>, Lüscher derives a relation which in our present circumstances takes the form

    M-m/m=-1/2m^2∫dp/2π1/2Ee^-β EF +e^-2mβ.

Here F is the sum of the two-body forward scattering amplitudes with the other particles of the theory, which works out to be -8π m^2 in the case of the large N limit of the O(N) non-linear sigma model. Expanding the gap equation (<ref>) to first order in fugacity we recover this result. Higher order in fugacity corrections to M^2 can be derived from thermal self-energy diagrams calculated using the Feynman rules of the following section, and these correspond to multi-particle  forward scattering amplitudes in the same way.



 §.§ Diagrams in the large N theory


The equations (<ref>) and (<ref>) give a compact expression for the free energy, but actually solving them order-by-order quickly leads to a tedious calculation. We will organize the calculation of the free energy by an approach involving thermal field theory diagrams, which then can be quickly related to forward scattering amplitudes through the cutting rules discussed in Sec <ref>.

Let us return to the form of the Lagrangian involving the zero-temperature mass m^2 (<ref>). We may keep finite λ dependence. There is a free boson contribution f_1 coming from the quadratic action of the ϕ field

    f_1 = N/2∫_βd^2p/(2π)^2log(p^2+m^2)=-Nm^2/π∑_j=1^∞K_1(jβ m)/jβ m.

There are further corrections coming from thermal field theory diagrams involving internal α propagators. Since each α propagator introduces a factor of 1/N, at leading order it must be paired with a ϕ loop, which introduces a compensating factor of N. This implies that at leading order there will be no momentum flowing through the α propagators.

So to calculate the leading order diagrams, we must consider a class of integrals which may be derived from the basic tadpole integral I in (<ref>),
	
    I_,n≡∫_βd^2p/(2π)^21/(p^2+m^2)^n+1=(-1)^n/n!d^n I/d(m^2)^n .

These integrals may be calculated by taking derivatives of the formula,

    I= 1/4πlogM_UV^2/m^2+1/π∑_j=1^∞ K_0(jβ m),

where M_UV is a UV cutoff. Using the Bessel function identity K_α+1(x) =( α/x-x)K_α(x), we can show

    I_,n=1/4π n m^2n[1+x_n/2^n-2(n-1)!],
    
    	x_n≡∑_j=1^∞(jβ m)^n K_n(jβ m).

The integrals I_,n neatly decompose into an `uncut' piece I_0,n=(4π n m^2n)^-1, and a `cut' piece involving the quantity x_n. The uncut bubbles I_0,1 will be used to form effective propagators for α.

An effective action for the zero mode of the α field may be found by integrating out the ϕ field in (<ref>),

    _eff = -N(λ^-1+I_,1)α^2 + N∑_j=3^∞(-2)^j-1/jI_,j-1α^j.
 
The effective propagator may be written in terms of a geometric series of cut bubbles,

    -λ/2N1/1+λ I_,1=-2π m^2 Λ/N∑_j=0^∞(-2π m^2Λ/NN x_1/π m^2)^j,    Λ≡λ/λ+4π m^2.

The renormalized coupling Λ is simply equal to 1 in the non-linear sigma model limit.



Now we may present the Feynman rules for large N free energy diagrams:

	
  * α propagators are drawn as dashed lines, each contributing -2π m^2 Λ/N.	
	
  * All diagrams are tree diagrams in terms of the α propagators.
	
  * ϕ loops are drawn as solid lines, and are always taken to be `cut.'
	
  * An uncut α^j vertex is drawn as a solid point, and contributes a factor N(-2)^j-1I_0,j-1/j.
	
  * A cut α^j vertex is drawn as a ϕ loop, and contributes an additional factor of 2^3-jx_j-1/(j-2)!.
	
  * A ϕ tadpole contributes N x_0/π. 
	
  * ϕ bubbles may be inserted in an α propagator, contributing factors of N x_1/(π m^2).


The free energy diagrams up to four ϕ loops are shown in Fig <ref>. Together with the `one-body' contribution (<ref>), they may be expanded up to fourth-order in fugacity to match with the calculation of f_(4) via the gap equation (<ref>).






 §.§ On-shell contributions from the DMB formula



In general, a free energy diagram with n ϕ loops corresponds to the n-body forward scattering amplitude where each loop is cut. Diagrams that only involve ϕ tadpoles are straightforward to match to the DMB formula, as in Sec <ref>. However, a bubble insertion in an α propagator will correspond to a single internal propagator that goes on-shell, much like the three-body part of the bubble diagram in Sec <ref>. And a ϕ loop vertex with p attached α propagators will involve p-1 internal propagators going on-shell.

These are all what we have been referring to as on-shell contributions, and they are not specific to the large N theory. From the previous section it is clear that a forward scattering diagram with j-1 internal propagators going on-shell should involve a factor of x_j-1, which is a sum of Bessel functions K_j-1. So the problem of dealing with on-shell contributions in the DMB approach amounts to deriving this sum of Bessel functions from the DMB formula.

To do this it is helpful to use another expression for x_s, which is derived in Appendix <ref>,

    x_s = ∫dk/2Em^2s/E^s∑_l=0^s (s+l)!/(2E)^l(s-l)!l!(-d/dE)^s-ln(E).

In particular, the simplest case is

    x_1 = ∫ dk m^2/2E^2(-d/dE+1/E)n(E),

which is clearly involved in the three-body part of the bubble diagram (<ref>). We will first sketch how the bubble diagram is derived from the DMB formula, and then move on to the general case.

For simplicity we will ignore particle exchange corrections in the discussion that follows. This amounts to truncating the summation in x_s to the first term and replacing n(E) by e^-β E. For example (<ref>) becomes

    β m 	K_1(β m)=∫ dk m^2/2E^2(β+1/E)e^-β E.




  §.§.§ DMB calculation of the bubble diagram


Given the second-order DMB formula (<ref>), the bubble diagram will involve the amplitude ⟨α| (G_0-G̅_0) V 𝒫 G_0 V|α⟩. The three-particle state is |α⟩=a^†(p_1)a^†(p_2)a^†(p_3)|0⟩ and the interaction V=m^2g^2/4!∫ d^1xϕ(x)^4. This involves the free field operator

    ϕ(x)=∫dk/2π1/√(2E_k)(a^†(k)e^ikx+a(k)e^-ikx).

This is essentially an old-fashioned perturbation theory calculation, and the rightmost copy of V can act on the state α in two different ways leading to an intermediate state with either three or five particles, see Fig <ref>. The diagram with a three-particle intermediate state is the source of the forward scattering divergence, and it involves a propagator with the same energy difference E-E_α that appears in the would-be delta function G_0-G̅_0.



As noted already in <cit.>, this is equivalent to a derivative of the energy delta function

    (G_0-G̅_0)𝒫G_0    = -2iϵ/(E-E_α)^2+ϵ^2E-E_α/(E-E_α)^2+ϵ^2=-1/2E(G_0-G̅_0).

The E derivative will produce a factor of β by acting on e^-β E after integration by parts, so effectively the propagator is replaced by 
    𝒫G_0→ -1/2β.


On the other hand, in the diagram with a five-particle intermediate state, all of the combinatorics end up being the same but the propagator never goes on-shell so we can safely set E→ E_1+E_2+E_3,

    𝒫G_0→𝒫1/E-(E_1+E_2+3E_3)= -1/21/E_3.

This is how the factor β+1/E_3 in (<ref>) arises from the DMB perspective.



  §.§.§ Higher-order on-shell contributions



The basic large N free energy diagram involving a single cut α^p vertex and p tadpoles is shown in Fig <ref>. It corresponds to a forward scattering amplitude among p+1 particles, where p-1 internal propagators all go on-shell with momentum k'. Following the Feynman rules in Sec <ref>, the free energy f_α^p associated to this diagram is

    f_α^p=-Nm^2/π(-2Λ)^p/p!x_0^p x_p-1.





This may be calculated in old-fashioned perturbation theory as in the previous subsection. The α propagators lead to an effective interaction V= Λπ m^2/N∫ d^1x(ϕ(x)^2)^2 which may be used to calculate T̂ in the DMB formula. For any given diagram with p interaction vertices, the field operators may be contracted in p! different orderings, which involve different energies of intermediate states.

 The most singular contribution is due to the monotonic ordering such that each intermediate state has the same energy as the external particles E_α. Each propagator G_0 evaluates to (E-E_α+iϵ)^-1. This means that for the purposes of calculating this contribution we can treat G_0 like a c-number in the DMB formula,

    logŜ =log(	1+	(G_0-G̅_0)V/1-G_0 V) = log(1-G̅_0 V)-log(1-G_0 V).

Now the logarithms may be expanded up to pth order in V,

    (logŜ)^(p)= V^p1/p(G_0^p-Ĝ_0^p)=V^p(-1)^p-1/p!d^p-1/dE^p-1(G_0-G̅_0).


So the DMB formula for the most singular ordering leads to a (p-1)th derivative of an energy delta function. This will correspond to the l=0 term in the expansion (<ref>) for x_p-1. The other orderings which are less singular will lead to lower-order derivatives of a delta function, and these will correspond to the l>0 terms in the expansion. This is shown in detail for p=4 in Appendix <ref>.

In the remainder of this section we will continue to focus on the most singular ordering, and show agreement with the expression (<ref>) for the free energy f_α^p. After integrating by parts, the (p-1)th derivative leads to a factor of β^p-1, and the DMB formula reduces to

    f_α^p=  1/LN^p+1/(p+1)!∫(∏_α^p+1dk_α/(2π)2E_α)e^-β∑_α E_α⟨ V^p⟩_α^p(-1)^p-1/p!β^p-1.

⟨ V^p⟩_α^p indicates the factors arising from the contraction of V^p as in the forward scattering diagram in Fig <ref>. This factor is the same for all orderings,

    ⟨ V^p⟩_α^p=(8×Λπ m^2/N)^p(p+1)!/(2E')^p-1L.

There are p vertex factors from the effective interaction V, each introducing a combinatorial factor of 8. There are (p+1)! permutations of the large N indices of the particles, a factor of L from the momentum conservation delta function, and factors of (2E')^-1 from contracting the fields in the p-1 internal propagators.

In total

    f_α^p=-Nm^2/π(-2Λ)^p/p!(∫ dk/2E e^-β E)^p ∫dk'/2E'm^2(p-1)/(E')^p-1β^p-1e^-β E'

which indeed agrees with (<ref>) and the l=0 term of (<ref>) at lowest order in fugacity.

	

§ THE TBA AND BOUND STATES

	Although our motivation is coming from more general non-integrable theories like the ϕ^4 theory, it is useful to consider a perturbative expansion of the TBA in order to have an extra check on the validity of the DMB approach. In Sec <ref> we will consider the TBA for the attractive Lieb-Liniger model and show that the perturbative expansion is consistent for both signs of the coupling. The collinear contributions to the free energy cancel delicately with the contributions coming from the bound states that appear for attractive coupling. In Sec <ref> the DMB approach is used to calculate the density of states in the two-particle sector of the attractive LL model, and it is shown how the two-body bound state arises.
	 
	
	

 §.§ Perturbative expansion of the TBA

	
	We have already compared the DMB formula and the TBA at second order in fugacity back in Sec <ref>. Now to consider the details of three-body forward scattering contributions it is useful to instead expand the TBA to second order in the coupling but to all orders in fugacity. 
	
	The two-body and three-body parts of the free energy density can be distinguished by the number of appearances of the kernel. The two-body part (<ref>) has already been suggested from the fugacity expansion,
	
    f_2^(2)=+T∫dp/2π1/e^β E-1[log(1-e^-β E)⋆ K^(2)].

The second-order three-body part is
	
    f_3^(2)   = -T∫dp/2π1/e^β E-1(1/2(log(1-e^-β E)⋆ K^(1))^2/1-e^-β E-[1/e^β E-1(log(1-e^-β E)⋆ K^(1))]⋆ K^(1))   =+T/2β∫dθ/2π1/e^β E-1(log(1-e^-β E)⋆ K^(1))^2.

	The second, simpler form of f_3 is valid for SG but not LL.
	

  §.§.§ Perturbative expansion of the kernel

		Some care is required in defining the perturbative expansion of the kernel. Recall that the kernel for both SG and LL is proportional to 
    K_LL =1/πc/p^2+c^2≃c/π p^2∑_j=0^∞(-c^2/p^2)^j.

	
	This perturbative expansion is valid for p≠ 0, but it fails at p=0 and so can not be used for calculating the collinear contributions, such as the melon diagram. The divergence of the K^(1) term at p=0 must be regularized, and there will also be additional terms involving derivatives of delta functions for even powers of c.
	
		A valid perturbative expansion of the kernel that takes the behavior near p=0 into account may be found by simply Fourier transforming before expanding. This is easy to see for the LL kernel,
	
    K_LL(p)=∫du/2πe^-c|u|e^-ipu=∑_j=0^∞(-1)^jc^j/j!∫du/2π|u|^je^-ipu.

	Even powers of j are just even derivatives of delta functions. Odd powers of j involve odd derivatives of the inverse Fourier transform of sgn u, which may be understood as the principal value of 1/p,
	
    𝒫1/p=p/p^2+ϵ^2=π i∫du/2πsgn u e^-ϵ|u|e^-ipu.

	So this gives a definite regularization at p=0 for the inverse powers of p^2 in the naive expansion of the kernel (<ref>).
	
	The situation is qualitatively the same for the SG kernel.
	
    K_SG(θ)=1/2π2(c/m)coshθ/sinh^2θ+(c/m)^2=∫dν/2πe^-γ|ν|+e^-(π-γ)|ν|/1+e^-π|ν|e^-iνθ,

	where γ≡arcsin (c/m). This may be expanded to c^2,
	
    e^-γ|ν|+e^-(π-γ)|ν|/1+e^-π|ν|=1-ctanhπν/2(ν/m)+c^2/2(ν/m)^2+c^3.

	Recall that c= m g^2/8-m g^4/64π+g^6, 	so the total second-order SG kernel is,
	
    K_SG^(2)=-g^2/8πK_SG^(1)-g^4/128δ^”(θ).

	
	When this is substituted in (<ref>), the first term in K^(2)_SG leads to the two-body part of the bubble diagram (<ref>) and the second term leads to the two-body part of the melon diagram (<ref>).
	
	

  §.§.§ The bubble and melon from the TBA

	
	The three-body contribution f_3^(2) involves the first-order kernel
	
    K^(1)_SG= -g^2/8πθ𝒫1/sinhθ.

	The principal value regularization appears since the function π itanh (πν/2) in (<ref>) may be understood as the Fourier transform of  𝒫(sinhθ)^-1.[The exact Fourier transform for finite ϵ is sinhθ/sinh^2θ +ϵ^2=	π i ∫e^-δν-e^-(π-δ)ν/(1+e^-πν)coshδe^-iνθ with δ≡arcsinϵ.]
	
First let us naively calculate f_3^(2) neglecting the details of the principal value regularization.  We calculate (<ref>), using the notation p_12≡ msinh (θ_1-θ_2),

    f_3^(2)   ≃T/2(mg^2/4)^2β∫dθ_1dθ_2dθ_3/(2π)^3(e^β E_1-1)^-11/p_12p_13∂^2/∂θ_2∂θ_3log(1-e^-β E_2)log(1-e^-β E_3)      =T/2(mg^2/4)^2β∫dθ_1dθ_2dθ_3/(2π)^3n_1n_2n_3p_2 p_3/p_12p_13β^2.

Now if the momentum factor is symmetrized it can be shown to reduce to 1,

    p_2 p_3/p_12p_13+p_3 p_1/p_23p_21+p_1 p_2/p_31p_32=1,


    f_3^(2)   ≃T/3!(mg^2/4)^2β∫dθ_1dθ_2dθ_3/(2π)^3n_1n_2n_3β^2.

The β derivative acting on β^2 produces the free energy due to the sextic vertex diagram (<ref>), and the β derivative acting on one of the occupation numbers produces the free energy due to the bubble diagram (<ref>).


	The melon diagram is due to the details of the principal value prescription. Using identities like 𝒫1/p_12=1/p_12+iϵ+iπδ(p_12), all momentum denominators may be adjusted so that they have the same sign of iϵ. The regularized version of (<ref>) is
	
    p_2 p_3𝒫1/p_12𝒫1/p_13+p_3 p_1𝒫1/p_23𝒫1/p_21+p_1 p_2𝒫1/p_31𝒫1/p_32       =p_12p_13p_23+ϵ/(p_12+iϵ)(p_13+iϵ)(p_23+iϵ)+2π ip_1p_2δ(p_13)𝒫1/p_12+p_1^2/4(2π)^2δ(p_12)δ(p_13).

	The first term reduces to 1, and leads to (<ref>) as before. The second term is odd under the transformation p_1, p_2 → -p_1, -p_2 and thus vanishes upon integration. It is the third term that leads to the three-body part of the melon diagram (<ref>),
	
    f_melon, 3   = T/3!(mg^2/4)^2β∫dθ/2πn(θ)^3 p^2/4β^2=-1/24m^2g^4/16∫dθ/2πn(θ)^3.

	

	

 §.§ The attractive Lieb-Liniger model

	The perturbative expansion of the TBA raises a minor puzzle. Upon flipping the sign of the coupling c, the kernel K_LL flips sign, and thus the two-body correction f_2 in (<ref>) flips sign. But in the thermal field theory approach, the even orders of perturbation theory do not depend on the sign of c, so how is this consistent?
	
	The resolution of course is that the theory with negative c has bound states, and these must explicitly be taken into account in the formulation of the TBA. The contribution from the bound states should compensate the change in sign of the collinear contributions, so that the odd orders of perturbation theory flip sign and the even orders are invariant.
	
	So far we have been focusing mostly on the sinh-Gordon model, and a change in the sign of c corresponds to considering the sine-Gordon model instead. The TBA for the sine-Gordon model <cit.>  is complicated by the presence of the soliton sector. In the following sections we will focus on the simpler LL model instead (see Appendix <ref> for the LL analogue of calculations in Sec <ref>)
	
The LL model with attractive coupling is not stable since there are bound states with arbitrarily large negative energy. However we may still make sense of the model by truncating at some finite order in the fugacity expansion. The TBA may be constructed using standard methods (see e.g. <cit.>). There is a distinct pseudoenergy ϵ_j for each j-body bound state,
	
    f=-T∑_j j ∫dp/2πlog(1+e^-βϵ_j)
    ϵ_j = E_j-T∑_k log(1+e^-βϵ_k)⋆K_kj,     K_kj≡1/2π i.ulog S_kj(u)|_u=p_k - p_j.

Here E_j is the energy of the j-body bound state, and S_kj is the two-body S-matrix between a k-body bound state and a j-body bound state. Only the lowest order quantities will be needed for our purposes

    E_1=p^2/2m,  	E_2=2p^2/2m-c^2/4m,   E_3=3p^2/2m-c^2/m,
    
    S_11(u)=u+i|c|/u-i|c|,    S_21(u)=S_11(u-ic/2)S_11(u+ic/2).

By expanding (<ref>), the free energy at second-order in fugacity is

    f_(2)=-T∫dp/2π[-1/2e^-2β E_1+2e^-β E_2+e^-β E_1(e^-β E_1⋆ K_11)].

In the following section this will be derived in the DMB approach, but first let us consider the perturbative expansion in c.

At zeroth order in c, this ought to be given by the free boson gas (<ref>),

    f_(2)^(0)=-T∫dp/2π1/2e^-2β E_1.

Indeed E_2^(0)=2E_1 and K_11^(0)(p) =-δ(p), so this works out. Note that even though the theory is not interacting at zeroth order, the bound state contribution is needed to get the appropriate statistics of the free boson gas. The contributions to the free energy (<ref>) due to ϵ_1 and ϵ_2 alone do not have the appropriate statistics.

At second order in c, the free energy is given by the non-relativistic analogue (<ref>) of the two-body part of the melon diagram (<ref>)

    f_(2)^(2)=-T∫dp/2πβ c^2/4me^-2β E_1.

For positive c this is entirely due to the contribution of the kernel K^(2), but for negative c this flips sign

    -T∫dp/2πe^-β E_1(e^-β E_1⋆ K_11^(2))=+T∫dp/2πβ c^2/4me^-2β E_1.

In order to get the appropriate free energy it must cancel with the binding energy of the two-body bound state (e^-β E_2)^(2)= β c^2/4me^-2β E_1. In this sense, the collinear contributions to the free energy have an intimate relation to the contributions due to bound states for negative c.

At higher orders in fugacity similar cancellations occur with the binding energy for higher order bound states. Introducing notation z_j ≡ e^-β E_j, the next-order contribution is

    f_(3)=-T∫dp/2π   [z_1(1/3z_1^2+(z_1(z_1⋆ K_11))⋆ K_11+1/2(z_1⋆ K_11)^2).      + z_1(-z_1(z_1⋆ K_11)-1/2(z_1^2⋆ K_11))     . +3 z_3+2z_2(z_1⋆ K_21)+z_1(z_2⋆ K_21)].

The first line does not depend on the sign of c, and the second line flips sign. The third line involves the two-body and three-body bound states, and it is only present for negative c. Using the expansion of the kernel K_21,
    K_21(p)=-2δ(p)+2K_11^(1)(p)+5/4c^2δ^''(p)+c^3,

it can be shown that the perturbative expansion of f_(3) indeed has the appropriate dependence on the sign of c at each order.
	

 §.§ Bound states in the DMB approach

	Now we will treat the two-particle sector of the attractive LL model from the DMB approach. This is arguably the simplest example involving bound states. The Hamiltonian is just
	
    H =p^2/m+c/mδ(x)+P^2/4m,

where the the relative momentum p=(p_1-p_2)/2 and position x=x_1-x_2 form a canonically conjugate pair, and the total momentum P=p_1+p_2 decouples. For the moment, we will ignore the decoupled P degree of freedom.
	
	Using the notation |x_0⟩ for the state localized at x=0, the delta function operator in the interaction may be written as (c/m)|x_0⟩⟨ x_0|. The T-matrix operator (<ref>) may be explicitly calculated
	
    T̂=c/m-⟨ x_0| G_0 |x_0⟩|x_0⟩⟨ x_0|,

and the trace-log of the S-matrix becomes

    TrlogŜ=log(1+c ⟨ x_0| G_0-G̅_0|x_0 ⟩/m-⟨ x_0|G_0|x_0⟩).

For these calculations, note that  ⟨ x_0|p⟩=1, and thus ∫dp/2π⟨ p| G_0|x_0 ⟩=⟨ x_0| G_0|x_0 ⟩.

The expectation of the propagator is

    ⟨ x_0| G_0|x_0 ⟩ = ∫dp/2π1/E-p^2/m+iϵ = -im/2√(mE^+),

where √(mE^+) is defined to be the square root of mE+iϵ with positive imaginary part. Similarly, ⟨ x_0| G̅_0|x_0 ⟩= im /(2√(mE^-)), where √(mE^-) is defined as the square root of mE-iϵ with negative imaginary part.

So (<ref>) becomes,

    TrlogŜ=log(1-ic/2/√(mE^+)+ic/2(1+√(E^+)/√(E^-))).

In the limit of small ϵ, the factor (1+√(E^+)/√(E^-)) evaluates to 2 for E>0, and vanishes for E<0. However if c is negative, then there is a pole at √(m E^+)=-ic/2, and this can compensate the vanishing of the factor. The argument of the logarithm winds around the complex plane in the vicinity of the pole, and this will lead to a delta function localized at the bound state energy after taking the E derivative in (<ref>),

    1/2π iETrlogŜ= dp/dE1/2πc/p^2+c^2/4θ(E)+δ(E+c^2/4m).


Let us now reinstate the total momentum P degree of freedom in the problem. All appearances of E become E-P^2/4m, and there is an overall factor of L and an additional P integral due to the trace,

    δ Z = L∫ dP[e^-β(P^2/4m-c^2/4m)+∫ dp  e^-β(p^2/m+P^2/4m)1/2π2c/4p^2+c^2].

After changing variables to p̅≡ P/2 in the bound state contribution, to p_1, p_2 in the scattering contribution, and multiplying by -(β L)^-1 to obtain the free energy density, this takes the form of δ f_(2) calculated from the TBA in <ref>.
	

§ DISCUSSION

	
	In this paper we have investigated the DMB formula (<ref>) for some massive scalar field theories in two spacetime dimensions. In particular we focused on integrable theories for which we may compare with the TBA or a large N expansion. For multi-particle scattering it was shown that there are two issues that complicate a naive treatment of the DMB formula. The first issue has to do with the presence of forward scattering divergences. This issue has been discussed from the outset of the DMB formula <cit.>, but we show here how by simply applying the DMB formula consistently the forward scattering divergences can be associated to derivatives of energy delta functions, and how the sum over all vertex time orderings corresponds to an easily calculable loop diagram in the thermal field theory approach.
	
	The second issue is perhaps specific to two spacetime dimensions and has not to my knowledge been previously discussed in the literature. The issue has to do with the fact that the trace in the DMB formula has large contributions from states with collinear momenta. This issue arises already in a calculation of the melon diagram (<ref>) at second order in the ϕ^4 theory. It is not just a difficulty of the DMB approach, and calculations of the melon diagram using thermal field theory or the TBA also involve subtleties at collinear momentum. The lesson is that regions of integration with collinear momenta should also be considered carefully for higher order diagrams, and for more complicated 1+1 dimensional theories.
	
	There have not been many concrete examples of application of the DMB approach, so there are a number of possible directions to build on the initial investigations in this paper. With an eye towards the effective string theory program, clearly it would be illuminating to consider scattering in massless 1+1 dimensional theories. Also, one advantage of the TBA for that program is that higher energy levels in finite volume may be calculated <cit.>, not just the ground state. It would be interesting to consider higher energy levels in the DMB approach as well. Something similar has already been done for the thermal self-energy <cit.>, which gives information on the first excited state in finite volume. Indeed in section <ref> the energy of first excited state M(β) was shown to be given at lowest order by the two-body forward scattering amplitude, as in the work of Lüscher <cit.>, and it is straightforward to extend this to multi-particle scattering by calculating higher order corrections to the thermal self-energy using the rules of Sec <ref>.
	
	We have mostly considered theories with one type of particle in the spectrum which corresponds neatly to the field in the Lagrangian, but of course the possibilities in quantum field theory are much richer. The particles of the free theory may form bound states or may decay into other stable particles when an interaction is included. We have taken a preliminary look at bound states in the Lieb-Liniger model and seen that they may indeed be calculated in the DMB approach. The analysis of Sec <ref> could be further extended to treating the breather bound states in the sine-Gordon model, although that model is more complicated due to the presence of solitons, and relatedly the presence of non-trivial semi-classical saddles in the thermal field theory approach.
	
	Of course we may avoid these problems for sine-Gordon by considering the dual massive Thirring model instead. Treating fermions in the DMB approach is no major difficulty (see e.g. <cit.>), and although the calculation was not included here, indeed the lowest order correction to the free energy may be derived from the massive Thirring two-body forward scattering amplitudes. Even so, it may be enlightening to consider the bosonic description of the theory and investigate how the non-perturbative corrections due to the solitons can possibly arise in the DMB framework.
	
	

§ ACKNOWLEDGEMENTS

	I would like to thank Vladimir Rosenhaus for introducing reference <cit.>, and collaboration on an early version of this project. I would also like to thank Balázs Pozsgay for discussions. 
	This work  is supported in part by NSF grant PHY-2209116  and by the ITS through a Simons grant. 
	

§ MORE ON THE MELON DIAGRAM

		This section deals more with the melon diagram integral
		M_4≡∫ dτ dx D(τ,x)^4,
		where D(τ,x) is the finite temperature propagator (with 0≤τ <β)
			
    D(τ,x)   =1/β∑_n∫dk/2πe^-i(kx+ω_n τ)/ω_n^2+k^2+m^2   =∫dk/2πe^-ikx(e^- E |τ|+e^- E(β- |τ|)/2E(1-e^-β E)).

	
	For a theory with interaction term m^2g^2/4!ϕ^4, the contribution to the free energy density is
	
    f_melon=-1/3m^4g^4/16M_4-counterterms,

	where the counterterms subtract the zero-body and one-body parts.
	
	The full result for M_4 is
		
    M_4 = 7/8m^2ζ(3)/(2π)^3+1/16m^2∫dθ/2π[n(θ)+3n(θ)^2+2n(θ)^3].

	The two-body part was calculated in (<ref>). The three-body part (<ref>) was calculated indirectly via the TBA in Sec <ref>, and will be calculated directly here. 
		
		

 §.§ Zero- and one-body parts

		For completeness we will also present the parts of the integral M_4 that are canceled by counterterms.
		
		At zero temperature (β→∞), the propagator is just given by a Bessel function,
		
    D_∞(τ,x)=1/2πK_0(m√(τ^2+x^2)).

		The `zero-body' part is equivalent to the zero-temperature limit,
		
    M_4, 0=1/m^2(2π)^3∫_0^∞ du u K_0(u)^4 =7/8m^2ζ(3)/(2π)^3.

		
		The one-body part involves the self-energy diagram given by cutting one line of the melon diagram. This self-energy diagram is often referred to as the `sunset' diagram and it is given by
		
    Π_sunset(p)   =
    			∫ d^2x e^ip· xD_∞(x)^3 = 1/(2π)^2∫_0^∞ dr  r J_0(|p|r)(K_0(mr))^3.

		The one-body part of M_4 is then given by the on-shell (p^2=-m^2) value of the sunset diagram,
		
    M_4, 1   =∫dθ/2πn(θ).4Π_sunset|_p^2=-m^2   =∫dθ/2πn(θ)(4/m^2(2π)^2∫_0^∞ du  u  I_0(u)(K_0(u))^3)   =1/16m^2∫dθ/2πn(θ).

		

 §.§ Cutting the melon diagram

		
		For the sake of finding the two- and three-body parts, cutting the melon diagram according to the method of Sec <ref> is not very convenient due to the singularities in ℳ_π. In Sec <ref> we nevertheless considered the forward scattering amplitudes corresponding to cutting a line of the melon diagram, and regularized them by the principal value prescription suggested by the second-order DMB formula (<ref>).
		
		In this subsection we will justify this directly from the integral M_4. This involves integrating the vertices of the diagram over Euclidean time, rather than summing over Matsubara frequencies. There is a generalization of this method <cit.> based on the work of Baym and Sessler <cit.>, but M_4 is particularly simple, and it has been considered in <cit.>.
		
		The propagator (<ref>) may be written
		
    D(τ,x)=∫dk/2πe^ik· x∫dω/2E(δ(ω-E)-δ(ω+E))(e^- ω|τ|/1-e^-βω),

	and a melon diagram M_N with N propagators going between two vertices  may be calculated,
		
    M_N≡∫ dτ dx D(τ,x)^N=∫dk_1/2π…dk_N/2π2πδ(∑_i k_i) M̃_N,
    M̃_N=∫∏_j=1^N( dω_j/2E_jδ(ω_j-E_j)-δ(ω_j+E_j)/1-e^-βω_j)1-e^-β∑_iω_i/∑_iω_i.
 
		
		Let us focus on M̃_4,
		
    M̃_4   =2(∏_j=1^4n_j/2E_j)[e^β∑_iE_i-1/∑_iE_i+4e^β(E_1+E_2+E_3)-e^β E_4/E_1+E_2+E_3-E_4+3e^β(E_1+E_2)-e^β(E_3+E_4)/E_1+E_2-E_3-E_4].

	The final term of M̃_4 is actually analytic, since the numerator vanishes when E_1+E_2=E_3+E_4. Thus it is valid to adjust the denominator by an arbitrary infinitessimal ± iϵ, after which we may exchange integration variables 1,2 and 3,4,
	
    3e^β(E_1+E_2)-e^β(E_3+E_4)/E_1+E_2-E_3-E_4± iϵ→𝒫6 e^β(E_1+E_2)/E_1+E_2-E_3-E_4.
 
	This is ultimately how the principal value prescription (<ref>) arises in the thermal field theory approach.
	
		The exponentials in the numerators may be written as e^β E_j= n_j^-1+1 and so some of the occupation numbers in the product ∏_j n_j will be canceled. The r-body part of the diagram is just taken to be the part of the diagram with r overall occupation number factors. The three-body part is
		
    M̃_4, 3   =(∏_j=1^3n_j/E_j)[1/-E_4^2+(E_1+E_2+E_3)^2+𝒫3/-E_4^2+(E_1+E_2-E_3)^2].

		This obviously agrees with the DMB expression (<ref>).
		
		The two-body part is
		
    M̃_4, 2   =3/4n_1 n_2/E_1 E_2 E_3 E_4[1/∑_iE_i+2/E_3+E_4+E_1-E_2+𝒫1/E_3+E_4-E_1-E_2].

		This also agrees with the DMB expression (<ref>) after we write it in terms of the integral J(p), which may be written in terms of energy denominators
		
    J(p)=∫dk_3/2πdk_4/2π2πδ(k_3+k_4-p^1)/2 E_3  2E_4[1/ip^0+E_3+E_4+1/-ip^0+E_3+E_4].

	
		

 §.§ Direct calculation of the three-body part

		This section builds on the discussion of Sec <ref>, but the iϵ terms are considered carefully.
		
		For brevity introduce notation such as 
    e_12≡ e^θ_1-e^θ_2,    _12≡ 2πδ(e_12),      H_12≡sgn(e_12)+1/2.

		As usual the Heaviside function H_12 is taken to equal 1/2 when θ_1=θ_2.
		
		From (<ref>), the principal value of the scattering amplitude corresponding to the middle diagram on the lower row of Fig <ref> is
		
    𝒫e^θ_1+θ_2+θ_3/m^2 (e^θ_1+e^θ_2)e_13e_23   =e^θ_1+θ_2+θ_3/m^2(e^θ_1+e^θ_2)Re1/(e_13+isgn(e_23)ϵ)(e_23+isgn(e_13)ϵ)   =e^θ_1+θ_2+θ_3/m^2(e^θ_1+e^θ_2)Re[(1/e_13+iϵ+i H_32_13)(1/e_23+iϵ+i H_31_2 3)]   =e^θ_1+θ_2+θ_3/m^2(e^θ_1+e^θ_2)[(1/e_13+iϵ)(1/e_23+iϵ)+1/4_13_2 3+i/2(_13𝒫1/e_23+_23𝒫1/e_13)].

		The last terms involving a single delta function will vanish upon integration since for instance,
		
    e^θ_1+θ_2+θ_3/e^θ_1+e^θ_2_13𝒫1/e_23=e^θ_1+θ_2/e^θ_1+e^θ_22πδ(θ_1-θ_3)𝒫e^-θ_1+θ_2/2/2sinhθ_1-θ_2/2=2πδ(θ_1-θ_3)𝒫1/2sinh(θ_1-θ_2),

		and this is an odd under the interchange of the signs of θ_1, θ_2.
		
		The first term in (<ref>) will be symmetrized in order to cancel in the same way as in (<ref>), but since the signs of the iϵ terms must be adjusted to be the same this will also produce some additional terms,
		
    1/3[1/e^θ_1+e^θ_2(1/e_13+iϵ)(1/e_23+iϵ)+(3↔ 1) + (3↔ 2)]       =1/3[…-1/e^θ_3+e^θ_2i_13(1/e_21+iϵ)+1/e^θ_1+e^θ_3(i_12(1/e_23+iϵ) +i_23(1/e_21+iϵ)-_12_23)]       =1/3[…-1/21/e^θ_3+e^θ_2_13_12],

		where the ellipsis includes both the terms which will cancel as in (<ref>) and the terms with a single delta function that were shown to vanish immediately above.
		
		So in total the non-vanishing part of M_4, 3 in (<ref>) involves a double delta function,
		
    M_4, 3 = ∫dθ_1 dθ_2 dθ_3/(2π)^3n_1^33e^3θ_1/2m^2e^θ_1(1/4-1/6)_13_2 3=2/16m^2∫dθ/2πn(θ)^3.

	
	

§ MORE ON LARGE N DIAGRAMS

	

	

 §.§ An expansion of x_s

Recall that x_s is defined as a sum of Bessel functions,
	
    x_s≡∑_j=1^∞(jζ)^sK_s(jζ),    ζ≡β m.

	There is a formula for x_s in terms of energy derivatives of n which was used in Sec <ref> to relate thermal field theory diagrams to the DMB formula,
	
    x_s = ∫dk/2Em^2s/E^s∑_l=0^s (s+l)!/(2E)^l(s-l)!l!(-d/dE)^s-ln(E).


This formula may be proven inductively. The s=0 case involves a well-known integral expression for Bessel functions

    x_0=∑_j=1^∞ K_0(jζ)=∑_j=1^∞1/2∫ dθ  e^-jζcoshθ=∫dk/2E  n(E).


To prove (<ref>) for x_s+1, first note that[Incidentally, the s=0 case of this formula can be used to make the simplification (<ref>).]

    -ζ^2s+1ζζ^-2sx_s=∑_j=1^∞(jζ)^s+1(s/jζ-(jζ))K_s(jζ) = x_s+1.

The same operator may be written as a derivative of m^2 at fixed β,

    -ζ^2s+1ζζ^-2s=-2 m^2s+2m^2m^-2s.


Now act with this operator on both sides of (<ref>),

    x_s+1   =-m^2s+2∫ dkm^2[1/E^s+1∑_l=0^s (s+l)!/(2E)^l(s-l)!l!(-d/dE)^s-ln(E)].

Inside the integrand, the operator .m^2.=1/2EE for fixed k. It is now a simple exercise to show that the acting with d/dE reproduces the terms of (<ref>) for x_s+1, completing the proof.

Also note that the summation over j involved in x_s is not essential to the proof. If n(E) is expanded like ∑_j=1 e^-jβ E, then the formula holds order by order in j. In particular it is true at the leading order in fugacity, j=1.



 §.§ Calculation of f_α^4


Recall from Sec <ref> that the free energy f_α^p due to p+1 particles scattering in the manner of Fig <ref> is given by (<ref>),

    f_α^p=-Nm^2/π(-2Λ)^p/p!x_0^p x_p-1.


The leading order (l=0) term of this formula was matched to the DMB formula for arbitrary p in Sec <ref>. In this section we will fix p=4, but match the full l expansion in (<ref>) to the DMB formula. We will continue to work at lowest order in fugacity, corresponding to the leading order (j=1) terms in all appearances of x_s.

f_α^4 involves the factor x_3, which is given by (<ref>),

    x_3^(1)=	m^6 ∫dk'/2E'e^-β E'/E'^3(β^3+6β^2/E'+15β/E'^2+15/E'^3).

Since the overall factors in f_α^4 have already been established for the leading order β^3 term in Sec <ref>, it is sufficient here to reproduce the expansion in parentheses.

We can expand logŜ in the DMB formula up to fourth order. Each term will take the schematic form 

    Δ G  V O_3 V O_2 V O_1 V

where Δ G≡ G_0-G̅_0, and the operator O_i at position i represents either a propagator  G_0 or a delta function Δ G. When the trace is taken, the H_0 in leftmost copy of Δ G will evaluate to the energy of the incoming or outgoing state E_α, but the three operators O_1, O_2, O_3 may involve some other intermediate energy which will depend on the `time ordering' in which the four copies of V are contracted (see e.g. Fig <ref> for the p=2 case).


There are 4!=24 total time orderings of the vertices of the forward scattering diagram corresponding to f_α^4.



  * Of these, only the single ordering considered in Sec <ref> will involve all three intermediate energies evaluating to E_α. According to (<ref>) this leads to a term proportional to
	
    -1/4!d^3/dE^3δ(E-E_α).

	
  * There are 3 time orderings where two intermediate energies are on-shell, and the remaining energy is E_α+2E', where E' is the energy of the privileged particle in Fig <ref>. Any delta functions Δ G vanish if they are evaluated at an `off-shell' energy (i.e. an energy not equal to E_α), and the terms in the fourth-order expansion of logŜ that survive are just proportional to a third-order expansion of logŜ with all intermediate energies set to E_α. So using (<ref>) again, these 3 time orderings lead to a term
	
    +3(1/E-E_α-2E')1/3!d^2/dE^2δ(E-E_α).

	
  * Proceeding similarly, there are 7 time orderings with one energy equal to E_α and two equal to E_α + 2E',
	
    -7(1/E-E_α-2E')^21/2!d/dEδ(E-E_α).

	
  * There are 9 time orderings where all energies equal E_α + 2E', and 4 time orderings where two energies equal E_α + 2E', and one energy equals E_α + 4E',
	
    +[9(1/-2E')^3+4(1/-2E')^2(1/-4E')]δ(E-E_α).



All of these terms are added together in the density of states, where they multiply e^-β E in the integrand of (<ref>). After integrating the derivatives of delta functions by parts (being careful to also act on the appearance of E in the off-shell propagator), the sum of the above terms becomes

    -1/4!(β^3+6β^2/E'+15β/(E')^2+15/(E')^3)δ(E-E_α).

which gives the factor in parentheses in (<ref>), which we set out to show.






	
	

§ THE LIEB-LINIGER MODEL

	
	In the text the discussion has mostly focused on diagrams in the relativistic ϕ^4 and sinh-Gordon models, but much the same story occurs for the non-relativistic Lieb-Liniger (LL) model, where the two-body interaction involves a repulsive delta function potential of strength c/m.
	
	We may calculate with a coherent state path integral with Lagrangian
	
    _LL=∑_ka^†_k ( ∂_τ + E_k) a_k +c/2m∑_k+l=m+na^†_m a^†_n a_k a_l,

	where the indices k on the c-number field a_k refer to spatial momentum. No counterterms are needed as long as the a^† fields in the interaction vertex are understood to be at an infinitesimally later time than the a fields. The propagators ⟨ a(τ)a^†(0)⟩^(0) are directed, and at zero-temperature the propagator vanishes for negative τ. This greatly simplifies the perturbation theory since at zero-temperature all non-vanishing diagrams must involve arrows pointing in the same direction.
	
	The diagrams correcting the free energy at c^2 are shown in Fig <ref>. The strict requirements on the propagator at zero-temperature means that the bubble and melon diagrams correspond to far fewer scattering amplitudes than in Fig <ref>.
	
	
	
	
	The free energy density due to the bubble diagram may be calculated straightforwardly in thermal field theory,
	
    f_bubble   =-2β c^2(∫dp/2πn(p)/m)^2∫dq/2πe^-β E_q/(1-e^-β E_q)^2.

	From the DMB perspective this involves only the on-shell divergent three-body scattering amplitude depicted on the lower left in Fig <ref>. The result is a straightforward non-relativistic limit of the relativistic result (<ref>).
	
	Note that the bubble diagram in LL does not have a two-body part. Recall that in the relativistic theory the two-body part of the melon was associated with expanding the first-order in c correction to the free energy density to the order g^4, but in the present case there is no g^2 parameter at all.
	
	As in the relativistic case, the bubble diagram may be found from the three-body part of the TBA (<ref>) if the principal value signs in the first order expansion of the kernel are ignored. The free energy density calculated from the TBA is
	
    f_3^(2)=-2βc^2/m^2∫dp_1dp_2dp_3/(2π)^3e^β E_1n_1^2n_2n_3[p_2 p_3𝒫1/p_12𝒫1/p_13+…_3].

	In this case quantities such as p_12 should be understood as p_1-p_2, and once again the identity (<ref>) holds.
	
	The melon diagram involves a single two-body and a single three-body scattering amplitude depicted on the lower right of Fig <ref>. These can either be calculated directly, using the method of Appendix <ref>, or indirectly from the perturbative expansion of the TBA using (<ref>) and (<ref>). The result is a straightforward non-relativistic limit of the relativistic case (<ref>) and (<ref>),
	
    f_melon   =-c^2/12m∫dp/2π(3n(p)^2+2n(p)^3).
 
	










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