Paper Title: Nanohertz gravitational waves from a quasar-based supermassive black hole binary population model as dark sirens.











Key Laboratory of Cosmology and Astrophysics (Liaoning) & College of Sciences, Northeastern University, Shenyang 110819, China


Key Laboratory of Cosmology and Astrophysics (Liaoning) & College of Sciences, Northeastern University, Shenyang 110819, China


School of Physics and Optoelectronic Engineering, Hainan University, Haikou 570228, China



Key Laboratory of Cosmology and Astrophysics (Liaoning) & College of Sciences, Northeastern University, Shenyang 110819, China


College of Physical Science and Technology, Shenyang Normal University, Shenyang 110034, China
Key Laboratory of Cosmology and Astrophysics (Liaoning) & College of Sciences, Northeastern University, Shenyang 110819, China






Key Laboratory of Cosmology and Astrophysics (Liaoning) & College of Sciences, Northeastern University, Shenyang 110819, China

Corresponding author
zhangxin@mail.neu.edu.cn
Key Laboratory of Cosmology and Astrophysics (Liaoning) & College of Sciences, Northeastern University, Shenyang 110819, China
Key Laboratory of Data Analytics and Optimization for Smart Industry (Ministry of Education), Northeastern University, Shenyang 110819, China
National Frontiers Science Center for Industrial Intelligence and Systems Optimization, Northeastern University, Shenyang 110819, China





§ ABSTRACT 

Recently, several pulsar timing array (PTA) projects have detected evidence of the existence of a stochastic gravitational wave background (SGWB) in the nanohertz frequency band, providing confidence in detecting individual supermassive black hole binaries (SMBHBs) in the future. Nanohertz GWs emitted by inspiraling SMBHBs encode the luminosity distances of SMBHBs. They can serve as dark sirens to explore the cosmic expansion history via a statistical method to obtain the redshift information of GW sources' host galaxies using galaxy catalogs. The theoretical analysis of the dark siren method relies on the modeling of the population of SMBHBs. Using a population model consistent with the latest SGWB observations is essential, as the SGWB provides significant information about the distribution of SMBHBs. In this work, we employ a quasar-based model, which can self-consistently account for the SGWB amplitude, to estimate the population of SMBHBs. We constrain the Hubble constant using the mock GW data from different detection cases of PTAs in the future. 
Our results show that a PTA consisting of 100 pulsars with a white noise level of 20 ns could measure the Hubble constant with a precision close to 1% over a 10-year observation period, and a PTA with 200 pulsars may achieve this goal over a 5-year observation period. The results indicate that modeling the SMBHB population significantly influences the analysis of dark sirens, and SMBHB dark sirens have the potential to be developed as a valuable cosmological probe.




















Nanohertz gravitational waves from a quasar-based supermassive black hole binary population model as dark sirens
    Xin Zhang
    
================================================================================================================





§ INTRODUCTION

The study of cosmology has entered the era of precision cosmology <cit.>. 
Five of the six basic parameters of the ΛCDM model have been precisely constrained by the Planck cosmic microwave background (CMB) data with constraint precisions better than 1% <cit.>.


However, as the precision of cosmological parameter measurements improves, some puzzling tensins have come to light. The Hubble constant (H_0) is a crucial cosmological parameter describing the universe's current expansion rate. Its values inferred from the Planck CMB observation (assuming the ΛCDM model) <cit.> and the model-independent local-universe distance ladder measurement <cit.> are in more than 5σ tension, which has become a critical crisis in cosmology <cit.> (see Refs. <cit.> for related discussions). 
While the Hubble tension indicates the potential for new physics beyond the ΛCDM model, there remains a lack of consensus on a valid extended cosmological model to address this tension. Additionally, it is imperative to explore new cosmological probes capable of independently measuring H_0.


Compact binary coalescence (CBC) events, a category of gravitational wave (GW) sources, have been discovered by the LIGO-Virgo-KAGRA collaboration in recent years, with approximately a hundred events detected.
The luminosity distances of CBC events can be determined through the analysis of GW waveforms, and these events are commonly referred to as “standard sirens” <cit.>. 
Standard sirens can provide absolute distance measurements, avoiding poorly understood calibration processes inherent in the distance ladder method. If the redshifts of GW sources are also determined, the distance-redshift relation could be established to study the expansion history of the universe (see Refs. <cit.> for related discussions).  
The binary neutron star coalescence event, GW170817, gives H_0=70^+12_-8  km s^-1 Mpc^-1 with a precision of 14% <cit.>. Its redshift was determined by directly identifying the electromagnetic (EM) signals produced when the binary merged <cit.>. Such standard sirens are known as bright sirens, with only one case detected to date. 


For the standard sirens without EM counterparts (usually referred to as dark sirens), we can cross-match them with galaxy catalogs and statistically infer their redshifts (see Refs. <cit.> for related works). Recently, using 46 dark sirens with signal-to-noise ratios (SNRs) over 11 from GWTC-3 and the GLADE+ K-band galaxy catalog <cit.>, researchers achieved a ∼19% constraint on H_0. Combined with the bright siren GW170817, the constraint precision of H_0 improves to ∼10% <cit.>. 

The frequencies of GWs generated by the GW sources with varying masses are hugely different, which requires GW detectors sensitive to diverse frequency bands. The current LIGO-Virgo-KAGRA detector network <cit.> can detect only stellar-mass CBCs at around hundreds of hertz. In the future, precise measurements of GWs in various bands will become possible. In the frequency band from a few hertz to several thousand hertz, the third-generation ground-based GW detectors, e.g., the Cosmic Explorer <cit.> and the Einstein Telescope <cit.>, will significantly enhance the detection capabilities for stellar-mass CBCs. In the decihertz band, the Decihertz Interferometer Gravitational-Wave Observatory <cit.> and lunar-based detectors <cit.> could detect intermediate-mass black hole binaries and binary white dwarfs. In the millihertz band, GWs from massive black hole binaries and extreme mass ratio inspirals could be detected by space-borne GW detectors such as the Laser Interferometer Space Antenna <cit.>, Taiji <cit.>, and TianQin <cit.>. In the nanohertz (nHz) band, the pulsar timing arrays (PTAs) could detect continuous GWs from the inspiraling supermassive black hole binaries (SMBHBs) resulting from galaxy mergers.

The pulsars constructing PTAs are millisecond pulsars (MSPs) and have highly stable rotation periods. Their radio signals can be periodically detected by radio telescopes on the Earth with extremely high precision. GWs passing between the MSPs and the Earth will alter the proper distances between them. This leads to a discrepancy between the actual and expected arriving times of radio signals, which is known as the timing residuals. By analyzing the time residuals of PTAs, we can extract information about nHz GWs.
Recently, North American Nanohertz Observatory for Gravitational Waves <cit.>, European PTA + Indian PTA <cit.>, Parkes PTA <cit.>, and Chinese PTA <cit.> have reported substantial evidence for the existence of a stochastic GW background (SGWB) <cit.>. 

Such evidence boosts confidence in making more advances in detecting nHz GWs. For instance, detecting individual SMBHBs has become one of the next important missions of PTA observations, which has been widely discussed in Refs. <cit.>. Although extracting an individual GW signal from the superimposed GW background is challenging, this goal may be achievable with the addition of more advanced radio telescopes, especially in the era of the Square Kilometre Array (SKA).

Recently, Refs. <cit.> indicated that utilizing SMBHBs as standard sirens could help to improve the measurement precisions of cosmological parameters. 
The authors of Ref. <cit.> pointed out that the EM counterparts of SMBHBs can be observed during its inspiraling stage and forecast the constraint on the equation of state of dark energy with 154 SMBHB bright sirens observed by SKA-era PTAs.

The authors of Ref. <cit.> investigated the dominant factors affecting the detection capability of PTAs for SMBHBs and found that the root mean square (RMS) of white noise, σ_t, is more dominant than the number of pulsars in affecting the detections of SMBHBs. They also proposed that detecting nHz GWs from individual SMBHBs has significant implications for exploring the properties of dark energy and precisely determining the Hubble constant.

In the aforementioned papers, the SMBHB standard siren mock data are simulated based on the galactic major merger model proposed by Mingarelli et al. <cit.> (hereafter the M17 model).
Recently, Casey-Clyde et al. <cit.> indicated that most major merger models predicted characteristic amplitudes of SGWB lower than that derived from the NANOGrav 12.5-yr data. In this context, they proposed a quasar-based SMBHB population model (hereafter the C21 model), considering that galaxy mergers could trigger quasars, and SMBHBs arising from galaxy mergers might be linked to quasar populations. This model can self-consistently predict the local number density of SMBHBs given the amplitude of the SGWB.


Based on the C21 model and the common-process signal in the NANOGrav 12.5-yr dataset, the number of SMBHBs in the local universe is roughly five times higher than that predicted by the M17 model. 
It is evident that there is a significant difference in the number of SMBHBs predicted by these two models. Notably, the number of SMBHB dark sirens significantly influences the constraints on cosmological parameters, especially when the localization of dark sirens is poor <cit.>. Nevertheless, previous studies have not forecast the capability of SMBHB dark sirens of constraining cosmological parameters based on the C21 model. Consequently, a crucial question arises: Based on the C21 model that is more consistent with SGWB observations, what role can SMBHB dark sirens actually play in measuring cosmological parameters?

To answer this question, we simulate the mock SMBHB dark siren data based on the C21 quasar-based model and the SGWB signal inferred from the NANOGrav 15-yr dataset <cit.>.
We analyze the ability of SKA-era PTAs to detect the local-universe SMBHBs based on various PTA detection cases, and then combine the mock dark siren data with the 2 Micron All Sky Survey (2MASS) Extended Source Catalog <cit.> to perform cosmological parameter estimations. 
In this work, we only focus on the ability of SMBHB dark sirens to measure H_0. This is because our aim is to investigate how SMBHB dark sirens can contribute to resolving the Hubble tension, and we primarily mock SMBHB dark sirens in the local universe, where the distance-redshift relationship is only sensitive to H_0. Therefore, we use the SMBHBs dark sirens to infer H_0 and fix all other cosmological parameters to the Planck 2018 values. 

This paper is organized as follows. Section <ref> introduces the methodology used in this work. In Section <ref>, we report our constraint results and make relevant discussions. The conclusion is given in Section <ref>. Unless otherwise stated, we adopt the system of units in which G = c = 1 throughout this work.



§ METHODOLOGY




 §.§ Subcatalog of 2MASS galaxy catalog

Massive galaxies are hypothesized to have a higher probability of hosting SMBHBs <cit.>. To select massive galaxies in the local universe, we first estimate their K-band absolute magnitudes by M_K=m_K-5 ×log _10(d_ L)-25-0.11 × A_v, where d_ L represents the luminosity distance to the galaxy. Ref. <cit.> considered the impact of peculiar motions on redshift measurements, allowing for precise calculations of distances for all galaxies in the 2MASS Redshift Survey (2MRS) catalog <cit.>. The apparent magnitude m_K and the extragalactic extinction A_v can be obtained from the 2MRS catalog. We implement a K-band cutoff with M_K≤ -25 for selecting galaxies with stellar mass M_⋆≥ 10^11 M_⊙. 
 


Based on this criterion, we select 13172 galaxies from the 2MRS catalog as potential host galaxies for SMBHBs. These candidate host galaxies constitute a new catalog, referred to as the 2MASS catalog in the following text.


Since the 2MASS catalog is incomplete at higher redshifts, we further consider its completeness.
The completeness fraction, P_com(𝒮), can be expressed as

    P_com(𝒮) ≡N_cat(𝒮)/n̅_gal V_ c(𝒮),

where 𝒮 = 𝒮(z, Ω̂; Δ z, ΔΩ) represents a volume that is a portion of a cone. This cone is defined by the central redshift z with a range from z - Δ z to z + Δ z, and the central angle Ω̂ with a solid angle ΔΩ. 
n̅_gal represents the density of galaxies per comoving volume.
V_ c is the comoving volume of 𝒮, and N_cat is the number of galaxies in the catalog within 𝒮.
We assume that the completeness of the 2MASS catalog is isotropic and count the number of galaxies in each redshift bin. The redshift range from 0 to 0.1 is equally divided into 30 bins. Additionally, we assume that galaxies are uniformly distributed within the comoving volume and consider the 2MASS catalog complete at z < 0.04. Therefore, we can derive the completeness of the 2MASS catalog as a function of redshift using Eq. (<ref>), which is shown in Fig. <ref>.







 §.§ Population models of SMBHBs

The differential comoving number density of SMBHBs can be described as d^3Φ_BHB/d M_BH, 1d z d q, where M_ BH,1 is the mass of the larger SMBHB, and q is the mass ratio defined as q=M_ BHB/M_ BH,1-1 with M_BHB being the total binary mass. q is assumed to follow a log-normal distribution, p(q), centered at log_10q = 0 with a standard deviation of σ_ log_10q = 0.5dex, within the range [0.25, 1.0], similar to the “hiq” model described in Sesana et al. <cit.>.


While the galactic major merger models can effectively estimate the distribution of SMBHBs, these models are based on the assumptions about the evolution processes of the galaxies and black holes, the galaxy merger rates, and the relationship between the galaxy mass and the black hole mass, etc. However, these assumptions vary greatly among different major merger models, leading to significantly different SMBHB number densities in simulations. It should be noted that the SGWB characteristic amplitude predicted by the M17 model was lower than that of NANOGrav 12.5-yr data. This issue must be addressed if the SGWB signal is considered to be contributed by SMBHBs. 
We note that the C21 model <cit.> can self-consistently incorporate the SGWB characteristic amplitude as a parameter to infer the distribution of SMBHBs, and the number of nearby SMBHBs inferred from the NANOGrav 12.5-yr data is more than five times the number estimated by the M17 major merger model. Therefore, we choose the C21 model to estimate the distribution of SMBHBs and compare it with the M17 major merger model <cit.>.




  §.§.§ Major merger model

We convert M_K to the stellar masses for early-type galaxies using the empirical equation, log_10M_⋆ = 10.58-0.44(M_K + 23) <cit.>. The masses of SMBHBs in these galaxies are determined via the M_ BHB-M_ bulge relationship from Ref. <cit.>, where the bulge mass, M_ bulge, is approximated to the stellar mass M_⋆. The mass ratio q of the binary is drawn from a log-normal distribution, p(q), centered at log_10 q = 0 and with the standard deviation of σ_ log_10q = 0.5 dex, within the range [0.25, 1.0]. Then, the probability of the j-th galaxy hosting an SMBHB in the PTA band, p_j, is given by


    p_j(M_⋆, z)=t_ c,j/T_ life∫_μ_⋆≥ 0.25 dμ_⋆d N/ d t(M_⋆, μ_⋆, z') T_ life,

where t_ c, j = (5/256) (π f_ low)^-8/3 (GM_ c/c^3)^-5/3 is the time to coalescence of the SMBHB in the j-th galaxy, with f_ low=1 nHz being the lower limit of the PTA band. T_ life = T_ df + T_ sh is the lifetime of the SMBHB, where T_ df and T_ sh are the dynamical friction timescale <cit.> and the stellar hardening timescale <cit.>, respectively. 
d N / d t(M_⋆, μ_⋆, z') represents the galaxy merger rate derived from the Illustris cosmological simulation project <cit.>, where M_⋆ denotes the stellar mass of the galaxies, μ_⋆ denotes the progenitor stellar mass ratio, and z' denotes the redshift at which the galaxies merge, calculated at T_z with Planck cosmological parameters <cit.>.

The number of SMBHBs in the galaxy catalog is N_ SMBHB = ∑_j p_j for one simulation. We perform numerous simulations to obtain a distribution of N_ SMBHB. From this distribution, we select the simulation that yields the N_ SMBHB value closest to the maximum likelihood. We find that there are 213 sources emitting GWs in the PTA band in the 2MASS catalog. Then, we randomly select 213 galaxies from the 2MASS catalog as the host galaxies of SMBHBs according to the probability p_j.




  §.§.§ Quasar-based model

The characteristic amplitude h_ c of the GW signal generated by the population of merging SMBHBs can be expressed as <cit.>

    h_ c^2(f)=4/π f^2∭   d M_BH, 1d z d q 
       ×d^3Φ_BHB/d M_BH, 1d z d q1/1+zd E_gw/dln f_r,

where d^3Φ_BHB/d M_BH,1 dz dq represents the differential comoving number density of SMBHBs, dE_gw/ dln f_ r is the GW energy emitted per unit logarithmic frequency change, and f_ r = f(1 + z) is the GW frequency in the rest frame of the source. For a population of SMBHBs in circular orbits, the GW energy per unit logarithmic frequency change is <cit.>

    d E_gw/ dln f_r    = (π f_ r)^2 / 3/3M_ c^5 / 3,

where the SMBHB chirp mass is given by M_ c^5/3 = q/(1+q)^2M^5/3_BHB.
Therefore, the differential comoving number density of SMBHBs is the main parameter influencing the value of h_ c. 


Since some quasars could be driven by the mergers of black holes, the populations of quasars and black holes might be correlated. Therefore, the C21 model makes a simplified assumption that the ratio of the differential comoving number density of quasars to that of black holes remains constant:

    d^3Φ_BHB/d M_BH, 1d z d q∝d^2Φ_Q/d M_BHd z p(q).

That is to say, the differential comoving number density of SMBHBs is proportional to the differential comoving number density of quasars, d^2Φ_Q/d M_BH d z, and an assumed mass ratio distribution p(q). The number density of quasars is derived from the quasar luminosity function and Eddington ratio distribution of the lifetime of the quasar from the galaxy merger simulations in Ref. <cit.>. 
To estimate the number density of quasars, we convert the differential quasar triggering rate <cit.>, d^2Φ_Q/d M_BHd t_ r, into a redshift distribution using the ΛCDM model with WMAP9 data,

    d^2Φ_Q/d M_BHd z=d^2Φ_Q/d M_BHd t_ rd t_ r/d z.


According to Eqs. (<ref>) and (<ref>), we can renormalize the SMBHB population with both the SGWB characteristic amplitude and the quasar number density distribution,

    d^3Φ_BHB/d M_BH, 1d z d q=Φ_BHB, 0/Φ_Q, 0d^2Φ_Q/d M_BHd t_ rd t_ r/d zp(q)/∫_q ≥ 0.25d q p(q),

with


    Φ_BHB, 0 =    . ∬_M_BH, 1≥ M_BH, det, 
     q ≥ 0.25d M_BHd q 
    d^3Φ_BHB/d M_BH, 1d z d q|_z=0,



    Φ_Q, 0=.∫_M_BH≥ M_BH, detd M_BHd^2Φ_Q/d M_BHd t_ rd t_ r/d z|_z=0.


The best-fit value of the SGWB characteristic amplitude from the NANOGrav 15-yr data set is h_ c = 2.4 × 10^-15 <cit.>. Assuming that this SGWB signal is contributed by numerous SMBHB inspirals, according to our calculations using Eqs. (<ref>)–(<ref>), there are approximately 688 SMBHBs in the redshift range (0, 0.1). 
Considering the completeness of the 2MASS catalog, we use Eq. (<ref>) to calculate the total completeness fraction of the 688 SMBHBs. This completeness fraction actually refers to the proportion of the host galaxies included in the 2MASS catalog relative to the total host galaxies. We find that it is approximately equal to 490. Therefore, we randomly select 490 galaxies based on their P_ com values from the 2MASS catalog as SMBHBs’ host galaxies. We assume that the host galaxies of the other 198 SMBHBs are not included in the 2MASS catalog and ignore their contributions to the subsequent calculations for dark sirens.







 §.§ GW detection and Fisher analysis


The timing residuals induced by a single GW source located in the direction represented by the unit vector Ω̂ and measured at time t on Earth can be expressed as

    s(t, Ω̂)=F^+(Ω̂) Δ A_+(t)+F^×(Ω̂) Δ A_×(t).

Here, F^+(Ω̂) and F^×(Ω̂) are the antenna pattern functions <cit.>, which can be expressed as

    F^+(Ω̂)=1/4(1-cosθ){(1+sin ^2δ) cos ^2δ_pcos[2(α-α_p)]. 
       .  -sin 2 δsin 2 δ_pcos(α-α_p)+cos ^2δ(2-3 cos ^2δ_p)}, 
       F^×(Ω̂)=1/2(1-cosθ){cosδsin 2 δ_psin(α-α_p). 
       .  -sinδcos ^2δ_psin[2(α-α_p)]},

where (α, δ) and (α_ p, δ_ p) represent the right ascension and declination of the GW source and the pulsar, respectively. θ represents the opening angle between the GW source and the pulsar,

    cosθ=cosδcosδ_pcos(α-α_p)+sinδsinδ_p.

In Eq. (<ref>), Δ A_{+, ×} (t)=A_{+, ×}(t)-A_{+, ×}(t_p) <cit.> represents the difference between the Earth term A_{+, ×}(t) and the pulsar term A_{+, ×}(t_p).
t_p is the time when the GW passes the pulsar and can be expressed as t_p=t-d_p(1-cosθ)/c, with d_p representing the pulsar distance.

For evolving SMBHBs in circular orbits, we have

    A_+(t)=     h(t)/2 π f(t){(1+cos ^2ι) cos 2 ψsin [ϕ(t)]
       +2 cosιsin 2 ψcos[ϕ(t)]},


    A_×(t)=     h(t)/2 π f(t){(1+cos ^2ι) sin 2 ψsin [ϕ(t)] 
       -2 cosιcos 2 ψcos[ϕ(t)]}.

Here, ψ is the GW polarization angle, ϕ is the phase of the GW, ι is the inclination angle of the GW, and the GW strain amplitude h(t) can be formulated as

    h(t) = 2 (G M_c)^5 / 3/c^4(π f(t))^2 / 3/d_L,

where the frequency of GW signals is calculated by the following formula <cit.>,

    f(t) = π^-1(GM_ c/c^3)^-5/8[256/5(t_ c-t)]^-3/8.

Here, t_ c is uniformly sampled from [100 yr, 26 Myr]. This range represents the time to coalescence of a SMBHB with M_ BH,1 = 10^9 M_⊙ and q = 1 from 1 nHz to 100 nHz.
h_0 and f_0 represent the values of h(t) and f(t) when t=0, respectively.
Fig. <ref> shows the distribution of mock SMBHBs on the f_0-h_0, f_0-M_ BHB, z-M_ BHB planes.

The inclination angle, the polarization angle, and the initial phase are randomly sampled in the ranges cosι∈ [-1, 1], ψ∈ [0, 2π], and ϕ_0 ∈ [0, 2π], respectively.
The GW phase, ϕ(t), can be expressed as

    ϕ(t)=ϕ_0 + 1/16(G M_c/c^3)^-5 / 3{(π f_0)^-5 / 3-[π f(t)]^-5 / 3}.

The square of SNR of a GW signal detected by a PTA is defined as

    ρ^2=∑_j=1^N_p∑_i=1^N[s_j(t_i)/σ_t, j]^2,

where N is the total number of the data points of timing residuals for each pulsar, N_ p is the number of pulsars, s_j(t_i) is the timing residual induced by the GW signal in the j-th pulsar at time t_i, and σ_t,j is the RMS of the white noise of the j-th pulsar. The SNR threshold is set to 10 in the simulation.

It should be noted that in this work, we did not consider red noise, nor did we take an unresolved SGWB as red noise into account. We assume that the pulsar noise is primarily dominated by white noise and that individual GW sources can be disentangled from the SGWB through specialized methodologies. 

We utilize the Fisher information matrix for parameter estimations. For a PTA comprising N_ p independent pulsars, the Fisher information matrix can be expressed as

    Γ_a b=∑_j=1^N_ p∑_i=1^N∂ s(t_i)/σ_t, j∂θ_a∂ s(t_i)/σ_t, j∂θ_b,

where θ denotes the free parameters to be estimated.

For each GW source, the timing residuals depend on N_p+8  system parameters, including eight parameters of the GW source (i.e., M_ c, α,δ, ι, ψ, ϕ_0, f_0, d_ L) and the distances of pulsars d_p,j ( j = 1, 2,⋯, N_ p). With additional EM information priors, the estimation precision of parameter can be improved by adding to the appropriate diagonal element of the Fisher information matrix, i.e., Γ_a b→Γ_a b-⟨∂^2ln P(θ_i)/∂θ_a∂θ_b⟩. Here, P(θ_i) is the prior distribution of the parameter θ_i <cit.>. Since the disk direction is randomly distributed in a 4π solid angel, the prior distribution of the inclination angle ι is given by P(ι) ∝sin (ι). Consequently, we can modify the Fisher information matrix element Γ_i i to Γ_i i→Γ_i i+1/sin ^2ι with θ_i = ι.  


The 1σ absolute error of the parameter θ_a can be estimated by 

    Δθ_a=(Γ^-1)_a a^1 / 2.


The angular resolution ΔΩ can be calculated according to the uncertainties of the parameters,

    ΔΩ=2 π√((cosδΔδΔα)^2-(cosδ(Γ^-1)_δα/ΔδΔα)^2).

For the parameter d_ L, we first estimate its instrumental error by Eq. (<ref>). Besides, we also consider the error arising from weak lensing, which is expressed by the fitting formula from Refs. <cit.>,

    Δd_L^lens(z)     = d_L(z) × 0.066[1-(1+z)^-0.25/0.25]^1.8.

The total error of d_ L is expressed as

    Δd_L    = √((Δd_L^ inst)^2+(Δd_L^ lens)^2).

It should be added that we assume the distance to each pulsar has been precisely determined. Under this assumption, the derivatives of GW frequencies can be obtained, and the chirp masses of the SMBHBs can be further measured. This enables us to break the degeneracy between M_ c and d_ L, and therefore precisely infer d_ L from GW amplitudes. 
The distributions of the d_ L relative errors and the angular resolutions of simulated SMBHBs versus SNR based on 20, 100, and 200 MSPs with σ_t = 20 ns and 10-year observation are displayed in Fig. <ref>. The sky locations of MSPs are randomly selected from the Australia Telescope National Facility pulsar catalog <cit.>.








 §.§ Inference of the Hubble constant


In this work, we adopt the flat ΛCDM model to make the cosmological analysis. The Hubble parameter, which quantifies the rate of expansion of the universe at redshift 
z, can be expressed as

    H(z) = H_0√(Ω_ m(1+z)^3+1-Ω_ m),

where H_0 is the Hubble constant and Ω_ m is the current matter density parameter. The luminosity distance at redshift z is given by

    d_ L(z) = c(1+z) ∫_0^z1/H(z^')d z^',

where c is the speed of light in vacuum. Throughout this work, 
we adopt the flat ΛCDM model to generate the dark siren data with the best-fit values of Ω_ m=0.3111 and H_0=67.66  km s^-1 Mpc^-1 given by the Planck 2018 TT, TE, EE+lowE+lensing+BAO data <cit.>. Based on the flat ΛCDM model, we utilize the GW signals from SMBHBs as dark sirens to infer H_0, fixing the other cosmological parameters to the Planck 2018 results.

Using the Bayesian method to infer the cosmological parameters, the posterior probability distribution of H_0 can be expressed as

    p(H_0|𝒟_GW, 𝒟_EM) 
       = ∏_i=1^N_GW p(𝒟_GW, i, 𝒟_EM, i|H_0)p(H_0)/γ(H_0),

where 𝒟_GW and 𝒟_EM represent the detected GW data and the EM data, respectively. p(H_0) is the prior distribution of H_0, and N_GW is the number of GW events. p(𝒟_GW,i, 𝒟_EM,i|H_0) is the likelihood function of a single GW event, which can be written as

    p(𝒟_GW, i, 𝒟_EM, i| H_0) 
       = ∫ p(𝒟_GW, i, 𝒟_EM, i, d_L, α, δ, z | H_0) dd_Ldαdδd z 
       = ∫ p(𝒟_GW, i|d_L, α, δ) p(𝒟_EM, i| z, α, δ) p(d_L| z, H_0) 
           × p_0(z, α, δ| H_0) dd_Ldαdδd z 
       = ∫ p(𝒟_GW, i|d_L, α, δ) p(𝒟_EM, i| z, α, δ) 
           ×δ.(d_L-d̂_L(z, H_0).) p_0(z, α, δ| H_0) d d_Ldαdδd z 
       = ∫ p.(𝒟_GW, i|d̂_ L(z, H_0), α, δ.) p(𝒟_EM, i| z, α, δ) 
           × p_0(z, α, δ| H_0) dαdδd z.


p.(𝒟_GW, i|d̂_L(z, H_0), α, δ.) is the likelihood of the GW data, which can be expressed as

    p.(𝒟_GW, i|d̂_L(z, H_0), α, δ.) ∝ e^-χ^2 / 2,

with χ^2 = (x-x_ gw)^T𝒞^-1(x-x_ gw), where 𝒞 is the 3 × 3 covariance matrix only relevant to (d_L, α, δ), obtained from the Fisher information matrix. x = (d̂_L(z,H_0), α, δ) represents the three-dimensional (3D) position in the sky.
x_ gw=(d_ L,i, α_ i, δ_i) represents the 3D position of the GW source. 

Considering the bias between the true value and the central value of the posterior distribution from the actual observation, we use the Fisher matrix to calculate the Gaussian distribution of x_ gw, and randomly select the value of x_ gw according to this Gaussian distribution.

We establish the boundary of the GW source’s localization volume based on the 3D measurement errors of x_ gw. The galaxies whose positions satisfying χ^2 ≤ 11.34  (corresponding to 99% confidence) can be considered as the potential host galaxies of the GW source. The number of potential host galaxies for a GW source is defined as N_ in, which is depend on the prior values of H_0. We display N_ in as a function of SNR in Fig. <ref>, fixing H_0=67.66  km s^-1  Mpc^-1. Due to the lack of EM signals in dark sirens, p(𝒟_EM,i| z, α, δ) can be set to a constant value <cit.>.

For the potential candidate host galaxies, p_0(z, α, δ| H_0) in Eq. (<ref>) can be expressed as

    p_0(z, α, δ| H_0) =     f_ com p_cat(z, α, δ)+(1-f_ com) 
       × p_miss(z, α, δ| H_0),

where f_ com is the completeness fraction of the catalog, which can be expressed as

    f_ com = N_cat/n̅_gal V_ max,

where N_cat is the number of galaxies in the galaxy catalog, and V_ max is the entire comoving volume bounded by the upper limit of redshift, i.e., z_ max= 0.1. p_cat(z, α, δ) is the observed galaxy distribution function provided by the galaxy catalog, which is expressed as 

    p_cat(z, α, δ)=1/N_in∑_j=1^N_in   𝒩(z_j, σ_z, j) δ(α-α_j)
       ×δ(δ-δ_j),

where 𝒩 represents the Gaussian distribution with mean z_j and the standard deviation σ_z,j = (1+z) √(⟨v_ p^2⟩)/c <cit.>. 
√(⟨v_ p^2⟩) is the standard deviation of the radial peculiar velocity of a galaxy, set to 500  km s^-1 <cit.>.   
And p_ miss is given by 

    p_miss(z, α, δ| H_0) ∝ (1 - p_com(z, α, δ)) d V_ c/d z dαdδ.



The normalization term γ(H_0) in Eq. (<ref>) can be written as 

    γ(H_0)= ∫   p_det^GW(d_L(z, H_0), α, δ) p_det^EM(z, α, δ) 
       × p_0(z, α, δ) d z dαdδ,

where p_det^GW and p_det^EM represent the GW selection effect and the EM selection effect, respectively. The GW selection effect can be expressed as

    p_det^GW(d_L(z, H_0), α, δ)=
       ∫_𝒟_GW>𝒟_GW^th p(𝒟_GW|d_L(z, H_0), α, δ) d𝒟_GW,

where 𝒟^ th_ GW denotes the SNR threshold for GW detections and is set to 10. The EM selection effect can be expressed as p_det^EM(z, α, δ) ∝ℋ(z_max-z), where ℋ is the Heaviside step function and z_max=0.1.




§ RESULTS AND DISCUSSIONS








In this section, we compare the two SMBHB population models, discuss the GW parameter estimations, and present the constraint results for H_0 from various PTA detection cases.

As shown in Fig. <ref>, the h_0, z, M_ BHB, and f_0 distributions of SMBHBs simulated from the two population models are essentially similar. Nevertheless, the numbers of simulated SMBHBs for the two population models are 490 and 213, respectively, showing that the C21 model has a noticeably larger number of SMBHBs than the M17 model. We consider the 10-year observations based on the C21 model as the representative example of PTA observations for the following discussions.











 
In Fig. <ref>, we present the distributions of ρ, ΔΩ, and Δ d_ L/d_ L based on the 20, 100, and 200 MSPs with σ_t = 20 ns. Note that the calculations are based on 213 (M17 model) and 490 (C21 model) simulated SMBHBs at z < 0.1.
As shown in the left panel of Fig. <ref>, fewer available pulsars result in lower SNRs, thus showing worse localization abilities. We note that the localization capability with 100 MSPs is much better than that of 20 MSPs, but 200 MSPs are slightly better than 100 MSPs. This indicates that 100 MSPs seem to be sufficient and more economical, as further increasing the number of pulsars does not significantly enhance the localization ability.









For the mock SMBHBs from the two population models, we consider PTA cases of monitoring 20, 100, and 200 MSPs with σ_t = 20 ns to detect them. The observation time spans are T_ span = 5, 10, 15, 20 years, respectively, with a cadence of 2 weeks that is the typical cadence used in current PTAs. The detected SMBHBs are used as dark sirens to constrain H_0. 


Figure <ref> illustrates the relative errors for H_0 as a function of T_ span. Table <ref> shows the relative error of H_0, defined as ε(H_0)=σ(H_0)/H_0. We observe that the PTAs with N_ p=20 give the poorest measurement precision for H_0. Based on the C21 model, the constraint precision of H_0 is 2.44% for a 5-year observation, improving to 1.71% over a 20-year observation. These values are worse than the constraint precision obtained from current distance ladder measurements. Increasing the number of MSPs to 100 significantly enhances the precision of H_0 compared to using 20 MSPs. With a 5-year observation, the constraint precision of H_0 is 1.33% and can be improved to 0.85% with a 20-year observation; both values are close to or better than 1%. Furthermore, monitoring 200 MSPs provides even better constraints on H_0, achieving precisions ranging from 1.07% to 0.71% over 5 to 20 years of observations.




Figure <ref> presents the constraints on H_0 obtained from individual GW events as well as the combined results. The colored lines represent the constraints from individual events and the black line represents the combined result of all events.

There is a notable discrepancy between the two SMBHB population models in terms of the H_0 posterior distributions given by the individual events. In the M17 model, there are several events whose SNRs far exceed the others, but in the C21 model, there is no significant difference in the SNRs given by all the events. This discrepancy can also be seen in Fig. <ref>: Several SMBHBs (represented by the orange circles) in the M17 model have significantly higher masses and GW strains than the others. This is due to the fact that the mechanisms of the two models in predicting the SMBHB populations are substantially different.
The M17 model predicts the masses of SMBHBs based on the apparent magnitude of the galaxies in the galaxy catalog. 
A few of these galaxies have significantly larger masses, and the SMBHBs inhabiting them could produce the GW
signals with notably higher SNRs than the other SMBHBs. However, for the C21 model, its SMBHB distribution is derived from the SGWB signal, which is the overall superposition of many GW events and does not reflect the information of individual sources. As a result, the SMBHB distribution predicted by the C21 model is statistical rather than individualistic. Therefore, in the C21 model, no event exhibits a SNR that is significantly higher than other events.







Compared with the dark siren analysis in Ref. <cit.>, under the same detection conditions and population models, the constraint precision of H_0 is improved in this paper. This is mainly due to the fact that we simulate SMBHBs over a broader range of redshift (0<z<0.1), resulting in a larger number of GW sources available as dark sirens. We further consider the effect of incompleteness for galaxies at z>0.04.
The authors of Ref. <cit.> also indicated some advantages of nHz GW standard sirens over the standard sirens in other frequency bands. Here, we want to emphasize another advantage: Since SMBHBs are all hosted in supermassive galaxies, we can directly exclude those galaxies with low masses when using the dark siren method, and only need to consider the supermassive galaxies within the localization volume of the GW source. Compared with the stellar-mass dark sirens, it can greatly reduces the computational costs and the uncertainties caused by spurious host galaxies in Bayesian inference.




 










§ CONCLUSION

Several PTA projects have reported substantial evidence for the existence of a SGWB in the nHz band. In addition to SGWB, the GW signals from individual SMBHBs may also be observed in the future. Such individual GW events can be used as dark sirens to constrain cosmological parameters. In the previous studies on dark sirens, the simulations of SMBHBs were based on the M17 model, which cannot fit the observed SGWB signal well. In this paper, we use the C21 model, which can self-consistently predict the local number density of SMBHBs given the amplitude of the SGWB, to investigate the potential of SMBHB dark sirens in measuring the Hubble constant. According to our results, the following conclusions can be drawn:

(i) In the C21 model, a PTA with 100 MSPs over a 10-year observation period could measure H_0 with a precision close to 1%. If the number of MSPs increases from 100 to 200, a 5-year observation could measure H_0 with a precision close to 1%.

(ii) The C21 model predicts a larger number of SMBHBs than the M17 model. The numbers of SMBHBs derived from the M17 and C21 models are 490 and 213, respectively.  



(iii) The SMBHB distribution in the C21 model is derived from the SGWB signal, which is the overall superposition of the GW signals from individual events. Therefore, the mass distribution of SMBHBs predicted by the C21 model is more statistical than those predicted by the M17 model.

Our conclusions indicate that modeling the SMBHB population has a great impact on the analysis of SMBHB dark sirens. In the future, more accurate measurements of the SGWB signal will not only give us a better understanding of the distribution and evolution history of SMBHBs, but also have important implications for measuring cosmological parameters via the dark siren method.






















We thank Shang-Jie Jin for helpful discussions.
This work was supported by the National SKA Program of China (Grants Nos. 2022SKA0110200 and 2022SKA0110203), the National Natural Science Foundation of China (Grants Nos. 11975072, 11875102, 11835009, 12305069, and 12305058), the National 111 Project (Grant No. B16009), the Program of the Education Department of Liaoning Province (Grant No. JYTMS20231695), and the Natural Science Foundation of Hainan Province (Grant No. 4240N215).






 Paper Title: Electromagnetic $|G_E/G_M|$ ratios of hyperons at large timelike $q^2$.







+3pt

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    et al.#1#2#1/#2∫_κ k
        0
             LFTC-24-08/91
            
            ^1Department of Physics and OMEG Institute, Soongsil University, 
            
            Seoul 06978, Republic of Korea 
            ^2LIP, Laboratório de Instrumentação e Física 
            Experimental de Partículas, 
            
            Avenida Professor Gama Pinto, 1649-003 Lisboa, Portugal
            
            ^3Departamento de Física e Departamento de Engenharia e Ciências Nucleares, 
            
            Instituto Superior Técnico (IST),
            Universidade de Lisboa, 
            Avenida Rovisco Pais, 1049-001 Lisboa, Portugal 
            ^4Laboratório de 
            Física Teórica e Computacional – LFTC,
            Universidade Cidade de  São Paulo,  
            01506-000,  São Paulo, SP, Brazil
            0§ ABSTRACT 
            In recent years, it has become possible to
            measure not only the magnitude of the electric (G_E)
            and magnetic (G_M) form factors of spin 1/2 baryons,
            but also to measure the relative phases
             of those quantities in the timelike kinematic region.
             Aiming to interpret present |G_E/G_M| data on hyperons of the baryon octet,
             as well as to predict future data, we present model calculations of
             that ratio for large invariant 4-momentum square q^2 in the timelike region (q^2>0). 
             Without any further parameter fitting, we extend to the timelike region a covariant quark model previously developed
              to describe the kinematic spacelike region (q^2 ≤ 0)
              of the baryon octet form factors.
              The model takes into account
              both the effects of valence quarks and the excitations
            of the meson cloud which dresses the baryons.
             We report here that our results are in good agreement with very recent data from BESIII,
             within the uncertainty limits of our model. We also make 
            new predictions which may be tested in the near future,
             when the range of the present measurements is expanded
             to the 20–50 GeV^2 region.
            
             
              Electromagnetic |G_E/G_M| ratios of hyperons
            at large timelike q^2
                G. Ramalho^1, M. T. Peña^2,3,  
            K. Tsushima^4 and Myung-Ki Cheoun^1
                August 5, 2024
            =======================================================================
            
            
            
            § INTRODUCTION
            
            
            In recent years it became possible to experimentally probe the structure
            of short-lived baryons through
            e^+ e^- → B B̅ and p p̅→ B B̅ experiments in
            the timelike kinematic region above the production threshold
            q^2 ≥  4 M_B^2 (M_B is the mass of the
            baryon) <cit.>.
            
            These timelike studies complement the two last decades knowledge
            on the spacelike region[In this work,  for simplicity,  we include into the spacelike region the photon point q^2 = 0,
            due to the physical continuity between the regions q^2 < 0 and q^2=0.]
            (q^2 ≤ 0) based on results from electron scattering experiments <cit.>.
            
            
            The electromagnetic structure of the baryons is represented
            by spin (J) and parity (P)
            dependent structure form factors.
            In the timelike region these form factors are complex functions of the 
            e^+ e^- or p p̅ square center-of-mass energy 
            s= q^2 <cit.>.
            
            Of particular interest are the spin 1/2 baryons (J^P =1/2^±),
            including the nucleon and hyperons 
            characterized by two form factors only, the electric and magnetic form factors.
            
            We focus here on hyperons because the experimental information
            about the nucleon is at the moment less complete.
            Also, our model uncertainty decreases with increasing q^2,
            and the convergence of the nucleon data to the asymptotic large q^2
            behavior is expected to emerge at much larger q^2 values, due
            the dominance of light quark dynamics.
            
            Given the short lifetime of hyperons, the challenges in the determination
            of their electromagnetic properties
            in the timelike region, where their weak decays can be analyzed
            in detail <cit.>
            are less significant than in the spacelike region.
            
            
            The first experiments measured the total integrated cross section
            of the e^+ e^- → B B̅ reactions for spin 1/2 baryons,
            which determine the magnitude of the  "effective"
            form factor G defined as <cit.>
            |G (q^2)|^2 = 2 τ |G_M (q^2)|^2 + |G_E (q^2)|^2/2 τ + 1,
            
            where τ = q^2/4M_B^2.
            
            These experiments provided information only about
            a combination of the electric and magnetic form factors,
            and did not reveal their relative phases.
            
            The separate determination of |G_E| and |G_M| or of the ratio
            |G_E/G_M| for spin 1/2 baryons in the timelike region 
            can be obtained by further measuring  the differential
            cross section <cit.>, while
            the knowledge of the relative phase between the two form
            factors demands in addition the determination of polarization cross sections
            (which involve complete spin separation) <cit.>.
            Today, indeed it is possible to measure the ratio |G_E/G_M|
            and the relative angle between the form factors
            for a few baryons <cit.>.
            Most measurements to date have been done
            for ground state hyperons, but in the future excited states
            may be accessed. In a first experimental period the proton and the neutron form factors
            were extracted under some assumptions about
            the relation between G_E and G_M <cit.>.
            
            
            Although the initial measurements were performed near threshold,
            it is expected that with increased accelerator power,
            the experimental energy will go higher up to the 20–50 GeV^2
            range. This extension will allow us to test the predictions described in this work.
            
            Measurements of the effective elastic form factors
            of the Λ, Σ^0,±, Ξ^0,-
            as well as the transition form factors Σ^0 →Λ
            have been performed at BaBar <cit.>, CLEO <cit.>,
            Belle <cit.> and BESIII <cit.>.
            Measurements of the ratios  |G_E/G_M| for the Λ and Σ^+
            have also been determined recently at BESIII <cit.>.
            
            Complete experiments, that measure  |G_E|, |G_M|, 
            and the relative phase between G_E and G_M have been performed
            for the Λ and Σ^+ at BESIII <cit.>, 
            and even more measurements are planned for a near future at BESIII
            and PANDA <cit.>.
            
            
            In this work, we enlarge a previous theoretical study of the effective form factors
            of J^P= 1/2^+ hyperons <cit.> to calculate the |G_E/G_M| ratios 
            for large q^2.
            
            Calculations in this momentum transfer region are important,
            given that at very large q^2, |G_M (q^2)| dominates
            the effective form factor |G (q^2)|, allowing us to probe where that regime sets in.
            
            
            The numerical calculations are based on the
            covariant spectator quark model formalism 
            for the spacelike region <cit.>
            extended to the timelike region.
            The model has been used in the study
            of electromagnetic structure of 
            nucleon resonances <cit.>
            and other baryons
            systems <cit.>.
            
            
            
            
            In the case of the nucleon is it possible to
            compare the magnitude of G_E/G_M in the spacelike
            and timelike regions <cit.>.
            
            Timelike data on the nucleon electromagnetic
            form factors can be found
            in Refs. <cit.>
            for the proton and in Refs. <cit.> for the neutron.
            
            Theoretical studies of the nucleon electromagnetic form factors
            in the timelike region can be found in Refs. <cit.>.
            
            At the moment the measurements of the nucleon form factors in
            the timelike region are incomplete, since their relative phases
            have not been measured.
            The data of the proton and neutron effective form factors
            revealed an oscillatory dependence on the center-of-mass
            energy <cit.>.
            Different explanations have been proposed <cit.>.   
            In the present work we do not discuss this subject because
            we are focused on the asymptotic region, where
            the oscillatory component is expected to be suppressed.
            
            
            Beyond the members of the baryon octet, there are also measurements
            of the effective form factors of the Ω^-, Δ
            and Λ_c^+ <cit.>.
            Theoretical works on hyperon form factors
            in the timelike region can be found in 
            Refs. <cit.>.
            
            
            
            
            
            
            § FORMALISM
            
            
            
            Within the one-photon-exchange approximation, <cit.>
            we can write the e^+ e^- → B B̅ differential
            cross section, where B stands for
            a spin 1/2 baryon B( 1/2^±), as
                  d σ_ Born/d Ω (q^2) =
            
                  α^2 β C/4 q^2[ |G_M (q^2)|^2 (1 + cos^2 θ)
                + 1/τ |G_E (q^2)|^2 sin^2 θ],
              
            where θ is the angle between the baryon
            and the initial photon in the center-of-mass frame,
            α≃ 1/137 is the fine structure constant,
            β= √(1 - 4M_B^2/q^2) is a kinematic factor
            (the baryon speed in c=1 units) and 
            C is the Sommerfeld-Gamow factor.
            This  factor takes into account the Coulomb interaction
            and can be written as C = y/1-exp(-y)
            with y= πα/β2 M_B/√(q^2)
            for charged particles, and C=1 for 
            neutral particles <cit.>.
            For charged particles the factor C converges rapidly to C ≃ 1 
            when q^2 increases.
            
            By integrating the previous relation, we obtain
            σ_ Born (q^2) =
            4 πα^2 β C/3 q^2( 1 + 1/2 τ) |G (q^2)|^2,
            
            using Eq. (<ref>).
            We can conclude then that the effective form
            factor can be calculated directly from the integrated cross section.
            However, it does not give individual information on  G_E or G_M.
            
            
            
            To untangle information on the two form factors G_E and G_M
            one uses  the differential cross section (<ref>) by
            determining the coefficients of the angular functions
            1 + cos^2 θ and sin^2 θ <cit.>.
            
            Combining the two results, one can extract the ratio |G_E/G_M|.
            
            The measurement of the phase between the form factors
            requires in addition the measurement of cross section
            with defined polarizations of the baryons <cit.>.
            
            
            
            
            
            
            
            Finally, at threshold, where q^2= 4 M_B^2, the functions
            G_E and G_M are such that G_E = G_M,
            as a consequence of the definition
            of these Sachs from factors in terms of the kinematic independent
            Dirac and Pauli form factors. Therefore one expects that G_E/G_M ≃ 1
            for measurements 
            near threshold, and the non-zero phases between the form factors are to be sought 
            above the threshold  only (q^2 > 4 M_B^2) <cit.>.
            
            
            The main conclusion of this section is that in the case
            of a spin 1/2 baryon, to obtain 
            the absolute values of the two form factors from
            experiments of the  differential cross section
            it is necessary to measure accurately angular distributions
            of the differential cross section
            besides the effective form factor |G(q^2)|.  However, as we will see in the next section the range for non-zero imaginary parts cannot extend to indefinitely large |q^2|.
            
            
            
            
            
            
            
            
            
            
            § CALCULATION OF ELECTROMAGNETIC FORM FACTORS AT LARGE Q^2
            
            
            
            
            In this work the electromagnetic form factors G_E and G_M
            in the timelike region  are obtained from extrapolating the results
            from the region q^2=-Q^2 ≤  0 to the region q^2 > 0.
            
            
            
            
            
            For that purpose we consider the large-|q^2| relations 
            
            G_ℓ (q^2)= G_ℓ (-q^2)
            
            for ℓ=E,M. 
            These asymptotic relations are a consequence of
            two general mathematical principles: 
            unitarity as well as the Phragmén-Lindelöf theorem,
            which is valid for analytic functions <cit.>.
            
            
            We notice that this theorem implies that the imaginary parts of
            the form factors in the timelike region 
            must be negligible for large q^2,
            since the form factors are real in the spacelike region.
            
            
            
            
            
            
            
            
            
            
            Due to the character of this approximation,
            we expect our estimates here to be accurate only for large q^2,
            deviating  from the condition G_E = G_M at the threshold.
            
            Since the relations (<ref>) are valid for
             the large-|q^2| mathematical limit, and we want to make predictions
            to finite q^2, we consider finite corrections to these
            asymptotic relations.
            
            We start noticing that the elastic form factors
            are divided into two regions of  q^2: (- ∞, 0] (spacelike region)
            and  [4M_B^2, + ∞),  with a gap of 4M_B^2 length
            between the photon point (q^2=0) and
            the threshold of the baryon-antibaryon production (q^2 = 4 M_B^2).  
            
            The interval (0,4M_B^2) defines an unphysical region
            where no physical baryons can be detected <cit.>.
            
            
            
            There is then an ambiguity about the center of reflection
            for the asymptotic symmetry relation of Eq. (<ref>).
            
            To take into account the ambiguity of the location of the center of symmetry 
            we replace
            q^2 → q^2 ( 1 - 2M_B^2/q^2) in (<ref>), 
            which introduces finite corrections (-2M_B^2)
            to the limit q^2= + ∞.
            
            Therefore we use <cit.>
            G_ℓ (q^2)= G_ℓ^ SL (q^2 - 2 M_B^2)
            
            for ℓ =E,M. 
            The label SL on the r.h.s. indicates that 
            theG_ℓ^ SL (Q^2) is calculated
            in the spacelike region (Q^2 = -q^2 ≥ 0).
            
            The expression (<ref>) estimates the reflection
            symmetry point underlying (<ref>)
            exactly at the center of the unphysical interval (0,4M_B^2).
            
            
            
            
            In summary, the measured form factors in the timelike region are complex functions
            characterized by their magnitudes, |G_E|, |G_M|, and their different phases.
            
            Our model calculation is based on results obtained in the q^2 ≤ 0
            region and the application of the large-q^2 relation (<ref>).
            This way, by construction, we obtain real numbers for G_E
            and G_M that carry ± signs.
            
            In principle, for very large q^2 the comparison of our results with the
            experimental values for G_E/G_M is possible
            since in that region the relative phase can only have two possibilities,
            ΔΦ =0 (when the form factors have the same sign)
            or  ΔΦ = π (if they have different signs).
            
            
            
            
            For a control of the approximation we explored the 
            band of variation of the form factors with respect to a reasonable variation
            of the symmetry point around this central value:
            when the functions G_ℓ (q^2) are positive (negative), since
            they are expected to monotonically decrease (increase)
            with increasing q^2, we take
            the upper (lower) limit values to be G_ℓ (q^2)= G_ℓ^ SL (q^2 - 4M_B^2)
            and the lower (upper) limit  G_ℓ (q^2)= G_ℓ^ SL (q^2).
            
            This procedure provides a band of variation of the
            finite corrections to the asymptotic symmetry identity.
            
            When q^2 is very large, the variation of the results becomes
            very narrow and the three estimates (central, upper and lower limits)
            converge to the same value, providing accurate predictions
            that can be tested by experimental data. Moreover, the comparison with data
            in that region provides information of the onset of the regime of
            the asymptotic symmetry (<ref>).
            
            
            
            
            
            
            
            
            § NUMERICAL RESULTS FOR |G_E/G_M|
            
            
            
            In this section we present our results of the ratio |G_E/G_M| for large q^2 values.
            The calculation method of the form factors G_E and G_M
            for the baryon octet family in the timelike region is discussed
            in detail in Ref. <cit.> which uses the
            covariant spectator quark model formalism
            for the spacelike region <cit.>. 
            In its nutshell, we use impulse approximation for the electromagnetic
            quark current, which is justified for large Q^2, since then
            the interacting photon has sufficient resolution to interact with a single quark at a time.
            As a result we can integrate over the relative internal variables
            of the spectator quarks  and therefore we end up with the
            baryon three-quark system regarded 
            as a quark-diquark configuration.
            In the model, the SU(3) symmetry quark flavor symmetry
            is broken at the level of the baryon radial wave functions
            and the constituent quark current <cit.>,
            and gluon and quark-antiquark dressing
            are effectively taken into the constituent quark structure.
            
            
            The different baryon radial wave functions are parametrized
            by two parameters, a short range scale and a long range scale,
            consistent with the expected size of the systems
            (baryons with strange quarks are more
            compact than baryons with light quarks) <cit.>.
            The quark current is parametrized by
            vector meson dominance regulated by
            light (ρ, ω) and intermediate (ϕ)
            vector meson mass poles <cit.>.
            
            
            
            
            
            
            In particular, we use the parameterization
            of the baryon octet from Refs. <cit.>.
            To the valence quark contributions
            we add also an effective description of meson cloud
            dressing processes <cit.>,    
            which are inevitably present in the low-Q^2 region.
            
            In the large-|Q^2| region, the meson cloud contributions are suppressed
            and their effects are manifest on the rescaling of
            the bare contributions to the form factors due to
            the normalization of the hyperon wave functions <cit.>.
            
            This formalism has been used to successfully describe the spacelike electromagnetic
            structure data of baryons including the low-lying nucleon resonances <cit.>.
            The free parameters of the model are fixed by the
            data for the nucleon,
            baryon octet and baryon decuplet <cit.>.
            
            
            Our calculations of the ratio |G_E/G_M| and their comparison with the data
            are shown in Figs. <ref> and <ref>,
            that depict the results for Λ, Ξ^-,0 and  the Σ^0,±, respectively.
            In the graphs for the Λ and Σ^+ we
            include the available data for |G_E/G_M|.
            How we estimate the uncertainty of the results of our model for each form factor
            was defined in the previous section.
            
            The bands of variation for the magnitude of G_E/G_M are determined
            from the independent band intervals of variation for the G_E and G_M functions.
            
            
            
            
            
            The results and their uncertainty bands near the threshold provide only 
            qualitative estimates since our results are expected to be valid only for large q^2,
            as discussed in the previous section.
            
            The deviation from the result  |G_E/G_M|=1 at threshold
            is a consequence of the approximation used.
            
            The signs of our results for the form factors in the asymptotic
            large q^2 values are indicated on the
            right side in Figs. <ref> and <ref>. 
            
            
            
            
            The separated calculations of the form factors G_E and G_M
            for Λ, Σ^0,±  and Ξ^-,0 are presented
            in the Supplementary Material <cit.>.
            
            Note that we are not attempting to predict the zeros
            of G_E/G_M, rather, we are focused on determining its magnitude at sufficiently large 
            q^2.
            In the current formalism, the zeros can only be determined within an uncertainty band of width 
             4 M_B^2 for q^2.  
            
            
            
            We discuss now how we can take into account
            the information of the relative angle
            between the form factors in the analysis of the ratio |G_E/G_M|.
            
            In complete experiments
            the ratio  G_E/G_M is determined,
            and can be written in the form
            G_E/G_M = |G_E|/|G_M| e^i ΔΦ,
            
            where |G_ℓ| is the magnitude of
            the complex form factor G_ℓ and ΔΦ
            is the phase between G_E and G_M.
            
            Once the phase is measured,
            we can extract the real part of the ratio from
            |  Re(G_E/G_M) |=
            | G_E/G_M| |cos (ΔΦ)|,
            
            which is a fraction of the measured value to |G_E/G_M|.
            This does not only show that model calculations
            of |G_E/G_M| based on real form factors 
            agree with the absolute value of the real part of G_E/G_M
            when the imaginary parts are negligible.
            
            The result (<ref>) allows also us to infer from the comparison between the
            experimental data for the r.h.s. and our model results
            for the ratio |G_E|/|G_M| (where G_E and G_M are real functions)
            if the regime where the imaginary parts of
            the form factors vanish has been attained. 
            
            The comparison of our model results for |G_E/G_M|
            with the experimental
            values for | Re(G_E/G_M)| from Refs. <cit.>
            that measure phases, are displayed in Fig. <ref>.
            
            In the graphs we notice the agreement of our results for the form factors
            (considering the model uncertainty  band) and the experimental
            data point at the largest q^2 point measured,
             q^2 ≃ 14.3 GeV^2 for the Λ
            and q^2 ≃ 8.4 GeV^2 for the Σ^+.
            
            In the graphs we ignore the uncertainties
            of the angles ΔΦ, for simplicity, and used
            their central value.
            
            The consideration of the uncertainties on ΔΦ
            only increases the error bars of the | Re(G_E/G_M)| data.
            Hopefully, future experiments will be more precise,
            which will help to determine the point were our
            model calculations provide a good description of the data
            and determine the large q^2 region where reflection symmetry sets in.
            
            
            
            
            
            
            § EFFECTIVE FORM FACTORS 
            
            
            
            In a previous work <cit.> we presented predictions
            for the effective form factors (<ref>) for
            the baryon octet members Λ, Σ^0,± and Ξ^0,-
            for q^2=10–60 GeV^2.
            At the time, only Λ, Σ^+, Σ^-, Ξ^0 and Ξ^-
            could be tested in the region of 12–15 GeV^2 by the data from CLEO <cit.>.
            
            Our estimates were on average consistent with the high q^2 data. 
            
            Since then more data for |G(q^2)| became available.
            
            In particular data for Σ^+ and Σ^- became available from BESIII <cit.>
            for the range 5.7–9.1 GeV^2.
            Surprisingly, the data are in excellent agreement with our estimates 
            (namely with the central line values, see Supplementary Material <cit.>)
            even outside the region where model validity was expected to be justified.
            
            A rough estimate of the lower limit for the application
            of our model based on the asymptotic
            relations, can be q^2 ≈ 8 M_B^2  or  q^2 ≈ 10 GeV^2.
            
            
            Recently, data from BESIII became available for
            Λ up to q^2= 21 GeV^2 <cit.>,
            and for Ξ^- up to q^2= 23 GeV^2 <cit.>.
            
            The new data are compatible with the CLEO data and extend the previous
            range covered by the experiments.
            The comparison of the new data with the model results
            from Ref. <cit.> is presented in
            Figs. <ref> and <ref> in a linear scale and in the logarithm scale for |G(q^2)|.
            
            It is clear from the figures that there is an excellent agreement between
            our predictions and the data, indicating that
            the range of the asymptotic form for the effective form factors
            may not be too far from the present day experimentally studied region.
            Future data for other hyperons may confirm or deny this tendency.
            
            
            
            
            
            
            Calculations of the effective form factors
            for all the hyperons of the baryon octet are presented in Supplementary Material <cit.>
            and compared with the world data.
            
            
            
            
            
            
            
            
            
            
            § OUTLOOK AND CONCLUSIONS 
            
            
            Experiments in new facilities (BaBar, CLEO, Belle, PANDA
            etc.) allow us to access
            the electromagnetic structure of hyperons in the elastic
            timelike region (q^2 ≥ 4 M_B^2).
            
            Till recently we had access only to
            the integrated cross section which can be used
            to determine the effective form factor |G|,
            and encode only the information
            of a certain combination of the electric and magnetic
            form factors |G_E| and |G_M|, 
            for spin 1/2 hyperons.
            
            No separated information about  |G_E| and |G_M|
            was possible till a few years ago.
            Even the separation between |G_E| and |G_M|
            for the proton and neutron was difficult
            and affected by significant errors.
            
            In the last few years it became possible
            to determine  |G_E| and |G_M| and the relative phase
            between them for Λ and Σ^+.
            
            We expect that the ratio |G_E/G_M| became
            accessible in the following years for most
            ground state hyperons (Σ^0,- and Ξ^0,-),
            as well as for charmed baryons (starting with Λ_c^+).
            
            Our calculations show a partial 
            agreement with the data in the range
            q^2 ≃ 14 GeV^2 (√(s)≃ 3.8 GeV),
            which may be an indication that our model
            calculations may be compared with future measurements
            in the range of q^2 =20–30 GeV^2
            (√(s)= 4.5–5.5 GeV)  or higher.
            
            Also the separation between the real and imaginary
            components may be tested in that range of energies.
            
            
            
            
            
              
            G.R. was supported the Basic Science Research Program 
            through the National Research Foundation of Korea (NRF)
            funded by the Ministry of Education  (Grant No. NRF–2021R1A6A1A03043957).
            M.T.P was supported by the Portuguese Science Foundation FCT
            under project CERN/FIS-PAR/0023/2021, and FCT computing project 2021.09667.
            K.T. was supported by Conselho Nacional de Desenvolvimento 
            Científico e Tecnológico (CNPq, Brazil), Processes No. 313063/2018-4 
            and No. 426150/2018-0, and FAPESP Process No. 2019/00763-0
            and No. 2023/07313-6, and his work was also part of the projects,
            Instituto Nacional de Ciência e 
            Tecnologia - Nuclear Physics and Applications 
            (INCT-FNA), Brazil, Process No. 464898/2014-5.
            The work of M.K.C. was supported by the National
            Research Foundation of Korea
            (Grant Nos. NRF–2021R1A6A1A03043957 and NRF–2020R1A2C3006177).
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            § FORM FACTORS G_E AND G_M
            
            
            The absolute values of the form factors G_E and G_M
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            The signs of G_E and G_M can be inferred from
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            § EFFECTIVE FORM FACTORS G
            
            
            
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 Paper Title: Neutron Star Collapse From Accretion: a Probe of Massive Dark Matter Particles.







liuning@njnu.edu.cn
Department of Physics and Institute of Theoretical Physics, Nanjing Normal University, Nanjing, 230021, China



arvindm215@gmail.com
Department of Physics and Institute of Theoretical Physics, Nanjing Normal University, Nanjing, 230021, China


		

	

§ ABSTRACT 


We explore the multi-scatter capturing of the massive dark matter (DM) particle inside the neutron star via a momentum-dependent dark matter-nucleon scattering cross-section. We find that the capturing enhanced for the positive velocity and momentum transfer dependent DM-nucleon scattering in comparison with the constant cross-section case. Further, a large capture of the DM particles can be thermalized and lead to black hole formation and, therefore, destroy the neutron star. Using the observation of the old neutron star in the DM-dominated region, we obtain strong constraints on massive DM parameters.  

		

		
		
		Neutron Star Collapse From Accretion: a Probe of Massive Dark Matter Particles
    
	Arvind Kumar Mishra 
	
	https://orcid.org/0000-0001-8158-6602
    < g r a p h i c s > 

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



	


§ INTRODUCTION


The various observations suggest that dark matter (DM) exists from its gravitational interaction with the baryons  <cit.>). The ongoing DM searches experiment assumes some form of the DM-baryon interactions; however, irrespective of current technological progress, the micro-physical properties of the DM particles remain an enigma <cit.>. 
The direct DM-detection experiments are sensitive for 𝒪(1-10^4) GeV DM masses. However, if the DM particles are massive, i.e., m_χ> 10^6 GeV, or lighter, m_χ< GeV, then during DM-nucleon collision, the energy transfer to the nucleus becomes small; therefore the detection experiment becomes insensitive to these masses.


In this work, we aim to explore the constraints on the massive dark matter, m_χ> 10^6 GeV, from the neutron star observations. The neutron star provides a lucrative place to study the DM-nucleon interaction, thanks to their strong gravity and density, see a recent review <cit.>. NS situated in the DM-dominated regions can be used to investigate the constraints in two ways: First, after capturing, DM particles become thermalized in the NS (via DM-nucleon scattering or DM-annihilation to the visible particles), which leads to heating of the NS and, therefore, observation of isolated of NS with high temperature can constraint the DM parameters <cit.>. Second, after thermalization, the DM particles can form a self-gravitating core, which can collapse, form a black hole, and ultimately destroy the NS; therefore, observation of the old neutron star in a DM-dominated region (central region of galaxies) can used to probe the DM microphysics <cit.>. However, most of the previous work has been done using a constant cross-section, without form factor, and the rest frame of the DM halo. Furthermore, incorporating the above effects into account, we obtained a strong constraint on the low-mass DM particles from the survival of NS <cit.>.


The constraint on the massive DM has been explored from heating for both a single and multiscatter capture formalism <cit.>. In previous works, <cit.>, the single scatter capture formalism has been applied to put constraints on the massive DM particles from the survival of NS, specifically using constant DM-nucleon cross-section without form factor. However, for actual estimation, one needs to consider the multiscatter capture formalism for massive DM particles <cit.>.  For other methods to explore massive DM particles, please see Refs <cit.>. 


In this work, we explore the constraint on the massive DM particles from the neutron star survival. To explore the capture of massive DM particles on the neutron star, we adopt from the multiscatter capture approach. Here, we focus on the velocity-dependent and momentum-transfer dependent DM-nucleon cross-section, which, to our knowledge, was not explored for massive DM particles in the context of black hole formation (see, for low mass DM constraint in our previous work <cit.>). In addition, we have also considered the general relativistic effect, the finite size effect of the nucleon, and the relative motion of the neutron star and galaxy frame in the capture rate. 


Further, considering the power law parametrization of momentum and velocity dependent DM-nucleon cross-section, we found that the capture rate is enhanced with a positive power of velocity and momentum. We obtained that pulsar data provide the strongest constraint on DM parameter space (both for bosonic and fermionic DM case) for the model σ∝ v^2_rel, where v_rel is the relative velocity of DM and nucleon.
Using this model, pulsar observation strongly constrain bosonic DM mass range, 10^6 GeV≤ m_χ≤ 6× 10^8 GeV, and fermionic DM masses, 10^6 GeV≤ m_χ≤  10^10 GeV. We also report that although the DM accretion is higher for σ∝ v^4_rel model, a large thermalization time puts weak constraints on the DM parameters. Interestingly, a cross-section model with a large momentum dependency, i.e., σ∝ q^4 (q represent the momentum transfer), provides no constraint on the DM parameters. 


The arrangement of our work is as follows: In section <ref>, we present a brief discussion on the DM particle capture rate using a multiscatter capture process. In Section <ref>, we discuss the black hole formation from DM accretion. Further, we present DM-Nucleon cross-section and estimate the captured particles <ref>. In Section <ref>, we present and discuss our main results. Finally, in Section <ref>, we conclude our work.



§ MULTISCATTER CAPTURE OF DM PARTICLES ON NEUTRON STAR 



We assume that the neutron star is situated in a dark matter-rich environment and explore the capture of the massive DM particles to the neutron star. In the case of massive DM particles, an energy transfer via a single interaction between the DM particles and nucleons will be insufficient to capture the DM particles. Thus, a multiple scattering will be required <cit.>. Inside the NS, the number of times by which DM scatter with the nucleons can be obtained as the ratio of the typical diameter of the NS, 2R_NS with the mean free path of the DM particles, λ_MFP, given by  N∼ 2R_NS/λ_MFP, where R_NS is neutron star radius. Assuming λ_MFP∼ 1/(n_Nσ), one can obtain, N∼ 2n_Nσ R_NS, where σ is DM-nucleon scattering cross-section, and n_N is the nucleon density <cit.>. 


Further, the probability of DM particles, p_N(τ) participating in N collisions with an optical depth, τ  can be defined via Poisson distribution (τ,N) and given by <cit.>

    p_N(τ)=2∫^1_0 y e^-yτ(yτ)^N/N! dy  ,

where 
 y is related to the incidence angle subtended by DM particles while entering the star.
The optical depth is defined as τ=3σ/2σ_sat, where σ is the DM-nucleon cross-section and σ_sat is the saturation cross-section allowed during the scattering between the DM-nucleon interaction. In the context of NS, a saturation limit is dictated via geometrical cross-section, given by  σ_geo=π R^2_NS/N_B where, R_NS is the radius of NS, and N_B is total number of the baryons, respectively. For NS, considering, R_NS=10.6 km and N_B=1.7× 10^57, we obtain σ_sat=2× 10^-45 cm^2, which we assumed to remains a constant with DM mass    [Although we have assumed the saturation limit to be a constant quantity throughout the DM mass range, in general, it depends on the DM mass, see Refs. <cit.>. Therefore our estimation are somewhat conservative.].





In the multiscatter capture process, the capture rate for N scatter, C_N is given by <cit.> [The derivation has been done  is the assumption of multi-capture with v_esc (r)≃ v_esc (R_NS) and number density of the baryons are constant inside the neutron star, n_B(r)=n_B see Ref. <cit.>]

    C_N=  π R^2_NS p_N(τ)∫^∞_v_escf(u)/u^2  w^3 g_N(w) dw  .

where g_N(w) corresponds to the probability of DM particles reducing their velocity lower than the escape velocity, v_esc after N collisions, and f(u) is the DM velocity distribution function.  In Eq. (<ref>), u is the DM velocity at very large distance, and w=√(u^2+v^2_esc). 
Therefore, the total capture rate after the N collision is obtained as

    C_total=∑^∞_N=1 C_N .

However, summation can be cut off at some value N_max corresponding to condition, p_N_max(τ)≈ 0. In our calculation, 
we approximate, g_N(w)=Θ[v_esc(1-⟨ z_i⟩β_+/2)^-N/2-w], where, β_+=4m_N m_χ/(m_N+m_χ)^2 in which, m_χ and m_N, represents the mass of the DM particles and nucleons, respectively <cit.>. Here we assume, ⟨ z_i⟩=1/2, <cit.> however for more general estimation of g_N(w), see Ref. <cit.>. 
Furthermore, the DM particle velocity distribution function is considered to be Maxwellian, given in the neutron star frame as <cit.> 

    f_0(u) = (3/2)^3/24/√(π)ρ_χ/m_χu^2/u^3_0e^-3/2u^2/u^2_0 ,

 where ρ_χ, and u_0 represent energy density, mass, and average velocity of the DM particles, respectively.


Furthermore, as on the surface of the neutron star, the DM velocity is high, so during the interaction, DM imparts large energy to the nucleons, see Ref.<cit.>. Therefore, in this analysis, we consider the finite-size effect of the nucleon during the DM-Nucleon interaction. Here, we adopt a conservative approach and assume the dipole form factor for DM-nucleon interaction (considered in the next subsection), given by <cit.>

    F(q^2)=Λ^4/(q^2+Λ^2)^2

where q≃√(2)(m_χm_N/m_χ+m_N)v_esc is momentum transfer during the DM-nucleon collision and m_N is the nucleon mass. Further, we assume Λ≃ 0.25 GeV throughout our analysis <cit.>. Furthermore, in our computation, we use the general relativistic (GR) correction, which modifies the capture rate and escapes velocity. So, using the GR correction, the capture rate and escape velocity is modifies via <cit.>

    C_N→C_N/(1-v^2_esc)  and    v_esc→√(2[ 1-(1-v^2_esc)^1/2]).

 





§ BLACK HOLE FORMATION FROM DM ACCRETION



After the accretion, the DM particles start interacting with the neutron star material and may get thermalized. In the context of the multi-capture process, the interaction of the DM particles with the neutron star is complex because, at each step, the DM particle transfers less momentum, and hence, the estimation of the thermalization time is complex.  Now, we consider the simplified approach and assume the form of the thermalization time scale as <cit.>

    t_th≃   10700μ/(1+μ)^2(10^5K/T_NS)^2(10^-45cm^2/σ)   years.

where, μ=m_χ/m_N, and σ is DM-nucleon scattering cross-section. When the thermalization time scale is smaller than the lifetime of the NS (t_th<t_NS), DM particles will be thermalized inside the NS and lie inside the  thermal radius given by 

    r_th=
    ( 9T_NS/8π G_Nρ_B m_χ)^1/2  .

where G_N is the Newton's gravitational constant,  and ρ_B is the baryon energy density. For a given density and temperature of the NS, r_th∝√(1/ m_χ), which implies that the thermal radius is smaller for massive DM particles and vice versa. 
Further, DM particles start self-gravitating whenever

    ρ_χ>ρ_B  for    r ≤ r_th.

Therefore, with the help of the above condition, we define the DM particles dictated via N_self, necessary for the self-gravitation of the DM particles inside the NS as

    N_self≃  4.8× 10^41( 1.4× 10^14gm/cm^3/ρ_B)^1/2( 100 GeV/m_χ)^5/2( T_NS/10^5K)^3/2  .

So the condition for DM self-gravitation suggest, N_χ≥ N_Self.



Furthermore, when the number of DM particles becomes higher than those defined by the Chandrashekhar limit, then self-gravitating DM particles collapse and form a black hole. In the case of fermionic dark matter (FDM) particles, gravitational collapse requires gravity to surpass the fermi pressure. For non-interacting fermionic DM particles, the Chandrashekhar  limit is obtained as <cit.>

    N^Fermions_Chandra≃ 1.8× 10^51( 100 GeV/m_χ)^3 .

From the above equation, it is important to emphasize that the massive DM particles form low-mass BH and vice versa.
In this case, the BH formation occurs when DM particles start self-gravitating, i.e., N_χ≥ N_Self, and captured DM particles are higher than the Chandrashekhar limit,i.e.,N_χ>N^Fermions_Chandra. The mass of formed BH is obtained as M_BH∼ m_χN^Fermions_Chandra, and given by

    M_BH≃    1.8× 10^53( 100 GeV/m_χ)^2GeV.

From the above equation, it is important to emphasize that the massive DM particles form low-mass BH and vice versa. 
Furthermore, to form a BH in the bosonic dark matter (BDM) particle case, gravity must overcome pressure originating from the uncertainty relation. For non-interacting bosons, the Chandrashekhar limit is given by <cit.>

    N^Bosons_Chandra≃ 1.5× 10^34( 100 GeV/m_χ)^2 .
 
In bosonic DM case, BH forms as soon as DM starts self-gravitating, 

    N_χ≥ N_Self,  for  m_χ≤ 10^21( ρ_B/1.4× 10^14gm/cm^3)  ( 10^5K/T_NS)^3 GeV  .

In this case, the mass of formed BH is obtained as M_BH∼ m_χN_Self, and explicitly given by

    M_BH≃ 4.8× 10^43( 1.4× 10^14gm/cm^3/ρ_B)^1/2( 100 GeV/m_χ)^1/2( T_NS/10^5K)^3/2  GeV.

From the above equation, we find that the massive DM particles and low neutron star temperature form low-mass BH and vice versa. 


After its formation, BH starts growing via accreting the surrounding material and also reducing its mass via Hawking evaporation. The BH mass evolves  as <cit.>

    dM_BH/dt≃ 4πλ_s( G_NM_BH/v^2_s)^2ρ_B v_s+(dM_BH/dt)_DM -1/1560π G^2_N M^2_BH  ,

where λ_s and v_s are the accretion eigenvalue and sound speed. In the r.h.s, the first and second term corresponds to the accretion rate obtained from baryon (Bondi-Hoyle accretion rate) and DM particles, and the last term represents Hawking evaporation. Assuming the DM accretion contribution is small, Eq. (<ref>) suggests an initial critical BH mass given as M^crit_BH=1.2× 10^37 GeV. Whenever formed BH with initial mass,M^ini_BH is larger than the critical mass, i.e., M^ini_BH>M^crit_BH, BH will grow via accretion and destroy the host NS. However, in the opposite case, i.e., M^ini_BH<M^crit_BH, black hole will evaporate and decrease its mass with the time. 


Furthermore, the value of the critical mass of the formed BH can be applied to estimate an upper limit on the mass of DM particles, which can be probed from the BH formation in the core of NS. The initial formed BH mass formed by fermionic DM particles,M_BH∼ m_χN^Fermions_Chandra, and bosonic DM particles via M_BH∼ m_χN_Self. Therefore assuming that BH grows via accreting the NS materials, we obtain the m_χ<2.5× 10^8 GeV (for T_NS=10^7 Kelvin) for BDM particles, and m_χ<10^10 GeV for the FDM particles. We emphasize that these constraints are obtained using the assumption that formed BH is growing via accreting the NS material and not considered DM accretion. Therefore, when DM accretion is taken into account (or NS temperature is changed for the BDM case), the constraint will be modified.    


 

§ DM ACCRETION FROM NON-CONSTANT DM-NEUCLEON ELASTIC SCATTERING

 

In this work, we explore the DM-Nucleon interaction, which is dependent on relative velocity and momentum transfer during the collision <cit.>, which are investigated in contest of the DM capturing on sun <cit.>, white dwarfs <cit.> and neutron stars <cit.>. 
 In this work, we parameterize the relative velocity-dependent cross-section (VDCS) as a power law form, given by <cit.>

    σ^vd_χN(v_rel)=σ_χN(v_rel/u_0)^2α≃σ_χN(w/u_0)^2α,

where, v_rel is the DM-nucleon relative velocity, and u_0 represent the normalized velocity. Here α depends on the DM models, and we specifically focus on the positive velocity dependence cases, i.e., α=1, and 2.  This kind of interaction is called p-wave (α=1), and d-wave (α=2) can be originated for the cases when initial state particles have relative angular momentum of 1 or 2 units, see Refs. <cit.>,<cit.>.  Here we consider the typical value, u_0=220 km/sec <cit.>. 
  

Further, we parameterize the momentum-dependent cross-section (MDCS) in the power law form as<cit.>

    σ^md_χN(q)=σ_χN(q/q_0)^2β,

where q is the momentum transfer and  q_0 is a normalized momentum. The momentum dependent cross-section can originate from various mechanisms such as dipole interaction between the DM and baryons, parity violating DM-baryon interaction, see Refs. <cit.>. Further, for calculation purposes, we assume, q_0=40 MeV <cit.>). 



Having been equipped with the form of the DM-baryon interaction, we will now calculate the probability and DM capture rate using the multicapture process.  This can be obtained by using Eq.(<ref>) and Eq.(<ref>). We have estimated the multiplescatter capture rate using constant, MDCS and VDCS in Appendices <ref>,<ref>, and <ref>, respectively. Below, we will briefly summarize the  results:


Capture rate in constant cross-section case: For constant cross-section case, the expression of probability is obtained after using Eq.(<ref>) as

    P^cont_N(τ_0)=2/τ^2_0 N![Γ(2+N)-Γ(2+N,τ_0)]

where, τ_0=3σ_χN F(q^2)/2σ_sat, and Γ(a,b) represent the incomplete Gamma function. Therefore using the above probability, C_N can be obtained as
 
    C^cont_N(τ_0)=   R^2/1-v^2_escn_χ/6u_0P^cont_N(τ_0)f(v_N,η)Θ(v_N-v_esc)

where, v_N=v_esc(1-β_+/2)^-N/2 is the DM velocity after N scattering <cit.>. Further, the total capture rate after N collision is given by C^cont_tot(τ_0)=∑^∞_N=1C^cont_N(τ_0). In this case, an analytic expression of the total capture rate is obtained using various analytic approximations, see Refs. <cit.>. Further, to calculate the total collected DM particles, we considered DM particles to be asymmetric (neglecting DM annihilation). In this assumption, the number of collected DM particles by neutron star throughout its lifetime is given by N_χ(t_NS)=C^cont_tot(τ_0) t_NS. Here, we will not apply the analytical approximations and calculate N_χ(t_NS) numerically.  


Capture rate in MDCS case: For MDCS, the optical depth is given by τ_md=τ_0(q/q_0)^2β. So, the probability after N collision is obtained as

    P^MDCS_N(τ_0,q)=2/τ^2_0 N!( q/q_0)^-4β[Γ(2+N)-Γ(2+N,τ_0 ( q/q_0)^2β)] .

The form of C_N in the MDCS case is given by 

    C^MDCS_N(τ_0,q)= R^2/1-v^2_escn_χ/6u_0P^MDCS_N(τ_0,q)f(v_N,η)Θ(v_N-v_esc)

  Further, the total capture rate after N collision is given by C^MDCS_tot(τ_0,q)=∑^∞_N=1C^MDCS_N(τ_0,q). Further, assuming the DM particles to be asymmetric, the total captured DM particles in the MDCS case is given by N_χ(t_NS)=C^MDCS_tot(τ_0,q) t_NS. In this case, we will calculate N_χ(t_NS) numerically.  

Capture rate in VDCS case: For VDCS, the optical depth is given by τ_vd=τ_0(w/u_0)^2α. So, the probability after N collision is obtained as

    P^VDCS_N(τ_0,w)=2/τ^2_0 N!( w/u_0)^-4α[Γ(2+N)-Γ(2+N,τ_0 ( w/u_0)^2α)] .

Using the above probability after simplification (see appendix <ref>), we obtain an analytic expression for the capture rate as
  
    C^VDCS_N(τ_0,w_0)= R^2/1-v^2_escn_χ/6u_0P^VDCS_N(τ_0,w_0)f(v_N,η)Θ(v_N-v_esc)

In this case, the total capture rate after N collision is obtained as C^VDCS_tot(τ_0,w_0)=∑^∞_N=1C^VDCS_N(τ_0,w_0). Further, in the assumption that the DM particles are asymmetric, the total captured DM particles in this case is given by N_χ(t_NS)=C^VDCS_tot(τ_0,w_0) t_NS. In this case, we will calculate N_χ(t_NS) numerically. 






Furthermore, the estimation of total accreted DM particles using the general expression of capture rate and probability is tedious. Therefore, we try to restrict our calculation up to a certain accuracy as follows:  We calculate the total sum up to N=N_max, where N_max≤τ <cit.>. Further, we use a simplified expression of probability in some extreme cases of optical depth. For a single scatter (τ≪ 1) and multiscatter (τ≫ 1), the probability is approximated as  <cit.>

P_N (τ)≈ 
2τ^N/N!(N+2)+O(τ^N+1),   if   τ≪1

2/τ^2(N+1) Θ(τ-N),   if   τ≫1 

Therefore, using the capture rate expression estimated above, we calculate the accreted DM particles for constant, MDCS, and VDCS cases numerically.





§ RESULTS AND DISCUSSIONS



In this Section, we will present our results. For calculation purposes, we assume the NS parameters as M_NS=1.44 M_⊙, R_NS=10.6 km, N_B=1.7× 10^57, and v_esc=1.8× 10^5 km/sec <cit.>, otherwise specified explicitly. 



 §.§ Effect of DM-Nucleon interactions on the multiscatter probability


In Fig. <ref>, we plot the probability of the DM multiscatter capture as a function of a number of scattering, N, (left panel) and cross-section (right panel). It can be seen that the probability decreases as the number of scattering N increases and becomes zero for some value of N, say N_max, i.e., p_N_max(τ)≈ 0. Further, fixing the DM parameters, m_χ=10^6 GeV,ρ_χ=0.3GeV/cm^3 and N_max takes values 1 and 2, and 3 corresponding to σ_χN=10^-47cm^2, σ_χN=10^-46cm^2, and σ_χN=10^-45cm^2, respectively. 
 Also, increments in the cross-section lead to an enhanced probability and, therefore, enlarge N_max. 

Further, the right panel of Fig <ref> suggests that
depending on the N, the probability becomes max at some σ_crit and vanishes on low DM-Nucleon scattering cross-section, σ_min such that σ_min<σ_crit . Nevertheless, increasing the number of scattering will shift the peak probability position and σ_min on a comparably higher cross-section.  Furthermore, we also find that increasing the DM masses decreases the probability.






In Fig. <ref>, we plot the probability of the DM capture using MDCS, σ∝ q^2β (Left panel) and VDCS, σ∝ v^2α_rel (Right panel) cases.  From the Left panel of Fig. <ref>, it can be seen that the probability decreases as the number of scattering N increases and becomes zero for some value N_max as discussed previously. However, β=2 shows an interesting feature that probability increases first and decreases sharply after a large scattering. For our parametrization and fixing the other DM parameters, m_χ,ρ_χ and σ_χN, N_max takes values 6,30 and 1000 corresponding to β= 0,1, and 2, respectively.   Further, in the VDCS case (Left panel of Fig. <ref>), DM capturing inside the NS requires more scattering, N_max. For example, keeping DM parameters fixed, m_χ=10^6 GeV, ρ_χ=0.3GeV/cm^3 and σ_χN=10^-45cm^2, DM capturing in α=2 case required ∼ 10^9 scattering. Further, for constant cross-section, the probability decreases promptly, leading to a smaller N_max value in comparison with the MDCS and VDCS cases. It is important to emphasize that the value of N_max depends on the DM parameters; therefore, changing it will modify the N_max value.






 §.§ Accreted DM particles via multi-capture process

 Fig. <ref> shows the accreted DM particles via multi-scatter capture for MDCS (Left panel), and VDCS (Right panel) cases. We found that for both cases, the number of the accreted DM particles, N_χ, enhanced with the DM scattering, but decreased with the DM mass. For the MDCS case, N_χ increases with the positive power of cross-section, β>0 and number of scattering, N. Here we see that for β=1, N_χ is same for N=1 and N=10. This is because for such a small DM-neutron scattering and also parameters used in the calculation (see Left panel of Fig. <ref>), DM capturing happens through single scattering, and therefore  N_χ remains constant for N>1. For the β=2 case, DM capturing becomes possible via multiscattering, and therefore, N_χ increases with N until it gets saturated at N_max=1000.  Furthermore, in the VDCS case (Right panel of Fig. <ref>) and for α=1, the number of captured DM particles is equal for N=1 and N=1000 (same reason as previously). For the strong VDCS case, α=2, DM particles get captured via multiscattering, and therefore, N_χ increases till its saturation at N_max=5000. From  Fig. <ref>, we also point out that for positive MDCS and VDCS cases, the number of captured DM particles is higher than the constant cross-section case.








 §.§ Thermalization time scale


To estimate a DM thermalization time for non-constant DM-nucleon interaction, we assume as follows: In VDCS and MDCS case, we replace σ via Eq.(<ref>), and Eq.(<ref>), and the relative velocity is assumed to be the thermal velocity inside the NS, i.e., v_rel≃ v_th=√(3T_NS/m_χ). With this simplification, we plot the thermalization time scale for our various parameterizations in Fig. <ref>. We find that for constant and σ∝ v^2_rel, σ∝ q^2, t_th is remains constant but increases linearly with mass, i.e., t_th∝ m_χ. This behavior is evident from Eq.(<ref>), and Eq.(<ref>). It can be seen that for a few cases, i.e. 
σ∝ v^2_rel,σ∝ v^4_rel and σ∝ q^2, DM quickly thermalized inside the NS, and therefore, these cases are interesting in the context of the BH formation. However, for the σ∝ q^4 case, the thermalization scale is quite large; therefore, DM does not thermalize in the typical lifetime of the NS. 






 §.§ Constraining massive DM from NS collapse





Now, we will constrain the DM-nucleon parameter space represented via σ_χN -m_χ using the survival of the neutron star situated in the DM-dominated region. We consider the cold and old neutron stars whose distance from the galaxy center, central temperature, and age are estimated accurately. For this purpose, we consider pulsar J2124-3858 situated at a distance 270 pc with surface temperature, T_e<4.6× 10^5K (central temperature, T_c=2.5× 10^7K [The central temperature can be estimated using the surface data from the analytic formula discussed in Ref. <cit.> .]), t_age∼ 7.81×10^9 years. As this pulsar is close to the sun, we may approximate u_s≃ u_⊙= 230 km/sec <cit.>, and ρ_χ=0.3GeV  cm^-3, where u_⊙ represent the sun's speed relative to galactic halo. 



Fig <ref> shows a constraint on the DM-nucleon scattering cross-section for a constant cross-section using the pulsar data J2124-3858. Here, orange and red colors are excluded from the BH formation via bosonic and fermionic DM particles. The olive and brown color regions represent the BH evaporation regions for bosonic and fermionic DM particles, and in these regions, the constraint is relaxed. Further, sky-blue regions are excluded via non-thermalization of the captured DM particles, i.e., t_th>t_NS. The black dashed line is the saturation limit of the DM-Nucleon scattering cross-section, and above this line, the constraints are not valid.   

 
From Fig <ref>, we see that for BDM, the number of the accreted DM particles is high enough to form the BH across all the DM mass ranges. The higher DM particles form low mass BH, so for sufficiently large DM masses, formed BH masses become smaller than the critical mass, i.e., M^ini_BH>M^crit_BH, so BH start evaporating. Due to evaporation, constraints are relaxed; hence, the collapse of the neutron star may not constrain quite massive DM particles, i.e., m_χ>6× 10^8 GeV for bosonic DM. 
Therefore, most of the DM parameter space constraints are relaxed from the evaporation constraint. The BH formation from BDM  constrains very limited parameter spaces 10^-45 cm^2≤σ_χN≤ 4× 10^-46 cm^2 of BDM masses 5× 10^7 GeV≤ m_χ≤ 5× 10^8 GeV. Further, we find that for FDM particles, BH formation happens on high DM masses, i.e. m_χ>10^9 GeV. This is expected because overcoming the degeneracy pressure of the FDM requires more mass. Furthermore, BH formation in FDM does not constrain the DM microphysics as all the parameter space is relaxed from the saturation limit and evaporation constraints.       


 Fig. <ref> shows the constraint on the DM-nucleon scattering cross-section for the MDCS, case from the pulsar observation. The Left and Right panel corresponds to σ∝ q^2 (N_max=10) and  σ∝ q^4 (N_max=1000) cases, respectively. In this case, the DM accretion is enhanced (in comparison to the constant cross-section); therefore, the pulsar data constrain the massive DM particles. For σ∝ q^2, the pulsar data constraint  on cross-section 10^-48 cm^2≤σ_χN≤ 10^-45 cm^2 for BDM masses 10^6 GeV≤ m_χ≤ 6× 10^8 GeV and 
 2× 10^-47 cm^2≤σ_χN≤ 10^-45 cm^2 for FDM masses 10^8 GeV≤ m_χ≤ 10^10 GeV. 
 Although the pulsar data also constrains more massive DM particles, those parameter spaces are relaxed by Hawking evaporation. Furthermore, for σ∝ q^4 case, DM accretion is even higher than  σ∝ q^2 case.
 However, in this model, the thermalization time is higher than the lifetime of the pulsar, which makes the BH formation highly unlikely, and therefore, this model does not constrain the DM parameters.  






Furthermore, we constrain the velocity dependent DM-Nucleon scattering cross-section from the pulsar observation in Fig. <ref>. The Left and Right panel corresponds to σ∝ v^2_rel (N_max=1000) and  σ∝ v^4_rel (N_max=5000) cases, respectively.  For σ∝ v^2_rel, the captured DM particles are higher than the constant case; therefore, it constrains the DM parameter space more strongly. This model constrain 10^-52 cm^2≤σ_χN≤ 10^-50 cm^2 for BDM masses, 10^6 GeV≤ m_χ≤ 6× 10^8 GeV, and 10^-50 cm^2≤σ_χN≤ 2× 10^-47 cm^2 for FDM masses, 10^6 GeV≤ m_χ≤  10^10 GeV. Furthermore, due to large positive power, the number of accreted DM particles is highest for model σ∝ v^4_rel (among all models discussed previously).  Therefore we expect that this model will put the tightest constraint. However, as seen in Fig. <ref>, the thermalization constraint strongly restricts the DM parameter space and limits the cross-section, 10^-50 cm^2≤σ_χN≤ 10^-47 cm^2 for the BDM masses, 10^6 GeV≤ m_χ≤ 6× 10^8 GeV, and 10^-50 cm^2≤σ_χN≤ 3× 10^-46 cm^2 for the FDM masses, 10^6 GeV≤ m_χ≤  10^10 GeV. 

From the above discussion, we find that the pulsar data provide the strongest constraint on DM parameter space (both for bosonic and fermionic DM case) for the model σ∝ v^2_rel. 






§ CONCLUSION



 A neutron star situated in a DM-dominated environment can efficiently capture the dark matter particles, and for large capturing, dark matter forms a black hole and destroys the host neutron star. In this work, we constrain the massive dark matter particles from the survival of the neutron star.

 To estimate the DM capture rate of massive dark matter particles m_χ>10^6 GeV, we adapted the multiscatter capture formalism to a non-constant DM-nucleon scattering cross-section. Adapting a model-independent approach, we assumed a power law parametrization of the velocity dependent and momentum dependent DM-Nucleon scattering cross-section, σ^vd_χN(v_rel)=σ_χN(v_rel/u_0)^2α and σ^md_χN(q)=σ_χN(q/q_0)^2β.
  Further, using the form factor and incorporating the general relativistic correction, we obtain analytic expressions for the capture rate in the neutron frame.
 Our estimation shows that the capture rate enhanced for the positive velocity and momentum dependent cross-section, and a number of the DM-Neucleon scattering, which leads to a black hole formation in a typical lifetime of the neutron star.    

 Further, applying the observation of pulsar data J2124-3858, we explored the constraint on the dark matter parameter space, i.e.,  σ_χN -m_χ for non-constant cross-section. We obtained that pulsar data provide the strongest constraint on DM parameter space (both for bosonic and fermionic DM case) for the model σ∝ v^2_rel. 
This model constrain 10^-52 cm^2≤σ_χN≤ 10^-50 cm^2 for bosonic DM masses, 10^6 GeV≤ m_χ≤ 6× 10^8 GeV, and 10^-50 cm^2≤σ_χN≤ 2× 10^-47 cm^2 for fermionic DM masses, 10^6 GeV≤ m_χ≤  10^10 GeV. Interestingly, we also found that although for model σ∝ v^4_rel, the DM accretion is higher, but due to a large thermalization time, it put weak constraints on the DM parameters. Furthermore, a large momentum dependent cross-section model, i.e., σ∝ q^4 provides no constraint on the DM parameters. 

To the best of our knowledge, this is the first constraint on the massive DM particles from the neutron star survival using the velocity and momentum dependent cross-section (in the multiscatter capture formalism). Nevertheless, we also point out that our calculation of capture rate has been done under certain assumptions, such as non-relativistic scattering of DM-nucleon, saturation cross-section is constant with DM mass (however, varies for massive DM particle), nucleon density  (also escape velocity) is constant. After inclusion, these effects will change the constraint, and we leave these interesting aspects for the future. 




§ ACKNOWLEDGEMENT


This work is supported by the National Natural Science Foundation of China (NNSFC) under grants No. 12275134, No. 12335005, and No. 12147228. AKM would like to thank Prof. Lei Wu and Prof. Chih-Ting Lu for the useful discussions. 




§ CALCULATION OF MULTISCATTER CAPTURE OF DM PARTICLES VIA CONSTANT CROSS-SECTION



After having equipped with the probability expression (see Section), we now estimate the multiscatter capture rate using Eq.(<ref>). The derivation of Eq.(<ref>) has been done is the assumption of multi-capture with v_esc (r)≃ v_esc (R_NS) and number density of the baryons are constant inside the neutron star, n_B(r)=n_B see Ref. <cit.>. To estimate the capture rate, we apply the Maxwell distribution function in the neutron star frame, Eq. (<ref>).


In a constant cross-section case, the probability, P^cont_N(τ_0) is independent of the DM velocity. So, applying Eq.(<ref>), Eq.(<ref>), and Eq.(<ref>) the analytic expression of C_N is obtained as
 
    C^cont_N(τ)=   π R^2/1-v^2_esc1/6π u_0ρ_χ/m_χP^cont_N(τ_0)f(v_N,η)Θ(v_N-v_esc)

 where 

    f(v_N,η)= 1/η[e^-3(v^2_N-v^2_esc)/2u^2_0+2√(v^2_N-v^2_esc) η/u_0 +η^2]
    ×[-√(6π)u_0( -1+2√(v^2_N-v^2_esc)η/u_0)√(v^2_N-v^2_esc)
       (1+e^√((v^2_N-v^2_esc))η/u_0-2e^-3/2u^2_0[(v^2_N-v^2_esc)+2√((v^2_N-v^2_esc)u_0η])u_0η 
    +π(3v^2_esc +u^2_0(1+3η^2))e^3/2u^2_0[v^2_N-v^2_esc+2√(v^2_N-v^2_esc)u_0η+u^2_0η^2]
       { ( 2 Erf√(3/2η) + Erf(√(3/2)(√(v^2_N-v^2_esc)-u_0η)/u_0) 
     + Erf(√(3/2)(√(v^2_N-v^2_esc)+u_0η)/u_0)}] ,

 where, v_N=v_esc(1-β_+/2)^-N/2 is the DM velocity after N scattering, and erf(z)=2/√(2)∫^z_0e^-y^2dy represent the error function.
In the above equation, ρ_χ is the DM energy density surrounding the NS, and N_B is the total number of baryons inside the NS. We checked that in the rest frame of DM halo, i.e., η→ 0, and point-like structure of nucleon (neglecting the form factor), our expression of C^const_χN matches with the Ref. <cit.>.  





§ DERIVATION OF MULTISCATTER CAPTURE OF DM PARTICLES VIA MDCS



In this case, σ^md_χN∝ q^2β, where q≃√(2)(m_χm_N/m_χ+m_N)v_esc.  So, P^MDCS_N(τ_0,q) is independent of the initial DM velocity. Using this, we obtain the analytic form of the capture rate after N scattering for the arbitrary momentum dependence cross section. The form of C_N in the MDCS case is given by

    C^MDCS_N(τ_0,q)=   π R^2/1-v^2_esc1/6π u_0ρ_χ/m_χP^MDCS_N(τ_0,q)f(v_N,η)Θ(v_N-v_esc)





§ DERIVATION OF MULTISCATTER CAPTURE OF DM PARTICLES VIA VDCS



In this case, σ^md_χN∝ w^2β, where w=√(u^2+v^2_esc). Therefore the probability P^VDCS_N(τ_0,w) depends on the DM velocity, see Eq. (<ref>). Using this probability expression, no reliable analytic (or numerical) solution was obtained. Therefore, to calculate the capture rate, we follow a conservative approach and approximate  w≃ w_0=√(u^2_0+v^2_esc) in the probability expression. After applying this approximation, we can easily integrate and obtain an analytic expression for the capture rate as

    C^VDCS_N(τ_0,w_0)=   π R^2/1-v^2_esc1/6π u_0ρ_χ/m_χP^VDCS_N(τ_0,w_0)f(v_N,η)Θ(v_N-v_esc)



utphys

 


 Paper Title: An alternative way to decipher the nature of the doubly charmed tetraquark $T_{cc}(3875)^+$: its antiparticle $T_{{\bar c}{\bar c}}(3875)^-$ photoproduction off nuclei near threshold.









An alternative way to decipher the nature of the doubly charmed tetraquark T_cc(3875)^+: its antiparticle
T_c̅c̅(3875)^- photoproduction off nuclei near threshold
    E. Ya. Paryev

Institute for Nuclear Research of the Russian Academy of Sciences

Moscow, Russia
    August 5, 2024
==================================================================================================================================================================



§ ABSTRACT 
We study the inclusive photoproduction of T_c̅c̅(3875)^- mesons (which are the antiparticles of the doubly charmed tetraquarks T_cc(3875)^+ discovered recently by the LHCb Collaboration) from nuclei in the near-threshold energy region within the nuclear spectral function approach by considering incoherent direct (γp(n) → D^+(D^0)T_c̅c̅(3875)^-Λ^+_c) photon–nucleon T_c̅c̅(3875)^- creation processes as well as five possible different scenarios for their internal structure with the main goal of clarifying the possibility to decipher this structure (and, hence, that of T_cc(3875)^+) in photoproduction via integral and differential observables. We calculate the absolute and relative excitation functions for T_c̅c̅(3875)^- production off ^12C and ^184W target nuclei at near-threshold photon beam energies of 30–38 GeV, the absolute differential cross sections for their production off these target nuclei at laboratory polar angles of 0^∘–10^∘ as well as the A and momentum dependences of the relative (transparency ratios) cross sections for T_c̅c̅(3875)^- production at photon energy of 35 GeV within the adopted scenarios for the T_c̅c̅(3875)^- meson intrinsic structure. We demonstrate that the absolute and relative observables considered show a certain sensitivity to these scenarios. Therefore, the measurement of such observables in future experiments at the proposed high-luminosity electron-ion colliders EIC and EicC in the US and China in the near-threshold energy region might shed light on the T_cc(3875)^+ internal structure.






§ 1. INTRODUCTION


 The study of exotic heavy hadrons, which exhibit properties incompatible with the
predictions of the conventional quark model for quark-antiquark mesons and three-quark baryons and which
are composed of four or five quarks (and antiquarks) and are named as tetraquark and pentaquark
states, respectively, has received considerable interest in recent years and is one of the most exciting
topics of the nuclear and hadronic physics nowadays owing to the expectation to better understand perturbative
and nonperturbative aspects of QCD and to extend our knowledge about the world of "elementary" particles.
(for a recent overview of the multiquark states see Refs. [1–5]). This interest was triggered by
the discovery of X(3872) resonance (also known as χ_c1(3872)) by the Belle Collaboration in
2003 [6] as a narrow peak in the vicinity of the D^0D̅^*0 (D̅^0D^*0) mass threshold in the J/ψπ^+π^- invariant mass distribution in exclusive B^±→ K^±X(3872) → K^±(J/ψπ^+π^-) decays.

Recently (in 2021), the LHCb Collaboration announced the first observation of the isoscalar doubly charmed tetraquark state T_cc(3875)^+ in the D^0D^0π^+ mass spectrum in the prompt production of the proton-proton
collisions at LHC [7, 8]. This state has a quark content ccu̅d̅, a mass of approximately 3875 MeV and
manifests itself as a very narrow peak in this spectrum just below the D^0D^*+ mass threshold
[^)Thus, its mass with respect to this threshold (m_D^0+m_D^*+=3875.1 MeV) and width are
δm_ exp=-360±40^+4_-0 keV, Γ=48±2^+0_-14 keV when the T_cc(3875)^+ is fitted
with a unitarized Breit-Wigner shape [8].]^)
.
Its spin-parity quantum numbers were estimated as J^P=1^+ by the experiment. The T_cc(3875)^+ is the second
doubly charmed hadron that has been experimentally observed up to now. In 2017, 2018, the LHCb Collaboration reported the observation of the doubly charmed baryon Ξ^++_cc with quark content ccu through the non-leptonic decay modes
Ξ^++_cc→Λ^+_cK^-π^+π^+ [9] and Ξ^+_cπ^+ [10].
After the experimental discovery of the T_cc(3875)^+ meson, its internal structure has been intensively debated.
Since the mass of T_cc(3875)^+ lies just below the nominal D^0D^*+ and D^+D^*0 mass thresholds
[^)The latter one lies at a slightly higher energy of m_D^++m_D^*0=3876.51 MeV than the former and difference between them is only 1.41 MeV.]^), it is natural to treat it as an
extremely narrow D^+D^*0-D ^0D^*+ weakly bound molecular state in a relative S-wave with a very small width and binding energy, dynamically generated from the interaction of the coupled channels D ^0D^*+ and D^+D^*0 in the isospin I=0 sector (see, for example, Refs. [11–14] and those cited below). On the other hand, the T_cc(3875)^+ has possibilities to be either a compact multiquark state where the quarks and antiquarks form compact configurations with the size of a typical hadron [15–17] or a mixture of a molecular and compact tetraquark states [18, 19]
[^)It is worth noting that the large value of the ratio of the partial radiative decay widths of the
χ_c1(3872) into ψ(2S)γ and J/ψγ final states measured very recently by the
LHCb Collaboration makes the interpretation of the χ_c1(3872) state as a pure
D^0D̅^*0+D̅^0D^*0 molecule questionable and provides a strong argument in favor of a compact
component also in its structure [20].]^).
All these plausible interpretations of the T_cc(3875)^+ exotic state were supported by many current theoretical studies (see references herein below). But, in spite of large efforts the full understanding of the T_cc(3875)^+ structure remains a challenging open problem.

Most of the above various interpretations are based on the analysis of the three-body hadronic and radiative decays of
T_cc(3875)^+ via intermediate states D^0D^*+ and D^+D^*0 into the channels D^0D^0π^+, D^0D^+π^0
and D^0D^+γ, using the experimental values of its mass and total decay width as inputs [21].
However, the internal structure of T_cc(3875)^+ could also be studied in high-energy proton–proton and heavy–ion collisions at hadron colliders where a large number of charm quarks is produced, since its abundance in these collisions could reflect its structure [22]. In these collisions the T_cc(3875)^+ mesons can be produced as a compact tetraquarks from the four-quark coalescence in the QGP phase at the critical temperature or as a hadronic molecules from the coalescence of charm mesons D and D^* formed at the end of the mixed phase [22] (cf. Ref. [23] containing the similar statement for the exotic hadron X(3872)) and in them their such observables as the transverse momentum distributions and total production yields are expected to be completely different for both these configurations [24].
Alternatively, the structure of T_cc(3875)^+ can also be investigated through its antiparticle
T_c̅c̅(3875)^- photoproduction off nuclei at near-threshold energies in the same manner as it was done for the X(3872) resonance in Ref. [25].
The advantage of photoproduction compared to the hadronic collisions is that the interpretation of data from such future experiments is clearer due to insignificant strength of initial-state photon interaction and since in it a
fewer individual exclusive elementary T_c̅c̅(3875)^- meson production channels need to be accounted for.

In this work, we present the detailed predictions for the absolute and relative excitation functions for T_c̅c̅(3875)^- production off ^12C and ^184W target nuclei as well as for the A and momentum dependences of the relative (transparency ratios) cross sections for T_c̅c̅(3875)^- production from γA reactions at threshold energies obtained in the framework of the first collision model within the possible different scenarios for the T_c̅c̅(3875)^- (and, hence, for the T_cc(3875)^+) meson intrinsic structure. The predictions can be tested by future high-precision measurements at electron-ion colliders EIC [26] and EicC [27] with the aim of discriminating between these scenarios.



§ 2. FRAMEWORK: DIRECT T_C̅C̅(3875)^- PHOTOPRODUCTION MECHANISM


 Direct T_c̅c̅(3875)^- photoproduction off nuclei in the near-threshold photon beam
energy region E_γ≤ 38 GeV of interest
[^)Which corresponds to the center-of-mass energies W of the photon-proton system W ≤ 8.5 GeV,
or to the relatively "low" excess energies ϵ above the D^+T_c̅c̅(3875)^-Λ^+_c production threshold W_ th=m_D^++m_T_c̅c̅(3875)^-+m_Λ^+_c (m_D^+, m_T_c̅c̅(3875)^- and m_Λ^+_c are the D^+, T_c̅c̅(3875)^- mesons and Λ^+_c hyperon free space masses, respectively ) 0 ≤ϵ≤ 0.47 GeV and in which the doubly anticharmed tetraquark T_c̅c̅(3875)^- can be observed in γp and γA reactions at the proposed electron-ion colliders EIC [26] and EicC [27] in the US and China.]^)
can occur in the following elementary processes with the lowest free production threshold (≈ 33.9 GeV) [28, 29]:


    γ+p → D^++T_c̅c̅(3875)^-+Λ^+_c,



    γ+n → D^0+T_c̅c̅(3875)^-+Λ^+_c.

The D^+, D^0, T_c̅c̅(3875)^- mesons and Λ^+_c hyperons, produced in these processes, are sufficiently energetic.
Thus, for example, the kinematically allowed D^+, T_c̅c̅(3875)^- mesons and final Λ^+_c
hyperon laboratory momenta in the direct process (1), proceeding on the free target proton at rest, vary within the momentum ranges of 5.749–11.125 GeV/c, 13.748–20.066 GeV/c and 7.335–13.061 GeV/c, respectively, at incident photon beam energy of E_γ=35 GeV.
Since the neutron and D^0 meson masses are approximately equal to the proton and D^+ meson masses, respectively,
the kinematical characteristics of final particles, produced in the reaction (2), are close to those of
final particles in the process (1). The binding of target nucleons and their Fermi motion will distort the
distributions of the outgoing high-momentum D^+, D^0, T_c̅c̅(3875)^- mesons and Λ^+_c
hyperons as well as lead to a wider accessible for them momentum intervals compared to those given above
(cf. Figs. 8 and 9 given below). Since the medium effects are expected to be reduced for high momenta [30, 31],
we will ignore the medium modifications of these final particles in the case when the reactions (1), (2)
proceed on a nucleons embedded in a nuclear target
[^)It should be pointed out that the properties of the D, T_cc(3875)^+ and T_c̅c̅(3875)^- mesons and Λ^+_c hyperons in a strongly interacting nuclear environment at low energies have been theoretically investigated, respectively, in the works [32–42], [43, 44] and [45, 46].]^)
.

  Then, neglecting the distortion of the incident photon at energies of interest
and describing the T_c̅c̅(3875)^- meson absorption by intranuclear nucleons by the absorption cross section σ_T_c̅c̅(3875)^-N, we represent the total cross section for the production of
T_c̅c̅(3875)^- mesons off nuclei from the direct processes (1), (2) as follows [25]:


    σ_γA→T_c̅c̅(3875)^-X^( dir)(E_γ)=I_V[A,σ_T_c̅c̅(3875)^-N]
    <σ_γp → D^+T_c̅c̅(3875)^-Λ^+_c(E_γ)>_A,

where


    I_V[A,σ]=2π∫_0^Rr_dr_∫_-√(R^2-r_^2)^√(R^2-r_^2)dz
    ρ(√(r_^2+z^2))
    exp[-σ∫_z^√(R^2-r_^2)ρ(√(r_^2+x^2))dx],



    ρ(r)=ρ_p(r)+ρ_n(r),   r=√(r_^2+z^2)    or r=√(r_^2+x^2);



    <σ_γp → D^+T_c̅c̅(3875)^-Λ^+_c(E_γ)>_A=
    ∫∫
    P_A( p_t,E)d p_tdE
    σ_γp → D^+T_c̅c̅(3875)^-Λ^+_c(√(s^*))

and


    s^*=(E_γ+E_t)^2-( p_γ+ p_t)^2,



    E_t=M_A-√((- p_t)^2+(M_A-m_p+E)^2).

Here, σ_γp→ D^+T_c̅c̅(3875)^-Λ^+_c(√(s^*)) is the "in-medium"
total cross section for the production of T_c̅c̅(3875)^- mesons in process (1)
[^)In Eq. (3) we assume that the T_c̅c̅(3875)^- production cross sections
in γp and γn reactions are the same.]^)
at the "in-medium" γp center-of-mass energy √(s^*);
ρ_p(r),  ρ_n(r) (r is the distance from the nucleus center)
and P_A( p_t,E) are normalized to the numbers of protons Z, neutrons
N and to unity the local proton, neutron densities and the
spectral function of target nucleus with mass number A (A=Z+N), having mass M_A and radius R
[^)The information about the spectral function of target nucleus, used in the subsequent calculations,
is given in Refs. [47, 48] and the local nucleon densities, employed in them, will be defined below.]^);
p_t  and E are the internal momentum and binding energy of the struck target proton
just before the collision; p_γ and E_γ are the momentum and energy of the incident photons;
m_p is the proton bare mass.

As in Ref. [25], we suppose that the "in-medium" cross section
σ_γp → D^+T_c̅c̅(3875)^-Λ^+_c(√(s^*)) for T_c̅c̅(3875)^- production in process (1)
is equivalent to the free space cross section
 σ_γp → D^+T_c̅c̅(3875)^-Λ^+_c(√(s(E_γ))), in which
the free space center-of-mass energy squared s(E_γ) for given photon energy E_γ, presented by the formula


    s(E_γ)=W^2=(E_γ+m_p)^2- p_γ^2=m_p^2+2m_pE_γ,

is replaced by the in-medium expression (7).
For the free total cross section σ_γp → D^+T_c̅c̅(3875)^-Λ^+_c(√(s(E_γ))) no data are available up to now. Therefore,
in the photon energy range W ≤ 8.5 GeV of interest we have adopted for it the following parametrization of the results of the calculations of this cross section here (for a cutoff parameter α=1.7) within the approach [28], in which the γp → D^+T_c̅c̅(3875)^-Λ^+_c reaction is studied by considering the central diffractive mechanism and T_c̅c̅(3875)^- as a D̅D̅^* molecule:


    σ_γp → D^+T_c̅c̅(3875)^-Λ^+_c(√(s(E_γ)))=2.409(1-s_ th/s(E_γ))^1.745 [ pb],

where


    s_ th=W_ th^2=(m_D^++m_T_c̅c̅(3875)^-+m_Λ^+_c)^2.

In the energy regime W ≤ 8.5 GeV, the cross section of the γp → D^+T_c̅c̅(3875)^-Λ^+_c reaction can reach the values of the order of 0.05 pb [28]. Although, this cross section
is quite small here, one may hope that its measurements cold be performed in the future at the electron-ion colliders in the US [26] and China [27] (cf. also [28]). The empirical formulas (10), (11) will be used in our subsequent calculations
as a guideline for a reasonable evaluation of the T_c̅c̅(3875)^- yield in γA reactions.

We discuss now the local nucleon densities, employed in our calculations of the T_c̅c̅(3875)^- photoproduction on the target nuclei ^12_6C, ^27_13Al, ^40_20Ca, ^63_29Cu, ^93_41Nb, ^112_50Sn, ^184_74W, ^208_82Pb and ^238_92U considered in the present study.
For lightest nucleus ^12_6C we use the same proton and neutron density profiles of the harmonic oscillator
model [49, 50]. For nuclei ^27_13Al, ^40_20Ca and ^63_29Cu for the proton and neutron densities, ρ_p(r) and ρ_n(r), we have employed in our present calculations the Woods-Saxon distributions with the same radial parameters for protons and neutrons given in Ref. [25].
For medium-weight ^93_41Nb, ^112_50Sn and heavy ^184_74W, ^208_82Pb, ^238_92U target nuclei, we adopted for protons and neutrons the two-parameter Fermi density distributions with proton and neutron density parameters also reported in Ref. [25]. As in Ref. [25], we use here the neutron density ρ_n(r) in the
'skin' form.

To go further, we have to determine the T_c̅c̅(3875)^-–nucleon absorption cross section
σ_T_c̅c̅(3875)^-N. It is a crucial to estimate this cross section in different theoretical pictures describing  the possible underlying structure of T_c̅c̅(3875)^- for its interpretation
through the T_c̅c̅(3875)^- photoproduction on nuclei since there is a direct dependence of the rate of this production on the absorption cross section σ_T_c̅c̅(3875)^-N
(cf. Eq.(3)). This rate is expected to be different for different options for the inner structure of the
T_c̅c̅(3875)^-. That is, in our approach the rate of the T_c̅c̅(3875)^- meson
photoproduction off nuclei reflects the sensitivity to this structure. This remarkable feature can be used to better
determine/constrain the inner structure of this exotic state in the future dedicated experiments. In our present study we consider three scenarios for the hypothetical T_c̅c̅(3875)^- meson based on the proposed in literature three popular configurations for the discovered doubly charmed tetraquark T_cc(3875)^+:
i) a molecular state - a loosely bound, with large spatial size characterized by the r.m.s. radius of the order of 5–7 fm or more,  S-wave state in the D̅^0D^*-(D^-D̅^*0) system consisting of an anticharmed D̅^0 and D^*- or D^- and D̅^*0 mesons having the quarks contents c̅u and c̅d or c̅d and c̅u, respectively, [51–65]
[^)Loosely bound S-wave charm-meson molecules like X(3872) and T_cc(3875)^+ have universal
properties  determined by their binding energies [66, 67]. Accordingly, within the molecular hypothesis, the r.m.s. size r_T of the separation of the D̅^0 and D^*- constituents in the T_c̅c̅(3875)^- molecule
can be estimated from the relation r_T=1/√(4μ_0δ_T_c̅c̅(3875)^-)
by knowing the binding energy δ_T_c̅c̅(3875)^-=m_D̅^0+m_D^*--m_T_c̅c̅(3875)^- of the T_c̅c̅(3875)^- molecule [66, 67]. Here, μ_0 is the D̅^0D^*- reduced mass. Assuming that this binding energy is equal to that δ_T_cc(3875)^+ of the T_cc(3875)^+
state relative to the D^0D^*+ mass threshold and taking δ_T_cc(3875)^+=360±40^+4_-0 keV as input
in the case when an unitarized Breit-Wigner shape is used to fit the T_cc(3875)^+
[8], we find that the anticharmed mesons in the T_c̅c̅(3875)^- have a sufficiently large r.m.s. separation: r_T=5.3±0.3 fm. The typical scale for the relative momentum between the D̅^0 and
D^*- mesons [66, 68], √(2μ_0δ_T_c̅c̅(3875)^-)≃25–30 MeV/c, is much smaller than their average laboratory momenta ∼ 8.1 and 8.8 GeV/c, respectively (see above). With these, we expect that the application of the considered below quasi-free approximation to evaluate the T_c̅c̅(3875)^-–nucleon absorption cross section in molecular scenario is sufficiently well justified.]^),
ii) compact, ∼ 1 fm, diquark [ud] and antidiquark [c̅c̅] tetraquark state [24, 69–72],
and iii) a hybrid state - a state in which the T_c̅c̅(3875)^- wave function contains a molecular and
compact four-quark components [18, 73].

In the "pure" molecular picture of the T_c̅c̅(3875)^-, whose wave function takes the
following charge-conjugation form


    |T_c̅c̅(3875)^->_ mol=1/√(2)(|D^-D̅^*0>-|D̅^0D^*->)

compared to that


    |T_cc(3875)^+>_ mol=1/√(2)(|D^+D^*0>-|D^0D^*+>)

for the T_cc(3875)^+ in the isoscalar channel [52, 64, 74, 75]
[^)It should be pointed out that in a molecular picture (13) the T_cc(3875)^+ couples equally
to the D^+D^*0 and D^0D^*+ channels. However, in some works (see, for example, Refs. [53, 64, 65]) it was predicted that the T_cc(3875)^+ flavor wave function contains about 70% of the D^0D^*+ and 30% of the D^+D^*0
molecular components. Since in our model the specific flavor content of the T_cc(3875)^+ (and, respectively, of the T_c̅c̅(3875)^-) has no influence on the determination of the T_c̅c̅(3875)^-–nucleon absorption cross section in a molecular scenario, we will treat the T_c̅c̅(3875)^- below in this
scenario as a D^-D̅^*0-D̅^0D^*- molecule.]^)
and in which the constituents (anticharmed mesons) are weakly bound and have large spatial separation in coordinate space (see above)
the T_c̅c̅(3875)^-–nucleon absorption cross section σ_T_c̅c̅(3875)^-N^ mol can be naturally estimated in the quasi-free approximation [22, 76, 77]. In this approximation, the binding energy of
the anticharmed mesons is ignored. 
They are taken to be on-shell and they are considered to fly together with an average laboratory momentum of the order of 8 GeV/c each. In this approximation the
T_c̅c̅(3875)^- is absorbed when a nuclear nucleon (proton or neutron) interacts (elastically or inelastically) with the D̅^0 or with the D^*- (or with the D^- or with the D̅^*0).
In each of these interactions the other anticharmed meson is a spectator.
Then, assuming that the total cross sections of the free D^*-N and D̅^*0N interactions are the
same as those for the D^-N and D̅^0N ones [78, 79], we can evaluate the cross section for T_c̅c̅(3875)^- absorption, σ_T_c̅c̅(3875)^-N^ mol, as [25, 77]:


    σ_T_c̅c̅(3875)^-N^ mol=σ_T_c̅c̅(3875)^-p^ mol=σ_T_c̅c̅(3875)^-n^ mol,

where the T_c̅c̅(3875)^-p and T_c̅c̅(3875)^-n absorption cross sections, σ_T_c̅c̅(3875)^-p^ mol and σ_T_c̅c̅(3875)^-n^ mol,
are given by


    σ_T_c̅c̅(3875)^-p(n)^ mol≈σ_D̅^0p(n)^ el+σ_D̅^0p(n)^ in+σ_D^-p(n)^ el+σ_D^-p(n)^ in+σ_D^-p →D̅^0n(σ_D̅^0n → D^-p).

Here, σ_D̅^0p(n)^ el(in) and σ_D^-p(n)^ el(in) are the elastic (inelastic) cross
sections of the free D̅^0p(D̅^0n) and D^-p(D^-n) interactions, respectively. And,
σ_D^-p →D̅^0n and σ_D̅^0n → D^-p are the total cross sections of the free charge-exchange reactions D^-p →D̅^0n and D̅^0n → D^-p, correspondingly. In the calculations we employ for them the following constants, which are relevant to the momentum regime above of 1 GeV/c of our interest:
σ_D̅^0p(n)^ el=σ_D^-p(n)^ el=10 mb,
σ_D̅^0p(n)^ in=σ_D^-p(n)^ in=0,
σ_D^-p →D̅^0n=σ_D̅^0n → D^-p=12 mb [78, 79]. Adopting these values, we get that
σ_T_c̅c̅(3875)^-p^ mol=σ_T_c̅c̅(3875)^-n^ mol=32 mb and, in line with Eq. (14), σ_T_c̅c̅(3875)^-N^ mol=32 mb.

In the second scenario, in which the T_c̅c̅(3875)^- meson is treated as a compact c̅c̅ud
tetraquark state, its breakup cross section σ_T_c̅c̅(3875)^-N^ 4q can be approximated
by the following relation [77]:


    σ_T_c̅c̅(3875)^-N^ 4q≈1/3σ_T_c̅c̅(3875)^-N^ mol.

For σ_T_c̅c̅(3875)^-N^ mol=32 mb we obtain that σ_T_c̅c̅(3875)^-N^ 4q=10.7 mb. It is interesting to note that this value for the quantity
σ_T_c̅c̅(3875)^-N^ 4q is similar to that of 13.3 mb, adopted in Ref. [25]
for the high-momentum X(3872)–nucleon absorption cross section σ_X(3872)N^ 4q considering
the X(3872) meson as a compact tetraquark with radius r_4q=0.65 fm
and approximating the latter cross section by a geometrical cross section σ_4q^ geo=πr_4q^2.

In the hybrid scenario, it is assumed that the T_c̅c̅(3875)^- wave function is a linear superposition of the compact c̅c̅ud (or 4q) and the molecular D̅D̅^* components [18, 73]:


    |T_c̅c̅(3875)^->_ hyb=α|c̅c̅ud>+β/√(2)(|D^-D̅^*0>-|D̅^0D^*->).

Here, α^2 and β^2 represent the probabilities to find a compact and non-compact hadronic configurations,
respectively, for the normalization


    α^2+β^2=1.

The limiting case of α^2=0, β^2=1 corresponds to the pure molecular interpretation of the T_c̅c̅(3875)^- state, while the case of α^2=1, β^2=0 refers to its pure compact four-quark treatment.
Following Refs. [18, 43, 44, 73], we assume that the T_c̅c̅(3875)^- wave function (17) contains 80% of the non-molecular 4q component and 20% of the molecular D̅D̅^* component ((α^2,β^2)=(0.8,0.2)), 50% of the compact component and 50% of the non-compact component
((α^2,β^2)=(0.5,0.5)) as well as 20% of the non-molecular component and 80% of the molecular component
((α^2,β^2)=(0.2,0.8)).
These five options for the non-molecular and molecular probabilities of the T_c̅c̅(3875)^- cover the bulk of theoretical and experimental information presently available in this field and they will allow us to see the sensitivity of the T_c̅c̅(3875)^- meson photoproduction cross sections from the direct processes (1), (2) on nuclei to its internal structure. In the hybrid picture (17) of the T_c̅c̅(3875)^-, we can represent  the T_c̅c̅(3875)^-–nucleon absorption cross section σ_T_c̅c̅(3875)^-N^ hyb in the following incoherent probability-weighted sum [25]:


    σ_T_c̅c̅(3875)^-N^ hyb=α^2σ_T_c̅c̅(3875)^-N^ 4q+β^2σ_T_c̅c̅(3875)^-N^ mol.

In view of the above, we set σ_T_c̅c̅(3875)^-N^ 4q=10.7 mb and σ_T_c̅c̅(3875)^-N^ mol=32 mb.
With these values, the T_c̅c̅(3875)^- absorption cross section (19) is σ_T_c̅c̅(3875)^-N^ hyb=15.0, 21.35 as well as 27.7 mb for the non-molecular and molecular probabilities of the T_c̅c̅(3875)^- 80% and 20%, 50% and 50% as well as 20% and 80%, respectively. We summarize below the results obtained above for the T_c̅c̅(3875)^-–nucleon absorption cross section σ_T_c̅c̅(3875)^-N in the considered schemes for the T_c̅c̅(3875)^- tetraquark inner configuration:


    σ_T_c̅c̅(3875)^-N={[  10.7  mb         ,;          ;  15.0  mb         ,;          ; 21.35  mb         ,;          ;  27.7  mb         ,;          ;    32  mb         . ].

Therefore, the T_c̅c̅(3875)^- as a D̅D̅^* molecule has the largest absorption cross section and, hence, it is expected to be more easily absorbed than its other configurations in a nuclear medium.
We will use these values for the quantity σ_T_c̅c̅(3875)^-N in our calculations.

Neglecting the nuclear effects considered: the target nucleon binding and Fermi motion, from Eq. (3)
we obtain the following simple expression for the total cross section
σ_γA→T_c̅c̅(3875)^-X^( dir)(E_γ):


    σ_γA→T_c̅c̅(3875)^-X^( dir)(E_γ)=I_V[A,σ_T_c̅c̅(3875)^-N]
    σ_γp → D^+T_c̅c̅(3875)^-Λ^+_c(√(s(E_γ))),

where the elementary cross section σ_γp → D^+T_c̅c̅(3875)^-Λ^+_c(√(s(E_γ))) is given above by Eq. (10).
This expression is valid only at photon energies well above threshold
[^)In our case, the difference between the expressions (3) and (21) is small at
incident photon energies around 37 GeV and more, and it becomes substantial at photon energies close to the
threshold energy of 33.9 GeV as our calculations have shown
(compare magenta short-dashed and cyan short-dashed-dotted curves in figures 1–4 given below).]^)
.
As a measure for the
T_c̅c̅(3875)^- absorption cross section σ_T_c̅c̅(3875)^-N in nuclei and, hence, for the T_c̅c̅(3875)^- intrinsic configuration we will use the so-called T_c̅c̅(3875)^- transparency ratio defined as [80, 81]:


    S_A=σ_γA →T_c̅c̅(3875)^-X(E_γ)/A σ_γp → D^+T_c̅c̅(3875)^-Λ^+_c(√(s(E_γ))),

i.e. the ratio of the inclusive nuclear T_c̅c̅(3875)^- photoproduction cross section divided by A times the same quantity on a free proton. It should be noted that this relative observable is more
favorable compared to those based on the absolute cross sections for the aim of obtaining the information
on the T_c̅c̅(3875)^- nuclear absorption [25] (and, hence, on the T_c̅c̅(3875)^- internal configuration).
It is expected that the direct processes (1), (2) will be dominant in the T_c̅c̅(3875)^- production in γA interactions close to threshold
[^)At least, at incident photon energies ≤ 35 GeV, where
the elementary processes γN → DT_c̅c̅(3875)^-Λ^+_cπ
with one pion in the final states are expected to be suppressed in T_c̅c̅(3875)^- production in γA reactions compared to those of (1), (2) due to their larger production threshold energy (≈ 35.1 GeV) in free γN collisions. Moreover, since the main inelastic channel in γN
collisions at these photon energies is the multiplicity production of pions with rather
low energies the secondary pion-induced T_c̅c̅(3875)^- production processes are energetically suppressed at them as well.
We expect that the full inclusive nuclear T_c̅c̅(3875)^- photoproduction cross section σ_γA →T_c̅c̅(3875)^-X(E_γ), entering into Eq. (22), is entirely exhausted by that of Eq. (3) from the direct T_c̅c̅(3875)^- production processes (1), (2) not only at photon energies E_γ≤ 35 GeV, but also at higher energies considered in the present study.]^).
Then, according to Eqs. (3) and (4), we have:


    S_A=σ_γA →T_c̅c̅(3875)^-X^( dir)(E_γ)/A σ_γp → D^+T_c̅c̅(3875)^-Λ^+_c(√(s(E_γ)))=
    I_V[A,σ_T_c̅c̅(3875)^-N]/A<σ_γp → D^+T_c̅c̅(3875)^-Λ^+_c(E_γ)>_A/σ_γp → D^+T_c̅c̅(3875)^-Λ^+_c(√(s(E_γ))).

Usually, the transparency ratio of the meson is normalized to a light nucleus like ^12C [80, 82].
With this and Eq. (23), for T_c̅c̅(3875)^- we have:


    T_A=S_A/S_C=12 σ_γA →T_c̅c̅(3875)^-X^( dir)(E_γ)/A σ_γ C→T_c̅c̅(3875)^-X^( dir)(E_γ)=
    12 I_V[A,σ_T_c̅c̅(3875)^-N]/A I_V[ C,σ_T_c̅c̅(3875)^-N]<σ_γp → D^+T_c̅c̅(3875)^-Λ^+_c(E_γ)>_A/<σ_γp → D^+T_c̅c̅(3875)^-Λ^+_c(E_γ)>_ C.

In the case when direct T_c̅c̅(3875)^- production processes (1), (2) proceed on a free target
nucleons being at rest the expression (24) for the quantity T_A is reduced to the following simpler form:


    T_A ≈12 I_V[A,σ_T_c̅c̅(3875)^-N]/A I_V[ C,σ_T_c̅c̅(3875)^-N].

The formulas (21)–(25) will be used in calculations of the integral observables considered at photon energies of our interest.







Before going further, we consider now the momentum-dependent inclusive differential cross section for production
of T_c̅c̅(3875)^- mesons with momentum p_T_c̅c̅(3875)^- and total energy
E_T_c̅c̅(3875)^- (or with momentum p_T_c̅c̅^- and total energy E_T_c̅c̅^- for brevity) on nuclei in the direct processes (1) and (2) in γA reactions. According to Eqs. (3), (6) and Ref. [83], we represent this cross section as follows:


    dσ_γA→T_c̅c̅(3875)^-X^( dir)(E_γ, p_T_c̅c̅^-)/d p_T_c̅c̅^-=
    I_V[A,σ_T_c̅c̅(3875)^-N]
    <dσ_γp → D^+T_c̅c̅(3875)^-Λ^+_c( p_γ, p_T_c̅c̅^-)/d p_T_c̅c̅^->_A,

where the off-shell elementary differential cross section
dσ_γp → D^+T_c̅c̅(3875)^-Λ^+_c( p_γ, p_T_c̅c̅^-)/d p_T_c̅c̅^- for T_c̅c̅(3875)^- production in process (1)
is given by (cf. Ref. [83])


    dσ_γp → D^+T_c̅c̅(3875)^-Λ^+_c( p_γ, p_T_c̅c̅^-)/d p_T_c̅c̅^-=π/4σ_γp → D^+T_c̅c̅(3875)^-Λ^+_c(√(s^*))/I_3[s^*,m_D^+,m_T_c̅c̅(3875)^-,m_Λ^+_c]E_T_c̅c̅^-λ[s^*_D^+Λ^+_c,m_D^+^2,m_Λ^+_c^2]/s^*_D^+Λ^+_c,

assuming for simplicity that the T_c̅c̅(3875)^- angular distribution in reaction (1) is
isotropic in the γp c.m.s. at photon beam energies of interest.
Here,


    I_3[s^*,a,b,c]=(π/2)^2∫_(b+c)^2^(√(s^*)-a)^2λ[x,b^2,c^2]/xλ[s^*,x,a^2]/s^*dx,



    λ(x,y,z)=√([x-(√(y)+√(z))^2][x-
    (√(y)-√(z))^2]),



    s^*_D^+Λ^+_c=s^*+m_T_c̅c̅(3875)^-^2-2(E_γ+E_t)E_T_c̅c̅^-+2( p_γ+ p_t)
     p_T_c̅c̅^-

and the quantity s^* is defined above by Eq. (7).
At the initial photon energies of interest the T_c̅c̅(3875)^- mesons are produced at very small
laboratory polar angles
[^)Thus, for example, the maximum angle of T_c̅c̅(3875)^- meson production off a free proton at rest in reaction (1) is about 2.5^∘ at photon energy of 35 GeV.]^)
. Therefore, we will calculate their momentum distribution from target nuclei considered
for the laboratory solid angle ΔΩ_T_c̅c̅^- = 0^∘≤θ_T_c̅c̅^-≤ 10^∘ and 0 ≤φ_T_c̅c̅^-≤ 2π.
Then, in line with Eq. (26), we can get the following expression for this distribution:


    dσ_γA→T_c̅c̅(3875)^-X^( dir)
    (E_γ,p_T_c̅c̅^-)/dp_T_c̅c̅^-=
    ∫_ΔΩ_T_c̅c̅^-^dΩ_T_c̅c̅^-dσ_γA→T_c̅c̅(3875)^-X^( dir)
    (E_γ, p_T_c̅c̅^-)/d p_T_c̅c̅^-p_T_c̅c̅^-^2


    =2πI_V[A,σ_T_c̅c̅(3875)^-N]
    ∫_cos10^∘^1dcosθ_T_c̅c̅^-<dσ_γp→ D^+T_c̅c̅(3875)^-Λ^+_c(p_γ,
    p_T_c̅c̅^-,θ_T_c̅c̅^-)/dp_T_c̅c̅^-dΩ_T_c̅c̅^->_A.




























§ 3. RESULTS AND DISCUSSION


 We discuss now the results of our calculations in the framework of the approach outlined above.
First of all, we consider the excitation functions for production of T_c̅c̅(3875)^- mesons off
^12C and ^184W nuclei. They were calculated in line with Eq. (3)
for five adopted options for the T_c̅c̅(3875)^- absorption cross section σ_T_c̅c̅(3875)^-N in nuclear medium (cf. Eq. (20)) as well as in line with formula (21) for the free target nucleon
at rest with the value of σ_T_c̅c̅(3875)^-N=32 mb corresponding to the molecular picture for the T_c̅c̅(3875)^-, and are given in Figs. 1 and 2, respectively.
One can see that the difference between calculations with and without
accounting for the target nucleon Fermi motion (between magenta short-dashed and cyan short-dashed-dotted curves)
is sufficiently small at well above threshold photon energies ∼ 37–38 GeV, while at lower incident energies its impact on the T_c̅c̅(3875)^- yield is substantial.
It can be seen yet that the T_c̅c̅(3875)^- meson total cross sections show a definite sensitivity to the
absorption cross section σ_T_c̅c̅(3875)^-N and, hence, to the T_c̅c̅(3875)^- internal structure, for both target nuclei and for all photon energies considered.
But their absolute values are quite small even at above threshold photon energies of ∼ 35–38 GeV.
At these energies, they are ∼ 0.1–0.5 and 1–5 pb for carbon and tungsten target nuclei, correspondingly,
which is accessible at the proposed electron-ion colliders [26, 27] (see below).
For the heavy target nucleus ^184W, there are experimentally distinguishable
differences between the considered options:
σ_T_c̅c̅(3875)^-N=32 mb and σ_T_c̅c̅(3875)^-N=27.7 mb, σ_T_c̅c̅(3875)^-N=27.7 mb and σ_T_c̅c̅(3875)^-N=21.35 mb, σ_T_c̅c̅(3875)^-N=21.35 mb and σ_T_c̅c̅(3875)^-N=15.0 mb, σ_T_c̅c̅(3875)^-N=15.0 mb and σ_T_c̅c̅(3875)^-N=10.7 mb
or between the considered scenarios:
pure molecular state and hybrid state with the non-molecular and molecular probabilities of 20% and 80%,
hybrid state with the non-molecular and molecular probabilities of 20% and 80% and
hybrid state with the non-molecular and molecular probabilities of 50% and 50%,
hybrid state with the non-molecular and molecular probabilities of 50% and 50% and hybrid state with the non-molecular and molecular probabilities of 80% and 20%, hybrid state with the non-molecular and molecular probabilities of 80% and 20% and pure 4q compact state for the T_c̅c̅(3875)^-–nucleon absorption cross section or for the T_c̅c̅(3875)^- intrinsic configuration. They are ∼ 12, 21, 25, 20% for all considered photon energies. For the light target nucleus ^12C, the sensitivity of the cross sections to these quantities becomes somewhat lower and, respectively, the same differences as above become somewhat smaller. They are ∼ 6, 11, 11, 8%. Measurement of such small effects is a difficult experimental challenge.
On the other hand, the differences between calculations corresponding to the molecular configuration and hybrid configuration with the non-molecular and molecular probabilities of 50% and 50%, hybrid configuration with the non-molecular and molecular probabilities of 50% and 50% and pure 4q compact configuration for the T_c̅c̅(3875)^- are yet not so small. They are ∼ 18, 20%. This means that, at least, these configurations could be tested in the future T_c̅c̅(3875)^- measurements on carbon and light target nuclei like carbon as well.
To motivate such measurements at the future colliders EicC and EIC, it is desirable to estimate the T_c̅c̅(3875)^- production rates (the event numbers) in the γ^12C and γ^184W reactions.
For this purpose, we translate the T_c̅c̅(3875)^- photoproduction total cross sections, reported above, into the expected event numbers for the T_c̅c̅(3875)^- from the reactions
γ^12 C(^184 W) →T_c̅c̅(3875)^-X, T_c̅c̅(3875)^- →D̅^0D̅^0π^-, D̅^0 → K^+π^-, D̅^0→ K^+π^-.
To estimate the total numbers of the T_c̅c̅(3875)^- events in a one-year run at EicC, one needs
to multiply the above T_c̅c̅(3875)^- photoproduction total cross sections of 0.1–0.5 and 1–5 pb
on the carbon and tungsten target nuclei, respectively, by the integrated luminosity of 60 fb^-1 [27, 84]
as well as by the detection efficiency and by the appropriate branching ratios
Br[T_c̅c̅(3875)^- →D̅^0D̅^0π^-]≈ 60% and
Br[D̅^0 → K^+π^-]≈ 4%
[^)We suppose here that Br[T_c̅c̅(3875)^- →D̅^0D̅^0π^-]=Br[T_c c(3875)^+ → D^0D^0π^+]≈ 60% [18, 55, 58, 85] and Br[D̅^0 → K^+π^-]=Br[D^0 → K^-π^+]≈ 4% [86].]^)
.
With a realistic 50% detection efficiency [27],
we estimate about of 3–15 and 30–150 events per year for the T_c̅c̅(3875)^- signal in the cases of the ^12C and ^184W target nuclei, respectively. The event numbers for
T_c̅c̅(3875)^- →D̅^0(→ K^+π^-)D̅^0(→ K^+π^-)π^- at EIC
are about of 15–75 and 150–750 with an integrated luminosity of 300 fb^-1 [26, 84].
We see that a sufficiently large number of T_c̅c̅(3875)^- events could be observed (especially at
EIC collider) at well above threshold photon energies of 35–38 GeV.
Thus, the electron-ion colliders EicC and EIC provide a good platform to study the nature of the T_c̅c̅(3875)^- (and, respectively, of the T_cc(3875)^+) in near-threshold photonuclear reactions.

To gain further insight, we show in Figs. 3 and 4 the energy dependences of the
ratios of the cross sections, presented in Figs. 1 and 2, to the cross section, calculated in the molecular picture for the T_c̅c̅(3875)^- for an off-shell target nucleons, on the same linear scale for ^12C and ^184W target nuclei, respectively. It is clearly seen that for the heavy ^184W target nucleus
the sensitivity of the T_c̅c̅(3875)^- meson yield to variations in its intrinsic structure is
significantly stronger than that in the case of light ^12C target nucleus and it would definitely be possible for ^184W nucleus to distinguish between the results corresponding to all considered choices for this structure at all initial photon energies. At the same time, for light ^12C target nucleus we see a big enough distinctions between
the curves corresponding to the pure molecular configuration, hybrid configuration with the compact and non-compact probabilities of 50% and 50% and pure compact tetraquark configuration for the T_c̅c̅(3875)^-. On the basis of the foregoing, we can conclude that the structure of T_c̅c̅(3875)^- mesons could be studied at the future electron-ion colliders via the energy dependence of their absolute (and relative) total production cross sections in inclusive near-threshold photonuclear reactions. Figures 3 and 4 also show clearly that the difference between the absolute T_c̅c̅(3875)^- photoproduction cross sections, calculated in the pure molecular picture for the T_c̅c̅(3875)^- with allowance for the impact of the binding of intranuclear nucleons and their Fermi motion on the direct processes (1), (2) and without it (cf. Eqs. (3) and (21)) becomes indeed small only for above-threshold photon energies ∼ 37–38 GeV.

Additional information about the sensitivity of the cross sections to the internal structure of the
T_c̅c̅(3875)^- meson is contained in Figs. 5 and 6, where the A–dependences of the transparency ratios S_A and T_A from the direct reaction channels (1), (2) in γA
(A=^12C, ^27Al, ^40Ca, ^63Cu, ^93Nb, ^112Sn, ^184W, ^208Pb, and ^238U)
collisions, calculated for the photon beam energy of 35 GeV in line with Eqs. (23) and (24), respectively,
and for the values of the T_c̅c̅(3875)^-–nucleon absorption cross section σ_T_c̅c̅(3875)^-N given above by Eq. (20).
It is seen that the transparency ratio S_A depends more strongly on this cross section
or on the adopted interpretation of T_c̅c̅(3875)^- than that for the quantity T_A.
Thus, there are a sizeable changes ∼ 8, 13, 17, 11% in the ratio S_A
between calculations corresponding to the cases when the loss of T_c̅c̅(3875)^- mesons in nuclear
matter is determined by their absorption cross sections of 32 and 27.7 mb, 27.7 and 21.35 mb, 21.35 and 15 mb, 15 and 10.7 mb, respectively, (cf. Eq. (20)) already for comparably "light" nuclei (^27Al, ^40Ca).
For the medium-mass (^93Nb, ^112Sn) and heavy (^184W, ^238U) target nuclei these changes are
larger and already are measurable. They are about 10, 18, 22, 18% and 13, 21, 27, 21%, respectively.
We see that the highest sensitivity of the quantity S_A to the T_c̅c̅(3875)^-
structure is observed, as is expected, for heavy target nuclei.
For the quantity T_A the analogous changes are smaller and are almost insignificant. They are about
2, 3, 5, 3%, 4, 7, 10, 10% and 6, 10, 15, 12%, correspondingly, in the cases of "light", medium-mass
and heavy target nuclei indicated above.
Therefore, we conclude that the observation, at least, of the A dependence of the transparency ratio S_A, especially for large mass numbers A, in the future high-precision photoproduction experiments at electron-ion colliders
opens the possibility to discriminate between all considered inner configurations of the T_c̅c̅(3875)^- tetraquark state.
At the same time, the precise T_c̅c̅(3875)^- photoproduction data on the A dependence of the transparency ratio T_A in the range of large A, obtained in such experiments, could also additionally help to distinguish, at least, between a pure compact tetraquark state and compact tetraquark-molecule mixture with the compact and molecular probabilities ∼ 50% and 50%, compact tetraquark-molecule mixture with the compact and molecular probabilities ∼ 50% and 50% and a pure molecular configuration of the T_c̅c̅(3875)^- exotic state.

In addition to the A dependence of the transparency ratio T_A at photon energy of 35 GeV presented in Fig. 6, we have investigated the initial photon energy dependence of this ratio for the ^184W/^12C combination. This dependence is shown in Fig. 7. It was calculated in line with Eq. (24) for five adopted options for the T_c̅c̅(3875)^- meson intrinsic structure and for the off-shell target nucleons as well as in line with simple formula (25) for its pure molecular interpretation and for the free target nucleons at rest. One can see that there is a certain
sensitivity of the transparency ratio T_A to various interpretations of T_c̅c̅(3875)^- considered
at all studied photon energies. At these energies, the sensitivity is similar to that available in Fig. 6 for heavy nuclei. Inspection of this figure tells us yet that the simple formula (25) describes well the quantity T_A at above
threshold photon energies (at energies E_γ > 33.9 GeV), where (and at subthreshold energies E_γ < 33.9 GeV) it exhibits practically a flat behavior. This behavior of the transparency ratio T_A can also be used for discriminating between possible scenarios for the T_c̅c̅(3875)^- internal structure.
Thus, it is optimistic that this relative (also integral) observable can, additionally, be used to help
determine this structure.

We discuss now the T_c̅c̅(3875)^- differential observables.
Figs. 8 and 9 show the absolute T_c̅c̅(3875)^- meson momentum distributions from the direct
processes (1), (2), respectively, in γ^12C and γ^184W reactions, determined in line with
Eq. (31) for laboratory polar angles of 0^∘–10^∘, for photon beam energy of 35 GeV and
for five adopted values of the T_c̅c̅(3875)^-–nucleon absorption cross section
in the considered interpretations of T_c̅c̅(3875)^-.
Our results indicate that the differential cross sections for producing the T_c̅c̅(3875)^- mesons
on ^12C nucleus are roughly one order of magnitude smaller than those on ^184W nucleus and, therefore,
they have a lesser chance to be observed (at EIC).
The latter ones have, according to estimates given above, a rather measurable (at EIC) strength ∼ 0.1–0.3 pb/(GeV/c) in the central momentum region of 13–19 GeV/c and also show a rather sizeable variations
[^)Which are similar to those shown in Fig. 2.]^)
, when going from the T_c̅c̅(3875)^-–nucleon absorption cross section value of 10.7 to 32.0 mb.
Such behavior of the T_c̅c̅(3875)^- production differential cross sections on tungsten target nucleus
can also be used to decipher the T_c̅c̅(3875)^- inner structure from comparison our model
calculations with future experimental data.

 Finally, in Fig. 10 we show the "differential" transparency ratio T_A for target combination W/C for
T_c̅c̅(3875)^- mesons produced in the direct reaction channels (1), (2) at laboratory angles
of 0^∘–10^∘ by 35 GeV photons as a function of the T_c̅c̅(3875)^- laboratory
momentum. This ratio is calculated in line with Eq. (24) by using in it the differential cross sections presented
in Figs. 8 and 9 instead of the total cross sections. One can see that the sensitivity of the transparency ratio
T_A to the adopted configurations of T_c̅c̅(3875)^- is similar to that available in Fig. 7 for
the "integral" quantity T_A considered as a function of photon energy. The transparency ratio T_A exhibits dips
at momenta ∼ 9–10 GeV/c. This can be explained by the fact that the differential cross sections for
T_c̅c̅(3875)^- production on ^184W nucleus "are bent down" at these momenta (see Fig. 9)
due to the off-shell kinematics of the direct photon-nucleon collision and the role played by the nucleus-related
effects such as the target nucleon binding and Fermi motion, encoded in the nuclear spectral function
P_A( p_t,E). The spectral functions for ^12C and ^184W, employed in the calculations,
are different [48]. This behavior of the "differential" transparency ratio T_A can additionally be adopted for discriminating between possible schemes for the T_c̅c̅(3875)^- internal structure.

In conclusion, the absolute total and differential cross sections for production of T_c̅c̅(3875)^-
mesons in near-threshold photonuclear reactions as well as their relative (transparency ratios) yields might allow one to shed light on the T_c̅c̅(3875)^- (and, respectively, on the T_cc(3875)^+) intrinsic configuration.



§ 4. CONCLUSIONS


 Motivated by the experimental discovery of a doubly charmed tetraquark state T_cc(3875)^+, in this work we have studied theoretically the inclusive photoproduction of T_c̅c̅(3875)^- mesons (which are the antiparticles of the T_cc(3875)^+) from nuclei in the near-threshold energy region within the nuclear spectral function approach by considering incoherent direct (γp(n) → D^+(D^0)T_c̅c̅(3875)^-Λ^+_c) photon–nucleon T_c̅c̅(3875)^- creation processes as well as five possible different scenarios for their internal structure with the main goal of clarifying the possibility to decipher this structure (and, hence, that of T_cc(3875)^+) in photoproduction via integral and differential observables. We have calculated the absolute and relative excitation functions for T_c̅c̅(3875)^- production off ^12C and ^184W target nuclei at near-threshold photon beam energies of 30–38 GeV, the absolute differential cross sections for their production off these target nuclei at laboratory polar angles of 0^∘–10^∘ as well as the A and momentum dependences of the relative (transparency ratios) cross sections for T_c̅c̅(3875)^- production at photon energy of 35 GeV within the adopted scenarios for the T_c̅c̅(3875)^- meson intrinsic structure. We show that the absolute and relative observables considered reveal a certain sensitivity to these scenarios. Hence, they may be an important tool to clarify it and that of the T_cc(3875)^+. The measurements of these observables could be performed in the future experiments at the proposed high-luminosity electron-ion colliders EIC and EicC in the US and China.



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 Paper Title: Brane-vector dark matter and its connection to inflation and primordial gravitational waves.








nam.caohoang@phenikaa-uni.edu.vn (corresponding author) 
Phenikaa Institute for Advanced Study and Faculty of Fundamental Sciences, Phenikaa University, Yen Nghia, Ha Dong, Hanoi 12116, Vietnam

hung.tranngoc@phenikaa-uni.edu.vn  
Phenikaa Institute for Advanced Study and Faculty of Fundamental Sciences, Phenikaa University, Yen Nghia, Ha Dong, Hanoi 12116, Vietnam


§ ABSTRACT 
The scalar mode describing the fluctuation of the 3-brane (the observable universe) in a five-dimensional bulk spacetime compactified on a circle is absorbed by the Kaluza-Klein U(1) gauge field, leading to a massive brane-vector living on the 3-brane. The brane-vector can be responsible for dark matter because it is odd under a Z_2 symmetry, neutral under the Standard Model (SM) symmetries, and couples extremely weak to the SM particles due to its gravitational origin. Interestingly, the brane-vector dark matter could leave particular imprints on the cosmic microwave background (CMB) and the primordial gravitational waves. Hence, the precise measurements of the CMB and the observations of the primordial gravitational waves generated during the inflation can provide a potential way to probe the extra-dimensions and branes which are the main ingredients of string/M theory.



Brane-vector dark matter and its connection to inflation and primordial gravitational waves
    Tran N. Hung
    August 5, 2024
===========================================================================================



§ INTRODUCTION


Extra-dimensions and branes (hypersurfaces embedded in a higher-dimensional spacetime or the bulk) play a fundamental role in string/M theory which has been considered as a promising candidate for quantum gravity. String/M theory predicts that the spacetime has spatial extra-dimensions wrapped upon a compact internal space and open strings describing the non-gravitational sector with their endpoints are attached to branes. Whereas, closed strings describing the gravitational sector can propagate freely in the bulk. The low energy regime of string/M theory hence can imply a scenario that our observable universe is actually a 3-brane on which the Standard Model (SM) fields are confined, the directions transverse to the 3-brane are compact, and only the gravitational degrees of freedom and other exotic ones can move freely in the bulk <cit.>. In such a scenario, the constraints on the size of the extra-dimensions could be obtained from deviations from the usual Newton's law <cit.> and the experiments of gravitational waves (GW) <cit.>. Therefore, the experimental bounds of the size of the extra-dimensions can be relaxed, where the size of the extra-dimensions is as large as the micron scale. Interestingly, the compactified extra-dimensions can provide a possible solution to the hierarchy problem where the fundamental Planck scale can be much lower than the observed Planck scale <cit.>.

The brane tension is an important parameter that characterizes the typical size of the brane fluctuations along the extra-dimensions. If the brane tension is large enough, corresponding to the branes being too heavy, the branes can be considered as rigid objects. On the contrary, when the brane tension is much lower than the fundamental scale, the branes can fluctuate along the extra-dimensions. The fluctuations corresponding to the oscillating brane are realized as the Nambu-Goldstone bosons, also well-known as the branons, which arise as a result of the spontaneously broken isometry of the internal space due to the presence of the brane <cit.>. The branons are the physical scalar excitations in the low energy spectrum of the effective field theory generically only in the global case where the physical degrees of freedom describing the higher-dimensional gravitation are negligible. 

From the Kaluza-Klein (KK) theory, we know that the isometry of the internal space leads to the gauge transformations where the gauge fields are the off-diagonal components of the bulk metric <cit.> or the connection on the spacetime fiber bundle with the typical fiber being the internal manifold <cit.>. This implies when the isometry of the internal space is broken spontaneously by the presence of the brane, the KK gauge fields would absorb the branons as their longitudinal modes, corresponding to a gravitational Brout-Englert-Higgs mechanism. This leads to massive vector fields appearing on the brane, as well-known as brane-vector <cit.>. The mass of the brane-vector is determined by the brane tension, and hence it is a free parameter. Because the brane tension is much lower than the fundamental Planck scale for the case of the flexible branes, we expect that the mass of the brane-vector can range from the eV regime to the TeV regime. The brane-vector interacts with the SM fields via their energy-momentum tensor with the coupling determined by the observed Planck scale as a result of the fact that the brane-vector has a gravitational origin as new dynamic degrees of freedom describing the behavior of the spacetime at the short-distances <cit.>.

In this paper, we explore the brane-vector as a candidate for dark matter (DM) whose stability is guaranteed by
a Z_2 parity as a result of the 4D general coordinate invariance on the brane. The coupling of the brane-vector to the SM particles is strongly suppressed by the observed Planck scale, hence the relic abundance of the brane-vector DM could not generated by the thermal processes that appeared after the inflation. On the other hand, the brane-vector DM which couples extremely weak to the SM particles could be produced by the gravitational mechanism due to the quantum fluctuations in the inflationary stage <cit.>. The transverse polarizations of a massive vector field that couples minimally to 4D gravity are conformally invariant in analogy to the massless vector field, since its production is suppressed and could be negligible during the inflation <cit.>. In this sense, the observed relic abundance of the brane-vector DM should result from the production of its longitudinal polarization.

The Friedmann equation describing the evolution of the brane universe consists of linear and quadratic terms in terms of the energy density. The quadratic term is inversely proportional to the mass of the brane-vector or the brane tension since it plays an important role in the early universe. (Whereas, the linear term is dominant when the energy density is much smaller than the brane tension corresponding to the late time of the universe evolution.) This novel feature of the modified Friedmann equation combined with the production of the brane-vector DM during the inflation leads to a prediction of the tensor-to-scalar ratio and the reheating temperature in terms of the brane-vector mass. Note that, the reheating temperature can be estimated by the primordial GW spectrum <cit.>. In this way, the brane-vector DM could leave specific imprints on the cosmic microwave background (CMB) and the primordial GWs. Therefore, the precise measurements of the CMB as well as the future detections of the primordial GWs and the B-mode polarization in the CMB can allow us to probe the extra-dimensions and branes.

This paper is organized as follows. In Sec. <ref>, we study the four-dimensional effective field theory derived from a flexible 3-brane which propagates in a bulk spacetime compactified on a circle S^1 and on which the SM and inflaton fields are confined. We show that the low-energy spectrum of the effective theory includes a brane-vector that possesses a discrete Z_2 parity and hence could be responsible for the DM. Then, we compute the relic abundance of the brane-vector DM based on the production of its longitudinal mode during the inflation. In Sec. <ref>, we study the imprints of the brane-vector on the spectrum of the CMB and the primordial GWs carrying the information of the inflation era for two typical potentials of the inflation: i) α-attractor potential and ii) natural potential. Finally, we present a brief conclusion in Sec. <ref>.





§  BRANE-VECTOR DM


For simplicity without loss of generality, we consider the dynamics of a 3-brane propagating in a five-dimensional bulk spacetime compactified on a circle S^1. The action of the system is given by a sum of the bulk action and the brane action as follows

    S   =   S_bulk+S_brane,
    
    S_bulk   =   M^3_5/2∫ d^5X√(-G)(ℛ^(5)-2Λ_5),
    
    S_brane   =   ∫ d^4x√(-g)[-f^4+ℒ_SM+1/2g^μν∂_μϕ∂_νϕ-V(ϕ)].

In the bulk action (<ref>), M_5 is the five-dimensional Planck scale, ℛ^(5) is the Ricci scalar of the bulk spacetime, G is the determinant of the bulk metric, X^M with M=0,1,2,3,4 are the coordinates of the points on the bulk spacetime, and Λ_5 refers to the bulk cosmological constant which is necessarily negative to restore the conventionally Hubble expansion law in the low-energy regime <cit.>. In the brane action (<ref>), f is the tension of the 3-brane, the coordinates of the points on the 3-brane are parameterized by x^μ with μ=0,1,2,3, and g is the determinant of the induced metric g_μν on the 3-brane. The SM fields that are described by the Lagrangian ℒ_SM and the inflaton field ϕ are confined to the 3-brane. 

The presence of the 3-brane could be created spontaneously by a proper physical process, such as the tachyon condensation when turning on the magnetic fluxes <cit.>. This configuration should be realized as the ground state. As a result, the formation of the 3-brane would break spontaneously the local isometry of the bulk spacetime in the compactified extra dimension. As we see below, the Nambu-Goldstone (NG) mode describing the excitation of the 3-brane would be eaten by the KK gauge field, corresponding to the local U(1) isometry of the circle S^1, hence it becomes massive. 



 §.§ Dimensional reduction of the bulk action


Let us study the compactification of the five-dimensional bulk spacetime on the circle S^1 that is parametrized by an angle θ. The bulk metric can be decomposed into the four-dimensional fields as follows

    G_MN=(
    [ g_μν+ρ^2 A_μ A_ν          ρ^2 A_μ;          ρ^2 A_ν              ρ^2;                  ]
    ),

where g_μν is the four-dimensional metric tensor, A_μ is the KK gauge field corresponding to the local isometry in the fifth compactified dimension, and ρ is a scalar field whose vacuum expectation value (vev) determines physically the radius of the circle S^1. With this decomposition, we can write the bulk action (<ref>) in terms of the four-dimensional fields as follows.

    S_bulk   =   M^2_P/2⟨ρ⟩∫ d^4x√(-g)ρ(ℛ^(4)-2Λ_5-ρ^2/4F_μνF^μν+2/3∂^μρ∂_μρ/ρ^2),

where g is the determinant of the four-dimensional metric g_μν, ℛ^(4) is the four-dimensional Ricci scalar, F_μν≡∂_μ A_ν-∂_ν A_μ is the field strength tensor of the KK gauge field, and M_P is the observed Planck scale determined as M^2_P=2π⟨ρ⟩ M^3_5 with ⟨ρ⟩ denoting the vev of the scalar field ρ or the radius of the circle S^1. We can check that the gravitational action (<ref>) is invariant under the following transformation 

    θ   ⟶   θ+α(x),
    
    A_μ   ⟶   A_μ-∂_μα(x).

A set of such transformations forms the local isometry in the compactified dimension associated with a particular class of general coordinate transformations. In particular, the off-diagonal element of the bulk metric transforms as a gauge field of the U(1) local symmetry.

Reducing the bulk action (<ref>) on the internal manifold S^1, we obtain an effective action that consists of the four-dimensional Einstein gravity plus a cosmological constant coupled to the massless U(1) gauge field and the scalar field as follows

    S_4D   ⊃   ∫ d^4x√(g)[M^2_P/2(ℛ^(4)-2Λ_5)-1/4κ^2F_μνF^μν+1/2∂^μ r∂_μ r+⋯],

where we have expanded the scalar field ρ around its vev as ρ=⟨ρ⟩(1+√(3)r/2) with r to be the radion field, κ is the KK gauge coupling given as 

    κ≡√(2)⟨ρ⟩^-1/M_P,

and the ellipse refers to the terms related to the self-couplings of the radion field and its couplings to the four-dimensional metric and the KK gauge field.



 §.§ Effective action on the 3-brane


The induced metric on the 3-brane can be determined by the embedding geometry as ds^2_brane=g_μνdx^μ dx^ν where g_μν=G_MN∂_μ Y^M∂_ν Y^N with Y^M(x) being the embedding function which describes the position of the 3-brane in the bulk spacetime. In the static gauge, the embedding function is given by Y^M(x)=(x^μ,Y^θ(x)). With the ansatz of the bulk metric as given by Eq. (<ref>), the induced metric g_μν is found as follows

    g_μν   =   g_μν-ρ^2(A_μ+∂_μ Y^θ)(A_ν+∂_ν Y^θ),
       =   g_μν-⟨ρ⟩^2(A_μ+∂_μ Y^θ)(A_ν+∂_ν Y^θ)+𝒪(r).


When the 3-brane is created at a certain point Y^θ_0 in the fifth compactified dimension which minimizes the brane action (<ref>) and corresponds to the ground state. In this situation, the presence of the 3-brane breaks spontaneously the U(1) local isometry in the fifth compactified dimension. The fluctuations of the 3-brane around the ground state in the fifth compactified dimension can be parametrized as Y^θ(x)=Y^θ_0+π(x) where π(x) is the NG mode associated with the excitation of the 3-brane along the direction of the U(1) local isometry breaking. As a result, the KK gauge field would absorb this NG mode as its longitudinal mode, leading to the brane-vector. This brane-vector is expressed by the definition X_μ≡ A_μ+∂_μπ. The field strength tensor of the brane-vector is given by 

    X_μν≡∂_μ X_ν-∂_ν X_μ=F_μν.


It should be noted that the NG mode describing the fluctuation of the 3-brane in the fifth compactified dimension would be the physical excitation in the decoupling limit of the KK gauge field corresponding to take the limit κ→0. This physical excitation is well known as the branon which is massive in the situation that the U(1) global isometry in the fifth compactified dimension is realized as an approximate symmetry. Interestingly, it was indicated that the branon can be responsible for the DM candidate <cit.>. However, as indicated in Ref. <cit.>, this limit leads to the violation of the Sublattice Weak Gravity Conjecture <cit.>, the Swampland Distance Conjecture <cit.>, and the Festina Lente bound <cit.> which are recognized as the constraints of quantum gravity on the low-energy effective theories. On the other hand, this limit would belong to the Swampland which consists of the effective field theories that cannot be ultraviolet completed in quantum gravity. 

In this way, we can rewrite the induced metric on the 3-brane in terms of the four-dimensional metric and the brane-vector as follows

    g_μν=g_μν-⟨ρ⟩^2X_μ X_ν+𝒪(r).

Using this result, we can expand the square root of the determinant of the induced metric on the 3-brane as follows

    √(-g)=√(-g)(1-⟨ρ⟩^2/2X^μ X_μ+⋯),

where g refers to the determinant of the four-dimensional metric g_μν. With Eqs. (<ref>) and (<ref>), we can find an effective action on the 3-brane as follows

    S_4D   ⊃   ∫ d^4x√(-g)(-f^4+m^2_X/2κ^2X_μ X^μ+λ/κ^2X^μ X^ν T^mt_μν+ℒ_SM+1/2g^μν∂_μϕ∂_νϕ-V(ϕ)+⋯),

where m_X=√(2)f^2/M_P is the mass of the brane-vector which is acquired due to the spontaneous breaking of the U(1) local isometry in the fifth compactified dimension, T^mt_μν is the energy-momentum tensor of the matte fields (the SM and inflation fields) which is given by

    T^mt_μν=2δℒ_mt/δ g^μν-g_μνℒ_mt,

with ℒ_mt denote the Lagrangian of the matter fields living on the 3-brane, and λ=1/M^2_P is the coupling constant of the brane-vector to the matter fields.

From the actions (<ref>) and (<ref>), we observe that first, the brane-vector is odd under a discrete symmetry Z_2. This means that the effective action (<ref>) is invariant under the transformation X_μ→-X_μ, which means that the brane-vector always appears in the pairs. Hence, this discrete symmetry would guarantee the stability of the brane vector. Second, in analogy to the graviton, the brane-vector couples to the energy-momentum tensor of the matter fields with the coupling strength suppressed strongly by the observed Planck scale. This is due to the fact that the brane vector has a geometric or gravitational origin associated with the off-diagonal component of the bulk metric. Because the brane-vector is stable, neutral, and couples very weakly to the matter fields, it behaves as the DM. 



 §.§ Production of brane-vector particles during the inflation


Because the brane-vector is a massive vector field, it has three physical polarizations corresponding to two transverse components and one longitudinal component which is the NB mode describing the fluctuation of the 3-brane in the fifth compactified dimension. However, the production of the transverse polarizations of the brane-vector by the quantum fluctuations is suppressed during the inflation because the transverse polarizations are conformally invariant. This implies that the contribution of the transverse polarizations to the production of the energy density of the brane-vector can be negligible. Whereas, unlike transverse polarizations, the production of the longitudinal polarization is unsuppressed because it is not conformally invariant. Therefore, the relic abundance of the brane-vector DM which is observed today is essentially due to the production of its longitudinal polarization during the inflation.

The relic abundance of the brane-vector observed today is

    Ω_X=ρ_X/ρ_c,

where ρ_c is the critical density and ρ_X is the present energy density of the brane-vector. As discussed above, the contribution to the present energy density of the brane-vector is approximately that of its longitudinal mode and it is calculated as follows <cit.>

    ρ_X≃ρ_X,L=H^2_i/8π^2a_*k^2_*/a^3_0∫ dln kk^2/k^2_*[a_0/a_*|X_L(k,t_0)/X_0(k)|^2+a^3_0/a_*(k^2+a^2_0m^2_X)|∂_tX_L(k,t_0)/X_0(k)|^2],

where H_i refers to the Hubble parameter during the inflation, a_0 and a_* are the values of the scale factor at the present time and the moment of H=m_X, respectively, the wavenumber k_* is defined as k_*=a_*m_X. X_L(k,t_0) is the solution of the equation of motion for the longitudinal polarization of the brane-vector at the present time, given by

    (∂^2_t+3k^2+a^2m^2_X/k^2+a^2m^2_XH∂_t+k^2/a^2+m^2_X)X_L(k,t)=0,

with an initial condition at the horizon exit given as

    X_L(k,t)|_exit=X_0(k)(1-ik/aH_i)exp{ik/aH_i}.

By solving numerically Eq. (<ref>), we can find X_L(k,t) and then substitute it into the integral given in Eq. (<ref>) to obtain an approximate result which is equal to 1.2 <cit.>. As a result, we derive the energy density of the brane-vector observed at the present time as follows

    ρ_X≃3m^2_XH^2_i/20π^2(a_*/a_0)^3.

From the expansion law along the 3-brane as <cit.>

    H^2+k^2/a^2=(ρ_m+M^2_Pm^2_X/2/6M^3_5)^2+Λ_5/6,

and the relation a_*≃ a_mre√(H_mre/m_X) with a_mre and H_mre denoting the scale factor and the Hubble parameter at the time of the matter-radiation equality, we can determine the ratio a_*/a_0 in Eq. (<ref>) as

    a_*/a_0=(ρ_r,0/(√(7)-1)M^2_P)^1/41/√(m_X),

where ρ_r,0 is the energy density of the radiation at the present time. In addition, using the tensor power spectrum amplitude in the brane-world scenario as A_t≃12x_0[H_i/(2π M_P)]^2 with x_0≡2√(3H^2_i M^2_P/(2f^4)) <cit.> and the relation between the tensor power spectrum amplitude A_t and the scalar power spectrum amplitude A_s given by the tensor-to-scalar ratio r as r=A_t/A_s, we can determine the Hubble parameter H_i in terms of the inflationary quantities that can be observed in the CMB spectrum as

    H^3_i=π^2A_sr/6√(3)m_XM^2_P.

Finally, we find an expression for the relic abundance of the brane-vector DM in terms of its mass, the tensor-to-scalar ratio and the scalar power spectrum amplitude as

    Ω_Xh^2   ≃   0.16×(rA_sM^2_P/10^26GeV^2)^2/3(m_X/10^3 GeV)^7/6.


Using the constraints from both the Planck data on the observed DM relic abundance Ω_CDMh^2≃0.12 <cit.> and the 95% CL upper limit on the tensor-to-scalar ratio as r<0.056 <cit.>, we depict the allowed parameter space for the tensor-to-scalar ratio r and the brane-vector mass m_X in Fig. <ref>.

We observe that these constraints impose a lower bound on the brane-vector mass as m_X≳262.45 GeV. This bound can be transferred to an upper bound on the radius of the compactified extra-dimension. First, we need to determine the brane-vector mass in terms of the radius of the compactified extra-dimension. We observe that the expansion law on the 3-brane as given by Eq. (<ref>) would lead to the standard expansion law in the low energy limit of ρ_m≪ M^2_Pm^2_X/2 and the bulk cosmological constant satisfying 

    Λ_5=-6M^6_5/M^4_P.

Using this relation, we find an expression for the brane-vector mass in terms of ⟨ρ⟩ as follows

    m_X=√(3)/π⟨ρ⟩.

Finally, we determine the constraint on the radius of the compactified extra-dimension as ⟨ρ⟩≲4.14×10^-19 m in order for the brane-vector responsible for the DM candidate. Obviously, this upper bound satisfies the constraint coming from the fifth-force experiments <cit.> which set an upper bound ⟨ρ⟩≲𝒪(40) μm. In addition, we find that the brane-vector mass is about a few hundred GeV for the region r∈[𝒪(10^-3),𝒪(10^-2)] which is predicted in some potential inflation models and is experimentally accessible in the near future/present experiments.




§  CONNECTION OF BRANE-VECTOR DM TO INFLATION AND PRIMORDIAL GRAVITATIONAL WAVE

As discussed above, the brane-vector DM has the geometric origin by which its coupling to the SM particles would be strongly suppressed by the observed Planck scale. Hence, it is extremely hard to detect the signatures of the brane-vector DM at the colliders. However, the inflation and the primordial GWs generated during the inflation are very sensitive to the presence of the brane-vector DM. This can be seen from the non-standard expansion law given by Eq. (<ref>) with the allowed mass region of m_X≳262.45 GeV. The very different behavior of the expansion rate compared to the usual one due to the presence of the brane-vector DM becomes significant in the energy density region around/above 5.32×10^39 GeV^4. Therefore, the production of the brane-vector DM due to the quantum fluctuations during the inflation would leave imprints in the spectrum of the CMB and the primordial GWs whose accurate measurements allow us to probe the brane-vector DM and hence the compactified extra dimension and the brane universe. 

In the following, we assume that the brane-vector particles constitute almost the observed relic abundance by which the tensor-to-scalar ratio can be expressed in terms of the brane-vector mass as follows

    r≃(50.55 GeV/m_X)^7/4.

From this relation, we observe that if the tensor-to-scalar ratio is not too small, the energy density during the inflation is much larger than M^2_Pm^2_X/2. As a result, the expansion law during the inflation is given by

    H_i≃V(ϕ)/√(3)M^2_Pm_X.

The equation of motion which plays the behavior of the inflaton during the inflation in the slow-roll approximation reads

    3H_iϕ̇≃-V'(ϕ).

Also, we can determine the slow-roll parameters in terms of the inflation potential, its derivatives, and the brane-vector mass as follows

    ϵ≃M^4_Pm^2_X/V(ϕ)( V'(ϕ)/V(ϕ))^2,   η≃M^4_Pm_X^2/V(ϕ)V”(ϕ)/V(ϕ).

These slow-roll parameters are related to the scalar spectral index n_s and the tensor-to-scalar ratio r which can be measured in the CMB spectrum as

    n_s-1=-6ϵ+2η,   r=16ϵ.

The number of e-folds between the moment that the comoving mode with the wave number k exits the horizon and the end of the inflation is computed as 

    N_k   =   ∫^ϕ_e_ϕ_kH_i/ϕ̇dϕ
       =   -1/M^4_Pm^2_X∫^ϕ_e_ϕ_kV^2(ϕ)/V'(ϕ)dϕ,

where ϕ_k and ϕ_e refer to the values of the inflaton at the time of the mode k exiting the horizon and the end of the inflation, respectively. We can determine ϕ_e from ϵ≈1 which occurs at the end of the inflation. 

With the equations related to the inflation and the brane-vector DM production during the inflation, we can predict the scalar spectral index n_s and the tensor-to-scalar ratio r in terms of the number of e-folds N_k. The relation between r and N_k is given by

    r^15/7/1078.47=M^4_PGeV^2/V(ϕ_k)( V'(ϕ_k)/V(ϕ_k))^2,

where ϕ_k=ϕ_k(N_k,r) which is a function of the number of e-folds and the tensor-to-scalar ratio and can be determined from Eqs. (<ref>) and (<ref>).
Whereas, the relation between n_s and N_k is determined by 

    n_s=1-3/8r(N_k)+2η(N_k).




 §.§ α-attractor potential

The inflation with the α-attractor potential is given by <cit.>

    V(ϕ)=Λ^4[ 1-exp( -ϕ/M̅) ] ^2n,

where ϕ is the inflaton field coupled minimally to gravity on the 3-brane, Λ is related to the inflationary energy scale, M̅≡√(3α/2)M_P with α to be a parameter determining the inverse curvature of the Kähler manifold, and n is a parameter. Note that, in the case of α=1 and n=1, the α-attractor potential leads to the Starobinsky model <cit.>.

By substituting the potential (<ref>) into Eq. (<ref>), we obtain the slow-roll parameters as follows

    ϵ   =   [2nβ e^-ϕ/M̅/(1-e^-ϕ/M̅)^n+1]^2,
    η   =   (1-e^ϕ/M̅/2n)ϵ,

where

    β≡M_P^2m_X/M̅Λ^2.

At the moment that the mode k crosses the comoving Hubble horizon corresponding to X_k=exp(-ϕ_k/M̅)≪ 1, we obtain the following approximation

    X_k   ≃   √(ϵ)/2nβ,
    η   ≃   ϵ-β√(ϵ).

As a result, we can find approximate expressions for the scalar spectral index n_s and the tensor-to-scalar ratio r as follows

    ϵ   ≃   1/16[√(4(1-n_s)+β^2)-β]^2,
    
    r   ≃   [√(4(1-n_s)+β^2)-β]^2.

Note that, from Eqs. (<ref>) and (<ref>), we can express the parameter β in terms of the scalar spectral index n_s and the brane-vector mass m_X as follows

    β≃ 0.02[4(1-n_s)-958.48(GeV/m_X)^7/4](m_X/GeV)^7/8.


The value of the inflaton at the end of the inflation corresponding to ϵ≃ 1 reads

    X_e≃1+n+2nβ-√(1-n^2+4n(1+n)β+4n^2β^2)/n(1+n),

where 
X_e≡exp(-ϕ_e/M̅). Then, the number of e-folds is calculated by

    N_k=1/2nβ^2[B(1-X_k,2+2n,-1)-B(1-X_e,2+2n,-1)],

where B(z,p,q)=∫^z_0t^p-1(1-t)^q-1dt is the incomplete beta function.


In order to find the relation between N_k and n_s, we solve numerically Eq. (<ref>) for various values of the number of e-folds. The result is shown in Fig. <ref> where n is taken the values from 1 to 10 and the mass of the brane-vector DM is considered as 350 GeV and 1000 GeV.

We observe that the brane-vector DM would be consistent with the inflation which requires 50 - 60 e-folds for its high mass region. The low mass region of the brane-vector DM may be favorable to the inflation if the parameter n in the inflation potential is large enough.

By solving numerically Eq. (<ref>) and using Eq. (<ref>), we obtain the relation between the number of e-folds N_k and the mass of the brane-vector DM, as depicted in Fig. <ref>, with n=1 and n=10. Our prediction is shown by the purple bands with the scalar spectral index taking the value from 0.961 to 0.969. In adition, we show the 68% and 95% CL experimental constraints for n_s and r <cit.> corresponding to the (dashed) blue and (dashed) red contours.  

We see that the number of e-folds required by the inflation is satisfied in the high mass region of the brane-vector DM. Whereas, in order for the low mass region to meet this criterion, the parameter n in the inflation potential needs to be large enough. These are completely consistent with what is observed in Fig. <ref>.

We arrive to investigate the reheating constraint for the brane-vector DM produced during the inflation. The connection between the parameters of the inflation and those of the reheating can be obtained by the following relation 

    ρ_re/ρ_e=e^-3N_re(1+ω_re),

where ρ_re and ρ_e are the energy density at the end of reheating (re) and the end of inflation, respectively, and N_re is the number of e-folds during the reheating corresponding to the equation-of-state parameter denoted by ω_re. The energy density at the end of reheating is determined in terms of the reheating temperature T_re and the effective number of relativistic degrees of freedom g_re at the reheating epoch as ρ_re=π^2g_reT^4_re/30. With the energy density at the end of inflation given by the inflation potential at the end of the inflation V_e as ρ_e≃3V_e/2, we express the reheating temperature as follows

    T_re   =   [(45/π^2g_re)^3π^2A_s/2]^1/12(M^2_Pm_X)^1/3(50.55 GeV/m_X)^7/48e^-3/4N_re(1+ω_re)
          ×[n(n-2β)-1+√(1-n^2+4n(1+n)β+4n^2β^2)/n(n+1)]^n/2.


In order to determine N_re, we use the relation k/(a_0H_0)=a_kH_k/(a_0H_0) with k=0.002 Mpc^-1 being the pivot scale and a_k (H_k) is the scale factor (the Hubble parameter) at the moment which the mode k crosses the horizon, which leads to <cit.>

    k/a_0H_0=e^-(N_k+N_re)a_re/a_mrea_mreH_mre/a_0H_0H_k/H_mre.

From the conservation of entropy, we can find the ratio of a_re to a_0 as follows

    a_re/a_0=(43/11g_s,re)^1/3T_0/T_re,

where g_s,re is the effective number of relativistic degrees of freedom that contribute to the entropy and T_0 is the present temperature of the universe. Then, the number of e-folds at the end of reheating is given by

    N_re   =   4/1-3ω_re[67.38-N_k-ln( k/a_0H_0)-1/12ln g_re +1/3lng_re/g_s,re+1/4ln(π^2A_s/3).
          .+7/16ln(50.55 GeV/m_X)+n/2ln(n(n+1)/n(n-2β)-1+√(1-n^2+4n(1+n)β+4n^2β^2))].


From Eqs. (<ref>) and (<ref>), we can find the relation between the reheating temperature T_re and the mass of the brane-vector DM. In Fig. <ref>, we depict this relation for n=1 (the left panel) and n=10 (the right panel). The region is bounded by the solid and dashed purple curve represents our prediction with the 68% CL value of the scalar spectral index as n_s=0.965±0.004.

We see that the present observation of n_s leads to the large uncertainty for the prediction of the reheating temperature in terms of the mass of the brane-vector DM. The CMB experiments in the future which will be able to decrease the uncertainty of the n_s measurement will provide a more accurate prediction for this relation.


The spectrum of inflationary GWs <cit.>, which is a function of the frequency f, is connected to the brane-vector DM as

    Ω_GWh^2(f)   ≃   A_s/3(πΩ_mhf/a_0H_0)^2(50.55 GeV/m_X)^7/4exp{-1/8(50.55 GeV/m_X)^7/4log(f/f_0)}
          ×g_in/3.36[3.9/g_s,in]^4/3[3j_1(z_k)/z_k]^2T^2_1(f/f_eq)T^2_2(f/f_re),

where “in" denotes the moment that the mode k crosses the horizon, j_1(z) is the spherical Bessel function of the first kind, z_k≡4π f/(a_0H_0), the functions T_1,2(x) describe the change of the primordial GW spectrum at the frequency f_re≃0.26Hz× T_re/(10^7GeV) <cit.> due to the presence of the reheating and are given as follows

    T_1(x)   =   (1+1.57x+3.42x^2)^1/2,
    
    T_2(x)   =   (1-0.32x+0.99x^2)^-1/2.

In Fig. <ref>, we show the prediction of the primordial GW spectrum for various values of the brane-vector mass and the parameter n. In addition, we include the projected sensitivities of the different GW observations, represented by the colored regions, which are the future SKA <cit.>, the space-based LISA <cit.>, BBO <cit.>, DECIGO <cit.>, and Ultimate-DECIGO <cit.>.

We observe that Ultimate-DECIGO can probe the reheating temperature and the brane-vector DM in its allowed mass region if the parameter n in the inflation potential is large enough. On the contrary, if n is low, Ultimate-DECIGO only allows us to probe the low mass region of the brane-vector DM. In this sense, the observation of the primordial GWs produced during the inflation provides a potential way to probe the extra-dimensions and branes: the reheating temperature and the mass of the brane-vector DM which are determined by the spectrum of the primordial GWs are compared to the prediction which is given in Eqs. (<ref>) and (<ref>).




 §.§ Natural potential

We study the connection of the brane-vector DM to the inflation with the natural potential or the axion potential <cit.> which appears naturally in the compactifications of M/string theory <cit.> and provides a plausible solution to the strong CP problem <cit.>. The inflation potential in the natural inflation is given as follows

    V(ϕ)=Λ^4[1-cos(ϕ/f)],

where f is a parameter. Using Eq. (<ref>) with the inflation potential (<ref>), we find the slow-roll parameters as follows

    ϵ = γ^21+cos(ϕ/f)/[1-cos(ϕ/f)]^2
    η = γ^2cos(ϕ/f)/[1-cos(ϕ/f)]^2

where

    γ≡M_P^2m_X/fΛ^2.

At the horizon crossing of the mode k, corresponding to ϵ≪ 1, we find the value of the inflaton field in terms of the spectral index n_s and the brane-vector mass as follows

    cos(ϕ_k/f)≃ -1 + 4/25γ^2[5(1-n_s)-γ(γ+√((5-5n_s)/2+γ^2))]≡ X_k,

where the parameter γ can be expressed as

    γ/√(2)≃1-n_s-300(GeV/m_X)^7/4/√(1-n_s-240(GeV/m_X)^7/4).


The number of e-folds is

    N_k=1/γ^2[X_k-X_f - 2log( 1+X_k/1+X_f)  ],

where X_f≡cos(ϕ_f/f) with ϕ_f being the value of the inflaton field at the end of inflation which is determined by the condition ϵ≃ 1 leading to

    X_f≃ 1+γ/2(γ-√(γ^2+8)).


In Fig. <ref>, we show the dependence of the scalar spectral index n_s on 
the number of e-folds N_k.

We can see that the brane-vector DM is only consistent with the natural inflation in the 2σ region for the value of n_s and its low mass region around 280 GeV. However, as we see above, the low mass region of the brane-vector DM is hard to detect in the spectrum of the primordial GWs in the near future.



§ CONCLUSION


In the present paper, we have explored dark matter and its connection to the inflationary observations and the spectrum of the primordial gravitational waves in a scenario in which our observable universe is realized as a flexible 3-brane that could oscillate in a five-dimensional bulk spacetime compactified on a circle. (Such a scenario can be considered as a low-energy limit of ten-dimensional string theory which is reduced on a five-dimensional internal manifold.) The off-diagonal components of the bulk metric that is the gauge field corresponding to the U(1) isometry of the internal space would absorb the scalar mode describing the fluctuation of the 3-brane in the bulk. As a result, this leads to a massive vector field living on the 3-brane, so-called the brane-vector. Because the coupling of the brane-vector is strongly suppressed by the observed Planck scale, the observed relic abundance of the brane-vector dark matter is essentially due to the production of its longitudinal mode by the quantum fluctuations in the inflationary stage. In this way, the tensor-to-scalar ratio is determined in terms of the brane-vector mass or the brane tension. In addition, the mass of the brane-vector enters into the Friedmann equation describing the evolution of the brane universe, which is very sensitive with respect to the region of the high energy density such as the inflation scale. Therefore, we have studied the imprints of the brane-vector dark matter in the inflationary observables and the spectrum of the primordial gravitational waves. This can open a promising observation window to detect the extra-dimensions and branes which play a fundamental role in string/M theory.


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 Paper Title: Pion polarizabilities from a dispersive analysis of $γγ\to ππ$.











Institut für Kernphysik, Johannes Gutenberg Universität, D-55099 Mainz, Germany


We present results for the charged and neutral pion polarizabilities, obtained through a dispersive analysis of photon-fusion reactions with two pions in the final state. This analysis is motivated by current and future measurements at COMPASS, JLab (Hall D), and the BESIII measurement of single-virtual photon-fusion reactions in the space-like region, offering insight into generalized pion polarizabilities. To improve predictions within an unsubtracted dispersion formalism with only the pion left-hand cut, one needs to consider the influence of heavier left-hand cuts. We parametrize the latter using an expansion in a suitably constructed conformal variable, capturing the analytical structure of the left-hand cuts. The coefficients in this expansion are determined from the matching with the Lagrangian-based approach, the Adler zero constraint for γγ→π^0π^0, and a fit to the cross-section data. The extension to the single-virtual case is also discussed.


Pion polarizabilities from a dispersive analysis of γγ→ππ
    Viktoriia Ermolina1vermolin@uni-mainz.de, 
Igor Danilkin1 Marc Vanderhaeghen1
        

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





§ INTRODUCTION


The γγ→ππ reaction is a promising way to study the electromagnetic characteristics of the pion, such as its polarizabilities. Utilizing dispersive techniques, one can analytically continue the γγ→ππ amplitude into the unphysical region (t=m_π^2 and s=0), and be close enough to the two-pion threshold where cross section data is available. More direct experimental extraction of pion polarizabilities is challenging and is typically carried out through various Primakoff processes <cit.> or by radiative pion photoproduction on the proton <cit.>. Among these extractions, only the latest result for the charged pion polarizability from COMPASS <cit.> confirms the chiral prediction <cit.>. Additional measurements with higher statistics are planned or under discussion at Jefferson Lab's Hall D and at AMBER@CERN.

Our approach is based on the partial wave (p.w.) dispersion relations, which account for the analyticity and unitarity properties of the S-matrix. The unsubtracted dispersion relations with only pion-pole left-hand cut given in <cit.> offer a reasonable description of the photon fusion cross-section data and predict the charged pion dipole polarizability in line with recent COMPASS measurements and ChPT. Meanwhile, the neutral pion dipole polarizability appears far from the two-loop ChPT result <cit.>, exhibiting a different order of magnitude and a different sign. It is reasonable to assume that the neutral pion polarizability pattern will also hold for the single-virtual extension of the unsubtracted dispersion relation with only pion left-hand cut. Its impact on the f_0(500) contribution to hadronic light-by-light (HLbL) scattering in (g-2)_μ remains to be shown.


The neutral pion dipole polarizability discrepancy can be potentially resolved by taking into account corrections from heavier left-hand cuts, which are dominated by the ω exchange <cit.>. However, in the conventional scheme, this requires the introduction of subtraction constants that compromise the high-energy behavior of the p.w. amplitudes. The latter complicates utilizing cross-section data across an extended energy range. Instead, in order to model-independently capture the effect of the heavier left-hand cuts, we employ the conformal mapping technique. With experimental data in both neutral and charged channels, the unknown coefficients in the expansion can be directly fitted to the data. In the absence of precision data close to the threshold (which is the present case), we need to rely on a few assumptions - specifically, the Adler zero of γγ→π^0π^0 amplitude and a resonance exchange model for heavier left-hand cuts. 








§ FORMALISM

The kinematically unconstrained p.w. amplitudes rely on a decomposition of the hadronic tensor of γ^(*)γ^(*)→ππ into a suitable set of invariant functions. For the real photon case, they are given by <cit.>

    h̅^(0)_1=h̅^(0)_++/s ,  h̅_1^(2)=h̅^(2)_+-/s(s-4m_π^2) ,  h̅_2^(2)=h̅^(2)_++/s^2(s-4m_π^2) ,

where h_λ_1λ_2^(J) are the isospin p.w. helicity amplitudes (the isospin index I is not explicitly written for simplicity), and "bar" stands for the Born subtracted amplitude, h̅≡ h-h^Born. For the single and double-virtual cases, the transformation to the kinematically unconstrained amplitudes is more complicated and detailed in <cit.> (also see <cit.>). The new set of amplitudes h_i(s) contains only dynamical singularities, specifically the right- and left-hand cuts, and satisfies the standard dispersion relations. As pointed out in <cit.>, the solution to this dispersion relation can be expressed using different types of Muskhelishvili-Omnès (MO) representations, involving either only right-hand cut integrations or both, which are exactly equivalent <cit.>. However, since typically the left-hand cuts are not well known, the so-called modified MO representation for h̅^(J)_i(Ω^(J))^-1 <cit.> is a preferable choice and is given by

    h̅^(J)_i=Ω^(J)(
    -∫_Rds'/π Im (Ω^(J)(s'))^-1 h^(J),Born_i(s')/s'-s+∫_Lds'/π (Ω^(J)(s'))^-1 Im h^(J),res_i(s')/s'-s) ,

where under the left-hand cut integral, we approximate the imaginary part of the Born-subtracted amplitude by the resonance exchanges <cit.>. In Eq.(<ref>), Ω^(J) represents the Omnès functions, which in the single-channel approach are given as,

    Ω_I^(J)=exp(s/π∫_4m_π^2^∞d s'/s'δ_I^(J)(s')/s'-s).

Here the corresponding phase shifts for D-wave are taken from the Roy analysis <cit.>. For the S-wave amplitudes with isospin I=0,2, we utilize results obtained from a single-channel partial-wave dispersion relation <cit.> (see also mIAM <cit.> and DIA <cit.>)


From the analysis of the radiative couplings, it becomes apparent that the dominant contribution arises from the ω-exchange. Therefore, in <cit.> only light vector mesons were considered. Contributions from other heavier resonances were effectively incorporated by slightly adjusting the  VPγ coupling (which is defined in a Lagrangian governing the exchange of a vector meson), as detailed in <cit.>:

    h^(J),res_i≈ h^(J),V_i,     C_ρ^±,0π^±,0γSU(3)≃C_ωπ^0γ/3≈  g^eff_VPγ .

The coupling g^eff_VPγ is the only parameter used to dispersively describe the cross-section data around the f_2(1270) resonance, resulting in g^eff_VPγ =0.326(4) GeV^-1 <cit.>. This value falls within 15% of the PDG average g^PDG_VPγ=0.38(2) GeV^-1 <cit.>, thus justifying the approximation of left-hand cuts by vector mesons. Given that the dominant resonance contribution comes from ω exchange, forthcoming single-virtual BESIII data around the f_2(1270) resonance <cit.>, will enable us to better constrain the ω transition from factor.

This is particularly crucial for virtualities above 1 GeV^2, where the correct perturbative QCD asymptotic behavior may begin to play a significant role - a factor not presently accounted for in dispersive analyses <cit.>.


While it is possible to utilize the unsubtracted dispersion relation in the form of (<ref>) with (<ref>) for the D-wave, this does not apply to the S-wave. The dispersive integral along the vector left-hand cuts is formally convergent, owing to the asymptotically bounded behavior of the S-wave Omnès functions and the imaginary part of the amplitude h^(0),V_++:

    h^(0)_++   =h^(0),Born_+++Ω^(0) (-s/π∫_Rds'/s' Im (Ω^(0)(s'))^-1 h^(0),Born_++(s')/s'-s+ℒℋ𝒞)
    ℒℋ𝒞   ≡s/π∫_Lds'/s' (Ω^(0)(s'))^-1 Im h^(0),V_++(s')/s'-s .

However, the left-hand contribution acquires significant corrections from the integration over large negative s. Therefore, in <cit.>, the results for the S-wave were produced based only on the pion-pole (Born) rescattering. To improve convergence under the dispersive integral over the left-hand cut, the conventional scheme necessitates the introduction of subtraction constants, compromising the high-energy behavior of the partial wave amplitudes. Instead, we propose utilizing the conformal mapping technique and approximating the contribution from heavier-than-pion left-hand cuts as follows

    h^(0)_++=h^(0),Born_+++Ω^(0) (-s/π∫_Rds'/s' Im (Ω^(0)(s'))^-1 h^(0),Born_++(s')/s'-s+∑_n=1^∞ C_n ξ^n
    ) ,

with the property that ξ(s=0)=0, in order to ensure the soft-photon limit. The conformal mapping variable is chosen such that it maps the left-hand cut plane onto the unit circle. The form of ξ depends on the cut structure of the reaction. For the real photon fusion reaction, all left-hand cuts lie on the real axis. Therefore, ξ is given by

    ξ=√(s-s_L)-√(s_E-s_L)/√(s-s_L)+√(s_E-s_L) ,

with s_E=0. In Eq.(<ref>) s_L is the position of the closest left-hand cut branching point beyond the one pion exchange. Strictly speaking, it is given by the two-pion exchange in the t and u channels, s^(2π)_L=-9m_π^2/4. However, the two-pion discontinuity along these cuts concentrates around t(u)=M_ρ^2. A similar story applies to the three-pion left-hand cut s^(3π)_L=-64 m_π^2/9 and t(u)=M_ω^2, which provides an even better approximation due to the narrow width of the ω-meson. Therefore, for our central result, we take

    s_L ≃ s^(V)_L=-(m_V^2-m_π^2)^2/m_V^2 .

For the single-virtual case with space-like photon virtuality, the left-hand cut consists of the two pieces (-∞,s_L^(-)] and [s_L^(+),0]. 







The cut [s_L^(+),0] is tiny and very close to the physical region. Therefore, we propose explicitly incorporating its contribution evaluating the integral in Eq.(<ref>), while expanding the contribution from the dominant cut using the conformal variable with s_L≃ s_L^(-) and s_E=-Q^2. The latter is needed for the soft-photon limit.

Given the form of ξ(s), the series in Eq.(<ref>) truncated at any finite order is bounded asymptotically, consistent with the once-subtracted dispersion relation in (<ref>). In general, the unknown coefficients C_n can be determined from fitting cross-sectional data in the γγ^(*)→π^+ π^- and γγ^(*)→π^0 π^0 channels. In the absence of very precise data in both channels, additional constraints from χPT and Lagrangian-based models for the heavier left-hand cuts can be used. Particularly, chiral dynamics requires the S-wave γγ→π^0 π^0 amplitude to have an Adler zero

    h^n,(0)_++(s_A)=0 ,   s_A=(1± 0.2)  m_π^2 ,

where s_A=m_π^2 corresponds to NLO.

At NNLO, this position shifts to s_A=1.175 m_π^2 <cit.>. Therefore, we consider a range similar to that in <cit.>. The Adler zero lies between the soft-photon zero and the threshold, thus accommodating the negative dipole polarizability for the neutral pion <cit.>, which corresponds to a negative slope at s=0 for the corresponding amplitude. Note also that at NLO, the Adler zero for γ^*γ^* →π^0 π^0 does not depend on the virtualities of the photons. Another constraint arises from the exchange of vector mesons. We propose the following matching condition

    (∑^∞_n=1 C_n  ξ^n)”_|s=0=ℒℋ𝒞”_|s=0= 2/π∫_-∞^s_Lds'/s'^3 (Ω^(0)(s'))^-1 Im h^(0),V_++(s') ,

where the right-hand side of the above equation is essentially a sum rule (multiplied by a factor of 2) for the over subtracted ℒℋ𝒞(s) at s=0. The integrand in Eq.(<ref>) is now suppressed by an additional power of s in the denominator compared to Eq.(<ref>). For the single-virtual case Eq.(<ref>) modifies to

    (∑^∞_n=1 C_n(Q^2)  ξ^n)”_|s=-Q^2=ℒℋ𝒞”_|s=-Q^2= 2/π∫_-∞^s_L^(-)ds'/(s'+Q^2)^3 (Ω^(0)(s'))^-1 Im h^(0),V_++(s') ,

with the corresponding MO representation

    h^(0)_++   =h^(0),Born_+++Ω^(0) (-s+Q^2/π∫_Rds'/s'+Q^2 Im (Ω^(0)(s'))^-1 h^(0),Born_++(s')/s'-s
       +s+Q^2/π∫_s_L^(+)^0ds'/s'+Q^2 (Ω^(0)(s'))^-1 Im h^(0),V_++(s')/s'-s+∑_n=1^∞ C_n(Q^2) ξ^n) .







§ RESULTS

Both the γγ→π^0π^0 and γγ→π^+π^- channels have been experimentally measured. However, in the low-energy region, there is only old data from Crystal Ball <cit.> and Mark II <cit.> with relatively large errorbars. For the π^0π^0 channel, starting from 0.64 GeV, there is also relatively precise data from Belle <cit.> and preliminary BESIII <cit.>. In the current analysis of the cross-section data, we perform a single-channel description of the S-wave isoscalar channel, therefore limiting ourselves to energies below √(s)=0.85 GeV. The pion-pole post-diction of the cross-section, shown in Fig. <ref>, corresponds to χ^2/N_data=6.5(1.5), where in brackets we show the result excluding preliminary BESIII data. The pion dipole and quadrupole polarizabilities are given in Table <ref>, indicating a significant discrepancy, particularly for the dipole (and to a lesser extent, quadrupole) polarizability for the neutral pion. 



In the scheme with heavier than pion-pole left-hand cuts, we truncate the series in (<ref>) at n_max=3, i.e., take the first two dominant terms into account in both isospin channels. This corresponds to 4 unknown parameters. If one fixes these parameters using ChPT values given in Table <ref>, then the description of the neutral cross-section improves to χ^2/N_data=1.0(0.7), showing the consistency between the ChPT values and experimental data in a dispersive approach. A similar result was shown in <cit.> using the Roy-Steiner equations. We next want to explore the possibility of extracting polarizabilities directly from the current real photon data. Clearly, we found out that the current cross-section data cannot constrain 4 parameters from a direct fit, therefore we rely on a few assumptions - specifically, the Adler zero of γγ→π^0π^0 amplitude (see Eq.(<ref>)) and resonance exchange model for heavier left-hand cuts (see Eq.(<ref>)). One free parameter left in the expansion scheme can then be adjusted to the available cross-section data. The results of the fit correspond to χ^2/(N_data-1)=0.8(0.6) and given in Table <ref> and Fig. <ref>. The obtained charged dipole polarizability turned out to be somewhat smaller than the ChPT result, but still consistent with the COMPASS measurement. Clearly, γγ→π^+π^- data, planned at Jefferson Lab's Hall D, are needed to obtain the charged polarizability more accurately. 











The forthcoming step is to extend this analysis to the coupled-channel cases (γγ→ππ/KK,γγ→πη/KK) and access kaon polarizabilities. Furthermore, the current analysis can be applied to the finite virtuality of one of the photons Q^2 in combination with future data from BESIII.



§ ACKNOWLEDGMENTS

This work was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) within the Research Unit [Photon-photon interactions in the Standard Model and beyond, Projektnummer 458854507 - FOR 5327].








 Paper Title: Quasilocal Newtonian limit of general relativity and galactic dynamics.


apsrev4-1



marco.galoppo@pg.canterbury.ac.nz
School of Physical & Chemical Sciences, University of Canterbury, 
 Private Bag 4800, Christchurch 8140, New Zealand


federico.re@unimib.it
Dipartimento di Fisica Giuseppe Occhialini, Università di Milano Bicocca,
 Piazza dell'Ateneo Nuovo 1, 20126, Milano, Italy
 &
INFN, sezione di Milano, Via Celoria 16, 20133, Milano, Italy


david.wiltshire@canterbury.ac.nz
School of Physical & Chemical Sciences, University of Canterbury, 
 Private Bag 4800, Christchurch 8140, New Zealand

§ ABSTRACT 
A new Newtonian limit of general relativity is established for stationary axisymmetric gravitationally bound differentially rotating matter distributions with internal pressure. The self-consistent coupling of quasilocal gravitational energy and angular momentum leads to a modified Poisson equation. The coupled equations of motion of the effective fluid elements are also modified, with quasilocal angular momentum and frame–dragging leading to novel dynamics. The solutions of the full system reproduce the phenonomenology of collisionless dark matter for disc galaxies, offering an explanation for their observed rotation curves. Halos of abundant cold dark matter particles are not required. 


Quasilocal Newtonian limit of general relativity and galactic dynamics
    David L. Wiltshire
    August 5, 2024
======================================================================



The Newtonian limit of general relativity is crucially important to both fundamental physics and astrophysical phenomenology. In cosmology two key features consistent with observation are: (i) far from any localised sources the Universe is not empty; (ii) the Universe is not globally rotating on the largest scales. 

Conventionally, one invokes an asymptotic Minkowski space, with respect to which additive Newtonian potentials are defined, despite observation (i). Moreover, even though known rotating asymptotically flat solutions of Einstein's equations—the Kerr and Kerr-Newman geometries—possess global non-zero angular momenta at spatial infinity, it is implicitly assumed that superposition of the effectively quasilocal angular momenta of such pointlike sources can be reconciled with observation (ii).

In this Letter we show that such implicit assumptions are misguided, even at nonrelativistic orbital speeds. The conventional limit neglects an important coupling of the quasilocal energy and angular momentum defined by the regional time–averaged motion of matter sources. We set out a new, self-consistent quasilocal Newtonian limit of general relativity, which does not assume a Minkwoskian spacetime background a priori. It introduces novel, first-order features that fundamentally modify the dynamics considered. A direct consequence is the elimination of the need for abundant cold dark matter particles in the modelling of galactic dynamics.

The prescription of a fixed, global spacetime background is not a requirement of general relativity. Indeed, the nonlinearity of Einstein's field equations (EFE), which arises from the self–interaction of matter and geometry, results in dynamical, regional backgrounds in which spacetime itself carries its own energy and angular momentum <cit.>. Moreover, the highly nonlinear nature of general relativity gives rise to a noncommutativity of averaging and limiting procedures in spacetime <cit.>. This has fundamental consequences for the dynamics of non-pointlike sources of large spatial extent, such as disc galaxies <cit.>.

The crucial ingredient of the differentially rotating models <cit.>, and in particular of the exact solutions of Ref. <cit.>, is that essential nonlinearity is retained before any low energy limit is considered. No global background is assumed. This contrasts with: (a) approaches that investigate frame–dragging on asymptotically flat <cit.> or asymptotically Friedmann <cit.> backgrounds, necessitating dark matter halos; (b) dark matter alternatives which modify the laws of gravitation, e.g., modified Newtonian dynamics (MOND) <cit.>.

In this Letter we demonstrate how insights from the conventional Newtonian framework may nonetheless be embedded in a quasilocal framework with an effective fluid. Furthermore, Einstein was never fully satisfied by the extent to which the EFE embody Mach's principle, viz., “Local inertial frames are determined through the distributions of energy and momentum in the universe by some weighted average of the apparent motions” <cit.>. The quasilocal Newtonian limit exhibits such a suitable weighted average, which we apply to disc galaxies showing that the EFE may actually contain an understanding of inertia consistent with MOND phenomenology.

The spacetime metrics can be written in the generalized Lewis–Papapetrou–Weyl form <cit.>,

    
    s^2     =-c^2 e^2Φ(r, z)/c^2( t+A(r, z)  ϕ)^2 +e^-2Φ(r, z)/c^2[W(r,z)^2 ϕ^2 +e^2k(r, z)/c^2( r^2+ z^2)]  ,



The energy-momentum tensor takes the form

    T^μν=((r, z)+p(r,z)/c^2)   U^μU^ν  + p(r,z) g^μν,

where (r,z) is the local matter density, p(r,z) is the effective pressure, and each element of the fluid possesses a 4–velocity U^μ, given by

    U^μ∂_μ=(-H(r,z))^-1/2(∂_t +Ω(r,z)  ∂_ϕ)  .

Here dϕ/dt=Ω(r, z) uniquely defines the angular speed of rotation at any point, and H(r, z) is a normalization factor. Since U^μU_μ=-c^2, it follows that

    H=-e^2Φ/c^2(1+A  Ω)^2+e^-2Φ/c^2W^2   Ω^2/c^2  .

Relevant physical velocities have to be identified to correctly implement any low-velocity limit. In the present case, we define the kinetic and dragging velocities

    := r Ω  , 
        := r χ  ,

where χ := -g_tϕ/g_ϕϕ is the frame–dragging term. For systems in the low-energy regime—i.e., nonrelativistic relative local velocities v ≪ c, weak pseudo-Newtonian potential Φ v^2, nonrelativistic frame–dragging, and small pressure p  v^2—then to leading order  coincides with the special relativistic interpretation of the redshift, whilst  follows by analogy <cit.>.

Let us then write the EFE as

    R_μν = 8 π G/c^4(T_μν-1/2T g_μν)  ,

where T=T^μ_μ = 3p- c^2. With (<ref>)–(<ref>) this yields the partial differential equations (PDEs) <cit.>

   
    Φ^,a_,a + W^,aΦ_,a/W + c^4A^,aA_,a/2W^2e^4Φ/c^2 = 4 π G e^2(k-2Φ)/c^2[(+p/c^2)(1+AΩ)^2e^2Φ/c^2+c^-2W^2Ω^2e^-2Φ/c^2/-H+2p/c^2]   , 
       A^,a_,a - W^,aA_,a/W + 4/c^2Φ^,aA_,a = 16 π G/c^4 W^2Ω1+AΩ/H(+p/c^2)e^2(k-2Φ)/c^2  , 
       W^,a_,a = 16 π G/c^4p   , 
        W_,rr-W_,zz+2/c^2(k_,zW_,z-k_,rW_,r)+2W/c^4(Φ_,r^2 - Φ_,z^2) + c^2/2We^4Φ/c^2(A_,z^2 - A_,r^2) = 0   , 
       c^2/2We^4Φ/c^2A_,z A_,r-W_,rz-2W/c^4Φ_,rΦ_,z+1/c^2(k_,zW_,r-k_,rW_,z) = 0   , 
       1/c^2(Φ^,a_,a-k^,a_,a)-1/2WW^,a_,a + 1/c^2W^,aΦ_,a/W - 1/c^4Φ^,aΦ_,a+ c^2/4W^2e^4Φ/c^2(A_,z^2 + A_,r^2) = 4 π G/c^2e^2(k-2Φ)/c^2(-p/c^2)  ,
 

where a∈{r,z}. Eq. (<ref>) and (<ref>) are the tt and rzcomponents of (<ref>) respectively, (<ref>) and (<ref>) are obtained from the R_zz± R_rr equations, (<ref>) is found by taking the combination W^-1c^-2(g_ϕϕR_tt-2 g_tϕR_tϕ+g_ttR_ϕϕ), and finally (<ref>) follows from the combination 2c^-2(R_tϕ-A R_tt). 

The system of PDEs is completed by the perfect fluid elements' equations of motion, which give 

    
    Hp_,a/e^2(k-Φ)/c^2=[Φ_,a+(c^2 A_,a+2AΦ_,a)Ω+(c^2AA_,a+A^2Φ_,a)Ω^2]e^2Φ/c^2+Ω^2(W^2Φ_,a/c^2-WW_,a)e^-2Φ/c^2  ,


We now take an expansion in powers of v/c to implement the nonrelativistic limit, where v is any relevant local velocity. We apply the self-consistent ansatz 

    W(r,z) = r + 𝒪(v^4/c^4)   ,

to solve (<ref>) up to fourth order. From (<ref>), and the definition of χ, we find A = r  /c^2+𝒪(v^3/c^3). Thus, by (<ref>) it follows that 

    H = -1 + ^2/c^2 - 2/c/c + 2 Φ/c^2 + 𝒪(v^4/c^4) .

Hence, by direct substitution of (<ref>) into (<ref>)–(<ref>) we find

    ΔΦ + 1/2r^2||∇⃗Ł_D||^2 = 4 π G  + 𝒪(v^2/c^2)  , 
       Δ̂Ł_D = 0 + 𝒪(v^2/c^2)   , 
        k_,r = 1/4r(Ł_D,z^2-Ł_D,r^2)+ 𝒪(v^2/c^2)    , 
        k_,z = -1/2r(Ł_D,z Ł_D,r) + 𝒪(v^2/c^2)   ,

where Ł_D := r is the quasilocal angular momentum per unit of mass associated with spacetime rotation, Δ is the standard laplacian and Δ̂ := ∂^2_r -(1/r)∂_r + ∂^2_z, is the Grad–Shafranov laplacian <cit.>. The number of equations reduces by one, as (<ref>) is equivalent to (<ref>) to the order considered. Furthermore, we see that the fluid effective pressure does not enter the EFE as a source term at this order of approximation. 

Eq. (<ref>) is a general relativistic generalization of the Poisson equation, and coincides with it in the absence of spacetime rotation. The extra term in (<ref>) can be interpreted as the rotational energy associated with the quasilocal angular momentum of the averaged background spacetime. This term is absent in the conventional Newtonian approximation in which the background is taken to be Minkowski space a priori. Furthermore, when moved to the r.h.s. of (<ref>), it can be identified as an effective density, ρ𝒬=-||∇⃗Ł_D||^2/(8π G  r^2).  This represents a negative gravitational binding energy relative to the background <cit.>.

Even for nonrelativistic dragging velocities spacetime rotation gives first-order corrections to the field equations. For fixed , Eq. (<ref>) shows that the strength of the pseudo-Newtonian potential needed to sustain equilibrium decreases as the local rotational energy of the background geometry increases. Furthermore, the pseudo-Newtonian potential in (<ref>) is essential to accommodate nonrigid rotation of the perfect fluid source. Indeed, if we set Φ=0 then (<ref>)–(<ref>) are the same equations that characterize a system of rigidly rotating dust <cit.>. As such this choice is unphysical, requiring an identification of  purely with rotational energy.

At the same order, equations (<ref>) reduce to 

    -p_,r/=Φ_,r+Ω Ł_D,r - ^2/r + 𝒪(v^2/c^2)   , 
       -p_,z/=Φ_,z+ΩŁ_D,z+𝒪(v^2/c^2)  .

Hence, the effective pressure of the fluid still plays a crucial role in determining the physics of the system, as expected. In Eq. (<ref>) and (<ref>), we recognize the gradient of the pseudo-Newtonian potential and the centrifugal acceleration in the term ^2/r. The conventional Newtonian equations follow in the case of negligible dragging (≪). However, more generally Eqs. (<ref>), (<ref>) once again show the crucial role that the quasilocal angular momentum of the regional background plays in determining the dynamics. For fixed  and p, the magnitude of the pseudo-Newtonian potential needed to match the l.h.s. of (<ref>) and (<ref>) decreases linearly with the quasilocal angular momentum of the underlying averaged geometry.

The quasilocal Newtonian limit defined here is thus realized by Eqs. (<ref>)-(<ref>). As a first application of the quasilocal Newtonian limit, we derive the dragging velocity profile for a disc galaxy rotation curve supported exclusively by baryonic matter. The baryonic mass distribution in a disc galaxy can be written as =  +, where  and  are the densities of the bulge and the disc, respectively. 

We assume a spherical bulge component given by the Plummer density profile <cit.>

    (R) = 3^2/4π(R^2+^2)^5/2 ,

where R = √(r^2+z^2),  is the scale parameter of the bulge and  is the total bulge mass. The resulting circular velocity in the conventional Newtonian limit is

    (R) = √(G R^2(R^2+^2)^-3/2)  .


The baryonic disc matter distribution is usually taken as (r, z)=Σ(r)Z(z). The surface density of the disc is well approximated by the exponential profile

    Σ(r)=/2π^2e^-r/  ,

where  is the total disc mass and  is the scale length in the radial direction for the galactic disc. The distribution along z, with ∫ Z(z)dz=1, is assumed to be

    Z(z)=1/2e^-|z|/  ,

where  is the thickness scale length and ≪. When the thickness  tends to zero, we apply the conventional infinitely thin disc approximation, i.e., Z(z)=δ(z)<cit.>. The Newtonian circular velocity field in the thin disc approximation, , is given by <cit.>

    
    (r) = √(G/2^3r^2[I_0(r/2)K_0(r/2) -I_1(r/2)K_1(r/2)])  ,


where I_{0,1} and K_{0,1} are modified Bessel functions of the first and second kind.

To explain the observed rotation curves in the conventional Newtonian framework, a spherical cold dark matter (DM) halo must be included, so that the matter density is =  +, where  is typically given by the isothermal profile <cit.>

    (R)=ρDM0/1+(R/)^2  .

Here  is the halo scale parameter, and ρDM0 is the maximum DM density. The DM halo contribution to the rotation curve, , is then <cit.>

    (R)^2=4π G ^2 ρDM0(1- arctan(R/)/R/)   .

Thus the circular velocity of the matter in Newtonian modelling of the disc galaxy, , on the equatorial plane, is given by

    (r,0)=√((r,0)^2+(r)^2+(r,0)^2)  .

In Fig. <ref>, we consider a Milky Way-like galaxy with = 0.8 · 10^10M_⊙, = 0.8 kpc, = 8.1 · 10^10M_⊙, = 2.1 kpc, = 5.69 kpc and ρDM0= 1.88·10^-21 kg/m^3. We plot the predicted rotation curve (r,0), and the respective (r,0), (r) and (r,0). 


To apply the general relativistic solution, let us now assume that =, i.e., that the inferred Newtonian rotation curve corresponds to that of a disc galaxy, and that =. We will demonstrate that the quasilocal energy and angular momentum of the galaxy produce observations equivalent to those of a conventional cold dark matter halo.

We begin by writing the pseudo-Newtonian potential as Φ=-, so that  is given by the Newtonian solution for the chosen , while  is defined as the solution of the Poisson equation sourced purely by the quasilocal rotational energy of the spacetime 

    Δ=||∇⃗Ł_D||^2/2r^2  .


We now solve (<ref>), (<ref>), and (<ref>) for Ł_D by noting that in the thin disc approximation the pressure term is exactly the same for both the Newtonian and the general relativistic cases. Moreover, a similar ansatz can be applied to the spherical bulge, where dark matter is not required to explain the observed dynamics, i.e., it is well captured by purely Newtonian models with purely baryonic matter. Thus we can take p(r,z):=p N (r,z) for the whole system, where p N (r,z) is the expected effective Newtonian pressure. We then use Eqs. (<ref>),(<ref>) to deduce

    = ΩŁ_D,r -(^2-^2)/r   , 
        = ΩŁ_D,z ,

where := √(^2+^2) is the Newtonian contribution to the observed rotation curve from the baryonic mass.

Substituting (<ref>), (<ref>) in (<ref>) it follows that

    Δ =Ω_,rŁ_D,r +2ΩŁ_D,r/r +Ω_,zŁ_D,z -(^2-^2)_,r/r  ,

where we have also used the quasilocal Newtonian limit of the (tϕ) Einstein Eq. (<ref>). Eqs. (<ref>), (<ref>) then give a first-order nonlinear PDE for Ł_D

    ∇⃗Ł_D·[∇⃗Ł_D - 2∇⃗(r)] + 2r(^2-^2)_,r = 0  .

We evaluate Eq. (<ref>) on the galactic plane, z=0, where thanks to the plane symmetry we have Ł_D,z(r,0)=vK,z(r,0) = 0. Thus, the nonlinear PDE (<ref>) is reduced to an algebraic equation for Ł_D,r on the galactic plane, whose solutions are given by 

    Ł_D,r=(r)_,r±√([(r)_,r]^2 - 2r(^2-^2)_,r)  .


We choose the minus sign in (<ref>) so that it reduces to the expected Newtonian case for =. Furthermore, given the baryonic density distribution and the conventional Newtonian profile for (r,0), we can numerically solve (<ref>) for (r,0). In Fig. <ref> we show the resulting frame–dragging velocity profile generating the simulated galaxy rotation curve, . The frame–dragging speed reaches a maximum 42.2km/s at r=24.4kpc, well within our nonrelativistic approximation. Furthermore, as we would expect: (i)  decreases with increasing radius, tending to zero at infinity; and (ii)  is negligible in the bulge, showing the self-consistency of the pressure ansatz used in the calculations. 


The new quasilocal Newtonian limit applies to stationary axisymmetric matter distributions with geometry (<ref>). These symmetry assumptions mean that for a Milky Way-like disc galaxy the approximations here, along with the exact solutions <cit.> hold on time scales, 10^7 t10^9yr. Eqs. (<ref>)–(<ref>) will not apply during galaxy formation, and therefore do not explain how disc galaxies form in a cold dark matter free environment. While such complex problems must be left to future work, the precise definition of finite infinity <cit.> provides a mathematical framework for further exploring the quasilocal Newtonian limit, in settings from the early Universe through to highly dynamical astrophysical systems. The interplay of quasilocal gravitational energy and angular momentum must play a key crucial role.

In summary, we have shown that via a novel self-consistent quasilocal Newtonian limit, general relativity can explain the observed rotation curves of disc galaxies without the need for halos of cold dark matter particles. The phenonomenology of collisionless dark matter is reproduced by the regional gravitational energy and angular momentum of the average spacetime geometry.

This discovery has far-reaching consequences for cosmology, astrophysics and particle physics. Indeed, many interesting questions are now opened up. Within the given realization of the quasilocal Newtonian limit, disc galaxy stability, gravitational lensing and fits to observational datasets must be addressed to definitively test the new paradigm. Comparisons with MOND phenomenology potentially opens the prospect not only of placing MOND within the theoretical framework of general relativity, but also of providing insights into the development of this important new physical limit of Einstein's theory. Finally, in light of our results, the phenomenology of dark matter in the early universe, in the dynamics of galaxy clusters, and in other astrophysical environments should now be reconsidered and further investigated.

Acknowledgments  DLW is supported by Marsden Fund grant M1271 administered by the Royal Society of New Zealand, Te Apārangi. We thank Davide Astesiano, Luca Ciotti, Sergio Cacciatori, John Forbes, Vittorio Gorini, Christopher Harvey-Hawes, Frederic Hessman, Morag Hills, Emma Johnson, Zachary Lane, Ryan Ridden-Harper, Antonia Seifert, Chris Stevens, Shreyas Tiruvaskar and Michael Williams for useful discussions. 






 Paper Title: Real-time chiral dynamics at finite temperature from quantum simulation.


Methodological Foundations of Modern Causal Inference in Social Science Research
    Guanghui Pan[Department of Sociology, University of Oxford. 41 Park End Street, Oxford, the United Kingdom, OX1 1HH. Email: guanghui.pan@sociology.ox.ac.ukguanghui.pan@sociology.ox.ac.uk. Manuscript in progress, do not cite or redistribute without permission.]
    August 5, 2024
========================================================================================================================================================================================================================================================================





§ INTRODUCTION

In recent years, the exploration of quantum simulations for field theories <cit.> has garnered significant attention across many subfields of physics <cit.>. This growing interest is largely driven by the advancements in quantum computing technology, which promises to revolutionize our ability to tackle complex and computationally intensive problems. Traditional classical computers, despite their power, often fall short when dealing with the intricate dynamics and properties of quantum systems due to the exponential growth of computational resources required. On the other hand, quantum simulations hold the potential for breakthrough discoveries in areas such as high-energy physics, material science, and chemistry, aiding in understanding phase transitions <cit.>, solving quantum many-body problems <cit.>, and discovering new materials with unique properties <cit.>.

To explore these potential developments, we focus on the Schwinger model <cit.>, quantum electrodynamics (QED) in (1+1) dimensions. 
The Schwinger model provides an excellent testing ground for studying various phenomena, including confinement, dynamical mass generation, and anomaly-induced symmetry breaking. It offers the simplicity of fewer degrees of freedom while still encapsulating the rich physics of gauge theories, making it an ideal benchmark model for quantum simulations on both quantum and classical hardware <cit.>.


In this study, we explore the real-time dynamics of the chiral magnetic effect (CME) at finite temperature within the massive Schwinger model. The CME refers to the generation of an electric current along a magnetic field in systems where there is an imbalance between left-handed and right-handed fermions <cit.>. This imbalance is typically produced by the presence of a chiral chemical potential. To investigate the CME within the context of the Schwinger model, we employ a quench protocol as outlined in our previous study <cit.>. By introducing a chiral chemical potential through this quench process, we drive the system out of equilibrium and analyze the resulting vector currents and how they evolve over time. The Hamiltonian is adjusted to include the time-dependent chiral chemical potential, enabling the investigation of the CME within a quantum computing framework.

The interest in the study of the temperature dependence of the real-time CME stems from the following consideration. The CME is intrinsically a non-equilibrium, real-time phenomenon. However, when the external fields and the chiral chemical potential vary slowly with time, the CME response is driven by the fermion zero modes, and should thus be temperature-independent <cit.>. This independence of CME current on the temperature has been established in analytical computations <cit.>. However, under a rapid quench, the CME current receives contributions from excited fermion modes <cit.>, and can thus develop a dependence on the temperature. This temperature dependence  has not been evaluated before from first principles. In this paper, we will evaluate it using the methods of quantum simulation in massive Schwinger model at finite temperature.  

We employ the quantum imaginary time evolution (QITE) algorithm to study the thermal states and utilize the Suzuki-Trotter decomposition for real-time evolution. The QITE algorithm is essential for efficient quantum simulation on quantum circuits for several reasons, and various related algorithms have been developed <cit.>.
It enables the simulation of thermal states, crucial for studying phenomena like phase transitions at finite temperatures <cit.>. Since imaginary time evolution involves non-unitary operations, which cannot be directly implemented on quantum hardware, QITE approximates these operations with unitary ones, making it feasible on quantum circuits. Additionally, traditional classical algorithms struggle to prepare thermal states of many-body systems, while QITE allows for efficient exploration of these state spaces. Its versatility makes it applicable to a wide range of quantum systems and models, bridging theoretical models and practical implementations on quantum hardware. By iteratively applying small imaginary time steps and recalibrating the system, the QITE ensures the system gradually reaches the desired state where subsequent measurements are performed to extract physical observables.

Our motivation follows the successful quantum simulations performed on the (1+1)-dimensional Nambu–Jona-Lasinio (NJL) model <cit.>, where quantum algorithms provided insights into the chiral phase transition at finite temperature and chemical potential. Extending this line of research, our previous work on the quantum computational study of field theories has demonstrated the feasibility and efficacy of these simulations in delivering significant theoretical insights. This work on the Schwinger model bridges the study of simple models accessible by current quantum hardware and the more complex theories underpinning the standard model of particle physics. By exploring the unique properties and dynamics of the Schwinger model, we aim to provide insights into chiral dynamics in low-dimensional quantum field theories.

The rest of this article is organized as follows. In Sec. <ref>, we present the model and the quench protocol that we study in this work. In Sec. <ref>, we describe our algorithm of the quantum imaginary time evolution to implement the thermal states and subsequent expectation evaluation to extract physical observables at finite temperature. In Sec. <ref>, we show the results from our quantum simulations, where we discuss the real-time chiral dynamics of the current at finite temperatures, the phase diagram of the Schwinger model, and the temperature dependence of the electric charge (axial current). Finally, the article is concluded in Sec. <ref>. 




§ THE MASSIVE SCHWINGER MODEL WITH THE TOPOLOGICAL TERM
 


 §.§ The Schwinger Hamiltonian

The action of the massive Schwinger model <cit.> with θ term in (1+1)-dimensional Minkowski space is

    S = ∫ d^2x[-1/4 F^μν F_μν + gθ/4πϵ^μνF_μν + ψ̅(iD-m)ψ],

with D=γ^μ(∂_μ + i gA_μ).
Here, A_μ is the U(1) gauge potential, E=Ȧ_1 is the corresponding electric field, ψ is a two-component fermion field, m is the fermion mass, and γ^μ are two-dimensional γ-matrices satisfying the Clifford algebra. When g≠0, the θ-term induces the classical background electric field, which breaks the parity symmetry. The gauge field A_μ and the coupling constant g have mass dimensions 0 and 1, respectively.
By a chiral transformation, ψ→ e^iγ^5θψ and ψ̅→ψ̅e^iγ^5θ, the action is transformed to 

    S = ∫ d^2x[-1/4 F^μν F_μν  + ψ̅(iγ^μ D_μ-me^iγ^5θ)ψ].

The action is invariant under this transformation only up to the boundary term.
It is evident from eq. (<ref>) that the massive theory with a positive mass (m>0) at θ=π is equivalent to the theory at θ=0 but with a negative mass (-m).  

The Hamiltonian in the temporal gauge A_0=0 is given by

    H=∫ dz [    
        E^2/2
        -ψ̅(iγ^1∂_1 - gγ^1A_1 - me^iγ^5θ)ψ]  ,

where the space-time coordinate is labeled by x^μ=(t,z). We denote the Pauli matrices as X, Y, and Z, and use the following convention for the Dirac matrices: γ^0 = Z, γ^1 = i Y, γ^5=γ^0 γ^1 = X.
In (1+1) dimensions, the axial charge density Q_5(x)≡ψ̅(x)γ^5γ^0ψ(x) 
and the vector current density J(x)≡ψ̅(x)γ^1ψ(x) are related 
by Q_5(x) = -J(x). Moreover the vector charge density Q(x)≡ψ̅(x)γ^0ψ(x) and the 
axial current density J_5(x)≡ψ̅(x)γ^5γ^1ψ(x) are related by Q(x) = J_5(x).



 §.§ Real-time chiral dynamics at finite temperature


As we have mentioned in the introduction, the chiral magnetic effect refers to the generation of an electric current along a magnetic field in a system with an imbalance in the number of left-handed and right-handed fermions, induced by the presence of a chiral chemical potential. In the Schwinger model, when the chiral quench is applied, the system is driven out of equilibrium, creating a non-zero chiral chemical potential μ_5. This chiral chemical potential represents an asymmetry between left-handed and right-handed fermions.

Here, we consider the μ_5-quench <cit.> protocol: The system Hamiltonian with θ=0 is prepared at time t<0. Then, starting at t=0, the Hamiltonian rotates the θ angle according to θ=-2μ_5t corresponding to constant chiral chemical potential (i.e. θ̇ = -2μ_5.). The Hamiltonian after the quench is modified as

    H'= dz [    
        E^2/2
        -ψ̅(iγ^1∂_1 - gγ^1A_1-γ^1θ̇/2 - me^iγ^5θ)ψ]  .

The real-time evolution of an operator O can be evaluated by the time-ordered integral 

    O(t)=𝒯[e^i∫_0^tdt'H'(t')]O𝒯[e^-i∫_0^tdt'H'(t')],

and the thermal expectation value of O evaluated at time t at finite temperature T is

    ⟨ O(t)⟩_β=[e^-β HO(t)]/ [e^-β H],

where H is the unquenched Hamiltonian in eq. (<ref>) at θ=0 and β=1/T is the inverse of the temperature of the system.

Notably, this evolving θ = -2μ_5 t after the quench at t = 0 indicates a dynamical chiral chemical potential, driving the system into a non-equilibrium state where left-handed and right-handed fermions are imbalanced. In the presence of this chiral chemical potential, a vector current J (which is typically zero before the quench) is induced. The vector current's time evolution at a finite β, influenced by the quench, aligns with the characteristic behavior observed in the CME at a finite temperature.





 §.§ The lattice Hamiltonian on the qubits


To discretize our Hamiltonian, we use staggered fermions <cit.> on a lattice such that

    ψ(x≡ na) = 1/√(a)(
        [          χ_2n;        χ_2n+1 ]) ,

where a is the finite lattice spacing. Then the lattice Hamiltonian corresponding to eq. (<ref>) is <cit.> 

    H    = a g^2/2∑_n=1^N-1 L_n^2
     -i/2a∑_n=1^N-1[χ^†_n+1 U_nχ_n-χ^†_nU_n^†χ_n+1]
    + mcosθ∑_n=1^N (-1)^n χ^†_nχ_n
       + imsinθ/2∑_n=1^N-1 (-1)^n [χ^†_n+1 U_nχ_n-χ^†_nU_n^†χ_n+1]-θ̇/4a∑_n=1^N-1[χ^†_n+1χ_n+χ^†_nχ_n+1] ,

where L_n is the electric field operator satisfying the Gauss' law constraint

    L_n-L_n-1 =  χ_n^†χ_n-1-(-1)^n/2.

Using the Gauss' law constraint, we eliminate the link fields U_n by the gauge transformation,

    χ_n→ g_nχ_n,  χ^†_n→χ^†_n g_n^†,  
        U_n → g_n+1U_n g_n^†,

with

    g_1=1,    g_n = ∏_i=1^n-1U^†_i.


For the purpose of quantum simulation, we rewrite the lattice Hamiltonian in the spin representation using the Jordan–Wigner transformation <cit.>:

    χ_n=X_n-iY_n/2∏_i=1^n-1(-i Z_i).

The full Hamiltonian in the qubit representation considered in this study is

    H=   ∑_n=1^N-1(1/4a-m/4(-1)^nsinθ)[X_n X_n+1+Y_n Y_n+1]
         +mcosθ/2∑_n=1^N(-1)^n Z_n
       +a g^2/2∑_n=1^N-1L^2_n-θ̇/8a∑_n=1^N-1[X_n Y_n+1-Y_n X_n+1].

Moreover, the local vector and axial charge densities are, respectively <cit.>, 

    Q_n ≡    ψ̅γ^0ψ = Z_n+(-1)^n/2a,
    
        Q_5,n≡    ψ̅γ^5γ^0ψ = X_nY_n+1-Y_nX_n+1/4a .

For later convenience, we introduce the average vector and axial charge operators that sum over all lattice sites,

    Q≡a/N∑_n=1^NQ_n,     
        Q_5≡a/N-1∑_n=1^N-1Q_5,n.

Notably, Q commutes with the Hamiltonian while Q_5 does not. With the boundary condition L_0=0, the Gauss' law constraint, eq. (<ref>), leads to the solution L_n =  a∑_j=1^n Q_j.

Notably, one can project the Hamiltonian to different charge sectors to work with states with specified quantum numbers. In the case of the Schwinger model, the projection operator P to the charge neutral sector (Q=0) can be written as P_Q=0 = ∑_n|n⟩⟨n| where n are all the bitstrings with even number of 1s and 0s. On the qubits, P_Q=0 can be always written as the sum of Pauli-Z operators such that H_Q=0=P_Q=0 H P_Q=0^† is the projected Hamiltonian to the Q=0 sector[For example, at N=4, the projection operator is

    P_Q=0 = P_Q=0^† = 0.375 (1 + Z_1 Z_2 Z_3 Z_4) - 0.125 ( Z_3 Z_4 + Z_2 Z_4 + Z_2 Z_3 + Z_1 Z_4 + Z_1 Z_3 + Z_1 Z_2 )
]. Here the coefficients are determined in such a way that P_Q=0^2=P_Q=0. For the numerical simulation results presented in this paper, we always work in the Q=0 sector. This is because 
the Schwinger model spectrum does not have charged states due to confinement at all scales.
Therefore, when we say Hamiltonian, in what follows, we always mean the charge neutral Hamiltonian H=H_Q=0. 



§ QUANTUM SIMULATION OF QED_2 AT FINITE TEMPERATURE



 §.§ Quantum imaginary time evolution

Quantum imaginary time evolution (QITE) is an efficient way to investigate eigenstates and thermal states on a quantum computer <cit.>. Here, we briefly review the QITE algorithm used in this work for preparing the thermal state and extracting the finite temperature observable.

In QITE, one attempts to approximate the action of the imaginary time evolution
operator on a state | Ψ⟩ with a parameterized unitary
operation:

    c(Δβ)^-1/2 e^-ΔβĤ | Ψ⟩≈ e^-i ΔβÂ(a⃗)| Ψ⟩,

where Δβ is a small time step and c(Δβ)^-1/2 is a normalization coefficient approximated
by 1-2Δβ⟨Ψ | Ĥ | Ψ⟩. The real-time evolution operator Â(a⃗)
is given by a linear combination of N_μ Pauli string:

    Â(a⃗) = ∑_μ^N_μ a_μP̂_μ.

Here, P̂_μ=∏_lσ_μ_l,l is a Pauli string and the subscript μ of a_μ labels the various Pauli strings, and their coefficients a_μ are solved from the linear equation ( S+ S^T)  a= b, where the matrix S and vector b are defined by

    S_μν   =⟨Ψ(β)|P̂_ν^†P̂_μ|Ψ(β)⟩ ,
    
    b_μ   =-i/√(c(Δβ))⟨Ψ(β)| (ĤP̂_μ+P̂_μ^†Ĥ)|Ψ(β)⟩ .

The QITE procedure is repeated for a specified
number of time steps to reach a target total evolution time. Using this iterative approach, we are able to evolve an initial quantum state |Ψ_k(0)⟩ under the unitary operator e^-iΔβ A to any imaginary time β by trotterization:

    |Ψ_k(0)⟩→ |Ψ_k(β)⟩ +𝒪(Δβ) .

Note the sum of |Ψ_k(β)⟩ over all basis states approximates e^-βĤ/(e^-βĤ) at finite temperature T=1/β in the sense that it gives the thermal expectation of an observable Ô:

    1/Z∑_k⟨Ψ_k(β/2)|Ô|Ψ_k(β/2)⟩≈⟨Ô⟩_β=1/Z (e^-βĤÔ),

where Z= (e^-βĤ)=∑_k⟨Ψ_k(β/2)||Ψ_k(β/2)⟩ is the canonical partition function, which is also the sum of the normalizations over all the quantum states.





 §.§ Real-time evolution at finite temperature


Studying the dynamics of the quantum system at finite temperatures is usually complicated. In this work, we are interested in the real-time evolution of the thermal mixed state at finite temperatures of the Schwinger Hamiltonian. Using another evolution Hamiltonian H' that is not equal to the Hamiltonian H of the thermal system, we define a quench process by the real-time evolution of the operator. See eq. (<ref>) for our quench protocol.

To implement this protocol on the quantum circuit, we first use the QITE algorithm to obtain the collection of the |Ψ_k(β/2)⟩ states to represent the thermal state at finite temperature T=1/β. Then, we simply evolve each |Ψ_k(β/2)⟩ in real time as usual:

    |Ψ_k(t,β/2)⟩=𝒯[e^-i∫_0^tdt'Ĥ'(t')]|Ψ_k(0,β/2)⟩.

The final thermal expectation at real-time t is

    ⟨Ô(t)⟩_β=1/Z∑_k⟨Ψ_k(t,β/2)|Ô|Ψ_k(t,β/2)⟩.

Compared with eq. (<ref>), we see that thermal expectation evaluated on the quantum circuit is associated with the states instead of observables. Nevertheless, the two methods are entirely equivalent, which can be seen in our comparison between the quantum simulation and exact diagonalization results. To perform the real-time evolution, we trotterize the time into small time steps using second-order Suzuki-Trotter product formula <cit.>, which gives good convergence.




§ NUMERICAL RESULTS




 §.§ Thermal state preparation

To study the real-time evolution of physical properties at finite temperatures, we must first prepare the initial thermal state at the desired temperatures. In this work, we use the QITE algorithm developed using Qiskit <cit.> in a proceeding work <cit.> for the thermal medium preparation. Specifically, the unitary evolution is implemented using PauliEvolutionGate library with SuzukiTrotter method, and the thermal expectation is evaluated with the Sampler class on the statevector quantum simulators. In particular, we use the QITE to prepare thermal states with four qubits from β/a=0 (aT=∞) to β/a=4.0 (aT=0.25) for the three masses ma=0.25, 0.5, 4.0, and at coupling g=a^-1. A universal trotter step size Δβ = 0.01 is used.



To measure the validity of the quantum simulation, we use the fidelity between two density matrices σ and ρ:

    F(σ,ρ)=(√(√(σ)ρ√(σ)))^2.

Here, we take σ=e^-β H/Z to be the exact density matrix calculated via exact diagonalization and ρ=∑_k|Ψ_k(β/2)⟩⟨Ψ_k(β/2)|/Z for the thermal mixed state prepared by the QITE algorithm. In Fig. <ref>(left), we show the fidelity of these two density matrices as a function of the QITE imaginary evolution parameter β (i.e., inverse temperature). 
For all three cases of different masses, the fidelity is between 0.999 and 1.0, very close to 1, suggesting that the thermal state prepared by QITE is a good practical approximation of the corresponding thermal equilibrium state. In particular, we see a comparatively slight decrease of fidelity in the early β (high temperature) region for the heaviest mass ma=4.0, which is due to the insufficiently small step size Δβ = 0.01 used in the simulation for the heavy mass.

In Fig. <ref>(right), we show the purity of the mixed quantum state ρ at different temperatures and masses. Purity is defined by [ρ^2] and is an index used to determine how much a quantum state is mixed, with values equal 1 indicating a pure state, and values approaching 0 signifying a mixed state. For every mass, we expect that the purity to approach 1 as the temperature approaches 0 (equivalently, as β approaches infinity), since the state is pure at zero temperature (i.e., the ground state). For smaller mass values, such as ma = 0.25, the curve shows that the purity decreases more slowly with temperature, indicating smaller mass states transition to mixed states more gently as the temperature rises. In constrast, as the mass ma increases, the purity drops more rapidly. Despite these differences, all curves will eventually approach a lower purity limit as the temperature continues to increase, showing the natural tendency of quantum systems to shift towards mixed states under a finite temperature. 




 §.§ Real-time evolution of the charge and the current at finite temperature


Now that the thermal state is prepared using the QITE protocol, we study real-time evolution of the total electric and axial charge densities for the system at a selected temperature with both light and heavy masses. Our simulations are conducted using the exact statevector quantum simulator provided by Qiskit <cit.> and compared to the T = 0 limit obtained from exact diagonalization. Specifically, we perform our simulation using eqs. (<ref>) and (<ref>), following the quench protocol provided in Sec. <ref> at various chiral chemical potentials.


The real-time evolution is compared against the zero-temperature limit using exact diagonalization, serving as a benchmark for the results obtained from the quantum simulator at finite temperatures. We first note that the results at T=0 are consistent with the results in <cit.>, which performed the study under the periodic boundary condition at g=0 while we work with the open boundary condition at g≠0. We also observe that ⟨ Q⟩ =0 at all finite temperatures and it does not evolve over time, which is expected. The quantum simulation uses four qubits and applies the QITE algorithm for thermal state preparation. These methods allow for accurate modeling of thermal states and real-time evolution, offering a robust framework for understanding the underlying physics. The second order Suzuki-Trotter product formula <cit.> is used for the real-time evolution, with step sizes dt varying based on the masses: 0.1 for light masses and 0.01 for heavy masses. 



Fig. <ref> shows the real-time evolution of the thermal average of the axial charge ⟨ Q_5(t) ⟩_β for various temperatures, aT = 10.0,1.0,0.25. The mass takes values of ma=0.25, 1.0, 4.0 to examine how light and heavy masses affect the evolution of the vector and axial charge densities, showing how it evolves under different temperatures. The results highlight the role of a chiral chemical potential μ_5 and include both exact solutions and approximate results using quantum simulation. To confirm consistency with theoretical expectations, we show that the total vector charge is conserved and does not depend on time. 

In studying the chiral dynamics in the Schwinger model, our results share several important similarities with our previous results of nonlinear chiral magnetic waves (CMWs) <cit.>, particularly in observing how the oscillation of axial charges is influenced by mass. Here, chiral magnetic waves are collective excitations in a (1+1)-dimensional chiral medium under the influence of a magnetic field. These waves are a direct consequence of the interplay between axial (chiral) and vector (electric) currents in the presence of magnetic fields. Notably, the tendency that heavier mass results in faster oscillations of the axial charge is common <cit.>.
In both studies, it is observed that the axial charge density oscillates more rapidly with increasing mass. In the nonlinear CMWs, the nonlinearity arises because, in massive Schwinger model, the frequency of axial charge oscillations becomes significantly higher than that of the vector charge. This results in the axial charge oscillating rapidly while being nearly confined within static electric dipoles, a phenomenon referred to as the “thumper” solution  <cit.>. The interplay between the excited state (causing rapid oscillations) and the ground state (with almost static behavior) creates a non-linear dynamic in the system.




In Fig. <ref>, we present a comparison between our simulation results and exact diagonalization, observing good agreement when sufficiently small time steps are used. This figure illustrates how accurately our simulation can predict the system's behavior, validating our approach. Furthermore, the rapid oscillations observed in the simulation at a larger mass (ma = 4) also verified our expectation. These oscillations are attributed to the interplay between different energy eigenstates, and their amplitude increases as the temperature rises. The consistency between the simulation in Fig. <ref> underscores the accuracy of our model in capturing the dynamic properties of the system at varying conditions.

To study the dynamical interplay between finite temperature and real-time evolutions, we also present the density plots for ⟨ Q_5(t) ⟩_β in Fig. <ref>. The axial charge oscillates as a function of time t at different temperatures T, with increasing frequency for the heavier mass. Notably, the thermal state is first obtained using the QITE algorithm at finite β/a=1/(aT) values in the range of [0, 4], and then we obtain the density plot versus aT to the range of [∞, 0.25] with interpolation to T=0 temperature. The number of modes for the time oscillations is directly related to the mass in the Schwinger Hamiltonian. 



With the goal of extrapolating our results in the continuum, we perform the exact diagonalization with N=10 qubits in Fig. <ref>. Importantly, we have direct access to the finite temperature thermal state with exact matrix exponentiation using eq. (<ref>), and our exact results are in terms of aT and t. Here, we show the density plot of the thermal expectation value of ⟨ Q_5(t)⟩_T (with temperature in units of aT in the range of [0, 4]). Comparing to results simulated on the quantum simulator shown in Fig. <ref>, we observe the same behavior for both ma=1.0 and 4.0 cases, despite having many more qubits. Quantitatively, we can also observe good agreement between the quantum and classical simulations by counting the oscillation modes and the peak intensity values in the density plots.



§ CONCLUSION

This paper investigates the real-time dynamics of the chiral magnetic effect (CME) at finite temperature in the context of the massive Schwinger model, quantum electrodynamics (QED) in (1+1) dimensions. The study utilizes quantum simulation techniques, combining the quantum imaginary time evolution (QITE) algorithm for thermal state preparation and trotterization for real-time evolution, to examine the thermal properties of the system and the transitions induced by a chiral chemical potential via a quench process.

Our results illustrate the dynamic interplay between the fermion mass and the real-time dynamics of the axial charge. At zero temperature, the corresponding dynamics has been investigated in our previous studies   <cit.>. 
The new result in this paper is the temperature dependence of the real-time CME response. Specifically, at temperatures small compared to the fermion mass, the CME response is similar to the one found previously at zero temperature. However, at larger temperatures, the oscillations of the axial charge induced by the CME are dampened by thermal effects. This can be understood as a result of the contribution from excited fermion modes to the real-time CME response, that may depend on the temperature <cit.>. 

In addition, at large fermion mass the axial charge oscillations become very non-linear. The origin of this  non-linearity can be explained in the following way: in bosonized description of the massive Schwinger model, fermion mass induces non-linear interactions among the bosons, so the chiral magnetic waves become highly non-linear <cit.>.

Finally, our study of CME dynamics suggests that quantum simulations can be used to study other real-time phenomena at finite temperatures in gauge theories that form the basis of the Standard Model. 




§ ACKNOWLEDGEMENT

We thank Meijian Li for helpful comment on the manuscript. We are also grateful to Andrea Palermo and 
Ismail Zahed for useful discussions. This work was supported by the U.S. Department of Energy, Office of Science, National Quantum Information Science Research Centers, Co-design Center for Quantum Advantage (C2QA) under Contract No.DE-SC0012704 (KI, DK, FZ), the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, Grants Nos. DE-FG88ER41450 (DK), DE-SC0012704 (DK) and DE-SC0011090 (FZ), and the  National Science Foundation under grant No. PHY-1945471 (ZK). WQ is supported by the European Research Council under project ERC-2018-ADG-835105 YoctoLHC; by the Spanish Research State Agency under project PID2020-119632GB-I00; by Xunta de Galicia (Centro singular de investigacion de Galicia accreditation 2019-2022), by European Union ERDF, and by the Marie Sklodowska-Curie Actions Postdoctoral Fellowships under Grant No. 101109293.


JHEP




 Paper Title: Numerical integration of the double- and triple-box integrals using threshold subtraction.



A Simple Low-bit Quantization Framework for Video Snapshot Compressive Imaging
    Miao Cao1,2 Lishun Wang2Huan Wang2Xin Yuan2()
    August 5, 2024
==============================================================================



->-/.style=decoration=
  markings,
  mark=at position #1+0.1 with latex,postaction=decorate


§ INTRODUCTION


The construction of locally finite amplitudes of refs. <cit.> calls for an efficient method to compute loop integrals numerically in momentum space. 
A viable approach is based on the subtraction of threshold singularities, which originated in <cit.> and was later extended in <cit.> for the intersecting singularity structures.
In conjunction with the local infrared (IR) and ultraviolet (UV) subtraction of <cit.>, threshold subtraction has proven to be effective for the computation of N_f-contributions to the virtual correction at NNLO for the production of massive electroweak vector bosons at the LHC <cit.>.
In this contribution, we apply the threshold subtraction method to scalar two- and three-loop integrals: the four-point ladder diagrams with massive external legs and massless internal propagators.
These integrals are finite in IR and UV limits and can be evaluated directly in d=4 spacetime dimensions.
Furthermore, these integrals allow us to test the threshold subtraction method  for the first time for two- and three-loop type thresholds without introducing the added complexity of multi-channelling, as we will show below.
These calculations pave the way for future applications, such as extending the work in ref. <cit.> to the complete virtual cross section at NNLO.


§ METHOD

We consider the double- and triple-box integrals

    G^(2)= ∫^4k_1/(2π)^4^4k_2/(2π)^41/A_1⋯ A_7 ,      G^(2)= ∫^4k_1/(2π)^4^4k_2/(2π)^4^4k_3/(2π)^41/B_1⋯ B_10 ,

with 

    A_1   =B_1= k_1^2 ,          A_2=B_2= (k_1-p_1-p_2)^2 ,

    A_3   =B_3= (k_1-p_2)^2 ,          A_4=B_4= (k_1-k_2)^2 ,

    A_5   =B_5= k_2^2 ,          A_6=B_6= (k_2-p_1-p_2)^2 ,

    A_7   =(k_2-q_1)^2 ,          B_7=(k_2-k_3)^2 ,    B_8=k_3^2 ,

    B_9   =(k_3-q_1)^2 ,          B_10=(k_3-p_1-p_2)^2 ,

where the ϵ causal prescription is left implicit and 
p_1+p_2=q_1+q_2, p_i^2=m_i^2≠0, q_i^2=m_i^2≠0 .
Despite being IR- and UV-finite, these integrals remain difficult to evaluate using Monte Carlo integration.
The  ϵ prescription, which formally shifts the on-shell poles of the propagators away from the contour of integration, is impractical for numerical integration.
We summarise how this issue is resolved, following refs. <cit.>. In section <ref>, we analytically integrate the energy component of the loop momenta, in order to expose the threshold singular surfaces. 
In section <ref>, we regularise the corresponding threshold singularities.



 §.§ Integration of loop energies

Time-ordered perturbation theory (TOPT) <cit.>, loop-tree duality (LTD) <cit.>, causal LTD <cit.>, the cross-free family (CFF) representation <cit.>, and partially time-ordered perturbation theory (PTOPT) <cit.> all provide a systematic way of performing the energy integration, 
but they yield different algebraic representations (of the same expression). 
Their derivation exploits that each (quadratic) propagator denominator

    D_i = q_i^2 - m_i^2+
        ϵ =(q_i^0 - E_i)(q_i^0 + E_i)

has two poles at positive and negative on-shell energies q^0_i=± E_i .
After the integration over the energy component of the loop momenta, the integrand is given by

    ∫(∏_j=1^n k_j^0/(2π))
        𝒢^(n)({k_i})
        =  ( -)^n
        ^(n)({k⃗_i}) ,

where 𝒢^(n) denotes the integrand of G^(n).
^(n) is a rational function of the on-shell energies, whose integral over the spatial loop momentum space we denote as I^(n). 
The remaining poles are precisely the threshold singularities that are addressed in the next section.
We will implement our expressions as in ref. <cit.>.
Since CFF, unlike LTD, has no spurious singularities, it is more numerically stable and it is therefore our preferred representation for numerical evaluation.
On the other hand, the LTD representation is more compact and we use it for the residues needed for the threshold counterterms presented below in eq. (<ref>).
We generate these expressions using Form <cit.> and Python, as described in <cit.>. 



 §.§ Subtraction of threshold singularities

For our scalar integrals, we find the threshold singularities listed as Cutkosky cuts <cit.> in figs. <ref> and <ref>. Alternatively, they can be determined from the denominator structure in the CFF or LTD expression.


  

The threshold singularities illustrated in fig. <ref> are 



	_1    =  E_2+E_1-p^0_2-p^0_1             ,            _2 =  E_6+E_5-p^0_2-p^0_1             ,
 
 	_3    =  E_2+E_4+E_5-p^0_2-p^0_1         ,           _4 =  E_6+E_4+E_1-p^0_2-p^0_1         , 
 
 
	_5    =  E_2+E_3-p^0_1                   ,          _6 =  E_6+E_4+E_3-p^0_1              ,

		_7     =  E_3+E_1-p^0_2  ,          _8 =  E_4+E_3+E_5-p^0_2              ,

 _9    =  E_7+E_5-q^0_1  ,          _10 =  E_7+E_4+E_1-q^0_1               ,

 	 _11   =  E_6+E_7+q^0_1-p^0_2-p^0_1       ,          _12=  E_2+E_7+E_4+q^0_1-p^0_2-p^0_1   ,
  

where

    E_1   =|k⃗_1|  ,   E_5=|k⃗_2| ,   E_3=|k⃗_1-p⃗_1| ,   E_4=|k⃗_1-k⃗_2|
     ,
    
    E_2   =|k⃗_1-p⃗_1-p⃗_2|
     ,   E_6=|k⃗_2-p⃗_1-p⃗_2|
     ,   E_7=|k⃗_2-q⃗_1| .

Analogously for ^(3), we find the twenty-one threshold singularities represented in fig. <ref>, which we omit listing explicitly for brevity.
We refer to a threshold that contains a sum of n+1 on-shell energies E_i as an n-loop type threshold. 
The real solutions of =0, where  is one of the above threshold singularities, correspond to poles of the integrand inside the integration domain.
They exist because p^2>0, where p is the momentum flowing through the cut.
If p^2=0 , they would describe a pinch singularity, which we do not have in our examples.
Following refs. <cit.>, we will parameterise the threshold singularities  as r=r_(), where r is the radial variable of hyperspherical coordinates (k⃗_1,…,k⃗_n)==r. We will remove these poles at r=r_() using local counterterms.
This approach allows us to determine the dispersive and absorptive parts of G^(n) through separate Monte Carlo integrations.
More details on the hypercube mapping in the Monte Carlo integration can be found in ref. <cit.>.
We can express the dispersive part of I^(n) as an integral over n spatial loop momenta,

    I^(n)   = 
        ∫[3n]/(2π)^3n(
        ^(n)()
        - ∑_∈_[^(n)]()
        ) ,

and the absorptive part as 

    2 I^(n)   =
        ∫[3n-1]/(2π)^3n-1∑_∈ 
        R_t[^(n)]() ,

where  denotes the set of all threshold singularities.
The threshold counterterm _t is defined as

    _[
        ^(n)(r)
        ]
           =
        R_[^(n)]/r^3n-1(
        χ(r-r_)
        /
            r-r_ -ϵ
        +
        χ(-r-r_)
        /
            -r-r_ -ϵ),

where we choose

    χ(x) = exp[-(x/E_CM)^2]
        Θ[(3 E_CM)^2-x^2] ,

although any function χ would suffice to regulate the UV behaviour of _, provided that χ satisfies χ(0)=1 and χ(x)→ 0 for x→±∞ and is symmetric around the origin.
With these requirements, the integrated counterterm only contributes to the absorptive part of the integral since the Cauchy principal value vanishes by construction, i.e.

    ∫_0^∞r r^3n-1_[^(n)(r)] = π R_[^(n)]() .

The residue R_t reads

    R_[^(n)()] = [r^3n-1^(n)(r),r=r_()]
           = 
        . 
        r^3n-1// rlim_→ 0(^(n))
        |_r=r_() ,

where the threshold parametrisation r_() is defined implicitly through the on-shell condition

    (r_()) = 0 ,     where     r_()>0 .

r_() is the location of the threshold singularity along the direction .
For the one-loop type thresholds, the on-shell condition of eq. (<ref>) can be solved analytically in the radial variable as explained in ref. <cit.>.
In general, the solution to the on-shell condition of eq. (<ref>) for higher-loop type thresholds can always be found numerically, for example using Newton's or Brent's method <cit.>.
However, in our examples and in the centre-of-mass (COM) frame, all threshold singularities collapse to a quadratic equation in r, which can easily be solved analytically at fixed .
Each threshold singularity describes the locus of points such that ()=0, hence it defines a hyper-surface in (spatial) loop momentum space.
The above equations are only guaranteed to be correct and the ϵ causal prescription can be safely removed if the origin  of the spherical coordinate system lies inside all threshold surfaces, i.e.  t()<0 for all ∈ . 
Otherwise, the resulting parametrisation would introduce integrable singularities at the tangents of the threshold surfaces, where / r()=0, in the residues of eq. (<ref>). 
More importantly, if the origin lies inside all the threshold surfaces, it is assured that the counterterms are also correct where the integrand develops higher-order poles, namely at intersections of thresholds surfaces. 
At these intersections, the corresponding residues will also have poles, which, however, locally cancel among each other, as shown in ref. <cit.>.
If no point inside all intersecting threshold surfaces can be found, one can employ a multi-channelling procedure, as in ref. <cit.>.
Fortunately, in our examples, the point 0⃗ lies inside all threshold surfaces simultaneously. We can therefore use the above equations directly without the need for multi-channelling for the double- and triple-box in the COM frame, by setting =0⃗.
We remark that eq. (<ref>) is a local manifestation of the Cutkosky rule <cit.>, in the same way as was shown for its one-loop analogue in ref. <cit.>.



§ RESULTS

Our results are presented in table <ref>.
We use the Vegas adaptive Monte-Carlo algorithm  <cit.> of the Cuba library <cit.> for multidimensional numerical integration. 
The solution to the on-shell condition, r_() , upon which the threshold counterterms (and residues) depend, is found analytically for all one-loop type thresholds <cit.>.  The remaining on-shell conditions we solve numerically using Brent's method implemented in Rust <cit.>, since we have not yet implemented higher-loop type analytic solutions that are valid only in specific reference frames.
However, with the analytic solution, we expect an improvement in runtime and stability.


The reference results in table <ref> are from the analytic computation of ref. <cit.>. 
The example kinematic configurations , , and  are taken from <cit.>, where these integrals were evaluated using contour deformation. 
For completeness, they are listed in table <ref>.
The three-loop ladder diagram's dispersive part evaluation requires the rescue of unstable samples in quadruple precision. 
For this diagram, we checked the stability of the integrand evaluation using a rotation test, as described in <cit.>. This test involves comparing the integrand evaluation at the original kinematic configuration and at a rotated one. If a specified minimum number of digits match, the evaluation is deemed stable. Additionally, the rotation test can be repeated in quadruple precision to potentially recover an unstable sample. If the evaluation remains unstable, it is set to zero. The number of unstable and rescued samples is tracked, and the count of unstable samples should not exceed a certain bound to avoid distorting the Monte Carlo estimate.
It remains to be investigated why the errors are large in the dispersive integral of  for kinematic configurations  and  , a curiosity that was already observed in ref. <cit.>. It  may be attributed to their larger scale hierarchy with respect to . Potentially the method can be adjusted to tackle these kinematic points more efficiently.





§ CONCLUSION

We demonstrated the threshold subtraction method in difficult two- and three-loop finite scalar integrals, where a large number of thresholds are present.
In the future, for the extension of ref. <cit.> to the full NNLO virtual cross section, we will encounter similar two-loop diagrams. 
However, since these diagrams will have massless initial particles, 
some threshold singularities will be traded for pinch singularities, which will instead be subtracted by the local infrared counterterms of ref. <cit.>.
M.V. thanks the organisers of LL2024. 
This work was supported by the Swiss National Science Foundation through its Postdoc.Mobility funding scheme (grant number 211062) and project funding scheme (grant number 10001706).




JHEP




 Paper Title: Gravitational form factors of pion from top-down holographic QCD.






[]daisuke@rcnp.osaka-u.ac.jp
Advanced Science Research Center, Japan Atomic Energy Agency (JAEA), Tokai, 319-1195, Japan
Research Center for Nuclear Physics, Osaka University, Ibaraki 567-0048, Japan


[]iwanaka@rcnp.osaka-u.ac.jp
Research Center for Nuclear Physics, Osaka University, Ibaraki 567-0048, Japan


[]tanaka@hken.phys.nagoya-u.ac.jp
Department of Physics, Nagoya University, Nagoya 464-8602, Japan

§ ABSTRACT 

The gravitational form factors (GFFs) of pions has been calculated from a top-down holographic QCD approach with momentum transfer dependence for the first time. It is important because the GFFs of hadrons have information on the internal stress distribution that may provide insight into the mechanism of how QCD forms hadrons. The forward limit value of this GFFs, i.e. the D-term, was also obtained. Furthermore, in this approach, we observe the so-called glueball dominance, in which pions have gravitational interactions via infinite glueball spectra. 



Gravitational form factors of pion from top-down holographic QCD
    Mitsuru Tanaka
    
================================================================



Introduction.—
How forces act inside hadrons to confine quarks and gluons forming hadrons has not been solved. 
Addressing this question is inseparable from developing an insight into the non-perturbative properties of quantum chromodynamics (QCD), such as confinement and spontaneous chiral symmetry breaking, which is a crucial problem in modern physics. 

Recently, the stress inside hadrons has attracted attention (see a recent review <cit.>) due to the fact that experimentally it has been measured. The stress distribution inside nucleons, extracted from the matrix elements of the nucleon energy-momentum tensor (EMT), is obtained from deeply virtual Compton scattering <cit.>. 
The form factors that characterize the matrix elements are called gravitational form factors (GFFs), which are calculated theoretically from their coupling with gravity. 

The forces inside nucleons were analyzed experimentally in Ref. <cit.> for the quark part (2018) and in Ref. <cit.> for the gluon part (2022). Experimental results show that around the center of the nucleon, the force pushes the quarks outwards, while around the surface it reverses into a confining force, indicating an extremely anisotropic behavior. As a result, the pressure near the center of the nucleon may reach the highest value currently observed in nature. The pion GFFs have also been analyzed experimentally in Ref. <cit.>. The GFFs of nucleons <cit.> and pions <cit.> have also been measured by lattice QCD analysis, a first-principles numerical simulation of QCD. 

In this letter, we are the first to calculate the momentum transfer dependence of the pion GFFs from a top-down holographic QCD approach. We use the Sakai-Sugimoto model, a top-down holographic QCD model based on the D4-D8 brane construction <cit.>. With this approach, the spectroscopy <cit.>, decay widths and electromagnetic form factors <cit.> of mesons, baryons and glueballs have been investigated with strong coupling and large N_c limit. Recently, using the same approach, nucleon GFFs have been formulated and D-terms have been estimated <cit.>. The GFFs of pions <cit.> and other hadrons <cit.> have also been calculated using bottom-up holographic QCD, which has no direct connection with QCD through string theory. 

It is interesting that stress distributions and GFFs of pions<cit.>, which are the focus of this letter, show anomalous behavior. The stress distribution inside hadrons is very sensitive to interactions in the system. The forward limit of the GFFs giving the stress distribution is called the D-term, which takes the value -1 for spin-zero particles in the free Klein-Gordon theory. On the other hand, even an infinitesimally small interaction causes a significant deviation from -1 <cit.>. 


However, the D-term of the Nambu-Goldstone (NG) bosons in the strongly coupled system becomes -1 like the free particles, in the soft pion limit because of the restriction by chiral symmetry <cit.>. 
This fact may provide some insight into the role of chiral symmetry breaking in hadron formation since the D-term carries information about the forces inside hadrons. 
These insights are expected to be validated in more detail via analysis of hadron GFFs from future hadron experiments <cit.>. 




Formulation.—
In this section, we provide the method to derive the GFFs of pions in the top-down holographic QCD <cit.>. 
The method for calculating the GFFs of nucleons with spin 1/2 has been formulated in Ref. <cit.> and will be applied in the following. 


The matrix elements of EMT of a spin-0 particle are expressed as

    <π^a(p_2)|T̂^μν(x)|π^b(p_1)> 
       =δ^ab[2P^μ P^ν A(t)+1/2(Δ^μΔ^ν-η^μνΔ^2)D(t)]e^ix·Δ,

where a,b are the indices for isospin and the kinematic variables are defined as P^μ=(p^μ_1+p^μ_2)/2, Δ^μ=p^μ_2-p^μ_1 and t=Δ^2 with Lorentz indices μ,ν <cit.>. 
In this paper, we employ the mostly plus metric η^μν= diag(-1,1,1,1) and the normalization condition as <p⃗_⃗2⃗|p⃗_⃗1⃗>=2E(2π)^3δ^(3)(p⃗_⃗1⃗-p⃗_⃗2⃗),    E=√(p⃗_⃗1⃗^2+m^2).


The method for calculating the matrix elements of the EMT by holographic QCD is as follows. 
Within the framework of holographic QCD, perturbations to the metric caused by matter fields in the bulk propagate to the boundary, allowing us to extract the expectation values of the EMT in the boundary theory from these perturbations. The top-down holographic QCD used in this study, the Sakai-Sugimoto model, consists of N_f D8 branes embedded as a probe in black 4-brane background <cit.>, a holographic dual of N_c D4 branes. This background is a 10-dimensional curved spacetime of type IIA supergravity, derived from 11-dimensional supergravity related to M-theory on doubly Wick rotated AdS_7 black hole × S^4, 

    ds^2=   r^2/L^2[f(r)dτ^2-dx^2_0+dx_1^2+dx^2_2+dx^2_3+dx^2_11] 
       +L^2/r^2dr^2/f(r)+L^2/4dΩ^2_4 , 
    
        f(r)=    1-R^6/r^6,    R = L^2M_KK/3,    L^3=8π g_s N_c l^3_s,    
    
        R_11=   
        λ/2π N_c M_KK,    λ N_c=L^6M_KK/32π g_s l^5_s=216π^3κ,

where M_KK is the Kaluza-Klein mass in the τ direction, g_s string coupling, l_s string length, λ 't Hooft coupling, and R_11 the radius of the circle compactified in the x_11 direction.

The expectation values of EMT on the boundary, <T_μν>, can be extracted from the boundary values of the metric fluctuations in the bulk in the background spacetime (<ref>) propagated to the boundary as follows, 

    ∫_x_11,τδ g_μν∼2κ^2_7 L^5/6<T_μν>/r^4+...,

where κ^2_7 is related to the 11-dimensional gravitational constant as κ^2_11= Vol(S^4)κ^2_7=8π^2/3(L/2)^4κ^2_7.
Note that, to obtain the above, we need to apply the holographic renormalization, developed by Ref. <cit.>. 


The metric fluctuations δ g_MN of the bulk obey the following linearized Einstein equations:

    ℋ̅_MN=   ℋ_MN-g_MN/2ℋ^P_P=-2κ^2_7𝒯_MN, 
    ℋ_MN≡   ∇^2δ g_MN+∇_M∇_Nδ g^ P_P 
       -∇^P(∇_Mδ g_NP+∇_Nδ g_MP)-12/L^2δ g_MN,

where M, N and P are the Lorenz indices of the 7-dimensional space-time after integration over S^4, as in M,N,P=0,1,2,3,τ,r,11, and the source 𝒯_MN is the EMT of the matter fields which here considered only the pion field. 
We solve Eq. (<ref>) under the axial gauge, δ g_Mr = 0. 

In order to examine the EMT 𝒯_MN in the bulk, we embed the D8 branes into the 11-dimensional space, considering the action as

    S_D8= 
       -C/2π R_11g_s∫ dτ dx_11d^4xdrdΩ_4(δ(τ)+δ(τ-π/M_KK)) 
       ×√(-G_(11))/√(G_ττ)1/4G^MNG^PQ tr[F_MPF_NQ]+...,

with C=(64π^6l_s^5)^-1 and the metric tensor G_MN in Eq. (<ref>). We define the EMT as

    𝒯_MN=-2/√(-G_(11))δ S_D8/δ G^MN_(11) Vol(S^4) ,

which leads to the following expression:



    𝒯_MN(Δ⃗,r) ≡ 2π R_11∫ dτ^'𝒯_MN(τ^',k⃗,r), 
       𝒯_μν (Δ⃗,r)=L^5/r^5π^2Cr^2 L^2/3g_s √(f(r))) tr[F_μρF_ν^ρ+(1+z^2)^4/3M_KK^2F_μ zF_ν z-η_μν/4(F_ρσF^ρσ+2(1+z^2)^4/3M^2_KKF_ρ zF^ρ_z)], 
       𝒯_11,11(Δ⃗,r)=L^5/r^5π^2 Cr L^2/3g_s√(f(r)) tr[-1/4F_ρσF^ρσ-(1+z^2)^4/3M^2_KK/2F_ρ zF^ρ z], 
       𝒯_μ z(Δ⃗,r)=L^5/r^5π^2 Cr L^2/3g_s√(f(r)) tr[F_μρF_z^ρ],    𝒯_zz(Δ⃗,r)=L^5/r^5π^2 Cr L^2/3g_s√(f(r)) tr[1/2F_zρF_z^ρ-1/4(1+z^2)^4/3M^2_KKF_ρσF^ρσ],


where the radial coordinate r  (R≤ r<∞) is related to z  (-∞<z<∞) as z=±√(r^6/R^6-1).
In this study, we consider the case where the pion field is the only matter field as the source of the gravitational waves and decompose A_M(x^ν,z) as follows

    A_μ(x^ν,z)=0,     A_z(x^ν,z)=π(x^μ)ϕ_0(z), 
       ϕ_0(z)=c_0(1+z^2)^-1,

where c_0^-2=2M^2_KKκ∫ dz(1+z^2)ϕ_0(z)^2 is the normalization constant of the eigenfunction ϕ_0(z) <cit.>. Since the pion field is an external line, it can be replaced by a plane wave as π(x^ν)∼ e^ip_ν x^ν with incoming momentum p_ν.

Since we are interested in the EMT of QCD at the boundary such as to satisfy Δ^μ<T_μν>=0, we consider the metric fluctuations that are transverse Δ^μδ g_μν∼0 near the boundary, which can be parametrized as δ g_ij∼δ g^ TT_ij+(δ_ij-Δ_iΔ_j/Δ⃗^2)/5δ g^α_α with the transverse-traceless (TT) part δ g^ TT_ij, i,j=1,2,3 and α=μ,τ,11. Under the axial gauge, the Einstein equation requires the six-dimensional traceless condition δ g^α_α∼0 in the asymptotic AdS region (r→∞) labeled by (x^μ,τ,x^11). 
Therefore, we consider TT modes in the metric fluctuation, which is nothing but glueball spectra.
Among the 14 TT modes named in Ref. <cit.> (see also Ref. <cit.>), only T_4 and S_4 contribute to the EMT. Actually, substituting the pion field (<ref>) into equation (<ref>), we confirm that δ g^ other_μν falls faster than T_4 and S_4 mode at r→∞. 
By decomposing the metric fluctuations into each mode, δ g_μν=δ g^ T(2)_μν+δ g^ T(0)_μν+δ g^ S(0)_μν+δ g^ other_μν, the Einstein equations reduce to 

    r^2/L^2[1/L^2r^5∂_r((r^7-rR^6)∂_r)-L^2Δ^2/r^2]L^2/r^2δ g^ T_μν=-2κ^2_7𝒯^ T_μν, 
       r^2/L^2[1/L^2r^5∂_r((r^7-rR^6)(∂_r+144r^5R^6/(5r^5-2R^6)(r^6+2R^6))) 
       -L^2Δ^2/r^2]L^2/r^2δ g^ S(0)_μν|_r→∞=-2κ^2_7𝒯^ S(0)_μν,

where the source of each mode is decomposed as 𝒯_μν=𝒯^ T_μν+𝒯^ S(0)_μν+𝒯^ other_μν (𝒯^ T_μν≡𝒯^ T(2)_μν+𝒯^ T(0)_μν), and the two T_4-modes are combined as δ g^ T_μν=δ g^ T(2)_μν+δ g^ T(0)_μν, since they obey the same equation. 

Following the same manner as in Ref. <cit.>, each part of EMT is written as 


    2κ^2_7𝒯^ T_μν=2κ^2_7𝒯_μν-2κ^2_7𝒯^ S_μν-2κ^2_7𝒯^ other_μν, 
       2κ^2_7𝒯^ S_μν=r^6+2R^6/4(r^6-R^6)(η_μν-Δ_μΔ_ν/Δ^2)[1/L^4r^3∂_r(r(r^6-R^6)∂_r(a+b)-3R^6(a+b))-Δ^2b], 
       2κ^2_7𝒯^ other_μν=1/L^4r^3∂_r[(η_μν-Δ_μΔ_ν/Δ^2)r(r^6-R^6)∂_r a-3R^6η_μνb],  
       ∂_r a=-3R^6/(5r^6-2R^6)b+9r^11/(5r^6-2R^6)R^62κ^2_7𝒯_zz,    b=-iΔ^μ/Δ^2r^6√(r^6-R^6)/R^9 2κ^2_7 𝒯_μ z,


with the ansatz, 

    δ g^ other_μν(Δ⃗,r)=r^2/L^2Δ_μΔ_ν/Δ^2a(r,Δ), 
       δ g^ other_11,11(Δ⃗,r)=r^2/L^2b(r,Δ), 
       δ g^ other_ττ(Δ⃗,r)=r^2f(r)/L^2(-b(r,Δ)).



We determine the perturbation to the metric caused by the pion field as the bulk EMT source using Green's function method as follows, 

    δ g^ T/S_μν(Δ⃗,r)∼r^2/L^2∫ dr^'√(-G_(7)(r^'))G^ T/S(Δ⃗,r,r^') 
       ×L^2/r^'^22κ^2_7𝒯^ T/S_μν(Δ⃗,r^'),

where, for S_4 mode, we can write as this form in the r→∞ limit, and ∼ implies this point. 
The Green's function G^ T/S is written as 

    G^ T/S(Δ⃗,r,r^')=∑^∞_n=1Ψ^ T/S_n(r)Ψ^ T/S_n(r^')/(m^ T/S_n)^2+Δ⃗^2,

by expanding in the complete system {Ψ^ T/S_n(r)} which obeys the eigenvalue equation, 

    1/r^3L^4∂_r((r^7-rR^6)∂_rΨ^ T_n)=-(m^ T_n)^2Ψ^ T_n, 
       1/r^3L^4∂_r((r^7-rR^6)(∂_r+144r^5R^6/(5r^6-2R^6)(r^6+2R^6))Ψ^ S_n) 
       =-(m^ S_n)^2Ψ^ S_n,

where m^ T/S_n is n'th glueball mass of T/S mode and Ψ^ T/S_n(r) satisfy the completeness conditions,

    r^3/L^3∑^∞_n=1Ψ^ T/S_n(r)Ψ^ T/S_n(r^')=δ(r-r^'), 
       ∫^∞_R drr^3/L^3Ψ^ T/S_n(r)Ψ^ T/S_m(r)=δ_mn.








Results.—
Substituting the pion field into the EMT of the bulk, we read off the matrix elements of the boundary EMT from Eq. (<ref>) with the dictionary Eq. (<ref>) as follows, 

    A(t)=6/L^7∫^∞_Rdr^'∑^∞_n=1α^ T_nΨ^ T_n(r^')/(m^ T_n)^2+Δ⃗^2r^'^3/L^3P^μν_ t𝒯_μν(Δ⃗,r^'), 
       D(t)= 
       -6/L^7∫^∞_Rdr^'∑^∞_n=1α^ T_nΨ^ T_n(r^')/(m^ T_n)^2+Δ⃗^2r^'^3/L^3P^μν_ s(𝒯_μν+𝒯^ S_μν+𝒯^ other_μν) 
       +6/L^7∫^∞_Rdr^'∑^∞_n=1α^ S_nΨ^ S_n(r^')/(m^ S_n)^2+Δ⃗^2r^'^3/L^3P^μν_ s𝒯^ S_μν, 
       α^ T/S=(m^ T/S_n)^2L^4/6∫^∞_R dr r^3Ψ^ T/S_n(r),
 
where P^μν_ t and P^μν_ s are defined as P^μν_ t=(3P^μ P^ν/P^2-η^μν)/P^2 and P^μν_ s=(P^μ P^ν/P^2-η^μν)/Δ^2. 

To calculate infinite sums, we expand as 

    6/L^7∫^∞_Rdr^'∑^∞_n=1α^ T/S_nΨ^ T/S_n(r^')/(m^ T/S_n)^2+Δ⃗^2=∑^∞_k=0F_k^ T/S(r^')(-Δ⃗^2)^k, 
       F^ T/S_k(r^')=∫_R^∞ drr^3/L^3∑_n=0^∞Ψ_n^ T/S(r)Ψ_n^ T/S(r^')/(m_n)^2k.

From the eigenvalue equations (<ref>), F^ T/S_k(r) satisfies the following equations:

    1/r^3L^4∂_r((r^7-rR^6)∂_rF^ T_k)=-F^ T_k-1, 
       1/r^3L^4∂_r((r^7-rR^6)(∂_r+144r^5R^6/(5r^6-2R^6)(r^6+2R^6))F^ S_k) 
       =-F^ S_k-1,

where we can prove F^ T/S_0=1 by using Eq. (<ref>). 
We determine F^ T/S_k(r) by solving it as the recurrence formula with boundary conditions on ∂_r F^ T/S_k(r)|_r→ R=1,  F^ T/S_k(r)|_r→∞=0. 

We show the results of GFFs that truncate the expansion at k=10 using Eq. (<ref>) in FIG. <ref> (also including results that truncate at k=8,9), where we set κ =0.00745 and M_KK =949  MeV which are determined by ρ meson mass 776   MeV and the pion decay constant f_π=92.4  MeV <cit.>. 
In the current calculation, D(t) begins to deviate from around t∼0.5   GeV^2 while A(t) converges well in the range 0<t<0.8   GeV^2 on the expansion up to k=10 with momentum transfer. The current approach begins to deviate from QCD in the region where t is close to the M_KK, so further efforts are unnecessary. 

In the top-down holographic QCD approach, the amplitude of D(t) dumps more rapidly than that of A(t), which is qualitatively consistent with the results of lattice QCD <cit.>. 
These differences are due to the contribution of S_4 modes to D(t), which is not observed in the bottom-up holographic QCD framework <cit.>. 
Also in this approach, as in previous studies <cit.>, we observe the so-called glueball dominance, in which pions have gravitational interactions via infinite glueball spectra. 

As you can see in FIG. <ref>, our result shows D(0)=-1. 
Note that this is a result of our calculations with loop effects in the large Nc limit, unlike the fact A(0)=1, given by the general constraints. 
This is consistent with the low-energy theorem obtained in the soft pion limit <cit.>. The result is as expected since in our present work we consider the Sakai-Sugimoto model, which is the holographic dual of QCD in the chiral limit. Several methods have been proposed on how to introduce quark masses into this model <cit.>, thus analysis beyond the chiral limit is a future work. 










Summary and outlook.—
We calculated the GFFs of pions from the top-down holographic QCD. 
We solve the Einstein equations in bulk theory with pion fields as matter fields and derived the GFFs using the dictionary (<ref>).
Our results are qualitatively compatible with those from lattice QCD <cit.> and successfully reproduce the feature where the behaviors of A(t) and D(t) are different, which is distinct from bottom-up holographic QCD <cit.>.
Additionally, we obtained the D-term as D(0)=-1 in the chiral limit. 

Finally, this study opens up several avenues for future research. First, it is possible to generalize to other hadrons, such as nucleons and vector mesons with non-zero spin. Also, it is very interesting to investigate beyond the chiral limit, because the connection between the Gell-Mann–Oakes–Renner relation and the stability of the stress distribution has been discussed in Ref. <cit.>. It is also important to investigate the behavior of the GFFs and the stress distribution inside hadrons under finite temperatures, densities, and magnetic fields, which change the nature of the interaction in the system. 




Acknowledgements.—
We are grateful to the authors of Ref. <cit.> for generously supplying the data tables utilized in FIG. <ref>.
We also thank Atsushi Hosaka, Kei Suzuki and Shigeki Sugimoto for useful discussions. This work was supported by the Japan Society for the Promotion of Science (JSPS) KAKENHI (Grant Nos. JP24K17054, JP24KJ1620). 





*










 Paper Title: Quantum Computing Universal Thermalization Dynamics in a (2+1)D Lattice Gauge Theory.



=5

IQuS@UW-21-085,UMD-PP-024-08




niklasmu@uw.edu
InQubator for Quantum Simulation (IQuS), Department of Physics,
University of Washington, Seattle, WA 98195, USA

Department of Physics, Duke University, Durham, NC 27708
Duke Quantum Center, Duke University, Durham, NC 27701
The NSF Institute for Robust Quantum Simulation, University of Maryland, College Park, Maryland 20742, USA

Duke Quantum Center, Duke University, Durham, NC 27701
Department of Electrical and Computer Engineering, Duke University, Durham, NC 27708
School of Applied and Engineering Physics, Cornell University, Ithaca, NY 14853

Department of Physics and Maryland Center for Fundamental Physics, University of Maryland, College Park, MD 20742 USA
Joint Center for Quantum Information and Computer Science, NIST and University of Maryland, College Park, MD 20742 USA
The NSF Institute for Robust Quantum Simulation, University of Maryland, College Park, Maryland 20742, USA
National Quantum Laboratory (QLab), University of Maryland, College Park, MD 20742 USA

Department of Physics, Duke University, Durham, NC 27708
Duke Quantum Center, Duke University, Durham, NC 27701
Department of Electrical and Computer Engineering, Duke University, Durham, NC 27708
The NSF Institute for Robust Quantum Simulation, University of Maryland, College Park, Maryland 20742, USA

§ ABSTRACT 
Simulating non-equilibrium phenomena in strongly-interacting quantum many-body systems, including thermalization, is a promising application of near-term and future quantum computation.
By performing experiments on a digital quantum computer consisting of fully-connected optically-controlled trapped ions, we study the role of entanglement in the thermalization dynamics of a Z_2 lattice gauge theory in 2+1 spacetime dimensions. 
Using randomized-measurement protocols, we efficiently learn a classical approximation of non-equilibrium states that yields the gap-ratio distribution and the spectral form factor of the entanglement Hamiltonian. 
These observables exhibit universal early-time signals for quantum chaos, a prerequisite for thermalization.
Our work, therefore, establishes quantum computers as robust tools for studying universal features of thermalization in complex many-body systems, including in gauge theories.


Quantum Computing Universal Thermalization Dynamics 


in a (2+1)D Lattice Gauge Theory 

    Marko Cetina
    February 2024
==========================================================================================








§ INTRODUCTION


Thermalization of isolated quantum many-body systems
in e.g., ultra-cold atomic gases <cit.>, trapped ions <cit.>, condensed matter physics <cit.>, cosmology <cit.>, and nuclear and high-energy physics <cit.>, remains a vibrant frontier.  
Most quantum systems thermalize according to the Eigenstate Thermalization Hypothesis (ETH) <cit.>, which
posits that observables eventually attain thermal values after long unitary evolution <cit.>. Yet, probing large quantum many-body systems undergoing thermalization is inherently challenging due to the non-equilibrium nature of the process, which precludes using first-principle classical computational techniques. Recent advancements in quantum information theory and experiment <cit.> have brought this topic within immediate experimental reach, potentially enabling, via quantum simulation <cit.>, verification of important thermalization paradigms, and discovery of novel principles grounded in quantum information science.


While it is commonly posited that quantum chaos and ergodicity are prerequisites for thermalization <cit.>, their demonstration in the context of quantum many-body systems remains somewhat elusive. Indicators of chaos and ergodicity involve measures associated with the eigenvalues of a given Hamiltonian <cit.>, or the properties of its eigenstates <cit.>. A connection has been drawn recently between 

the entanglement of quantum states and quantum chaos, via the so-called entanglement Hamiltonians (EHs) <cit.>. This connection can, in principle, be leveraged to experimentally probe thermalization via quantum simulation. Most studies to date, nonetheless, have stayed in the realm of theoretical exploration <cit.>.



Analog quantum simulation allows one to monitor a quantum system continuously in time. However, current analog quantum simulators have limited programmability and are restricted to probing specific physical models and a limited set of observables <cit.>. In contrast, universal digital quantum computers allow, in principle, the probing of the dynamics of any physical model. The universal control in digital quantum computers enables the use of tomographic techniques to extract a wide range of observables, including entanglement.  
Digitized time evolution, via 

Trotterization <cit.> or other schemes <cit.>, limits the near-term simulations to times that are short compared to those of thermalization. Nonetheless, there exist universal phenomena that are indicative of quantum chaos and ergodicity at earlier times. Probing these phenomena, therefore, is an attractive near-term opportunity for digital quantum computers.

Gauge theories and their lattice formulations are among prime physical models whose simulations will benefit from quantum-computing technology <cit.>. Gauge theories are key in high-energy and nuclear physics <cit.>, condensed and synthetic quantum matter <cit.>, local fermion-to-qubit mappings <cit.>, and quantum-error correction  <cit.>. Studying thermalization dynamics of gauge theories, e.g., in early universe and in high-energy particle collisions, remains challenging using first-principles simulation methods <cit.>. As a first step in experimentally probing thermalization dynamics of gauge theories, we study a Z_2 lattice gauge theory (LGT) in 2+1 spacetime dimensions <cit.> using a digital trapped-ion quantum computer <cit.>. 
We use a chain of fifteen ^171Yb^+ ions to realize a general-purpose fully-connected digital quantum computer with twelve qubits, and use this computer to natively and accurately encode system's initial state, evolve it in time, and measure observables.

Our analysis relies on EH tomography <cit.> in combination with randomized-measurement protocols <cit.> to learn representations approximating non-equilibrium states. The EH is defined as H_ E≡ -log(ρ_A), where ρ_A denotes the reduced density matrix of subsystem A formed by bipartitioning a (pure) quantum state. The utility of EH as a theoretical and experimental tool stems from the observation that, in many cases, it consists of approximately local operators <cit.>. This resemblance to conventional Hamiltonians, i.e., energy operators governing system dynamics, facilitates simple theoretical analysis: If an EH is (k-)local, it can be described by parameters whose number scales polynomially with the system size, unlike the matrix ρ_A which requires an exponential number of parameters. 

Randomized-measurement protocols <cit.> can be used to learn the EH: after repeatedly preparing and then measuring the quantum state in a randomly chosen basis, one can fit a  parameterized EH to the obtained measurement outcomes.

Because the spectrum of H_ E, known as the entanglement spectrum, scales logarithmically with the Schmidt eigenvalues of ρ_A, it is difficult to learn an EH and reproduce the entire entanglement spectrum with arbitrary accuracy. Randomized-measurement protocols may require resolving exponentially small probabilities even for relatively small systems, which is out of reach of current quantum devices and perhaps fault-tolerant devices, too. 
The present experiments can, therefore, only aim at approximating H_ E and accurately reproducing its low-energy spectrum. Applications of this approach include the verification of topological phases <cit.>.

Nonetheless, what if the precise quantitative structure of H_ E were not crucial, but only its statistical properties were? This viewpoint is that of random matrix theory <cit.>, where metrics like level distribution and spectral form factor differentiate between integrable and chaotic dynamics, while being indifferent to quantitative details, with the notable exception of symmetries. This statistical perspective underpins our study. Guided by physical insights regarding the expected operator content of a non-equilibrium EH, we ask whether one can learn, from experimental data, a classical representation of a state to answer a simpler question: does a quantum state exhibit universal signatures of quantum chaos evident in the statistical properties of its EH? Crucially, can we discern this scenario from one where the state lacks chaotic behavior?

To answer these questions, we focus on two observables indicative of quantum ergodic and chaotic behavior: the entanglement(-Hamiltonian) gap-ratio distribution (EGRD) <cit.> and the entanglement spectral form factor (ESFF) <cit.>. The EGRD is predicted to exhibit level repulsion for chaotic states, in contrast to uncorrelated levels in non-chaotic scenarios. Similarly, the ESFF is predicted to display a plateau-ramp structure in chaotic states
 [Strictly speaking, both indicate quantum ergodicity which usually implies quantum chaos.]. To experimentally test these predictions, we initialize our system in a product state and perform a quantum quench by digitally evolving the system for a variable length of time. At short evolution times, we observe an EGRD that is characteristic of uncorrelated states. With increasing evolution time, we observe the onset of level repulsion and a ramp-plateau structure in the ESFF, indicative of quantum chaos. Detailed analysis, including comparison with emulated data, suggests that the observed behavior primarily arises from the thermalization dynamics of the isolated quantum system under simulation, with experimental inaccuracies and shortcomings of our ansatz playing a small, though non-negligible, role. Our work, therefore, establishes present-time quantum computers as robust tools for studying universal feature of thermalization dynamics in complex many-body systems.

This manuscript is organized as follows. In Secs. <ref> and <ref>, we introduce, respectively, the model and the experimental setup, including state preparation, real-time evolution, and measurement. We discuss our tomography scheme and classical post-analysis in Sec. <ref>. Finally in Sec. <ref>, we present our main experimental results, reporting the observation of a behavior consistent with quantum chaos in the structure of the EH of time-evolved states. Several Appendices offer additional insights: Appendix
<ref> provides further details of 
the quantum circuits employed;
Appendix <ref> investigates finite-size, Trotterization, and late-time effects; and Appendix <ref> offers a comprehensive explanation of our EH-tomography protocol.







§ MODEL


We focus on a Z_2 LGT in 2+1 D with the Hamiltonian


    H = ∑_□ W_□ +g ∑_ℓσ_ℓ^z .


Here, W_□≡∏_ℓ∈□σ_ℓ^x is a magnetic-field operator, where □ denotes the elementary (square) plaquettes of the lattice [see fig:model(a)]. σ^x_ℓ and σ^z_ℓ are Pauli operators representing gauge link and electric fields on edge ℓ, respectively. These act on spin-1/2 hardcore bosons residing on the edges of a two-dimensional spatial square lattice. The first term in eq:Z2Hamiltonian corresponds to the magnetic energy, and the second to the electric energy, with coupling g controlling the relative strength of the two non-commuting contributions. In this formulation, the z basis, therefore, represents the electric basis, while the x basis corresponds to the magnetic basis. The Hamiltonian is expressed in dimensionless units and ħ is set to one throughout.

We consider the joint +1 eigenspace of the Gauss-law operators G_j≡∏_ℓ∈ +_jσ^z_ℓ as the physical Hilbert space, where ℓ∈ +_j denote the four links adjacent to a lattice site j. In the subsequent discussion, we consider a quasi-1D chain composed of L_x plaquettes along the x direction with periodic boundary conditions. Fixed boundary conditions are applied along the short side of the chain (with L_y=1 plaquettes) as illustrated in fig:model(b). In this configuration, all Gauss laws contain three links at each site. Consider a short-ribbon operator V_y≡σ^z_ασ^z_β, where α and β are the two opposing links along the inner and the outer circumference at an arbitrary position along the closed chain shown in fig:model(b) [Note that V_y is independent of α and β because, for Gauss law eigenstates, the operator can be acted on by Gauss law operators in such a way that it can be moved around the lattice.]. This operator commutes with the Hamiltonian and defines superselection sectors of the model; we work in the V_y=1 sector (i.e., short-ribbon operators possess eigenvalue one).

As initial conditions, we randomly select Gauss-law–respecting electric eigenstates (z-basis product states) of the form  | Ψ(t=0)⟩≡ | Ψ_0⟩ = ∏_ℓ | ℓ⟩ with | ℓ⟩∈{| ⟩ ,| ⟩}. We then define a subsystem A consisting of L_A ≤ L_x/2 plaquettes. The reduced density matrix ρ_A(t) of A after evolution time t is related to the density matrix ρ(t) of the whole system as


    ρ_A (t) ≡Tr_A̅[ ρ(t)],   ρ(t)≡ U(t) ρ_0 U(t)^†,


where A̅ is the complement of subsystem A shown in fig:model (c) and (d), ρ_0 ≡ |Ψ_0 ⟩⟨Ψ_0 |,  and U(t)≡ e^-iHt is the time-evolution operator.

To directly encode the degrees of freedom of the model described onto the qubits, one requires one qubit per link degree of freedom, or N_q=3L_x qubits in total. To adapt the simulation to the available hardware, we consider a dual formulation of the model that requires only N_q=L_x + 2 qubits. In the dual formulation, all operators, except those containing operators having support  at the boundaries between the subsystems, labeled ℓ_A̅A and ℓ_AA̅, are entirely represented using gauge-invariant (Ising dual) spin-1/2 variables which act solely within the physical Gauss-law subspace; they are denoted by μ_i^x,z Pauli matrices [see purple circles in fig:model(c)]. Concretely, μ_i^x → W_i= ∏_ℓ∈□_iσ_ℓ^x, while electric variables are represented by μ^z_i μ_i-1^z ≡σ_ℓ^z where ℓ is the link between plaquettes i and i-1. The corresponding “bulk” Hamiltonian terms for the subsystem A then reads


    H_A^ dual =  ∑_i∈A_ bulkμ^x_i  + g (∑_⟨ i,j ⟩∈ Aμ^z_iμ^z_j
    +κ∑_i∈ Aμ^z_i),


where κ≡ 1+V_y. Here, i ∈ A_bulk in the sum over μ_i^x indicates that the original plaquette □_i is entirely in A (i.e., does not touch the boundaries). ⟨ i,j ⟩∈ A denotes nearest-neighbor bulk-spin pairs i,j in A. Finally, the sum over μ_i^z runs over all bulk-spin indices i. The terms in the Hamiltonian acting on the complement are defined identically. Importantly, at the boundaries between the subsystems, ℓ_A̅A and ℓ_AA̅ [see the orange circles in Fig. <ref>(c)], the gauge-variant variables of the LGT, as described by eq:Z2Hamiltonian, are retained. Denoting the Ising-spin index at the A side of one of the boundaries to be k, the Hamiltonian terms coupling the subsystems at this boundary are


    H^ dual_A̅A≡μ^x_
        kσ^x_ℓ_A̅A + μ^x_
        k-1σ^x_ℓ_A̅A + gσ^z_ℓ_A̅A ,


with a similar definition for the other boundary. Note that on the plaquette k containing one boundary link, μ_k^xσ^x_ℓ_A̅A≡ W_k, where μ_k^x≡∏_ℓ∈□̃_kσ_ℓ^x, with □̃_i referring to all but non-boundary links of plaquette k, and similarly for the plaquette k-1. The Gauss laws at the boundaries are not eliminated by the duality. The two Gauss laws in the dual model that are independent (one at each boundary) are


    G^ dual_A̅A≡μ^z_
    
    kσ^z_ℓ_A̅Aμ^z_
    k-1 ,


and similarly for the boundary at ℓ _AA̅.
The dual Hamiltonian of the model is the sum of the terms above:


    H^ dual =H_A^ dual +H_A̅^ dual + H^ dual_A̅A+ H^ dual_AA̅ .



This formulation ensures that all gauge-invariant variables of the LGT and its dual have the same expectation values. Importantly, by maintaining the Gauss laws at the boundaries, the entanglement properties of the dual formulation are identical to those of the LGT <cit.>. This is a subtle but crucial distinction from the standard Ising duality <cit.>, which does not preserve the entanglement structure.

The entanglement structure depends on  the symmetries of the reduced density matrix, which stem from the remaining Gauss laws.
Specifically, [S_j,ρ_A(t)] =0,
where in LGT variables, S_j≡∏_ℓ_j ∈ A σ^x_ℓ_j,
with ℓ_j being the two links originating from a boundary site with lattice coordinates j=(j_x,j_y) within A. This is a direct consequence of the state being a Gauss-law eigenstate <cit.>. In addition, [V_x,A,ρ_A(t)] =0 where the long-ribbon operator within A is defined as V_x,A≡∏_ℓ∈𝒫σ^x_ℓ where σ^x_ℓ act on the original degrees of freedom along the path 𝒫 connecting both boundaries within A, see Fig. <ref>(d). Not all symmetry operators are independent.  For the quasi-1D chain of plaquettes shown in fig:model, only one of the two S_j is independent at each boundary. We place one of the boundaries at sites (0,0) and (0,1) and the other at sites (L_A,0) and (L_A,1). We then choose the two independent S_j operators to be S_(0,0) and S_L_A,0, henceforth referred to as  S_AA̅ and S_A̅A, respectively. Note that, when acting on a Gauss-law–respecting state, V_x,A =S_AA̅S_A̅A. Consequently, there are four independent symmetry blocks ρ_A(t)≡⊕_s=1,⋯,4ρ_A,s(t) that will play a role in our analysis.  In the dual formulation, these symmetry operators are simply S_AA̅^ dual≡μ_k^z σ^z_ℓ_A̅A and S_AA̅^ dual≡μ_k'^z σ^z_ℓ_A̅A where k and k' are the first and last plaquettes within A [This discussion is general, independent of the choice of a V_y eigensector.]. We denote the eigenvalue {1,1}, {1,-1}, {-1,1}, and {-1,-1} sectors of {S_A̅A^ dual,S_AA̅^ dual} operators with labels s=1,2,3,4, respectively. 






§ EXPERIMENTAL SETUP


We use a universal trapped-ion digital quantum computer, schematically shown in fig:introoverview(a), to simulate the time evolution of the dual formulation of the Z_2 LGT. We consider L_x=10 plaquettes, focusing on the entanglement structure of a subsystem consisting of L_A=4 plaquettes, as shown in fig:model(c). 

Our experimental system consists of a linear chain of fifteen ^171Yb^+ ions trapped along the 𝐱 axis of a microfabricated ion trap  <cit.> and spaced by approximately 3.7 μm. The twelve LGT spins are mapped to the ions as shown on fig:model(c). The | ⟩ and | ⟩ spin states are respectively mapped to the first-order magnetic-field–insensitive hyperfine | F=1, m_F=0⟩ and | F=0, m_F=0⟩ states of the ^2S_1/2 ground electronic manifold of the ions [Here, we use the quantum-information convention for the spin states, i.e., |↑⟩≡|0⟩ and |↓⟩≡|1⟩.]. The spin states are initialized in the |⟩ state by optical pumping and are measured by coupling to the excited ^2P_1/2, |F=0,m_F=0⟩ state using a 369-nm laser, wherein the presence (absence) of emitted photons differentiates between the bright state | ⟩ and the dark state | ⟩ with an average 0.3% infidelity, limited by off-resonant photon scattering.

The ions are individually addressed by an equispaced array of 1 μm-diameter, 355-nm, laser beams oriented perpendicular to the trap surface (𝐳 axis), together with a global 300 μm × 30 μm beam oriented along the 𝐲 axis, which is parallel to the trap surface (see fig:introoverview(a) and Ref. <cit.>). These beams drive stimulated Raman transition via the ^2P_1/2 and ^2P_3/2 excited electronic states, where a photon is absorbed from the global beam and emitted into an individual beam to flip the qubit from | ⟩ to | ⟩ and vice versa. All spin-manipulation light is produced by a single pulsed laser, modified to control its repetition rate to null 4-photon Stark shifts. The phase and amplitude of the 355-nm beams is controlled by single-channel and 32-channel acousto-optic modulators [Provided by L3 Harris Corporation.].




To minimize addressing errors, crosstalk, and stray coupling to axial motion of the ions <cit.>, single-qubit operations are realized using compound SK1 pulses <cit.> with Gaussian-shaped sub-segments, with typical infidelities of 0.2%. Entangling operations between any two qubits are realized via variable-angle pairwise Mølmer-Sørensen (MS) gates <cit.> employing laser-induced state-dependent forces on the ≈ 3-MHz motional modes of the ion chain oriented along the 𝐲+𝐳 axis. Robust decoupling from the ion motion at the end of each entangling gate is accomplished using amplitude-modulated, detuning-robust pulse waveforms <cit.>. Typical non-unitary errors of fully-entangling MS gates are 1% and consist predominantly of gate-angle errors and Z-flips of the individual qubits <cit.>.

Our ultimate goal is to study thermalization dynamics in real-time non-equilibrium conditions by learning system's entanglement structure. To this end, we first demonstrate the capability of our digital computer to compute the dynamics of the Z_2 LGT model, including initial-state preparation, time evolution, and subsequent measurements. We initialize the system in a randomly chosen electric eigenstates that respects Gauss law (|Ψ_0⟩=|↓↓↓↑↓↓↑↑↑↓↑↑⟩, see fig:model(c) for the qubit mapping, indicated in blue numbers, read from left to right) and simulate its time evolution for g=0.85 using four Trotter steps that separately implement the non-commuting terms of H^dual, i.e., those diagonal in electric (z) or magnetic (x) bases. Our choice of coupling is such that magnetic and electric terms are of similar magnitude, making the model sufficiently non-integrable. For each  evolution time t, we apply single-qubit gates to measure all qubits in either the Bloch x or the z bases and repeat each experiment N_shots=500 times. The measurement results are used to compute single- and two-qubit observables in the Bloch x and z bases and are plotted in Figure <ref> for select observables [For simplicity, the operators whose expectation values are measured are generally noted as Pauli σ operators. When acting on the bulk dual spins, they should be thought of as the Pauli μ operators, while when acting on the boundary spins, they are the Pauli σ operators in the original LGT.]. The statistical errors are determined by a standard bootstrap resampling analysis with N_boots=1000 bootstrap samples. Quantum circuits implementing Trotterized time evolution and measurements are provided in Appendix <ref>. Each Trotter step of evolution uses 12 variable-angle MS gates. 

For most evolution times, the expectation values of the physical operators (dark blue and red) show good agreement with the zero-parameter theory prediction (solid lines). We ascribe the ∼10% deviation in the evolution rate of the system to miscalibrations of the MS-gate angles. Importantly, the expectation values of gauge-invariance–violating operators (i.e., those that do not commute with the Gauss laws at the two boundaries), shown in light blue, are consistent with zero within measurement errors. Figure <ref> of Appendix <ref> provides a close-up of these observables, showing that gauge-invariance violation is about 5% throughout the time evolution. 

To establish the connection between these simulations and the signatures of chaos and thermalization, we now introduce a measurement and analysis protocol that allows the determination of the system's entanglement structure, going beyond low-order correlation functions.






§ ENTANGLEMENT-HAMILTONIAN SPECTRUM AND TOMOGRAPHY


Our main interest lies in the statistical properties of the Schmidt decomposition of the reduced density matrix of the subsystem A, given by ρ_A(t) ≡∑_λ p_λ(t) | λ(t) ⟩⟨λ(t)|, and its associated EH <cit.>


    H_ E(t) ≡ -log(ρ_A(t)) .


The EH's eigenspectrum, called the entanglement spectrum, is defined by {ξ_λ(t) ≡ -log(p_λ(t))}.





The statistical properties of the entanglement spectrum directly inform the thermalization dynamics of the system. One statistical probe is the distribution, P(r), of the gap ratios, r_λ, of the entanglement spectrum <cit.>,


    r_λ≡min(δ_λ , δ_λ-1)/max(δ_λ , δ_λ-1)  ,


where δ_λ≡ξ_λ - ξ_λ-1≥ 0 are the gaps between the eigenvalues ξ_λ of H_ E and r ≡{r_λ}. According to random matrix theory of Hermitian matrices, this quantity distinguishes chaotic from nonchaotic behavior depending on whether the distribution is centered away from zero (level repulsion) or centered around zero (uncorrelated levels) <cit.>. The gap ratios, nonetheless, only capture local correlations of the entanglement spectrum and, owing to their logarithmic nature, treat entanglement eigenvalues ξ_λ≡ -log(p_λ) associated with very small probabilities p_λ≪ 1 no differently than those where p_λ≈ O(1). 

A complementary view of the statistics of the entanglement spectrum that captures global correlations in the level distribution is afforded by the entanglement spectral form factor (ESFF) <cit.>,


    ℱ(θ ;t) ≡⟨1/ℛ^2_A(t)∑_λ,λ' e^iθ[ξ_λ(t) -ξ_λ'(t) ]⟩ ,


where ⟨·⟩ denotes the average over initial states |Ψ_0⟩. Here, ℛ_A ≡lim_α→ 0exp{1/1-αlog(∑_λ p_λ^α)} <cit.>  is the effective rank of H_ E, whose value depends on the state and lies in the range [1, 2^L_A] [In practice, both statistical measures are typically used to describe the properties of physical Hamiltonians, whose rank equals the Hilbert-space dimension. For EHs, we employ regularization suitable for non-full–rank matrices. To address levels with probabilities close to zero within machine precision, we define an effective rank. Only the ℛ_A lowest levels of the EH are included in the analysis. However, experimental constraints, such as finite statistics and device errors, typically lead to EHs that are reconstructed as nearly or exactly full rank, rendering any regularization obsolete. Nonetheless, non-full–rank EHs occur for states computed in exact diagonalization, particularly at early times, that we contrast our data with.].  Starting from an initial flat behavior, a ramp-plateau structure in the ESFF as function of the `Fourier time' θ is known to indicate ergodic behavior and imply quantum chaos <cit.>. 

While in classical computation of a tractably small system, the state and its entanglement are readily available, in a quantum-simulation experiment, these must be inferred from measurements. To tackle this challenge, we constrain the operator content of the EH to obtain an approximate H_E, and utilize an EH-tomography scheme based on randomized measurements <cit.>.   
Concretely, we perform a single-layer, single-qubit randomized-measurement <cit.> of the time-evolved state. 
At the end of the time evolution, this protocol applies one of N_𝒰 different gates 𝒰_j = ⊗_i=0,⋯,N_q-1 u_i,j for j=1,⋯,N_𝒰 consisting of independent single-qubit gates u_i,j sampled from a unitary 2-design <cit.>. For each 𝒰_j, N_shots experiments are performed, measuring all qubits in the Bloch z basis. 

We compare the relative frequencies of the different bitstrings b to the prediction based on an approximate EH inspired by the Bisognano-Wichmann (BW) theorem <cit.>. 
Explicitly, we assume that H_ E(t) is a linear combination of k-local terms 𝒪_i, i.e.


    H_ E (t ; {β_i} ) ≡∑_iβ_i(t) 𝒪_i .


To find a suitable set of operators { O_i }, we proceed as follows. Starting from the operator content of the physical Hamiltonian in eq:Z2Hamiltonian, containing at most 4-local terms (2-local in the dual formulation), new operators are generated iteratively by forming non-trivial commutators from the existing set. This process can be halted after two iterations (`commutators of commutators') resulting in a maximum of 7-local operators within the LGT framework (or 3-local in the dual formulation). We exclude operators that violate the Gauss laws and those that can be related to others upon application of the Gauss-laws. To ensure Hermiticity, we adjust prefactors and create linear combinations with conjugates [If this iteration were continued, eventually one would end up with an exponentially large set of all possible operators. Our approach is based on a purely heuristic observation that, in practice, only polynomial many (quasi)local operators are required to accurately reproduce the state ρ_A(t) with exponential accuracy <cit.>.]. To find the parameters β_i, a global optimization routine, based on standard Matlab libraries <cit.>, is used to determine the parameterization that most closely reproduces all measurements (refer to Appendix <ref> for details).






§ EXPERIMENTAL STUDY OF THERMALIZATION DYNAMICS OF Z_2 LGT


To study the evolution of the entanglement spectrum in the Z_2 LGT, we repeat the time-evolution experiments shown in Fig. <ref> starting from 6 randomly chosen initial electric eigenstates that satisfy the Gauss laws [We numerically checked that the results remain valid for other randomly chosen states in this basis]. Since these are not the eigenstates of the full Hamiltonian H, the system undergoes non-trivial time evolution when evolved under H. For each evolution time t, symmetry sector, and initial state, we subsequently perform randomized-measurement tomography with g=0.85, N_𝒰=24 bases, and N_ shots=750 bitstring measurements in each basis, and use these measurements to reconstruct the EH as described in the previous section. This procedure yields a set of gap ratios for each initial state, symmetry sector, and evolution time.

In fig:gap-ratio-exp-analysis(a), we plot in orange points the average gap ratio ⟨ r⟩≡∑_r rP̅(r) of the reconstructed EH  as a function of the scaled evolution time g t. The plotted gap ratios are averaged over both the symmetry sectors and the 6 randomly chosen initial states. The black lines correspond to the predicted exact distributions following the Trotterized time evolution. The cyan lines correspond to the EH obtained from an optimal BW-inspired ansatz, where we numerically minimize the
the relative entropy (Kullback–Leibler divergence) between the exact state and the ansatz, corresponding to the limit of infinitely many measurements. A buildup of the level repulsion is discernible as the observed average gap ratio ⟨ r ⟩ evolves from ≈ 0.4 predicted for a non-repulsive Poisson distribution (blue dashed line) towards ≈ 0.6 characteristic of repulsive level statistics of a Gaussian Unitary Ensemble (GUE) <cit.>. Three time regimes (I), (II), and (III) are identified that correspond, respectively, to the evolution of the predicted average gap ratio toward the Poisson-distribution value, toward the GUE value, and to saturation at the GUE value.

In fig:gap-ratio-exp-analysis(b) we plot, for the three ranges of evolution times, the corresponding normalized distribution of the gap ratios, P̅(r), combined over 6 random initial states and 4 symmetry sectors. We observe a clear transition from early-time non-repulsion (Poisson distribution, blue dotted line) in regime (I) to level repulsion (GUE, red) in regime (III). At intermediate times, a distribution is observed between the initial absence of level repulsion and the subsequent emergence of level repulsion.  Error bars and bands denote the variance resulting from averaging over symmetry sectors and initial states. The predicted exact distribution (black line) exhibits a sharper peak around zero at the earliest times due to the reduced density matrix not being full rank. Conversely, the BW-inspired ansatz (cyan line) generally parametrizes a full-rank matrix unless couplings are very finely tuned, resulting in an overestimation of level repulsion in the initial stages. Calculations indicate that the onset of the three regimes remains consistent for any randomly chosen initial states and depends only on the (subsystem-)size when measured in coupling-independent units gt, indicating the universality of thermalization dynamics.






In fig:ESFF-simulator-analysis
we plot the ESSF reconstructed from the data used in fig:gap-ratio-exp-analysis. The three panels of the plot correspond to the three time regimes discussed in relation to the EGRD evolution. The displayed theory curves are averaged over initial states, symmetry sectors, and each of the three time ranges. Starting from an initial flat behavior of the ESSR as a function of θ in panel (I), our data in panels (II) and (III) clearly show the buildup of a ramp-plateau structure, indicating ergodic behavior. In panel (III), we show a fit (purple dotted line) to the observed ramp, indicating a power-law behavior θ^κ where κ = 0.6 ± 0.2, with the fit error determined by changing the fit regime. This exponent is consistent with the results obtained from the numerical analysis of a significantly larger  system in Appendix <ref>. 


Our results demonstrate that, using observables that reveal universal statistical or global features of entanglement spectrum, the onset of chaotic behavior can be robustly traced in the entanglement dynamics of arbitrary initial states. Importantly, time digitization and experimental infidelities in current systems do not significantly impact qualitative features of thermalization dynamics. 






§ CONCLUSIONS


Using a digital trapped-ion quantum computer, we observe early-stage thermalization dynamics in a Z_2 lattice gauge theory in 2+1 dimensions by generating nonequilibrium quantum states and measuring their entanglement structure. As performing quantitative state tomography is extremely challenging and resource intensive, we focused on a simpler question: can a randomized-measurement–based entanglement-Hamiltonian
tomography recover universal properties of the EH that indicate quantum chaos? We experimentally determine the entanglement spectrum of time-dependent quantum states and use two properties of this spectrum, namely the entanglement gap-ratio distribution [Eq. (<ref>)] and the  entanglement spectral form factor [Eq. (<ref>)], as indicators of the emergence of quantum chaos. Our data indicate that both quantities behave as expected: the initially non-repulsive level distribution transitions to a repulsive one. Likewise, a ramp-plateau feature of the ESFF is initially absent but builds up with time. Our results indicate that the timescales for these entanglement-based phenomena mainly depend on subsystem size.

The key technique enabling our analysis is a Bisognano-Wichmann–inspired ansatz for the EH, which allows the parameterization of a nonequilibrium state using a polynomial number of parameters. While this classical parameterization optimally reproduces the observed (randomized) measurement results, in Appendix <ref>, we show that the higher-lying part of the EH spectrum is not quantitatively recovered. Indeed, the measurement cost for precise state tomography still scales exponentially with the subsystem size. Fortunately, to detect the presence of quantum chaos, one needs to only distinguish repulsion from non-repulsion in the EGRD  or identify a ramp-plateau structure in the ESFF. Our work indicates that selecting a sufficiently small subsystem of a much larger, potentially classically nonsimulable system, can sufficiently constrain these observables. 

Our randomized-measurement procedure remains classically simulable because the lattice sizes we considered are manageable with exact diagonalization. Our procedure is also heavily tailored at minimizing the computational load on the quantum computer, avoiding deep quantum circuits and the need for error mitigation, at the cost of classical post-processing. To extend our approach to larger systems, several steps can be implemented in future work:



  –
The employed single-qubit one-layer randomization strategy is symmetry ignorant, i.e., it randomizes regardless of the known symmetry structure of the subsystem density matrix ρ_A(t). Although our scheme is tomographically complete, and enables state reconstruction along with its symmetries, it is inefficient.  Symmetry-conscious protocols, for the LGT in this work and for other models, have been developed <cit.> and could be used in future work.


  –
A major methodological uncertainty is the BW-inspired parameterization of the EH, which is entirely heuristic. While there is research exploring the operator content of the EH of ground, excited, and thermal states <cit.>, further investigations are needed into the applicability and limitations of EH-based schemes for far-from-equilibrium states. Generally, the BW(-inspired) ansätze describe the low-energy regime of the EH well, and incorporating progressively more nonlocal terms  improves convergence into the bulk <cit.> which the EGRD, and to a lesser extent the ESFF, predominantly rely on.  Notably, our analysis reveals that despite the quantitative discrepancy in the bulk, the statistical distribution of the EH appears to be accurately reproduced. This suggests that it may be unnecessary to perform precise state tomography when one's interest is solely in statistical properties. 


  –
Trotterization corresponds to time evolution with an effective, rather than the desired, Hamiltonian. We observe that employing too few Trotter steps leads to poor convergence of our optimization procedure for the BW-inspired ansatz. In this case, more nonlocal operators must be included in the ansatz; constraining these becomes challenging, especially when relying on a limited number of measurements. Since using a large number of Trotter steps increases 
experimental errors, the optimal Trotterization and its interplay with the BW ansatz should be further investigated.


  –
Detailed comparison between the emulator and the experimental data (see Appendix <ref>) highlights the influence of device errors, primarily for the time evolution and tomography steps of our algorithm. The dominant errors in our study are Z-flip errors and over- and under-rotations due to mechanical motion of the ions <cit.>.
Our tomography protocol is especially susceptible to small single-qubit rotation errors during randomization <cit.>.  
Recently demonstrated sympathetic cooling during circuit execution <cit.> would allow the needed measurement circuits to be executed with higher fidelity. The same technique could extend our study to larger systems, later times, and more Trotter steps, e.g., allowing us to directly test the eigenstate thermalization hypothesis in a digital quantum-computing set-up. 



In summary, our results demonstrate that entanglement structure is a measurable quantity in present-day LGT quantum-simulation experiments, and illustrates the potential value of our approach to probe thermalization dynamics and its robust universal features in strongly coupled isolated quantum many-body systems. A compelling future direction is to extend the investigation to later times, to probe aspects such as pre-thermalization <cit.> or fluctuation-dissipation relations <cit.>, once experimental capabilities permit. 
This would also allow for probing the applicability of the Eigenstate Thermalization Hypothesis, e.g., in systems with non-Abelian symmetries <cit.> and other gauge theories <cit.>. Furthermore, the experimental and theoretical tools of this study can be applied in a number of other applications, including obtaining thermodynamic quantities such as work and heat exchanged during nonequilibrium processes <cit.>, and detecting phases of matter, including topological phases <cit.>, in quantum-simulation experiments.






§ ACKNOWLEDGMENT

N.M. thanks H. Froland, A. Polkovnikov, M. Savage, M. Srednicki, X. Yao, T. Zache, P. Zoller, and the participants of the InQubator for Quantum Simulation (IQuS) workshop “Thermalization, from Cold Atoms to Hot Quantum Chromodynamics ”(<https://iqus.uw.edu/events/iqus-workshop-thermalization/>) at the University of Washington in September 2023 for many valuable discussions leading to this work. M.C. thanks L. Feng for valuable discussions and help with the experimental setup leading to this work.
N.M. acknowledges funding by the Department of Energy (DOE), Office of Science, Office of Nuclear Physics, IQuS (<https://iqus.uw.edu>), via the program on Quantum Horizons: QIS Research and Innovation for Nuclear Science under Award DE-SC0020970. Z.D., M.C., and T.W. were supported by the National Science Foundation's Quantum Leap Challenge Institute for Robust Quantum Simulation under Award OMA-2120757. Z.D. further acknowledges support by the DOE, Office of Science, Early Career Award DE-SC0020271. This work is further supported by a collaboration between the US DOE and other Agencies. This material is based upon work supported by the DOE, Office of Science, National Quantum Information Science Research Centers, Quantum Systems Accelerator. 






§ COMPETING INTERESTS

M.C. is a co-inventor on  patents that are licensed from the University of Maryland to IonQ, Inc.













§ CIRCUIT REPRESENTATION FOR STATE PREPARATION, EVOLUTION, AND MEASUREMENT



 


In this Appendix, we provide circuits employed to experimentally realize state preparation, time evolution, and randomized measurement in the Z_2 LGT in (2+1)D within the dual Isnig formulation.

We label qubits with indices q=0,…,L-1, with L=12. Qubit q=0 corresponds to the spin at ℓ_A̅A positioned at the interface between subsystem A and its complement, identifiable by an orange circle in fig:model(c). Qubits q=1,…,4 represent the succeeding dual variables arranged counterclockwise at the center of plaquettes denoted by purple circles in fig:model(c). Following this sequence, qubit q=5 resides at the opposite boundary marked as ℓ_AA̅, and so forth. 










Working in the electric eigenbasis and starting from an all-up spin state, we first perform single-qubit σ^x bit-flip operations to select a randomly chosen product state in the σ^z basis that is consistent with Gauss laws, i.e., G_j |ψ(0)⟩=|ψ(0)⟩ for all j. Recall that in the dual formulation, there are only two Gauss-law operators each located at one of the subsystem's boundaries: G^ dual_A̅A = μ^z_11σ^z_0 μ^z_1 and G^ dual_AA̅ = μ^z_4σ^z_5 μ^z_6. Importantly, any spin configuration in either bulk, A and A̅, is physical, as in the dual formulation, only gauge-invariant degrees of freedom are kept in the bulk.


For time evolution, we utilize a first-order Trotter scheme with time-evolution operator


    U(t)≡ e^-itH^ dual≡∏_δ t U(δ t).


We take δ t to be a variable time step while keeping the Trotter depth fixed. In the simulations conducted in this work, t/δ t= 4, resulting in states and observables closely resembling the exact time-evolved states at early times. However, for late times, Trotter effects become more pronounced.

U(δ t) in Eq. (<ref>) can be written as U(δ t)≡∏_a∈ X,Z,XX,ZZ U_a with U_a ≡  e^-iδ tH^ dual_a; H^ dual_a are the respective 1- and 2-local operators of the dual Hamiltonian in Eq. (<ref>), sorted in X and Z operations, as well as single- and two-qubit entangling gates. While XX, X, and Z are native operations, ZZ entangling operations are realized via basis transformation and usage of the native MS gate,


    R^zz_ij(α) ≡ R^y_i(-π/2) R^y_j(-π/2)    R^xx_ij(α)   R^y_i(π/2) R^y_j(π/2).


Here, R^x/y/z_i(α) ≡ e^-iα/2σ_i^x/y/z and R^xx_ij(α) ≡ e^-iασ_i^xσ_j^x. The experimental errors of our MS gates increase with the absolute value of the gate angle. To minimize the gate error, all MS gates are optimized as follows. First, all angles are mapped to the regime α∈ [-π,π]. Within this range, we make the following substitution


    R^xx_ij(α) →
             R^xx_ij(α)    if |α| ≤π/4
    
             R^xx_ij(α+π/2) R^x_i(π) R^x_j(π)    if |α| ∈ (π/4,3π/4]
        &  α<0 
    
             R^xx_ij(α-π/2) R^x_i(π) R^x_j(π)    if |α| ∈ (π/4,3π/4]
        &  α>0
    
              R^xx_ij(α+π)     if α < -3π/4
    
              R^xx_ij(α-π)     if α > 3π/4


so that all MS operations are restricted to the range |α| ≤π/4. 

An example study of the initial-state preparation, time evolution, and measurement in a fixed (Bloch x or z) basis is shown in fig:experimentbasis (b) and (c) of the main text, where we presented measurement of several gauge-invariant and non-gauge–invariant
one- and two-qubit observables. In addition, we present the measurement of several other non-gauge–invariant observables in fig:gauge_invariance. These observables are expected to be zero, but remain non-vanishing because of finite measurements and device errors. A main source of errors are likely coherent errors, related to the over- and under-rotation of single- and two-qubit gates. While the time-evolution circuit only contains gauge-invariant operations for any gate angle employed, the final single-qubit rotations R^y_i(-π/2) and R^x_i(π/2) (that transform all qubits i from the Bloch z basis into the Bloch x and y bases, respectively) can introduce gauge-invariance violation if their rotation angle is set inaccurately. Additionally, errors stemming from initial-state preparation and readout processes can contribute to this violation.






§ FINITE-SIZE DEPENDENCE, TROTTERIZATION, AND LATE-TIME BEHAVIOR


In this Appendix, we perform various tests aiming to extend our study toward larger systems, and to provide an outlook on the expected late-time behavior of non-equilibrium states, features that are inaccessible in current quantum computers. We present a detailed classical  computation of a fairly large system (L=22, L_A=8 with coupling g=1), evolving the state continuously, and computing its entanglement structure using exact diagonalization, hence avoiding a Trotterization error. Such an analysis is possible for the model under consideration, owing to its relative simplicity. This simplicity enables us to achieve results that are fairly insensitive to finite-size effects, even with moderately large systems.  A similar classical computation will be impossible for more involved models or in higher dimensions.

We compute the same entanglement-related observables indicative of quantum chaos as in the main text, starting from several randomly drawn initial product states. The EGRD, i.e., the (normalized) distribution of the gap ratios P̅(r) with r defined after Eq. (<ref>), is computed similarly as in the main text: we consider ten randomly chosen initial states, and combine the gap ratios that are separately computed for each symmetry sector of the respective reduced states and for all initial states. Additionally, we combine the distribution of gap ratios in three time regimes (I), (II), and (III). The result is shown fig:gap-ratio-theory-analysis(a). Further, the mean of the gap ratio, combining initial states and symmetry sectors but not times, ⟨ r ⟩, is shown in fig:gap-ratio-theory-analysis(b) as a function of scaled time gt. 
The distribution in fig:gap-ratio-theory-analysis(b) is peaked near zero first, closely resembling a Poisson distribution. In this regime, the rank ℛ_A≪d_A, where d_A is the Hilbert-space dimension of the subsystem. The effective rank ℛ_A is defined after Eq. (<ref>) and plotted for the present example in fig:gap-ratio-theory-analysis(c) starting from a product state with ℛ_A=1 at t=0. The largest ξ_λ values in this regime correspond to extremely small probabilities p_λ = exp(-ξ_λ), close to or at the level of machine precision. A regularization of the smallest p_λ is required: we manually cut off probabilities below 10^-15 before computing ℛ_A, varying this limit by two orders of magnitude in each direction to provide the blue bands in panel fig:gap-ratio-theory-analysis(c), then using only the ℛ_A lowest levels in the analysis of the EGRD and the ESFF. Gray bands represent the variance with respect to the randomly chosen initial states. In regime (II), ⟨ r ⟩ quickly jumps to a value associated with a Gaussian Orthogonal Ensemble (GOE), until continuing, albeit at much slower pace, towards a  Gaussian Unitary Ensemble (GUE) indicating quantum chaotic behavior. At the end of this stage, ℛ_A= d_A is maximal, and all probabilities p_λ are well above machine precision, eliminating the need for any regularization. Finally,  regime (III) sees the saturation of the gap ratio to GUE level statistics (up to minimal finite lattice-size effects). The beginning and end of the time regimes (I-III) depend on the subsystem size and are chosen as follows, (I): 0 ≤ gt <1.8, (II): 1.8 ≤ gt <5.0, and (III): 5.0 ≤ gt<10.0.





An analogous picture is evident in the time evolution of the ESFF defined in eq:defESFF, and shown in Figs. <ref> (d) and (e). Panel (e) shows the ESFF for various times 0 ≤ gt ≤ 10, starting from a flat distribution at earliest time and showing a plateau-ramp structure at late times. Similarly as in the main text, the ESFF is computed separately for every symmetry sector of the EH to avoid contamination from uncorrelated levels in different sectors, and then averaged over these sectors and over the randomly chosen initial states. Panel (d) shows the time average over regime (III). In this regime, the ramp-plateau is evident. A gray band marks the statistical deviation from the initial-state and symmetry-sector averages. We fit the ramp to the form ∼θ^0.6± 0.1 which is consistent with the experimental data shown in the main text for a much smaller system. The fit error is determined by varying the fit range.

Because the late-time behavior, i.e., g t≫ 1, is inaccessible to current Trotter-based digital quantum simulation, we extend our analysis towards this regime, aiming to elucidate the potential outcomes that future quantum simulators might uncover. A similar analysis has been performed previously in Ref. <cit.>. At late times, it is expected that the entanglement entropy of a subsystem becomes equal to the thermal entropy contained in that subsystem, corresponding to a global Gibbs state with a temperature corresponding to the average energy of the initial state. In fig:gap-ratio-theory-analysis (f), we plot the von Neumann entanglement entropy, for a variety of randomly chosen initial states. The individual curves are color coded relative to their initial average energy density ∼ E_0 = ⟨ψ(0)| H | ψ(0) ⟩, normalized relative to the energy bandwith Δ E,  i.e., the difference between the highest and lowest eigenvalues of the EH. Displayed are only initial states whose energies lie within the interquartile range, representing the central 50% energy spectrum of states. States highlighted in yellow denote the highest energies, while those in blue represent the lowest. We were not able to extend this study to even later times to observe the expected saturation of the von Neumann entanglement entropy, owing to the fact that eventually finite-size effects become large. These results demonstrate that entanglement-entropy saturation is not a practical measure of thermalization in present experiments, demanding long evolution times and exhibiting lack of initial-state insensitivity (i.e., universality).

Shown in fig:gap-ratio-theory-analysis(g) is a separation of the von Neumann entanglement entropy, S_vN, into components, S_vN,s, related to the symmetry structure of  ρ_A = ⊕_s ρ_A,s,


    S_vN = -∑_s p_s log(p_s) + ∑_s p_s S_vN,s,


where the first term is the symmetry component, and the second is the distillable entanglement <cit.>. Here, p_s≡Tr[ρ_A,s]≤ 1 with ∑_s p_s=1, and S_vN,s≡ - Tr[ρ̅_A,slog(ρ̅_A,s)] is the sector-wise entanglement entropy with ρ̅_A,s≡ρ_A,s / p_s. Dashed lines in the plot denote the distillable component, while dotted lines denote the symmetry part. It is evident that both components saturate on different time scales: the symmetry component saturates fairly quickly (albeit still later than the build-up of level repulsion) to its maximal value log(1/p_s) where p_s =1/4 corresponds to equal mixing of the four symmetry sectors of ρ_A. In contrast, the distillable components dominates the late-time behavior of the von Neumann entropy.




While experimental constraints prevent us from accessing late times, fig:vonNeumanExpt illustrates the von Neumann entanglement entropy derived from our experimental data, utilizing the same dataset as in the main text. In this figure, exact classically computed results are denoted by black lines, while cyan crosses represent the optimal BW-inspired parameterization assuming infinite measurements, and orange circles are the experimental data points. The right-hand side of the figure displays two panels: the top panel depicts the symmetry component, while the bottom panel depicts the distillable component. Notably, the observed behavior closely mirrors the (classically computed) findings of the much larger system in fig:gap-ratio-theory-analysis(g). Specifically, the symmetry component saturates at its maximum value of log(4), while the distillable entanglement demonstrates continued growth.  Generally,  the entanglement measured in our experiment via BW-inspired tomography overshoots the exact result. We attribute this discrepancy primarily to  over- or under-rotations within the single-qubit random-bases changes. 






§ ENTANGLEMENT-HAMILTONIAN TOMOGRAPHY


In this section, we outline the randomized-measurement and analysis strategy behind extracting approximate EHs from experiment. We also systematically investigate the performance of the approach with regard to the number of bases sampled, the number of shots performed, and the influence of device errors.




Our tomography protocol consists of two parts: randomizing the state via single-qubit random unitaries drawn from a circular unitary ensemble, followed by classical post-analysis of the extracted measurements to determine an EH that best describes all observables. We use single-qubit random circuits, 𝒰≡⊗_i u_i where u_i is the following single-qubit unitary 



width=0.36

u_i ^     

=

    R^z(γ^1_i)     R^y(γ^2_i)      R^z(γ^3_i)       ,




and for each qubit, the angles γ^1_i, γ^2_i, γ^3_i are drawn according to a circular unitary ensemble (an overall phase is ignored) following Refs. <cit.>. A drawback of this approach is that it does not maintain the symmetry structure of ρ_A, as it randomizes over the whole Hilbert space instead of each symmetry block of ρ_A. Such a symmetry-ignorant randomization mixes different symmetry sectors, resulting in an outcome that is inaccessible to any physical time-evolved quantum state. We choose it, nonetheless, to avoid the larger circuit depth associated with a symmetry-conscious randomization such as that proposed in Ref. <cit.>. We note that a symmetry-conscious scheme would significantly reduce the measurement cost, as the sampling cost would only scale as the size of the symmetry block instead of the Hilbert-space size. It would thus simplify the classical optimization significantly. However, because the single-qubit scheme is tomographically complete, one can still reconstruct, approximately, the symmetry structure from the data.

The ansatz for the EH is a sum of local operators 𝒪_i,


    H_A (t ; {β_i} ) ≡∑_iβ_i(t) 𝒪_i ,


which are either the physical-Hamiltonian terms or are chosen iteratively from commutators of those terms. β_i are real parameters to be constrained from data. While the BW theorem hints at an optimal choice of operators 𝒪_i for ground states, our ansatz is heuristic, as a systematic ansatz for non-equilibrium states is not known. Our criteria for the operators are that i) they are local, involving operators with support on at most two neighboring plaquettes, ii) they are compatible with the symmetries of the reduced state ρ_A (which will be discussed below), and iii) they are independent, meaning they cannot be transformed into each other through Gauss laws when acting on a physical state (although they are certainly not independent in the algebraic sense).




The operators we use are pictorially represented in Figs. <ref> plus all operators that are generated from those depicted upon translation. Here for the sake of generality, we represent them in terms of Z_2 variables, albeit, in practice, they are represented in the dual formulation in our algorithm. Figure <ref> shows a special operator; this operator (and related examples) relates operators within the subsystem to operators in the complement via Gauss laws. Further, it commutes with ρ_A, and is hence connected to the symmetry blocks of ρ_A. All terms in Figs. <ref> are obtained by considering the physical-Hamiltonian operators and by recursively commuting the physical-Hamiltonian operators, stopping at two recursions (i.e., commutators of commutators). Other selection strategies are also feasible.







Given the ansatz and following Ref. <cit.>, we then minimize the functional


    ⟨∑_b[P_𝒰(b) - Tr[ 𝒰^† |b ⟩⟨ b|  𝒰ρ_A(t ; {β_i}) ] ]^2 ⟩_𝒰


where P_𝒰(b) is the probability to measure a bitstring b in the basis determined by 𝒰 and


    ρ_A(t ; {β_i}) ≡ e^-H_A (t ; {β_i} )/Tr[e^-H_A (t ; {β_i} )] .


is the normalized reduced density matrix parameterized by H_A.
Here, ⟨·⟩_𝒰 is the average over random circuits.
Note that optimizing eq:optimziaton effectively weighs more favorably the largest Schmidt eigenvalues of ρ_A because they, on average, contribute the most to any random observable. Because of this, the optimization more accurately reproduces the low-energy part of H_A. 
The optimization is performed using Matlab's <cit.> non-linear fmincon optimization package with the `sqp' algorithm <cit.>. Convergence and uniqueness of the obtained minimum have been cross checked for several data sets using Matlab's GlobalSearch routine with default parameters <cit.>. All EH couplings are confined to the range β_i ∈ [-50,50]. We check explicitly that the routine does not come close to the boundary of the parameter regime. The EH that is obtained in this way is projected into symmetry sectors, which we then separately analyze. For this projection, the symmetry sector can be read off directly from the row and column numbers of the EH when they are interpreted as binary. The corresponding subblock H_ E,s is then selected for further analysis, where s labels one symmetry sector. For the EGRD analysis, we analyze the gap-ratio distribution separately for every sector, then combine the distributions and re-normalize the total distribution. For the ESFF analysis, we average over the sectors. Finally, EGRD and ESFF shown in the main text and appendices involve a combination/average over initial states. The error band and the error bars in the plots are the standard deviation for the symmetry-sector and initial-state averages. 

An example of the eigenvalue spectrum of the EH, comparing the exact values with those obtained from our procedure, is shown in fig:overviewES. Panel (a) shows the reconstructed entanglement spectrum for several times, comparing exact data (black lines), infinite-measurement results (cyan crosses), emulator data (blue squares), and quantum-computed data (orange circles), for one given initial state. Panel (b) shows the symmetry-resolved spectrum for gt=1.02. Here, s  are the symmetry sectors of ρ_A, see the discussion at the end of Sec. <ref> in the main text. While our analysis effectively describes the low-lying part of the entanglement spectrum, it fails to quantitatively reproduce the higher part associated with very small probabilities. Additionally, the reconstructed spectrum appears more mixed (more entangled) compared to e.g., the emulator data. This discrepancy primarily is due to finite-measurement statistics and device errors. Comparison with the infinite-(ideal-)measurement results also reveals that the BW-inspired parameterization falls short in capturing even quantitatively higher-lying components of the spectrum. It is noteworthy that the limitations of the parametrization become more pronounced at later times where the Trotter-step size is large and even the low-lying part deviates. This is because the effective Trotter Hamiltonian at late times deviates from the known target Hamiltonian from which the BW-inspired ansatz is derived.

In fig:BWgapratiobases, we study the accuracy of our tomography scheme concerning the number of random bases. Each basis is probed with N_ shots=750 samples, employing emulated data to eliminate the effect of device errors. Applying N_𝒰=4 to N_𝒰=128 random bases shows convergence in the lower-lying range of the entanglement spectrum relatively fast. However, the higher-lying part is not reconstructed even with a large sample size. The top panel displays the average EGRD over time, showing consistent behavior across the sample range. Error bars indicate the spread from combining initial states and symmetry sectors, which decrease with increasing N_𝒰. Concretely the error bars are obtained by computing the mean of the gap ratios for every sector and initial states separately, and then computing the standard deviation. 

Finally in fig:BWgapratioshots, we investigate the dependence on the number of measurement of probabilities P_𝒰(s) in each basis (i.e., the number of shots), maintaining a constant number of measurement bases, N_U=24. Once more, convergence is evident in the lower segment of the ES in the lower panel, while the higher portion remains beyond reach even with infinite shots (red octagons). The top panel illustrates the EGRD, indicating consistency, within error bars, with the optimal BW-inspired ansatz result (cyan band).

In principle, given that the scheme is tomographically complete, we anticipate the ability to precisely reconstruct the entire ES with exponential resources. However in practice, we were not able to do so because of the significant bias of the cost function eq:optimziaton towards the low-energy portion of the entanglement spectrum. This bias places considerable strain on the numerical minimization routine, pushing it beyond its numerical-accuracy threshold. Despite this, its accuracy is satisfactory for analyzing experimental data, and we abstained from further optimization attempts. Exploring advanced optimization routines, including machine-learning techniques <cit.>, holds promise for EH tomography in future studies.



