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







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             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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            SuppM
              Supplementary material.
            
            
            
              
            
             Supplemental Material
            
            14cmWe present here calculations of the form factors
            G_E and G_M and the effective form factor G
            in the timelike region for the hyperons
            Λ, Ξ^-,0 and  the Σ^0,±.
            
            
            § FORM FACTORS G_E AND G_M
            
            
            The absolute values of the form factors G_E and G_M
            are presented in Fig. <ref>.
            The signs of G_E and G_M can be inferred from
            the labels included in the figures.
            The representation -G_E or -G_M indicate
            that the functions are negative in that region.
            The cusps indicate the zero of G_E.
            
            
            
            § EFFECTIVE FORM FACTORS G
            
            
            
            The effective form factors are presented in Fig. <ref>.
            The limits of the theoretical calculations
            are represented by the dashed lines.
            
             DataΣ^+:  Belle <cit.>,
            BESIII <cit.>,
            and CLEO <cit.>;
             DataΣ^0: BaBar <cit.>, BESIII <cit.>, 
            CLEO <cit.> and Belle <cit.>;
             DataΣ^-: BESIII <cit.>; 
             DataΛ: BaBar <cit.>, BESIII <cit.>
            and CLEO <cit.>;
             DataΞ^0: BESIII <cit.> and CLEO <cit.>;
             DataΞ^-: BESIII <cit.> and
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 Paper Title: Pairing phase transition in the odd-A nuclei: identification and classification.


 UTF8





School of Science, Jiangnan University, Wuxi 214122, China.


 
School of Science, Jiangnan University, Wuxi 214122, China.


 
liulang@jiangnan.edu.cn
School of Science, Jiangnan University, Wuxi 214122, China.


§ ABSTRACT 

The investigation into the pairing phase transition in the odd-A nucleus ^161Dy utilizes a sophisticated blend of covariant density functional theory and the shell-model-like approach. It is discerned that variations in thermodynamic quantities at the critical temperature do not exclusively align with pairing phase transitions. The presence of an S-shaped heat capacity curve, often interpreted as indication of such transitions, does not offer a definitive confirmation. Additional factors, including the blocking effect, can modify the heat capacity curve and impede the transition process. The pairing phase transition in ^161Dy, occurring around 0.7 to 1.0 MeV, is unequivocally characterized as a first-order transition. Furthermore, an exploration into the impact of varying strengths of pairing correlations on these transitions reveals a nonlinear relationship, adding complexity to the transition dynamics.



Pairing phase transition in the odd-A nuclei : identification and classification 
    Lang Liu (gbsn刘朗)
    August 5, 2024
=================================================================================





§ INTRODUCTION


The pairing phase transition is widely recognized as essential for comprehending the thermodynamic properties of atomic nuclei <cit.>. Due to the precise measurement of level densities at low excitation energies, thermodynamic quantities have been deduced within the canonical ensemble framework, thereby confirming the S-shaped curve of heat capacity as a function of temperature <cit.>. The significance of this S-shaped curve has been extensively explored across various theoretical models, including the nuclear shell model <cit.>, the mean field model <cit.>, and others <cit.>. It is considered the hallmark of the pairing phase transition from the superfluid phase to the normal phase. However, in the superfluid phase, particle number conservation is violated by the BCS theory or the Bogoliubov transformation within the mean field model. The importance of conservation laws in studying phase transitions of finite systems has been elucidated in Refs. <cit.>. To address these drawbacks, we employ the shell-model-like approach, which ensures particle number conservation and accurately deals with the blocking effect <cit.>. Additionally, covariant density functional theory (CDFT) has effectively described numerous nuclear phenomena <cit.>. The shell-model-like approach has been integrated within the CDFT framework <cit.>. Previous studies have examined the thermodynamic properties of even-even nuclei and odd-A nuclei using CDFT and the shell-model-like approach <cit.>.

Although the pairing and pairng phase transition has been extensively investigated in many aspects <cit.>, few studies have focused on the order of these transitions, and it remains to be more clearly defined. The initial classification of general types of phase transitions, introduced by Paul Ehrenfest in 1933, has played a pivotal role in the thermodynamic study of critical phenomena <cit.>. Subsequently, Lee and Yang proposed a theorem on the distribution of roots of the grand partition function, predicting its broad applicability even in small systems <cit.>. In Refs. <cit.>, a classification scheme for phase transitions in finite systems was proposed, based on the distribution of zeros (DOZ) of the canonical partition function in complex temperature. This method has been successfully applied to determine the order of phase transitions in ^162Dy <cit.>. To our knowledge, it has not yet been utilized to study the order of the pairing phase transition in odd-A nuclei.

This work aims to further investigate the microscopic mechanism underlying the S-shaped heat capacity curve, examining whether its appearance entirely corresponds to the occurrence of pairing phase transitions. If a pairing phase transition does occur, the order of the transition in odd-A nuclei remains unclear, and the influence of the blocking effect on the phase transition needs to be studied. Additionally, this research will explore whether and how the strength of the pairing correlation affects the pairing phase transition, an aspect that has not been addressed in previous studies.

In this work, we use ^161Dy as an example. By employing the shell-model-like approach and covariant density functional theory, combined with the effective interaction PC-PK1 <cit.>, we investigate the thermodynamic properties and the influence of the blocking effect on the pairing phase transition. The order of the pairing phase transition is determined according to a classification scheme based on the distribution of zeros of the pairing partition function in the complex temperature plane. Furthermore, we construct an ideal odd system and an ideal even system to explore the impact of pairing correlation strength on the pairing phase transition, as evidenced by changes in the order of the transition.

This paper is organized as follows: In Sec.II, we briefly introduce the theoretical framework. Sec.III presents the results and discussion. Finally, Sec.IV contains the summary.




§ THEORETICAL FRAMEWORK



In this section, we introduce briefly the basic formalism of the CDFT theory, which has been summarized in Ref.<cit.>. CDFT begins with a Lagrangian. From this Lagrangian, the Dirac equation for nucleons with local scalar S(r) and vector V^μ(r) potentials can be deduced as 

    [ γ_μ (i ∂^μ - V^μ) - (m + S) ] ψ_ξ = 0,

where

    S(r) = Σ_S,  V(r) = Σ^μ + τ⃗·Σ⃗^μ_TV,

and ψ_ξ is Dirac spinor. 


In this study, SLAP is applied within the CDFT framework to deal with the pairing correlations by diagonalizing the Hamiltonian in the multi-particle configurations (MPCs) space, which is constructed by the single-particle levels obtained through the CDFT method. The total Hamiltonian reads

    H    = H_ s.p. + H_ pair
       = ∑_iε_ia^+_ia_i -G∑^i≠ j_i,j>0 a^+_ia^+_i̅a_j̅a_j,

where ε_i is the single-particle energy gained from the Dirac equation (<ref>), i̅ is the time-reversal state of i, and G represents constant pairing strength.

For a system with an odd particle number N = 2n+1, it is necessary to block the state occupied by the odd nucleons. The MPCs for s = 0 states can be expressed as,

    |c_bc_1c̅_1⋯ c_nc̅_n⟩=a_b^+(a_c_1^+a_c̅_1^+⋯ a_c_n^+a_c̅_n^+)|0⟩,

in which b denotes the single particle level blocked by the odd nucleon.

During the calculation process, a cutoff energy E_c is needed to truncate the MPCs space. The configurations with energies E_m-E_0 ≤ E_c are used to diagonalize the Hamiltonian (<ref>), in which E_m and E_0 are the energies of the mth configuration and the ground-state configuration, respectively. 

After the diagonalization of the Hamiltonian (<ref>), the nuclear many-body wave function is shown as

    |ψ_β⟩ =   ∑_c_1⋯ c_nv_β, c_1⋯ c_n|c_1c̅_1⋯ c_nc̅_n⟩
        +∑_i,j∑_c_1⋯ c_n-1v_β(ij), c_1⋯ c_n-1|ij̅c_1c̅_1··· c_n-1c̅_n-1⟩
        + ⋯,

where β = 0 corresponds the ground state, β = 1, 2, 3, … correspond the excited states with the excitation energy E_β, and v_β means the coefficient after diagonalization.


	



Moreover, the pairing gap can be calculated by Refs. <cit.>

    Δ_β=G[-1/G⟨Ψ_β|H_p|Ψ_β⟩]^1/2,

The thermodynamic quantities of the pairing interaction are calculated in the canonical ensemble <cit.>. The partition function Z can be calculated by

    Z = ∑^∞_β=0η(E_β) e^-E_β/T.

where E_β is the excitation energy which could be acquired through the CDFT + SLAP method, and the corresponding level density η (E_β) is taken as 2^s, i.e., the degeneracy of each state.


The classification scheme for phase transitions relies completely on Ref. <cit.>. Only the main formulas of the classification scheme will be given below. The scheme relies on the DOZ of the canonical partition function in the complex temperature plane.

The inverse complex temperature is defined as:

    ℬ = β + iτ,

where β=1/T and τ corresponds the imaginary part of the inverse complex temperature which is measured in MeV^-1. The zeros of the canonical partition function typically line up on curves in the complex temperature plane. The zeros are denoted by (β_j,τ_j). If there are n zero points, j = 1 … n and j increases with the increasing distance from the real axis.

If the first three zeros closest to the real axis have been determined, the average inverse distance between zeros can be calculated as

    Φ(τ_j)=1d_j,

where τ̃_j=(τ_j+τ_j+1)/2, d_j=√((β_j+1-β_j)^2+(τ_j+1-τ_j)^2). The function Φ can then be approximated in the vicinity of the real axis by a power law of τ_j only,

    Φ(τ_j)∝τ_j^α,

then α can be calculated by means of

    α=lnΦ(τ_3)-lnΦ(τ_2)lnτ_3-lnτ_2.

According to the definition in Ref. <cit.>, 
when α < 0, the system displays a first-order phase transition. When 0 < α < 1, the phase transition is of second order. When α > 1, higher-order phase transition occurs.






§ RESULTS AND DISCUSSION








 §.§ ^161Dy 


For neutrons and protons, 16 single-particle levels and 9 valence nucleons are selected to construct a multi-particle configuration space, including 5 levels on and below the Fermi surface and 11 energy levels above the Fermi surface. PC-PK1 is chosen as the effective interaction. For neutrons, the Fermi surface is occupied by a single particle, and the maximum seniority number s is 8. The span of the upper and lower energy levels of the model space is 7.56 MeV. The corresponding pairing strengths of neutrons and protons are 0.29 MeV and 0.32 MeV, which can reproduce the experimental odd-even mass differences well. In this paper, the treatment of blocking is consistent with the method in Ref. <cit.>. “Bk1” corresponds to the blocking of the Fermi surface, while “bkn” corresponds to the blocking of the (n-1)th energy level above the Fermi surface.







Figure <ref> illustrates the behavior of the pairing gap (a), heat capacity (b), average energy (c), and partition function (d) of neutrons (indicated by black solid lines) and protons (represented by red dashed lines) in ^161Dy as functions of temperature. Subfigure (a) reveals obvious increase in the neutron curve at temperature below 0.3 MeV, contrasting with the almost constant trend observed in the proton curve within this temperature range. This behavior in the neutron curve indicates the pairing re-entrance phenomenon, wherein the pairing correlation in nuclei with an odd number of neutrons or protons is bolstered as temperature rises. This enhancement is primarily attributed to the influence of the single-particle level structure near the level occupied by the odd nucleons, which weakens the blocking effect and reinforces the pairing correlation.
Regarding heat capacity, both protons and neutrons exhibit an S-shaped curve. The initial turning point occurs at approximately 0.3 MeV for protons and 0.2 MeV for neutrons, aligning with the turning points of the pairing gap and average excitation energy. A second turning point emerges at around 1 MeV for protons and 0.7 MeV for neutrons, corresponding to the critical temperature of the phase transition. Additionally, the neutron curve demonstrates fluctuations in the low-temperature region. In the average energy diagram (c), both curves are nearly zero at low temperatures (around 0.25 MeV for neutrons and 0.3 MeV for protons), gradually increasing in a near-linear way thereafter.








Figure <ref> (a) displays the heat capacity of neutrons in ^161Dy, considering total and different blocking scenarios, as functions of temperature. The fluctuations observed at low temperatures primarily stem from the contribution of “bk1", attributable to the single-particle level structure. In the case of “bk2", corresponding to the blocking of the first level above the Fermi surface, Figure <ref> (b) illustrates that no levels exist around this energy level. Consequently, an energy input is required to excite nucleons, resulting in near-zero heat capacity at low temperatures. Subsequent blockings involve energy levels occupied by odd nucleons, which have less impact on the shape of the heat capacity curve. Hence, accurate treatment of Fermi surface blocking and its neighboring levels is crucial for investigating the relevant properties of odd-A nuclei.









To gain a detailed microscopic understanding of the nuclear pairing transition, it is valuable to examine the number of Cooper pairs that become broken with increasing temperature in ^161Dy. In this analysis, we follow the definition of the seniority component as outlined in Ref. <cit.> 

    χ_s=Z^-1∑_β∈{s}η(E_β)e^-E_β/T.

The seniority component illustrates the contribution of excited states with specific seniority numbers to the total average energy. Figure <ref> presents the seniority components for different seniority numbers s = 0, 2, 4, 6, 8 of neutrons (a) and s = 0, 2, 4, 6, 8, 10 of protons (b) in ^161Dy as functions of temperature.
It is observed that s = 0 states contribute nearly 100% below T ≈ 0.3 MeV for protons and 0.2 MeV for neutrons, aligning with the vanishing heat capacity illustrated in Fig. <ref>. As the temperature exceeds 0.3 MeV for protons and 0.2 MeV for neutrons, the contribution of s = 0 states begins to decrease. Simultaneously, the curve for s = 2 states starts to rise, corresponding to the first turning point in the heat capacity curve and the temperature where the pairing gap curve begins to decline and the average energy curve begins to rise. The curves for s = 2 peak at 0.8 MeV for neutrons and 1 MeV for protons, respectively, corresponding to the second turning point of the heat capacity curve, indicating the pairing phase transition from the superfluid phase to the normal phase. This transition is responsible for the S-shape of the heat capacity curve. Furthermore, around T ≈ 0.5 MeV, the contribution of s = 4 states begins to increase.








The level density serves as a valuable tool for characterizing pairing transitions in hot nuclei. To further elucidate the microscopic mechanisms underlying the S-shaped heat capacity curve, we investigate the number of excited states of nuclei in terms of different seniority contributions. Figure <ref> illustrates the level densities with different seniority numbers s = 0, 2, 4, 6, 8 for neutrons (a) and s = 0, 2, 4, 6, 8, 10 for protons (b) in ^161Dy as functions of excitation energies.
From the curves, distinct protuberances are evident for s = 2, 4, 6, and 8 for neutrons, and s = 2 and 4 for protons. Notably, these protuberances are absent at higher seniority numbers. This observation suggests that the initial fluctuations around 4 MeV are primarily attributed to the contribution of one-pair-broken states. The number of excited states increases rapidly due to the emergence of one-pair-broken states with increasing excitation energies. However, when T ≥ 4 MeV, the growth rate of the curves diminishes because no additional one-pair-broken states can be formed. This phenomenon corresponds to the first S-shaped feature in the heat capacity curve.
Subsequently, with the emergence of the second fluctuation, the appearance of two-pair-broken states leads to rapid increase, followed by a decline after 8 MeV excitation energy for neutrons and 7 MeV for protons. Other protuberances may indicate additional S-shaped features in the heat capacity curve at higher energies.



















Figure <ref> illustrates contour plots of the partition function of neutrons and protons in the complex temperature plane. Zero points at β≈ 1.05  MeV^-1 for protons and 1.3 MeV^-1 for neutrons effectively reflect the critical temperature of the phase transition. The corresponding critical temperature of protons and neutrons are 0.95 MeV and 0.77 MeV, and the experimental value for ^161Dy is 0.52 MeV <cit.>. The contour plots of the partition function of neutrons for ^161Dy with different blocking scenarios in the complex temperature plane are presented in Fig. <ref>. It is evident that the partition functions of “bk1" and “bk2" contribute the most to the overall partition function. Conversely, patterns of blocking higher energy levels are similar, contributing minimally to the overall partition function. This observation aligns with earlier discussions regarding the effects of different blocking scenarios on heat capacity.
Furthermore, in cases of blocking higher energy levels, the partition function predominantly occupies the zero-point domain. This suggests weakening of the nuclear superfluidity, with the nucleus primarily existing in the normal phase.






According to Fig. <ref>, the minimums are determined iteratively as the zero points of the partition function. The distributions of zero points of the partition function for neutrons and protons in the complex temperature plane are displayed in Fig. <ref>, with the zero points of the pairing phase transition indicated by the solid line.
By calculation using Eq. <ref>, the α values corresponding to the neutrons and protons of ^161Dy are -2.04 and -5.66, respectively. Based on the phase transition classification method, this indicates a first-order phase transition.










Figure <ref> illustrates the contour plots of the partition function for ^161Dy in the complex temperature plane with different seniority numbers s = 0, 2, 4, 6, 8, and 10, where the upper and lower lines correspond to neutrons and protons, respectively.
In the first column of the seniority number s = 0 (fully paired states), zero points exist only in the region where β is very small. Additionally, there are no zero points in the region of 0 to 2 MeV (β > 0.5 MeV^-1), indicating that the nucleus is entirely in the superfluid phase.
As we move to the s = 2 (one pair broken) state, the zero point domain begins to appear with a clear boundary, signifying the coexistence of the superfluid and normal phases. Subsequently, differences emerge between neutrons in odd systems and protons in even systems. The zero domain for neutrons is smaller than that for protons in the s = 2 state, primarily due to the blocking effect of the odd nucleon hindering the evolution of the pairing phase transition.
With increasing seniority number, the zero point domain gradually expands, and the normal phase gradually dominates. The evolution of the pairing phase transition can still be effectively demonstrated. By the time we reach the seniority number s = 8 or 10 (completely broken) state, the image is nearly entirely covered by the zero point domain, indicating that the nucleus is entirely in the normal phase.




 §.§ The Toy Model


The study above found that the structure of single particle levels affects the heat capacity curve. Under the SLAP framework, single particle levels are derived by Eq. <ref>. In fact, it may be affected by many factors, such as nuclear deformation, shell structure, etc. Since this paper only focuses on the influence of pairing correlation, it is necessary to remove the influence of other factors, so an ideal model can be established. In this study, the ideal models with odd particles and even particles are constructed with controllable pairing strength, and thermodynamic quantities are investigated under the same framework. 
For the even system, a configuration space with 10 levels and 10 particles is selected, with 5 levels above the Fermi surface. The energy levels range from 1 to 10 MeV, and the pairing strength varies from 0 to 2 MeV in increments of 0.1 MeV. The cut-off energy is set to 45 MeV, resulting in a configuration space dimension of 10^4.
In the case of the odd system, a configuration space with 10 levels and 11 particles is chosen, with 4 levels above the Fermi surface. The single particle levels, pairing strength, and cut-off energy are consistent with the even system. The configuration space dimension is 1.4×10^4.By employing these ideal models, the study aims to investigate the thermodynamic quantities under the same framework, thus allowing for clearer understanding of the sole influence of pairing correlation on the system.






Figure <ref> illustrates the heat capacity with varying pairing strengths for both the ideal even system (a) and the ideal odd system (b) as functions of temperature. As the pairing strength G increases, the temperatures corresponding to the first and second turning points of the curves, as well as the peak values of the curves, also increase.
In Fig. <ref> (b), fluctuations are observed in the low-temperature region, similar to the situation discussed for ^161Dy. These fluctuations are primarily attributed to the pairing re-entrance phenomenon in the odd-nucleon system.








According to the contour plots of the partition function with varying pairing strengths for the ideal even system in the complex temperature plane, the Distribution of Zeroes (DOZ) of the partition function is displayed in Fig. <ref>.
When the pairing strength is 0.1, a zero point is observed outside the plotted range, as indicated in the insert figure. In contrast, when the pairing strength is 0 (indicating no pairing correlation), there is no zero point in the corresponding β region. However, the heat capacity curve for G = 0 still exhibits an S shape, suggesting that the appearance of an S-shaped heat capacity curve does not solely correspond to the pairing phase transition.
As the pairing strength G increases, the branch of zero points near the real axis undergoes significant changes. The zero point corresponding to the peak value of the heat capacity curves gradually shifts to the region of smaller β due to the increasing temperature.

The corresponding α values are calculated and presented in Table <ref>. For pairing strengths G ≤ 0.7, a first-order phase transition is observed. When G is between 0.8 and 0.9, a higher-order phase transition is identified. Finally, for G = 1, a second-order phase transition is noted. This analysis reveals a nonlinear relationship between the pairing strength and the order of the pairing phase transition, consistent with findings in Ref. <cit.>.














Figure <ref> illustrates the Distribution of Zeroes (DOZ) of the partition function with varying pairing strength for the ideal odd system in the complex temperature plane. Notably, a branch is observed when the pairing strength is 0. Presence of this branch is primarily attributed to changes in the zero point distribution caused by the blocking effect, and it does not correspond to a real pairing phase transition. This behavior contrasts with the case of even systems.




Based on the zero distribution depicted in Fig. <ref>, the corresponding α values are calculated and presented in Table <ref>. For pairing strengths G ≤ 0.7, a first-order phase transition is identified. However, for G between 0.8 and 1, a higher-order phase transition is observed. This analysis demonstrates that the pairing strength also influences the pairing phase transition in the odd system.



§ SUMMARY



In summary, this work investigates the pairing phase transition in the odd-A nucleus ^161Dy using the covariant density functional theory and the shell-model-like approach. Thermodynamic quantities are evaluated within the canonical ensemble theory, and the phase transition order is determined using a classification scheme based on the distribution of zeros of the partition function in the complex temperature plane. The impact of different pairing strengths on the pairing phase transition is also examined.

The heat capacity curve of ^161Dy exhibits an S shape, indicating a pairing phase transition from the superfluid phase to the normal phase. Extending the calculation to the complex temperature plane reveals that the pairing phase transition is of first order. Blocking the Fermi surface and the first level above it significantly contribute to the total partition function, highlighting the importance of considering the blocking of the nearest levels above the Fermi surface in the study of odd-A nuclei. The contour plots of the partition function with different seniority show that the blocking effect of the odd nucleon hinders the evolution of the pairing phase transition.

Furthermore, ideal models with odd and even particles are constructed to explore the influence of different pairing strengths on the pairing phase transition in the complex temperature plane. In the ideal even system, the absence of pairing correlation does not lead to a zero point corresponding to the critical temperature in the large β region, yet the heat capacity curve still exhibits an S shape. This suggests that the appearance of an S-shaped heat capacity curve does not exclusively indicate a pairing phase transition. Conversely, in the ideal odd system, the presence of a zero point branch in the partition function plot is due to the blocking effect. The analysis also demonstrates that the pairing strength has nonlinear influence on the pairing phase transition. In the ideal even system, the phase transition order changes from first order to higher order and then to second order as the pairing strength increases. For the ideal odd system, the phase transition order changes from first order to higher order with increasing pairing strength. In the near future research, we will focus more on the experimental spectroscopic properties and other properties to study the pairing.


This work was supported by National Natural Science Foundation of China under Grant No. 11775099 and the Jiangnan University Basic Research Program (JUSRP202406002).


apsrev4-1





 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: 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: 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.



 Paper Title: Hot and Dense QCD Shear Viscosity at (almost) NLO.


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    𝒞#1Appendix <ref>#1Section <ref>#1Ref. <cit.>#1Fig. <ref>Institut für Kernphysik, Technische Universität Darmstadt
    
    Schlossgartenstraße 2, D-64289 Darmstadt, Germanyidanhoni@theorie.ikp.physik.tu-darmstadt.de,guy.moore@physik.tu-darmstadt.deThe next-to-leading order weak-coupling shear viscosity of QCD was computed 6 years ago.
     However, these results have never been applied at finite baryon chemical potential μ, even though intermediate-energy heavy ion collisions and merging neutron stars may explore the Quark-Gluon Plasma in a regime where baryon chemical potentials are large.
     Here, we extend the next-to-leading order shear viscosity calculations to finite μ, and we show that, while the convergence of the weak-coupling expansion is questionable for achievable plasmas, it is somewhat better at μ > T than at μ=0.
    
    Hot and Dense QCD Shear Viscosity at (almost) NLO
        Isabella Danhoni, Guy D. Moore
        August 5, 2024
    =================================================
    
    
    
    § INTRODUCTION
    
    
    The deconfined phase of hot and dense QCD is known as the quark-gluon plasma (QGP). Evidence shows that this phase was present in the early universe and that it has been recreated at microscopic scales in heavy ion collision experiments<cit.>.
    These results have sparked large interest in this subject over the past few decades, and experiments indicate that the QGP in such collisions is a strongly coupled fluid<cit.> with behavior close to that of a perfect fluid<cit.>.
    On the other hand, sufficiently early in the history of the early Universe, the running coupling should have been small enough that a perturbative treatment may be valid.
    For thermodynamical quantities, suitably resummed perturbative calculations work starting at temperatures a few times that of the chiral symmetry restoration transition<cit.>.
    
    
    
    
    
    In lower-energy collisions, quark-gluon plasma is generated with a significant net baryon number, such that the baryon chemical potential μ_B is several times higher than the temperature.
    This regime cannot be directly studied on the lattice due to the sign problem (or complex action problem).
    For a chemical potential μ_B≠ 0 the fermion determinant (or Monte Carlo action) becomes complex, which ruins the probabilistic interpretation of this quantity used in lattice QCD calculations<cit.>.
    For this reason, the high-density region of the QCD phase diagram is much less well understood than the μ_B=0 axis.
    As experimental efforts focus on bringing clarity to this region, such as RHIC-BES <cit.>, HADES<cit.> and CBM at
    FAIR<cit.>, the need for parallel theoretical development becomes clear.
    
    At high temperatures, we claim that the perturbative series should work better at a high chemical potential, such that μ_B > T > T_c, than in the case of μ_B=0<cit.>. This is a consequence of the fact that scattering from quarks is enhanced by a factor of μ_B^2 / T^2.
    The failure of perturbation theory in hot QCD occurs before the coupling becomes large, due to the strongly-coupled nature of soft gluons.
    These play a central role in the breakdown of the perturbative expansion for the shear viscosity <cit.>.
    But at high chemical potential, where quarks play a larger role, one expects the perturbative series to be better behaved.
    This means that perturbation theory has a better chance of working at high μ/T than at μ=0.
    Although the (μ,T) values which can be explored in experiment are unlikely to extend into the region where perturbation theory is directly predictive, the perturbative results can still be used in modeling the (μ,T) behavior of the shear viscosity phenomenologically, as was recently done using leading-log results in Ref. <cit.>.
    Essentially, perturbation theory serves as a limiting behavior so that η/s can be interpolated based on high-density perturbative, low-density hadron-resonance gas, and intermediate density data-driven estimates.
    
    Previously Arnold, Moore, and Yaffe have computed the shear viscosity at leading perturbative order in the absence of a chemical potential
    <cit.>, and Ghiglieri et al have extended this calculation to “almost” next-to-leading order (NLO) <cit.>.
    Separately, we have presented the shear viscosity at finite chemical potential at leading-log order <cit.>.
    The goal of this paper is to work to almost next-to-leading order at finite chemical potential, both to obtain the most reliable estimates for the viscosity in this regime that we can, and to test how perturbation theory converges in the high-μ regime.
    We start in Section <ref> by reviewing the kinetic theory and computational approach that Arnold, Moore, and Yaffe used in <cit.>.
    In Section <ref>, we review how to extend the collision operator to next-to-leading order and describe the computational tools required. 
    In Section <ref>, we show what changes must be made to the calculation to include chemical potentials.
    The analysis of our calculations and final results are shown in
    Section <ref>.
    We end in Section <ref> with a short discussion and summary.
    But we can present the main conclusions very briefly now.
    The shear viscosity, rescaled as η T/(ε + P), is a weak function of μ_q/T (with μ_q=μ_B/3 the chemical potential of a quark).
    The convergence of the perturbative series improves when
    μ_q^2 > N_c π^2 T^2, but the improvement is somewhat modest.
    For more complete details, see Section <ref>.
    
    
    
    
    
    § KINETIC THEORY
    
    In this section, we briefly summarize how shear viscosity is defined and how it can be computed to leading order (LO) in the Effective Kinetic Theory (EKT).
    Shear viscosity is a property relevant for systems with space-nonuniform flow velocity, meaning that the energy and momentum densities vary through space, but in a manner which is sufficiently smooth that the system is locally near equilibrium.
    In this case we can exchange the conserved densities T^μ 0 for the local 4-velocity and energy density, defined with the Landau-Lifshitz convention <cit.>:
    u^μ(x,t) is the timelike unit eigen(four)vector of T^μν and ε is its eigenvalue,
    T^μν u_ν = ε u^μ
    with[We use a mostly-positive metric tensor g_μν = Diag[-1,+1,+1,+1],
    and natural units, c=1, ħ = 1. Capital letters P,K are 4-vectors, lower case letters p⃗,k⃗ or p_i,k_i are the space components, p^0,k^0 are the time components, and p,k are the magnitudes of the space components.]u^μ u_μ = -1.
    For uniform u^μ,ε the system will be in equilibrium with
    T^μν = ε u^μ u^ν + P g^μν,
    where P the pressure is determined from ε by the equilibrium equation of state.
    
    However, for nonuniform u^μ the system will depart from equilibrium and T^μν will not follow its ideal form.
    Defining the projection operator into the local rest frame,
    
        Δ^μν≡ g^μν + u^μ u^μ ,
    
    the shear tensor is defined as
    
        σ^μν≡∇^μ u^ν + ∇^ν u^μ
            - 2/3Δ^μν∇_α u^α .
    
    For small and slowly varying σ_ij, linear response theory predicts that the stress tensor should change by an amount linear in the shear tensor:  in the local rest frame,
    
        T_ij = P g_ij - η σ_ij
    
    with P the pressure and η, the shear viscosity, defined as the coefficient of the linear response of T_ij to nonvanishing σ_ij.
    If the fluid flow also has a divergence θ = ∂_i u_i, then an additional term T_ij = … - ζθΔ_ij would be present, with ζ the bulk viscosity.
    Bulk viscosity requires a different analysis <cit.> which we will not attempt here.
    
    Many (not all) theories have a kinetic description, and this should include nonabelian gauge theories in the weak coupling regime.
    In this case one can give a coarse-grained description of the state of the system in terms of the occupancies of each field type at each point in time and phase space, f^a(p⃗,x⃗,t).
    In equilibrium and at lowest order in the coupling, we have
    
        f_0^a(p⃗,x⃗,t) = 1/exp(γβ ( p^0 - u_i p^i - μ_a ) ∓ 1 
            = 1/exp (β (-u_μ P^μ - μ_a)) ∓ 1.
    
    Here u^μ = (γ , γv⃗) is the local velocity 4-vector as before, γ is the associated relativistic gamma-factor, and β=β(x⃗,t) = 1/T is the local temperature.
    The sign ∓ 1 is - for bosons and + for fermions (the upper sign will always apply to bosons), and μ_a is the chemical potential for species a, which will be nonzero for quark species and opposite for a particle and its antiparticle.
    In this work, we will consider SU(3) gauge theory with  vectorlike quark species each with the same chemical potential.
    
    
    Shear viscosity, as well as other transport coefficients, is theoretically defined in a Kubo-type formulae relating them to the zero-frequency transport limit of the spectral functions of two-point correlators of the appropriate operators.
    Performing a perturbative evaluation at leading order using the diagrammatic approach based directly on these Kubo formulas would require complex resummations of numerous sub-diagrams to all orders.
    This has been done successfully for scalar field theory
    <cit.> and for QED for electrical conductivity
    <cit.> and shear viscosity <cit.>,
    and in each case the result proved to be the same as a kinetic description.
    In the case of full QCD such a diagrammatic calculation has never been successfully attempted, but physical arguments indicate that the result should again justify a kinetic description <cit.>.
    Therefore, we adopt the linearized version of the EKT developed in <cit.>.
    
    One starts as usual with a Boltzmann equation
    
        [ v⃗_p ·∂/∂x⃗] f^a(p⃗,x⃗,t) = - ^a[f],
    
    where f^a refers to the nonequilibrium form of the statistical function,
    which we linearize about the equilibrium form
    
        f^a(p⃗,x⃗) = f_0^a(p⃗, x⃗) + f_0(1 ± f_0)  f_1^a(p⃗,x⃗)  ,
    
    according to the Landau-Lifshitz conventions under which f_0 contributes all of the energy, momentum density, and net particle numbers in the system.
    The choice to normalize f_1 with a factor of f_0(1± f_0) will be convenient in what follows.
    For a more detailed description one can refer to <cit.> which developed the leading-log application of Boltzmann equations to the evaluation of the QCD shear viscosity,
    <cit.>, which extended the treatment to leading order and established most of our notation and techniques,
    and <cit.>, which extended the treatment to finite chemical potentials, albeit at leading-logarithmic order.
    We will save the discussion of the collision operator ^a[f] for the next section.
    
    The departure from equilibrium must first arise at linear order in σ_ij.
    Using rotational invariance, one can then write the departure from equilibrium as
    
        f_1^a(p⃗,x⃗) = β^2 X_ij(x⃗) χ^a_ij(p⃗) = β^2 X_ij(x⃗) I_ij(p⃗) χ^a(p)  .
    
    Here χ^a(p) is a species-dependent, pure scalar function of p which characterizes the departure from equilibrium in the presence of shear stress, and
    
        X_ij   ≡1/√(6)σ_ij
        
            I_ij(p⃗)    ≡√(3/2)( p̂_i p̂_j - 1/3δ_ij)  .
    
    The normalization is explained in Ref. <cit.> and is chosen so that I_ij is a properly normalized projection operator.
    Evaluating the space derivatives on the lefthand side of boltz
    one finds to lowest order,
    
        v⃗_p ·∂/∂x⃗
            f^a_0(p⃗,x⃗,t) = β^2 X_ij S^a_ij
    
    with
    
        S^a_ij(p⃗)  ≡ p^0   T   f_0^a(1± f_0^a) I_ij(p⃗)  .
    
    Using this and inserting chidef into the RHS, boltz reduces to
    
        β^2 X_ij  S^a_ij(p⃗) = ^a[β^2 X_ij χ^a_ij(p⃗)]
                
            S^a_ij(p⃗) = χ^a_ij(p⃗) ,
    
    where χ^a_ij(p⃗) means the result of factoring β^2 X_ij out of the collision operator.
    
    This equation is solved using a variational method.
    This strategy was used in <cit.> in leading log and leading order calculations. 
    First one defines an inner product on departures from equilibrium:
     
        (f,g) = β^3 ∑^_a∫_p⃗ f^a(p⃗)g^a(p⃗)
    
    It can be shown that the collision operator is Hermitian with respect to this inner product.
    We define
     
        𝒬[χ] = (χ_ij, S_ij)- 1/2(χ_ij,χ_ij)  ,
    
    which can be viewed as a functional over the space of χ^a(p) values.
    That is, the functional 𝒬 takes its maximum value when the Boltzmann equation is satisfied.
    Furthermore, its value at this maximum,
     
        =1/2(χ_ij, S_ij)=1/2 (χ_ij,χ_ij) = 1/2 ( S_ij,^-1 S_ij )  ,
     
    determines the viscosity:
     
        η = 2/15.
    
    As known from other variational problems, one can obtain highly accurate approximate results by performing a restricted extremization within a well-chosen finite dimensional subspace.
    For this purpose, we expand the χ(p) functions into a finite basis set:
    
        χ(p) =∑_m=1^N a_m ϕ^(m)(p) .
    
    We will use the variational Ansatz proposed
    [To our knowledge this variational Ansatz was first used in Ref. <cit.>, but it is not actually presented there; we thank Larry Yaffe for private communications on this point.]
    in Ref. <cit.>:
    
        ϕ^(m)=p(p/T)^m/(1+p/T)^N-1,  m=1,...,N  .
    
    The subspace spanned by this basis strictly increases as N is increased.
     
    
    
    
    § COLLISION OPERATORS FOR AN NLO EVALUATION
    
    
    At vanishing chemical potential, the perturbative framework suggests that the QGP predominantly comprises quasiparticle excitations with momenta of the order of the temperature.  
    Their dispersion and occupancy are corrected only at (g^2), which for an (g)-subleading calculation means that we can ignore corrections both to the lowest-order entropy density s and to the lowest-order particle dispersion relations (except for splitting, see below).
    Therefore NLO corrections will only be needed in the collision integral, which we turn to next.
    The LO calculation <cit.> requires two types of scattering processes, the  scatterings with all hard ((T)) external particles and  effective splitting processes between hard participants. 
    
    At NLO, all the scatterings and corrections to be added are the ones suppressed by a single power of g.  As shown in detail in <cit.>, there are only a
    few such (g) subleading effects, which can be described as follows. The rate of soft
     scattering is modified to be considered as an
    additional momentum-diffusion coefficient δq̂.  This along with an (g) correction to the medium corrections to dispersion provides an (g) shift in the  splitting rate.  Next, the  splitting rate must be corrected wherever one participant of the process is "soft", p ∼ gT, or the opening angle is less collinear.  Finally, the numerical implementation of the LO scattering kernel <cit.> already resums a small amount of these NLO effects, requiring a subtraction to δq̂ and the longitudinal momentum diffusion, δq̂_L. 
    This section will briefly review the collision operator at LO and NLO at vanishing chemical potential.
    
    
    
     §.§ Collision Integrals at LO
    
    
    The leading-order collision operator encodes the contribution of tree-level 
    scattering processes, with Hard-Loop resummed
    propagators in the soft-sensitive channels,
    as well as collinear, effective  processes resumming the effect
    of splitting caused by any number of soft scatterings.
    Both processes contribute to order g^4T to the collision operator; a subset of C_a^[f]
    is logarithmically enhanced, g^4Tln(1/g), due to the sensitivity
    to the soft scale gT.
    C_a^[f] and C_a^[f] are described in detail
    in <cit.>. Therefore, the collision term has the form:
    
        𝒞_a[f] = 𝒞_a^2↔ 2[f] + 𝒞_a^1↔ 2[f]
    
    
    
    The matrix elements that contribute to 2 ↔ 2 processes at leading order come from the diagrams shown in Fig. <ref>. In terms of the collision operator, these contributions can be written as:
    
        ^a[f](p⃗) =1/2∑_bcd∫_k⃗,p⃗,k⃗'⃗   |ℳ_abcd(P,K,P',K')|^2/2p^0  2k^0  2p'^0  2k'^0
            (2π)^4 δ^4(P+K-P'-K') 
           ×{ f^a(p⃗)f^b(k⃗)[1± f^c(p⃗'⃗)][1± f^d(k⃗'⃗)] 
           ×{ -  f^c(p⃗'⃗)f^d(k⃗'⃗)[1± f^a(p⃗)][1± f^b(k⃗)]}
    
    where the incoming/outgoing momenta p⃗,k⃗ and p⃗',k⃗' are all on shell, p^0=p. The squared matrix element 
    |ℳ_cd^ab| is hard thermal loop (HTL) resummed
    <cit.>
    and is summed over all spin polarizations and colors.  We write ∫_k⃗ = ∫d^3k⃗/(2π)^3 to simplify the notation. 
    
    
    Diagrams (D) and (E) are processes involving t (or u) channel fermion exchange.
    These processes have a logarithmic infrared divergence when computed using free propagators.
    To cut off this divergence, one can include the retarded HTL fermionic thermal self-energy, as was originally done in Ref. <cit.>.
    It was later shown in Ref. <cit.> that there is a simpler treatment which is also correct at the leading order.
    The most important feature of the HTL self-energy insertion is that it correctly reproduces the total scattering rate for this process; since the process is nearly isotropic, the detailed angle dependence is not relevant.
    A mass-like insertion, with the correctly chosen coefficient, reproduces this rate and is much simpler to use in calculation.
    Specifically for a soft fermion exchange in the t or u channel, one can simply use:
    
        u-s/t→u-s/tq^2/q^2 + ξ_q^2 m^2_∞,q
    
    where q is the exchanged momentum, ξ_q = e/2 ≃ 1.359, and m^2_∞,q is the medium-induced effective quark mass-squared for quarks, given by
    
        m^2_∞,q    = 2g^2∫d^3 p/2p^0(2π^3)[2 f_g(p) +  (f_q(p)+ f_q̅(p) )]
         
            = g^2 /4( T^2 + μ^2/π^2) = g^2/3( T^2 + μ^2/π^2).
    
    We have allowed ourselves to make this simplification – also because the cumulative effect of these diagrams is rather small and is exponentially suppressed ∝ e^-μ/T in the large μ/T region.
    
    On the other hand, diagrams (A),(B), and (C) represent processes involving t or u-channel gauge boson exchange that require the inclusion of the thermal
    gauge boson self-energy on the internal propagator to cut off the infrared sensitivity of these processes. The matrix element of interest is (s^2+u^2)/t^2 which can be written as:
    
        1/8D_μα^retD_νβ^adv(pγ^μkγ^ν)(p'γ^αk'γ^β)
    
    where D_μα^ret and D_μα^adv, are the retarded and advanced boson propagators, respectively.
    Following the same notation as <cit.>, the plasma frame frequency and momentum
    carried by the gauge boson propagator are denoted as ω and q. The retarded gauge boson propagator can be conveniently expressed in Coulomb gauge as:
    
        D_00^ret(ω,q)=-1/q^2 - ^ret(ω,q)
           D_ij^ret(ω,q)=δ_ij- q̂_iq̂_j/q^2 - ω^2 +^ret(ω,p)
           D_0i^ret(ω,q)=D_i0^ret(ω,q)=0 .
    
    The equilibrium transverse and longitudinal gauge boson self-energies in the HTL approximation are <cit.>
        ^ret(ω,)     =  ^2
        	{ω^2/2q^2 
        	+
        	    ω  ( q^2 -ω^2 )/4 q^3[
        		ln( q+ω/q-ω) - i  π]
        	}  , 
        
        Π^ret(ω,)     =  ^2
        	{
        	    1
        	    -
        	    ω/2q[
        		ln(q+ω/q-ω) - i  π]
        	}  ,
    
    where we have assumed |ω| < q,
    which is the only case of relevance. 
    Here ^2 is the squared Debye mass, which we will give for the case of nonzero μ,T in mDebye.
    The advanced propagators are the complex conjugates of the retarded ones.
    
    The remaining diagrams and the interference terms are all finite and can be computed directly. One should refer to <cit.> for more details on these calculations.
    
    The second type of contribution to leading order comes from 1 ↔ 2 nearly collinear processes. These can be approximated as strictly collinear, which only produces relative changes suppressed by at least one power of g.  Corrections at this order will be treated in the next subsection.
    The leading-order computation proceeds as follows
    <cit.>:
    
        𝒞_a^1↔ 2[f](p⃗) =    (2π)^3/2|p⃗ |^2ν_a∑_bc∫_0^∞ dp' dk δ(|p|-p'-k')γ^a_bc(p;p',k') 
           {f^a(p)[1± f^b(p')][1 ± f^c(k)]-f^b(p') f^c(k)[1 ± f^a(p)]} 
        
            +   (2π)^3/2|p⃗ |^2ν_a∑_bc∫_0^∞ dk dp' δ(|p|+k-p')γ^c_ab(p';p,k) 
           {f^a(p)f^b(k)[1 ± f^c(p')]-f^c(p') [1 ± f^a()][1 ± f^b(k)]},
    
    where γ_bc^a represents the splitting rate, and is given by <cit.>:
    
        γ_bc^a(p;p-k,k)= g^2d_R_bC_R_b/64π^4{[ p^4+k^4+(p-k)^4/p^3k^3(p-k)^3                        g ↔ gg;     p^2+(p-k)^2/p^2k^3(p-k)^2                        q ↔ qg;     p^2+(p-k)^2/p^3k^2(p-k)^2                        g ↔ qq;                               ]}×∫dh^2/(2π)^22 h·F_b,
    
    where h= p×k is a transverse, two-dimensional vector related to the transverse momentum picked up during the splitting process. And F_b can be obtained through the following integral equation:
    
        2=     iδ E(h)F_b(h) + ∫dq_⊥^2/(2π)^2C(q_⊥) {(C_R_b-C_A/2)[F_b(h)-F_b(F-kq_⊥)]
            C_A/2[F_b(h)-F_b(h+pq_⊥)] + C_A/2[F_b(h)-F_b(h-(p-k)q_⊥)]}.
    
    
    For the case of g ↔ q q̅, C_R_b - C_A/2 =  - C_A/2 multiplies the term with F_b(h-pq_⊥)
    rather than F_b(h-kq_⊥). Here C(q_⊥) represents the  leading-order transverse scattering kernel:
    
        C(q_⊥)=g^2T^2/q_⊥^2(q_⊥^2+^2),
    
    and δ E(h) is the energy difference between initial and final collinear particles, expanded to first nontrivial order in thermal masses and transverse momenta:
    
        δ E(h) = h^2/2pk(p-k)+m^2_∞,c/2k+m^2_∞,b/2(p-k) - m^2_∞,a/2p.
    
    We encountered m^2_∞,q already in thermalquarkmass.
    For gluons, m^2_∞,g = ^2/2.
    
    
    
    
     §.§ Next-to-Leading Order corrections
    
    
    We now focus on the next-to-leading order corrections.
    Treating kinetic theory to NLO was pioneered in
    Ref. <cit.> and applied to QCD shear viscosity in Ref. <cit.>.
    In this work, the authors observed that the analytically complex parts of the collision operator do not receive (g) corrections, which instead only modify collinear exchange, incorporated by changing γ^a_bc,
    and soft exchange, which can be modeled as momentum-diffusion and identity-changing processes.
    We briefly review the extended treatment that captures these O(g) corrections, and then we summarize it and present an explicit form of the collision operator at NLO.  
    
    
    
    
      §.§.§ Momentum diffusion
    
    
    The small momentum exchange part of diagrams (A), (B), (C) can be well approximated by momentum diffusion for the particles on each line.
    Beyond leading order, there are additional small-momentum-exchange effects which are only suppressed by (g) relative to LO, which must be incorporated.
    One such effect is the possibility for the gluon in diagrams (A), (B) to carry a soft momentum.
    Other effects involve the interference between multiple soft scatterings, soft gluon emission during scattering, and nonabelian interactions between scattering processes, such as those depicted in Figure <ref>.
    
    
    
    Because the momentum exchanges involved in these processes are small (that is the reason they can occur at (g) and not (g^2)), these terms can be adequately described in terms of Langevin dynamics, that is, in terms of momentum drag and longitudinal and transverse momentum diffusion.
    This is analyzed in the context of heavy quark dyamics in Ref. <cit.> and jet energy loss in Ref. <cit.>, and applied to shear viscosity in Ref. <cit.>.
    As shown there, the effective collision term due to soft scatterings, cut off at a scale μ_⊥, is
     
        (C_diff^ f_1)^a() =
            -1/2∂/∂ p^i q̂^ij_a    f_0^a(p) (1 ± f_0^a(p))
            ∂ f_1^a()/∂ p^j  +   ,
    
    where
    
        q̂^ij_a = q̂_a,Lp̂^i p̂^j
          +  12q̂_a (δ^ij - p̂^i p̂^j )
    
    represents the momentum diffusion parallel and perpendicular to the particle's momentum through q̂_L and q̂ respectively.
    
    Contracting with the departure from equilibrium and integrating over , the contribution to the inner product of Qdef takes the form
    
     (f_1, C^2 → 2_diff f_1)  ≡. (f_1, C^2 → 2_diff f_1)|_ loss
     +
     . (f_1, C^2 → 2_diff f_1)|_ gain  ,
     
     where
     . (f_1, C^2 → 2_ diff f_1) |_ loss
    = 1/2β^3 ∑_aν_a  ∫_  f_0^a(p)(1 ± f_0^a(p))  q̂^ij_a ∂ f_1^a() /∂ p^i∂ f_1^a() /∂ p^j   ,
    
    
    The gain collision term takes the form
    
        =    1/2∂/∂ p^i (
           f_0^a(p) (1 ± f_0^a(p))   ∑_bν_b ∫_^ij_ab (·̨̂)   f_0^b(k) (1 ± f_0^b(k))
           ∂ f_1^b()̨/∂ k^j )
        
        . (f_1, C^2 → 2_diff f_1)|_ gain 
        =      -β^3/2∑_abν_a ν_b ∫_
        f_0^a(p)(1 ± f_0^a(p)) f^b_0(k) (1 ± f_0^b(k))
        
           ×_ab^ij (·̨̂) ∂ f_1^a() /∂ p^i∂ f_1^b()̨/∂ k^j,
    
    where the angular function is
    _ab^ij(·̨̂)
    =  g^4 C_R_a C_R_b/d_A∫d^4Q/(2π)^4   |G_μν^R(Q) v_^μ v_^̨ν|^2     2πδ(v_· Q) 2πδ(v_·̨Q) q^i q^j   ,
    
    which is related to q̂^ij_a through
    q̂^ij_a =  ∑_b ν_b ∫_f̨_0^b(k) (1 ± f_0^b(k))  _ab^ij(·̨̂)   .
    
    Beyond leading order it turns out to be much easier to determine q̂^ij_a than ^ab_ij(·̨̂),
    since light-cone techniques work for the former but not the latter.
    Therefore, in practice we will not be able to compute this gain term.
    Since our best estimates based on leading-order analyses indicate that angular cancellations make this term small, we will instead drop it.
    This is why we generally refer to our results as “almost next-to-leading order.”
    
    Examining the expression for
    . (f_1, C_diff^f_1) |_ loss,
    it becomes evident that it involves a single light-like vector denoted as v_. 
    Using the causality and KMS properties of such light-like correlators <cit.>, these leading-order
    soft contributions to q̂ and q̂_L  can be evaluated
    in closed form <cit.>,
    
        q̂^a|_soft=g^2C_R_a T
        	^2/2πlnμ_⊥/,	    q̂_ L^a|_soft=g^2C_R_a T
        	^2/4πln√(2)μ_⊥/.
    
    In these expressions, gT≪μ_⊥≪ T is a cutoff on the q_⊥≡√(q^2 - ω^2)
    integration, which separates the soft from the hard scale. In the region where ω,q≳ T, and the bare matrix elements can be used to compute the hard contribution to the collision operator C^, there is a cancellation on this cutoff dependence. Since light-like separated points are effectively causally disconnected
    as far as the soft gauge fields are concerned,  q̂ and q̂_L have a simple form.
    Using the explicit form of the angular dependence of
    f_1^a() = β^2 χ_ij() X_ij,
    
    straightforward analysis shows that the loss term  reduces to
    
        . ( χ_ij, C^_lossshort χ_ij) |_ loss =
         	 β^3/2∑_a ν_a∫_ f^a_0(p)
         	 (1± f_0^a(p)) ×
        [(χ^a(p)')^2  q̂^a_ L|_soft
         	+
         	ℓ(ℓ+1)χ^a(p)^2/2p^2q̂^a|_soft]   ,
    
    which is the most useful form for evaluating the transport coefficients
    numerically.
    
    The next-to-leading order value of q̂^a has been computed <cit.>.
    This result can be simply included by adding it to the above results <cit.>:
      
        δq̂^a=g^4C_R_a C_A  T^2 /32π^2(3π^2+10-4ln 2)  .
    
    the shift included in the soft gluon exchange diffusion rate gives
     
        ( χ_ij, C^δq χ_ij) =
         	 β^3/2∑_aν_a δq^a
         	 ∫_   f^a_0(p)
         	[1± f_0^a(p)]
         	 ℓ(ℓ+1)χ^a(p)^2/2p^2 .
    
    
    As discussed in Ref. <cit.>, the precise procedure described in the last subsection for including HTL effects in the leading-order calculation also introduces some NLO effects in that calculation.
    In particular, the phase space and statistical functions for scattering from soft gluons incorporate some of the (g) effects which we have just described.
    Therefore, the leading-order procedure we use will not in fact reproduce qhatql, but will reproduce that expression plus some (g) additional terms.
    These contributions must then be subtracted from the (g) contribution we include here, in order to avoid overcounting such effects.
    Fortunately, since the effects arise from scattering off of gluons, they are unmodified by the inclusion of quark chemical potentials, and we can take over the analysis of
    Ref. <cit.>.
    The resulting counterterm can be written as
     
        ( χ_ij, C^_(g) finite g χ_ij)    =   d_AC_Ag^4 /32π^5 T∑_a T_R_a∫_0^∞ dp  p^2 f^a_0(p)[1± f_0^a(p)]
              ×{4.2695[(χ^a(p)')^2+(χ^g(0)')^2]
        	  +7.1769ℓ(ℓ+1)/2p^2[(χ^a(p))^2+(pχ^g(0)')^2]
              +
        	  18.0669 δ_ℓ2[χ^g(0)']^2
        	  },
    
    and the soft fermion exchange counterterm is given by
    
        ( χ_ij, C^_(g) finite q χ_ij) =     d_F ^2 g^4
        m_∞/32π^5T^2 9.95268 ∫_0^∞ dk k 
         f^q_0(k)	[1+ f_0^g(k)]
           ×[(χ^q(k)-χ^g(k))^2+(χ^q(0))^2] .
    
    One should refer to <cit.> for a detailed derivation of these counterterms.
    
    
    
    
      §.§.§ Collinear and semi-collinear contributions 
    
    
    The collinear splitting process, collop21 to deltaEeq, involves the same momentum diffusion effects we just described.
    And as we just described, the strength of momentum diffusion receives (g) corrections.
    Furthermore, deltaEeq shows that collinear splitting is sensitive to thermal effective masses.
    These also turn out to receive (g) corrections.
    Therefore we need to analyze how these two corrections impact the collinear splitting rate.
    To conduct the analysis, we will follow Ref. <cit.> and Fourier transform collop21 from transverse-momentum _⊥ to transverse impact parameter _⊥ space.
    This not only diagonalizes the collision kernel C(k_⊥), but also turns an integral equation into a differential equation.
    Furthermore,  the source on the left-hand side becomes a boundary condition at b=0
    and the desired final integral becomes a boundary
    value of the ODE solution.
    Hence, defining
    
        F() = ∫d^2h/(2π)^2 e^i· hF(h) ,
    
    we have
    
        ∫d^2h/(2π)^2 2  ·F(q)
        = ( 2 ∇_ b·F(0)) ,
    
    and split_int becomes
    
        -2iδ^2()   =   i/2pω(p-ω)(p(p-ω)m_∞ ω^2+
        p ω m_∞ p-ω^2-ω(p-ω)m_∞ p^2-∇_^2)F()
              +
        ('_R(|ω| b)-'_A(|ω| b)/2
        +'_A(| p| b)/2+'_A(| p-ω| b)/2)F(),
    
    with
    
        _R'(|ω| b) ≡∫d^2k_⊥/(2π)^2( 1 - e^i ω·k_⊥) _R(k_⊥) .
    
    As noted above, we need to incorporate the change in the collision operator,
    
        '_R  , LO+NLO(b) = _R'(b) + δ_R'(b),
    
    where the _⊥-space correction of _R is the unintegrated version of δq̂^a detailed in qhatnlo.
    The details appear in Appendix C of Ref. <cit.>.
    We must also incorporate a shift in the thermal masses:
    
        m^2_∞ a, LO+NLO    = m^2_∞ a +δ m^2_∞ a,     δ m^2_∞ q    = -gT/2π,    δ m^2_∞ g    = -gC_AT/2π.
    
    These shifts are incorporated into bspace by solving for their effects perturbatively, by treating F()
    formally as an expansion in powers of δ m_∞, δ;
    F() = F_0()+F_1()+…  to first order. For details on the boundary conditions and the numerical evaluation of these
    equations, one should refer to 
    <cit.>
    
    The above analysis considers (g) corrections to the regime in which the external states are collinear up to (g^2) corrections.
    This allowed for simplifications in the analysis, in particular in expressing deltaEeq in terms of the leading mass and transverse-momentum corrections.
    But this is not always the case.
    Integrating over opening angles, deltaEeq breaks down if the external momenta P,K obey P· K ∼ gT^2,
    and this regime represents an (g) part of the splitting processes considered.
    Similarly, for hard scattering, if the transfer momentum Q becomes nearly lightlike, our treatment also breaks down, and this again represents an (g) effect.
    Both regimes feature small angles, but not as small as in the collinear splitting processes; so we refer to them as semi-collinear.
    The analysis of this kinematic region was carried out in Ref. <cit.>, and we will follow the development there and in Ref. <cit.>.
    
    The good news is that, in treating this regime, the kinematic variable h becomes large, namely h≫ gT^2.
    The collision operator then becomes subdominant, ≪δ E, and one can perturb in it to first order.
    The lowest-order and first nontrivial-order solutions are
    
        F_b1(h)     = 2/iδ E(h)  , 
        F_b2(h)     = i/δ E(h)∫d^2 q_⊥/(2π)^2C̅(q_⊥) {( C_R_b - C_A/2)
               [F_b1(h) - F_b1(h-kq_⊥)] .
               
             . + C_A/2 [F_b1(h) - F_b1(h+pq_⊥)]
               + C_A/2 [ F_b1(h) - F_b1 (q-(p-k)q_⊥)]
               } .
    
    This is the same as treating the emission in the Bethe-Heitler limit.
    However, at large h, q^- = δ E ∼ gT ∼ q_⊥, which changes the kinematics of the scattering. Therefore,
    C̅(q_⊥) must be recomputed.  A more accurate form for
    C̅(q_⊥) in this regime is
    <cit.>
        C̅_NLO(q_⊥,δ E) =
          g^2 T ^2/(q_⊥^2 + δ E^2)(q_⊥^2 + δ E^2 + ^2)
               + 2 g^2 T δ E^2/q_⊥^2(q_⊥^2 + δ E^2) .
    
    Physically, this is a consequence of the
    splitting being induced by an elastic scattering or by the
    absorption of a soft on-shell particle, as observed in <cit.>. Therefore,
    the already-computed LO  contribution, corresponding to the small δ E
    limit, and the already-included LO  contribution <cit.> need to be subtracted as:
    
        δC̅(q_⊥,δ E) =
          g^2 T ^2/(q_⊥^2 + δ E^2)(q_⊥^2 + δ E^2 + ^2)
               - g^2 T ^2/q_⊥^2(q_⊥^2+^2) .
    
    The second term is the LO collinear form for C̅ (the small
    δ E limit of CLONLO).
    
    The semi-collinear contribution is found by substituting CNLO
    into Fseries and using it to evaluate split_int and finally
    collop21, which results in
    
        γ^a_bc|_semi(p;p-k,k)
              = g^2/32π^4{[ d_AC_Ap^4+k^4+(p-k)^4/p^3k^3(p-k)^3                               g↔ gg;       d_F p^2+(p-k)^2/p^2(p-k)^2k^3                              q↔ q g;       d_F (p-k)^2+k^2/(p-k)^2k^2p^3                              g↔ qq̅ ].
        ∫d^2h/(2π)^2∫d^2/(2π)^2δC̅(,δ E)
           ×[(C_R-C_A/2)(h/δ E(h)-
        h-kq_⊥/δ E(h-kq_⊥))^2+C_A/2(h/δ E(h)-
        h+pq_⊥/δ E(h+pq_⊥))^2.
           .+C_A/2(h/δ E(h)-
        h-(p-k)q_⊥/δ E(h-(p-k)q_⊥))^2],
    
    with (C_R-C_A/2) appearing on the h + pq_p term for
    g ↔ qq̅ processes.  This is then inserted into
    collop21, resulting in
    
        (f_1,𝒞^semif_1)
        	   ≡   2π/T^3∑_abc∫_0^∞ dp
                ∫_0^p dk γ^a_bc|_semi (p;p-k,k)
        	f^a_0(p)[1± f^b_0(k)][1± f^c_0(p-k)]
              ×[f_1^a()-f_1^b(k)-
        	f_1^c((p-k))]^2.
    
    
      §.§.§ Summary
    
    
    In summary, the corrections to be added to the strict leading-order contributions
    of the previous section produce a collision operator that is fully
    NLO.  At NLO, the collision term will be given as
    
        ( f_1, C_NLO  f_1 )
            =
            ( f_1, C_LO  f_1 ) +
            ( f_1, δ C   f_1 )  ,
    
    where
    
        ( f_1, δ C   f_1 )
          ≡   ( f_1, C^δq  f_1 )-
          ( f_1, C^_(g) finite  f_1 )		  +
          ( f_1, C^  f_1 )  +
          ( f_1, δ C^  f_1 ).
    
    Here, the first two terms are the contribution from the soft gluon exchange and its counterterm. The last two terms are the contribution from semi-collinear processes and the correction to the splitting rate, respectively. All of these corrections were carefully studied in <cit.>, and what was found is that the main difference between LO and NLO results arises from the soft gluon exchange. 
    
    
    
    § FROM ZERO TO NONZERO CHEMICAL POTENTIAL
    
    
    In this section, we delve into the differences between the description of the QGP at vanishing chemical potential, μ=0, and at high densities, μ>0.
    Temperature, density, and screening enter the calculation, and we must make sure that we include them correctly each place they both arise.
    The structure fits within the general effective kinetic framework of Ref. <cit.>, for which finite μ and T case is a special case.
    
    When introducing a chemical potential we must shift the equilibrium occupancies from their simple thermal values; we have already done this in f0.
    This change propagates into the Boltzmann equation and the definition of the extremization problem, Qdef, in a straightforward way.
    There is no need to change the basis of variational functions, but more terms may be needed because the most important momentum range is likely to be higher, p ∼μ rather than p ∼ T.
    
    The matrix elements depend on the chemical potential indirectly through the strength of screening effects.
    As shown in Ref. <cit.>, the form of the HTLs are identical for high density as for high temperature.
    The two effects simply combine in the Debye screening mass squared,
    
        ^2 = N_c/3 g^2T^2 + /6 g^2(T^2 + 3/π^2μ^2).
    
    The last term represents the increase in the screening due to the quark abundance induced by the chemical potential.
    The strength of quark screening is similarly enhanced; one uses the occupancies with chemical potentials in evaluating thermalquarkmass.
    The resulting expression is already written in that equation.
    
    In evaluating splitting rates, a pivotal role was played by the transverse scattering rate, described by the kernel C̅(_⊥).
    Remarkably, scattkernel still holds at finite chemical potential, provided one uses the adjusted value of the Debye mass squared.
    This is discussed in Ref. <cit.>.
    Essentially, the shifted ^2 value accounts for the change in the number of scattering “targets” in the medium and therefore correctly adjusts the strength of medium scattering which induces the splitting.
    Similarly, the calculation of δ E for splitting, deltaEeq, must use the form of the asymptotic masses which accounts for chemical potentials.
    
    Similarly, at NLO we must correctly account for the strength of scattering in the medium, but generally the calculations presented so far remain true after one does so.
    The light-cone methods which were used to derive qhatql and qhatnlo are based on integrating out the quarks and incorporating their effects in the HTLs or equivalently within the effective theory of EQCD.
    This proceeds the same with μ as without, except that the value of ^2 must be adjusted using mDebye.
    By similar reasoning, mshift and CLONLO apply as written, but using the corrected form of ^2.
    
    Next, let us briefly analyze the formal convergence of perturbation theory.
    At weak coupling ≪ 1 and μ = 0 hot QCD has three scales: the hard scale π T, the soft scale gT, and the ultrasoft scale g^2 T.
    At leading order the scale π T participates as the dominant propagating degrees of freedom, while the gT scale is relevant as the momentum-exchange scale for soft scattering relevant in both  and  processes.
    NLO corrections occur primarily because these gT scale degrees of freedom have occupancies f(q) ∼ 1/g and loop corrections are only suppressed by a single power of g.
    Therefore the structure of NLO (g) corrections relative to LO is often as a ratio g^2 T/.
    Including color and other numerical factors, these corrections often prove to be quite large.
    
    The inclusion of a chemical potential with μ > π T raises the hard scale from π T to μ, and it increases the Debye scale as one sees in mDebye:
    ∼ g μ rather than gT.
    Therefore the separation between the soft and ultrasoft scales is increased, and (g) corrections are expected to scale as g^2 T / gμ instead of g^2 T/gT.
    This underlies the expectation that NLO effects should be milder in the large-μ/T case.
    
    In practice, the largest NLO effect is the shift in the transverse momentum diffusion coefficient, qhatnlo.
    This effect is linear in , while the leading-order part is proportional to ^2.
    The ratio δq̂ / q̂∼ g^4  T^2/g^2 ^2 T is therefore of order g^2 T/, as anticipated above.
    For parametrically large μ/T, one indeed finds an improvement in the NLO convergence by a single factor of T/μ.
    In the next section we will see to what extent this really improves the convergence in practice.
    
    
    
    
    § RESULTS
    
    In the last sections, we have presented the ingredients necessary for the computation of shear viscosity at leading order, discussed the relevant next-to-leading-order corrections, and presented a good basis set for these calculations.
    Using this computational apparatus, we compute the shear viscosity as a function of μ/T at both leading order and (almost) next-to-leading order.
    
    
    
    In figure <ref> we plot the shear-viscosity to entropy density ratio η/s for LO and NLO, and in figure <ref> we
    show the kinematic viscosity, that is, the ratio of the shear viscosity to the enthalpy density, η T /(e + P ).
    [Raw data for all plots are included in the ancillary materials.]
    Note that the chemical potential μ we refer to is that for a quark, and we treat it to be the same for all quark species.
    The baryon number chemical potential is then 3 times larger.
    We have used the EQCD-inspired scale setting by choosing the MS renormalization point to be
    =2.7T, following the treatment from <cit.>, and we run the coupling using the 2-loop beta function and the value = 0.118 at the Z-pole.
    We assume that when  reaches a quark mass, one crosses the quark-mass threshold and the quark becomes active both in the medium and in the running of the coupling constant.
    So for temperatures below 2.7T=m_b we use =4 and for temperatures above 2.7T=m_b we use =5 (and similarly for the charm and top quarks).
    The plots clearly show that the difference between LO and NLO is smaller at large μ than at μ=0, particularly at high temperatures where the coupling is smaller and  is larger.
    
    
    
    
    Since one motivation to study high-density regimes comes from the expectation that perturbation theory should show better convergence than at μ=0, we present a more detailed comparison between LO and NLO results in Fig. <ref>.
    We plot the ratio between LO and NLO calculations as a function of the temperature for each fixed chemical potential.
    We reproduce the previous result <cit.> that the NLO corrections at μ=0 are large, more than a factor of 2, up to 10^3 GeV temperatures.
    Once one goes to higher densities, the NLO effects become less dominant, falling below a factor of 2 by T=500 MeV for μ/T = 6.
    This suggests that the expansion in g is better behaved at large μ/T.
    Nevertheless, NLO effects are rather important at all temperatures and chemical potentials we consider.
    
    
    
    
    
    
    
    
    
    
    § SUMMARY AND DISCUSSION
    
    
    We have extended the investigation of shear viscosity for hot QCD with chemical potentials to both leading and (almost) next-to-leading order.
    We find that the viscosity-to-entropy ratio η/s is larger at finite μ than at zero μ for all temperatures we consider.
    However, the kinematic viscosity η T/(e+P) is more weakly dependent on μ, falling below the μ=0 result except at NLO below T=1 GeV, where the μ/T dependence almost vanishes.
    We argued previously that, at a given temperature or coupling strength, the convergence of perturbation theory should be better in the large μ/T regime than for μ=0.
    Our results show that this is indeed the case, but the improvement is not dramatic.
    NLO effects are a factor of 2 at μ/T=0 for
    temperatures of hundreds of GeV,
    whereas for μ=6T they are a factor of 2 at about 500 MeV temperature.
    Perturbation theory does not converge at experimentally achievable combinations of (μ,T), but it comes closer at high μ than at small μ.
    
    It would be straightforward to apply the techniques developed here to the more physical case in which different quark species experience different chemical potentials.
    This could allow the development of an interpolating estimate for the viscosity based on experimental data, hadron-resonance modeling, and NLO perturbative input, similar to what was recently developed by one of us <cit.>.
    
    
    
    § ACKNOWLEDGEMENTS
    
    The authors acknowledge the support by the State of Hesse within the Research Cluster ELEMENTS (Project ID 500/10.006), and support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through the CRC-TR 211 'Strong-interaction matter under extreme conditions'– project number 315477589 – TRR 211. The authors also thank Nicolas Wink, Jacopo Ghiglieri, and Kevin Ingles for instructive conversations.
    
    
    
    
    
    
    § SELF-ENERGIES BEYOND THE HARD THERMAL LOOP APPROXIMATION
    
    As discussed in <cit.>, the t-channel exchange process is subtle, since the use of vacuum matrix elements leads to logarithmic divergences in the “transport” cross-section and in the inverse viscosity.
    Baym et al already identified the correct procedure to fix this problem in the context of viscosity <cit.>; one should amend the matrix element through the inclusion of the thermal self-energy.
    At the leading-log level, relevant in Refs. <cit.>, it is sufficient to identify the energy scale where this occurs.
    At leading order it is necessary to really include the self-energy, at least within an approximation which is adequate in the formal kinematic regime (p,k) ∼ T but (q,ω) ∼ gT ≪ T.
    This is precisely the regime where the HTL approximation should be valid.
    
    At large chemical potentials we can try to do better.
    In this regime, the self-energy is dominated by fermionic loops.
    For these, the self-energy is gauge invariant and the full expression, not just the hard-loop limit, is known <cit.>.
    Specifically, the medium-induced part of the fermion-loop contribution to the gluonic self-energy is
    <cit.>
        = g^2  T^2/3H(ω,q),  = g^2  T^2/3(-1/2H(ω,q) + q^2/2(q^2-ω^2)G(ω,q))
    
    where H(ω,q) and G(ω,q) are, writing 2ω_+=ω+q and 2ω_-=ω-q:
    
        G(ω,q)   = 3/π^2T^2∫_0^∞ dk f(k)(4k+q^2-ω^2/2qln[(k+ω_-)(k-ω_+)/(k-ω_-)(k+ω_+)]) 
         H(ω,q)   = 3/π^2T^2∫_0^∞ dk f(k)(2k-4k^2+ω^2-q^2/4qln[(k+ω_-)(k-ω_+)/(k-ω_-)(k+ω_+)] 
           -2kω/qlnω_+/ω_--kω/qln[(k+ω_-)(k-ω_+)/(k-ω_-)(k+ω_+)]) 
         G(ω,q)   = 3/π^2T^2∫_0^∞ dk f(k)q^2-ω^2/2q[Θ(k+ω_+)Θ(-k-ω_-)-Θ(-k+ω_+)Θ(k-ω_-)] 
         H(ω,q)   = 3/π^2T^2∫_0^∞ dk f(k)[-(2k+ω)^2-q^2/4qΘ(k+ω_+)Θ(-k-ω_-)
           +(2k-ω)^2-q^2/4qΘ(k-ω_-)Θ(-k+ω_+)+2kω/qΘ(ω_+)Θ(-ω_-)]
    
    The references only present the real parts, but the imaginary parts are easily obtained by using the retarded prescription ω→ω+i0 and carefully interpreting the logarithms.
    The hard thermal loop approximation corresponds to expanding the k integrals in the approximation k ≫ (q,ω), in which case the functions simplify:
    
        H_HTL(ω/q) = 1 - ω/2qlnω_+/ω_-+iπω/2qΘ(ω_+)Θ(-ω_-)
           G_HTL(ω/q) = 1 .
    
    
    One concern regarding the AMY leading-order calculation <cit.> has always been the use of hard thermal loop self-energies, rather than full thermal self-energies.
    At high chemical potential, where the hard loop is dominated by fermions, the above self-energies accurately represent the correct loop screening effect at any energy and momentum, without the need for an HTL approximation.
    Therefore we can find out how well the HTL approximation works by comparing how the leading-order shear viscosity changes when we “improve” the screening effects by replacing the HTL approximation, GH-HTL, with the full self-energy, GH.
    
    We make this comparison in figure <ref>.
    The figure considers a single collision process, diagram (C), and computes the collision integral (χ_ij,χ_ij) for the single-parameter Ansatz.
    We then take the ratio of this collision rate for the case of the full self-energy over the collision rate using the HTL self-energy.
    The Debye screening strength is chosen to vary with μ as it would if the coupling strength g^2 is held fixed.
    
    The figure shows, first and foremost, that the effect of replacing the HTL self-energy with the full self-energy is not that large.
    Also, the HTL approximation actually becomes more reliable as the chemical potential is increased.
    This indicates that the use of the HTL approximation for the self-energies is justified and represents a smaller effect than the NLO (and, presumably, unknown NNLO) corrections.
    
    
    JHEP2
    
    § TESTS AND CHECKS 
    
    Since previous results were computed using different numerical integration techniques, it is convenient to compare our results with AMY leading order results. This is done in figure <ref>. The plot shows that there are small differences due to different integration methods and base choice, but they are compatible, especially at small values of , where one expects that the consequences of these effects should have a small role on the transport coefficient calculation.
    
    We also verified that a basis of size N=4 is enough to give accurate results, especially at high chemical potential, where the most important contribution comes from the diagram (C). This is shown in figure <ref>, where we compare the results given by a basis of size N=1 and N=4 to a basis of size N=6, which was considered to give accurate results for LL calculations.


 Paper Title: Investigating the crust of neutron stars with neural-network quantum states.




Lifelong Person Search
    Jae-Won Yang, Seungbin Hong, and Jae-Young Sim, Member, IEEE

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





empty



§ INTRODUCTION

Low-density nuclear matter found in the crust of neutron stars exhibits a complex and fascinating structure that varies significantly with density. In the outer crust, nucleons are bound to fully ionized nuclei. As density increases within this region, these nuclei become increasingly neutron-rich, and consequently, terrestrial experiments can only directly determine the dominant nuclei species that are present at lower densities. When the neutron drip density is reached, about 2×10^-4 fm^-3, the neutron chemical potential becomes too large for nuclei to bind additional neutrons. This leads to the formation of an unbound neutron component, marking the boundary between the outer and inner crust of the star. As the density increases, the proportion of unbound neutrons grows significantly, and at the highest densities, the nuclei undergo phase transitions into various “pasta” phases where they deform into non-spherical shapes. Eventually, upon reaching nuclear saturation density, these nuclei coalesce, transitioning into a uniform distribution of nucleons <cit.>.


Modeling inner crust material poses significant challenges due to the inherent complexity of clustering phenomena, the emergence of superfluidity, and the existence of both free neutrons and neutron-rich nuclei. Various phenomenological approaches have been developed to address some of these challenges, including the compressible liquid-drop model <cit.> and self-consistent mean-field models <cit.>. In the former, the energy is parameterized as a function of global properties, and the nucleons within clusters are treated separately from the free neutrons, all of which are assumed to be uniformly distributed. While this treatment has a low computational cost, it neglects quantum mechanical shell effects, which are critical for determining the equilibrium composition of the crust. In contrast, self-consistent mean-field models are fully quantum mechanical, constructing the many-body state in terms of single-particle or quasi-particle wave functions. However, since they neglect correlations beyond the mean-field approximation, they have limited applicability for low-density nuclear matter, where strong correlations can lead to significant deviations from the mean-field behavior.

Ab initio methods offer a more fundamental approach, as they solve atomic nuclei and neutron-star matter starting from the individual interactions among nucleons <cit.>. Most of them, however, suffer from significant limitations when modeling the transition from neutron-rich nuclei to uniform matter <cit.>. Their applicability is presently limited to the uniform case—a notable exception in this regard is the lattice effective field theory approach <cit.>. Even continuum diffusion Monte Carlo methods, which are well known for their accuracy across different physical systems, are limited to the uniform phase of neutron-star matter because of the fermion-sign problem <cit.>. 

We overcome the limitation of diffusion Monte Carlo approaches by leveraging a variational Monte Carlo method based on neural-network quantum states (NQS) <cit.>. Owing to their flexibility, NQS enable an accurate description of the ground-state of quantum many-body systems. This feature is particularly relevant for modeling the different phases of matter comprising the crust of neutron stars, which is inaccessible by conventional ansätze. This approach also circumvents the fermion-sign problem, as Monte Carlo techniques are only employed to sample the modulus squared of the wave function, which is positive even for systems of fermions. 

In a recent application <cit.>, NQS based on the hidden-nucleon architecture have demonstrated success in modeling the normal and superfluid phases of dilute neutron matter within a unified framework. A more compact and flexible neural Pfaffian-Jastrow architecture, featuring a message-passing neural network to efficiently capture pairing and backflow correlations, was shown to outperform state-of-the-art diffusion Monte Carlo methods in modeling ground-state properties of ultracold Fermi gases <cit.>. Chiefly, the neural Pfaffian-Jastrow approach is designed to capture strong pairing effects, which characterize both cold atoms and the matter found in the crust of neutron stars. 

In this work, we put forward a generalization of the neural Pfaffian-Jastrow ansatz and adapt it to nuclear potentials, which strongly depend on the spin-isospin degrees of freedom of the interacting particles. We provide evidence that this ansatz is capable of modeling the different phases matter found in the crust of neutron stars, including the self-emergence of nuclear clusters and their dissolution. In particular, we find dramatically lower energies than those obtained with state-of-the-art auxiliary-field diffusion Monte method in the low-density regime. In addition to ground-state energies, we compute two-body distribution functions and the proton fraction of beta-equilibrated matter. Our findings underscore the viability of utilizing neural Pfaffians to achieve an ab initio description of the crust of neutron stars from the nuclear physics perspective. 




§ RESULTS




 §.§ Simulation setup


We model infinite nuclear matter using periodic boundary conditions and N=28 nucleons in a cubic simulation cell with side length L, where L is determined by the baryon density n_B=N/L^3. 
For each density investigated, we run three separate simulations with different proton fractions x ≡ n_p/n_B = N_p/N, where n_p is the proton density and N_p is the number of protons in the simulation. Specifically, we run simulations with N_p = 0, 4, and 14 at each density, allowing us to study pure neutron matter (PNM), isospin-symmetric nuclear matter (SNM), and an intermediate isospin-asymmetric case. The simulated nucleons are confined within the cell, with potential contributions from image particles considered up to a distance of 3L/2. Each nucleon has a continuous three-dimensional position vector and discrete spin and isospin projections on the z-axis. As common in quantum Monte Carlo calculations of nuclear systems <cit.>, we make the approximation that protons and neutrons have equal masses. 

The dynamics of neutrons and protons is modeled according to a non-relativistic Hamiltonian based on a leading-order pionless effective field theory expansion that contains two- and three-nucleon interactions. This “essential” Hamiltonian has been shown to reproduce properties of various nuclei across the nuclear chart, including ground-state energies, radii, and magnetic moments, within few-percent from their experimental values <cit.>. It also reproduces well the equation of state (EOS) of dilute neutron matter, as the predicted values agree with highly realistic phenomenological and chiral-EFT interactions <cit.>. 
Additionally, we assume the repulsive Coulomb interactions among finite-size protons to be completely screened by the electrons present in the system. 




 §.§ Energy per particle




We begin our analysis by comparing the energy per particle of SNM obtained from NQS with that obtained from state-of-the-art auxiliary-field diffusion Monte Carlo (AFDMC), as shown in Figure <ref>. Details on the AFDMC calculations are provided in the Auxiliary-field diffusion Monte Carlo subsection of the Methods section. For all simulated densities, n_B = 0.001 fm^-3, 0.01 fm^-3, 0.04 fm^-3, and 0.08 fm^-3, the NQS energies are lower than those predicted by AFDMC, especially at the lowest densities. Additionally, the curves exhibit distinctly different characters, with the NQS energies reaching a plateau at lower densities, while AFDMC energies continue to rise. This behavior highlights an important limitation of the AFDMC calculations, as they fail to capture the formation of nuclear clusters. This is ultimately due to the fermion sign problem, which is exacerbated by using a variational ansatz that is designed for the uniform phase. 
Therefore, the zero-density limit of the AFDMC results corresponds to an infinitely dilute gas of protons and neutrons, whose ground-state energy vanishes. As an additional point of reference for the zero-density ground state in the fully clustered system, we present the NQS results for ^28Si with no electromagnetic contribution to the potential, to be consistent with the SNM case. 
The similarity between the NQS energies of ^28Si and SNM at n_B=0.001 fm^-3 suggests that all nucleons in low-density SNM coalesce to form ^28Si. We stress that ^28Si self-emerges as the minimum of the quantum-mechanical energy of the system and is not a priori encoded in the Pfaffian ansatz. 

In the outer crust of neutron stars, nucleons are bound to fully ionized nuclei. At higher densities within the outer crust, these nuclei become increasingly neutron-rich. 
While terrestrial experiments can determine the dominant nuclei species at lower densities, at higher densities these nuclei are too neutron-rich to be stable and cannot be directly measured. Nonetheless, the dominant nuclei species in this regime can be reliably inferred through extrapolations from experiments using semiempirical nuclear mass formulae<cit.>.

When the neutron drip density is reached, about 2×10^-4 fm^-3, the neutron chemical potential becomes too large for nuclei to bind additional neutrons, leading to the formation of an unbound neutron component. This marks the boundary between the outer and inner crusts of the star. 
As the density increases, the proportion of unbound neutrons grows significantly, and at the highest densities, the nuclei undergo phase transitions into various “pasta” phases where they deform into non-spherical shapes. 
Modeling matter at these densities presents several challenges. Not only are the extremely neutron-rich nuclei impossible to measure directly, but they also coexist with a sea of unbound neutrons. A commonly used approach at this density is the compressible liquid drop model (CLDM). However, being a classical model, CLDM does not account for the shell effects within nuclei<cit.>. 

Check my understanding, they only mention this for outer crust material but CLDM seems obviously inferior there. However it seems to be commonly used for the inner crust and I would imagine the shell effects still matter... Probably need a more relevant citation too. The end of the inner crust and beginning of the core of the star is marked by the transition to uniform matter from the full dissociation of the nuclei, around half nuclear saturation density.

mention superfluid. mention core modeling is different and its difficult to have a consistent model over the crust-core boundary<cit.>, maybe mention this method can do that but we don't here? Even more thorough description of inner crust densities as that's what we have, especially as it relates to the plots I show.








Figure <ref> shows the ground-state energy of nuclear matter as a function of the baryon density n_B and proton fraction x. To generate this surface, the NQS energies for SNM and PNM, E(n_B, 1/2) and E(n_B, 0) respectively, were fitted to the following function based on a cluster expansion of the energy<cit.>,

    E(n_B, x) = a_0 
        + a_2/3( n_B/n_0)^2/3
        + a_1 ( n_B/n_0)
        + a_2 ( n_B/n_0)^2.

Here, n_0=0.16 fm^-3 represents the nuclear saturation density and the constant a_0 was fixed at zero for PNM. These fits enabled the computation of the symmetry energy S(n_B) as a function of baryon density, defined as the energy difference between PNM and SNM, 

    S(n_B)=E(n_B, 0)-E(n_B, 1/2).

The symmetry energy is an important quantity for characterizing nuclear matter, as it directly affects its neutron richness.
Finally, the energy surface between the two extremes of x=0 and x=1/2 was approximated through a well-known expansion in neutron richness, 1-2x, around symmetric matter<cit.>.
The energy computed from this expansion is given by

    E(n_B,x)=E(n_B, 1/2)+(1-2x)^2 S(n_B),

where we have already assumed that higher-order terms (which would come in as a fourth power of neutron richness) are negligible, hence the quadratic coefficient is simply the symmetry energy S(n_B) from Eq. (<ref>). Note that other expansions have been proposed <cit.>, and found to be more accurate in the low-density regime. We plan to employ them in future NQS calculations of isospin-asymmetric matter, as controlling finite size effects at small x requires using more than N=28 nucleons — see the Appendix on Finite-size effects for additional details. 

From Figure <ref>, we observe that the energy of nuclear matter decreases as the proton fraction increases, regardless of density. As expected, the change in energy from x=0 to x=1/2 is more pronounced in the high-density region compared to the low-density region. However, the onset of nuclear clustering makes the energy per particle of SNM at low-density negative by about -10 MeV per particle, as shown in Figure <ref>. Thus,
even at very low densities, the curvature of the energy per particle as a function of x is positive.
Finite-size effects are discussed in more detail in the Finite-size effect corrections subsection of the Methods section. 





 §.§ Beta-equilibrated nuclear matter


Cold nuclear matter at densities found in the inner crust of neutron stars is mostly composed of neutrons, protons, and electrons. Assuming the matter is neutrino transparent, the relative abundance of these particles at a specific density can be determined using a model for the energy as a function of proton fraction, under the well-supported assumptions of charge neutrality, n_p = n_e, and beta equilibrium, μ_e=μ_n-μ_p <cit.>. We will use this model of beta-equilibrated nuclear matter and our simulation results for SNM and PNM to construct an equation of state for our Hamiltonian. 



The proton fraction in beta-equilibrated matter as a function of baryon density is displayed in Figure <ref>. The NQS predictions align closely with the phenomenological Skryme parameterizations of the equation of state (Sly4 EOS) for the inner crust of neutron stars <cit.>. Figure <ref> underscores the flexibility of the NQS: by leveraging identical neural network architectures across all densities, we successfully capture the complexity of nuclear structures relevant to the inner crust of neutron stars. In contrast, the AFDMC yields proton fractions that tend towards those of non-interacting matter. This behavior is related to the inability of AFDMC to capture the onset of nuclear clusters at low density, making the presence of protons less energetically favorable compared to more accurate NQS calculations.

Due to the self-formation of nuclei in the low-density region, the NQS energy per particle of SNM obtained for N=28 would likely be even lower if we could carry out NQS calculations with N=40 particles or more. For example, the experimental energy per particle of ^28Si is 8.45 MeV, while that of ^40Ca is 8.55 MeV. These differences would be even larger if Coulomb interactions were turned off. 
On the other hand, the PNM simulations, once corrected for finite-size effects arising from the kinetic energy<cit.>, are expected to provide results close to the infinite case <cit.>. Thus, finite-size effects might underestimate the symmetry energy, yielding too small a proton fraction.




 §.§ Clustering

The Pfaffian NQS ansatz allows us to capture the self-emergence of nuclei and nuclear clusters from first principles, when their formation corresponds to a lower minimum of the variational energy than the uniform case. To better quantify the impact of clustering, we compute the two-body pair distribution functions, which provide the probability of finding two nucleons with defined isospin at a given distance from each other. The analytical expressions for the distributions are given by

    g_α(r) = 1/2π r^2 n_B N∑_i < j^N ⟨Ψ | δ (r_ij-r) P_α| Ψ⟩/⟨Ψ | Ψ⟩,

where P_α denotes the projector onto either neutron-neutron (nn), proton-neutron (pn), or proton-proton (pp) pairs. 





Instead of showing g_α(r) for SNM, we find it more informative to focus on an isospin-asymmetric case, specifically x = 4/28, as it yields simpler scenarios. The first possibility is the uniform case, in which the distributions should resemble those of the free Fermi Gas (FG), as the input Hamiltonian is relatively soft. A second possibility is the formation of four deuterons, which would yield a peak in the pn distribution but not in the pp one, as the small clusters would float in the simulation box. Finally, if at least one ^4He cluster is formed, the pp distribution should exhibit a peak at short distances. The two-body distributions displayed in Figure <ref> indeed seem to favor the latter case. In fact, compared to the non-interacting NQS case, obtained turning off the two- and three-body potentials in the input Hamiltonian, the distributions show a significant enhancement of pairs at short distances across all channels. For comparison, Figure <ref> also displays the FG distributions in the thermodynamic limit, whose expressions can be derived analytically. 

Unfortunately, the analysis of two-body distribution functions alone does not reveal whether heavier nuclei are present, including ^8Be or more neutron-rich isotopes. To overcome this limitation, we show a three-dimensional snapshot of Monte Carlo configurations for n_B =0.01 fm^-3 and x = 4/28 in the left panel of Figure <ref>. Clearly, the four protons (represented by solid orange circles) are always close to each other, indicating the presence of ^8Be. While at least four neutrons (represented by solid blue circles) are consistently close to the protons, there is also evidence of a free neutron component. This behavior is a microscopic indication of the neutron-drip, which is expected to occur in neutron-star matter at these densities. We note that these features are not specific to the Monte Carlo configurations displayed in Figure <ref>; rather, analogous structures are evident in all the samples we examined. 


In the right panel of Figure <ref>, we visualize Monte Carlo configurations for SNM at n_B = 0.01 fm^-3. Consistent with the energy per particle plotted in Figure <ref>, the 14 protons and 14 neutrons coalesce to form ^28Si. As a critical improvement compared to the state-of-the-art AFDMC method, the NQS effectively accommodates the self-emergence formation of a medium-mass nucleus in dilute isospin-symmetric matter. In addition, since the NQS does not require the a priori specification of free neutrons, it provides a more realistic and unbiased representation of the physical system as determined by our Hamiltonian.








The summation portion of this calculation is accomplished in our Monte Carlo simulation by a histogram over pair distances. This requires that the spherical shell area is replaced by the volume of a thin shell with thickness corresponding to the histogram bin size, 4π r^2 → (4/3)π (r_o^3-r_i^3).






§ DISCUSSION

In this work, we introduce a highly flexible Pfaffian NQS that leverages backflow transformations through a message-passing neural network to solve the nuclear quantum many-body Schrödinger equation with high accuracy. Notably, fermion anti-symmetry, parity, and translation invariance are respected by construction. As a crucial improvement over conventional variational wave functions used in state-of-the-art AFDMC calculations, a single Pfaffian ansatz is able to capture the different phases of matter across a wide range of densities corresponding to the crust of neutron stars. These phases, including extremely neutron-rich nuclei surrounded by superfluid neutrons and a uniform gas of protons and neutrons, are extremely difficult to model within conventional nuclear quantum many-body approaches, with the exception of lattice effective field theory <cit.>.

We demonstrate how the formation of nuclei dramatically decreases the energy of SNM in the low-density region, thereby lowering the value of the symmetry energy. This lowering has a tangible consequence on the proton fraction in beta-equilibrated, neutrino-transparent matter. The Pfaffian NQS yields a proton fraction that agrees well with the one computed using the phenomenological SLy4 functional. The predicted fraction is considerably larger than that obtained from AFDMC calculations in the low-density regime, while it agrees with this method in the uniform case, corresponding to n_B = 0.08 fm^-3 and beyond.

To probe the formation of nuclei heavier than the deuteron, we computed isospin-dependent two-body distribution functions at n_B = 0.04 fm^-3, simulating 4 protons and 24 neutrons with periodic-box boundary conditions. The appearance of a peak at short distances in all channels is a clear indication of the presence of nuclei heavier than the deuteron, which is corroborated by Monte Carlo samplings of the many-body wave functions at n_B = 0.01 fm^-3. The latter indicate the formation of neutron-rich Beryllium isotopes, surrounded by free neutrons. The flexibility of the ansatz is such that when simulating SNM at n_B = 0.01 fm^-3, the 14 protons and 14 neutrons self-organize to form ^28Si. 

In future calculations, we will leverage leadership-class GPU clusters to simulate a larger number of particles. Reducing finite-size effects will allow us to accurately simulate systems with proton fractions between SNM and PNM, which currently suffer from large systematic uncertainties. Concurrently, we plan to include screened Coulomb repulsion between protons, a fundamental component for investigating signatures of the “pasta” phase. A first approach in this direction involves treating the Coulomb lattice of nuclei using the Wigner-Seitz approximation <cit.>.

Finally, we are incorporating high-resolution Hamiltonians, both phenomenological and derived from chiral effective field theory, into our variational Monte Carlo method based on NQS. This step is crucial for achieving a unified description of neutron-star matter, from the crust to the inner core.



§ METHOD




 §.§ Neural-network quantum states

NQS have proven effective in solving the continuous-space Schrödinger equation for nuclear many-body systems, ranging from the deuteron to ^20Ne <cit.>. NQS based on the hidden-nucleon architecture have also successfully modeled both the normal and superfluid phases of dilute neutron matter using a single unified ansatz, demonstrating their flexibility in capturing a wide range of quantum many-body phenomena from first principles. Additionally, a more compact and flexible neural Pfaffian-Jastrow (PJ) architecture, which utilizes a message-passing neural network (MPNN) to efficiently encode pairing and backflow correlations, has been shown to outperform state-of-the-art diffusion Monte Carlo methods in modeling ultracold Fermi gases <cit.>. This ansatz is particularly effective at capturing strong pairing effects relevant to both cold atoms and neutron star crust matter.

In this work, we build upon the PJ+MPNN ansatz introduced in Ref. <cit.>,

    Ψ_PJ(X) = 
        e^J(X)× pf[           0 ϕ(x_1, x_2)           ⋯ ϕ(x_1, x_N); ϕ(x_2, x_1)           0           ⋯ ϕ(x_2, x_N);           ⋮           ⋮           ⋱           ⋮; ϕ(x_N, x_1) ϕ(x_N, x_2)           ⋯           0;             ].

To ensure skew-symmetry, the pairing orbital ϕ(x_i, x_j) is defined as 

    ϕ(x_i, x_j) 
    = ν(x_i, x_j) 
    - ν(x_j, x_i),

where ν(x_i, x_j) is a dense feedforward neural network (FNN). 
Consistent with Ref. <cit.>, we enforce translational invariance and parity in the ansatz. However, to encompass nuclear potentials, the MPNN is modified to take single-nucleon isospin coordinates through the one-body stream and isospin coordinates for each pair of nucleons through the two-body stream. Additionally, the pairwise spin inputs are modified to use the individual spins of each pair instead of their spin products.

We make a notable improvement to the PJ+MPNN ansatz by allowing the Pfaffian and Jastrow correlator to be complex, while keeping the MPNN real-valued. Generalizing the architecture proposed in Ref. <cit.> for spin systems, we make the Pfaffian complex-valued by feeding the outputs of the MPNN into two separate, real-valued FNNs, which represent the phase and amplitude of ν(x_i, x_j). Similarly, the MPNN outputs are fed into two real-valued, permutation-invariant Deep Sets <cit.>, representing the phase and amplitude of the Jastrow correlator J(X). 


To help mitigate runaway instabilities <cit.>, the logarithms of the real-valued amplitudes of all the neural networks, generally denoted as x, are regularized as

    x ↦ a tanh(x/a),

where a > 0 is a constant that sets the minimum and maximum allowed values of x. For all of our simulations, a was fixed to 8. All the individual FNNs in our ansatz have two hidden layers, each with a width of 16 nodes. The output dimension of the FNNs is either 1 or 16 nodes, depending on whether the FNN maps to a latent space or the output space. The activation function used throughout this ansatz is hyperbolic tangent and the pooling operation is logsumexp. In total, this NQS consists of 10,772 trainable parameters.



 §.§ Optimization

The optimization of the NQS is performed using a version of the stochastic reconfiguration (SR) algorithm, enhanced with RMSProp-like regularization <cit.>. For a complex wave function that depends on real parameters, the SR algorithm is modified such that only the real parts of the energy gradient g≡∇_pE and the quantum Fisher information matrix S are used to update the parameters p <cit.>, 

    p_t = p_t-1 - η(S_t-1)^-1(g_t-1).

Here, η is the learning rate and t is the optimization step. The RMSProp-inspired gradient accumulation is computed as

    v_t = βv_t-1 + (1-β) (g_t-1)^2,

with β = 0.99. Then, the quantum Fisher information matrix is regularized as

    S_t ↦ S_t + ε diag(√(v_t) + 10^-8),

where the small constant ε > 0 is typically set to 10^-3. 

All simulations are executed in parallel across 8 NVIDIA A100 GPUs, with a total of 16,000 configuration samples from 800 independent chains. Each optimization step, including Metropolis sampling and observable computation, takes approximately 80 seconds, with convergence generally requiring between 1,000 and 3,000 iterations. For the lowest baryon density examined, n_B = 0.001 fm^-3, transfer learning was employed to improve convergence and stability. Unlike the other densities, which were initialized with random parameters, the optimization for this density utilized parameters previously trained for n_B = 0.01 fm^-3 with the same number of protons and neutrons, significantly reducing the convergence time. 



 §.§ Finite-size effects

To mitigate finite-size effects, we adjust the NQS energies for a given n_B and x obtained with N=28 nucleons by subtracting the free kinetic energy of the finite system and adding the energy of the free FG in the thermodynamic limit <cit.>. These corrections, listed in Table <ref>, apply to the energies shown in Figure <ref>, Figure <ref>, and Table <ref>, as well as those used for calculating the proton fractions in Figure <ref>. 



It should be noted, however, that the above procedure is a rather crude one, as finite-size effects are not limited to the kinetic energy expectation value alone <cit.>. Furthermore, leveraging the free FG in the thermodynamic limits is only applicable when clustering is not as prominent as we found. Illuminating in this regard are simulations with low but nonzero proton fractions, as the number of simulated protons may be insufficient to form clusters large enough to significantly lower the energy. To better understand this effect, consider the hypothetical formation of stable nuclei in a sea of free neutrons at rest for a proton fraction of x=4/28 in the zero-density limit. Using experimental data for ground-state energies, we can infer that the binding energy per particle for the formation of a ^8Be cluster is 2.02 MeV, which increases to 2.28 MeV for ^16O, and further increases to 2.44 MeV for ^40Ca. This effect becomes even more amplified if the formed nucleus is neutron-rich. For instance, the binding energy is 2.45 MeV for a ^12Be cluster, 3.02 MeV for an ^24O cluster, and 3.21 MeV for a ^56Ca cluster. Therefore, while we can extract useful information to investigate clustering by simulating of N_p=4 protons, it is more reliable to employ Eqs. (<ref>) and (<ref>) to interpolate the NQS energies between PNM and SNM.




 §.§ Beta-equilibrated matter solution

To model neutrino-transparent nuclear matter in beta equilibrium, 
we apply the charge neutrality condition, n_p = n_e, and the beta equilibrium condition, μ_e = μ̂, where μ̂≡μ_n - μ_p represents the difference between neutron and proton chemical potentials. Using the well-known expansion of the energy per particle in neutron richness from Eq. (<ref>), the difference in chemical potentials can expressed in terms of the symmetry energy as

    μ̂ = - ∂ E(n_B, x)/∂ x = 4 (1-2x) S(n_B).

Next, by calculating the relativistic electron density at zero temperature and applying the charge neutrality condition, we obtain

    x≡n_p/n_B = n_e/n_B = (μ_e^2-m_e^2)^3/2/3π^2 n_B.

Finally, the beta equilibrium condition couples Equations (<ref>) and (<ref>), enabling the numerical solution for the proton fraction x. 




 §.§ Auxiliary-field diffusion Monte Carlo



The AFDMC solves the ground-state component of a given Hamiltonian by performing an imaginary-time propagation upon a starting variational ansatz <cit.>

    |Ψ_0⟩ = lim_τ→∞ |Ψ(τ)⟩ = lim_τ→∞ e^-(H-E_V)τ |Ψ_V⟩  .

In the above equation, E_V is a normalization constant, which is chosen to be close to the true ground-state energy E_0. The variational ansatz is assumed to be of the Slater-Jastrow form: |Ψ_V⟩ = F̂ |Φ⟩. To keep a polynomial cost in the number of nucleons, only linear terms in the spin-isospin correlation operators F̂ are included <cit.>. The optimal values of the variational parameters defining the correlation operator are found by employing the same stochastic reconfiguration method used to optimize NQS <cit.>. 



The mean-field part | Φ⟩ of the starting variational ansatz is the ground state of the uncorrelated Fermi gas, built from a Slater determinant of plane waves. The infinite system is simulated with N nucleons in a cubic periodic box of volume L^3. The momentum vectors in this box are discretized as 

    𝐤 = 2 π/L{ n_x, n_y, n_z}  ,

where n_x, n_y, and n_z are integer numbers. 
In this work, we carry out AFDMC calculations of SNM, simulating 14 protons and 14 neutrons as these “magic” numbers fill the discretized momentum shells for each species. AFDMC calculations of PNM with 28 particles in box would require either employing multiple Slater determinants in |Φ⟩ or twist–averaged boundary conditions <cit.>, which goes beyond the scope of this work. On the other hand, there is excellent agreement between NQS and AFDMC calculations of the energy per particle of PNM carried out with N=14 particles within a periodic box <cit.>. Thus, to obtain the symmetry energy results labeled as “AFDMC” in Fig. <ref>, we use the PNM energies obtained with NQS. 



 §.§ Data

The ground-state energies of obtained from NQS simulations at different n_B and x are listed in Table <ref>. For the isospin-symmetric case, we also report the corresponding AFDMC values. 



§ ACKNOWLEDGMENTS

We would like to thank Omar Benhar and Anthea Fantina for illuminating discussions. AL and BF are supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, under contracts DE-AC02-06CH11357, by the 2020 DOE Early Career Award program, and by the NUCLEI SciDAC program. JK is supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, under grant No. DE-SC0021152. MHJ is supported by the U.S. National Science Foundation Grants No. PHY-1404159 and PHY-2013047.



§ AUTHOR CONTRIBUTIONS

B. F., J. K., and A. L. developed the code, designed the Pfaffian-Jastrow ansatz. B. F. and J. K. performed the numerical simulations. A. L. and M. H. J. provided overall supervision and guidance throughout the project. All authors contributed significantly to writing the manuscript.


 






 Paper Title: The Smile of Cheshire Cat At High Density.










=1ex plus0.5ex minus0.2ex

AŁLØO
⟨⟩

    3pt1pt∼1pt<
    3pt1pt∼1pt>
    ≤[ m_N^+m_N^-m_π( [ )trm_πm_ρMeVm_N_+m_N_-m_p^-m_p^+≤( [ )trF_π^2m_πeMeVχ̅ω̅^0-hls EFT_ bsHLS-hls Skyrmion_ sHLS Tr∂ 𝔸
    
     
    The Smile of Cheshire Cat  At High Density 0.5cm
    
    Mannque Rho[ e-mail:mannque.rho@ipht.fr]Université Paris-Saclay, CNRS, CEA, Institut de Physique Théorique, 
     91191, Gif-sur-Yvette, France  0.3cm
     (August 5, 2024)
     ABSTRACT  Baryons in finite nuclei, nuclear matter and dense compact-star matter are described in terms of Cheshire Cat for  QCD. A potential conceptual link, admittedly short in mathematical rigor, between their manifestations is  made by what's called Cheshire Cat Principle. Put in terms taken  “dual" to QCD variables, going to very high density exposes quantum Hall droplets – or pancakes –  at which the dilaton-limit fixed point with g_A→ 1, f_π→ f_χ– where f_π and f_χ are respectively the pion and dilaton decay constants – and the baryon parity-doubling are reached. This scenario suggests a thus-far totally unexplored structure of dual baryonic matter at high density which does neither require nor  rule out  (rapid) first-order phase transitions from hadrons to quarks in the core of compact stars on the verge of gravitational collapse.  
     
     1.0cm
    
    
    
    § INTRODUCTION
    
    With the advent of gravity-wave observations with neutron-rich compact stars that are being – and will be – discovered, a thus-far more or less unexplored domain of physics is opening up,  resurrecting the old question raised more than a half a century ago: What is the structure of dense baryonic matter stable against, but on the verge of gravitational collapse into black a black hole? Of course general relativity enters with its own multifacets, but what has been baffling in modern physics  is how are the baryons crunched into such high density in the standard model (SM) of particle physics? This question arises in specific terms of what is the equation of state (EoS) of the matter that figures in the TOV (Tolman-Oppenheimer-Volcoff)  equation to describe the properties of massive compact (predominantly neutron) stars stable against gravitational collapse? This question becomes highly topical as the on-going – and up-coming – gravitational wave measurements are to provide ultimately precision data on the properties of such massive stars.
    
    This question brings back the long-standing issue on the basic structure of nucleon in the fundamental theory of strong interactions  (now belonging to  the Standard Model),  widely discussed several decades ago, as a model of QCD,  the  MIT bag model.  There was a big dispute in early 1970's between  Stony Brook, Long Island and the nuclear and particle theorists at MIT in Cambridge where the “MIT Bag"  was conceived <cit.> on the structure of baryons in heavy nuclei.  Taking  the well-studied heavy nucleus ^208Pb as a specific example, the MIT theory argued that  this nucleus should be more properly described in terms of 3× 208=624  quarks instead of 208 nucleons given that the MIT bag for the single nucleon was to have the confinement size  R_conf∼ 1 fm. This would depict the nucleon bags more or less close packed in heavy nuclei, giving the picture of the nucleons resembling  “grapefruits in a salad bowl." A nucleon has the size of radius ∼ 1 fm and a nucleon in the MIT bag model contains N_c=3 quarks  confined in the bag with a confinement radius ∼ 1 fm. This structure was considered,  generally at the time,  in nuclear community and particularly by the Stony Brook theorists,   “glaringly" at odds with the then generally accepted description of  nucleons in the Pb nucleus treated in the shell model with the nucleons pictured as quasi-fermions moving in  an average potential.  The nucleon was considered as a small object of radius much less than 1 fm  surrounded by meson cloud with energy much larger than the nucleon mass m_N ∼ 1  GeV,  in consistency with the old Yukawa theory of nuclear forces with strong self-energy processes with the pion field – and subsequently heavier meson fields –  bringing down the nucleon mass to its known measured value by self-energy corrections.  Following the notion of bag-confinement  in QCD,  while I was spending a year at the Nuclear Theory Group with  Gerry Brown, he and I proposed “The Little Bag Model" <cit.> to contrary the Big MIT bag.  The little bag was depicted as confining three quarks,  massless due to large density restoring chiral symmetry within the bag.  During the whole  period of my stay I was witnessing  the daily on-going disputes taking place at Stony Brook with its Little Bag (LB) and MIT with its big Bag (BB)[Gerry used to commute, quite frequently, between Stony Brook and Cambridge giving talks, generating heated arguments with the local protagonists of the BB. Amusingly each time he returned from MIT, his little bag got a little bit bigger, visibly persuaded by the local theorists. But it shrunk back after a little while,  dissuaded by us in the Stony Brook group. At his office door Gerry had hung two brown bags, one a big (brown shopping)  bag and another a small (lunch) bag contained inside the big one, clearly indicating his ambivalence.]. This – what one might call –“dichotomy" between the Stony Brook LB and the MIT BB becomes more striking at higher densities probed in compact stars as illustrated in Fig. <ref> copied from  <cit.> where it was used to argue persuasively against rapid first-order phase transitions from baryons to quarks inside massive compact stars.
    
    
    
    The LB was constructed for the nucleons with three colored quarks coupled to the small-mass pions, recognized as the Nambu-Goldstone (NG) bosons resulting from spontaneous breaking of the nearly perfect chiral symmetry. In proceeding with the LB, the quark-pion coupling at the bag boundary was constrained to be consistent  with chiral symmetry with the quarks confined inside the bag and the pions clouded outside.   
    With a suitable coupling boundary condition imposed at a given “confinement" radius R_bag,  the “Chiral Bag Model" (CBM) <cit.> was applied with some success to certain class of phenomena of nuclear dynamics <cit.>.   But there was however an obvious technical difficulty  due to its too big a size of  the self-energy corrections required to bring down the mass to near its physical value.  
    
    
    The way out of this stumbling block came with what one could  say a striking “tongue-in-cheek" idea proposed by Nadkarni, Nielsen and Zahed  in 1985 <cit.>, inspired from the Cheshire Cat in  Lewis Carroll's  1865 fable  “Alice in the Wonderland."  The  baryon charge that leaks out from the bag is linked to the  smile that disappears from the face of the Cheshire Cat in the Wonderland.
    
    
    
    What's  magical in the Cheshire Cat solution  was  that the nucleon could be described as a soliton, nowadays referred to as skyrmion, discovered by Skyrme in early 1960s <cit.>. What happens is that the baryon charge  B=1 for the nucleons can leak out of the LBs into the pion cloud residing outside of the bag. The crucial point, elaborated below, is that this leakage is caused by a quantum anomaly lodged at the bag boundary that makes the baryon charge disappear from the bag. This anomaly, associated with a gauge symmetry,  turns out to make the  bag radius R_bag, or equivalently confinement radius R_conf, an unphysical quantity. This is the emergence of what is referred to as “Cheshire Cat Principle (CCP)" and the associated mechanism  “Cheshire Cat Mechanism (CCM)."  As will be mentioned below there could be an alternative approach with the quark-pion coupling made in the whole space invoking no boundary conditions. This approach, referred to as “Cloudy Bag Model" <cit.>, could be considered as a variant of CCP without bag boundary conditions and hence involves no leakage of baryon charge. In this model the baryon charge is strictly confined inside the bag. If the bag is taken sufficiently large,  it could be formulated as a particular gauge-fixed version as described in Sec. <ref>.
    
    
    I first recount, qualitatively,  what took place when the CCM was introduced into the LB and how the resulting CC model fared with Nature in nuclear processes. I then describe how  a conceptually new development took place in the efforts to access the properties of superdense neutron matter relevant to massive compact stars currently being observed in gravity-wave and associated measurements. 
    
    
    
    § THE CHESHIRE CAT (CC)
    
    Reconciling  the “grapefruits-in-the-salad (GFF)" dilemma in heavy nuclei, say, ^208Pb in the LB scenario turns out to come from how  topology figures in the baryon structure: It turns out that the BBs in ^208Pb can be shrunk to the LBs relying on that that baryons are  topological solitons. The nucleons, proton and neutron, belonging to the baryon flavor  symmetry SU (N_f ) with N_f=2, can be considered as skyrmions in the large N_c limit  of QCD <cit.>  where N_c is the number of colors, thereby resurrecting Skyrme's idea left dormant for ∼ 2 decades. The outcome was that the whole baryon octet, not just proton and neutron, can be related to the meson octet via topology.  This is stated mathematically in terms of  the homotopy group π_3 (SU(N_f))=Z.  But it holds only if N_f >1,  not for N_f=1. There is no-go theorem for  a single flavor, i.e., π_3 (U(1))=0, so there can be no flavor-singlet  skyrmion coming from the flavor singlet meson, η^'.  Somewhat surprisingly this is due to the color symmetry of  QCD <cit.>. However  the flavor singlet soliton can actually be present topologically but not as a skyrmion. We will come back to this issue which is one of what could constitute a surprising new development in nuclear physics. It leads to a fractional quantum Hall droplet involving Chern-Simons topology. It  seems to play no significant role at low density, but, I suggest,  can figure importantly at high density.  
    
    
    In addressing hadronic interactions in nuclear and astrophysical systems, a true Cheshire Cat Mechanism posits that “the bosonic and fermionic degrees of freedom  can  in no way be independently constrained at the boundary" <cit.>.  This would  imply that the boundary terms should be  necessarily determined by locality, renormalizability and hermiticity, plus appropriate symmetries of QCD. This means that the bag wall, therefore the confinement radius, cannot be a physically relevant quantity. 
    
    The principal idea for what the Cheshire Cat  is all about can be glimpsed in an amusing pedestrian presentation given in 1987 Les Houches  meeting <cit.> where it is stated that   “The Cheshire Cat point of view is to postulate that the bag itself of the hybrid bag model does not exist, in analogy to the non-existence of the Cheshire Cat in the Lewis Caroll fable. Only the grin of the Cheshire Cat and the formalism of the bag exist <cit.>."  The application of the idea to nuclear physics up to year ∼ 2000 is partly summarized in <cit.>. Here I will recount some of what has transpired since then, going into the regime of nuclear physics that has not yet been explored.
    
    
    
     §.§ The chiral bag
    
    The gist of the basic point can be illustrated with  a (1+1)-dimensional CC involving an “infinite hotel." . 
    
    The chiral bag action consists of three components 
    
    S=S_V+S_V̅+S_∂ V
    where the first term is of the bag of volume V containing N_c=3  massless quarks ψ for the baryon, the second is the outside of the bag of volume V̅ populated by mesons (massless pions π– for now,  later heavier mesons will be brought in) and the third is the boundary connecting the inside to the outside. 
    
    What enters in the first and second terms of the  action is known classically.  What crucially matters is the boundary term which makes the Cheshire Cat Mechanism (CCM) operative. As it can be seen easiest for the issue, we proceed in 1 space dimension focussing on a single quark but it applies to all N_c of them.  The argument given for (1+1) dimensions applies equally to 3d for the flavor number ≥  2. The subtlety for N_f=1, a totally new element, will be postponed to later.
    
    
    With the appropriate constraints, e.g.,    chiral symmetries etc.,  imposed in the action (<ref>), the problem arises in the boundary term because there is a quantum anomaly on the vector current that triggers the leakage of the baryon charge confined in V into V̅. In <cit.>, this scenario is, amusingly, depicted as a “jailed" quark trying to escape through the jail boundary exploiting the “infinite hotel," the anomaly.  The consistency of the theory requires the anomaly generated at the bag side be cancelled by the pion field in V̅ wherever the bag boundary might be located and this is done by the pion field “turning topological" and picking up the leaked charge.  This action is independent of the bag location  R_bag/con and this renders the bag/confinement size physically irrelevant. The baryon charge, carried by the quarks confined inside the bag, disappears into the the pion field, with the latter carrying the soliton topology that leaks out. The baryon charge therefore is the smile of the Cheshire Cat. 
    
    
    
     §.§ The   Smile of CC is topological
    
    In (1+1)d, there is an exact bosonization of the fermion quark to a boson so the baryon charge B  is 1 independently of the confinement wall R_bag. Neither the charge in the bag nor the charge in the pion field separately  is physical while the total is. When the bag is shrunk to zero radius, the baryon charge remains in the smile of the cat.  Given the exact bosonization in (1+1) dimensions, it is possible to see that not just the baryon charge but also certain other  conserved quantities remain independent of the  R_bag.  But this is not feasible in (1+3) dimensions.
    
    How this phenomenon occurs is nicely illustrated in <cit.>. Consider a massless quark ψ swimming in attempt to escape from the “jail" to the positive direction on top of the Dirac sea. It gets blocked when it reaches R_bag. This is because the symmetries do  not allow it. Since it cannot turn back on the surface,  it will  plunge into the sea  but the Dirac sea is completely occupied, so  it tries wiggle itself out below the Dirac surface. This is where the quantum anomaly kicks in at the boundary, allowing the quark to escape exploiting  the “infinite hotel."  The fermion number is conserved, so the quark disappears into the “smile" of the CC.  What takes place there can be  precisely described thanks to the exact bosonization that exists in (1+1) dimensions. 
    
    
    
     §.§ How CC works in real world
    
    Now what about in (1+3) dimensions where the exact bosonization is not known to exist?  The early efforts to address this issue in nuclear physics are reviewed in <cit.>.
    
    It was first seen in <cit.> at the pion'a magic chiral angle θ=π/2 that exactly half of the baryon charge B=1/2 leaks out of the bag which then is picked up by a half-skyrmion. This magic-angle phenomenon holding trivially – thanks to the topology involved – for the baryon charge in both 1 and 3 space dimensions has a very interesting phenomenon particularly at high density. We will return to it below. 
    
    It was shown by Goldstone and Jaffe <cit.> using the multiple reflection method that  the leaking baryon charge from the bag in (1+3) dimension is  precisely picked up at any chiral angle by the fractionalized skyrmion charge in the exterior of the bag so that the total baryon charge remains intact. As in the (1+1) dimensions what is in action is the anomaly. The leaking charge which of course depends on the bag radius R_bag is precisely carried by the pion field (skyrmion) also in (1+3)d. This must happen because the baryon charge is a  topologically conserved quantity and it is the gauge invariance that tackles the anomaly that does the job. 
    
    Now the question is: What about other physical quantities that are not subject to topology?
    
    
    
     §.§ CC Mechanism (CCM) as gauge invariance in action:
     Flavor-singlet axial charge of the proton
    
    There is no exact bosonizaton known in (1+3))d, so it is not very likely that apart from what's involved with topology, one can expect the CCP to hold for  all physical observables.  Surprisingly there are certain properties that are not directly controlled by topology but seem however to come rather close to having the CCP work. Let me describe one case where while topology is not evidently involved but there is an anomaly present at the boundary.  Here  a conserved charge does play a role, namely, the flavor-singlet axial coupling constant (FSAC) g_A^(0) that involves the U_A (1) anomaly.  With the modern progress toward “first-principles" approaches to nuclear structure, calculations could be done a lot more accurately but even at this early stage of development with the technique which is more or less crude it becomes surprisngly clear that intricate details, not obviously topologically connected, need to be as fully treated as possible to see how the CCP can be arrived at. The FSAC, described below, gives an excellent illustration of this case.
    
    
    
      §.§.§  The η^' from the U_A(1) anomaly
    
    One prominent illustration of what could be involved in the working of the CCP is the case of the flavor-singlet axial charge g_A^(0) of the proton measured in the EMC at CERN <cit.>– erroneously, as it turned out, linked to the proton spin crisis <cit.>.  It is an extremely interesting case where the leakage of color from the bag and the flavor-singlet η^' excitation outside can be described by a Cheshire Cat mechanism involving the U_A(1) anomaly <cit.>.  
    
    
    First let me first give a simple description of the η^' properties combining the U_A(1) anomaly with the CCP.  It illustrates  what the CCP states at low energy: “hadronic processes do not discriminate between QCD degrees of freedom (quarks, gluons) on one hand and meson degrees of freedom (pions, vector mesons etc.) on the other, provided all the necessary quantum effects are properly taken into account <cit.>." The basic idea is to consider a slowly-fluctuating  η^' configuration in the vacuum. Over a small volume of size V,  applying the CCP allows one then to describe η^' excitation in terms of QCD variables or effective bosonized degrees of freedom. As in the case of the baryon charge, the presence of η^' at the boundary makes the color charge of the quarks to leak-out of V <cit.>. This symmetry-breaking term which involves the η^' coupling to the Chern-Simons current K_5^μ
    
    K_5^μ=ϵ^μναβ (A_ν^a G^a_αβ-2/3 f^abcgA_ν^a A_α^bA_β^c), 
     
    where A_μ^a is the gluon field, 
    needs to be cancelled by quantum effects. The boundary term reproduces the Witten-Veneziano relation <cit.> relating the  η^' mass to the topological susceptibility χ <cit.>.  
    
    That the properties of the η^' emerge correctly from the CC is an important point for the g_A^(0) treated below.  
    
    
    
      §.§.§  A chiral bag with  η^'
    
    We are now equipped to address the FSAC g_A^(0) problem <cit.>. The chiral bag model we will treat is given pictorially in Fig. <ref>.  
    
    
    The action contains three components as in (<ref>), now in,  in (1+3)d  
    
    S=S_V+S_V̅+S_∂ V.
    The first term is the usual QCD action. The other two terms are
    
    
    
    
    S_Ṽ   =   f^2/4∫_Ṽ d^4x(_μ U^†^μ U +1/4N_f
    m^2_η^'( lnU- lnU^†)^2
    )
       +    S_WZW,
    
    S_ V   =   1/2∫_ V dΣ^μ{(n_μψ̅ U^γ_5ψ)
    +i g^2/16π^2K_5_μ ( ln U^†- ln U)}
    where N_f=3 is the number of flavors and 
    
    U   =   e^iη^'/f_0e^2iπ/f,
    
    f_0   ≡   √(N_f/2) f. 
    Here π stands for the octet pseudo-scalar mesons and η^' for the singlet pseudoscalar.
    It should be mentioned at this point in anticipation of what's to come later that hidden local symmetry bosons (ρ,ω), in particular, via the homogeneous Wess-Zumino term <cit.>,  will figure importantly at high density.[It will be seen that unlike the pseudo-scalar octet mesons π which give rise to all the octet baryons as skyrmions, the flavor singlet η^' does not give a flavor singlet skyrmion baryon B^(0). It will lead to a fractional quantum Hall droplet which will figure at high density.] In the present problem, they are not important.
    
    
    
    
    The matrix element of the FSAC gotten from the model (<ref>) consists of terms from the bagged degrees freedom and the η^' satisfying the U_A(1) anomaly. As detailed in <cit.>, there are (1) the contribution from the bagged quark, a^0_Bq, with the baryon charge leakage properly taken into account, (2) the η^' contribution a^0_η, (3) the contribution of static gluons due to quark source a^0_G,stat, and (4) the gluon Casimir a^0_G,vac.  The results obtained (with available parameters) for bag radius 0 < R < 1.5 fm are plotted together with  the sum a^0_total in Fig. <ref>. The shaded area corresponds to the EMC result for g_A^(0).
    It shows  that  the CC-ism is quite sensitive to the detailed account of various contributions. It involves significant cancellations for the  CCP to to work out. As far as I am aware this is the only calculation in the literature that shows how the CC feature is captured numerically for quantities that are not fully topological. It would be theoretically challenging and meaningful to repeat this calculation in a more rigorous treatment. 
     
    
    
    
     §.§ CC Mechanism for finite nuclei and nuclear matter
    
    The CCP states that when the bag radius is shrunk to zero, the CC smile remains and physics should not depend on the “confinement radius" in the sense implied in the MIT model. Therefore nuclei should be describable in terms of the  smiles of the multiple CCs with the bags shrunk to points. Let me remark only briefly on this issue which deserves a lot more space than available.
    
    Nuclear physics in terms entirely of skyrmions is currently becoming a new frontier in the field. It could be considered as  of a “first-principles" approach in nuclear physics on par with χ EFT in QCD at low energy <cit.>  in the sense of Weinberg's “Folk Theorem" of QCD <cit.>. There is a great deal of developments going on  and expected to come. How this picture works has been extensively studied with the Skyrme model with only the quartic derivative term (referred in the literature as Skyrme term) with some remarkable success in various observables <cit.>.  It is now well established that the Skyrme quartic term alone would fail to give nuclear binding energies correctly. It gives too big a binding energy and some of the ground-state properties rather poorly. This defect can be cured to certain extent by the role of heavy vector mesons incorporated into the mesonic sector <cit.> and holographic dual models with infinite tower of vector mesons <cit.>. But as will become clearer in what follows, there is a great deal of what degrees of freedom – and what subtleties – that need to be figured out.
    
    The early effort to access dense matter, particularly compact star matter, is found in <cit.>.  Only very recently are some challenging efforts  made in mathematical rigor in the skyrmion approach to  dense matter in compact stars etc. starting  to appear in the literature <cit.>.  For instance, conformal symmetry found to play a significant role in the CCP – and treated further below – is yet to be incorporated in these efforts and would certainly impact on some of the topics covered below. An example of what might be in store is given below. 
    
    
    
      §.§.§  Precocious pseudo-conformailty
    
    There is a precocious onset of conformality which begs to be clarified and is presumably connected to the CC smiles at high density.   It could be a macroscopic rendition of what's taking place in the microscopically fluctuating behavior as one encounters in the sound speed in the interior of compact stars appearing above the skyrmion-half-skyrmion  transition at n∼ n_1/2.   This is observed in a crystal-lattice simulation of the skyrmion matter above the density n_1/2 which takes place roughly at n_1/2 3 n_0 where skyrmions transform to half-skyrmions. It is one of the phenomena in skyrmion matter that remains quite mysterious and intriguing and not understood, but it is believed to be one of the essential features in probing topology in CCM. Let me make a comment,  somewhat detailed, on this matter. 
     
    The half-skyrmion phase in the skyrmion-crystal simulation of dense baryonic matter is, it seems most likely,  in a state that can be described  entirely by mean fields,  largely undistorted by strong nuclear interactions. This phenomenon, discussed in detail already in <cit.>, and given a brief reminder below in Sect. <ref>, is found in Landau-Fermi liquid fixed point theory where the β function for the quasiparticle interactions is suppressed. This striking feature was first found in the Skyrme model  with the Atiyah-Manton ansatz in <cit.>.  Here we illustrate the result from <cit.>.
    
    Write the chiral field U as U(x⃗) = ϕ_0(x,  y,  z ) + i ϕ^j_π(x,  y,  z )τ^j  with the Pauli matrix τ^j and j=1,2,3. Including ρ and ω, we write the fields placed in the lattice size L as ϕ_η,  L(x⃗ ) with η =0, π, ρ, ω and normalize them with respect to their maximum values denoted ϕ_η,L, max for given L.  It can be shown, as in <cit.>, that in the half-skyrmion phase with L L_1/2 where  L_1/2≃2.9 fm, the field configurations are invariant under scaling in density as the lattice is scaled from L_1 to L_2ϕ_η, L_1(L_1t⃗ )/ϕ_η, L_1,  max= ϕ_η, L_2(L_2t⃗ )/ϕ_η, L_2,  max. 
    Since other fields are quite similar, we only show in  Fig. <ref> the case of ϕ_0,π for  ϕ_0,π (t, 0,0) vs. t with t≡ x/L.  What is seen there is that denisty-scale invariance sets in for L L_1/2. One can see that the field is independent of density in the half-skyrmion phase with L L_1/2  whereas for the skyrmion phase with lower density with L >  L_1/2, it is appreciably dependent on density.
    
    
    What does this imply for the energy density?
    
    The energy density for the skyrmion matter put on the lattice of lattice size L can be written as
    
        ϵ = E/A/V(=L^3) = 1/L^3∫^L_0 d^3x ∑_n, m c_n,m  f_n,m( ∇⃗_x, ϕ_η, L(x⃗ )) ,
    
    where c_n,m is the coefficient of f_n,m which is the function of ∇⃗_x and ϕ_η, L(x⃗ ) having nth power of ∇⃗_x and mth power ϕ_η, L(x⃗ ) with ∇_x, j = ∂/∂ x^j. One can reduce it to <cit.>
        ϵ
        
        = ∑_n, m(1/L)^n (ϕ_η, L,  max)^m  A_n, m ,
    
     where A_n, m is a constant independent of the lattice size L.
    
     Calculating the energy density (<ref>) in skyrmion-crystal simulations involves field configurations satisfying their equations of motion. Hence (<ref>) is a mean field expression. It captures all essential dynamics in terms of the mean fields of each degrees of freedom involved, with residual interactions suppressed. This is in the same spirit as the Landau Fermi-liquid fixed approximation that underlies GnEFT.  The density dependence lies, apart from the (1/L)^n factor, in the maximum field configuration (ϕ_η, L,  max)^m. This implies that in the half-skyrmion phase,  considered to set in at high density, the mean-field structure dominates. This agrees with the lore that at high density – and in the large N_c limit, the skyrmion crystal picture becomes valid  in QCD. In clear contrast, however, as one can see in Fig. <ref>, the mean-filed structure breaks down in the lower-density phase with L > L_1/2. Taking the RMF approximation to be more or less equivalent to Fermi-liquid fixed point theory at some high density difficult to pin down numerically, one can take the breakdown of the mean-field structure  as perhaps a signal for non-Fermi liquid structure. 
     
    In short, while it is far from quantitative and short in rigor, it is intriguing that the precocious pseudo-conformality seems to set in the half-skyrmion property (<ref>) in GnEFT as  in the sound velocity in compact stars <cit.> and also in the long-standing puzzle of the quenched g_A in nuclei <cit.>. Whether or not these features survive in  “first-principles" calculations or in Nature remains to be seen. They are not yet ruled out. 
    
    
    
     §.§ CC as a gauge symmetry
    
     To reiterate what has been developed up to here and to be developed further,  the CCP implies that hadronic processes do not discriminate between QCD degrees of freedom (quarks, gluons) on one hand and meson degrees of freedom (pions, vector mesons etc.) on the other, provided all the necessary quantum effects are properly taken into account. 
     An appealing suggestion for what may be involved is an action of  gauge symmetry <cit.>. As a specific case is the MIT bag radius R_bag associated with confinement. By introducing the bag radius expressed in terms of  a collective field as  a gauge degree of freedom, a given value of R_bag in the sense of the MIT bag with confined quarks can be taken as a gauge-fixing as in the cloudy bag <cit.>. With chiral as well other symmetric couplings to appropriate degrees freedom π, vector mesons etc. without the space divided into inside and outside but suitably fitted to data, a model in the class of cloudy bag (with no boundary conditions) could potentially be formulated as a gauge-fixed CC gauge theory. Of course it would be more powerful if it could be not just for the confinement size but in a broader sense as suggested in <cit.>.  It would be interesting to see how far one can proceed in confronting nature.
     
    
    
    § Η^' AND FRACTIONAL QUANTUM HALL DROPLET
    
    
    
     §.§ CC for quantum Hall (QH) droplet
    
    We have seen that the flavor singlet meson η^' played a crucial role for the color symmetry for baryons and  for the FSAC g_A^(0).   In Sect. <ref> the properties of η^' were derived  by  CCP. Now the question is why there is no skyrmion baryon coming via CC from the η^' as there are  octet skyrmions coming the octet mesons π?  There is no description of flavor-singlet  baryon B^(0) in the baryon multiplet whereas the η^' figures as a singlet in the multiplet of the octet π. This may be  due to the role of the color symmetry in QCD <cit.>. It is surprising how this is implied by the homotopy group  π_3 (U(1))=0 as opposed to π_3(SU(2))=π_3 (S^3)=Z. 
    
    This however does not mean that there is no topological baryonic object coming from the η^'.
    
    It gas been recently pointed out by Komargodski <cit.>  that for N_f=1  the chiral effective theory is dominated by the axial U(1) anomaly, and  the topological charge cannot be identified. It was noted that the presence of stable superselection rules in the
    QCD  vacuum (instanton tunneling between vacua with different Chern-Simons number) implies  the existence of 1(time) +2 (space) dimensional domain walls. These walls connect vacua with different Chern-Simons number and are observed to be stable at large N_c. When these sheets are finite dimensional with a boundary, they are found to 
    carry massless edge excitations with baryon quantum numbers. They  are identified with fast spinning baryons. These sheets, unconnected to skyrmions,  are  described by
    a topological field theory through a level-rank duality argument, much like in the fractional quantum Hall (FQH) effect <cit.>. The
     baryons are analogous to the gapless edge excitations in quantum Hall (QH) droplets. 
     
     
     
     Most remarkably this quantum Hall droplet (variably called  pita or pancake) can be obtained by the CCP <cit.>.  A chiral bag with a single-quark species of charge b  (a fermion number) confined to a 1+2 dimensional annulus is found to leak the fermion charge in a way different from the skyrmion falling into an “infinite hotel" <cit.> . When the bag radius is shrunk to zero,  the “smile" of the cat  is left with spinning gapless  quarks running  luminally, accounting for the edge modes and their chirality <cit.>. A current transverse to the smile is shown to appear, embodying the Callan-Harvey anomaly out-flow <cit.>, well-known to string theorists. This transverse current is shown to be quite analogous to the Hall current typical of the QH effect through the emergence of an effective U(1)gauge field, say, Chern-Simons field. This U(1)  gauge field lives in the disc enclosed by the Cheshire cat  smile, and is described by a purely topological field theory in 1+2 dimensions. The quantum numbers of this baryon as a QH droplet follow readily from the chiral bag construction. The droplet outside the bag is illustrated  in Fig. <ref>. 
    
    
    The QH droplet is found to be captured by the  action 
    
    S_1+2= b^2/4π∫_1+2 AdA
    where A_μ is the abelian Chern-Simons field <cit.>. It will be noted, as elaborated later,  that this U(1) field is an  emergent  field dual to the ω field in HLS theory. What results for N_f=1 is a topological object but quite different from skyrmions for N_f≥ 2.  In constructing the CC model Fig. <ref>, why the anomaly out-flow takes place instead of the “infinite hotel" mechanism taken for the skyrmion. The mathematical constraint π_3 (U(1) is the singlet in the flavor space, with no other quantum numbers involved. That the QH droplet/pancake is not visible at low energy in nuclear processes could be because it's unstable or high-lying in energy. In QCD it is the color that could be responsible for the flavor singlet B^0 not to join the octet skyrmion baryons coming from the octet π <cit.>.
    
    
    
     §.§ Tale of two hotels
    
    In <cit.> a chiral bag model was constructed to prevent the color charge  from leaking from the bag following the CCP.  A boundary term was added to the chiral bag to seal  the leaking charge. This boundary term can be readily obtained by noting that for y-independent fields, Ref. <cit.>  describes the  outside of the bag as a line segment in 1+1 dimensions with
     b/2π∫_1+1Adθ= b/2π∫_1+1θ F- b/2π∫_BA_0θ
    after an integration by parts, clearly showing the leaking of the  b-charge through the boundary. To seal the leak, the  inside of the bag has to be supplemented
    by the opposite boundary term
     b/2π∫_B nAη^'/f_η≡ - b/2π∫_B  ϵ^μνn_ν A_μη^'/f_η
    with n^ν the spatial normal to the bag boundary, after enforcing covariance on the 2-form.
    This is exactly the surface term suggested in the Cheshire cat construction.  The present arguments illustrate the subtle relationship between the chiral bag in <cit.> for the baryon charge as well as  <cit.> for the color charge and the present chiral bag for the baryon as a FQH droplet. In the former the  b-charge is absolutely confined, while in the latter the  b-charge is allowed to flow transversely, both making use of a Chern-Simons term. This is the tale of two hotels: the infinite hotel in our world for the confined anomaly, and the finite hotel in the other world for the flowing anomaly. This tale is highly relevant for our formulation of nuclear and astrophysical processes involving hadron-quark continuity. For instance, the role of  the η^' for the color charge conservation is responsible for the Cheshire Cat mechanism for the  tiny flavor singlet axial charge for the proton g_A^(0) described in Fig. <ref>. Furthermore since the η^' is expected to become light at high density, it could have a strong impact on the stiffness of the equation of state in compact-star matter required for the observed massive 2 M_⊙ stars as discussed in <cit.>. This possibility has not  been considered up to date.
    
    
    Now the question is: Are the two hotels connected and if so how?
    
    
    
    
     §.§ Going from skyrmions to quantum Hall droplet
    
    Several interesting questions are raised at this point. First can the skyrmions in the N_f=2 baryons, say, p and n, be made to approach the QH droplet structure  by tweaking parameters of the chiral Lagrangian? 
    
    This issue is addressed by Karasik  in two ways <cit.>. 
    
    First is to start with the two-flavor skyrmions in the presence of η^' field with zero quark masses m_u=m_d=0 in the large N_c limit. The mass m_d is then continuously increased by fiat from 0 to ∞ while keeping m_u at zero. The tweaking makes the theory continuously move to  a chiral compact boson living on a boundary, leading to a Chern-Simons theory. The process is quite intricate, involving a certain amount of involved arguments and reproduces the pancake structure of <cit.>. It also predicts the spin of B^0 to be N_c/2, the same as the decuplet Δ. Of course this by itself is not a controllable operation in reality but it shows how the QH droplet is connected to the skyrmions.
    
    The second way to see the appearance of the QH droplet is to couple the HLS vector mesons <cit.> to the pions and the η^'. This way, as I will describe below, offers a possibility – the only way known (to the author) – to probe the QH droplet by going to high density.
    
    Concisely stated, the key observation Karasik makes is that the cusp singularity in the potential V_η^' in the effective Lagrangian for η^' generated as gluon fields jump from one vacuum to another
    
    V_η^'≈1/2 f_π^2 m_η^'^2   min_k∈ Z (η^' - 2π k)^2 +⋯
    can be eliminated by the vector mesons in hidden local symmetry <cit.>  in the effective theory which play the role of Seiberg dual to the gluons <cit.>.  This leads to the fractional quantum Hall structure on the η^' sheet as a configuration carrying a baryon charge that interpolates between η^'=0 and η^'=2π captured by a U(1)_N_c Chern-Simons theory <cit.>. This structure is also indicated in holographic dual QCD <cit.>. 
    
    The power of exploiting hidden local symmetry (HLS) and hidden scale symmetry (HSS) together with the notion – up to date lacking rigorous formulation – of vector dominance (VD) is that there is a possibility of probing the QH droplet, more precisely, fractional QH sheet structure via an effective field theory suitably applicable to dense nuclear matter relevant to massive compact stars.  What figures is the homogenous Wess-Zumino term gauge-invariant with the global symmetries and with time reversal symmetry that acts on the field U, U↔  U^†.  Limited to N_f=2 for convenience, the 5-d Wess-Zumino term does not enter.  Remarkably it is the ω field in HLS that will play the role of the Chern-Simons field A_μ in (<ref>).
    
    
    To discuss how this can be achieved, we need a brief discussion of the effective field theory generalized from Weinberg's χEFT <cit.> to apply to density n 3n_0 relevant to compact-star matter. We called it GnEFT and was mentioned above..
    
    
    
      §.§.§  Generalized nuclear EFT (GnEFT)
    
    It is now well established that the χEFT anchored on Weinberg's “Folk Theorem" – call it standard (S)χEFT – works quantitatively well up to baryon density 2n_0 where n_0≈ 0.16 fm^-2  is the nuclear equilibrium density. The central density of massive compact-star matter measured in astrophysical observations could reach ∼ (5-7)n_0. To access this density regime, however, degrees of freedom higher in scale than what figures in  SχEFT need to be introduced. Holographic dual approaches incorporate infinite towers of vector mesons. In GnEFT, the next most relevant degrees of freedom  taken into account are the HLS vectors V_μ=(ρ_μ,ω_μ) and the hidden scale symmetry dilaton D that we will represent by the “conformal compensator field" χ=f_χ e^D/f_χ. To this GnEFT,   is added the flavor singlet η^'.[The mixing between the octet scalar η and the flavor singlet η^' will be ignored.] What figures importantly in this approach is the degree of freedom associated with the spontaneous breaking of scale symmetry hidden in QCD. It is not usually considered explicitly but appears in SχEFT at some low order in chiral perturbation theory.
    
    There is a long history – and with controversy – on the scalar dilaton resulting from the broken scale symmetry that we will choose to avoid to get into details. For our problem, we  adopt the most recent approach called “Genuine Dilaton (GD)" by Crewther (with Tunstall) <cit.> and/or “QCD-Conformal Dilaton (QCD-CD)" by Zwicky <cit.>, both of which are anchored on assuming an IR fixed-point in QCD with N_f 3.[The controversy has to do with whether or not there can be an IR fixed point for N_f ≪ 8 in QCD, away from the conformal window, being discussed re:  dilatonic Higgs in particle theory. Some current developments are at odds with the long-standing controversy. The work discussed in this article is in favor of the recent  proposals <cit.> as a possibility for an emergence phenomenon in nuclear correlations.]   Applied to our case where scale symmetry is mist likely  “emergent" by strong baryonic correlations,  these two approaches are essentially the same except for certain fine details we are not concerned with. The IR fixed point we are concerned with is characterized by the presence of the Nambu-Goldstone bosons π and χ with  nonzero decay constants, f_π≠ 0 and f_χ≠ 0, thereby accommodating massive matter fields such as baryons ψ, vector mesons V etc. The GnEFT was formulated to apply, at least semi-quantitatively,  from normal nuclear matter to compact-star matter. 
    
    The effective Lagrangian that leads to the GnEFT will be denoted L_ψ, HS,χ  with HS standing for hidden symmetries, both local and scale,  and  with the η^' coupled in to account for the U_A(1) anomaly, it will be written L_ψ, HS,χ,η^'.  Now the key ingredient among the quantities that enter into L_ψ, HS,χ is topological as befits the CCP. It is the topology change from skyrmions to half-skyrmions assumed to take place at the density n_1/2∼ (2-3) n_0. It encodes a wide-range of density-dependent parameters in the Lagrangian connected logically to the topological structure consistent with the CCP <cit.>.  This many-nucleon approach is to encompass the  SχEFT and go up in density scale to the compact-star regime by implementing Landau Fermi-liquid (LFL) fixed point approximation in a generalized density functional (GDF) approach à la Hohenberg-Kohn theorem. In our notation, this is referred to as pseudo-conformal model, concisely  summarized  <cit.>.  There one finds that its prediction, with essentially no free parameters, encounters, so far, no serious inconsistency with observables from normal nuclear matter densities to the densities measured in the recent gravity-wave and associated data in compact-star matter[Just for comparison, see the most up-to-date analyses <cit.>.].
    
    
    
      §.§.§  From chiral EFT to Kohn-Sham density functional 
     and the dilaton-limit fixed-point (DLFP)
    
    It may be somewhat nontrivial to those unfamiliar to how a generalized density functional can be gotten from  χEFT with hidden symmetries minimally implemented. This follows from the work of Ref. <cit.> based on the strategy given in <cit.> on the formulation of renormalization-group approach to many-fermions on the Fermi surface.
    In fact one arrives at it by a simple formulation in terms of the Landau Fermi-liquid fixed-point interactions given by BR scaling <cit.>. This GDF formulation anchored on Hohenberg-Kohn theorem invoked above  is most likely closely connected to what is discussed in the fractional Quantum Hall phenomenon in condensed matter physics.   The Kohn-Sham density functional theory, widely applied in nuclear physics, turns to have a powerful impact in the fractional quantum Hall effect studied in mesoscopic devices as shown in <cit.>. It would be interesting to further elucidate the possible conceptual link between the two fields.  
    
    In approaching the FQH droplet  B^0,  L_ψ, HS,χ,η^' is treated in the mean-field which corresponds to the  LFL fixed-point approximation. Next    
    one redefines the field Σ=f_π/f_χ Uχ with U=e^iπ/f_π  and take the limit Tr (Σ^†Σ)→ 0. In order to avoid the singularities that  develope  in the limiting process, one is required to put what are referred to as the “dilaton-limit fixed-point (DLFP)" constraints <cit.>[In this reference the notion of DLFP was first formulated in terms of quark model but it has been verified to equally apply to quasi-nucleons as in GnEFT.]       f^∗_π→  f^∗_χ,   g^∗_ρ NN=g^∗_ρ g^∗_ COR→ 0, 
           g^∗_A→ 1, m^∗_N→ m_0∝ f_χ^∗
    where g^∗_ρ is the HLS gauge coupling in medium in RG analysis <cit.> and  g^∗_ COR stands for standard (many-body) nuclear correlation effects in dense matter. The ∗ stands for density dependence encoded in the condensates of the vacuum modified by the medium. 
    
    There are several remarkably powerful implications in the constraints (<ref>), some of which, it should be mentioned, have not yet been  given solid theoretical support. The first is that the DLFT indicates an approach to the IR fixed point à la  GD (and CD-QCD) scenario that our approach espouses. Indeed f^∗_π→ f^∗_χ deviates strongly from the conformal window scenario (relevant to dilaton Higgs model)  f_π^2/f_d^2  ≪ 1 <cit.>. What's directly relevant to our approach to the FQH droplet is the decoupling of the iso-vector vector mesons in HLS   before reaching the vector manifestation fixed point g^∗_ρ→ 0 at which the ρ becomes the genuine gauge field.  Furthermore g^∗_A→ 1 is linked to the long-standing quenching of g_A in nuclear matter <cit.> and the relation m_N^∗≈ f^∗_χ follows from the emergence of parity-doubling symmetry in dense medium responsible for  the sound speed in the core of compact stars approaching precociously the pseudo-conformal sound velocity v^2_pcs≈ 1/3 in the core of compact stars <cit.>.
    
    
    
      §.§.§  The role of “homogeneous"/“hidden" Wess-Zumino term
    
     Now when the HLS fields are “integrated in," apart from the 5-d Wess-Zumino-Witten term which is absent for N_f=2 that we are concerned with, there is a 4-d action that has the homogeneous solution to the  Wess-Zumino anomaly equation, called homogeneous[Called “hidden" by Karasik when combined with the VD.] WZ term <cit.>. This action, conserving parity and charge conjugation but violating intrinsic parity, contains four terms including an external field constructed with Maurer-Cartan 1-forms. 
     
     What's most significant for the issue concerned here is that the vector mesons in HLS are most likely Seiberg-dual gluons <cit.> and in the DLFP,  the iso-vector mesons decouple before reaching the IR fixed point as seen in (<ref>). That leaves the ω meson. As argued by Karasik <cit.> in terms of the domain wall with lim_z→ -∞η^' (z)=0 and  lim_z→ +∞η^' (z)=2π, one  then arrives at the (1+2) domain theory
     L_ domain-wall= N_c/4πω dω
    which is lodged in the bulk of the sheet. This reproduces the CC result (<ref>) relating the ω field to the Chern-Simons field A_μ. The  VD (vector dominance) emerges in the theory fixing the arbitrary parameters in the hWZ of HLS <cit.> to hidden(h) WZ <cit.> .
    
    What we have here is just one sheet. There is an ongoing work of sheet structure in  dense matter in the field of skyrmion crystals. There are mathematical treatments of multi-sheet structure actively studied in the field <cit.> and it is much too early to make contact with Nature. In fact such a sheet structure is already seen in skyrmion-crystal structure at finite density <cit.> and speculated in compact-star physics where fractionally charged quasi-baryons play the role of deconfined quarks in the interior of compact stars <cit.>.
    
    In the DLFP limit, the ρ mesons decouple. In Nature, the DLFP density can be  pinned down neither by theory nor by experiments. It is likely at a lower density than that of the vector manifestation. The question then arises what happens if the iso-vector mesons are not sufficiently decoupled. It seems plausible that one would then encounter non-abelian Chern-Simons droplets before reaching the abelian Chern-Simons droplet. In fact this is what one finds in the Cheshire Cat description <cit.>. 
    For arbitrary N_f the spin and statistics arguments do not change as they are solely fixed by N_c. However, the leaking flavor currents lead to a U(N_f) flavor-valued emergent gauge field A_μ. The CCP applies  mutatis mutandis. 
    The  emergent non-Abelian Chern-Simons action  would then be
    N_c/4π∫_1+2 Tr( A  d A+2/3 A^3).
    
    This suggests a wide range of stacked sheets of Chern-Simons droplets/pancakes before reaching the truly deconfined quarks which may not survive the gravitational collapse. As in condensed matter systems, the phase structure as viewed in this scenario could be much richer than what can be seen in QCD variables. It could be that ultimately which way is superior will depend on which is more efficient in capturing the vast variety of physics.
    
    
    
    § SOME COMMENTS AND OPEN ISSUES
    
    Even at the far-from-rigorous level that was adopted in this article, there are several issues that beg to be further elaborated. Let me mention a few of them here for possible future development. In the present status of poor understanding of highly dense matter, it would be premature to rule out any of  them.
    
    ∙ Multifaceted role of ω meson in nuclear matter
    
    In the vacuum the ω meson  is with the ρ meson in hidden local symmetry. But in nuclei, it starts deviating from U(3) in controlling the nuclear forces while the ρ meson decouples from the dense matter before the vector manifestation fixed point (VMFP) <cit.> is arrived at. It provides the crucial repulsion to counter the attraction due to the scalar (dilaton)-meson exchange as the density approaches the dilaton-limit-fixed point (DLFP). It gets transformed via Vector Dominance to the abelian Chern-Simons field. How this feature gets manifested in terms of QCD variables would be important for the physics of compact-star matter. This is an open issue that has not been addressed up to date.
    
    ∙ Half-skymion structure
    
    The half-skyrmion structure figures crucially in the framework of GnEFT in controlling the baryonic matter at high density (as reviewed in <cit.>). Although it does not appear explicitly, a single skyrmion may actually be a bound two half-skyrmions by a hidden monopole as suggested in <cit.>. One would wonder whether  the monopole is suppressed at high density, so that one can consider deconfined half-skyrmions interacting scale-invariantly as seen in skyrmion crystal configurations in Fig. <ref>. In condensed matter physics in lower dimensions, deconfined quantum critical phenomena could involve such deconfined half-skyrmions <cit.>[More recent developments do away in condensed matter physics with the deconfined half-skyrmions, going to local effective field theories directly without changing  essential physics.]. Also in CCM, the magic angle that involves half-skymions with the other half shared by the vector meson ρ provides an extremely simple account to the electromagnetic form factors of the proton at high momentum transfers <cit.>. It is not clear how to transform the half-skyrmions to deconfined quarks.
    
    ∙ Stacks of FQH droplets
    
    As mentioned, there are indications for the formation of the sheets of half-skyrmions in the skyrmion crystal simulations <cit.> at increasing density. The discussion in Sect. <ref> suggested the formation of a QH droplet with J=N_c/3 as density is increased  toward the DLFP. It seems that there can be a stack of  QH droplets mixed with, possibly unstable, half-skyrmion sheets before reaching the DLFP. That may give rise to something like the domain-wall structure with deconfined quarks on the sheets (or droplets) and confinement in the bulk <cit.>.  This could lead to the precocious pseudo-conformal EoS <cit.> resembling what's considered as evidence of quark-matter core in massive compact stars <cit.>.
    
    
    
     §.§ Acknowledgments
    
    
    I am grateful for the invitation by Su Houng Lee to write this article for Elsevier.
    
    
    
    
    
    
    
    
    
    
    
    
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