Symmetries of spatial correlators of light and heavy mesons in high temperature lattice QCD

The spatial $z$-correlators of meson operators in $N_f=2+1+1$ lattice QCD with optimal domain-wall quarks at the physical point are studied for seven temperatures in the range of 190-1540 MeV. The meson operators include a complete set of Dirac bilinears (scalar, pseudoscalar, vector, axial vector, tensor vector, and axial-tensor vector), and each for six flavor combinations ($\bar u d$, $\bar u s$, $\bar s s$, $\bar u c$, $\bar s c$, and $\bar c c$). In Ref. \cite{Chiu:2023hnm}, we focused on the meson correlators of $u$ and $d$ quarks, and discussed their implications for the effective restoration of $SU(2)_L \times SU(2)_R$ and $U(1)_A$ chiral symmetries, as well as the emergence of approximate $SU(2)_{CS}$ chiral spin symmetry. In this work, we extend our study to meson correlators of six flavor contents, and first observe the hierarchical restoration of chiral symmetries in QCD, from $SU(2)_L \times SU(2)_R \times U(1)_A $ to $SU(3)_L \times SU(3)_R \times U(1)_A $, and to $SU(4)_L \times SU(4)_R \times U(1)_A $, as the temperature is increased from 190 MeV to 1540 MeV. Moreover, we compare the temperature windows for the emergence of the approximate $SU(2)_{CS}$ symmetry in light and heavy vector mesons, and find that the temperature windows are dominated by the $(\bar u c, \bar s c, \bar c c)$ sectors.


I. Introduction
Understanding the nature of strongly interacting matter at high temperatures is crucial for uncovering the mechanisms governing matter creation in the early universe and elucidating the outcomes of relativistic heavy ion collision experiments such as those at LHC and RHIC, as well as those of electron ion collision experiments at the planned electron-ion colliders.A first step in this pursuit is to find out the symmetries in Quantum Chromodynamics (QCD) at high temperatures, which are essential in determining the properties and dynamics of matter under extreme conditions.First, consider QCD with N f massless quarks.Its action possesses the SU (N f ) L ×SU (N f ) R × U (1) A chiral symmetry.At low temperatures T < T 0 c (where T 0 c depends on N f , and the superscript "0" denotes zero quark mass), quarks and gluons are confined in hadrons, and the SU (N f ) L × SU (N f ) R chiral symmetry is spontaneously broken down to SU (N f ) V by the vacuum of QCD, with nonzero chiral condensate.Moreover, the U (1) A axial symmetry such that the theory possesses the SU (N f ) L ×SU (N f ) R ×U (1) A chiral symmetry for T > T 0 c1 .
Next, consider QCD with physical (u, d, s, c, b) quarks.Its action does not possess the SU (N ) L ×SU (N ) R ×U (1) A chiral symmetry for any integer N from 2 to 5, due to the explicit breakings of the nonzero quark masses.However, as T is increased successively, each quark acquires thermal energy of the order of πT , and eventually its rest mass energy becomes negligible when πT ≫ m q .Also, since the quark masses range from a few MeV to a few GeV, it follows that as the temperature is increased successively, the chiral symmetry is restored hierarchically from SU (2) L × SU (2) R × U (1) A of (u, d) quarks to SU (3) L × SU (3) R × U (1) A of (u, d, s) quarks, then to SU (4) L × SU (4) R × U (1) A of (u, d, s, c) quarks, and finally to SU (5) L × SU (5) R × U (1) A of (u, d, s, c, b) quarks.Since the restoration of chiral symmetries is manifested by the degeneracies of meson z-correlators (as well as other observables), we can use the splittings of the meson z-correlators of the symmetry multiplets to examine the realization of the hierarchical restoration of chiral symmetries in high temperature QCD.
Strictly speaking, these chiral symmetries should be regarded as "emergent" symmetries rather than "restored" symmetries, since the QCD action with physical quark masses does not possess chiral symmetries at all.In the following, it is understood that "restoration of chiral symmetries" stands for "emergence of chiral symmetries".Similar to (1), we define where T qQ c (T qQ 1 ) is the temperature for the manifestation of SU (2) L × SU (2) R (U (1) A ) chiral symmetry via the meson z-correlators with flavor content qQ.Then, for T > T qQ c1 , the theory possesses the SU (2) L × SU (2) R × U (1) A chiral symmetry of the qQ sector.
Note that since 1987 [2], there have been many lattice studies using the screening masses of meson z-correlators to investigate the effective restoration of U (1) A and SU (2) L × SU (2) R chiral symmetries of u and d quarks in high temperature QCD, see, e.g., Ref. [3] and references therein.However, so far, there seems no discussions in the literature about the hierarchical restoration of chiral symmetries in high temperature QCD, except for a brief mention in Ref. [1].
In this work, we investigate the hierarchical restoration of chiral symmetries in N f = 2+1+1 lattice QCD with optimal domain-wall quarks at the physical point.We first observe the hierarchical restoration of chiral symmetries from SU (2) L ×SU (2) R ×U (1) A of (u, d) quarks, to SU (3) L × SU (3) R × U (1) A of (u, d, s) quarks, and finally to SU (4) L × SU (4) R × U (1) A of (u, d, s, c) quarks, as the temperature is increased from 190 MeV to 1540 MeV.We compute the meson z-correlators for a complete set of Dirac bilinears (scalar, pseudoscalar, vector, axial vector, tensor vector, and axial-tensor vector), and each for six combinations of quark flavors (ūd, ūs, ss, ūc, sc, and cc).Then we use the degeneracies of meson z-correlators to investigate the hierarchical restoration of chiral symmetries in high temperature QCD.
The relationship between the SU (2) L × SU (2) R and U (1) A chiral symmetries and the degeneracy of meson z-correlators for (u, d) quarks in N f = 2 + 1 + 1 QCD has been outlined in Ref. [1], and we follow the same conventions/notations therein.In this study, following Ref.[1], we also neglect the disconnected diagrams in the meson z-correlators.With this approximation, one can straightforwardly deduce the relationship between the SU (N ) L × SU (N ) R and U (1) A chiral symmetries of N (2 ≤ N ≤ N f ) quarks and the degeneracy of meson z-correlators, in QCD with N f quarks, as follows.The restoration of SU (N ) L × SU (N ) R chiral symmetry of N quarks is manifested by the degeneracies of meson z-correlators in the vector and axial-vector channels, C qi q j V k (z) = C qi q j A k (z), (k = 1, 2, 4), for all flavor combinations (q i q j , i, j = 1, • • • , N ).The effective restoration of the U (1) A symmetry of N quarks is manifested by the degeneracies of meson z-correlators in the pseudoscalar and scalar channels, C qi q j P (z) = C qi q j S (z), as well as in the tensor vector and axial-tensor vector channels, C qi q j T k (z) = C qi q j X k (z), (k = 1, 2, 4), for all flavor combinations (q i q j , i, j = 1, • • • , N ).At this point, we recall the studies of the symmetries and meson correlation functions in high temperature QCD with N f massless quarks [4,5], in which one the salient results is that the correlator of the flavor non-singlet pseudoscalar meson qγ 5 λ a q is equal to that of the flavor singlet pseudoscalar meson qγ 5 q, for QCD with N f > 2 at T > T c .This implies that the disconnected diagrams do not have contributions to meson z-correlators in QCD with N f > 2 massless quarks at T > T c .However, at this moment, it is unknown to what extent the disconnected diagrams are suppressed in QCD with N f = 2 + 1(+1)(+1) physical quarks.We will address this question with noise estimation of all-to-all quark propagators, and will report our results in the future.
Besides the hierarchical restoration of chiral symmetries, we are also interested in the question whether there are any (approximate) emergent symmetries which are not the symmetries of the entire QCD action but only a part of it, e.g., the SU (2) CS chiral spin symmetry (with U (1) A as a subgroup) [6,7], which is only a symmetry of chromoelectric part of the quark-gluon interaction, and also the color charge.Since the free fermions as well as the chromomagetic part of the quark-gluon interaction do not possess the SU (2) CS symmetry, its emergence in high temperature QCD suggests the possible existence of hadron-like objects which are predominantly bound by chromoelectric interactions.The SU (2) CS symmetry was first observed to manifest approximately in the multiplets of z-correlators of vector mesons, at temperatures T ∼ 220 − 500 MeV in N f = 2 lattice QCD with domain-wall fermions [8].
In Ref. [1], we studied the emergence of SU (2) CS chiral-spin symmetry in N f = 2 + 1 + 1 lattice QCD with optimal domain-wall quarks at the physical point, and found that the SU (2) CS symmetry breaking in N f = 2 + 1 + 1 lattice QCD is larger than that in N f = 2 lattice QCD at the same temperature, for both z-correlators and t-corralators of vector mesons of u and d quarks.In this paper, we extend our study to vector meson z-correlators of all flavor combinations (ūd, ūs, ss, ūc, sc, cc) in N f = 2 + 1 + 1 lattice QCD at the physical point, and compare the emergence of approximate SU (2) CS chiral spin symmetry between different flavor sectors.
The outline of this paper is as follows.In Sec.II, the hybrid Monte-Carlo simulation of N f = 2 + 1 + 1 lattice QCD with optimal domain-wall quarks at the physical point is briefly outlined, and the essential features and parameters of the seven gauge ensembles for this study are summarized.In Sec.III, the symmetry-breaking parameters for measuring the precision of various symmetries with the splittings of the z-correlators of the symmetry partners are defined.The results of meson z-correlators for six flavor combinations and seven temperatures in the range of 190-1540 MeV are presented in Sec.IV, while the corresponding results of symmetry-breaking parameters are presented in Sec.V.The realization of hierarchical restoration of chiral symmetries in A , and to SU (4) L ×SU (4) R ×U (1) A , as the temperature is increased from 190 MeV to 1540 MeV, is demonstrated in Sec.V A. The temperature windows for the approximate SU (2) CS symmetry of six flavor combinations are presented in Sec.V B, which reveal the dominance of heavy vector meson channels of (ūc, sc, cc) sectors.In Sec.VI, we conclude with some remarks.

II. Gauge ensembles
The gauge ensembles in this study are generated by hybrid Monte-Carlo (HMC) simulation of lattice QCD with N f = 2 + 1 + 1 optimal domain-wall quarks [9] at the physical point, on the 32 3 × (16, 12, 10, 8, 6, 4, 2) lattices, with the plaquette gauge action at β = 6/g 2 = 6.20.This set of ensembles are generated with the same actions [10,11] and alogrithms as their counterparts on the 64 3 × (20, 16, 12, 10, 8, 6) lattices [12], but with one-eighth of the spatial volume.The simulations are performed on a GPU cluster with various Nvidia GPUs.For each ensemble, after the initial thermalization, a set of gauge configurations are sampled and distributed to 16-32 simulation units, and each unit performed an independent stream of HMC simulation.For each HMC stream, one configuration is sampled every 5 trajectories.Finally collecting all sampled configurations from all HMC streams gives the total number of configurations of each ensemble.The lattice parameters and statistics of the gauge ensembles for computing the meson z-correlators in this study are summarized in Table I.The temperatures of these six ensembles are in the range ∼ 190 − 1540 MeV, all above the pseudocritical temperature T c ∼ 150 MeV.
The lattice spacing and the (u/d, s, c) quark masses are determined on the the 32 3 × 64 lattices with 427 configurations.The lattice spacing is determined using the Wilson flow [13,14] with the condition {t 2 ⟨E(t)⟩}| t=t 0 = 0.3 and the input √ t 0 = 0.1416 (8) fm [15].The physical (u/d, s, c) quark masses are obtained by tuning their masses such that the masses of the lowest-lying states extracted from the time-correlation functions of the meson operators {ūγ 5 d, sγ i s, cγ i c} are in good agreement with the physical masses of π ± (140), ϕ(1020), and J/ψ(3097).The chiral symmetry breaking due to finite N s = 16 (in the fifth dimension) can be measured by the residual mass of each quark flavor [16], as given in the last three columns of Table I.The residual masses of (u/d, s, c) quarks are less than (1.5%, 0.04%, 0.001%) of their bare masses, amounting to less than (0.06, 0.05, 0.02) MeV/c 2 respectively.This asserts that the chiral symmetry is well preserved such that the deviation of the bare quark mass m q is sufficiently small in the effective 4D Dirac operator of optimal domain-wall fermion, for both light and heavy quarks.In other words, the chiral symmetry in the simulations are sufficiently precise to guarantee that the hadronic observables (e.g., meson correlators) can be evaluated to high precision, with the associated uncertainty much less than those due to statistics and other systematics.

III. Symmetry breaking parameters
In order to give a quantitative measure for the manifestation of symmetries from the degeneracy of meson z-correlators with flavor content qQ, we consider the symmetry breaking parameters as follows.To this end, we write the meson z-correlators as functions of the dimensionaless variable where T is the temperature.
In general, the degeneracy of any two meson z-correlators C A (zT ) and C B (zT ) with flavor content qQ (where subscripts A and B denote their Dirac matrices with definite transformation properties, and the flavor content qQ is suppressed) can be measured by the symmetry breaking parameter If C A and C B are exactly degenerate at T , then κ AB = 0 for any z, and the symmetry is effectively restored at T .On the other hand, if there is any discrepancy between C A and C B at any z, then κ AB is nonzero at this z, and the symmetry is not exactly restored at T .
Here the denominator of (4) serves as (re)normalization and the value of κ AB is bounded between zero and one.Obviously, this criterion is more stringent than the equality of the screening masses, m scr A = m scr B , which are extracted from C A and C B at large z.
Note that κ AB in (4) can be written as [1].Also, all z-correlators in (4), as well as those shown in Figs.1-7 are unnormalized, while those in Ref. [1] are normalized by their values at z/a = 1 (i.e., C Γ (zT ) = 1 at z/a = 1).The former avoids any "accidental" degeneracies due to the normalization.In the following, any symmetry breaking parameter to measure the degeneracy of two meson z-correlators is always defined according to (4).
A. SU (2) L × SU (2) R and U (1) symmetry breaking parameters According to (4), the SU (2) L × SU (2) R symmetry breaking parameter can be written as Due to the S 2 symmetry of the z-correlators, it only needs to examine k = 1 and k = 4 components of (5).In general, the difference between k = 1 and k = 4 components of ( 5) is negligible, thus in the following, we only give the results of (5) with k = 1.
In general, to determine to what extent the SU (2) L × SU (2) R chiral symmetry is manifested in the z-correlators, it is necessary to examine whether κ V A is sufficiently small.To this end, we use the following criterion for the manifestation of SU (2) L × SU (2) R chiral symmetry at where ϵ V A is a small parameter which defines the precision of the chiral symmetry.For fixed zT and ϵ V A , the temperature T c for the manifestation of the SU (2) L × SU (2) R symmetry is the lowest temperature satisfying (6), i.e., In this study, we set ϵ V A to two different values, 0.05 and 0.01, to study how T c depends on For the U (1) A symmetry breaking, it can be measured by the z-correlators in the pseudoscalar and scalar channels, with as well as in the tensor vector and axial-tensor vector channels, with Due to the S 2 symmetry of the z-correlators, it only needs to examine k = 1 and k = 4 components of (9).In practice, the difference between k = 1 and k = 4 components of (9) is almost zero, up to the statistical uncertainties, thus in the following, we only give the results of (9) with k = 4.
Similar to (6), we use the following criterion for the manifestation of U (1) A symmetry at T where ϵ T X is a small parameter which defines the precision of U (1) A symmetry.For fixed zT and ϵ T X , the temperature T 1 for the manifestation of U (1) A symmetry is the lowest temperature satisfying (10), i.e., In this study, we set ϵ T X to two different values, 0.05 and 0.01, to study how the temperature of restoration of U (1) A symmetry depends on ϵ T X .
Next, consider QCD with As discussed in Sec.
I, upon neglecting the disconnected diagrams in the meson z-correlators, the quarks is manifested by the degeneracies of meson z-correlators in the vector and axial-vector channels, ), for all flavor combinations of N quarks (q i q j , i, j = 1, • • • , N ).Thus, to determine the temperature T c for the manifestation of the SU (N ) L × SU (N ) R chiral symmetry of N quarks, it needs to measure κ qi q j V A for all flavor combinations of N quarks, and check whether they all satisfy the criterion (6) for fixed zT and ϵ V A .This amounts to finding the largest T qi q j c satisfying ( 6) among all flavor combinations of N quarks, i.e., About the U (1) A chiral symmetry of N (2 ≤ N ≤ N f ) quarks, upon neglecting the disconnected diagrams in the meson z-correlators, it is manifested by the degeneracies of meson z-correlators in the pseudoscalar and scalar channels, C qi q j P (z) = C qi q j S (z), as well as in the tensor vector and axial-tensor vector channels, C qi q j T k (z) = C qi q j X k (z), (k = 1, 2, 4), for all flavor combinations of N quarks (q i q j , i, j = 1, • • • , N ).Thus, to determine the temperature T 1 for the manifestation of the U (1) A symmetry via the k = 4 component of the tensor vector and axial-tensor vector channels, it needs to measure κ qi q j T X for all flavor combinations of N quarks, and check whether they all satisfy the criterion (10) for fixed zT and ϵ T X .This amounts to finding the largest T qi q j 1 satisfying (10) among all flavor combinations of N quarks, i.e., B. SU (2) CS symmetry breaking and fading parameters Following the discussion and notations in Ref. [1], the SU (2) CS multiplets for the zcorrelators with flavor content qQ are where the "2" components due to the S 2 symmetry have been suppressed.Thus the degeneracies in the above triplets signal the emergence of SU (2) CS chiral spin symmetry.
For T ≥ T qQ c1 , the SU (2) L × SU (2) R × U (1) A chiral symmetry is effectively restored, and , and the multiplets in Eqs. ( 14) and ( 15) become: This suggests the possibility of a larger symmetry group SU (4) for T > T qQ c1 which contains CS as a subgroup.For the full SU (4) symmetry, each of the multiplets in Eqs. ( 16) and ( 17) is enlarged to include the flavor-singlet partners of A k , T k and X k , while the flavor-singlet partners of V 1 and V 4 are SU (4) singlets, i.e., where the superscript "0" denotes the flavor singlet.
In general, to examine the emergence of SU (2) CS symmetry, one needs to measure the splittings in both (A 1 , X 4 ) and (T 4 , X 4 ) of ( 14).To measure the splitting of A 1 and X 4 , we use while the splitting of T 4 and X 4 is measured by κ T X (9) with k = 4. Then we use the maximum of κ AT and κ T X to measure the SU (2) CS symmetry breaking, with the parameter Note that for (ūd, ūs, ss, ūc) sectors, κ AT (zT ) > κ T X (zT ) for all z and the seven temperatures in this study, thus κ CS = κ AT .
As the temperature T is increased, the separation between the multiplets of SU (2) CS and U (1) A is decreased.Therefore, at sufficiently high temperatures, the U (1) A multiplet M 0 = (P, S) and the SU (2 then the approximate SU (2) CS symmetry becomes washed out, and only the SU (2 ) never merges with M 0 and M 2 even in the limit T → ∞, as discussed in Ref. [1].Thus M 4 is irrelevant to the fading of the approximate SU (2) CS symmetry.
Here we use the SU (2) CS symmetry fading parameter similar to that defined in Ref. [1], except for taking the absolute value and using the unnormalized z-correlators, i.e., where In general, κ(zT ) behaves like an increasing function of T for a fixed zT .If κ(zT ) ≪ 1 for a range of T , then the approximate SU (2) CS symmetry is well-defined for this window of T .
On the other hand, if κ(zT ) > 0.3 for T > T f , then the approximate SU (2) CS symmetry is regarded to be washed out, and only the U (1) A × SU (2) L × SU (2) R chiral symmetry remains.Thus, to determine to what extent the SU (2) CS symmetry is manifested in the z-correlators, it is necessary to examine whether both κ(zT ) and κ CS (zT ) are sufficiently small.For a fixed zT , the following condition serves as a criterion for the approximate SU (2) CS symmetry in the z-correlators, where ϵ cs is for the SU (2) CS symmetry breaking, while ϵ f cs for the SU (2) CS symmetry fading.For fixed zT , (23) gives a window of T for the approximate SU (2) CS symmetry.Obviously, the size of this window depends on ϵ cs and ϵ f cs .That is, larger ϵ cs or ϵ f cs gives a wider window of T , and conversely, smaller ϵ cs or ϵ f cs gives a narrower window of T .
IV. Meson z-correlators of (ūd, ūs, ss, ūc, sc, cc) Following the prescription proposed in Ref. [1] for the cancellation of the contribution of unphysical meson states to the z-correlators, we compute two sets of quark propagators with periodic and antiperiodic boundary conditions in the z direction, while their boundary FIG. 4. The spatial z-correlators of meson interplotors for six flavor combinations (ūd, ūs, ss, ūc, sc, and cc) in N f = 2 + 1 + 1 lattice QCD at T ≃ 385 MeV. -10 FIG. 6.The spatial z-correlators of meson interplotors for six flavor combinations (ūd, ūs, ss, ūc, sc, and cc) in N f = 2 + 1 + 1 lattice QCD at T ≃ 770 MeV.-14 The spatial z-correlators of meson interplotors for six flavor combinations (ūd, ūs, ss, ūc, sc, and cc) in N f = 2 + 1 + 1 lattice QCD at T ≃ 1540 MeV.-18 -18 -18 conditions in (x, y, t) directions are the same, i.e., periodic in the (x, y) directions, and antiperiodic in the t direction.Each set of quark propagators are used to construct the z correlators independently, and finally taking the average of these two z correlators.Then, the contribution of unphysical meson states to the z correlators can be cancelled configuration by configuration, up to the numerical precision of the quark propagators.
In each of Figs.1-7, the z-correlators for six flavor contents (ūd, ūs, ss, ūc, sc, cc) at the same T are plotted as a function of the dimensionless variable zT (3).Due to the degeneracy (the S 2 symmetry) of the "1" and "2" components in the z correlators of vector mesons, only the "1" components are plotted.In general, each panel plots ten C Γ (zT ) For the classification and notations of meson interpolators, see Table II.Vector (V k ) Tensor vector (T k ) For any flavor combination, if the SU (2) L × SU (2) R chiral symmetry is restored, then its ) and (V 4 , A 4 ) become degenerate, and the number of distinct z-correlators appears to be reduced to eight.Furthermore, if the U (1) A symmetry is also restored, then its (P, S), (T 4 , X 4 ) and (T 1 , X 1 ) also become degenerate, and the number of distinct z-correlators is further reduced to five.Thus one can visualize the effective retoration of SU (2) L ×SU (2) R × U (1) A chiral symmetry when the number of distinct z-correctors becomes five.This provides a simple guideline to look for the restoration of chiral symmetry from the panels in Figs.
In Fig. Next we look at the ss panels in Figs.1-7.In Fig. 2, at T = 257 MeV, it appears to have five distinct z-correlators in the channels of (P, S), (V 1 , A 1 ), (T 4 , X 4 ), (V 4 , A 4 ) and (T 1 , X 1 ), in spite of the small splittings at large z in the channels of (V 4 , A 4 ) and (T 1 , X 1 ).
Thus the SU (2) L × SU (2) R × U (1) A chiral symmetry of ss can be regarded to be restored at T ss c1 ∼ 257 MeV.This implies that the SU (3 This implies that T sc c1 is in the range of 385-512 MeV.In general, a more precise estimate of T c and T 1 can be obtained by the criteria ( 6) and ( 10), which will be given in the next section.
Finally, we look at the cc panels in Figs.1-7.The SU (2) L × SU (2) R × U (1) A chiral symmetry of cc seems to manifest at T = 770 MeV, and it becomes highly pronounced at T = 1540 MeV.This implies that T cc c1 is in the range of 770-1540 MeV, and also the restoratrion of the SU (4) L × SU (4) R × U (1) A chiral symmetry of (u, d, s, c) quarks at T cc c1 ∼ 770-1540 MeV, since the SU (2) L × SU (2) R × U (1) A chiral symmetry in other sectors (ūd, ūs, ss, ūc, sc) has already been restored at lower temperatures.This gives the second step of the hierarchical restoration of chiral symmetries in N f = 2 + 1 + 1 lattice QCD at the physical point, from the restoration of the SU (3) L × SU (3) R × U (1) A chiral symmetry of (u, d, s) quarks at T ss c1 ∼ 257 MeV to the restoratrion of SU (4) L × SU (4) R × U (1) A chiral symmetry of (u, d, s, c) quarks at T cc c1 ∼ 770 − 1540 MeV.A more precise estimate of T cc c and T barcc 1 can be obtained by the criteria ( 6) and ( 10), which will be given in the next subsection.
Besides the hierarchical restoration of chiral symmetries, we are also interested in visually identifying the emergence of the approximate SU (2) CS chiral spin symmetry in each of the six flavor sectors.To this end, we look for the appearance of three approximately distinct multiplets M 0 = (P, S), which become more pronounced at higher temperatures, and they are in the order The emergence of M 2 and M 4 is in agreement with the SU (2) CS multiplets of ( 14) and (15), and the SU (2) CS × SU (2) L × SU (2) R multiplets of ( 16) and ( 17).This suggests the emergence of the approximate SU (2) CS and SU (4) symmetries.Moreover, the separation between the multiplets M 2 and M 0 is decreased as the temperature is increased further.
Thus, at sufficiently high temperatures, say T > T f , M 2 and M 0 merges together to form a single multiplet, then the approximate SU (2) CS symmetry becomes washed out, and only the SU (2) L × SU (2) R × U (1) A chiral symmetry remains.In other words, the approximate SU (2) CS symmetry can only appear in a window of T above T c1 , i.e., T c1 < T cs ≲ T ≲ T f , where T cs (T f ) depends on ϵ cs (ϵ f cs ) in the criterion (23) for the emergence (fading) of the approximate SU (2) CS symmetry.Note that the multiplet M 4 never merges with the multiplets M 0 and M 2 , even in the limit T → ∞, as discussed in Ref. [1].Thus M 4 is irrelevant to the fading of the approximate SU (2) CS symmetry.The above provides a guideline to look for the emergence and the fading of the approximate SU (2) CS symmetry in Figs.1-7.
First, we look at the panels of ūd and ūs in Figs.1-7.We see that their z-correlators are almost identical for all seven temperatures.Furthermore, as T is increased from 192 MeV to 770 MeV, we see the emergence of three approximately distinct multiplets M 0 , M 2 , and M 4 , which become more pronounced at higher temperatures, while the separation of M 0 and M 2 become smaller.This suggests the emergence of the approximate SU (2) CS and SU (4) symmetries in the window T ∼ 308-770 MeV, for both ūd and ūs sectors.Finally, at T = 1540 MeV, M 0 and M 2 (for any flavor combination) merge together to form a single multiplet, and the approximate SU (2) CS symmetry has become completely washed out, and only the chiral symmetry remains.
Next, from the ss panels in Figs.1-7, we see that its window for the approximate SU (2) CS symmetry is almost the same as that of ūd and ūs, i.e., T ∼ 308-770 MeV.
Finally, we visually estimate the windows of the approximate SU (2) CS symmetry for heavy mesons with the c quark, which seem to be simlar to that of the light mesons.However, if one performs a more precise estimate with the criterion (23), one can reveal some salient features of the heavy vector mesons which cannot be easily observed by visual estimate, as shown in the next section.
V. Symmetry breaking parameters of (ūd, ūs, ss, ūc, sc, cc) In this section, we use the criteria ( 6), (10)        At each T , and for fixed zT , the chiral symmetry breakings due to the quark masses of the meson operator can be seen clearly from κ V A , κ P S , and κ T X , in the order of for each channel of α = (V A, P S, T X).Also, for each flavor content, κ α (zT ) at fixed zT is a monotonic decreasing function of T .Note that for the charmonium cc, the chiral symmetry breakings at T = 1540 are still not negligible, e.g., at zT = 4, 0.02 About the SU (2) CS symmetry breaking parameter κ CS = max(κ AT , κ T X ), for any flavor combination, it is a monotonic decreasing function of T at fixed zT , since both κ AT (zT ) and κ T X (zT ) are monotonic decreasing function of T .However, the flavor dependence of κ CS turns out to be rather nontrivial, and it is temperature dependent.Similarly, the flavor depenedence of the SU (2) CS symmetry fading parameter κ is also temperature dependent.
Nevertheless, it is interesting to point out that κ CS of the ūc sector is the smallest among all flavor sectors, while κ is almost the same for all flavor sectors, for all seven temperatures in the range of 190-1540 MeV.This suggests that the most attractive vector meson channels to detect the emergence of approximate SU (2) CS symmetry are in the ūc sector.This will be addressed more quantitatively in the subsection V B, in terms of the window of T for the approximate SU (2) CS symmetry.

A. Hierarchical restoration of chiral symmetries
Now we proceed to investigate the restoration of chiral symmetries in N f = 2 + 1 + 1 lattice QCD at the physical point.We use the criteria ( 6) and ( 10) to obtain T c and T 1 for each flavor combination.To this end, we collect the data of κ V A (zT ) and κ T X (zT ) at the same zT = (0.5, 1, 2), and plot them as a function of T , as shown in Figs. 15 and 16.According to (24), it follows that for any ϵ V A in ( 6) and any ϵ T X in (10), the flavor dependence of T c which immedidately gives Equations ( 25)-( 27) are the first results of lattice QCD.They immediately give the hierarachic restoration of chiral symmetries in N f = 2 + 1 + 1 QCD, i.e., from the restoration of SU (2) L × SU (2) R × U (1) A chiral symmetry of (u, d) quarks at T ūd c1 to the the restoration of SU (3) L × SU (3) R × U (1) A chiral symmetry of (u, d, s) quarks at T ss c1 > T ūd c1 , then to the restoration of SU (4) L × SU (4) R × U (1) A chiral symmetry of (u, d, s, c) quarks at T cc c1 > T ss c1 .
In the following, we demonstrate the hierarchical restoration of chiral symmetries explicitly, for ϵ V A = (0.05, 0.01) and ϵ T X = (0.05, 0.01) respectively.In Tables III and IV, for both ūd and ūs sectors, both T c and T 1 are less than 190 MeV, for any combinations of ϵ V A = (0.05, 0.01), ϵ T X = (0.05, 0.01), and zT = (0.5, 1, 2).For these cases, A is restored at a temperature lower than 190 MeV, for both ūd and ūs sectors, However, for the ūc sector, only for A is restored at a temperature lower than 190 MeV.  5) 235( 5) 320( 5) 255( 10) 350( 10) T sc 1 335( 5) 730( 5) 375( 5) 800( 5) 400( 5) 790( 5) T cc 1 835( 5) 1610( 10) 875( 5) 1395( 5) 865( 5) 1420( 5) Now we investigate the hierarchical restoration of chiral symmetries with ϵ V A = ϵ T X = 0.05 and zT = 1.From Tables III and IV Next, we study how T c (T 1 ) depends on ϵ V A (ϵ T X ).Since κ q1 q 2 V A (κ q1 q 2 T X ) at fixed zT is a monotonic decreasing function of T , it follows that T c (T 1 ) is monotonically increased as ϵ V A (ϵ T X ) is decreased (i.e., the precision of the chiral symmetry becomes higher).For example, if we set ϵ V A = ϵ T X = 0.01, then at zT = 1, the SU  (ūd, ūs, ss) and heavy mesons (ūc, sc, cc) respectively.In general, for any flavor content, at fixed zT , κ CS is a monotonic decreasing function of T , while κ is a monotonic increasing function of T .Thus, for any ϵ cs and ϵ f cs , the window of T satisfying the criterion (23) can be determined.Note that, if ϵ cs or ϵ f cs becomes too small, the window of T would shrink to zero (null).Using linear interpolotion and extrapolation of the data points in Figs.
17 and 18, we obtain the results of T window in Tables V-VI at zT = (1, 2) respectively, each for six flavor combinations, and for all combinations of ϵ cs and ϵ f cs sampling from (0.1, 0.15, 0.20, 0.25, 0.30).For visual comparison, we plot the windows of T in Fig. 19, for a range of values of (ϵ cs , ϵ f cs ) from large to small ones.Tables V-VI and Fig. 19 are the first results of lattice QCD.
It is interesting to see that the T windows of the approximate SU (2) CS symmetry are dominated by the channels of heavy vector mesons of (ūc, sc, cc).As the precision of SU (2) CS symmetry gets higher with smaller ϵ cs or ϵ f cs , the T windows of the light vector mesons (ūd, ūs, ss) shrink to zero, only those of heavy vector mesons survive.This suggests that the most attractive vector meson channels to detect the emergence of approximate SU (2) CS symmetry are in the (ūc, sc, cc) sectors, which may have phenomenological implications to the observation of the approximate SU (2) CS symmetry in relativistic heavy ion collision experiments such as those at LHC and RHIC.Moreover, the results of Tables V-VI and Fig. 19 also suggest that the hadron-like objects, in particular, in the channels of vector mesons with c quark, are likely to be predominantly bound by the chromoelectric interactions into color singlets at the temperatures inside their T windows of the approximate SU (2) CS symmetry, since the noninteracting theory with free quarks does not possess the SU (2) CS symmetry at all.MeV, as summarized in Table I.Our plan is to complete 21 gauge ensembles with three lattice spacings a ∼ (0.064, 0.069, 0.075) fm, which can be used to extract the continuum limit of the observables, for temperatures in the range of 160-1540 MeV.
Using seven gauge ensembles with a ∼ 0.064 fm, we computed the meson z-correlators for the complete set of Dirac bilinears (scalar, pseudoscalar, vector, axial vector, tensor vector, and axial-tensor vector), and each for six combinations of quark flavors (ūd, ūs, ss, ūc, sc,  These are the first results in lattice QCD.They immediately give the the hierarchical restoration of chiral symmetries in N f = 2 + 1 + 1 QCD, i.e., from the restoration of SU (2) L × SU (2) R × U (1) A chiral symmetry of (u, d) quarks at T ūd c1 to the restoration of or vice versa.In reality, for physical (u, s, c) quarks, we observe that Yet, in general, it is unclear to what extent (28) depends on the ratios of quark masses.
One of the phenomenological implications of the hierarchical restoration of chiral symmetries is the pattern of hadron dissolution at high temperatures, which leads to the hierarchical dissolution of hadrons, and the hierarchical suppression of hadrons in the quark-gluon plasma.
Theoretically, the meson with quark content qQ dissolves completely as q and Q become deconfined, i.e., when the screening mass of qQ is larger than its counterpart in the noninteracting theory with free quarks of the same masses.Presumably, m qQ scr ≥ m qQ(free) scr happens at the temperature T qQ d ≳ T qQ c1 , after the SU (2) L × SU (2) R × U (1) A chiral symmetry of qΓQ has been effectively restored.Thus, for N f = 2 + 1 + 1 lattice QCD at the physical point, one expects that the hierarchy of dissolution of mesons is exactly the same as that of the restoration of chiral symmetries (27), i.e., This leads to the hierarchical suppression of mesons in quark-gluon plasma, which could be observed in the relativistic heavy ion collision experiments such as those at LHC and RHIC.Here we recall the seminal paper by Matusi and Satz [18], in which it was proposed that the dissolution of J/ψ in the quark-gluon plasma would result in the suppression of their production in heavy ion collision experiments.To investigate whether (29) holds in N f = 2 + 1 + 1 lattice QCD at the physical point is beyond the scope of this paper.
Besides the meson z-correlators, the restoration of chiral symmetry in high temperature QCD can also be observed in the baryon z-correlators [2].For QCD with N f = 2(3) massless quarks, the chiral multiplets of baryon operators have been obtained by the group theoretical methods, see e.g., Ref. [19] and the references therein.Now, for QCD with physical (u, d, s, c, b) quarks, with quark masses ranging from a few MeV to a few GeV, we expect that the hierarchical restoration of chiral symmetries can be observed from the degeneracies of z-correlators of baryon chiral multiplets.It would be interesting to see whether the hierachy of chiral symmetry restoration from the baryon z-correlators is compatible with that from the meson z-correlators.
1) A chiral symmetry in both ūd and ūs sectors has been restored at T < 190 MeV.This is the first step of the hierarchical restoration of chiral symmetries in N f = 2 + 1 + 1 lattice QCD at the physical point, from the restoration of SU (2) L × SU (2) R × U (1) A chiral symmetry of (u, d) quarks at T ūd c1 < 190 MeV to the restoratrion of SU (3) L × SU (3) R × U (1) A chiral symmetry of (u, d, s) quarks at T ss c1 ∼ 257 MeV.Note that, as discussed in Sec.I and Sec.III, the restoration of SU (3) L × SU (3) R × U (1) A chiral symmetry of (u, d, s) quarks requires the SU (2) L × SU (2) R × U (1) A chiral symmetry for all six flavor combinations (ūd, ūs, ds, ūu, dd, ss), which are reduced to (ūd, ūs, ss) if m u = m d .Here we have assumed that in high temperature QCD, the contribution of the disconnected diagrams to the z-correlator of qΓq is negligible in comparison with that of the connected ones, as discussed in Sec.I. Similarly, the restoration of SU (4) L × SU (4) R × U (1) A chiral symmetry of (u/d, d, s, c) quarks requires the SU (2) L × SU (2) R × U (1) A chiral symmetry for all six flavor combinations ūd, ūs, ss, ūc, sc, and cc.Next, we look at the sc panels in Figs.1-7.The SU (2) L × SU (2) R × U (1) A chiral symmetry seems to manifest at T = 385 MeV, and it becomes highly pronounced at T = 513 MeV.

κ
AT , and κ, as defined in Sec.III.In Figs.8-14, the symmetry breaking parameters of six flavor combinations are plotted as a function of the dimensionless variable zT , for seven temperatures in the range of 190-1540 MeV.

TABLE I .
The lattice parameters and statistics of the seven gauge ensembles for computing the meson correlators.The last 3 columns are the residual masses of u/d, s, and c quarks.

TABLE II .
The classification of meson interpolators q1 Γq 2 , and their names and notations.
1, at T = 192 MeV, we see that both T ūd c (the temperature for the restoration of SU (2) L × SU (2) R chiral symmetry in the ūd sector) and T ud 1 (the temperature for the restoration of U (1) A symmetry in the ūd sector) are lower than 190 MeV, i.e., T ūd c < 190 MeV and T ūd 1 < 190 MeV.Thus the SU (2) L × SU (2) R × U (1) A chiral symmetry of ūd has been restored at some temperature lower than 190 MeV, i.e., T ūd c1 < 190 MeV.

TABLE III .
The temperature T q1 q 2

TABLE IV .
The temperature T q1 q 2

TABLE V .
The approximate ranges of T satisfying the criterion (23) at zT = 1 for six flavor contents.The table lists all nonzero windows of T for all possible combinations of ϵ cs and ϵ f cs sampling from (0.1, 0.15, 0.20, 0.25, 0.30).Each T window is in units of MeV, with uncertainties ±5 MeV on both ends of the window.

TABLE VI .
The approximate ranges of T satisfying the criterion (23) at zT = 2 for six flavor contents.The table lists all nonzero windows of T for all possible combinations of ϵ cs and ϵ f cs