Symmetries of spatial meson correlators in high temperature QCD

Based on a complete set of $J = 0$ and $J=1$ spatial isovector correlation functions calculated with $N_F = 2$ domain wall fermions we identify an intermediate temperature regime of $T \sim 250 - 550$ MeV where chiral symmetry is restored but the correlators are not yet compatible with a simple free quark behavior. More specifically, in the temperature range $T \sim 250 - 550$ MeV we identify a multiplet structure of the spatial correlators that suggests the emergent (approximate) symmetries $SU(2)_{CS}$ and $SU(4)$ which are not symmetries of the free Dirac action. Our results indicate that at these temperatures the chromo-magnetic interaction is suppressed and the elementary degrees of freedom are chirally symmetric quarks bound into color-singlet objects by the chromo-electric component of the gluon field. At even higher temperatures the emergent $SU(2)_{CS}$ and $SU(4)$ symmetries disappear smoothly and above $T \sim 1$ GeV only chiral symmetries survive, which then finally are compatible with quasi-free quarks.


I. INTRODUCTION
Understanding the physics of strongly coupled matter at high temperature is one of the great open challenges in high energy physics. Addressing this question is the subject of large-scale experimental and theoretical efforts. Initially it was assumed that above some pseudocritical temperature T c quarks deconfine and chiral symmetry is restored such that above T c the degrees of freedom (d.o.f.) are liberated quarks and gluons [1].
A flavor nonsinglet chiral restoration was indeed confirmed on the lattice, which is signalled by the vanishing quark condensate above the crossover region around T c and by degeneracy of correlators that are connected by chiral transformations.
The expected confinement-deconfinement transition turned out to be more intricate to define. Such a transition was historically assumed to be associated with a different expectation value of the Polyakov loop [2,3] below and above the critical temperature T c . In pure SUð3Þ gauge theory the Polyakov loop is connected with the Z 3 center symmetry and indeed a sharp first-order phase transition is observed [4], which indicates that the relevant d.o.f. below and above T c are different. Still, one may ask whether this Z 3 transition is really connected with deconfinement in a pure glue theory. Traditionally the answer was affirmative, because the expectation value of the Polyakov loop can be related to the free energy of a static quark source. If this energy is infinite, which corresponds to a vanishing Polyakov loop, then we are in a confining mode, while deconfinement should be associated with a finite free energy, i.e., a nonzero Polyakov loop. However, this argumentation is self-contradictory because a criterion for deconfinement in pure gauge theory, i.e., deconfinement of gluons, is reduced to deconfinement of a static charge (heavy quark) that is not part of the pure glue theory. The Polyakov loop is a valid order parameter but strictly speaking its relation to confinement is an assumption. And indeed, just above the first-order Z 3 phase transition the energy and pressure are quite different from the Stefan-Boltzmann limit, which is associated with free deconfined gluons [5].
In a theory with dynamical quarks the first-order phase transition is washed out and on the lattice one observes a very smooth increase of the Polyakov loop [6]. The reason for that behavior is rather clear: in a theory with dynamical quarks there is no Z 3 symmetry and the Polyakov loop ceases to be an order parameter. Considering the finite energy of a pair of static quark sources (Polyakov loop correlator) the resulting string breaking potential is due to vacuum loops of light quarks that combine with the static sources to a pair of heavy-light mesons. Lattice measurements of the energy density and pressure with dynamical quarks indicate a smooth transition, and at T ∼ 1 GeV the system is still quite far from the Stefan-Boltzmann limit [7,8].
In view of the absence of a reliable, generally accepted definition and order parameter for deconfinement-except for the most straightforward statement that confinement is the absence of colored states in the spectrum-a key to understanding the nature of hot QCD matter is information about the relevant effective d.o.f. in high temperature QCD. Several model and lattice studies suggest the possible existence of interquark correlations or bound states above T c ; see, e.g., Refs. [9][10][11]. While models may provide helpful intuitive understanding, it is important to attempt finding model independent ways to identify the d.o.f. in high T QCD.
Among other observables, relevant information is encoded in Euclidean correlation functions. At zero temperature hadron masses can be extracted from the exponential slope of correlators in the Euclidean time direction t. At nonzero temperature the temporal extent is finite by definition (it vanishes at T → ∞) such that there is no strict notion of an asymptotic behavior for t-correlators. Spatial correlators on the other hand are well defined and do provide detailed information about the QCD dynamics [12][13][14][15][16][17][18][19][20]. These spatial correlators can be analyzed with respect to the symmetries they exhibit, which in turn allows one to extract information about the relevant effective d.o.f.
In previous work [21] we have studied a complete set of J ¼ 0 and J ¼ 1 isovector correlation functions in the z direction for a system with N F ¼ 2 dynamical quarks in simulations with the chirally symmetric domain wall Dirac operator at temperatures up to T ∼ 380 MeV. Similar ensembles have been used previously for the study of the Uð1Þ A restoration in t-correlators and via the Dirac eigenvalue decomposition of correlators [22,23]. We have observed the restoration of both SUð2Þ L × SUð2Þ R and Uð1Þ A chiral symmetries above T c on a finite lattice of a given size.
However, by analyzing the formation of multiplets for the spatial correlators, even larger symmetries, referred to as SUð2Þ CS chiral spin and SUð4Þ symmetries [24,25], have been identified in the J ¼ 1 correlators in the region T ∼ 2T c . These symmetries, while not symmetries of the Dirac Lagrangian, are symmetries of the Lorentz-invariant fermion charge. In the given reference frame they are symmetries of the interaction between the chromoelectric field with the quarks while the interaction of quarks with the chromomagnetic field breaks them. These symmetries include as subgroups the chiral symmetries as well as rotations between the right-and left-handed components of quarks. Such symmetries have been found already earlier in the hadron spectrum at zero temperature [26][27][28][29] upon artificial truncation of the near-zero modes of the Dirac operator [30]. While the SUð2Þ L × SUð2Þ R and Uð1Þ A chiral symmetries are almost exact above T c , the SUð2Þ CS and SUð4Þ symmetries are approximate. In this paper we improve the analysis and extend the temperature range up to T ∼ 1 GeV, in order to further study the temperature evolution of the symmetries of correlators and thus the temperature evolution of the emergent effective d.o.f.
We stress that the SUð2Þ CS and SUð4Þ symmetries are not symmetries of the free Dirac action and therefore their emergence is incompatible with the notion of quasifree, deconfined quarks. The emergence of these symmetries in a range from T ∼ 220-500 MeV (1.2T c -2.8T c ), as reported in this article, suggests that the effective d.o.f. of QCD at these temperatures are quarks with definite chirality bound by the chromoelectric component of the gluon field into color-singlet objects, "stringlike" compounds.
While the lattice study is possible only at zero chemical potential, the observed approximate symmetries should persist also at finite chemical potential, due to the quark chemical potential term in the QCD action being manifestly SUð2Þ CS and SUð4Þ symmetric [31].
When increasing the temperature to T ∼ 1 GeV we observe that at very high temperature the SUð2Þ CS and SUð4Þ multiplet structure is washed out and the full QCD meson correlators approach the corresponding correlators constructed with free, noninteracting quarks. This indicates that at very high temperature the coupling constant is sufficiently small to describe dynamics of weakly interacting quarks and gluons. Preliminary results of this work were presented at the Lattice 2018 conference [32].

II. SPATIAL FINITE TEMPERATURE MESON CORRELATORS FOR NONINTERACTING QUARKS IN THE CONTINUUM
We begin our presentation with a summary of the calculation of the spatial correlators for free massless quarks in the continuum. This situation is the limiting case that should represent QCD at very high temperatures where, due to asymptotic freedom, the interaction via gluons can be neglected. We discuss the multiplet structure for this reference case which we later use to compare to our lattice calculation at high, but not asymptotically high temperature. In particular, we find that at moderately high temperatures above T c the spatial correlators of full QCD display a multiplet structure different from the limiting case of free quarks discussed in this section. We remark that some of the free spatial continuum correlators computed here were already presented in [16,17], but for a systematical and complete discussion we need the full set of all spatial meson correlators and thus briefly summarize their derivation in this section and the Appendix.
In the continuum the free spatial meson correlators in infinite spatial volume are given by We consider Euclidean space at finite temperature, i.e., x; y; z ∈ R, and t ∈ ½0; βÞ, where β is the inverse temperature. In the correlators (1) we look at correlation in one of the spatial directions, here chosen as z, while the other two, x and y, as well as the Euclidean time t are integrated over. The latter integration over all coordinates that are perpendicular to the direction of propagation, i.e., the z direction, fixes a "Euclidean rest frame" for our correlators.
The meson interpolators are given by where we use the abbreviations x ¼ ðx; y; z; tÞ and 0 ¼ ð0; 0; 0; 0Þ, and Γ is an element of the Clifford algebra, i.e., a product of γ matrices (see below). Note that choosing the negative sign for O † Γ is a definition, since in general the sign obtained from conjugation depends on Γ. Throughout the whole paper we use the set γ μ , μ ¼ 1, 2, 3, 4 of Euclidean γ-matrices that satisfy the anticommutation relations uðxÞ, uðxÞ,dðxÞ, dðxÞ are free massless Dirac spinors which obey antiperiodic boundary conditions in Euclidean time. We remark that for simplicity we here have already expressed the nonsinglet correlators in terms of the flavor spinors u and d, while in the next section we write them in terms of isospin doublets qðxÞ ≡ ðuðxÞ; dðxÞÞ. After contracting the fermions, the two forms for writing the nonsinglet bilinears of course give the same expressions. Performing these contractions we obtain where the trace is over Dirac indices and S denotes the free continuum Dirac propagator. We are interested in the physics near the chiral limit, and therefore we consider massless quarks in this section. In terms of Fourier integrals S is given by where p ¼ ðp x ; p y ; p z ; ω n Þ, with the Matsubara frequencies ω n ¼ πð2n þ 1Þ=β. Inserting (5) into (4) and this into (1) we find wherep ≡ ðp x ; p y ; p 0 z ; ω n Þ and we have already integrated over x, y and t in (1), which generated two Dirac deltas and a Kronecker delta that were used to get rid of two of the momentum integrals and one of the Matsubara sums.
As we see below, the trace in the integrand has the general form where s x , s y , s z and s τ are signs that depend on the choice of Γ. Thus for the pair of integrals over the z components we can distinguish two cases, depending on whether the factor p z p 0 z appears in the integrand or not, . The integrals I 0 and I 1 are straightforward to solve with the residue theorem, We find for the correlator C Γ ðzÞ, C Γ ðzÞ ¼ −½s x C s ðzÞ þs y C s ðzÞ þs z C z ðzÞ þs τ C τ ðzÞ; ð11Þ with the individual correlators given by The correlators C s ðzÞ; C z ðzÞ and C τ ðzÞ obey the obvious sum rule i.e., only two of them are independent. We choose C z ðzÞ and C τ ðzÞ to express all other correlators. The treatment of the Matsubara sums and the necessary integrals for evaluating C z ðzÞ and C τ ðzÞ are discussed in the Appendix, where we also discuss the asymptotic behavior of the correlators. We now come to the identification of multiplets, i.e., we identify the sets of Clifford algebra elements Γ that share the same decay properties for their corresponding correlators C Γ ðzÞ. For this we need to determine the signs s x , s y , s z and s τ in the traces (7) for the different choices of Γ, which in turn determine how the respective correlator C Γ ðzÞ is composed from the contributions C s ðzÞ, C z ðzÞ and C τ ðzÞ according to (11).
We first note that for chiral partners, i.e., correlators where Γ is replaced by Γγ 5 , the corresponding correlators C Γ ðzÞ and C Γγ 5 ðzÞ have opposite overall signs, and thus also opposite individual signs s x , s y , s z and s τ . This follows from the trivial relation This implies that we need to determine the signs s x , s y , s z and s τ in the traces (7) only for eight out of the 16 Clifford algebra generators Γ. Our results for the signs s x , s y , s z and s τ that determine the decomposition of Tr½= pΓ †= pΓ according to (7) are listed in Table I.
Having determined the signs s x , s y , s z , s τ we use them in (11) to work out the composition of C Γ ðzÞ from the building blocks C s ðzÞ, C z ðzÞ and C τ ðzÞ, and after eliminating C s ðzÞ we obtain the representation for the C Γ ðzÞ in terms of C z ðzÞ and C τ ðzÞ evaluated in the Appendix. We find (overall signs were chosen such that chiral partners have the same overall sign) The vanishing of the correlators C γ 3 ðzÞ and C γ 3 γ 5 ðzÞ is a direct consequence of the sum rule (13). From a more physical point of view this vanishing is a consequence of current conservation. Indeed, C γ 3 ðzÞ is the correlator for the 3 component of the conserved vector current J μ ðxÞ 1 uðxÞγ μ dðxÞ and concerning the propagation in z direction the integral R dxdydtJ 3 ðx; y; z; tÞ is a conserved charge. Thus the corresponding spatial correlator and its chiral partner vanish, which also implies that the sum rule (13) is directly linked to current conservation. Furthermore the sum rule (current conservation) means that the correlators C γ 4 ðzÞ ¼ C γ 4 γ 5 ðzÞ are not independent from the correla- We conclude this section by quoting the asymptotic behavior of our correlators, which is obtained by using (A5) from Appendix in the expressions (15)  I. The signs s x , s y s z and s τ that determine the trace Tr½= p Γ †= p Γ for different choices of Γ according to (7). For chiral partners, i.e., when Γ is replaced by Γγ 5 , all signs are reversed [compare to (14)]. To simplify the notation we chose the (irrelevant) overall signs equal for both chiral partners such that the relative signs s x , s y s z and s τ as listed in the table are used for both chiral partners. In the two columns on the right we give the names of the bilinears and their chiral partners, which we discuss in detail in the next section. Since the interpolators with γ 3 and γ 3 γ 5 vanish identically no name is assigned.
Here we have only listed half of the correlators in each chiral multiplet without their chiral partners, which have identical correlators (up to an overall sign which we dropped). The fact that on the rhs of (16) appears only the dimensionless combination zω 0 ¼ πz=β ¼ πzT reflects the absence of any physical scale in the conformal theory of massless noninteracting quarks.

III. FERMIONIC BILINEARS AND THEIR SYMMETRIES
Having summarized the explicit form of the spatial correlators for the free case, let us now come to the general (full QCD) discussion of the mesonic bilinears and their symmetries. We are interested in the spatial correlators of the local isovector mesonic bilinears which we now write using the isospin doublets qðxÞ ≡ ðuðxÞ; dðxÞÞ. The isovector structure of the bilinears is determined by the isospin Pauli matrices τ a . Again Γ may be any element of the Clifford algebra and the choice of Γ determines the symmetry properties of the respective bilinear. Two J ¼ 0 bilinears can be defined by the following choices for Γ: These two bilinears can be transformed into each other by global Uð1Þ A rotations, qðxÞ → exp ðiγ 5 θÞqðxÞ: For J ¼ 1 we consider bilinears with the following choices of Γ that define the vector bilinears V, As we have already seen for the free case that we discussed in the previous section, due to current conservation the 3 componentqðxÞγ 3 ⃗ τ 2 qðxÞ does not propagate in the z direction such that we omit the choice Γ ¼ γ 3 .
The vector bilinears are related to their chiral partners through flavor nonsinglet axial rotations Their chiral partners, the axial-vector bilinears A, are defined as ðaxial vectorÞ: ð22Þ At 0 (or sufficiently small) temperature the chiral partner of the nonpropagating third vector current component, i.e., the bilinear with the gamma structure Γ ¼ γ 3 γ 5 , does indeed propagate also in the z direction due to broken chiral symmetry and then couples to the pseudoscalar channel. After restoration of chiral symmetry, i.e., at the temperatures we consider here, it behaves like its chiral partner and does not propagate in the z direction. Thus, like Γ ¼ γ 3 , also the choice Γ ¼ γ 3 γ 5 can be omitted. The bilinears that correspond to the six tensor elements σ μν of the Clifford algebra can be organized into two vectorvalued objects, the tensor-vector T, ðtensor vectorÞ ð23Þ and the axial-tensor-vector X, The bilinears T and X can be transformed into each other by the Uð1Þ A rotations (19). Table II summarizes our bilinears and lists the Uð1Þ A and SUð2Þ L × SUð2Þ R relations among them.
Due to the restoration of the Uð1Þ A and SUð2Þ L × SUð2Þ R symmetries at high temperature we expect the emergence of degeneracies among correlators of bilinears related by these symmetries, and of course those degeneracies clearly must also be seen explicitly in the free continuum correlators (15) and (16). The degeneracies based on Uð1Þ A and SUð2Þ L × SUð2Þ R are the degeneracies required by chiral symmetries that emerge above T c .
However, in addition to those, at temperatures not too far above T c a larger group of symmetries, SUð2Þ CS and SUð4Þ that contain Uð1Þ A and SUð2Þ L × SUð2Þ R [24,25], has been observed in our previous study of correlators [21]. The SUð2Þ CS chiral spin transformations are defined by where ⃗ ϵ ∈ R 3 are the rotation parameters. For the generators ⃗ Σ one has four different choices ⃗ Σ ¼ ⃗ Σ k with k ¼ 1, 2, 3, 4, but, as we discuss below, only the cases k ¼ 1 and k ¼ 2 are of interest here. The generators are given by and the suð2Þ algebra is satisfied for any choice k ¼ 1, 2, 3, 4. While these are not symmetries of the Dirac Lagrangian, both in Minkowski and Euclidean space, the Lorentzinvariant fermion charge in Minkowski space, is invariant under SUð2Þ CS , where ψðxÞ can be either a single-flavor quark field or an isospin doublet. The Euclidean fermion charge is also SUð2Þ CS invariant.
In Minkowski space in a given reference frame the quark-gluon interaction can be split into temporal and spatial parts,ψ where The temporal term includes the interaction of the coloroctet charge densitȳ with the chromoelectric component of the gluonic field. It is invariant under SUð2Þ CS [25]. We emphasize that the SUð2Þ CS transformations defined in Eq. (26) via the Euclidean Dirac matrices can be identically applied to Minkowski Dirac spinors without any modification of the generators. The spatial part contains the quark kinetic term and the interaction with the chromomagnetic field. This term breaks SUð2Þ CS . In other words, the SUð2Þ CS symmetry distinguishes between quarks interacting with the chromoelectric and chromomagnetic components of the gauge field. It is important to note that discussing electric and magnetic components can be done only in Minkowski space and in addition one needs to fix the reference frame. However, at high temperatures Lorentz invariance is broken and a natural frame to discuss physics is the rest frame of the medium. The SUð2Þ CS transformations (26) with k ¼ 1 generate the following two SUð2Þ CS singlets and two SUð2Þ CS triplets of bilinears: ðV y Þ; ðA y ; T t ; X t Þ; ð32Þ ðV t Þ; ðA t ; T y ; X y Þ: These irreducible representations of SUð2Þ CS can be obtained by applying the SUð2Þ CS transformation (26) on any of the bilinears from the given representation and the result is a linear combination of all bilinears in the given representation. The observation of a degeneracy of the correlators built from the triplet bilinears in Eq. (32) would imply the emergence of the corresponding SUð2Þ CS symmetry. We stress that this is not a symmetry of deconfined free quarks, see Eq. (15), and the observation of a degeneracy within the triplet in Eq. (32) means that the quarks in the system interact exclusively via the chromoelectric field, without any chromomagnetic admixture. Since only color-singlet bilinears can propagate on the lattice at any temperature the systems represent color-singlet quarkantiquark objects bound by chromoelectric interactions.
Note that the observation of a degeneracy of correlators for the triplet bilinears in Eq. (33) would not discriminate between the confining mode and free quarks, because the current conservation in the free quark system also provides such a degeneracy, as follows already from the discussion in the previous section; see Eq. (15). 1 The transformations (26) with k ¼ 2 generate the following singlets and triplets: ðV t Þ; ðA t ; T x ; X x Þ: Again, a degeneracy of the correlators built from the triplet bilinears in Eq. (34) is a signal for the emergence of the SUð2Þ CS symmetry. This is different from the degeneracy of the correlators of the triplet bilinears from Eq. (35), which in the free quark case can be connected to current conservation and thus is not suitable for discriminating between the interacting mode and a system of free quarks. This discussion [as well as a structure of the SUð4Þ multiplets below] implies that only the study of a possible degeneracy among correlators of the bilinears (32), as well as the bilinears (34) is suitable for the analysis of the underlying dynamics and d.o.f. Note that only those SUð2Þ CS ; k ¼ 1, 2, 3, 4 transformations can be considered for a given observable that do not mix operators of different spin and thus respect rotational invariance at nonzero temperature. This requirement is met for our setup by the k ¼ 1, 2 transformations, as indicated above.
We remark that at zero temperature in the continuum there is a SOð3Þ symmetry in the x, y, t subspace and the z-correlators of the V x , V y , V t bilinears (20) coincide. The same is true for the z-correlators of the corresponding x, y and t components of the bilinears (22)- (24). At finite temperature this rotational symmetry is broken down to a residual SOð2Þ symmetry which connects the correlators of the spatial components V x ↔ V y and A x ↔ A y et cetera. On the lattice the reduced symmetry for the T > 0 case and the z ¼ const subspace is D 4h and the relevant symmetry is S 2 × SUð2Þ CS [21], 2 such that the multiplets are ðV x ; V y Þ; ðA x ; A y ; T t ; X t Þ; ð36Þ ðV t Þ; ðA t ; T x ; T y ; X x ; X y Þ: ð37Þ Finally we remark that the group SUð2Þ CS ⊗ SUð2Þ F , where SUð2Þ F is the isospin symmetry group, can be extended to SUð4Þ with fifteen generators, The corresponding transformations are a trivial generalization of Eq. (26) obtained by replacing the generators ⃗ Σ by those listed in (38). Also the group SUð4Þ is a symmetry of the quark-chromoelectric interaction terms of the QCD Lagrangian, while the quark-chromomagnetic interaction as well as the kinetic term break it. The S 2 × SUð4Þ transformations connect the following J ¼ 1 operators from Table II: ðV t ; A t ; T x ; T y ; X x ; X y Þ: ð40Þ These are the multiplets of the isovector operators that are discussed in the present paper. The SUð4Þ symmetry requires degeneracy within both the (39) as well as the (40) multiplets, while a degeneracy of the normalized correlators from the multiplet (40) is also consistent with free noninteracting quarks. Obviously the chiral multiplets of the PS and S bilinears are not subject to this degeneracy.
The complete S 2 × SUð4Þ multiplets in addition also include the isoscalar partners of A x , A y , T t and X t in Eq. (39) as well as the isoscalar partners of A t , T x , T y , X x and X y in Eq. (40). The isoscalar partners of V x , V y and V t are the SU(4) singlets.

IV. LATTICE TECHNICALITIES
The correlators discussed in the previous section are evaluated on the JLQCD configurations for full QCD with N F ¼ 2 flavors of domain wall fermions. Details concerning the gauge configurations are presented in [22,23]. In this setup we choose L 5 , the extent of the auxiliary fifth dimension, such that for all our ensembles the violation of the Ginsparg-Wilson condition is less than 1 MeV.
For measurements the IroIro software is used [33], and the relevant parameters are fixed in a zero temperature study [34]. The quark propagators are computed on point sources with the domain wall Dirac operator after three steps of stout smearing. The fermion fields are periodic in the spatial directions and antiperiodic in time.
We use the Symanzik-improved gauge action at inverse gauge couplings β g in a range between β g ¼ 4.1 and β g ¼ 4.5, and with the different temporal lattice extents in use, N t ¼ 4, 6, 8 and N t ¼ 12, we cover a range of temperatures between T ≃ 220 MeV and T ≃ 960 MeV. For the bare quark mass parameters m u ¼ m d ≡ m ud we 1 This is true for the correlators normalized to 1, which we study here. Without this normalization there is an overall factor of 2 between the free correlators built with the V t , A t and T x , T y , X x , X y bilinears [see, e.g., Eq. (16)] that would allow one to distinguish the results for free quarks from the full SUð2Þ CS case in an elaborated calculation with properly renormalized full QCD correlators. 2 S 2 here denotes the permutation or symmetric group for x ↔ y interchanges. use the value m ud ¼ 0.001, which corresponds to physical quark masses at our different temperatures in the range between 2 and 4 MeV. We have also performed simulations with m ud ¼ 0.01, m ud ¼ 0.005 and observed stability of our results against quark mass variation because in the temperature range we consider (220-960 MeV) these quark masses are essentially negligible due to temperature effects. Further details concerning the chiral properties for our set of parameters are given in [22,23]. The complete list of our ensembles and their parameters is provided in Table III.
As already discussed, we measure finite temperature spatial correlators in the z direction, as was first suggested in [12]. To compare the results from our different ensembles we plot the correlators as a function of the dimensionless combination where z is the physical distance in the correlators, T the temperature, a the lattice constant, n z the distance in lattice units and N t the temporal lattice extent. We project to zero momentum by summing over all lattice sites in slices orthogonal to the z direction, i.e., we consider C Γ ðn z Þ ¼ X n x ;n y ;n t hO Γ ðn x ; n y ; n z ; n t ÞO Γ ð0; Obviously this is the lattice version of the continuum form in Eq. (1).

V. RESULTS
In Fig. 1 we compare the spatial correlators for a wide range of temperatures from T ∼ 220 to 960 MeV to give an impression of the changing behavior observed for different values of T. The correlators are shown as a function of the dimensionless combination zT ¼ n z =N t [compare to Eq. (41)] using the full range of n z values-up to periodicity. In order to compare different correlators without a proper renormalization, our correlators are normalized to 1 at n z ¼ 1. Because of the degeneracy of x and y components in vector operators we show only the correlators for the x components.
The top left panel of Fig. 1 shows correlators at a temperature of T ∼ 220 MeV, i.e., 1.2T c . All correlation functions of chiral partners are degenerate within errors. In detail, these are the two pairs ðV x ; A x Þ and ðV t ; A t Þ, each of which reflects SUð2Þ R × SUð2Þ L symmetry. Uð1Þ A symmetry in the vector channel, represented by the operator pairs ðT x ; X x Þ and ðT t ; X t Þ, is manifest for all ensembles. For the scalar ðPS; SÞ pair we find the restoration of Uð1Þ A symmetry to be heavily dependent on the parameters. As is evident from the top left panel of Fig. 1, PS and S are degenerate within errors for our finest lattice. On the coarser 32 × 8 ensemble at 220 MeV we find a visible difference of PS and S correlators consistent with previous findings in literature, e.g., the data for staggered quarks presented in Fig. 7 of Ref. [19]. 3 For temperatures between T ∼ 220 and 500 MeV the correlators are grouped into three distinct multiplets, 4 Possible splittings within each of these multiplets are obviously much smaller than the distances between the multiplets. The multiplet structure reflects the symmetries as follows: The multiplet E 1 indicates the restoration of Uð1Þ A symmetry. Degeneracies within the multiplets E 2 and E 3 reflect the larger symmetries SUð2Þ CS and SUð4Þ as discussed in the previous section. The formation of the multiplet E 3 is not necessarily a consequence of the SUð2Þ CS and SUð4Þ symmetries as the same degeneracy of correlators is seen also for noninteracting quarks (15) and can be attributed to current conservation. Consequently from the observation of the E 3 multiplet alone we could not claim the emergence of the SUð2Þ CS and SUð4Þ symmetries. However, the E 2 degeneracy is not manifest in the free quark system (15) and indeed can be attributed to the emergent SUð2Þ CS and SUð4Þ symmetries.
We speak of separate multiplets when the splittings within the multiplets are much smaller than splittings between different multiplets. All correlators connected by chiral Uð1Þ A and SUð2Þ L × SUð2Þ R transformations TABLE III. Ensembles and their parameters: We list the lattice size, the inverse gauge coupling β g , the lattice constant a in fm, the statistics, the extent L 5 used for the domain wall fermions, the temperature T in MeV and the ratio T=T c (see [22,23]  For detailed studies of Uð1Þ A symmetry around T c , see e.g., [35] or [23]. The latter study uses the same simulation setup as the present work. 4 Note that in E 2 and E 3 we leave out the y components, which are exactly degenerate with the respective x components explicitly listed in E 2 and E 3 . are indistinguishable at all temperatures. At temperatures above T ∼ 600 MeV we observe that the distinct multiplet E 2 , related to emergence of the SUð2Þ CS and SUð4Þ symmetries, is washed out. The remaining E 3 multiplet structure can be attributed to quasifree quarks. In Fig. 2 we now focus on the E 1 and E 2 multiplets at three different temperatures. For comparison we also show the corresponding correlators computed for free quarks (dashed lines). The latter correlators are obtained with the same lattice Dirac operator and lattice size as used for the full QCD but now with a unit gauge configuration. We note that for free quarks only those degeneracies exist that are predicted by the chiral Uð1Þ A and SUð2Þ L × SUð2Þ R symmetries.  Table III for details). We label groups of correlators according to the multiplets E 1 , E 2 and E 3 as introduced in Eqs. (43)-(45).
For the lowest temperature T ∼ 220 MeV we still observe a small residual splitting within the E 2 multiplet, while at T ∼ 380 MeV the difference nearly vanishes. Furthermore, there is a clear splitting between the E 1 and E 2 multiplets indicating SUð2Þ CS and SUð4Þ symmetries. In addition all correlators are well separated from their free quark counterparts shown as dashed curves.
At the highest temperature of this study, T ∼ 960 MeV, the situation has changed considerably: All correlators almost perfectly coincide with the corresponding free correlators, as seen by the dashed lines on top of the data points for the full QCD correlators. Thus at T ∼ 960 MeV we have reached the region where only chiral Uð1Þ A and SUð2Þ L × SUð2Þ R symmetries exist and the coincidence with the free correlators suggests a gas of quasifree quarks.
In an attempt to discuss the observed evolution of symmetries more quantitatively, in Figs. 3 and 4 we study ratios of correlators, where the fully symmetric case corresponds to a constant ratio 1 for all z. In Fig. 3 we show ratios of normalized correlators for different bilinears from the E 2 multiplet. The ratios are plotted as a function of the dimensionless quantity zT ¼ n z =N t and we compare different temperatures.
In the lhs plot we show the ratio C X t =C T t . The two correlators are related by Uð1Þ A and a deviation from a constant ratio 1 indicates a violation of Uð1Þ A . The data show no breaking effects within errors.
In the rhs plot we show the ratio C A x =C T t . These two correlators are related by SUð2Þ CS and thus a deviation from 1 indicates a violation of exact SUð2Þ CS . Here the lowest temperature displays sizable residual violation, which gradually becomes smaller with increasing temperature. At T ∼ 380 MeV the deviation from 1 becomes minimal.
Finally, in Fig. 4 we analyze the SUð2Þ CS sensitive ratio C A x =C T t for all our ensembles in a wider range of temperatures. We observe an evolution from sizable deviation from 1 at the lowest temperature T ∼ 220 MeV towards a coincidence with the corresponding ratio of correlators for The lhs plot shows the ratio C X t =C T t , i.e., a ratio of correlators connected by Uð1Þ A . The rhs plot shows the ratio C A x =C T t , i.e., two correlators connected by SUð2Þ CS transformations. In both cases we show the corresponding ratios for free quarks as dashed curves.
free quarks at the highest temperature, i.e., T ∼ 960 MeV. For intermediate temperatures we observe small deviations from 1. Figures 3 and 4 demonstrate that-while the chiral symmetries are practically exact-the SUð2Þ CS symmetry is not exact. Let us introduce a measure for the symmetry breaking and find a temperature range where the symmetry is appropriate.
In general a symmetry is established via its multiplet structure. For any multiplet structure a crucial parameter is the ratio of the splitting within a multiplet to the distance between multiplets. The splitting within a multiplet by itself is irrelevant without a scale, and should be compared to a scale relevant for the given problem, e.g., the distance between multiplets. Consequently, in our case the breaking of SUð2Þ CS and SUð4Þ can be identified through the parameter If κ ≪ 1, then we can declare an approximate or, if 0, an exact symmetry. If κ ∼ 1, the symmetry is absent. The criterion of small κ corresponds to the existence of a distinct multiplet E 2 that should be well separated from the multiplet E 1 . From the free quark expression (16) one finds κ ∼ 1, which stresses again that there is no chiral-spin symmetry for free quarks.
In Fig. 5 we show the evolution of the symmetry breaking parameter κ as a function of temperature at zT ¼ 2. The value of κ is less than 5% for all ensembles with T ∼ 220-500 MeV. This implies that the symmetries that we observe in the range between T ∼ 220 and 500 MeV are well pronounced.
At temperatures between T ∼ 500 MeV and T ∼ 660 MeV we notice a drastic increase of the symmetry breaking parameter κ to values of the order ∼1. We conclude that QCD exhibits an approximate SUð2Þ CS symmetry in the temperature range between T ∼ 220 and 500 MeV (1.2T c -2.8T c ) with symmetry breaking less than 5% as measured with κ. This suggests that the SUð2Þ CS symmetric regime begins just after the SUð2Þ R × SUð2Þ L restoration crossover.
We stress once more that the SUð2Þ CS symmetry is related to different components of the strong interaction. As we have discussed, an exact SUð2Þ CS symmetry implies that the interaction is strictly chromoelectric. Thus the observed evolution of the SUð2Þ CS symmetry as a function of temperature suggests the following picture for the relevant d.o.f. in high temperature QCD: At T ∼ 220 MeV we find C A x =C T t > 1 and a small violation of SUð2Þ CS such that the interaction between the quarks must be mediated not only by the chromoelectric component, but also to some extent by the chromomagnetic components of the gluonic field. When increasing the temperature, the ratio C A x =C T t evolves towards 1. This implies that at T ∼ 380 MeV the chromomagnetic interaction has become washed out and quarks interact via the chromoelectric field. The remaining small breaking of SUð2Þ CS is due to the quark kinetic term. It suggests that in this regime the elementary objects are chirally symmetric quarks confined by the chromoelectric field. At even higher temperatures also the contribution of the chromoelectric interaction decreases and the system enters the region of quasifree quarks, as reflected by the fact that for our highest temperatures the ratio C A x =C T t approaches the corresponding curve for free quarks.
We stress that the emerging SUð2Þ CS and SUð4Þ symmetries, observed in the range of T ∼ 220 MeV to T ∼ 500 MeV, are incompatible with the picture of free deconfined quarks.
This view is also reflected in the exponential decay properties, i.e., the factors ∝ expð−czÞ, of the full QCD correlators. A system of two free quarks cannot have zcorrelators where the exponent c is smaller than twice the lowest Matsubara frequency 2ω 0 , due to the antiperiodic boundary conditions of fermions in time direction [compare to Eq. (16)]. If the exponent c is smaller for the interacting case, this suggests that the quark-antiquark system is still coupled into a bosonic compound, since periodic boundary conditions for bosons do allow for the exponent c to be smaller than 2ω 0 . Figure 2 shows that the full PS-and S-correlators have significantly smaller exponents c than their noninteracting counterparts, which suggests that these correlators correspond to coupled quark-antiquark compounds [12]. In the J ¼ 1 channels the difference of the exponents c for full and free correlators at temperatures T < 500 MeV is much smaller, but still visible, and suggests a residual binding also in this case.

VI. CONCLUSIONS
In this paper we have studied spatial correlators of all possible local J ¼ 0 and J ¼ 1 bilinears in high temperature lattice QCD. We use N F ¼ 2 flavors of domain wall fermions and study temperatures up to T ∼ 960 MeV. Above the chiral restoration crossover at a pseudocritical temperature T c ∼ 175 MeV, we observe restoration of chiral SUð2Þ L × SUð2Þ R symmetry for all studied temperatures. While Uð1Þ A symmetry is present in all ensembles above 260 MeV, its restoration at 220 MeV is observed on the finest lattice solely. In the range between T ∼ 220 MeV and T ∼ 500 MeV we observe the formation of multiplets in spatial correlators that indicate larger emergent symmetries described by the chiral spin SUð2Þ CS and SUð4Þ groups with the breaking effects below 5% as measured by κ. These symmetries include the chiral Uð1Þ A and SUð2Þ L × SUð2Þ R groups as well as transformations that mix the right-and left-handed components of quarks as subgroups. These are not symmetries of the free Dirac action but are symmetries of the fermionic charge. In a given reference frame, which in our case is the medium rest frame, the quark-chromoelectric interaction is invariant under both SUð2Þ CS and SUð4Þ transformations, while the quark-chromomagnetic interaction and the quark kinetic term break them.
The emergence of these symmetries in the T ∼ 220-500 MeV window (1.2T c -2.8T c ) suggests that the chromomagnetic interaction between quarks is screened at these temperatures, while the confining chromoelectric interaction is still active. The emergence of approximate SUð2Þ CS and SUð4Þ symmetries in the window T ∼ 220-500 MeV is the principal result of our study. These emergent symmetries are incompatible with the picture of free, deconfined quarks and suggest that the physical d.o.f. are chirally symmetric quarks bound by the chromoelectric interaction without chromomagnetic effects. The latter conclusion is based entirely on our lattice observations and the symmetry classification of the QCD Lagrangian; i.e., it is model independent. We remark that correlation functions with the SUð2Þ CS and SUð4Þ symmetries cannot be analyzed perturbatively because perturbation theory reflects the symmetries of the free Dirac equation.
While we do not advocate any microscopic description of these ultrarelativistic objects, they are reminiscent of "strings." A string is the only known mathematical description of purely electric, relativistic objects, though a consistent theory of a relativistic string with quarks at the ends is missing in four dimensions. We refer to the SUð2Þ CS and SUð4Þ symmetric regime at temperatures T ∼ 220-500 as the "stringy fluid" to emphasize the possible nature of the objects-chirally symmetric quarks bound by the electric field.
At temperatures above T ∼ 600 MeV these symmetries disappear and the QCD correlation functions approach the correlators calculated with free, noninteracting quarks. This suggests that only at temperatures T ∼ 1 GeV and above hot QCD matter can be approximately described as a gas of weakly interacting quarks and gluons-the quark-gluon plasma (QGP).
Our analysis of spatial correlators and their multiplet structure suggests the following three regimes of QCD when increasing the temperature: At low temperatures up to the pseudocritical temperature T c QCD matter is a hadron gas where all chiral symmetries are broken by the nonzero quark condensate. From the hadron gas regime below T c there is a crossover to a regime with approximate SUð2Þ CS chiral spin symmetry, where quarks are predominantly bound by the chromoelectric interaction. This crossover coincides or is close to the chiral SUð2Þ L × SUð2Þ R restoration crossover (while in our setup the chiral crossover is at T c ∼ 175 MeV, for three-flavor QCD the chiral crossover is at a somewhat lower temperature of 155 MeV [36]). In the range T ∼ 500-660 MeV (2.8T c -3.8T c ) there is a fast increase of symmetry breaking: the confining electric interaction becomes small relative to the quark kinetic term. Finally, up to T ∼ 1 GeV (5.7T c ) there is an evolution to a weakly interacting QGP, where the relevant symmetries are the full set of chiral symmetries. Figure 6 provides an illustrative sketch of this temperature evolution for the effective d.o.f. of QCD. We note that the temperature range, in which the most drastic changes of thermodynamical bulk quantities occur, coincides qualitatively with the stringy fluid regime; see, e.g., Fig. 4 of Ref. [8].