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 Tc quarks deconfine and chiral symmetry is restored such that above Tc 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 Tc 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 Tc. In pure SU(3) gauge theory the Polyakov loop is connected with the Z3 center symmetry and indeed a sharp first-order phase transition is observed [4], which indicates that the relevant d.o.f. below and above Tc are different. Still, one may ask whether this Z3 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 Z3 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 Z3 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 Tc; see, e.g., Refs. [9–11]. 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 [12–20]. 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=0 and J=1 isovector correlation functions in the z direction for a system with NF=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 U(1)A restoration in t-correlators and via the Dirac eigenvalue decomposition of correlators [22,23]. We have observed the restoration of both SU(2)L×SU(2)R and U(1)A chiral symmetries above Tc 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 SU(2)CS chiral spin and SU(4) symmetries [24,25], have been identified in the J=1 correlators in the region T∼2Tc. 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 [26–29] upon artificial truncation of the near-zero modes of the Dirac operator [30]. While the SU(2)L×SU(2)R and U(1)A chiral symmetries are almost exact above Tc, the SU(2)CS and SU(4) 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 SU(2)CS and SU(4) 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 (1.2Tc–2.8Tc), 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 SU(2)CS and SU(4) symmetric [31].

When increasing the temperature to T∼1  GeV we observe that at very high temperature the SU(2)CS and SU(4) 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].

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 Tc 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 CΓ(z)=∫-∞∞dx∫-∞∞dy∫0βdt⟨OΓ(x,y,z,t)OΓ(0,0,0,0)†⟩.(1)We consider Euclidean space at finite temperature, i.e., x,y,z∈R, and t∈[0,β), 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 OΓ(x)≡u¯(x)Γd(x),OΓ(0)†≡-d¯(0)Γ†u(0),(2)where we use the abbreviations x=(x,y,z,t) and 0=(0,0,0,0), 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 OΓ† is a definition, since in general the sign obtained from conjugation depends on Γ. Throughout the whole paper we use the set γμ, μ=1, 2, 3, 4 of Euclidean γ-matrices that satisfy the anticommutation relations γμγν+γνγμ=2δμν,γ5≡γ1γ2γ3γ4.(3)u¯(x), u(x), d¯(x), d(x) 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 q(x)≡(u(x),d(x)). After contracting the fermions, the two forms for writing the nonsinglet bilinears of course give the same expressions.

Performing these contractions we obtain ⟨OΓ(x)OΓ(0)†⟩=Tr[S(x,0)Γ†S(0,x)Γ],(4)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 S(x,x′)=1(2π)3β∫-∞∞dpx∫-∞∞dpy∫-∞∞dpz∑n∈Zipp2eip(x-x′),(5)where p=(px,py,pz,ωn), with the Matsubara frequencies ωn=π(2n+1)/β. Inserting (5) into (4) and this into (1) we find CΓ(z)=-1(2π)4β∫-∞∞dpx∫-∞∞dpy∑n∈Z∫-∞∞dpzeizpzp2⁢∫-∞∞dpz′e-izpz′p˜2Tr[pΓ†p˜Γ],(6)where p˜≡(px,py,pz′,ωn) 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 Tr[pΓ†p˜Γ]=4[sxpx2+sypy2+szpzpz′+sτωn2],(7)where sx, sy, sz and sτ 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 pzpz′ appears in the integrand or not, ∫-∞∞dpzeizpzpz2+Ω2∫-∞∞dpz′e-izpz′pz′2+Ω2=[∫-∞∞dpzeizpzpz2+Ω2]2≡I02,(8)∫-∞∞dpzeizpzpzpz2+Ω2∫-∞∞dpz′e-izpz′pz′pz′2+Ω2=-[∫-∞∞dpzeizpzpzpz2+Ω2]2≡-I12,(9)where we have defined Ω=px2+py2+ωn2. The integrals I0 and I1 are straightforward to solve with the residue theorem, I02=π2Ω2e-z2Ω,-I12=π2e-z2Ω.(10)We find for the correlator CΓ(z), CΓ(z)=-[sxCs(z)+syCs(z)+szCz(z)+sτCτ(z)],(11)with the individual correlators given by Cs(z)=1(2π)2β∑n∈Z∫-∞∞dpx∫-∞∞dpye-2zpx2+py2+ωn2px2+py2+ωn2px2,Cz(z)=1(2π)2β∑n∈Z∫-∞∞dpx∫-∞∞dpye-2zpx2+py2+ωn2,Cτ(z)=1(2π)2β∑n∈Z∫-∞∞dpx∫-∞∞dpye-2zpx2+py2+ωn2px2+py2+ωn2ωn2.(12)The correlators Cs(z),Cz(z) and Cτ(z) obey the obvious sum rule 2Cs(z)+Cτ(z)=Cz(z);(13)i.e., only two of them are independent. We choose Cz(z) and Cτ(z) to express all other correlators. The treatment of the Matsubara sums and the necessary integrals for evaluating Cz(z) and Cτ(z) 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 CΓ(z). For this we need to determine the signs sx, sy, sz and sτ in the traces (7) for the different choices of Γ, which in turn determine how the respective correlator CΓ(z) is composed from the contributions Cs(z), Cz(z) and Cτ(z) according to (11).

We first note that for chiral partners, i.e., correlators where Γ is replaced by Γγ5, the corresponding correlators CΓ(z) and CΓγ5(z) have opposite overall signs, and thus also opposite individual signs sx, sy, sz and sτ. This follows from the trivial relation Tr[p(Γγ5)†p˜Γγ5]=-Tr[pΓ†p˜Γ].(14)This implies that we need to determine the signs sx, sy, sz and sτ in the traces (7) only for eight out of the 16 Clifford algebra generators Γ. Our results for the signs sx, sy, sz and sτ that determine the decomposition of Tr[pΓ†p˜Γ] according to (7) are listed in Table I.

Having determined the signs sx, sy, sz, sτ we use them in (11) to work out the composition of CΓ(z) from the building blocks Cs(z), Cz(z) and Cτ(z), and after eliminating Cs(z) we obtain the representation for the CΓ(z) in terms of Cz(z) and Cτ(z) evaluated in the Appendix. We find (overall signs were chosen such that chiral partners have the same overall sign) C1(z)=Cγ5(z)=2Cs(z)+Cz(z)+Cτ(z)=2Cz(z),Cγ1(z)=Cγ1γ5(z)=Cγ2(z)=Cγ2γ5(z)=Cz(z)+Cτ(z),Cγ4(z)=Cγ4γ5(z)=2Cs(z)+Cz(z)-Cτ(z)=2(Cz(z)-Cτ(z)),Cγ1γ3(z)=Cγ1γ3γ5(z)=Cγ2γ3(z)=Cγ2γ3γ5(z)=Cz(z)-Cτ(z),Cγ4γ3(z)=Cγ4γ3γ5(z)=-2Cs(z)+Cz(z)+Cτ(z)=2Cτ(z),Cγ3(z)=Cγ3γ5(z)=2Cs(z)-Cz(z)+Cτ(z)=0.(15)The vanishing of the correlators Cγ3(z) and Cγ3γ5(z) 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, Cγ3(z) is the correlator for the 3 component of the conserved vector current Jμ(x)=u¯(x)γμd(x) and concerning the propagation in z direction the integral ∫dxdydtJ3(x,y,z,t) 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 Cγ4(z)=Cγ4γ5(z) are not independent from the correlators Cγ1γ3(z)=Cγ1γ3γ5(z)=Cγ2γ3(z)=Cγ2γ3γ5(z).

We conclude this section by quoting the asymptotic behavior of our correlators, which is obtained by using (A5) from Appendix in the expressions (15) C1(z)=2πβ3e-2zω02zω0[1+12zω0]+O(e-4zω0zω0),Cγ1(z)=Cγ2(z)=2πβ3e-2zω02zω0[1+1(2zω0)2+⋯]+O(e-4zω0zω0),Cγ4(z)=4πβ3e-2zω0(2zω0)2[1-12zω0+⋯]+O(e-4zω0zω0),Cγ1γ3(z)=Cγ2γ3(z)=2πβ3e-2zω0(2zω0)2[1-12zω0+⋯]+O(e-4zω0zω0),Cγ4γ3(z)=2πβ3e-2zω02zω0[1-12zω0+⋯]+O(e-6zω0zω0),Cγ3(z)=0.(16)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 zω0=πz/β=πzT reflects the absence of any physical scale in the conformal theory of massless noninteracting quarks.

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 OΓ(x)=q¯(x)Γτ→2q(x),(17)which we now write using the isospin doublets q(x)≡(u(x),d(x)). The isovector structure of the bilinears is determined by the isospin Pauli matrices τa. 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=0 bilinears can be defined by the following choices for Γ: Γ={γ5…PS(pseudoscalar),1…S(scalar).(18)These two bilinears can be transformed into each other by global U(1)A rotations, q(x)→exp(iγ5θ)q(x).(19)

For J=1 we consider bilinears with the following choices of Γ that define the vector bilinears V, Γ={γ1…Vx,γ2…Vy,γ4…Vt.(vector).(20)As we have already seen for the free case that we discussed in the previous section, due to current conservation the 3 component q¯(x)γ3τ→2q(x) does not propagate in the z direction such that we omit the choice Γ=γ3.

The vector bilinears are related to their chiral partners through flavor nonsinglet axial rotations q(x)→exp(i2γ5τ→θ→)q(x).(21)Their chiral partners, the axial-vector bilinears A, are defined as Γ={γ1γ5…Ax,γ2γ5…Ay,γ4γ5…At.(axial vector).(22)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=γ3γ5, 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=γ3, also the choice Γ=γ3γ5 can be omitted.

The bilinears that correspond to the six tensor elements σμν of the Clifford algebra can be organized into two vector-valued objects, the tensor-vector T, Γ={γ1γ3…Tx,γ2γ3…Ty,γ4γ3…Tt,(tensor vector)(23)and the axial-tensor-vector X, Γ={γ1γ3γ5…Xx,γ2γ3γ5…Xy,γ4γ3γ5…Xt.(axial-tensor vector).(24)The bilinears T and X can be transformed into each other by the U(1)A rotations (19). Table II summarizes our bilinears and lists the U(1)A and SU(2)L×SU(2)R relations among them.

Due to the restoration of the U(1)A and SU(2)L×SU(2)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 U(1)A and SU(2)L×SU(2)R are the degeneracies required by chiral symmetries that emerge above Tc.

However, in addition to those, at temperatures not too far above Tc a larger group of symmetries, SU(2)CS and SU(4) that contain U(1)A and SU(2)L×SU(2)R [24,25], SU(2)CS⊃U(1)AandSU(4)⊃SU(2)L×SU(2)R×U(1)A,(25)has been observed in our previous study of correlators [21]. The SU(2)CS chiral spin transformations are defined by q(x)→exp(i2Σ→ε→)q(x),q¯(x)→q¯(x)γ4exp(-i2Σ→ε→)γ4,(26)where ε→∈R3 are the rotation parameters. For the generators Σ→ one has four different choices Σ→=Σ→k with k=1, 2, 3, 4, but, as we discuss below, only the cases k=1 and k=2 are of interest here. The generators are given by Σ→k={γk,-iγ5γk,γ5},(27)and the su(2) algebra is satisfied for any choice k=1, 2, 3, 4. While these are not symmetries of the Dirac Lagrangian, both in Minkowski and Euclidean space, the Lorentz-invariant fermion charge in Minkowski space, Q=∫d3xψ†(x)ψ(x),(28)is invariant under SU(2)CS, where ψ(x) can be either a single-flavor quark field or an isospin doublet. The Euclidean fermion charge is also SU(2)CS invariant.

In Minkowski space in a given reference frame the quark-gluon interaction can be split into temporal and spatial parts, ψ¯γμDμψ=ψ¯γ0D0ψ+ψ¯γiDiψ,(29)where Dμψ=(∂μ-igt·Aμ2)ψ.(30)The temporal term includes the interaction of the color-octet charge density ψ¯(x)γ0t2ψ(x)=ψ(x)†t2ψ(x),(31)with the chromoelectric component of the gluonic field. It is invariant under SU(2)CS [25]. We emphasize that the SU(2)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 SU(2)CS. In other words, the SU(2)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 SU(2)CS transformations (26) with k=1 generate the following two SU(2)CS singlets and two SU(2)CS triplets of bilinears: (Vy);(Ay,Tt,Xt),(32)(Vt);(At,Ty,Xy).(33)These irreducible representations of SU(2)CS can be obtained by applying the SU(2)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 SU(2)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 quark-antiquark 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

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 Vt, At and Tx, Ty, Xx, Xy bilinears [see, e.g., Eq. (16)] that would allow one to distinguish the results for free quarks from the full SU(2)CS case in an elaborated calculation with properly renormalized full QCD correlators.

The transformations (26) with k=2 generate the following singlets and triplets: (Vx);(Ax,Tt,Xt),(34)(Vt);(At,Tx,Xx).(35)Again, a degeneracy of the correlators built from the triplet bilinears in Eq. (34) is a signal for the emergence of the SU(2)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 SU(4) 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 SU(2)CS,k=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=1, 2 transformations, as indicated above.

We remark that at zero temperature in the continuum there is a SO(3) symmetry in the x, y, t subspace and the z-correlators of the Vx, Vy, Vt 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 SO(2) symmetry which connects the correlators of the spatial components Vx↔Vy and Ax↔Ay et cetera. On the lattice the reduced symmetry for the T>0 case and the z=const subspace is D4h and the relevant symmetry is S2×SU(2)CS [21],²

S2 here denotes the permutation or symmetric group for x↔y interchanges.

Finally we remark that the group SU(2)CS⊗SU(2)F, where SU(2)F is the isospin symmetry group, can be extended to SU(4) with fifteen generators, {(τ→⊗1D),(1F⊗Σ→k),(τ→⊗Σ→k)}.(38)The corresponding transformations are a trivial generalization of Eq. (26) obtained by replacing the generators Σ→ by those listed in (38). Also the group SU(4) 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 S2×SU(4) transformations connect the following J=1 operators from Table II: (Vx,Vy,Ax,Ay,Tt,Xt),(39)(Vt,At,Tx,Ty,Xx,Xy).(40)

These are the multiplets of the isovector operators that are discussed in the present paper. The SU(4) 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 S2×SU(4) multiplets in addition also include the isoscalar partners of Ax, Ay, Tt and Xt in Eq. (39) as well as the isoscalar partners of At, Tx, Ty, Xx and Xy in Eq. (40). The isoscalar partners of Vx, Vy and Vt are the SU(4) singlets.

The correlators discussed in the previous section are evaluated on the JLQCD configurations for full QCD with NF=2 flavors of domain wall fermions. Details concerning the gauge configurations are presented in [22,23]. In this setup we choose L5, 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 βg in a range between βg=4.1 and βg=4.5, and with the different temporal lattice extents in use, Nt=4, 6, 8 and Nt=12, we cover a range of temperatures between T≃220  MeV and T≃960  MeV. For the bare quark mass parameters mu=md≡mud we use the value mud=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 mud=0.01, mud=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 zT=(nza)/(Nta)=nz/Nt,(41)where z is the physical distance in the correlators, T the temperature, a the lattice constant, nz the distance in lattice units and Nt 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 CΓ(nz)=∑nx,ny,nt⟨OΓ(nx,ny,nz,nt)OΓ(0,0)†⟩.(42)Obviously this is the lattice version of the continuum form in Eq. (1).

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=nz/Nt [compare to Eq. (41)] using the full range of nz values—up to periodicity. In order to compare different correlators without a proper renormalization, our correlators are normalized to 1 at nz=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., 1.2Tc. All correlation functions of chiral partners are degenerate within errors. In detail, these are the two pairs (Vx,Ax) and (Vt,At), each of which reflects SU(2)R×SU(2)L symmetry. U(1)A symmetry in the vector channel, represented by the operator pairs (Tx,Xx) and (Tt,Xt), is manifest for all ensembles. For the scalar (PS,S) pair we find the restoration of U(1)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 detailed studies of U(1)A symmetry around Tc, see e.g., [35] or [23]. The latter study uses the same simulation setup as the present work.

For temperatures between T∼220 and 500 MeV the correlators are grouped into three distinct multiplets,⁴

Note that in E2 and E3 we leave out the y components, which are exactly degenerate with the respective x components explicitly listed in E2 and E3.

The formation of the multiplet E3 is not necessarily a consequence of the SU(2)CS and SU(4) 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 E3 multiplet alone we could not claim the emergence of the SU(2)CS and SU(4) symmetries. However, the E2 degeneracy is not manifest in the free quark system (15) and indeed can be attributed to the emergent SU(2)CS and SU(4) 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 U(1)A and SU(2)L×SU(2)R transformations are indistinguishable at all temperatures. At temperatures above T∼600  MeV we observe that the distinct multiplet E2, related to emergence of the SU(2)CS and SU(4) symmetries, is washed out. The remaining E3 multiplet structure can be attributed to quasifree quarks.

In Fig. 2 we now focus on the E1 and E2 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 U(1)A and SU(2)L×SU(2)R symmetries.

For the lowest temperature T∼220  MeV we still observe a small residual splitting within the E2 multiplet, while at T∼380  MeV the difference nearly vanishes. Furthermore, there is a clear splitting between the E1 and E2 multiplets indicating SU(2)CS and SU(4) 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 U(1)A and SU(2)L×SU(2)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 E2 multiplet. The ratios are plotted as a function of the dimensionless quantity zT=nz/Nt and we compare different temperatures.

In the lhs plot we show the ratio CXt/CTt. The two correlators are related by U(1)A and a deviation from a constant ratio 1 indicates a violation of U(1)A. The data show no breaking effects within errors.

In the rhs plot we show the ratio CAx/CTt. These two correlators are related by SU(2)CS and thus a deviation from 1 indicates a violation of exact SU(2)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 SU(2)CS sensitive ratio CAx/CTt 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 free quarks at the highest temperature, i.e., T∼960  MeV. For intermediate temperatures we observe small deviations from 1.