Chiral-spin symmetry of the meson spectral function above T c

Recently, via calculation of spatial correlators of J = 0, 1 isovector operators using a chirally symmetric Dirac operator within NF = 2 QCD, it has been found that QCD at temperatures Tc− 3Tc is approximately SU(2)CS and SU(4) symmetric. The latter symmetry suggests that the physical degrees of freedom are chirally symmetric quarks bound by the chromoelectric field into color singlet objects without chromomagnetic effects. This regime of QCD has been referred to as a Stringy Fluid. Here we calculate correlators for propagation in time direction at a temperature slightly above Tc and find the same approximate symmetries. This means that the meson spectral function is chiral-spin and SU(4) symmetric. 1 ar X iv :1 90 9. 00 92 7v 1 [ he pla t] 3 S ep 2 01 9


I. INTRODUCTION
Artificial truncation of the near-zero modes of the Dirac operator at zero temperature results in the emergence of a large degeneracy in the hadron spectrum, larger than implied by the chiral symmetry of the QCD Lagrangian [1][2][3][4]. A symmetry group of this degeneracy, the chiral-spin SU (2) CS group and its flavor extension SU (2N F ), contains chiral symmetries as subgroups [5,6]. These symmetries are not symmetries of the Dirac Lagrangian. However they are symmetries of the electric interaction in a given reference frame, while the magnetic interaction as well as the quark kinetic term break them. Consequently these symmetries allow us to separate the electric and magnetic interactions in a given frame. The emergence of the SU (2) CS and SU (2N F ) symmetries in the hadron spectrum upon the low mode truncation means that while the confining chromoelectric interaction is distributed among all modes of the Dirac operator, the chromomagnetic interaction contributes only (or at least predominantly) to the near-zero modes. Some unknown microscopic dynamics should be responsible for this phenomenon.
At high temperatures, above the pseudocritical temperature T c , chiral symmetry is restored due to the near-zero modes of the Dirac operator being naturally suppressed by temperature effects [7][8][9][10]. Then one could expect a natural emergence of the SU (2) CS and SU (2N F ) symmetries in QCD above T c [11].
In [12,13] we have studied a complete set of J = 0 and J = 1 isovector correlation functions in z-direction for a system with N F = 2 dynamical quarks in simulations with the chirally symmetric domain wall Dirac operator at temperatures up to 5.5T c . 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 [9,14]. We have observed emergence of approximate SU (2) CS and SU (4) symmetries in the spatial correlators in the temperature range T c − 3T c . While the spatial correlators do not have a special physical meaning, their symmetries at T c − 3T c do reflect symmetries of the QCD action since the correlation functions are driven only by the action of the theory. Observation of approximate SU (2) CS and SU (4) symmetries at T c − 3T c suggests that the physical degrees of freedom in this temperature range are chirally symmetric quarks bound by the chromoelectric field into color-singlet compounds without the chromomagnetic effects. Such a system is reminiscent of a "string", that is why the corresponding regime of QCD at T c − 3T c is referred to as a Stringy Fluid.
The chemical potential term in the QCD action has precisely the same symmetries [15], so one can expect that the symmetries observed in the lattice calculations at zero chemical potential will persist at µ > 0 as well.
Correlators along the time direction have a direct physical meaning since they are connected to the spectral density in Minkowski space via an integral transformation. Observation of the SU (2) CS and SU (4) symmetries in t-correlators would imply that the spectra of the corresponding color-singlet states in Minkowski space have the same symmetry. The symmetries of the z-correlators do suggest the same symmetries in the spectra, albeit indirectly. A direct observation of these symmetries in t-correlators in practice is a priori not obvious since on the lattice one has only a few lattice sites along the time direction at high T and large discretization errors as well as a small evolution time can easily spoil the real picture. Here we use N t = 12 ensembles at T = 1.2T c and observe very clear SU (2) CS and SU (4) symmetries in t-correlators. This implies that the corresponding spectral functions in Minkowski space are also SU (2) CS and SU (4) symmetric.
The SU (2) CS and SU (2N F ) groups are not symmetries of the Dirac equation as well of the QCD Lagrangian as a whole. In a given reference frame the quark-gluon interaction Lagrangian in Minkowski space can be split into temporal and spatial parts: Here D µ is a covariant derivative that includes interaction of the quark field ψ with the gluon field A µ , The temporal term includes an interaction of the color-octet charge densitȳ with the electric part of the gluonic gauge field. It is invariant under any unitary transformation acting in the Dirac and/or flavor spaces. In particular it is a singlet under SU (2) CS and SU (2N F ) groups. The spatial part consists of a quark kinetic term and interaction with the magnetic part of the gauge field. It breaks SU (2) CS and SU (2N F ). We conclude that interaction of electric and magnetic components of the gauge field with fermions can be distinguished by symmetry.
In order to discuss the notions "electric" and "magnetic" one needs to fix a reference frame. The invariant mass of the hadron is the rest frame energy. Consequently, to discuss physics of hadron mass generation it is natural to use the hadron rest frame.
The spectral density ρ(ω) is an integral transform of the rest frame t-direction Euclidean correlator where O Γ (x, y, z, t) is an operator that creates a quark-antiquark pair for mesons with fixed quantum numbers. Transformation properties of the local J = 1 quark-antiquark bilinears O Γ (x, y, z, t) with respect to SU (2) L × SU (2) R and U (1) A are given on the left side of Fig. 1 and those with respect to SU (2) CS , k = 4 and SU (4) on the right side of Fig. 1 [6]. Emergence of the respective symmetries is signalled by the degeneracy of the correlators (9) calculated with operators that are connected by the corresponding transformations.

III. METHODOLOGY
The lattice data presented in the next section is calculated on JLQCD gauge configurations with N F = 2 fully dynamical domain wall fermions ( [9,16]). The length of the fifth dimension for the fermions is chosen as L 5 = 16, to ensure good chiral symmetry [14].
where the Γ structures from Fig. 1 determine the resulting quantum numbers. To extract correlation functions of states with definite, i.e. zero, momentum, we perform a momentum projection according to Eq.( 9).
Finally, the data shown in the next section is rescaled to a dimensionless variable where t is the measured lattice distance in time direction, T the temperature, a the lattice constant, and N t the overall temporal lattice extent. For spatial correlators in z-direction the same rescaling is done with z = n z a instead of t.

IV. RESULTS
On the right side of Fig. 2 we show t-correlators (9) that measures the splitting within the SU (2) CS multiplet relative to the distance between different multiplets. With this definition, good symmetry implies |κ| 1.
The degree of the symmetry breaking obviously depends on the dimensionless variable tT . At tT ∼ 0.5 the breaking is tiny, as can be seen from Fig. 3. For the noninteracting quarks there is no SU (2) CS symmetry and in infinite volume |κ| ∼ 1 [13]. It is instructive to compare the scale dependence of the symmetry breaking parameter κ extracted from the t-correlators and from the z-correlators [13] since the t-and z-correlators probe QCD at different dimensionless "distance" tT , zT (the time extent of the lattice is smaller than its spatial extent). Apriori one would expect that at the same tT = zT a degree of the symmetry breaking would be similar as measured by the t-and z-correlators. As it can be seen from Fig. 3, both measurements show the same level of symmetry breaking.

V. CONCLUSIONS
We have calculated meson rest-frame correlators of J = 0 and J = 1 isovector operators along the time-direction with N F = 2 QCD with physical masses with the chirally symmetric domain wall Dirac operator at T = 1.2T c . We have observed a very clear emergence of approximate chiral-spin SU (2) CS and SU (4) symmetries in these correlaros. The t-correlators are connected via an integral transform with the measurable spectral density in Minkowski space. Approximate SU (2) CS and SU (4) symmetries of the t-correlators imply the same symmetries of spectral densities. This result reinforces our findings in Refs. [12,13].
These symmetries are incompatible with free deconfined quarks and suggest that the physical degrees of freedom are chirally symmetric quarks bound into color-singlet compounds by the chromoelectric field without the chromomagnetic effects. This result is model-independent and relies solely on lattice results and symmetry classification of the QCD Lagrangian. Such relativistic objects are reminiscent of "strings" since they are purely electric and we refer to the corresponding regime of QCD as a Stringy Fluid.