^{1}

^{3}

^{4}

^{5}

^{1}

^{1}

^{2,6}

^{1}

^{7,8,9}

^{3}.

Based on a complete set of

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

A flavor nonsinglet chiral restoration was indeed confirmed on the lattice, which is signalled by the vanishing quark condensate above the crossover region around

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

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

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

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

In previous work

However, by analyzing the formation of multiplets for the spatial correlators, even larger symmetries, referred to as

We stress that the

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

When increasing the temperature to

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

In the continuum the free spatial meson correlators in infinite spatial volume are given by

The meson interpolators are given by

Performing these contractions we obtain

As we see below, the trace in the integrand has the general form

We now come to the identification of multiplets, i.e., we identify the sets of Clifford algebra elements

We first note that for chiral partners, i.e., correlators where

The signs

Having determined the signs

We conclude this section by quoting the asymptotic behavior of our correlators, which is obtained by using

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

Two

For

The vector bilinears are related to their chiral partners through flavor nonsinglet axial rotations

The bilinears that correspond to the six tensor elements

Fermion bilinears considered in this work and their transformation properties (last column). This classification assumes propagation in the

Due to the restoration of the

However, in addition to those, at temperatures not too far above

In Minkowski space in a given reference frame the quark-gluon interaction can be split into temporal and spatial parts,

The

Note that the observation of a degeneracy of correlators for the triplet bilinears in Eq.

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

The transformations

This discussion [as well as a structure of the

We remark that at zero temperature in the continuum there is a

Finally we remark that the group

These are the multiplets of the isovector operators that are discussed in the present paper. The

The complete

The correlators discussed in the previous section are evaluated on the JLQCD configurations for full QCD with

For measurements the IroIro software is used

We use the Symanzik-improved gauge action at inverse gauge couplings

Ensembles and their parameters: We list the lattice size, the inverse gauge coupling

As already discussed, we measure finite temperature spatial correlators in the

We project to zero momentum by summing over all lattice sites in slices orthogonal to the

In Fig.

Overview of our spatial correlators in a wide range of temperatures. The correlators are shown as a function of the dimensionless combination

The top left panel of Fig.

For detailed studies of

For temperatures between

Note that in

The formation of the multiplet

We speak of separate multiplets when the splittings within the multiplets are much smaller than splittings between different multiplets. All correlators connected by chiral

In Fig.

Correlation functions of the bilinears in the

For the lowest temperature

At the highest temperature of this study,

In an attempt to discuss the observed evolution of symmetries more quantitatively, in Figs.

Ratios of normalized correlators for different bilinears from the

The ratio

In the lhs plot we show the ratio

In the rhs plot we show the ratio

Finally, in Fig.

Figures

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

In Fig.

The symmetry breaking parameter

At temperatures between

We stress once more that the

We stress that the emerging

This view is also reflected in the exponential decay properties, i.e., the factors

In this paper we have studied spatial correlators of all possible local

In the range between

The emergence of these symmetries in the

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

At temperatures above

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

Illustrative sketch for the temperature evolution of the QCD effective d.o.f. as suggested by the changing symmetry content manifest in our spatial correlators.

Support from the Austrian Science Fund (FWF) through Grants No. DK W1203-N16 and No. P26627-N27, as well as from NAWI Graz is acknowledged. The numerical calculations were performed on the Blue Gene/Q at KEK under its Large Scale Simulation Program (Grant No. 16/17-14), at the Vienna Scientific Cluster (VSC) and at the HPC cluster of the University of Graz. This work is supported in part by JSPS KAKENHI Grant No. JP26247043 and by the Post-K supercomputer project through the Joint Institute for Computational Fundamental Science (JICFuS). S. P. acknowledges support from ARRS (Grants No. J1-8137 and No. P1-0035) and DFG (Grant No. SFB/TRR 55).

All free spatial continuum correlators that we discuss in Sec.

For solving the first integral

We conclude this Appendix by quoting the asymptotic forms for the integrals