Three regimes of QCD

TL;DR

This paper identifies three distinct regimes of QCD based on symmetries and degrees of freedom, revealing a stringy fluid phase characterized by electric interactions and chiral symmetry above the pseudocritical temperature.

Contribution

It demonstrates that QCD exhibits three regimes with different symmetries and properties, including a novel stringy fluid phase with electric interactions and chiral symmetry, above the pseudocritical temperature.

Findings

QCD has three regimes: Hadron Gas, Stringy Fluid, Quark-Gluon Plasma.

Emergence of SU(2)_{CS} and SU(2N_F) symmetries above Tc.

Stringy Fluid is composed of chirally symmetric quarks bound by chromoelectric fields.

Abstract

While the QCD Lagrangian as the whole is only chirally symmetric, its electric part has larger chiral-spin SU(2)_{CS} and SU(2N_F) symmetries. This allows separation of the electric and magnetic interactions in a given reference frame. Artificial truncation of the near-zero modes of the Dirac operator results in the emergence of the SU(2)_{CS} and SU(2N_F) symmetries in hadron spectrum. This implies that while the confining electric interaction is distributed among all modes of the Dirac operator, the magnetic interaction is located at least predominantly in the near-zero modes. Given this observation one could anticipate that above the pseudocritical temperature, where the near-zero modes of the Dirac operator are suppressed, QCD is SU(2)_{CS} and SU(2N_F) symmetric, which means absence of deconfinement in this regime. Solution of the N_F=2 QCD on the lattice with a chirally symmetric Dirac operator reveals that indeed in the interval Tc - 3Tc QCD is approximately SU(2)_{CS} and SU(2N_F) symmetric which implies that degrees of freedom are chirally symmetric quarks bound by the chromoelectric field into color-singlet objects without the chromomagnetic effects. This regime is referred to as a Stringy Fluid. At larger temperatures this emergent symmetry smoothly disappears and QCD approaches the Quark-Gluon Plasma regime with quasifree quarks. The Hadron Gas, the Stringy Fluid and the Quark-Gluon Plasma differ by symmetries, degrees of freedom and properties.

Equations

$$\left(\begin{array}[]{c}R\\ L\end{array}\right)\rightarrow\exp\left(i\frac{\varepsilon^{n}\sigma^{n}}{2}\right)\left(\begin{array}[]{c}R\\ L\end{array}\right)\;.$$

$$\psi\rightarrow\psi^{\prime}=\exp\left(i\frac{\varepsilon^{n}\Sigma^{n}}{2}\right)\psi=\exp\left(i\frac{\varepsilon^{n}\sigma^{n}}{2}\right)\left(\begin{array}[]{c}R\\ L\end{array}\right)\;,$$

$${\Sigma }^n=\{{\gamma }_0,-i{\gamma }_5{\gamma }_0,{\gamma }_5\},$$

$$[{\Sigma }^a,{\Sigma }^b]=2i{ϵ}^{abc}{\Sigma }^c.$$

$${\Sigma }^n=\{{\gamma }_k,-i{\gamma }_5{\gamma }_k,{\gamma }_5\},$$

$${\gamma }_i{\gamma }_j+{\gamma }_j{\gamma }_i=2{\delta }^{ij};{\gamma }_5={\gamma }_1{\gamma }_2{\gamma }_3{\gamma }_4.$$

$$\psi \to {\psi }^{\prime }=\exp \left(i\frac{{ϵ}^m{T}^m}{2}\right)\psi ,$$

$$T^m=\{({\tau }^a\otimes 1_D),(1_F\otimes {\Sigma }^n),({\tau }^a\otimes {\Sigma }^n)\}$$

$$\Psi =\left(\begin{array}{c}{u}_{\text{textsc}R}\\ u_{\text{textsc}L}\\ d_{\text{textsc}R}\\ d_{\text{textsc}L}\end{array}\right).$$

$$Q=\int d^{3}x\bar{\psi }(x){\gamma }_0\psi (x)=\int d^{3}x{\psi }^{†}(x)\psi (x).$$

$$\overline{\psi }{\gamma }^{\mu }{D}_{\mu }\psi =\overline{\psi }{\gamma }^0{D}_0\psi +\overline{\psi }{\gamma }^i{D}_i\psi ,$$

$$D_{\mu }\psi =({\partial }_{\mu }-ig\frac{\mathbf{t}\cdot {\mathbf{A}}_{\mu }}{2})\psi .$$

$$\bar{\psi }(x){\gamma }^0\frac{\mathbf{t}}{2}\psi (x)=\psi (x{)}^{†}\frac{\mathbf{t}}{2}\psi (x)$$

$$<\bar{q}q>=-\pi \rho (0).$$

$$i{\gamma }_{\mu }{D}_{\mu }{\psi }_n(x)={\lambda }_n{\psi }_n(x).$$

$$S=S_{Full}-\sum _{i=1}^k\frac{1}{{\lambda }_i}|{\lambda }_i⟩⟨{\lambda }_i|,$$

$$C_{\Gamma }(n_z)=\sum _{n_x,n_y,n_t}⟨{\mathcal{O}}_{\Gamma }(n_x,n_y,n_z,n_t){\mathcal{O}}_{\Gamma }(0,0{)}^{†}⟩$$

$$zT=(n_{z}a)/(N_{t}a)=n_z/N_t,$$

$$\kappa =\frac{|C_{A_x}-C_{T_t}|}{|C_{A_x}-C_S|}.$$

$$C_{\Gamma }(t)=\sum _{x,y,z}⟨{\mathcal{O}}_{\Gamma }(x,y,z,t){\mathcal{O}}_{\Gamma }(0,0{)}^{†}⟩,$$

$$S={\int }_0^{\beta }d\tau \int d^{3}x\overline{\psi }[{\gamma }_{\mu }{D}_{\mu }+\mu {\gamma }_4+m]\psi ,$$

Full text

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1 Introduction

In this minireview we overview recent results on regimes and symmetries of QCD at different temperatures. These results are based only upon a recently discovered new symmetry of electric interactions in the QCD lagrangian and its observation in the QCD correlation functions measured on the lattice at different teperatures. Hence all conclusions are model independent. It is a standard lore that very soon after the Big Bang the Universe was a very hot Quark-Gluon Plasma (QGP) that upon cooling is frosen into hadrons at a pseudocritical temperature of $T_{c}\sim 155$ MeV. Here this wisdom is challenged and it is argued that at temperatures $T_{c}-3T_{c}$ the hot QCD is not a QGP but a matter with elementary degrees of freedom being not liberated quarks and gluons but rather chirally symmetric quarks bound by the chromoelectric field into ”hadron-like” color-singlet objects.

Asymptotic freedom predicts that at very high temperatures the strong coupling constant should be sufficiently small so that the degrees of freedom would be liberated quarks and gluons [1]. Such a state of matter was called the Quark-Gluon Plasma (QGP) [2]. For free quarks chiral symmetry is manifest, which is in contrast to the hadronic phase where it is very strongly dynamically broken. Due to small coupling constant one can anticipate that the weak coupling expansion, i.e. the expansion around the point where quarks and gluons are free particles, should correctly predict properties of QGP at the sufficiently high temperatures. It was found however very soon by Linde [3] that at high temperatures even with the small coupling constant there cannot be completely free deconfined quarks and gluons because of the infrared divergences that inevitably appear at some order of perturbation theory. This problem has not been overcome even today. Inspite of this there was a great expectation of the community that above some critical temperature $T_{c}$ QCD is deconfined with restored chiral symmetry so that the degrees of freedom are free quarks and gluons - the QGP phase. A lot of experimental and theoretical efforts were (are) invested to discover the Quark-Gluon Plasma.

The chiral restoration around $T_{c}$ was indeed observed on the lattice, which is signalled by the vanishing of the quark condensate and by the degeneracy of correlators connected by the chiral transformation [4]. This restoration happens during a crossover around the pseudocritical temperature $T_{c}\sim 155$ MeV. The expected confinement - deconfinement transition turned out to be much more difficult to define. For many years such a transition was believed to be associated with different behavior of the Polyakov loop below and above $T_{c}$ [5, 6]. The Polyakov loop is a true order parameter for the $Z_{3}$ symmetry of pure gluodynamics and in this case a clear first-order phase transition is observed. If one interprets the Polyakov loop as a free energy of a static charge, then a vanishing Polyakov loop below $T_{c}$ would correspond to a charge with infinite free energy, while a finite Polyakov loop above $T_{c}$ would represent a charge with a finite free energy. In QCD with dynamical quarks there is no $Z_{3}$ symmetry and the Polyakov loop looses its meaning as an order parameter: a static charge interacts with dynamical quarks at any temperature and its free energy is not infinite. Indeed in QCD with light quarks one observes nowadays a very smooth increase of the Polyakov loop beginning from zero temperature, see Fig  1 [7].

Obviously no specific temperature of ”deconfinement transition” can be defined.

A flattening of a potential between the static sources (the Polyakov loop correlator), that is often considered as a sign of the Debye screening and deconfinement, is actually induced by the ”string breaking”, i.e. by production of two heavy-light mesons. This can be clearly seen from Fig.  2, where such a potential is shown for $N_{f}=2+1$ at physical quark masses [8]. The extracted potential with obvious flattening is exactly the same for temperatures below and above $T_{c}$. I.e. it cannot be related do ”deconfinement”.

There is no reliable and accepted definition and order parameter for deconfinement except for the most straightforward one: confinement is the absence of color states in the spectrum. It is also very well established on the lattice that only the color-singlet objects survive the gauge averaging and consequently only the color-singlet objects can freely propagate a macroscopical distance at any temperature.

Tremendous experimental efforts at RHIC and LHC have supplied us with evidences that indeed at temperatures above $T_{c}\sim 155$ MeV some properties of the QCD matter are different as compared to the hadronic matter at small temperatures (perfect fluidity, jet quenching). This, together with the observation on the lattice of the pseudocritical temperature was a basis for a declaration of discovery of the QGP [9]. Even though that the pressure and the energy density immediately above $T_{c}$ are very far from the Stefan-Boltzmann limit that is associated with the free quarks and gluons, a general belief was that above $T_{c}$ one has the QGP phase with the liberated quarks and gluons that still undergo some sufficiently strong interactions - the strongly interacting Quark-Gluon Plasma (sQGP). A general justification was that when the energy density is of the order that of matter inside a proton (i.e., hadrons overlap) the only possibly is that quarks and gluons deconfine in the sense that they freely propagate within a macroscopical piece of matter. Such a picture, that was motivated by a simple MIT bag model view of hadrons, is implicitly based on the assumption that no other possibilities (than hadrons with broken chiral symmetry or liberated quarks and gluons) exist. Is it however true?

One of the pillars of the QGP was a very much spread opinion that only free quarks (and no quark bound states) can exist in the chirally symmetric phase and that hadron masses, such as of nucleons etc, are made at least mostly from the quark condensate of the vacuum. The latter view was a motivation to perform the following experiment on the lattice [10, 11]. The quark condensate of the vacuum is connected to the density of the near-zero modes of the Dirac operator via the Bancs-Casher relation [12]. On the lattice one can artificially remove the near-zero modes from the quark propagators. Chiral symmetry will be restored by construction. But what will happen with hadrons: will they survive and if yes what will be their masses? It turned out that hadrons (except for pions) do survive this artificial restoration of chiral symmetry and their masses remain large.

However not only degeneracies in the chiral multiplets were observed, but a much larger degeneracy was seen [13, 14, 15, 16], which was very surprising. Given quantum numbers of degenerate hadrons and their chiral transformation properties the unknown symmetry group was reconstructed. It turned out to be the $SU(2)_{CS}$ chiral-spin group and its flavor extension $SU(2N_{F})$ [17, 18]. The $SU(2)_{CS}$ chiral-spin group includes $U(1)_{A}$ as a subgroup and its transformations mix the right- and left-handed components of quarks. The $SU(2N_{F})$ group includes both the chiral symmetry $SU(N_{F})_{R}\times SU(N_{F})_{L}\times U(1)_{A}$ and $SU(2)_{CS}$ as subgroups. Obviously, these are not symmetries of the Dirac equation or of the QCD Lagrangian. However, they are symmetries of the Lorentz-invariant fermion charge and of the electric interaction in QCD. The magnetic interaction as well as the quark kinetic term break them. Consequently these symmetries allow us to distinguish between the electric and magnetic interactions in a given reference frame. This is a general property of any gauge-invariant theory with Dirac particles as a matter field.

The emergence of $SU(2)_{CS}$ and $SU(2N_{F})$ symmetries in hadrons upon truncation of the near-zero modes of the Dirac operator 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.

A definition of the $SU(2)_{CS}$ and $SU(2N_{F})$ symmetries as well as results on hadron spectrum upon truncation of the near-zero modes at zero temperature will be discussed in detail in sections 2 and 3.

It is known that at high temperatures, above the pseudocritical temperature, the chiral symmetry is restored because the near-zero modes are naturally suppressed by temperature. Also the $U(1)_{A}$ symmetry is at least approximately restored [19, 20]. Then, given the experience of the emergent symmetries upon the artificial truncation of the near-zero modes at zero temperature, one can expect a natural emergence of these symmetries above $T_{c}$ without any truncation [21]. That would imply that there are no free deconfined quarks above $T_{c}$.

This expectation has been tested in lattice simulations of full $N_{F}=2$ QCD with the chirally symmetric Domain Wall Dirac operator. Namely, a full set of spatial correlators with $J=0,1$ local meson operators has been calculated at temperatures $T_{c}-5.5T_{c}$ [22, 23]. Approximate degeneracy of meson correlators predicted by the $SU(2)_{CS}$ and $SU(2N_{F})$ symmetries was clearly observed in the temperature interval $T_{c}-3T_{c}$. The same symmetries have also been seen in temporal correlators that are directly connected to observable spectral density [24].

The lattice simulations are possible only at zero chemical potential. The chemical potential term in the QCD action is manifestly $SU(2)_{CS}$ and $SU(2N_{F})$ symmetric [25] . In other words, the observed property of QCD at zero chemical potential should also persist at finite chemical potentials.

These symmetries are incompatible with free deconfined quarks and imply that at these temperatures degrees of freedom are chirally symmetric quarks bound by the chromoelectric field without the chromomagnetic effects into color-singlet compounds. Such a system is reminiscent of a ”string”, that is why this $SU(2)_{CS}$ and $SU(2N_{F})$ symmetric regime of QCD was named Stringy Fluid [25]. At larger temperatures $T>3T_{c}$ these symmetries smoothly disappear and at temperatures $T\sim 5T_{c}$ only chiral symmetries survive so the full QCD correlators approach correlators calculated with free noninteracting quarks [23]. This implies that only at such high temperatures the strongly interacting matter can be approximately considered as a Quark-Gluon Plasma with the quasifree quarks and gluons as appropriate degrees of freedom. The Hadron Gas, the Stringy Fluid and the Quark-Gluon Plasma regimes are different by symmetries, degrees of freedom and properties.

Symmetries of QCD at different temperatures and their implications are addressed in detail in sections 4 and 5.

2 Chiral-spin symmetry

In ref. [17] the $SU(2)_{CS}$ chiral-spin transformation was defined as a transformation that rotates in the space of the right- and left-handed Weyl spinors

[TABLE]

So the fundamental irreducible representation of $SU(2)_{CS}$ is two-dimensional. In terms of the Dirac spinors $\psi$ the same transformation can be written via $\gamma$-matrices [18]

[TABLE]

where the generators $\Sigma^{n}$ of the four-dimensional reducible representation are

[TABLE]

with the $su(2)$ algebra

[TABLE]

The $U(1)_{A}$ group is a subgroup of $SU(2)_{CS}$.

In Euclidean space with the $O(4)$ symmetry all four directions are equivalent and the $SU(2)_{CS}$ transformations can be generated by any Euclidean hermitian $\gamma$-matrix $\gamma_{k}$, $k=1,2,3,4$ instead of Minkowskian $\gamma_{0}$:

[TABLE]

[TABLE]

The $su(2)$ algebra is satisfied with any $k=1,2,3,4$.

The direct product of the $SU(2)_{CS}$ group with the flavor group $SU(2)_{CS}\times SU(N_{F})$ can be embedded into a $SU(2N_{F})$ group. This group contains the chiral symmetry $SU(N_{F})_{L}\times SU(N_{F})_{R}\times U(1)_{A}$ as a subgroup. Its transformations are given by

[TABLE]

where $m=1,2,...,(2N_{F})^{2}-1$ and the set of $(2N_{F})^{2}-1$ generators is

[TABLE]

with $\tau$ being the flavor generators (with the flavor index $a$) and $n=1,2,3$ is the $SU(2)_{CS}$ index.

The fundamental vector of $SU(2N_{F})$ at $N_{F}=2$ is

[TABLE]

While the $SU(2)_{CS}$ and $SU(2N_{F})$ symmetries are not symmetries of the Dirac Lagrangian, they are symmetries of the Lorentz-invariant fermion charge

[TABLE]

The latter important feature allows us to use the $SU(2)_{CS}$ and $SU(2N_{F})$ symmetries to distinguish the electric and magnetic interactions in a given reference frame because the electric interaction is influenced only by the charge while the magnetic interaction is dictated by the spatial current. The latter current is not $SU(2)_{CS}$ and $SU(2N_{F})$ symmetric. This can be made explicit as follows.

While in Euclidean space the electric and magnetic fields cannot be separated because of the $O(4)$ symmetry, in Minkowski space in a given reference frame they are different fields. Interaction of fermions with the gauge field in Minkowski space-time can be split in a given reference frame into temporal and spatial parts:

[TABLE]

where the covariant derivative $D_{\mu}$ includes interaction of the matter field $\psi$ with the gauge field $\boldsymbol{A}_{\mu}$,

[TABLE]

The temporal term contains interaction of the color-octet charge density

[TABLE]

with the electric part of the gluonic field. It is invariant under $SU(2)_{CS}$ and $SU(2N_{F})$. Note that the $SU(2)_{CS}$ transformations defined 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. It breaks $SU(2)_{CS}$ and $SU(2N_{F})$. We conclude that interaction of the electric and magnetic components of the gauge field with fermions in a given reference frame can be distinguished by symmetry. Such a distinction does not exist if the matter field is bosonic, because a symmetry of the Klein-Gordon Lagrangian and of charge of the $J=0$ field is the same.

Of course, in order to discuss the electric and magnetic components of the gauge field 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 it is natural to use the hadron rest frame. At high temperatures the Lorentz invariance is broken and again a natural frame is the medium rest frame.

3 Emergence of chiral-spin and $SU(2N_{F})$

symmetries in hadrons at zero temperature

The quark condensate of the vacuum is connected with the density of the near-zero modes of the Euclidean Dirac operator via the Banks-Casher relation [12]

[TABLE]

The hermitian Euclidean Dirac operator, $i\gamma_{\mu}D_{\mu}$, has in a finite volume $V$ a discrete spectrum with real eigenvalues $\lambda_{n}$:

[TABLE]

Consequently, removing by hands $k$ lowest lying modes of the Dirac operator from the quark propagators,

[TABLE]

one apriori expects restoration of chiral $SU(2)_{L}\times SU(2)_{R}$ and possibly of $U(1)_{A}$ symmetries, if hadrons survive. This should be signalled by degeneracy of hadrons connected by the $SU(2)_{L}\times SU(2)_{R}$ and $U(1)_{A}$ transformations, see Fig. 3.

Surprisingly a larger degeneracy than the $SU(2)_{L}\times SU(2)_{R}\times U(1)_{A}$ symmetry of the QCD Lagrangian was observed in actual lattice measurements based on JLQCD overlap gauge configurations [13, 14], see Fig. 4.

This large degeneracy, presumably only approximate, represents the $SU(2)_{CS},~{}k=4$ (5) and the $SU(2N_{F})$ symmetries since it contains irreducible representations of both groups, see Fig. 3. The $SU(2)_{CS},~{}k=4$ and $SU(2N_{F})$ transformations commute with the $O(3)$ transformations and consequently there is no contradiction between the chiral-spin symmetry and conserved spin of hadrons.

These results imply, given the symmetry classification of the QCD Lagrangian, that while the confining chromoelectric interaction is distributed among all modes of the Dirac operator, the chromomagnetic interaction, which breaks both symmetries, is located at least predominantly in the near-zero modes. Consequently artificial removal of the near-zero modes leads to the emergence of $SU(2)_{CS}$ and $SU(4)$ in hadron spectrum. Similar results are also observed in $J=2$ mesons [15] and baryons [16].

The highly degenerate level seen in Fig. 4 represents a $SU(2N_{F})$- symmetric level of the pure electric confining interaction.111In reality the degeneracy of Fig. 4 represents a larger $SU(4)\times SU(4)$ symmetry, see discussion in Ref. [26] The hadron spectra could be viewed as a splitting of the level of the QCD string by means of dynamics contained in the near-zero modes of the Dirac operator, i.e. dynamics of chiral symmetry breaking that also includes magnetic effects in QCD.

4 Emergence of chiral-spin and $SU(2N_{F})$

symmetries at high temperatures

Above the pseudocritical temperature the chiral symmetry is restored and consequently the near-zero modes of the Dirac operator are suppressed by temperature. Then, given the results on the artificial truncation of the near-zero modes at zero temperature, presented in previous section, one can expect natural emergence of the $SU(2)_{CS}$ and $SU(4)$ symmetries above $T_{c}$. This expectation has been verified in the lattice simulations of $N_{F}=2$ QCD with quarks of the physical mass with the chirally symmetric Domain Wall Dirac operator (JLQCD ensembles) at temperatures $T=(220-960)$ MeV [22, 23]. Namely the spatial ($z$ direction) rest-frame correlators of all local isovector $J=0,1$ meson operators

[TABLE]

have been calculated as functions of the dimensionless variable

[TABLE]

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.

In Fig. 5 we show such correlators. Here $PS$ and $S$ denote correlators calculated with the $J=0$ pseudosclar and scalar operators that are connected by the $U(1)_{A}$ transformation. Consequently their degeneracy would indicate restoration of $U(1)_{A}$ symmetry. $V_{x},A_{x},T_{t},X_{t}$ are $J=1$ isovector operators. $V_{x}$ ($\gamma_{1}$) and $A_{x}$ ($\gamma_{1}\gamma_{5}$) are the $x$-components of the vector and axial-vector currents. These two operators are connected by the axial part of the $SU(2)_{L}\times SU(2)_{R}$ transformation and their degeneracy would be a signal of restored $SU(2)_{L}\times SU(2)_{R}$ symmetry. $T_{t}$ ( $\gamma_{4}\gamma_{3}$) and $X_{t}$ ($\gamma_{4}\gamma_{3}\gamma_{5})$ operators are connected by the $U(1)_{A}$ symmetry and a degeneracy of the corresponding correlators is required by restored $U(1)_{A}$ symmetry. The operators $(A_{x},T_{t},X_{t})$ are connected by the $k=2$ $CS$-transformation (5) which does not mix these $J=1$ operators with operators of different spin and thus respects the rotational $O(3)$ symmetry in Minkowski space at nonzero temperatures. A degeneracy of correlators with the $(A_{x},T_{t},X_{t})$ operators would be the evidence of emerged $SU(2)_{CS}$ symmetry. All four operators $(V_{x},A_{x},T_{t},X_{t})$ are connected by the $SU(4)$ transformation (7) and a degeneracy of the corresponding correlators would indicate emergent $SU(4)$ symmetry. We show the full QCD results (solid lines) as well as the correlators calculated on the same lattice size with free noninteracting quarks. Note that the correlators calculated with free quarks reflect physics at a very high temperature and only chiral $SU(2)_{L}\times SU(2)_{R}$ and $U(1)_{A}$ symmetries are present in this case. No $SU(2)_{CS}$ and $SU(4)$ symmetries exist for free quarks.

All correlation functions of chiral partners are degenerate within errors. In detail, this is the pair $(V_{x},A_{x})$ which reflects the $SU(2)_{R}\times SU(2)_{L}$ symmetry. The $U(1)_{A}$ symmetry in the scalar $(PS,S)$ pair as well as in the vector channel $(T_{t},X_{t})$, is manifest. At the lowest available temperature $220$ MeV ($1.2T_{c}$) we observe an obvious multiplet structure: $(PS,S)$ multiplet and the approximately degenerate $(V_{x},A_{x},T_{t},X_{t})$ multiplet reflecting the $SU(2)_{CS}$ and $SU(4)$ approximate symmetries. A residual small splitting within the latter multiplet is visible. At larger temperature $T=380$ MeV ($2.2T_{c}$) this residual splitting nearly vanishes. At these two temperatures we see the multiplet structures of all $U(1)_{A}$, $SU(2)_{R}\times SU(2)_{L}$, $SU(2)_{CS}$ and $SU(4)$ symmetries. At the same time correlators calculated with free quarks demonstrate a very different multiplet pattern and only chiral symmetries are manifest.

At the highest temperature of this study, $T\sim 960$ MeV ($5.5T_{c}$), the situation has changed significantly: All full QCD correlators almost coincide with the corresponding free correlators (the dashed lines are almost on top of the data points for the full QCD correlators). Hence at $T\sim 960$ MeV we have reached the region where only chiral $U(1)_{A}$ and $SU(2)_{L}\times SU(2)_{R}$ symmetries exist and the near coincidence with the free correlators suggests a gas of quasi-free quarks.

These results demonstrate that – while the chiral symmetries are practically exact – the $SU(2)_{CS}$ and $SU(4)$ symmetries are not exact. Now we introduce a measure for the symmetry breaking and find the 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. Consequently, in our case the breaking of $SU(2)_{CS}$ and $SU(4)$ can be identified through the parameter

[TABLE]

If $\kappa\ll 1$, then we can declare an approximate or – if zero – an exact symmetry. If $\kappa\sim 1$, the symmetry is absent. From the free quark analytical calculations one finds $\kappa\sim 1$ [23], which stresses that there is no chiral-spin symmetry for free quarks.

In Fig. 6 we show the evolution of the symmetry breaking parameter $\kappa$ as a function of temperature at $zT=2$. The value of $\kappa$ is less than 5 % for all ensembles with $T\sim 220$ – $500$ MeV. This implies that the symmetries that we observe in the range between $T\sim 220$ MeV and $500$ MeV are well pronounced. These symmetries of spatial correlators reflect symmetries of the QCD action since correlators are driven only by the action of the theory.

At temperatures between $T\sim 500$ MeV and $T\sim 660$ MeV we notice a drastic increase of the symmetry breaking parameter $\kappa$ to values of the order $\sim 1$. We conclude that QCD exhibits approximate $SU(2)_{CS}$ and $SU(4)$ symmetries in the temperature range between $T\sim 220$ – $500$ MeV with symmetry breaking less than 5% as measured with $\kappa$. This suggests that the $SU(2)_{CS}$ symmetric regime begins just after the $SU(2)_{R}\times SU(2)_{L}$ restoration crossover.

On the right side of Fig. 7 we show temporal correlators

[TABLE]

at a temperature $T=1.2T_{c}$ [24] where $\mathcal{O}_{\Gamma}(x,y,z,t)$ is an operator that creates a quark-antiquark pair with fixed quantum numbers. Summation over $x,y,z$ projects out the hadron rest frame.

Transformation properties of the local $J=1$ quark-antiquark bilinears $\mathcal{O}_{\Gamma}(x,y,z,t)$ with respect to $U(1)_{A}$, $SU(2)_{L}\times SU(2)_{R}$, $SU(2)_{CS},k=4$ and $SU(4)$ are given in Fig. 3. Emergence of the respective symmetries is signalled by the degeneracy of the correlators (20) calculated with operators that are connected by the corresponding transformations.

On the l.h.s of Fig. 7 we demonstrate correlators calculated with noninteracting quarks on the same lattice. They represent the QGP at a very high temperature where due to asymptotic freedom the quark-gluon interaction can be neglected. Dynamics of free quarks are governed by the Dirac equation and only $U(1)_{A}$ and $SU(2)_{L}\times SU(2)_{R}$ chiral symmetries exist. A qualitative difference between the pattern on the l.h.s. and the pattern on the r.h.s of Fig. 7 is obvious. In the latter case we clearly see $SU(2)_{CS}$ and $SU(4)$ symmetries. The temporal correlators are directly connected to measurable spectral density. $SU(2)_{CS}$ and $SU(4)$ symmetries of the t-correlators imply the same symmetries of spectral densities.

The emergent $SU(2)_{CS}$ and $SU(4)$ symmetries seen at $T_{c}-3T_{c}$ suggest that the physical degrees of freedom at these temperatures are chirally symmetric quarks bound into color singlets by the chromoelectric field. The magnetic effects are either absent or strongly suppressed. Upon increasing temperature above $T_{c}$ a strength of the confining electric field smoothly diminishes and at $T>3T_{c}$ it becomes sufficiently small so that the emergent $SU(2)_{CS}$ and $SU(4)$ symmetries disappear.

5 Conclusions

In the range $T_{c}-3T_{c}$ we observe formation of multiplets in correlators that indicate emergent approximate symmetries described by the chiral-spin $SU(2)_{CS}$ and $SU(4)$ groups. These symmetries include the chiral $U(1)_{A}$ and $SU(2)_{L}\times SU(2)_{R}$ as subgroups. These are not symmetries of the free Dirac action. In a given reference frame, which in our case is the medium rest frame, the chromoelectric interaction is invariant under both $SU(2)_{CS}$ and $SU(4)$ transformations, while the chromomagnetic interaction as well as the quark kinetic term break them.

The emergence of these symmetries in the $T_{c}$ - $3T_{c}$ window suggests that the chromomagnetic field disappears or is strongly suppressed, while the confining chromoelectric field is still active. This implies that the physical degrees of freedom are chirally symmetric quarks bound by the chromoelectric interaction into color-singlet objects without chromomagnetic effects. While we do not advocate any microscopic description of these ultrarelativistic objects, they are reminiscent of “strings”. We refer the $SU(2)_{CS}$ and $SU(4)$ symmetric regime at temperatures $T_{c}$ – $3T_{c}$ as the “Stringy Fluid” to emphasize the possible nature of the objects - chirally symmetric quarks bound by the electric field [25, 23].

These results suggest the following three regimes of QCD. At temperatures below the pseudocritical temperature $T_{c}$ QCD matter is a Hadron Gas where all chiral symmetries are spontaneously broken by the non-zero quark condensate. From the Hadron Gas regime there is a crossover to a regime with approximate $SU(2)_{CS}$ chiral-spin symmetry, where quarks are predominantly bound by the chromoelectric field. This crossover either coincides or is close to the chiral $SU(2)_{L}\times SU(2)_{R}$ restoration crossover. In the range above $\sim 3T_{c}$ there is a fast increase of chiral-spin symmetry breaking: the confining electric interaction becomes small relative to the quark kinetic term. Finally, up to $\sim 5T_{c}$ GeV there is an evolution to a weakly interacting QGP, where the relevant symmetries are the full set of chiral symmetries and quarks behave as if they were free. Fig. 8 provides an illustrative sketch of this temperature evolution for the effective degrees of freedom of QCD. All three regimes of QCD, the Hadron Gas, the Stringy Fluid and the QGP differ by symmetries, degrees of freedom and physical properties. E.g., while the viscosity of the Hadron Gas and of QGP is very large, as in an ideal gas, the Stringy Fluid, as it follows from experiments, is a perfect one. One can speculate that this feature is related to a relatively large size of ”strings” in the Stringy Fluid.

Our conclusions about three regimes of QCD are based only on symmetry classification of the QCD Lagrangian and lattice results obtained at zero chemical potential. At a nonzero chemical potential the lattice Monte-Carlo is not efficient. The quark chemical potential term $\mu\psi(x)^{\dagger}\psi(x)$ in the QCD action

[TABLE]

is $SU(2)_{CS}$ and $SU(2N_{F})$ invariant [25]. Consequently $SU(2)_{CS}$ and $SU(2N_{F})$ symmetries observed on the lattice at $\mu=0$ should also persist in a medium with a finite chemical potential.

We will finish this presentation with a historical analogy. Christopher Columbus had a theory that moving to the west through the Ocean he would reach India. He was able to convince Ferdinand and Isabella of Castile to finance his ”experimental research”. And indeed after three months of adventures he reached a land that was perfectly seen to him as India. But actually he discovered America. Inspite of the erroneous theoretical prediction and interpretation of results his discovery was of great importance. endequation