# A finite box as a tool to distinguish free quarks from confinement at high temperatures

AA ﬁnite box as a tool to distinguish free quarksfrom conﬁnement at high temperatures

L.Ya. Glozman ∗ and C.B. Lang † Institute of Physics, University of Graz, 8010 Graz, Austria (Dated: July 22, 2020)Above the pseudocritical temperature T c of chiral symmetry restoration a chiral spin symmetry(a symmetry of the color charge and of electric conﬁnement) emerges in QCD. This implies thatQCD is in a conﬁning mode and there are no free quarks. At the same time correlators of operatorsconstrained by a conserved current behave as if quarks were free. This explains observed ﬂuctuationsof conserved charges and the absence of the rho-like structures seen via dileptons. An independentevidence that one is in a conﬁning mode is very welcome. Here we suggest a new tool how todistinguish free quarks from a conﬁning mode. If we put the system into a ﬁnite box, then ifthe quarks are free one necessarily obtains a remarkable diﬀractive pattern in the propagator of aconserved current. This pattern is clearly seen in a lattice calculation in a ﬁnite box and it vanishesin the inﬁnite volume limit as well as in the continuum. In contrast, the full QCD calculations in aﬁnite box show the absence of the diﬀractive pattern implying that the quarks are conﬁned. ∗ [email protected] † [email protected] a r X i v : . [ h e p - l a t ] J u l I. INTRODUCTION.

At temperatures between 100 and 200 MeV one observes in QCD a smooth chiral symmetry restoration crossover[1, 2]. Below this crossover QCD describes a hadron (meson) gas. Above the crossover another physics regime emergesthat is characterized by a nearly perfect ﬂuidity where there are no free quarks and gluons and QCD is still in theconﬁning regime. What are the physical degrees of freedom here and how do they induce the perfect ﬂuidity?Certainly it is not yet a quark-gluon-plasma (QGP) where the degrees of freedom are free (i.e., deconﬁned) quarksand gluons.There are observables that behave as if quarks were free particles soon above the pseudocritical temperature ofchiral symmetry restoration. These are ﬂuctuations of the conserved charges [3, 4] as well as a nonobservation ofthe ρ -like structures via dileptons in experiments. At the same time it was established in lattice calculations [5–7]that QCD in the range T c − T c is characterized by the chiral spin symmetry [8, 9] which is a symmetry of the colorcharge and of the chromoelectric interaction . This is not a symmetry of the Dirac action and hence inconsistent withfree deconﬁned quarks. This suggests that the degrees of freedom are the chirally symmetric quarks bound into thecolor-singlet objects by the chromoelectric ﬁeld. What are these objects? This symmetry was observed in latticecalculations at zero baryon (quark) chemical potential. It should also persist at a nonvanishing chemical potentialsince the quark chemical potential in the QCD action is manifestly chiral spin symmetric [12].While this symmetry was observed in lattice calculations, there are direct experimental consequences, too. Namely,the chiral spin symmetry prohibits a ﬁnite axial chemical potential and consequently the electric current induced byan external magnetic ﬁeld should vanish or be very small [13]. The experimental search of the chiral magnetic eﬀect[14, 15] in heavy ion collisions suggests that such a current is either absent or very small, indeed [16].Both temporal and spatial meson correlators exhibit this symmetry very clearly at T c − T c , which suggests achromoelectric conﬁning interaction. At the same time the correlators of operators constrained by a conserved currentalmost coincide with those for free quarks [17]. The latter circumstance explains why ﬂuctuations of conserved chargesabove T c demonstrate an ideal quark gas-like behavior and why no ρ -like structures are seen via dileptons. Thisintriguing behavior of the correlators was a motivation for a conjecture of a deconﬁnement in a SU (2) color subgroupof SU (3) color induced by a SU (2) color - SU (2) isospin locking [17]. This would explain both the chiral spin symmetryof the correlators and at the same time their free-like behavior in channels with conserved currents. Because of the SU (2) color - SU (2) isospin locking the conserved currents do not see the SU (3) color /SU (2) color part of dynamics whichis still conﬁning. So while the correlators of the conserved currents behave as if quarks were free, in reality thesequarks are still in the conﬁning mode because of the conﬁnement in SU (3) color /SU (2) color .Given this intriguing situation an independent evidence is welcome that quarks in channels with conserved currentsare still in the conﬁning mode, even though the respective correlators look as if the quarks were free. This questionis the subject of the present paper. We demonstrate that even if the correlators of conserved currents are free-like inthe continuum, we can distinguish really free quarks from the free-like behavior by putting the system into a ﬁnitebox. If quarks are really free, in a ﬁnite box this leads to a very speciﬁc and bright interference pattern that doesnot exist in inﬁnite volumes or in the continuum. While we do observe such patterns in a ﬁnite box in a free quarksystem, these patterns are absent in full QCD calculations in a ﬁnite box. This allows the conclusion that the quarksare in a conﬁning mode. Hence we have two independent and complementary evidences that QCD is in the conﬁningregime: the chiral spin symmetry of the correlators and the absence of very pronounced interference patterns requiredby free quarks in a ﬁnite box on the lattice. II. FREE QUARKS IN A FINITE BOX.

In Minkowski space the Feynman propagator of a Dirac particle depending on the chronological order is either aforward running particle ( ∼ exp( − iEt )) or a backward running antiparticle ( ∼ exp(+ iEt )). Upon a Wick rotation toEuclidean space the forward running particle has an ∼ exp( − Et ) dependence while the backward running antiparticleevolves with time as ∼ exp(+ Et ). If we put the system into a ﬁnite box, e.g., on the lattice, then the rest frame( p = 0) time-direction propagator of a free quark with the mass mC ( t ) = (cid:88) x,y,z (cid:104) ψ ( x, y, z, t ) ¯ ψ ( , (cid:105) (1) In QCD with light quarks only one consistent deﬁnition of conﬁnement is known: Conﬁnement is the absence of color states in thespectrum. Hence deconﬁnement should be accompanied by a free motion of colored quarks and gluons. This symmetry was reconstructed from a large hadron spectrum degeneracy observed on the lattice upon artiﬁcial subtraction of thenear-zero modes of the Dirac operator at zero temperature [10, 11]. Conditionally the regime in QCD above T c but below 3 T c was named a stringy ﬂuid to emphasize the fact that the degrees of freedomare the ultrarelativistic chirally symmetric quarks bound by the chromoelectric ﬁeld and the chromomagnetic eﬀects are at least stronglysuppressed. n s -1-0.8-0.6-0.4-0.200.20.40.60.81 C OV, p.b.c.OV, a.b.c.W, p.b.c.W, a.b.c.cosh, E=0.373945sinh, E=0.373945

Quark prop. 32^3x8

FIG. 1. A single quark massless propagator obtained on the 32 × E ( z − N s / E ( z − N s / E = 0 . has a C ( t ) ∼ cosh( m ( t − N t / C ( t ) ∼ sinh( m ( t − N t / At nonzero temperature the temporal direction becomes short compared to the spatial one. There are cases inwhich a study of the propagators along the long spatial direction can supply us with the information that cannot beobtained from the temporal propagators along the short time direction. We choose this direction to be z and studythe following spatial correlators: C s ( z ) = (cid:88) x,y,t (cid:104) ψ ( x, y, z, t ) ¯ ψ ( , (cid:105) . (2)This spatial single quark propagator can be straightforwardly calculated on a ﬁnite N s × N t lattice with given boundaryconditions. We choose antiperiodic boundary conditions (a.b.c.) along the time direction, periodic ones (p.b.c.) alongthe x, y axes and either periodic or antiperiodic along the propagation direction z . The results for Tr C s ( z ) obtainedat zero quark mass with the Wilson and overlap Dirac operators [18] are shown in Fig. 1.An eﬀective ”chirally symmetric mass” for propagation of a massless quark in z direction is very close to the lowestMatsubara frequency π/N t = π/ /D ( p x = 0 , p y = 0 , p z = i E, p t = π/N t ) → E = arcosh (cid:18) − π/N t − π/N t (cid:19) . (3)The propagator obtained for a single quark with Wilson or overlap Dirac action is very accurately described bycosh( E ( z − N s / E ( z − N s / f ) running quark with the ”mass” E and of a backward ( b ) running antiquark with the same ”mass”.Symbolically the propagator can be written as C ( z ) p.b.c. ∼ exp( − Ez ) + exp( − E ( N s − z )) ≡ f + ¯ b. (4)For the a.b.c. the propagator is C ( z ) a.b.c. ∼ exp( − Ez ) − exp( − E ( N s − z )) ≡ f − ¯ b. (5)Having discussed the structure of a single quark propagator in a ﬁnite box we next study propagators of quarkbilinears still keeping quarks to be noninteracting particles (I.e., due to a pure Dirac Lagrangian without any gauge On a discrete lattice x, y, z, t should be discrete ( n x , n y , n z , n t ); N t is the lattice size in t -direction. Name Dirac structure Γ Abbreviation

Pseudoscalar γ P S (cid:3) U (1) A Scalar S Axial-vector γ k γ A (cid:3) SU (2) A Vector γ k V Tensor-vector γ k γ T (cid:3) U (1) A Axial-tensor-vector γ k γ γ X TABLE I. Fermion isovector bilinears and their U (1) A and SU (2) L × SU (2) R transformation properties (last column). Thisclassiﬁcation assumes propagation in z -direction. The open vector index k here runs over the components 1 , ,

4, i.e., x, y and t . ﬁelds.) The spatial correlators of the isovector bilinear operators O Γ ( x, y, z, t ) = ¯ ψ ( x, y, z, t )Γ (cid:126)τ ψ ( x, y, z, t ) with Γbeing out of a set of γ -matrices are C Γ ( z ) = (cid:88) x,y,t (cid:104)O Γ ( x, y, z, t ) O Γ ( , † (cid:105) . (6)The isovector fermion bilinears are named according to Table I.A complete set of such propagators in the continuum (in inﬁnite volume) has been determined analytically inRef. [6]. There these correlators are given as superpositions of the decaying exponents exp( − πz/N t ) / (2 πz/N t ),exp( − πz/N t ) / (2 πz/N t ) , . . . and terms with higher Matsubara frequencies and represent the propagators of theforward propagating ”mesons” that are made from noninteracting quarks.In a ﬁnite box a quark propagator of a given ﬂavor is represented as a sum (for p.b.c) or diﬀerence (for a.b.c.)of the forward propagating quark and of the backward propagating antiquark. The same is true for the antiquarkpropagator, that is a sum (or diﬀerence) of the forward propagating antiquark and of the backward propagating quark.Consequently correlators of the bilinears should be superpositions of four terms: p.b.c. : ( f + ¯ b )( ¯ f + b ) = f ¯ f + b ¯ b + f b + ¯ b ¯ f , (7) a.b.c. : ( f − ¯ b )( ¯ f − b ) = f ¯ f + b ¯ b − f b − ¯ b ¯ f . (8)Note that the two terms ∼ ¯ f f and ∼ ¯ bb represent the forward and backward propagating meson-like system. Theother two terms ∼ f b and ∼ ¯ f ¯ b do not represent any meson-like system. These terms are necessarily present inthe correlators of the quark-antiquark bilinears if quarks are free particles that do not interact. They exist onlyin a ﬁnite box and vanish in the inﬁnite volume limit or in the physical continuum. If we put the system of freequarks into a ﬁnite box, then these ”unphysical” terms must be observable since they interfere with the ”physical”meson-like amplitudes. The interference should be clearly seen in cases when the ”physical” and ”unphysical” termsare of a similar magnitude and interfere destructively. Since the ”unphysical” terms are very small one should expectthis destructive interference to be clearly visible only when the ”physical” terms are also very small. The numericalresults for the propagators calculated with free noninteracting quarks [5] show that the largest slope of the decaytakes place with the operators V t , A t , T x , T y , X x , X y and all other operators V x , V y , A x , A y , ... have smaller decayrate. This suggests that the ”physical” meson-like amplitude become suﬃciently small at large z for the operators V t , A t , T x , T y , X x , X y and we can expect in this case well visible interference eﬀects of the ”physical” and ”unphysical”amplitudes.The correlators calculated with the overlap action on the 32 × V t operator are shown in Fig. 2.The correlators of the A t , T x , T y , X x , X y operators are similar. We also show in the same ﬁgure the correlators of the V x operator that demonstrates a smaller decay rate.We clearly see a typical diﬀractive structure for the correlator of the V t operator at large z and when p.b.c.are imposed the correlator becomes negative for z ∼ −

18. This was ﬁrst noted in Refs. [5, 19] but remainedunexplained. Now we realize that this structure is the result of the destructive interference of the ”physical” and”unphysical” amplitudes. It is an immanent property of a system of free quarks in a ﬁnite box. In contrast, thecorrelator of the V x operator does not show a diﬀractive structure because the ”physical” terms in this case are alwaysessentially larger than the ”unphysical” ones.How to check this picture of the destructive interference ? If we change from p.b.c. to a.b.c. one should expect aconstructive interference of the ”physical” and ”unphysical” terms. Hence the diﬀractive structure should disappear.This is precisely what happens.Numerical checks indicate that the diﬀractive structure disappears exponentially upon increase of N z (at ﬁxed N t ).Hence it vanishes both in large lattice volumes as well as in the continuum theory. n z C ( n z ) p . b . c . / C ( ) p . b . c . V x V t n z C ( n z ) a . b . c . / C ( ) a . b . c . V x V t FIG. 2. Correlators of the V t and V x bilinears on a 32 × z direction. The V t correlator in the left panel is negative for n z ∼ −

18. The correlators are normalized to 1 at n z = 1. III. COMPARISON OF THE FULL QCD AND FREE QUARKS CORRELATORS IN A FINITE BOX.

We have established in the previous section that if quarks are free, then the spatial correlators of the conservedcurrents V t , A t and of some other operators exhibit on a ﬁnite lattice remarkable diﬀractive patterns. These are aconsequence of the fact that for free quarks there are necessarily amplitudes that represent a ”meson-like” propagation,called ”physical”, and ”unphysical” amplitudes that do not correspond to any meson-like system. These ”physical”and ”unphysical” amplitudes interfere destructively. The ”unphysical” amplitudes vanish on the inﬁnite lattice aswell as in the continuum and the diﬀractive pattern disappears.At the same time the ”unphysical” terms are much smaller than the ”physical” ones for another set of operatorsand the diﬀractive pattern does not exist. These features are a solid prediction of a free quark system put on a ﬁnitelattice.In the continuum full QCD above T c the spatial and temporal correlators of the conserved currents behave as ifquarks were free [17]. In reality they cannot be free since these correlators are subject to the chiral spin symmetrythat is not a symmetry of the Dirac action. Is there another means to decide that the quarks are not free? Theanswer is aﬃrmative. When we solve QCD at high temperatures on the ﬁnite lattice if the quarks are not conﬁned(i.e., free), one should observe the diﬀractive pattern as described above. If such a pattern is missing, then we couldsafely conclude that the quarks are not free. This is demonstrated below.In Fig. 3 we show correlators normalized to 1 at n z = 1 built with the V t , A t , T x , X x operators calculated in N F = 2QCD with the domain wall Dirac operator at physical quark masses on 32 × T = 380 MeV (2 . T c ) [5].The boundary conditions for quarks are a.b.c. in time direction and p.b.c. in all spatial directions. The solid curvesrepresent the full QCD results while the dashed curves are correlators calculated on the same lattice with the sameDirac operator with free noninteracting quarks, i.e. computed with a trivial gauge ﬁeld conﬁguration ( U = 1). Thefree quark correlator of the V t operator corresponds to the results shown in Fig. 2. It is rather obvious that the freequark results obtained with the domain wall Dirac operator in Fig. 3 are similar to those obtained with the overlapDirac operator in Fig. 2. In both cases we see a remarkable diﬀractive structure around n z ∼ − V t in full QCD is practically identicalwith the free quarks propagator at n z <

11, it does not represent a free quark system but describes a propagationof a meson-like system with conﬁnement. All ”unphysical” terms that exist in the case of the free quark system arekilled by a conﬁning gluonic interaction between quarks. In contrast the propagator of the V x operator, that is notconstrained by a current conservation, demonstrates the absence of the diﬀractive structure both in full QCD as wellas for free quarks, see Fig. 4.We summarize this section with the principal result of the present paper. There are two independent evidences thata system with quantum numbers of a conserved current is in a conﬁning mode above T c . The ﬁrst evidence are the z -7 -6 -5 -4 -3 -2 -1 C ( n z ) / C ( n z = ) free Vtfree Txdressed Vtdressed Atdressed Txdressed Xx

380 MeV

FIG. 3. Correlators of the V t , A t , T x , X x operators in full QCD at T = 380 MeV ( ∼ . T c ) for 32 × dressed ) and with non-interacting quarks ( free ) on the same lattice. From Ref. [5]. z -6 -5 -4 -3 -2 -1 C ( n z ) / C ( n z = ) free PSfree Vxfree Ttdressed PSdressed Sdressed Vxdressed Axdressed Ttdressed Xt dressed PS, Sdressed Vx, Axfree Vx, Axfree Tt, Xtfree PS, S

380 MeV dressed Tt, Xt

FIG. 4. Correlators of the

P S, S, V x , A x , T t , X t operators in full QCD at T = 380 MeV ( ∼ . T c ) for 32 × dressed ) and with non-interacting quarks ( free ) on the same lattice. From Ref. [5]. very clear patterns of the chiral spin symmetry both in spatial and temporal correlators [5–7]. The second evidence,demonstrated in the present paper, is the absence of the diﬀractive pattern required by a system of free quarks. IV. DISCUSSION AND CONCLUSIONS.

We have demonstrated that on a ﬁnite lattice in a system of free noninteracting quarks the spatial propagators ofthe bilinear quark-antiquark operators exhibit in case of periodic boundary conditions along the propagation directiona diﬀractive pattern for operators that are constrained by a current conservation and for some other operators. Thisdiﬀractive pattern is a consequence of a destructive interference of the amplitudes that correspond to the propagationof a meson-like system made of a quark and an antiquark with amplitudes that do not describe any meson-like system.The latter amplitudes arise exclusively due to a ﬁniteness of a box and vanish on an inﬁnite lattice or in the continuum.The latter amplitudes as well as a diﬀractive pattern is an immanent property of the free quark system in a ﬁnitelattice.In QCD the correlators of conserved currents above a chiral symmetry restoration crossover behave as if quarkswere free, i.e. the correlators of these currents calculated in QCD coincide with correlators obtained with freenoninteracting quarks [17]. This explains why ﬂuctuations of conserved charges indicate a free quark gas-like behaviorin the chirally restored regime as well as absence of the rho-like structures observed via dileptons in heavy ioncollisions. At the same time these correlators as well as another ones are a subject to a chiral spin symmetry [8, 9] at T c − T c [5–7]. This symmetry is not a symmetry of the Dirac action and hence inconsistent with free noninteractingquarks. It is a symmetry of the color charge in QCD and it indicates that QCD is in the conﬁning regime where thechromoelectric interaction binds the chirally symmetric quarks into color-singlet objects (”strings”) and a contributionof the chromomagnetic interaction is at least strongly suppressed.An independent evidence conﬁrming that the quark-antiquark systems with a conserved current quantum numbersare indeed in the conﬁning regime is supplied by QCD on the lattice in a ﬁnite box. If the quarks are indeed free, thenthere must a diﬀractive pattern described above. In full QCD calculations above T c in a ﬁnite box such pattern isnot observed. It follows then that the quarks are not free and conﬁning chromoelectric dynamics kills all amplitudesthat do not correspond to propagating mesons. It was conjectured that a free-like behavior of the QCD correlatorsof conserved currents in continuum at T c − T c is due to a deconﬁnement in a SU (2) color subgroup of SU (3) color from a color-isospin locking [17]. The conserved currents do not see the conﬁning SU (3) color /SU (2) color dynamicsand consequently the correlators of conserved currents look as if quarks were free.Hence we have two independent and complementary evidences of conﬁnement in SU (3) color at T c − T c . These arethe chiral spin symmetry of correlators and the absence of a diﬀractive structure required by free quarks in a ﬁnitebox . This regime we have conditionally called ”stringy ﬂuid” [6, 12].At temperatures above 3 T c the chiral spin symmetry smoothly disappears [6] and correlators of all operatorsapproach correlators calculated with free quarks. This suggests that eventually the color charge and electric conﬁninginteraction is Debye screened within SU (3) color . Still the correlators of the conserved currents in a ﬁnite box do notshow the diﬀractive structure required by really free quarks [6]. This indicates that there are no free, noninteractingquarks and the system is still in the conﬁning regime (deﬁning conﬁnement as the absence of free quarks and gluons.)The latter fact can be explained by the presence of a weak magnetic conﬁnement at very high temperatures. It isknown that at very high temperatures QCD is dimensionally reduced to a weakly coupled 3-dimensional pure magnetictheory [20]. Even though the theory is weakly coupled, there is a pure magnetic weak ”conﬁning” interaction thatdoes not allow quarks to be completely free [21, 22]. In this regime all properties of QCD should be close to thequark-gluon-plasma regime. ACKNOWLEDGMENTS

We thank T. Cohen, C. Gattringer, O. Philipsen and R. Pisarski for careful reading of the ms. [1] Y. Aoki, S. Borsanyi, S. Durr, Z. Fodor, S. D. Katz, S. Krieg and K. K. Szabo, JHEP

088 (2009).[2] A. Bazavov et al. [HotQCD Collaboration], Phys. Lett. B , 15 (2019).[3] F. Karsch, S. Ejiri and K. Redlich, Nucl. Phys. A , 619 (2006).[4] A. Bazavov et al. , Phys. Rev. D , no. 5, 054504 (2017).[5] C. Rohrhofer, Y. Aoki, G. Cossu, H. Fukaya, L. Y. Glozman, S. Hashimoto, C. B. Lang and S. Prelovsek, Phys. Rev. D , 094501 (2017) Erratum: [Phys. Rev. D , 039901 (2019)].[6] C. Rohrhofer, Y. Aoki, G. Cossu, H. Fukaya, C. Gattringer, L. Y. Glozman, S. Hashimoto, C. B. Lang and S. Prelovsek,Phys. Rev. D , 014502 (2019).[7] C. Rohrhofer, Y. Aoki, L. Y. Glozman and S. Hashimoto, Phys. Lett. B , 135245 (2020).[8] L. Y. Glozman, Eur. Phys. J. A

27 (2015).[9] L. Y. Glozman and M. Pak, Phys. Rev. D , 016001 (2015).[10] M. Denissenya, L. Y. Glozman and C .B. Lang, Phys. Rev. D , 034505 (2015).[12] L. Y. Glozman, Eur. Phys. J. A , 117 (2018).[13] L. Y. Glozman, arXiv:2004.07525 [hep-ph]. We might refer to this second evidence as a ”Cheshire cat evidence”: If we study QCD at high T only with conserved currents, then itbehaves as if quarks were free, i.e., no conﬁnement. We see the smile of conﬁnement only when we put QCD into a ﬁnite box. [14] D. E. Kharzeev, L. D. McLerran and H. J. Warringa, Nucl. Phys. A , 227 (2008).[15] K. Fukushima, D. E. Kharzeev and H. J. Warringa, Phys. Rev. D , 074033 (2008).[16] J. Zhao [STAR Collaboration], arXiv:2002.09410 [nucl-ex].[17] L. Y. Glozman, Phys. Rev. D , 014509 (2020).[18] C. Gattringer and C. B. Lang, ”Quantum Chromodynamics on the Lattice”, Lect. Notes. Phys.,788, 1 (2010).[19] R. V. Gavai, S. Gupta and R. Lacaze, Phys. Rev. D , 014502 (2008).[20] T. Appelquist and R. D. Pisarski, Phys. Rev. D , 2305 (1981).[21] A. D. Linde, Phys. Lett. , 289 (1980).[22] A. Hart, M. Laine and O. Philipsen, Nucl. Phys. B586