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
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