Chiral density waves in quarkyonic matter
aa r X i v : . [ h e p - ph ] J un Chiral density waves in quarkyonic matter
Tomasz L. Partyka and Mariusz Sadzikowski
Smoluchowski Institute of Physics, Jagellonian University, Reymonta 4, 30-059 Kraków, Poland
We study the phase diagram of strongly interacting matter including the inhomogeneousphase of chiral density waves (CDW) within the Polyakov loop extended Nambu - Jona-Lasinio (PNJL) model. We discuss the phase structure taking into account density andflavour dependence of the Polyakov loop potential parameter and temperature dependenceof the four-point coupling constant of the NJL model. It is shown that the CDW phase ex-ists and that can be interpreted as a special realisation of quarkyonic matter. This fact is ofparticular interest because the existence of homogeneous quarkyonic matter is strongly con-strained. This also indicates that the study of inhomogeneous phases at finite temperaturesand baryon densities are of special importance.
I. INTRODUCTION
An understanding of structure of the phase diagram of strongly interacting matter at finite baryondensity is still an open problem despite of the many efforts that have been devoted to its study since thevery beginning of the QCD era. The lack of experimental data and reliable tools of the direct QCD basedcalculations are the main culprits of this situation. Nevertheless, the progress is still possible and ourexperience must rely on the interplay between general arguments, models calculations and QCD latticeresults accessible at the low chemical potential. A very good example of such approach is the PNJLmodel [1] which combines the chiral and the deconfinement order parameters linked to QCD at finitetemperature through the parameter fits to the lattice data [2–4].The new insight into the high baryon density domain based on large N c expansion has been proposedby McLerran and Pisarski [5]. In the limit of a large number of colours N c → ¥ the QCD phase diagramsimplifies substantially. The low baryon density and temperature region is occupied by the confined,chirally broken phase where the matter pressure scales with N c as P ∼ O ( ) . The high temperaturephase is deconfined and dominated by gluons with the P ∼ N c scaling. Finally, at the low temperatureand high baryon density there is a confined phase which scales as P ∼ N c . The latter phase has beennamed quarkyonic.It is very natural to interpret the quarkyonic phase within the PNJL model as deconfined and chirallyrestored phase. This is based on the PNJL picture of the dense matter which consists of a degenerateFermi sea of quarks with the colorless particle-hole excitations at the Fermi surface. Then the P ∼ N c scaling at high density naturally arises. It was shown in paper [6] that at N c = N c QCD [8], skyrmion crystals [9], LOFF phases [10], Overhausereffect [11] and chiral density waves [12]. The competition between chiral density waves and uniformcolour superconductors were also discussed [13]. The chiral spirals has been already considered in thequarkyonic phase [14]. In this letter we would like to consider the chiral density waves in quarkyonicmatter from the point of view of the PNJL model. It is an interesting problem to check how the chiraldensity waves are influenced by the presence of the Polyakov loop field and what kind of the feedbackit generates. We also pay a particular attention to the relation between the inhomogeneous chiral phaseand the quarkyonic matter.
II. CHIRAL DENSITY WAVES IN THE PNJL MODEL
We consider the PNJL model with two light quarks and three colours with the parametrisation of thePolyakov loop based on paper [3] and the NJL part based on paper [15] L = ¯ y i g µ D µ y + G [( ¯ yy ) + ( ¯ y i g ~ ty ) ] − U ( F , T ) (1)where y is a massless quark field and D µ = ¶ µ − iA µ is covariant derivative. The SU ( ) gauge field A µ = ( A ,~ ) , A = gA a l a / l a are the Gell-Mann matrices. The effective potential describing thetraced Polyakov loop F = N c Tr (cid:20) P exp (cid:18) i Z b d t A (cid:19)(cid:21) , in the fundamental representation takes the form [3] U ( F , T ) T = − a ( T ) F + b ( T ) ln [ − F + F − F ] , a ( T ) = . − . T T + . (cid:18) T T (cid:19) , b ( T ) = − . (cid:18) T T (cid:19) , (2)where A = iA is a colour gauge field and the temperature T =
270 MeV describes the deconfinementtransition in a pure gauge sector. In equation (2) we already used the constraint that the expectationvalue of the Polyakov loops h F i , h F ∗ i are real [16] which gives F = F ∗ at the mean field level [3, 4].Above treatment may be improved with fluctuations that lead to a difference between the expectationvalues of the traced Polyakov loops at non-zero baryon densities [17]. However, in the present work, asa practical procedure, we consistently keep F = F ∗ . Among the literature, one encounter also anotherapproach. Fields F and F ∗ are treated as independent in the minimalization of the grand thermodynamicpotential [18].The coupling constant G = .
024 GeV − and the three dimensional momentum cut-off L = . h ¯ yy i = M cos ~ q · ~ x , h ¯ y i g t y i = M sin ~ q · ~ x , (3)where the third direction in isospin space was chosen arbitrarily. The colour field in a Polyakov gaugecreates a constant background field A = l f which is related to the Polyakov loop through the equation F = ( + f ) / W V = M G + U ( F , T ) − T (cid:229) n Z d p ( p ) Tr ln [ S − ( i w n ,~ p ] (4)where w n = ( n + ) p T . The inverse propagator in Nambu-Gorkov space reads S − ( i w n ,~ p ) = i g ( w n + A + iµ ) + ~ g (cid:0) ~ p − g t ~ q (cid:1) + M i g ( w n − A − iµ ) + ~ g (cid:0) ~ p − g t ~ q (cid:1) + M (5)where µ is a quark chemical potential. After diagonalization of the propagator one finally arrive at theformulae W V = U ( F , T ) + M G + M F p ~ q M − Z L d p ( p ) E −− T (cid:229) i = ± Z L d p ( p ) n ln h + F e − ( E i − µ ) / T (cid:16) + e − ( E i − µ ) / T (cid:17) + e − ( E i − µ ) / T i + ln h + F e − ( E i + µ ) / T (cid:16) + e − ( E i + µ ) / T (cid:17) + e − ( E i + µ ) / T io (6)where M = .
301 GeV is a constituent quark mass at zero temperature and density. The regularizationthrough the 3-dim momentum cut-off was introduced after paper [13]. The energy eigenvalues are givenby the expressions E ± = r ~ p + M + ~ q ± q ( ~ q · ~ p ) + M ~ q , E = p ~ p + M . Let us notice that potential (6) reduces to NJL model prediction in the deconfined limit F = III. THE PHASE DIAGRAM
The global minima of the potential (6) as a function of temperature and baryon density describe thephase diagram of the strongly interacting matter. This prescription leads to the self-consistent equations ¶W¶ M = ¶W¶F = ¶W¶ | ~ q | = . FIG. 1: The phase diagram of the PNJL model with inhomogeneous chiral wave. The dashed curves are lines ofthe deconfinement phase transition. The left panel shows the diagram for the constant Polyakov loop potentialparameter T =
270 MeV, whereas the right panel for the density dependent parameter T ( µ ) given by equation(7). The gray colour describes the inhomogeneous CDW phase. The dotted curves are lines of the transition toquarkyonic phase. These lines essentially coincide with the lines of the first order phase transition to CDW phaseat higher density. Fig. 1 shows the the PNJL model phase diagram including the inhomogeneous chiral density wavephase (CDW) marked with the gray colour. This phase is surrounded by the line of the first orderphase transition where there are jumps in the values of the all order parameters M , F and | ~ q | . Thejump in F is induced by the first order phase transition in M and it is not a mark of the transition to adeconfinement phase. One expects rather a crossover here and the exact place of this transition is notfixed unambiguously. We define the transition line as a place where the derivative d F / dT reaches itsmaximum unless stated otherwise (see subsection IIIB). Using our definition the lines of deconfinementphase transition are shown in Fig. 1 as dashed curves. For the Polyakov loop potential parameter T =
270 MeV the chiral and deconfinement transitions almost perfectly coincide at zero density, however,they split around the point ( T , µ ) = ( . , . ) GeV (left panel of Fig. 1). It is interesting to note thatthis is also the place where the quarkyonic phase appears. The boarder line for the quarkyonic matter isdefined as a place where the value of the quark chemical potential exceeds the value of the constituentquark mass [6]. This line perfectly coincides with the first order phase transition line of the CDW phasefor µ > .
12 GeV. It happens because at the first order phase transition there is a large jump in the valueof the constituent quark mass M which drops below the value of µ . One can conclude that within thePNJL model the quarkyonic matter can be treated as confined, however, spatially inhomogeneous phase. A. Deconfinement and the baryon density
It is important to remember that the critical temperature of the deconfinement phase transition de-creases with increasing baryon number density. This results in T parameter dependence on the quarkchemical potential [19]. It can be intuitively understand in the picture of overlaping hadrons. At zerodensity the finite temperature causes fluctuations of mesons and baryon-antibaryon pairs up to somecritical value of T c . Above this temperature hadrons overlap with each other and there is a possibilityof a colour flow in space which one can interpret as the process of deconfinement. At higher baryonchemical potential there are already some number of hadrons present in a medium. Then a new criticaltemperature, lower then T c , is sufficient to create appropriate number of hadrons which start to overlap.In paper [20] K. Fukushima tried to estimate such dependence from the statistical model data which wealso adopted here T ( µ ) = T − bµ , (7)where b = . · − MeV − and for T ( µ ) < B. Deconfinement and the number of flavours
A value of the temperature parameter T depends also on the number of active flavours. For the modelwhich contains two degenerate flavours one should reduce the temperature parameter to T =
208 MeV[2, 19]. The phase diagram for this parameter is shown on the left panel of Fig. 2. First of all one loosesa good coincidence between the chiral and deconfinement phase transitions at zero density, nevertheless,the CDW phase remains intact. The confined (quarkyonic), inhomogeneous phase appears at chemicalpotential larger then µ = .
21 GeV and temperatures below T = .
16 GeV. For lower baryon densitiesand higher temperatures the CDW phase still exists but it is deconfined.
FIG. 2: The phase diagram of the PNJL model with inhomogeneous chiral wave for the Polyakov loop potentialparameter T =
208 MeV which corresponds to N f = G = .
024 GeV − . On the right panel the temperature dependence of the NJL parameter G ( T ) is taken into account according to equation (8). Behavior of the Polyakov loop is disturbed by the firstorder transitions into the inhomogeneous phase and consequently deconfinement transition is determined withuncertainty within the blue band. The phase diagram at T =
208 MeV is not satisfactory in a sense that at zero density there is a mis-match between the chiral and deconfinement phase transitions which contradicts the lattice results [21].One can consider a possibility that this problem is a consequence of the wrong value of the couplingconstant G at non-zero temperature. The effective coupling constant G is in principle a function of tem-perature and density. It contains contributions which follow from the integration of the gluonic degreesof freedom. At higher values of T gluons interaction with quarks weakens which in turn influences theeffective four-quark interaction in the same way. Thus the coupling G decreases with increasing tem-perature which lowers the critical temperature of the chiral phase transition. If one attributes the wholemismatch between the chiral and deconfinement transitions at zero density to the wrong value of G thenone can try, at least at the phenomenological level, to reestablish the agreement changing its value intothe new one G ( T c ) = G c in such a way, that both transitions has the same critical temperature T c . Weassume here that the coupling G is a linear function of temperature G ( T ) = G ( − ( T / T c )) + G c ( T / T c ) , (8) G = .
024 GeV − , G c = .
221 GeV − , T c =
167 MeV . According to our fit the change in the constant G between zero temperature and T c is of the order of 15per cent which is not much. Then one can treat equation (8) as a series expansion in the temperaturearound T =
0. The density dependence of G is neglected since it is less important at large N c limit wherequarks decoupled from the gluonic degrees of freedom.The right panel of Fig.2 describes the phase diagram where the NJL coupling constant G is given bythe linear function from eq. (8). It is clearly seen that the chiral and deconfinement phase transitionscoincides at zero density to a good approximation. Let us mention that the usual definition of thedeconfinement transition line as a place where the derivative d F / dT reaches its maximum is not quiteuseful in a situation where F is a discontinues function of temperature and the points of discontinuityare close to the expected maximum value of d F / dT . In such situations we define the deconfinementtransition as a place where the Polyakov loop F takes the value 0 . ± .
07. This is rather a modestvalue, however, such range is suggested when one considers the deconfinement phase transition for thehomogeneous phases where the transition line is defined in a standard way. Using our definition theline of deconfinement phase transition is shown in Fig. 2 (later in Fig. 4) as a dark (blue) band. Atlow temperature and high baryon density one recovers the results of the NJL model with a constant G coupling. FIG. 3: The chiral phases of the PNJL model with inhomogeneous chiral wave. The NJL model parameters are G = .
024 GeV − and L = .
653 GeV. The temperature dependent part of the potential (6) is also regularized bythe finite cut-off. In such a regularization the CDW phase is limited to the high density region of the diagram.
C. Triple point
The concept of a triple point in the QCD phase diagram refers to various situations. For instance,the existence of a triple point between hadronic, color-superconducting and quark-gluon sectors wasdiscussed in Ref. [22]. In the context of the present work, we refer to a very recent idea, namely, to thetriple point in which hadronic, quarkyonic and quark-gluon phases meets together. Such a possibility wasemphasized in paper [23]. Although in Figs. 1,2 the CDW phase exists even at zero density region (thereis no space for triple point), but the location of the inhomogeneous phase depends on the regularizationparameter L . In Figs. 1,2 the cut-off parameter in the temperature dependent part of the potential (6)was sent to infinity. If one keeps this parameter at the constant value L = .
653 GeV then the phasediagram changes and it is shown in Fig. 3. In this case the triple point actually appears at the pointwhere the inhomogeneous phase ends ( T , µ ) = ( . , . ) GeV, what is in agreement with the resultsof Ref. [24]. However, one has to remember that the location of this point is strongly dependent on themodel parameters and the regularization method.For the quark chemical potential below µ the phase line describes the continuous phase transitionwhereas above µ the line is of the first order. When a non-zero current quark mass is turned on then thecontinuous phase transition changes into a smooth crossover and only the island of CDW phase remainson the diagram. IV. SUMMARY AND CONCLUSIONS
We have discussed the phase diagram of the strongly interacting matter including the spatially de-pendent chiral density waves in the PNJL model. It was shown that the inhomogeneous CDW phaseexists and dominates over the chirally restored phase in a large domain of the phase diagram. We havealso pointed out that the chiral density wave can be interpreted as a special realisation of the quarkyonicmatter. This is an interesting possibility, particularly, when the homogeneous quarkyonic phase wasstrongly constrained by the results of the statistical model [20]. Indeed in a case of two flavours thephase diagram with only homogeneous phases is shown in Fig. 4 (left panel) where the effects of flavourand density dependence of the Polyakov loop potential parameter T ( N f , µ ) as well as the temperaturedependence of the NJL coupling constant G ( T ) were taken into account. This last dependence reflectsthe fact that at higher temperature the four-point quark interaction should weaken which in turn let thechiral and the deconfinement transitions stay at the same critical temperature at zero baryon density.In the diagram the deconfinement phase transition precedes the chiral phase transition almost in all thedomain. Only at high density and low temperature the transitions start to coincide. In such situationthere is no space left for quarkyonic matter in accordance with the conclusion of paper [20].The phase diagram in the right panel of Fig. 4 sums up all physical effects that we discussed in theprevious sections and is the best candidate for the QCD phase diagram. The left panel of Fig. 4 shouldbe compared with the right panel of the same figure. Let us notice a clear change in the order of the phasetransitions. The phase transition to the CDW phase precedes the deconfinement phase transition. Thisfact also opens a window for the quarkyonic matter. The transition into the quarkyonic phase is closeto or coincides with the first order phase transition into the CDW phase. Then the matter is spatiallyinhomogeneous but still confined. Let us remind that the details of the phase diagram depends on the FIG. 4: Left panel: the phase diagram of the PNJL model with homogeneous phases only. Above the point (T, µ )= (0.085, 0.263), the nature of the chiral transition changes from first to second oreder. Right panel: the phasediagram of the PNJL model with inhomogeneous chiral density wave. The Polyakov loop potential parameter T ( µ ) is given by (7) with T =
208 MeV. The finite 3-dim momentum cut-off L = .
653 GeV regularized thepotential (6) and the NJL coupling constant G ( T ) depends on temperature through the eq. (8). The grey colourdepicts the CDW phase. The dark (blue) band shows a location of the deconfinement phase transition. temperature parametrisation of the NJL coupling constant G ( T ) . Nevertheless, at low temperatures oneshould expect the same pattern of phase transitions as given in Fig. 4.For the model parameters we choose the CDW phase exists up to a zero density line. However, thelow density and high temperature region is strongly affected by the temperature fluctuations which areneglected in the mean field approximation. These fluctuations probably melt the "crystal" structure ofthe CDW phase and the triple point is expected to appear on the phase diagram eventually.It is an interesting task for the future work to compare the phases of the chiral density waves and thechiral spirals [14]. However, it requires the implementation of the chiral spirals within the NJL modelin the first place. This is a subject of certain importance because the quarkyonic matter, if exists, wouldbe most probably of an inhomogeneous nature. One should also check the dependence of the resultsagainst the regularization scheme and the influence of the finite current quark mass [25]. Finally, it is ofgreat interest to consider more general ansatz then (3) to study the possibility of the creation of differentcrystal structures. [1] K. Fukushima, Phys. Lett. B591 (2004) 277.[2] C. Ratti, M. A. Thaler and W. Weise, Phys. Rev.
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