The problem of repulsive quark interactions - Lattice versus mean field models
aa r X i v : . [ h e p - ph ] J a n The problem of repulsive quark interactions - Lattice versus mean field models
J. Steinheimer a,b , S. Schramm a,b,c a Institut f¨ur Theoretische Physik, Goethe-Universit¨at, Max-von-Laue-Str. 1, D-60438 Frankfurt am Main, Germany b Frankfurt Institute for Advanced Studies (FIAS), Ruth-Moufang-Str. 1, D-60438 Frankfurt am Main, Germany c Center for Scientific Computing, Max-von-Laue-Str. 1, D-60438 Frankfurt am Main
Abstract
We calculate the 2nd and 4th order quark number susceptibilities at zero baryochemical potential, using a PNJL approachand an approach which includes, in a single model, quark and hadronic degrees of freedom. We observe that thesusceptibilities are very sensitive to possible quark-quark vector interactions. Compared to lattice data our resultssuggest that above T c any mean field type of repulsive vector interaction can be excluded from model calculations.Below T c our results show only very weak sensitivity on the strength of the quark and hadronic vector interaction. Thebest description of lattice data around T c is obtained for a case of coexistence of hadronic and quark degrees of freedom. Many recent experimental programs in heavy ion physics,e.g. at the relativistic heavy ion collider and the plannedFAIR facility, are aimed at a better understanding of bulkproperties of QCD matter. In particular the experimentalconfirmation of the deconfinement and chiral phase transi-tions as well as the search for the critical end point (CeP)are of great interest to the community. Lattice resultsat µ B = 0 suggest that both phase transitions exhibit asmooth crossover. Suffering from the so called sign prob-lem at finite baryo chemical potential, lattice studies haveso far not been able to constitute a consistent picture of thephase diagram and the existence of the CeP [1, 2]. Whilea Taylor expansion of lattice results at µ b = 0 predict theexistence of a critical end point these results may stronglydepend on the order to which the Taylor coefficients can beevaluated [3]. Lattice calculations at imaginary chemicalpotential on the other hand show no indication for a CeP,though these calculations are done on on a coarse latticewith a fermion action that is known to have large dis-cretization errors. Other studies trying to locate the CePoften rely on the applicability of various effective modelswhere CeP’s existence and location seems very sensitiveon the vector coupling strength at and below T c [4].
1. The PNJL model
The PNJL model was introduced in [5, 6] as an ef-fective chiral quasi-quark model that incorporates a meanfield like coupling to a color background field. It has oftenbeen shown to reproduce many general features of latticeresults at µ B = 0 [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, Email address: [email protected] (J. Steinheimer)
4, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28]. In our com-parison we will use a very basic parametrization of thetwo-flavor PNJL model and extend it to incorporate a re-pulsive vector interaction. The thermodynamic potentialof our parametrization reads:Ω = U (Φ , Φ ∗ , T ) + σ / G S − ω / G V − Ω q (1)with Ω q = 2 N f Z d p (2 π ) n T ln h e − ( E p − µ ∗ q ) /T + 3Φ ∗ e − E p − µ ∗ q ) /T + e − E p − µ ∗ q ) /T i + T ln h ∗ e − ( E p + µ ∗ q ) /T + 3Φ e − E p + µ ∗ q ) /T e − E p + µ ∗ q ) /T i + 3∆ E p Θ(Λ − ~p ) (cid:9) (2)Here the grand canonical potential includes contributionsof sates with 1, 2 and 3 times the single quark energy. Notethat the 3 quark state does not couple to the Polyakovloop.The (traced) Polyakov loop Φ was introduced as:Φ = 1 / T r e iφ/T (3)with φ = A , a background color gauge field. The ther-modynamics of Φ (and Φ ∗ ) are controlled by the effectivepotential U (Φ , Φ ∗ , T ) [29]: U = − a ( T )ΦΦ ∗ + b ( T ) ln[1 − ∗ + 4(Φ Φ ∗ ) − ∗ ) ] (4) Preprint submitted to Elsevier October 7, 2018 ith a ( T ) = a T + a T T + a T T , b ( T ) = b T T .This choice of effective potential satisfies the Z (3) cen-ter symmetry of the pure gauge Lagrangian. In the con-fined phase, U has a minimum at Φ = 0, while above thecritical Temperature T its minimum is shifted to finitevalues of Φ. The logarithmic term appears from the Haarmeasure of the group integration with respect to the SU(3)Polyakov loop matrix. The parameters a , a , a and b arefixed, as in [29], by demanding a first order phase transi-tion in the pure gauge sector at T = 270 M eV , and thatthe Stefan-Boltzmann limit is reached for T → ∞ .The dynamical mass of the quarks m = m − σ = m − G S (cid:10) ΨΨ (cid:11) is the same as in the NJL model and thevector coupling induces an effective chemical potential forthe quarks µ ∗ q = µ q + ω = µ q + G V (cid:10) Ψ † Ψ (cid:11) . The two aux-iliary fields σ and ω are controlled by the potential termsand the last term includes the difference ∆ E p between thequasi particle energy and the energy of free quarks. TheNJL part of the model has 4 parameters, the bare quarkmass for the u - and d -quarks (assuming isospin symmetry),the three-momentum cutoff of the quark-loop integration λ and the coupling strengths G S and G V . To reproducerealistic values for the pion mass and decay constant aswell as the chiral condensate, we take these values to be[6]: m u,d = 5 . , G S = 10 .
08 GeV − , Λ = 651 MeV( G V will be left as a model parameter to study the influ-ence of the vector coupling on our results).The self consistent solutions are obtained by minimizingthe thermodynamic potential with respect to the fields σ , ω , Φ and Φ ∗ .
2. The QH model
To estimate the influence, of hadronic contributions,on the susceptibilities we will compare the results fromthe PNJL model with those obtained from a model wherequark and hadronic degrees of freedom are present in asingle partition function (Quark-Hadron model).In the following we will shortly describe the different com-ponents of the model, for a more detailed discussion werefer to [30]. The hadronic part is described by a flavor-SU(3) model, which is an extension of a non-linear repre-sentation of a σ - ω model including the pseudo-scalar andvector nonets of mesons and the baryonic octet [31, 32, 33].Besides the kinetic energy term for hadrons and quarks,the terms: L int = − P i ¯ ψ i [ γ ( g iω ω + g iφ φ ) + m ∗ i ] ψ i , (5) L meson = − ( m ω ω + m φ φ ) − g (cid:16) ω + φ + 3 ω φ + ω φ √ + ωφ √ (cid:17) + k ( σ + ζ ) − k ( σ + ζ ) − k (cid:16) σ + ζ (cid:17) − k σ ζ + m π f π σ + (cid:16) √ m k f k − √ m π f π (cid:17) ζ + χ − χ + ln χ χ − k χ χ ln σ ζσ ζ . (6) represent the interactions between baryons and vector andscalar mesons, the self-interactions of scalar and vectormesons, and an explicitly chiral symmetry breaking term.The index i denotes the baryon octet and the differentquark flavors. Here, the mesonic condensates (determinedin mean-field approximation) included are the vector-isoscalars ω and φ , and the scalar-isoscalars σ and ζ (strange quark-antiquark state). The last four terms of eqn.(6) were intro-duced to model the QCD trace anomaly, where the dilatonfield χ can be identified with the gluon condensate.The effective masses of the baryons and quarks are gen-erated by the scalar mesons except for an explicit massterm ( δm N = 120 MeV, δm q = 5 MeV and δm s = 105MeV for the strange quark), m ∗ i = g iσ σ + g iζ ζ + δm i . Vector type interactions introduce an effective chemicalpotential for the quarks and baryons, generated by thecoupling to the vector mesons: µ ∗ i = µ i − g iω ω − g iφ φ .The coupling constants for the baryons [34] are chosen toreproduce the vacuum masses of the baryons, nuclear sat-uration properties and asymmetry energy as well as the Λ-hyperon optical potential. The vacuum expectation valuesof the scalar mesons are constrained by reproducing thepion and kaon decay constants. For the quarks we chosethe following coupling parameters: g qσ = g sζ = 4 .
0, whilethe vector coupling strength g qω is left as free parameter.The coupling of the quarks to the Polyakov loop is intro-duced in the thermal energy of the quarks [30]. All ther-modynamical quantities, energy density e , entropy density s as well as the densities of the different particle species ρ i , can be derived from the grand canonical potential:Ω = − L int − L meson + Ω th − U (7)Here Ω th includes the heat bath of hadronic and quarkquasi particles. The effective potential U (Φ , Φ ∗ , T ) whichcontrols the dynamics of the Polyakov-loop has the formof eqn.(4).Since we expect the hadronic contribution to disappear, atleast at some point above T c , we included effects of finite-volume particles to effectively suppresses the hadronic de-grees of freedom, when deconfinement is achieved. Includ-ing these effects in a thermodynamic model for hadronicmatter, was proposed some time ago [35, 36, 37, 38]. Wewill use an ansatz similar to that in [39, 40], but modify itto also treat the point like quark degrees of freedom con-sistently. We introduce the chemical potential e µ i whichis connected to the real chemical potential µ ∗ i , of the i -th particle species by the relation: e µ i = µ ∗ i − v i P ( P isthe sum over all partial pressures and v i the volume of ahadron). All thermodynamic quantities can then be calcu-lated with respect to the temperature T and the new chem-ical potentials e µ i . To be thermodynamically consistent, alldensities have to be multiplied by a volume correction fac-tor f , which is the ratio of the total volume V and thereduced volume V ′ , not being occupied. In this configura-tion the chemical potentials of the hadrons are decreasedby the quarks, but not vice versa. As the quarks start ap-pearing they effectively suppress the hadrons by changing2 .4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.20.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.20.00.20.40.60.81.0 0.00.20.40.60.81.0 G V =0 G V =G S /2 G V =G S G V =G S *2Lattice data N = 4 N = 6 c T/ T c Figure 1: The second order quark number susceptibility from thePNJL model, with different strengths of the vector interaction, as afunction of T over T c . Black solid line: no vector interaction, reddashed line: G V = G S /
2, green dotted line: G V = G S , blue dashdotted line G V = 2 G S . their chemical potential, while the quarks are only affectedthrough the volume correction factor f . Our descriptionof the excluded volume effects is admittedly simplified andparameter dependent, but it enables us to describe a phasetransition from hadronic to quark degrees of freedom in aquite natural and thermodynamically consistent manner.As has been mentioned above, the Lagrangian of thechiral model contains dilaton terms to model the scaleanomaly. These terms constrain the chiral condensate, ifthe dilaton is frozen at its ground state value χ . On theother hand, as deconfinement is realized, the expectationvalue of the chiral condensate should vanish at some point.On account of this we will couple the Polyakov loop to thedilaton in the following way: χ = χ (1 − . ∗ )) (8)Assuming a hard part for the dilaton field which essentiallystays unchanged and a soft part, which vanishes when de-confinement is realized. Hence, allowing the chiral conden-sate to also approach zero.For a more detailed discussion of the model and compar-isons with lattice data we refer to [30].
3. Susceptibilities
Lattice results at finite chemical potentials are oftenobtained as Taylor expansion of the thermodynamic quan-tities in the parameter µ/T around zero chemical potential[41]. In the Taylor expansion of the pressure p = − Ω, thecoefficients, which can be identified with the quark numbersusceptibilities, follow from: p ( T, µ B ) T = ∞ X n =0 c n ( T ) (cid:16) µ B T (cid:17) n (9) Lattice data N τ = 4 N τ = 6 c T/ T c Figure 2: The second order quark number susceptibility from theQHM model, with different strengths of the vector interaction, as afunction of T over T c . Black solid line: g Nω = 0 , g qω = 0 , v = 1, reddashed line: g Nω = 0 , g qω = g Nω / , v = 1, green dotted line: g Nω =0 , g qω = 0 , v = 1, blue dashed dotted line: g Nω = 0 , g qω = 0 , v = 0,orange dash dot dot line: g Nω = 0 , g qω = 0 , v = 0. c n ( T ) = 1 n ! ∂ n ( p ( T, µ B ) /T ) ∂ ( µ B /T ) n (cid:12)(cid:12)(cid:12)(cid:12) µ B =0 (10)In our approach we explicitly calculate the pressure atfinite µ B and then extract the expansion coefficients nu-merically. The results for the second coefficient calculatedfor the PNJL model and QH model, compared to latticeresults [42], are shown in figures (1) and (2). One canclearly observe that the best description can be obtainedwhen quark vector interactions are turned off. Any formof repulsive interaction strongly decreases the value of thesecond order coefficient above T c . The lattice results onthe other hand quickly reach a value that is expected for anon-interacting gas of quarks, even right above T c . This isin fact surprising as other thermodynamic quantities tendto favor a picture with a wide region around T c where in-teractions are strong. This behavior was also reproducedby the PNJL model and the QH model while both fail todescribe the fast increase in c right above T c .Next we try to disentangle the hadronic contributionto the second coefficient. As can be seen in figure (2)the dependence on the strength of the repulsive interac-tion below T c is rather small and one can not exclude anyscenario. In the crossover region, around T c , differencesbecome obvious.The solid black line in Figure (2) displays the result for c using the standard parametrization of the QH model asdescribed in [30] without any repulsive quark-quark inter-actions. The value of c only slowly approaches 1, whichis mainly due to the fact that the value of the chiral con-densate drops to 0 rather slow in our model and therefore3 .8 1.0 1.2 1.4 1.6 1.80.8 1.0 1.2 1.4 1.6 1.80.00.10.20.30.40.50.6 0.00.10.20.30.40.50.6 G V =0 G V =G S /2 G V =G S G V =G S *2Lattice data N = 4 N = 6 c T/T c Figure 3: The forth order quark number susceptibility from thePNJL model, with different strengths of the vector interaction, asa function of T over T c . Black solid line: no vector interaction, reddashed line: G V = G S /
2, green dotted line: G V = G S , blue dashdotted line: G V = 2 G S . the quark masses do not decrease as fast as in the PNJLapproach.Including repulsive vector interactions for the quarksgives a result which is similar to the one obtained fromthe PNJL (red dashed line). Here the repulsive interac-tions strongly decrease the value of c .Turning off all repulsive interactions, vector interac-tions for quarks and hadrons as well as the excluded vol-ume corrections (orange dash dot dotted line), leads toan drastic overestimation of c . This is expected as allhadronic degrees of freedom are present at and above T c ifthe excluded volume effects are turned off. Therefore onelargely overestimates the effective degrees of freedom.On the other hand if the repulsive vector interactionsare turned on only for the hadrons one obtains a rathergood description of the lattice results (blue dash dottedline). In this parametrization the hadrons are also stillpresent in the system up to arbitrary high temperatures,as the excluded volume effects are still turned of, andtherefore all thermodynamic quantities are largely overpredicted.To remove the hadronic contributions from the systemwe introduced excluded volume corrections as describedabove. This leads to a pronounced dip in c above T c (green dotted line), indicating that our excluded volumeapproach is either to simplified or all hadronic contribu-tions are already vanishing completely right above T c orthat the lattice results are still not accurate enough to suf-ficiently resolve effects of hadron hadron interactions.The fourth order coefficients calculated for the PNJL Lattice data N τ = 4 N τ = 6 c T/T c Figure 4: The fourth order quark number susceptibility from theQHM model, with different strengths of the vector interaction, as afunction of T over T c . Black solid line: g Nω = 0 , g qω = 0 , v = 1, reddashed line: g Nω = 0 , g qω = g Nω / , v = 1, green dotted line: g Nω =0 , g qω = 0 , v = 1, blue dashed dotted line: g Nω = 0 , g qω = 0 , v = 0,orange dash dot dot line: g Nω = 0 , g qω = 0 , v = 0. and QH model are shown in figures 3 and 4. Although theerrors on the lattice results are still significant our resultsfor c support the statements made for c . At tempera-tures above T c a gas of free non interacting quarks, with-out any hadronic contribution, gives the only reasonabledescription of the data already slightly above T c . Whilethe height of the peak does depend on the strength of thevector coupling, we observe also a strong dependence onthe slope of the change of the order parameters around T c ,which is larger in the QH model as compared to the PNJLmodel. Notice that the parametrization shown as the bluedash-dotted line, which gave the best description for c ,overestimates c around T c drastically. This is simply be-cause all thermodynamic quantities are over predicted inthis case and therefore the densities as well as the orderparameters increase steeper as they do in the lattice cal-culations.
4. Conclusion
We have compared results for the quark number sus-ceptibilities at µ B = 0 obtained from two mean field mod-els to recent results from lattice calculations. Both modelsstrongly indicate that the EoS of QCD above T c seems tobe composed purely of a gas of non-interacting quasi par-ticles. This finding is in agreement with recent work by[43] where the extracted quark vector coupling strengthapproaches zero very fast around T c . On the other hand,lattice observables like the interaction measure and thenormalized Polyakov loop indicate a large region abovethe critical temperature where the hot QCD medium is far4rom being an ideal gas. Around T c repulsive hadronic in-teractions are supported by our results with the QH model.To describe the steep increase of c around T c we need tointroduce hadronic contributions up to right above T c . Ateven larger temperature the hadrons seem to be replacedby a almost non interacting gas of quarks. This offers anintriguing implication concerning the CeP. In mean fieldstudies with the PNJL [4] as well as in the HQ model [30],a general feature of including repulsive interactions is thatthe CeP is moved to larger chemical potentials or evendisappears completely. From this point of view, a vanish-ing quark vector interaction would favor the existence ofa CeP.On the other hand, the phase structure of QCD up to T c and likely even slightly above it could be determined byhadronic interactions. In model calculations without anyrepulsive quark interactions the quarks often appear atrather small chemical potentials (at T = 0) making a rea-sonable description of the nuclear ground state difficult.A repulsive hadronic vector interaction, which is requiredon order to reproduce the properties of a saturated nuclearground state, may move the CeP to larger chemical poten-tials or even remove it completely from the phase diagramin accordance to mean-field results [4, 43].
5. Acknowledgments
This work was supported by BMBF, GSI and the Hes-sian LOEWE initiative through the Helmholtz Interna-tional center for FAIR (HIC for FAIR). The computationalresources were provided by the Frankfurt Center for Sci-entific Computing (CSC).
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