Baryon Resonances in a Chiral Hadronic Model for the QCD Equation of State
aa r X i v : . [ h e p - ph ] F e b Baryon Resonances in a Chiral Hadronic Model for the QCD Equation of State
Philip Rau,
1, 2, ∗ Jan Steinheimer,
1, 2
Stefan Schramm,
1, 2 and Horst St¨ocker
1, 3 Institut f¨ur Theoretische Physik, Johann Wolfgang Goethe-Universit¨at,Max-von-Laue-Str. 1, 60438 Frankfurt am Main, Germany Frankfurt Institute for Advanced Studies (FIAS),Ruth-Moufang-Str. 1, 60438 Frankfurt am Main, Germany GSI Helmholtzzentrum f¨ur Schwerionenforschung GmbH, Planckstr. 1, 64291 Darmstadt, Germany
In this paper we study the influence of hadronic resonances on the phase diagram calculatedwith an effective chiral flavour SU(3) model. We show that varying the couplings of the baryonicresonances to the attractive scalar and the repulsive vector fields has a major impact on the orderand location of the chiral phase transition and the possible existence of a critical end point as wellas on the thermodynamic properties of the model. Furthermore, we study (strange) quark numberfluctuations and show the related susceptibilities both at zero baryochemical potential and whencrossing the phase transition line at three different points in the T – µ plane. We obtain the bestagreement with current lattice data if we choose a rather strong vector coupling which in our modellimits the phase transition to a smooth crossover and implies the non-existence of a critical endpoint. I. INTRODUCTION
Currently there are many efforts both on the theoreti-cal, as well as on the experimental side to gain knowledgeabout the phase diagram of strongly interacting nuclearmatter. Experiments with highly energetic colliding goldnuclei at the Relativistic Heavy Ion Collider (RHIC) sug-gest that a new state of matter, comparable to a nearlyperfect fluid, is created at high temperatures [1–4]. Whilethese experiments at RHIC and the Large Hadron Col-lider (LHC), performed at very high beam energies, aimat the high-temperature, low-density region of the phasediagram, there are also experiments in which higher bary-onic densities may be reached, as for example planned atthe upcoming Facility for Antiproton an Ion Research(FAIR) at GSI. Common to all these experiments is thesearch for the phase transition of strongly interactingmatter from a confined state of hadrons at low tempera-tures to a state with deconfined quarks and gluons (QGP)as it is predicted for high densities and high tempera-tures [5]. This phase transition is commonly referred toas deconfinement transition [6]. The symmetries of quan-tum chromodynamics (QCD), however, imply another, chiral phase transition at which the chiral symmetry isrestored and the masses of the baryons, or constituentquarks, vanish [7–9]. For this transition, on which wewill mainly focus our studies, the chiral condensate σ acts as the corresponding order parameter.Since QCD can not be treated perturbatively at lowtemperatures, most information we have about the phasediagram of QCD matter today comes from lattice QCDcalculations [10–19]. In the region of vanishing bary-ochemical potentials µ = 0 and finite temperature, latticeQCD yields reliable results that show a smooth crossovertransition [15]. For finite potentials µ >
0, however, re-sults from lattice QCD are not available because of the ∗ [email protected] failing of standard Monte Carlo sampling methods dueto the so-called sign-problem. Currently there are vari-ous methods to extend the lattice results from µ = 0 tothe region of small chemical potentials [11, 12, 20–29].Some lattice QCD groups suggest that the phase transi-tion becomes first order at a critical endpoint [13, 20, 30–32] with its coordinates varying considerably for differentcalculations. Other more recent results [33] favour the ex-clusive existence of a crossover transition for a wide rangeof baryon densities, implying the non-existence of such acritical point. Moreover, the exact position of the phasetransition in the phase diagram is subject of a lively andongoing debate.Other common theoretical approaches to QCD are ef-fective models to study specific properties of QCD mat-ter [7, 34–40]. In our approach we will use an effectivemodel for the QCD equation of state to study the prop-erties of the phase diagram of nuclear matter. Our modelis able to reproduce the well known saturation propertiesof nuclear matter [41, 42].The degrees of freedom in our model are all knownbaryons and baryonic resonances up to masses of m =2 . T ≤ T c ) and was often used to reproduce lat-tice QCD results for thermodynamics and fluctuations ofconserved charges at low temperatures [14, 43–48]. Theparticle production in heavy ion collisions may also bereasonably well described with use of the HRG [49–53].Therefore, our first approach to use only hadronic degreesof freedom seems to be appropriate up to at least tem-peratures in the range of T c . For higher temperatures, atthe latest when the deconfinement phase transition setsin, this approach breaks down and, in a future step, par-tonic degrees of freedom need to be taken into accountas it was done in Refs. [54, 55] for example.This paper is organised as follows. First we introducethe chiral SU (3) model in section II. In section III, wepresent results for the order parameter of the chiral con-densate σ at zero and nonzero baryochemical potentials,thermodynamic quantities from the model, and quarksusceptibilities at different points in the phase diagram.Susceptibilities are interesting quantities since they re-semble fluctuations of conserved charges which itself areclosely linked to phase transitions. They offer an effec-tive possibility of comparison between results from the-ory and experiment because the susceptibilities can berelated to measured fluctuations of particle production.Of particular interest are susceptibility ratios since theyare not dependent on the volume or the impact parame-ter of the underlying system [56]. This work closes witha conclusion in section IV. II. MODEL
In our model, a SU (3)-flavour sigma-omega model us-ing the non-linear realization of chiral symmetry (seeRefs. [37, 41, 42, 57] for a comprehensive review), theLagrangian in mean field approximation has the form L = L kin + L int + L meson . (1)Here, the first term represents the kinetic energy of thehadrons, the terms L int = − X i ¯ ψ i (cid:0) m ∗ i + g iω γ ω + g iφ γ φ (cid:1) ψ i (2)describe the interaction of the baryons with the scalarmesons σ , ζ (attractive interaction, see Eq. (5)) and thevector mesons ω , φ (repulsive interaction) respectively.The summation index i runs over the baryon octet ( N , Λ,Σ, Ξ), the baryon decuplet (∆, Σ ∗ , Ξ ∗ , Ω), and all heavierresonance states up to masses of m N ∗ = 2600 MeV. Weonly include hadronic resonances whose existence is con-sidered to be very likely according to the Particle DataGroup listings [58] where they are recorded with a min-imum three-star rating. Since the listed masses of someheavy resonances may cover a broad range, all particlesare included with their average mass.The third term of the Lagrangian L meson = L vec + L + L ESB (3)= + 12 (cid:0) m ω ω + m φ φ (cid:1) + g (cid:18) ω + φ ω φ + 4 ω φ √ ωφ √ (cid:19) − k ( σ + ζ ) + k ( σ + ζ ) (4)+ k (cid:18) σ ζ (cid:19) + k σ ζ + k ln σ ζσ ζ − m π f π σ + (cid:18) √ m k f k − √ m π f π (cid:19) ζ includes the self interactions of the vector mesons and thescalar mesons together with the last two terms describingthe explicit symmetry breaking.The effective masses of the baryons m ∗ i = g iσ σ + g iζ ζ + δm i (5)are created by the coupling of the baryons to the scalarmeson fields (i.e. the non-strange chiral condensate σ andits strange equivalent ζ ), together with an explicit massof at least δm N = 150 MeV. In this way at high temper-atures and baryonic densities the decreasing σ -field leadsto smaller baryon masses and thus to the restoration ofchiral symmetry. Thereby the nucleons which have thesmallest explicit mass have lost roughly 45% of their vac-uum mass at T c . Since our model only includes hadronicdegrees of freedom it can only be applied in the hadronicregime up to temperatures slightly above T c . This alsoprevents baryonic masses from getting to small so thatthe mean field approximation should remain valid and nofluctuations need necessarily to be taken into account asit was for example done in Refs. [59, 60].The effective masses of the pseudoscalar mesons andvector mesons are given by the second derivative of themesonic potential V meson = −L vec − L − L ESB with re-spect to the respective mesons ξ j at the minimum of thegrand canonical potential (see Eq. (9)) [61] m ∗ j = ∂ ∂ξ j V meson ( ζ j ) , (6) ξ j = π, η, η ′ , K, ¯ K ; ρ, ω, ϕ, K ∗ , ¯ K ∗ . The couplings of the baryon octet to the mesonic fieldsand the mesonic potential are chosen in such a way as toreproduce the well-known vacuum masses, the nuclearground state properties (e.g. the correct binding energy),and the asymmetry energy. The coupling strengths ofthe baryons of the decuplet and all heavier resonancesare scaled to the nucleon couplings via the parameters r s , r v according to g Bσ,ω = r s,v · g Nσ,ω , (7) g Bζ,φ = r s,v · g Nζ,φ . (8)In this paper we systematically analyse the influence ofthe baryonic resonances on the hadronic matter proper-ties. Therefor we adjust the vector coupling parameter r v , controlling the abundance of the baryonic resonancesat finite baryochemical potentials in the model, in orderto study the influence on the resulting phase diagramand the thermodynamic properties of the model. Weuse this one-parameter approximation in order not to beswamped by a plethora of of unknown coupling constantsof the various hadronic multiplets. In this work we setthe scalar coupling parameter fixed to r s = 0 .
97 whichensures a smooth crossover phase transition at zero bary-ochemical potential (see Fig. 1 (b) for a study of the ef-fect of r s ). In general the scalar couplings are fixed byreproducing the particles’ vacuum masses (except for theexplicit mass term δm i ).As a reference we also perform calculations for an idealHRG that is not interacting with the mesonic fields. Inthis particular case all baryon couplings to the fields g B are set to zero and the masses of all particles are fixed attheir tabulated vacuum expectation value.The grand canonical potential of our model takes theform Ω V = −L int − L meson + Ω th (9)with the thermal contribution of the hadrons in the modelΩ th = − T X i ∈ B γ i (2 π ) Z d k (cid:16) ln h e − T ( E ∗ i ( k ) − µ ∗ i ) i + ln h e − T ( E ∗ i ( k )+ µ ∗ i ) i(cid:17) (10)+ T X j ∈ M γ j (2 π ) Z d k ln h − e − T ( E ∗ j ( k ) − µ j ) i . Here the sums run over all baryons B (anti baryons areexplicitly included in the second term of the first inte-gral) and all mesons M in the model, γ i,j stands for thespin-isospin-degeneracy factor of the respective particlespecies i, j and E ∗ i,j ( k ) = q k + m ∗ i,j for the single par-ticle energies. The effective baryochemical potential isdefined as µ ∗ i = µ i − g iω ω − g iφ φ . The grand canonicalpotential then leads to the thermodynamic quantities ofthe system, i.e. the pressure p and the energy and entropydensity e , s , together with the densities of the particularparticle species ρ i .Another effect we include in our model (following theworks of Refs. [55, 62]) is the effective suppression ofbaryonic states at high densities due to excluded volumeeffects [63–66]. While so far all particles were regardedas point like, we now can introduce for every single par-ticle j an average finite volume v j ex excluded from thetotal volume of the system V . In a first and very basicconsideration this excluded volume in a non-relativisticdefinition takes the form v j ex = 12 a i π (2 r ) , (11)where r is the mean radius of all particles in the modeland the parameter a i defines the excluded volume for thedifferent particle species i . In our case we set a B , M = 1as a first approximation for all baryons and mesons. Thismodification leads to the altered chemical potential µ ′ j = µ j − v j ex P, (12)where P is the sum over all partial pressures. All ther-modynamic observables must now be expressed in termsof T and µ ′ j . Furthermore, a volume correction factor foreach particle species f i = V ′ i V = X j v j ex ρ j − (13) with V ′ i the volume not being occupied, has to be intro-duced in order to express the densities in a thermody-namically consistent way ρ ′ i = f i ρ i , (14) e ′ = X i f i e i , (15) s ′ = X i f i s i . (16)The inclusion of the excluded volume effects also has animpact on the nuclear ground state properties and thusthis approach is considered only as a first attempt tostudy the impact of the suppression of baryons on thephase transition and on the thermodynamic propertiesof the model. However, reasonable nuclear ground stateproperties can be achieved by recalibration of the param-eters as it was shown in Ref. [67]. III. RESULTSA. Order Parameters
We start our investigation of the model properties withthe calculation of the order parameter for the chiraltransition σ at zero baryochemical potential µ B . Fig-ure 1 (a) shows the normalised chiral order parame-ter as a function of the temperature together with datafrom lattice QCD calculations. Here and in the follow-ing, the lattice data we refer to were obtained by differ-ent collaborations using various lattice actions (asqtad,hisq, p4 and stout) and temporal spacings of the lattice( N τ = 4 −
12) [33, 46, 48, 68–73].With the scalar couplings fixed to r s = 0 .
97 we ob-tain a smooth crossover in σ for both the model withexcluded volume (green dashed line) effects due to fi-nite size of the particles and without it (red solid line).The critical temperature, defined as the point with thelargest increase in σ , is found to be T c = 164 MeV with-out the excluded volume effect and T c = 174 MeV ifthe finite size effects are taken into account. The resultsfor σ without the excluded volume effects are in quali-tatively good agreement with lattice QCD calculationswhich predict a smooth crossover at µ B = 0 and crit-ical temperatures in the range from T c = 155 MeV to200 MeV [15, 33, 46, 68–72, 75], where the newest con-tinuum extrapolated data from the Hot-QCD and theWuppertal-Budapest Collaborations predict critical tem-peratures close to T = 160 MeV consistently. For tem-peratures below T C the slope of the chiral condensatefrom our model is relatively steep and deviates from thosesuggested by lattice QCD calculations. This is mainlydue to neglecting contributions from pseudoscalar me-son self interactions which are important at low temper-atures [76]. The slope of σ could be leveled by includingthe π self interaction in the model [67, 77]. s / s p4, N t =8asqtad, N t =8asqtad, N t =12hisq, N t =8hisq, cont.stout, cont.excl vol 0excl vol 1HRG r s = 0.97, m B = 0 (a) s / s T [MeV]r s =0.0r s =0.3r s =0.6r s =0.9orig. mod. m B = 0 (b) Figure 1. (Color online) (a) Normalised order parameter forthe chiral condensate σ/σ as a function of T at µ B = 0with (green dotted line) and without (red solid line) excludedvolume corrections compared to lattice data. The blue dottedline at σ/σ = 1 shows the reference line of the non-interactingideal hadron resonance gas. Here and in the following plotsthe lattice data are shown as calculated with the asqtad [68],the hisq [69, 70], the p4 [48, 68, 71], and the stout action [46,72, 74] on lattices with different temporal extent N τ . Panel(b) shows σ/σ for different strength of the scalar coupling r s together with results from our model without any resonancestates. The impact of the heavy resonances states can be seenfrom Fig. 1 (b) where the black dots show the normalisedchiral order parameter as a function of the temperature ascalculated with our original model which does not includeany hadronic resonance states [41, 42]. It shows thatwithout the hadronic resonance states the transition ismuch smoother and exhibits a higher critical temperatureclose to T c = 200 MeV. The same effect is achieved bydecreasing the strength of the attractive coupling of thebaryons to the scalar σ -field. This behavior is also shownin Fig. 1 (b) for values of r s from 0 to 0 .
9. The weakerthe scalar coupling is, the flatter is the transition andthe critical temperature moves to higher temperatures.For vanishing scalar couplings (red line) the results arethe same as in the original model. Note that for smallvalues of r s one would have to introduce large valuesfor the explicit mass term as the sum of the meson-fieldgenerated mass and the explicit term have to reproducethe vacuum masses of the states.Next we extend this study of σ to nonzero baryochem-ical potentials, i.e. the whole T – µ B plane. One of themajor interests in the study of the phase diagram ofstrongly interacting matter is certainly to obtain infor-mation about the chiral and deconfinement phase transi-tions and the search for a possible critical end point which T [ M e V ] m B [MeV] r s =0.97 r v : 0.00.20.40.60.81.0 Figure 2. Phase transition lines of the chiral order parameter σ in the T – µ B plane for different values of r v together withthe nuclear ground state at µ B = 939 MeV, T = 0 and theliquid gas first order phase transition. The point on eachphase transition line depicts the critical end point, separatinga first order phase transition (solid lines) from a crossover(doted lines). Note that for values r v > . T – µ B plane. divides the crossover phase transition at vanishing bary-ochemical potential from a first order phase transition atfinite chemical potentials. Since lattice QCD calculationsare systematically limited to µ B = 0 due to the so-calledsign problem, different lattice QCD groups use variousmethods to extend their results to non-vanishing poten-tials. Unfortunately, this did not lead to a consistent pic-ture for the critical end point until now. While resultsfrom (2+1)-flavour QCD calculations suggested the criti-cal end point at a critical quark chemical potential in therange from µ crit B ≈
725 MeV [20] to µ crit q ≈
140 MeV [73](with 3 µ q = µ B ) and µ crit B ≈
360 MeV [13], more recentstudies of the Wuppertal-Budapest Collaboration [33]doubt the existence of a critical end point at all and sug-gest a broad crossover phase transition over the whole T – µ B plane.In Fig. 2 we show the results from our model with-out excluded volume effects for different values of thecoupling constant of the repulsive vector interaction r v .In this figure the lines represent the biggest increase ofthe σ -field when varying T and µ . While a discontin-uous first order phase transition in σ is depicted by asolid line, a smooth crossover in σ is drawn with a dot-ted line. The point on each phase transition line standsfor the critical end point for the specific choice of r v . Wefind that increasing the vector coupling strength leads toa stronger suppression of heavier particles and thus toa phase transition at higher chemical potentials. Notethat for all coupling strengths smaller than r v = 0 . µ B = 939 MeV, T = 0 and thefirst order liquid gas phase transition are located abovethe phase boundary in the chirally restored phase. There-fore, within our simple model approximations those smallvector couplings do not lead to physically reasonable so-lutions. Thus, we conclude according to our model thesmallest critical chemical potential (for r v = 0 .
4) is lo- p / T T [MeV] (b) e / T (a) excl vol 0excl vol 1HRGasqtad, N t =8p4, N t =8stout, cont. r s = 0.97, m B = 0 Figure 3. (Color online) Energy density (a) and the pressure(b) divided by T at µ B = 0 as a function of the temper-ature. Depicted are results for the interacting hadronic gaswith (green dashed line) and without (red solid line) excludedvolume corrections and for the non-interacting ideal hadronresonance gas (blue dotted line) together with lattice data. cated at µ crit B ≈
840 MeV which is two times higher thanthe value suggested in Ref. [73].For vector couplings of r v = 0 .
6, in our model, thereis a very small temperature range of first order phasetransition going up to T crit ≈
20 MeV, at higher temper-atures there is a broad-ranged crossover transition. Forall higher vector couplings, the first order phase transi-tions vanish completely and only a broad crossover regionremains.
B. Thermodynamics
The thermodynamic quantities, as predicted by ourmodel at µ B = 0 are depicted in Fig. 3. Panel (a) showsthe energy density and panel (b) shows the pressure bothdivided by T , in comparison to lattice data. Here andin the following plots, the solid red curve represents theresults from our regular model including all baryonic res-onances, the green dashed curve depicts the results fromthe model with the excluded finite size volume effectstaken into account and the blue dashed line stands forthe non-interacting ideal HRG without finite size effects.For the standard case in our model, at T c the energy ( e − ) / T T [MeV]excl vol 0excl vol 1HRGp4, N t =8asqtad, N t =12hisq, N t =8, 0.05 m s hisq, N t =8, 0.2 m s stout, cont. r s = 0.97, m B = 0 Figure 4. (Color online) The interaction measure, three timesthe pressure subtracted from the energy density over T , asa function of the temperature at µ B = 0 for the interactinghadronic gas with (green dashed line) and without (red solidline) excluded volume corrections and for the non-interactingideal hadron resonance gas (blue dotted line) compared tolattice data [19, 46, 48, 68, 69, 71, 72, 74, 78]. The data forthe hisq action is shown for two different values of the lightquark mass m l = 0 . m s and 0 . m s , where m s stands forthe strange quarks mass. density and the pressure rise rapidly due to the emerg-ing abundance of particles in the system because of theirdecreasing masses. Besides a slightly higher critical tem-perature, the suppression of heavier particles caused bythe finite size effects leads to a much smaller maximumof the energy density at T C and a decrease for highertemperatures. In this case, three times the pressure di-vided by T exhibits only a slight increase with a maxi-mum around T c and decreases smoothly again for highertemperatures. The HRG results rise monotonically, asexpected.Figure 4 shows the so-called interaction measure de-fined as the energy density minus three times the pres-sure divided by T as a function of the temperature.Again, the results from our regular model (red curve)show a rapid increase of this quantity at T c , the HRGrises monotonously being in qualitatively good agreementin the region around the critical temperature with latticedata [48, 68, 71] which shows a peak slightly above T c .Comparing our results for the thermodynamic quan-tities to lattice data, we state that a good agreementup to the critical temperature is achieved for our resultsfrom the interacting HRG without excluded volume ef-fects and the continuum extrapolated stout action withphysical quark masses. However, for temperatures evenbelow the critical temperature, we overshoot the resultsfrom all other lattice actions. c T/T C (b) c (a) excl vol 0excl vol 1HRGp4, N t = 4p4, N t = 6hisq, N t = 8 r s = 0.97, r v = 0.8, m B = 0 Figure 5. (Color online) Second order (a) and fourth order(b) quark number susceptibilities as a function of
T /T c at µ B = 0. Results for the interacting hadronic gas ( r v = 0 . C. Susceptibilities
As mentioned above, lattice QCD calculations arelargely restricted to zero chemical potential because ofthe sign problem. However, thermodynamic quantitiesat non-vanishing potentials may be obtained by Taylorexpansion methods as described in Refs. [22, 48, 73, 79].The pressure at a specific point in the phase dia-gram p ( T, µ B ) with preferably small chemical potentialsis calculated by expanding the pressure at p ( T, µ B = 0)around µ B /T p ( T, µ B ) T = ∞ X n =0 c n ( T ) (cid:16) µ B T (cid:17) n (17)with the Taylor coefficients c n ( T ) = 1 n ! ∂ n (cid:0) p ( T, µ B ) /T (cid:1) ∂ ( µ B /T ) n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) µ B =0 . (18)The coefficients c n ( T ) are proportional to the quarknumber susceptibilities reflecting quark number fluctu-ations [80].Similarly, by expanding the pressure with respect tothe strange chemical potential µ s , the first order strange c / c T/T c excl vol 0excl vol 1HRGp4, N t = 4p4, N t = 6 r s = 0.97, r v = 0.8, m B = 0 Figure 6. (Color online) Ratio of the fourth order to the sec-ond order quark number susceptibility as a function of
T /T c at µ B = 0 for the three different cases as described in Fig. 5together with lattice data [48, 68, 71, 73]. quark susceptibility χ s ( T ) can be found to be χ s ( T ) = T ∂ ( p ( T, µ s )) ∂µ s (cid:12)(cid:12)(cid:12)(cid:12) µ s =0 . (19)In contrast to lattice QCD, with our model we are able tocalculate the pressure at any point in the phase diagramand can therefore numerically calculate the correspond-ing susceptibilities.In Fig. 5 we show the second order c (a) and fourthorder c (b) quark number coefficients in comparison tolattice QCD data [48, 68, 71, 73] at µ B = 0. Both,the model with and without the excluded volume effectsshow a maximum of c at the critical temperature anda smooth decrease for higher temperatures. The fourthorder coefficient c exhibits a narrow peak at T c . As ex-pected, the ideal non-interacting HRG susceptibilities donot show any peak due to the absence of a phase transi-tion. The susceptibilities of the ideal HRG come closestto the lattice data with the p4-action. However, a com-parison to the not yet available susceptibilities from thecontinuum extrapolated stout action would be very in-teresting since the lattice results with the stout actiongives the best agreement for the thermodynamic quanti-ties and the strange quark susceptibility (Fig. 7).In our study, we found that the strength of the repul-sive vector coupling has a major influence on the extentof the fluctuations at the phase transition, even thoughthe derivative is performed at µ B = 0. This effect was al-ready pointed out in Refs. [81, 82]. A lower vector repul-sion leads to significantly higher fluctuations of conservedcharges at T c . In our study we set the vector coupling to r v = 0 . T ≈ T c . This vector coupling strength stands incontradiction to the one we found earlier to be suitableto reproduce a critical end point in the region suggestedby lattice QCD results ( r v ≤ .
2, cf. Fig. 2).In Fig. 6 we show the ratio of the fourth order to the c s / T T [MeV]excl vol 0excl vol 1HRGp4, N t = 8asqtad, N t =12hisq, N t =8stout, cont. r s = 0.97, m B = 0 Figure 7. (Color online) Strange quark number susceptibilitydivided by T at µ B = 0 as a function of T for the threedifferent cases as described in Fig. 5 together with latticedata [33, 46, 69, 70, 72]. The vector coupling strength is setto r v = 0. second order baryon number susceptibility. For the reg-ular model without finite size corrections the ratio is inline with the completely flat curve of the HRG up to thecritical temperature. The curve exhibits a discontinuityat T c with a sudden decrease above. For higher temper-atures the curve is mostly constant around c /c ≈ . T c .The strange quark number susceptibility χ s dividedby T is shown in Fig. 7. Again, the three cases ofour model yield significantly different results. The curvefor the non-interacting ideal HRG rises again monoton-ically as expected. The regular model with all interac-tions switched on shows a massive rise at T c . The resultsfrom our interacting model is once again in good agree-ment with results from lattice QCD with the continuumextrapolated stout action up to temperatures of T c . Inthe case with the excluded volume effects, in particularheavy strange particles are suppressed and thus the curveis much lower in this case showing a maximum at T c anda slow decrease for higher temperatures.In a next step we study the susceptibility coefficients atnon-zero baryochemical potentials. Therefore, we com-pare the coefficients in three different regions of the phasediagram, namely in the crossover region, at the criticalend point, and at the first order phase transition. For thispurpose, in a first attempt we set the vector coupling to r v = 0, because this leads to a clearly observable andwell defined phase transition line and a critical end pointat µ crit B ≈
220 MeV; later on we will also calculate sus-ceptibilities with more realistic and stronger vector cou-plings. Note, that for higher values of r v this subdivisionof the phase diagram into three clearly distinguishableregions no longer applies. For a vector coupling strength c T/T c (b) CEPCOPTHRG at CEPp4, N t = 4p4, N t = 6hisq, N t = 8 r s = 0.97, r v = 0.9 c (a) r s = 0.97, r v = 0 Figure 8. (Color online) Second order quark number suscep-tibilities c for vector couplings r v = 0 (a) and r v = 0 . µ B = 0.For r v = 0 the blue line depicts the crossing of the phaseboundary in the first order phase transition regime, the greenline in the crossover regime, and the red line shows c goingdirectly through the critical end point (see text for more in-formation). Shown is also the reference of the ideal hadronresonance gas at the critical end point and lattice data at µ B = 0 [48, 68, 71, 73] . of r v > .
6, our model exhibits only a broad crossovertransition in the whole T – µ B plane (cf. Fig. 2).With our model we then calculate the second andfourth order susceptibilities along straight lines perpen-dicular to the phase boundary going through the pointson the phase transition line µ (1) B = 29 . T (1) =161 . µ (2) B = 216 . T (2) = 150 MeV at the critical end point, and µ (3) B =489 . T (3) = 105 MeV for the first order phasetransition region.In Fig. 8 we show the second order coefficients for thethree different regimes for r v = 0 in panel (a) and for r v =0 . µ B = 0.Note, that choosing a vanishing vector coupling of thebaryon resonances leads to unreasonably high suscepti-bilities at T c and must be regarded as a limiting test case.The green dashed curve shows the susceptibilities in thecrossover region at low µ B . Due to the smooth transitionof the quantities, the susceptibility only shows a small c / c T/T c (b) CEPCOPTHRG at CEPp4, N t = 4p4, N t = 6 r s = 0.97, r v = 0.9 c / c (a) r s = 0.97, r v = 0 Figure 9. (Color online) Ratios of the fourth order to the sec-ond order quark number susceptibilities for vector couplings r v = 0 (a) and r v = 0 . maximum (rise) at T c for r v = 0 . r v = 0). The suscep-tibilities at the discontinuous first order phase transition(blue dashed line) show the highest maximum for bothvalues of r v , as expected. At the critical end point (redsolid line), the maximum of c is located in between theresults from the crossover and the first order phase transi-tion region. As shown previously, the susceptibilities forthe non-interacting ideal HRG (pink dashed line) showa monotonous rising behaviour not being affected by avaried vector coupling strength.The ratio of the fourth order to the second order coeffi-cients in the three different regions of the phase diagramis shown in Fig. 9 for vector couplings r v = 0 in panel (a)and r v = 0 . T c . At non-vanishing baryochemical poten-tials the ratios deviate significantly from the HRG resultswhat is similar to the findings of suppressed fluctuationsin higher order cumulants in the vicinity of the transitionregion at zero baryochemical potential [83]. In our casethe fluctuations are suppressed by the repulsive vectorinteractions. This fact could help to probe the locationof the phase transition and a possible critical end point experimentally by extraction of the susceptibilities fromdata.However, the good agreement of our results for thesusceptibilities in the crossover region given a vector cou-pling strength of r v = 0 . σ in the whole T – µ plane (Fig. 2), leads us to theconclusion, that the existence of a critical end point ac-cording to our model is very questionable. The resultsfrom our model suggest that for reasonable values of thevector coupling, throughout the entire phase diagram thephase transition is a smooth crossover, ruling out the ex-istence of a critical end point. This finding correspondswell with the results from most recent lattice data [33]. IV. CONCLUSIONS
We presented an effective chiral SU (3) model for theQCD equations of state. In this sigma-omega model weincluded all known hadrons up to resonances with massesof 2 . r v that controls thecoupling strength of the baryons to the repulsive vectormeson field. Furthermore, we include a finite size effectthat effectively suppresses heavy particles at higher den-sities.Using this model, at zero baryochemical potential µ B we found a smooth crossover phase transition for the or-der parameter of the chiral condensate σ with a criti-cal temperature in the range from T c = 164 MeV to T c = 174 MeV depending on the excluded volume ef-fects being taken into account. These results are in goodagreement with various data from lattice QCD. Extend-ing this study to finite µ B , we show the strong depen-dence of the phase transition, i.e. position and order, onthe vector coupling strength. For reasonable values of r v ,we find that the phase transition is a smooth crossoverin the whole T – µ B plane and that there is no critical endpoint.We also show the thermodynamic quantities from themodel and calculate the quark number and strange quarknumber susceptibility coefficients at different values of µ B . The susceptibilities show a good qualitatively agree-ment with lattice data if a sufficiently strong vector cou-pling is chosen. This finding underlines the model sug-gestion of the non-existence of the critical end point. Ifthe susceptibilities are extracted at different positions onthe phase boundary, we show that their significantly dif-ferent behavior may be used to distinguish the order ofthe phase transition at a given point. V. ACKNOWLEDGEMENTS
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