The existence and the location of the critical end point (CEP) between the crossover and the first order part of the chiral phase transition in the phase diagram of the strongly interacting matter is a heavily studied area of recent particle physics. The baryon number fluctuations and related quantities such as kurtosis and other susceptibility ratios, that are assumed to be good signatures of CEP, are calculated in an (axial)vector meson extended (2+1) flavor Polyakov linear sigma model (ELσM) at zero and finite μB . It is compared with the results of lattice as well as other effective model calculations. Divergence of the kurtosis is found at the critical end point

Kovacs, Gyozo

Kovacs, Peter

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BARYON FLUCTUATIONS IN EXTENDEDLINEAR SIGMA MODEL∗Gyo˝zo˝ Kova´cs1,2, Pe´ter Kova´cs1,21ELTE Eo¨tvo¨s Lora´nd University, Pa´zma´ny Pe´ter s. 1/A,H-1117, Budapest, Hungary2Institute for Particle and Nuclear Physics, Wigner Research Centre for Physics,H-1525 Budapest, HungaryThe existence and the location of the critical end point (CEP) betweenthe crossover and the first order part of the chiral phase transition in thephase diagram of the strongly interacting matter is a heavily studied areaof recent particle physics. The baryon number fluctuations and relatedquantities such as kurtosis and other susceptibility ratios, that are assumedto be good signatures of CEP, are calculated in an (axial)vector mesonextended (2 + 1) flavor Polyakov linear sigma model (ELσM) at zero andfinite µB . It is compared with the results of lattice as well as other effectivemodel calculations. Divergence of the kurtosis is found at the critical endpoint.PACS numbers: 12.39.Fe, 14.40.Be, 14.40.Df, 14.65.Bt, 25.75.Nq1. IntroductionThe existence and location of a possible critical end point (CEP) betweencrossover and first order chiral phase transition regions on the phase diagramof strongly interacting matter is conjectured by many theoretical models [1].If there is such a point it should be located in a region, which is unreachableneither by perturbative quantum chromodynamics (QCD) nor with latticecalculations, therefore, we only left with effective field theories. An effective,axial vector and vector meson extended (2 + 1) flavor Polyakov linear sigmamodel (ELσM) was built in [2, 3]. According to symmetry considerations∗ Presented at the Excited QCD 2020 Conference, Krynica Zdo´j, Poland, February 2-8,2020.(1)arXiv:2008.02535v1  [hep-ph]  6 Aug 20202 Proc˙for˙eQCD printed on August 7, 2020the Lagrangian of the model has the form:L =Tr[(Dµφ)† (Dµφ)]−m0Tr(φ†φ)− λ1[Tr(φ†φ)]2− λ2[Tr(φ†φ)2]+ c1(detφ+ detφ†)+ Tr[H(φ+ φ†)]− 14Tr [LµνLµν +RµνRµν ]+ Tr[(m212+ ∆)(LµLµ +RµRµ)]+h12Tr(φ†φ)Tr [LµLµ +RµRµ]+ h2Tr[(φRµ)† (φRµ) + (Lµφ)† (Lµφ)]+ 2h3Tr[Rµψ†Lµφ]− 2ig2 (Tr{Lµν [Lµ, Lν ]}+ Tr{Rµν [Rµ, Rν ]}) + ψ¯ [iγµDµ −M]ψ.(1)Here the covariant derivatives can be written in terms of the electromagneticfields Aµ, and the gluon fields Gµ = gsGµaT a, asDµφ = ∂µφ− ig1(Lµφ− φRµ)− ieAµ [T3, φ] ,Dµψ = ∂µψ − iGµψ, (2)and the field strength tensors of the left and right handed (axial) vectorfields areLµν = ∂µLν − ieAµ [T3, Lν ]− {∂νLµ − ieAν [T3, Lµ]},Rµν = ∂µRν − ieAµ [T3, Rν ]− {∂νRµ − ieAν [T3, Rµ]}, (3)The (pseudo)scalar and (axial) vector nonets are,φ =8∑i=0(Si + iPi)Ti =1√2(σN+a00)+i(ηN+pi0)√2a+0 + ipi+ K∗+0 + iK+a−0 + ipi− (σN−a00)+i(ηN−pi0)√2K∗00 + iK0K∗−0 +K− K¯∗00 + iK¯0 σs + iηsLµ =8∑i=0(V µi +Aµi )Ti =1√2ωN+ρ0√2+f1N+a01√2ρ+ + a+1 K∗+ +K+1ρ− + a−1ωN−ρ0√2+f1N−a01√2K∗0 +K01K∗− +K−1 K¯∗0 + K¯01 ωS + f1SµRµ =8∑i=0(V µi −Aµi )Ti =1√2ωN+ρ0√2− f1N+a01√2ρ+ − a+1 K∗+ −K+1ρ− − a−1 ωN−ρ0√2− f1N−a01√2K∗0 −K01K∗− −K−1 K¯∗0 − K¯01 ωS − f1Sµ(4)The {Ti}8i=0 are the generators of U(3), while Si, Pi, V µi and Aµi representsthe scalar, pseudoscalar, vector and axial vector fields, respectively. Eq. (4)also shows the assignment of the physical particles except in the 0-8 sector,for which we use the so-called nonstrange-strange basis defined as,ϕN =1√3(√2ϕ0 + ϕ8), ϕS =1√3(ϕ0 −√2ϕ8), ϕ ∈ (Si, Pi, Vi, Ai). (5)Proc˙for˙eQCD printed on August 7, 2020 3In the Lagrangian Eq. (1) two more constant external fields H and ∆appear, which have the formsH = H0T0 +H8T8 =h0N2 0 00 h0N2 00 0 h0S√2∆ = ∆0T0 + ∆8T8 =δ˜N2 0 00 δ˜N2 00 0 δ˜S√2 =δN 0 00 δN 00 0 δS ,(6)The model contains 15 parameters that are fitted with χ2 method by usingtree-level meson masses and decay widths. The fitting procedure and theparameters are detailed in [3].To go to finite temperature analytic continuation to imaginary timet → −iτ should be performed, thus the temporal part of the gluon fieldtransformed as G0(t, x) → −iG4(τ, x). Without going into the details thePolyakov loop [4, 5] (path ordered Wilson loop in the temporal direction)can be defined asL = P exp(i∫ β0dτG4(τ, x)), L† = (L)†. (7)The Polyakov loop variables defined with the color traced Polyakov loopsΦ(x) =1NcTrcL, Φ¯(x) =1NcTrcL† (8)as being their thermal expectation values 〈Φ〉 and 〈Φ¯〉, but for simplicitywe from hereafter leave the 〈 〉 notation.We want to study the thermodynamics of a symmetric quark matter(µu = µd = µs = µq = 1/3µB). For this a grand potential Ω(T, µq) shouldbe obtained. In the mean field level approximation that is used for theevaluation of the grand potential the vacuum and thermal fluctuations of thefermions are taken into account but those for the mesons are neglected. Thusthe meson potential is classical (tree-level), while the fermionic determinantis obtained after performing the functional integration over the quark field,which is evaluated for vanishing mesonic fluctuating fields. Finally the grandpotential readsΩ(T, µq) = U(〈M〉) + U(Φ, Φ¯) + Ω(0)q¯q (T, µq), (9)where U(〈M〉) is the tree level mesonic potential, Ω(0)q¯q (T, µq) is the fermioniccontribution calculated at nonvanishing scalar-isoscalar backgrounds and4 Proc˙for˙eQCD printed on August 7, 2020vanishing mesonic fluctuations. The U(Φ, Φ¯) term is the Polyakov looppotential, which can be simply added to the grand potential, since it wastreated at mean field level only so there is no integration over the gluons.One can get the field equations by minimizing the grand potential withrespect to the order parameters as ∂Ω∂Φ =∂Ω∂Φ¯= ∂Ω∂φN =∂Ω∂φS= 0.2. Baryon number fluctuationsThe baryon number fluctuations are characterized by the higher ordercumulants of the net baryon number. These can be expressed with thegeneralized susceptibilities , the derivatives of the pressure , which is p =ΩT=µB=0 − Ω, with respect to the (baryo)chemical potentialχBn =∂np/T 4∂(µB/T )n∣∣∣∣T= Tn−4∂np∂µnB∣∣∣∣T, (10)where dimensionless pressure and reduced chemical potential are used. Higherorder cumulants (moments) diverges rapidly with the diverging correlationlength, thus they can be good signatures of a critical endpoint [6].To be able to compare the experimental results to the theoretically cal-culated ones we should define quantities that are accessible from both sides.The ratios of the baryon number cumulants might be good candidates sincethe dependence on the finite volume cancels in these ratios. Note that fromexperimental point of view this dependence or independence is not obvious,as it is investigated in [7]. One of these ratios is the (excess) kurtosis, thatcan be defined as the ratios of 4th (k4) and 2nd (k2) order cumulants asκ =k4k22. (11)Sincek2 = V T∂2p∂µ2B= V TT 4T 2χB2 = V T3χB2 ,k4 = V T3 ∂4p∂µ4B= V T 3T 4T 4χB4 = V T3χB4 ,(12)and k2 = σ2 is the variance, it can been rewritten asκσ2 = σ2k4k22=χB4χB2(13)This quantity, κσ2 that we actually call kurtosis. We calculated the suscep-tibilities numerically with finite difference method.Proc˙for˙eQCD printed on August 7, 2020 53. ResultsNow we present our results in the ELσM both at zero and finite µB.For zero chemical potential the 2nd and 4th order susceptibilities and thekurtosis are shown in Figure 1.050100150200250300350 0.14  0.16  0.18  0.2  0.22  0.24  0.26  0.28χ 2BT[GeV]LQCDELSMχM102030405060708090100110120 0.14  0.16  0.18  0.2  0.22  0.24  0.26  0.28χ 4BT[GeV]LQCDELSMχM 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0.1  0.2  0.3  0.4  0.5     free quark gasHRG 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0.14  0.16  0.18  0.2  0.22  0.24  0.26  0.28  ELSMHRGfree quark gasχ 4B /χ 2BT[GeV] 0 0.2 0.4 0.6 0.8 1 1.2 140  160  180  200  220  240  260  280HRGfree quark gas   ms/ml=20 (open)27 (filled)χ 4B/χ2BT [MeV]cont. est.Nτ=68Fig. 1: χB2 as function of temperature in the ELσM, in the chiral matrix model, and onthe lattice [8] (top left). Temperature dependence of χB4 compared again to the chiralmatrix model and lattice data (top right). The kurtosis in the ELSM (bottom left) and onthe lattice [9] (bottom right). The inset of the bottom left figure contains the same curvedepicted on a larger scale to show how the HRG and free quark gas limits are approached.As it can be seen our results are compatible with the lattice results.It is worth to note that in the case of the ELσM there are a double peakstructure around the phase transition (see e.g. top right of Fig. 1). Thereason behind this behavior is that we have four order parameters. These,namely the Polyakov loop parameters Φ, Φ¯ and the scalar-isoscalar vacuumexpectation values φN , φS changes at slightly different temperatures, whichcause the separation of peaks for the different order parameters.It is also worth to investigate the behavior of kurtosis at finite µB, whichis shown on Fig. 2. Going towards the predicted critical end point (CEP)the value of the kurtosis at the phase transition increases significantly and– as it is expected – even diverges at the CEP.6 Proc˙for˙eQCD printed on August 7, 20200100200300400500600700800900 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0 2 4 6 8 10location of CEPµB [MeV]T [GeV]κσ2Fig. 2: The 3D plot of the kurtosis as a function T and µB . The arrow points towardour prediction of the CEP at µB = 0.885 GeV and T = 0.052 GeV .AcknowledgementGy. Kova´cs acknowledges support by the NRDI fund of Hungary, fi-nanced under the FK 19 funding scheme, project no. FK 131982, while hisconference participation was also supported by the framework of COST Ac-tion CA15213 THOR. P. Kova´cs acknowledges support by the Ja´nos BolyaiResearch Scholarship of the Hungarian Academy of Sciences and was alsosupported by the U´NKP-19-1 New National Excellence Program of the Min-istry for Innovation and Technology.REFERENCES[1] M. A. Stephanov, K. Rajagopal and E. V. Shuryak, Phys. Rev. D 60, 114028(1999)[2] D. Parganlija, P. Kovacs, G. Wolf, F. Giacosa and D. H. Rischke, Phys. Rev.D 87, no.1, 014011 (2013)[3] P. Kova´cs, Z. Sze´p and G. Wolf, Phys. Rev. D 93, no. 11, 114014 (2016)[4] K. Fukushima, Phys. Lett. B 591, 277-284 (2004)[5] C. Ratti, M. A. Thaler and W. Weise, Phys. Rev. D 73, 014019 (2006)[6] M. Stephanov, Phys. Rev. Lett. 102, 032301 (2009)[7] P. Braun-Munzinger, A. Rustamov and J. Stachel, Nucl. Phys. A 960, 114(2017)[8] R. D. Pisarski and V. V. Skokov, Phys. Rev. D 94, no. 3, 034015 (2016)[9] A. Bazavov et al., Phys. Rev. D 95, no. 5, 054504 (2017)

Baryon fluctuations in extended linear sigma model

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Baryon fluctuations in extended linear sigma model

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