Tree-level and complete one-loop parametrisation of the linear sigma model
(LSM) is performed and the phase boundary between first order and crossover
transition regions of the m_pi-m_K-plane is determined using the optimised
perturbation theory (OPT) as a resummation tool of perturbative series. Away
from the physical point the parameters of the model were determined by making
use of chiral perturbation theory (ChPT). The location of the phase boundary
for m_pi=m_K and of the tricritical point (TCP) on the m_pi=0 were estimated.Comment: 4 pages, 1 figure, uses espcrc1.sty; to appear in the proceedings of
  Strong and Electroweak Matter 2006 (SEWM06), BNL, May 200

Chiku

de Forcrand

Fodor

Herpay

Karsch

Lenaghan

Meyer-Ortmanns

Philipsen

Pisarski

T. Herpay

Zs. Szép

English

arXiv

HERPAY, T

SZÉP, ZS

ELTE Digital Institutional Repository (EDIT)

arXiv:hep-ph/0608318v1  30 Aug 2006 The boundary of the first order chiral phase transition in themπ −mK–plane with a linear sigma modelT. Herpay a b and Zs. Szép c ∗aDepartment of Physics of Complex Systems, Eötvös UniversitybResearch Group for Statistical Physics of the Hungarian Academy of Sciences,H-1117 Budapest, HungarycResearch Institute for Solid State Physics and Optics of the Hungarian Academy ofSciences, H-1525 Budapest, HungaryTree-level and complete one-loop parametrisation of the linear sigma model (LσM) isperformed and the phase boundary between first order and crossover transition regionsof the mπ −mK–plane is determined using the optimised perturbation theory (OPT) asa resummation tool of perturbative series. Away from the physical point the parametersof the model were determined by making use of chiral perturbation theory (ChPT). Thelocation of the phase boundary for mπ = mK and of the tricritical point (TCP) on themπ = 0 were estimated.1. IntroductionLattice studies show that at the physical point of the mu,d−ms–plane the transition asa function of the temperature is of crossover-type [ 1]. In the mu,d = ms = 0 limit effectivemodels predict a first order phase transition and for ms = ∞ and mu,d = 0 a second orderphase transition [ 2]. Theoretically it is interesting to know where the boundary of thefirst order region lies and where the TCP is located on the mu = 0 axis.Several lattice studies with degenerate quarks (mu,d = ms) show that the critical pionmass on the boundary decreases substantially, from mcπ ≈ 290 MeV to mcπ = 67(18) MeV,as finer lattices and improved actions are used (see e.g. [ 3, 4] ). A low pion mass allows forstudies in effective models based on the chiral symmetry because it is expected that theywork better closer to the chiral limit. Existing works performed in these models give alsoa low value for the critical pion mass [ 5, 6, 7]. In these works only the Goldstone masseswere tuned, the other parameters of the Lagrangian were kept at their values determinedat the physical point. We think that in this low mass regime effective model studies mightcomplement lattice investigations if coupling parameters can be determined accurately.Hence we let the coupling constants to vary with the pion and kaon masses when movingaway from the physical point and determined their values using the low energy results ofthe ChPT, namely the dependence of fπ, fK , mη and mη′ on mπ and mK .∗Supported by OTKA Postdoctoral Grant no. PD 050015.12 T. Herpay, Zs. Szép2. The parametrisation of the linear sigma modelThe Lagrangian is written in terms of a 3 × 3 matrix M = λa(σa + iπa)/√2 (λa : a =1 . . . 8 are the Gell-Mann matrices and λ0 :=√2/31) containing the nonet of scalar (σa)and pseudoscalar (πa) particles:L(M) =12Tr (∂µM†∂µM + µ2M †M)− f1(Tr (M †M))2 − f2Tr (M †M)2−g(det(M) + det(M †))+ ǫ0σ0 + ǫ8σ8. (1)The determinant term breaks the axial U(1) symmetry and the external fields are in-troduced in order to give masses to the pseudo-Goldstone bosons. From the original0-8 basis one can switch with an orthogonal transformation to a basis in which in thebroken symmetry phase the two non-vanishing expectation values are the non-strange(x = (√2〈σ0〉+ 〈σ8〉)/√3) and the strange (y = (〈σ0〉 −√2〈σ8〉)/√3) condensates.2.1. Tree-level parametrisationThe tree-level parametrisation of the model can be performed at zero temperature bysolving a set of coupled linear equations [ 8, 9]. Only the trace of the mass matrix isused in the mixing η − η′ sector. Unfortunately the values the quartic coupling f1 andthe mass parameter µ2 can be obtained using only the scalar sector which compared tothe pseudoscalar sector is experimentally less known. Besides, one knows nothing aboutthe mπ and mK-dependence of the scalar masses and as a consequence some assumptionsare needed. For example in [ 9] we required the Gell-Mann-Okubo mass formula to besatisfied in the scalar sector on the entire mπ −mK–plane with the same accuracy as atthe physical point. Performing a consistency check by comparing mη and mη′ calculatedin the LσM with their values given by the ChPT remarkable agreement was obtained upto a limiting value of mK ≈ 800 MeV, even away from the physical point.2.2. One-loop parametrisationThe parametrisation at the one-loop level of the perturbation theory is more compli-cated and there are many possible ways of selecting the set of non-linear equations. Onesuch set of equations is presented below (see [ 10] for details).Because in the broken symmetry phase the tree-level mass can be negative a resumma-tion is needed. We performed this using the optimised perturbation theory method of Ref.[ 11] where a mass term is added to and subtracted from the Lagrangian, the differencebetween the original and the new mass is treated perturbatively (first at one-loop level).The new mass is determined from the principle of minimal sensitivity which requires thatthe one-loop pole mass for the pion (Mπ) stays equal to its tree-level value (mπ). Thiscan be translated into a self-consistent equation for the pion massm2π = −µ2 + (4f1 + 2f2)x2 + 4f1y2 + 2gy + ReΣπ(p2 = m2π). (2)Two more equations are the one-loop pole mass of the kaon and η (smaller mass eigenvaluein the mixing sector):M2K = −µ2 + 2(2f1 + f2)(x2 + y2) + 2f2y2 −√2x(2f2y − g) + ReΣK(p2 = M2K), (3)Det(p2 −m2ηxx − Σηxx(p2) −m2ηxy − Σηxy(p2)−m2ηxy − Σηxy(p2) p2 −m2ηyy − Σηyy(p2))∣∣∣∣p2=M2η= 0. (4)Phase boundary in the (mπ − mK)–plane 3We require additionally the equality of the one-loop an tree-level kaon masses:M2K = m2K = m2π − 2gy + 4f2y2 −√2x(2f2y − g), (5)and also make use of the PCAC relations:fπ = Z− 12π M−2π (−iD−1π (p = 0))x, fK = Z− 12K M−2K (−iD−1K (p = 0))(x+√2y)/2. (6)Away from the physical point we use for the determination ofmη, fπ and fK the formulaeof the SU(3) ChPT in the large-Nc limit. Because in the one-loop parametrisation therenormalisation scale (l) appears as a new parameter we checked how the other parametersand masses depends on it. The wave function renormalisation constants have a plateauand the variation of the one-loop masses is the mildest in the range l ∈ (1000, 1400) MeV,which was selected to be used for the determination of the phase boundary.3. Thermodynamics of the LσMThe order of the phase transition is determined by solving simultaneously the self-consistent gap equation for the pion massm2π = −µ2 + (4f1 + 2f2)x2 + 4f1y2 + 2gy + Σπ(p2 = m2π, mi(mπ), l), (7)where Σπ(p,mi(mπ), l) is the pion self-energy, and the two equations of stateǫx = −µ2x+ 2gxy + 4f1xy2 + (4f1 + 2f2)x3 +∑i Jitxi (x, y)Itp[mi(mπ), T ],ǫy = −µ2y + 2gx2 + 4f1x2y + (4f1 + 4f2)y3 +∑i Jityi (x, y)Itp[mi(mπ), T ] . (8)The isospin multiplicity factor Ji and the coefficients txi and tyi are given in Ref. [ 9].The difference between the tree-level and one-loop parametrisation is that in the firstcase one are enforced to throw out entirely the vacuum fluctuations and use only thetemperature dependent part of the integrals, while in the second case the renormalisedvacuum fluctuations are taken into account. The difference between the two methods ofsolving the model was checked quantitatively in [ 10].4. The phase-boundary in the mπ −mK–planeIn the case of the tree-level parametrisation the result on the phase boundary is not tooconclusive because of the dependence of the phase boundary on the assumptions madefor the scalar sector in the process of parametrisation. We have not seen the trace ofthe tricritical point. We estimated that the phase boundary crosses the diagonal of themπ −mK-plane for critical pion mass in the range mcπ = 40± 20 MeV.As for the one-loop parametrisation, along the diagonal of the pion-kaon mass plane weestimate mcπ = 110±20 MeV (see Fig. 1). At high values of mK the phase boundary is notrenormalisation scale dependent. We can clearly see the scaling region of the trictiticalpoint as well. Unfortunately we were not able to locate it directly because the modelparametrisation breaks down just before arriving at the TCP along the mπ = 0 line, butbased on the scaling near TCP we estimate it to be in the interval mTCPK ∈ {1700, 1850}MeV. We can directly compare our result on the location of the TCP with the recent4 T. Herpay, Zs. Szép 0 200 400 600 800 1000 1200 1400 1600 1800 2000 0  20  40  60  80  100  120  140mK [MeV]mπ [MeV]Physicalpointdiagonalfit:m2K=m2K|TCP + 0.5 mπ2 - α mπ4/5for l=1200 MeVcrossover1st orderVfittedscalingcurvesl=1000 MeV1100        1200        1300        using Veneziano formula (V)Figure 1. Phase boundary of the LσM obtained using a one-loop parametrisation at T = 0and the formulae of large Nc ChPT for continuation on the mπ −mK–plane.lattice result of Ref. [ 12] by showing the phase boundary in the quark mass-plane. InRef. [ 12] the physical point is more close to the boundary than in our case and the valueof the mass of the strange quark at the TCP is mTCPs = 3ms (mphyss is the value of thestrange quark mass at the physical point). In our case mTCPs = 13− 15×mphyss . In Ref. [12] the location of the TCP was estimated also by fitting with the corresponding scalingequation (mu,d ≈ (mTCPs −ms)5/2), but only points with ms ≤ mphyss . The investigationperformed in the LσM shows that the scaling region sets in very close to the mu = 0(mπ = 0) axis. It would be interesting to know how the lattice estimate would change ifpoints more closer to the mu = 0 axis were available.REFERENCES1. Z. Fodor, S.D. Katz, JHEP 0404 (2004) 0502. R. D. Pisarski and F. Wilczek, Phys. Rev. D 29, 338 (1984).3. F. Karsch, E. Laermann, Ch. Schmidt, Phys. Lett. B520, 41 (2001)4. F. Karsch et al., Nucl. Phys. Proc. Suppl. 129-130, 614 (2004)5. H. Meyer-Ortmanns and B.J. Schaefer, Phys. Rev. D 53, 6586 (1996)6. C. Schmidt, PhD Thesis, Bielefeld University, 20037. J. T. Lenaghan, Phys. Rev. D 63, 037901 (2001)8. J.T. Lenaghan, D.H. Rischke and J. Schaffner-Bielich, Phys. Rev. D 62, 085008 (2000)9. T, Herpay, A. Patkós, Zs. Szép and P. Szépfalusy, Phys. Rev. D 71, 125017 (2005)10. T. Herpay and Zs. Szép, Phys. Rev. D 74, 025008 (2006)11. S. Chiku and T. Hatsuda, Phys. Rev. D 58, 076001 (1998)12. O. Philipsen, PoS LAT2005 (2005) 016; P. de Forcrand and O. Philipsen,hep-lat/0607017

The boundary of the first order chiral phase transition in the m_pi-m_K--plane with a linear sigma model

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The boundary of the first order chiral phase transition in the -plane with a linear sigma model

The boundary of the first order chiral phase transition in the
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The boundary of the first order chiral phase transition in the m_pi-m_K--plane with a linear sigma model

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