Decay Process σ→ππ and Chiral Phase Transition

Comtnun. Theor. Phys. (Beijing. China) 37 (2002) pp 431-434日Intornational Aradernic PublishersVnl 37. No.4. Aprl 15. 2002Decay Process σ→ππ and Chiral Phase Transition*ZHU Xiang-Lei and ZHUANG Peng-Fei04ADepartment of Physics, Tsinghua University, Beijing 10084, China(Rcceived September 5, 2001)A bstract The docay processσ→at high tempcrature and density and is rlation with chiral phasc transtiou民are discussed iu the framework of the Nambu -Jona Lasinio model. The decay ratc for the proccssσ +→π is calculntedin the whole temperature and densily rcgion. The contribution of the final statc pion statisties to the dcay rate isdiscussed. The maximum decay rate al diftrent chenical potentials is computed. Finally, we investigate the rclationbetween the starting point of the decay proccss and the critical point of the first-order chiral phase transition.PACS numbers: 11.30. Rd, 05.70.Jk, 14.40-n, 25.75.-qKey words: chiral phase transition, σ decay, relativistic heavy-ion collisions1 Introductioninvestigate spontaneous chiral symmetry breaking in theIt is generally believed(1l that there are two kinds ofvacuum and chiral synuletry restoration at finite ten-QCD phase transitions in relativistic heavy ion cllisions. perature and density.。. Although the model has no con-One of them is related to the deconinement process 0finement mechanism in the vacuuIn and therefore it canquarks, and the other is about chiral symmetry restora-not be used to study deconfinemnent,5l it describes chiraltion. When we investigate chiral properties, σ meson hasproperties very well7l In this article we calculate in theto be inclnded!2) While the 。mass and its width are framework of the NJL model the decay rate for 0→πvery large in the vacum, around the chiral phase tran- and investigate its relation with the chiral plhase transsition point σ becomes light and its width is also verytion.small.3) Therefore there will be numerous σ's in the early2TheDecayRateforσ十ππ.stage of a relativistic heavy-ion cllision if the systemThe Lagrangian density of SU(2) NJL model isundergoes a chiral phase transition. When the termper-ature and density of the system drop down, σ mass in-LL =西(in40 - m)4 + G(00)2 - (4i7sT问)引]，(1)creases but pion mass keeps as a constant.When the where ψ and ψ are the quark fields, t is the isospin gener-mass condition mo = 2mx is reached, the processo →ππ ator, G is the coupling constant with dimension GeV-2,starts immediately, From those considerations, if chiral and mo is the current quark mass.phasc transition happens in the systerm, the numerousTaking quark propagator in the mean field approximaσ's around the phase-transition point will affect remark- tion and meson propagator in the RPA approximnation.'ably the final state pions in magnitude and in momnentun we get the lowest order Feynman diagram for the decaydistribution.4 .processσ→ππ sketched in Fig.1. The decay rate for theNambu- Jona-Lasinio (NJL) model!s| is often used to, process at finite teruperature and density is3√m2/4-m幅r+zr(T,u)=r. +2n0(T,u)+To→π+π (I,4)=2m2where the triangle factor Aσπr is defined by[8]Atr(T,w)= 4mqNcNdq [e(E。-4)-f，(-Eq-4)8(证)- (2m? + 4m2)9.p+m2/2 -2m2E? (3)(27)32E。(m名-4B)(m; - 2.1)2- m字路)In Eq. (2), T and μ are the temperature and baryonic' fn(x)=(ez/T- 1)-' in Eq. (2) is the Bose-Einstein dischenical potential respectively, mo and m. are σ mass tribut中国煤化工: pions. In Eq. (3),and π mass, go and 9π are the coupling constants of qσ Nc =|YHCNMHGrandAavordegresand qqπ interactions. The temperature and density de- of feedomn jv(x)= (e*1T+1)-1 is the Ferni- Dirac dstripendence of these quantities ca be found in Ref. [3). bution function for quarks, Eg =√mg + q2 is the quark"The project supported in part by National Natural Science Foundaton of China (19925519). the 973 Project (200077407) and theNatijm防数擂tory of Heavy-lon Physics in Lanzhou432ZHU: XiangLi anl ZHLANG Peng Fei .Vol 57energv. T'he temperature alld density dlependeruce of (quark we can gu" the inalytic xprission (of the dery rat'at themass mq can also be found in Ref. [3}.critical point,Iowr(T→I )3nT。p1-p ! -1N.Ny sA dq1qi[Ur(- E,一川)- Jo(E,-间(4)7石[一1-”---- ! . iq;三二10:20Fig.1 Feynuann diagram for the process。→罚tothe lowust. order. The solid lines denotr quarks and anti-quarks. the dashed liucs denote Incsons.3 Relation with the Chiral Phase Transition012;)0710 12;17We first discuss the mnedium efect for the fiual stateT(ieV)pions in the decay process. The temperature dependenceFig. 2 The decay rate with dfesrent statisical fartors(μ二0} of the process σ→ππ was considlered in Ref. {91].in the chiral limit (u= 0).The difcrence betwccn Ref. [9| and Eq. (2) is in the con-{sideration of the statistics for the final state pions. It canThe pion mass is not zvero in the real physical world.be provedl that the temperature and density efects of The explicit chiral sylnmetry breaking. namely mo≠0.two final state mesons can be represented as a statistical nust be consilered. In this case, the three parametersfactor(I +f+ (h2). In the caseofσ→下, the correct in the NJL model can be determinedl by the quark con-statistical factor is therefore (1 + 2fu(mo/2)) in Eq. (2)，densate, pion coupling coustant and pion mass Ir二but not(1+ fa(n0/2))2 in Ref. {9]. Tlis diference in 0.134 GeV in the vacum, they areG = 4.93 GeV 2,the final state statistics will result in an esential change A = 0.6535 GeV and ms = 0.005 GeV. Figure 3 showsin the behavior of the decay rate at the critical poimut of the decay rate in the whole temperature and densily他chiral phase transition.gion beyond chiral limit. In the vacnun), the decay rateThe parametcrs of the .NJL mnodel in chiral limnit canis about 0.09 GeV. Keeping T二0 and increasing 1I,be determined by the quark condensate density (u) = the decay rate behaves as a constant until 1 is close to(dd) = (-0.25 GeV) and the pion decay constant j。the critical point of the first-order chiral phase transitiou.0.093 GoV in the vacuum (T =0, μ = 0), they are the The decay rate decreases slightly at μ = 0.3 GeV. thencoupling constant G = 5.02 GeV-2 and the quark mo-increases and reaches the maxinn vialue t the riticalmentum cutoff A = 0.653 GeV. The decay rateTonn asa point. Finally it jumps dow11 to zero ut the critical point.function of temperature T (μ= 0} is displayed in Fig. 2. In the high density region where the first. order phase tra)-The dlecay begins and reaches suddenly the maximun de~ sition exists, this jump relains at the chiral phase trausi-cay rate at the critical temperature T。= 190 MeV. Thetion points. Whcther the decay rate juinps to zero willsolid line represents the decay rate with statistical fac- be dscussed in a while. When the chemical putential μtor (1 +2fn) and the dashed line represents thal with is not so high that the first. order plase transition cannot(1+ fn)P. It can be found in Fig. 2 that, there is alnost happen (14 < 0.327 GeV) with increasing temperature theno diference between the two factors at low temperature.decay rale rises gradually from its vacuum value to theHowever, when the tenperature approaches to the critical maxinum at sone temperature, and this tenmperatur depoint of the second -order chiral phase transition, the firstcreases wAfter reachingstatistical factor makes the decay rate finite at the critical the maxim中国煤化工apidly btt (opoint; but the later one makes it infinite. From this we tinously.|YHC N M H Gture defined bycan conclude that the chiral phase transition leads tn a mo(T,u) = 2m(T,1u). Tle fact that the maximum valuestrong pion enlancement, but the singularity in the decay of r is not in the vacum but near the threshold pointrate conles from the wrong considerarion of the mediunl is attributed to the contribution of the final stiate pioneffect for the final state pious. By further computation. statistics which is importaut only ai high temnperatnre.No.4T)ecay Processo→前and Chural Phasc Transitinu33r(GeV C.050.IC0.150.200.100.075r (GeV]c. 050.0250. 00C.05μ(GeV)0.250.30Fig. 3 The decay rate in the whole T-μ region.For each chemnical potential μ, we can extract a nax-' or if Ta equals Teimum decay rate Tmax from Tr(T,1). Figure 4 shows the0.032Tmax as a function of μ. When the chemical potential isvery low or very high, namely in the limit of high temper-0.030ature or high density, r max is larger than the values in the :Eamiddle region. However, the variation is not big, we can0.028 tconsider Tmax as a constant approximately.~ 0.026 t0.120.02410.0180.326 0327 1328 0329 033 1) 31μ (GirV10.06Fig. 5 The solid line denotes the first