The Meson Loop Effect on the Chiral Phase Transition

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1 1 The Meson Loop Effect on the Chiral Phase Transition Tomohiko Sakaguchi, Koji Kashiwa, Masatoshi Hamada, Masanobu Yahiro (Kyushu Univ.) Masayuki Matsuzaki (Fukuoka Univ. of Education, Kyushu Univ.) Hiroaki Kouno (Saga Univ.) Contents 1. Introduction 2. The NJL model 2.1. Formalism 2.2. The mean field approximation 2.3. The meson loop effects 3. Summary 4. Future problem

2 2 1. Introduction The phase transition of QCD in finite temperature and density T, GeV QGP.1 E critical point CSC nuclear vacuum matter quark matter CFL quark matter 1 µ B, GeV the phase diagram of QCD [M. Stephanov, PTP. Suppl. (24)] 1. Experiment in RHIC Discovery of QGP and the critical point of the first order phase transition [I. Tserruya, Pramana, (23)] 2. Existence of various states in the CSC phase [J. Berges and K. Rajagopal, NPB, (1999)]

3 3 Lattice calculation [M. Asakawa et al., NPPS, (23)], the gauge/gravity correspondence in string theory [K. Ghoroku, T.S., N. Uekusa and M. Yahiro, PRD, (25)] the existence of mesonic correlation even above the deconfinement phase transition temperature Question How does the existence of mesonic correlation affect to the phase transition? The Nambu Jona-Lasinio (NJL) model the next-leading terms in the 1/N c expansion the meson loop effects Our Aim We investigate the meson loop effects of the NJL model in finite temperature and density, especially, about the effects on the phase transition curve and the critical point.

4 4 Importance of the meson loop effect 1. the existence of mesonic correlation even above the deconfinement phase transition temperature 2. The meson loop effect becomes very strong because pion mass is very light. [E. N. Nikolov et al., NPA (1996)] 3. In the auxiliary field method, We can improve its results by carrying out the loop expansion in the auxiliary field to higher orders, even if the loop expansion parameter (1/N) is not small. In other words, the addition of the higher orders gives a better result than the mean field approximation when the loop expansion parameter is not small. [T. Kashiwa and T. S., PTP, (23); PRD, (23)]

5 5 2. The NJL model 2.1. Formalism Lagrangian (two flavor and three color) L = q(i µ γ µ m )q + G (( qq) 2 + ( qiγ 5 τ a q) 2 ) 2N c the partition function Z[J σ, J π ] = L = q DqDq exp [ β dt V d 3 xl ( + m µγ J σ iγ 5 τ J ) π q + G [ (qq) 2 + (qiγ 5 τq) 2] + N c 2N c 2G the partition function after introducing the auxiliary fields Z[J σ, J π ] = DσD π exp ( N c I[σ, π]), ( 1 ( I[σ, π] = d 4 x σ 2 + π 2) + 1 ( σj σ + π 2G G J ) π ln det ( β the effective potential V(M, T, µ) = V (M, T, µ) + 1 2N c [ β ], d 4 p ln 1 σ (M, T, µ; p) + 3 β ( Jσ 2 + J ) π 2 + m µγ + σ + iγ 5 τ π) ] d 4 p ln 1 π (M, T, µ; p)

6 The mean field approximation We investigate the meson loop effects to the mean field approximation result of Scavenius et al., PRC, (21). parameter set the current mass : m = 5.5 MeV F π = 93 MeV and m π = 138 MeV (in vacuum) the four-fermi coupling constant : G/N c = GeV 2 the ultra-violet divergence cut-off : Λ = 631 MeV [ T. Hatsuda and T. Kunihiro, KEK Preprint, (1987) ]

7 7 The phase transition curve [Scavenius et al., PRC, (21)] Today s talk When the meson loop effects are included, 1. T dependence of the effective potential at µ = 34 MeV 2. µ dependence of the effective potential at T = 4 MeV

8 The meson loop effects T dependence of the effective potential at µ = 34 MeV V V T = 4MeV 4 27, 28, T = 2MeV M [MeV] M [MeV] the left figure : the mean field approximation [T = 2, 27, 28, 29, 4 MeV from the bottom] the right figure : the meson loop approximation [T = 4, 5, 1, 15, 2 MeV from the bottom]

9 9 µ dependence of the effective potential at T = 4 MeV V.1 V µ = 34MeV 333, 334, 335 µ = 33MeV M [MeV] M [MeV] the left figure : the mean field approximation [µ = 33, 333, 334, 335, 34 MeV from the bottom] the right figure : the meson loop approximation [µ = 34, 4, 425, 45, 5 MeV from the bottom]

10 1 3. Summary T dependence of the effective potential at µ = 34 MeV the mean field approximation : the first order phase transition the meson loop effects cross over µ dependence of the effective potential at T = 4 MeV the mean field approximation : the first order phase transition the meson loop effects cross over The meson loop effects drastically change the results of the mean field approximation.

11 11 4. Future problem 1. The estimates of the critical point and the first phase transition curve with the meson loop effects 2. The reevaluation of the four-fermi coupling constant and the cut-off with the meson loop effects the critical point and the first phase transition curve 3. The meson loop effects in the CSC phase 4. Extension to the three flavor NJL model