1 NJL model at finite temperature and chemical potential in dimensional regularization T. Fujihara, T. Inagaki, D. Kimura : Hiroshima Univ.. Alexander Kvinikhidze : Razmadze Math. Inst.

2 Motivation NJL model is not renormalizable in 4 space-time dimensions. The theory depends on the regularization method. Cut-off Dimensional Symmetry ○× Regularized region in 1/N expansion Poincare inv. (Gauge inv.) Problem in dimensional regularization: Can physics be represented itself as 4 dimensional space-time meaning? There are many studies for NJL and extended NJL models under the cut-off regularization. Here we apply it to the dimensional regularization and inspect it for a low energy effective model of QCD.

3 1. NJL model in the dimensional regularization 2. Extended NJL model in the dimensional regularization Contents 1-1 chiral symmetry breaking 1-2 meson mass 2-1 color symmetry breaking 2-2 energy momentum tensor 2-3 high dense star 3. Summary 1-3 finite temperature and chemical potential

4 1. NJL model in the dimensional regularization 1-1 chiral symmetry breaking 2-flavor NJL model where g is an effective coupling constant, represents the isospin Pauli matrices, and m=diag(m u, m d ) is the mass matrix of up and down quarks. dim 2D-2 operators At the massless limit of quarks this Lagrangian is invariant under the global flavor transformations, When develops a non-vanishing expectation value, the chiral symmetry is dynamically broken, SU(2) L ×SU(2) R / SU(2) V, and quark acquires a dynamical mass.

5 is found by solving the gap equation, where trace runs over flavour, colour and spinor indices, 2 < D < 4. The gap equation has three individual solutions. To find the stable solution we evaluate the effective potential V. D=2.8, m u =3MeV, m d =5MeV, g=-0.01MeV 2-D x

6 renormalization In the leading order of the 1/N expansion the radiative correction of the 4-fermion coupling for the scalar channel is given by the summation of all the bubble type diagrams, ＋ ＋ ＋ ・・・ where Π s (p 2 ) is the scalar self-energy. We renormalize g by imposing the following renormalization condition, where M 0 is the renormalization scale. c.f.

7 1-2 meson mass The propagator of the scalar channel can be also expressed as, The sigma meson mass is given by, To determine the parameters g, D, M 0, we have used the following relation and values. ・ Current algebra ・ Gell-Mann—Oarkes—Renner relation ・ Gap equation

8 1-3 finite temperature and chemical potential We introduce the temperature T ( ) and chemical potential. The gap equation is modified as, The behavior of V and the solution for the gap equation.

9 The behavior of as T is increased. phase diagram

10 Summary of section 1 1. NJL model in the dimensional regularization Results by the dimensional regularization has similar meson properties to cut-off regularization. From the phase diagram, But critical point is different T (MeV) ★ cut-off dim D= TcTc -

11 2. Extended NJL model in the dimensional regularization 2-1 colour symmetry breaking T μ Hadron 2SC Plasma CFL High dense star Phase diagram We consider 2-flavor system and investigate the 2-color superconductivity phase by using the dimensional and cut-off regularizations. Finally, we discuss the mass and radius relation of the high dense star.

12 2-flavor extended NJL model where a,b,c (=1,2,3) and i,j,k (=u,d) denote the color and flavor indices respectively. We introduce the auxiliary fields, consider a thermal system with chemical potential. When develops a non-vanishing expectation value, the color symmetry is dynamically broken [SU(3) C / SU(2)].

13 The solution of the gap equation The behavior of is different at large, because there is in cut-off regularization We compare calculations of the dim. regularization at D=2.24 with that of cut-off. We wish we had evaluated the extended NJL model at D~2.3 in dim. regularization. We did not have time to recalculate The phase transition of cut-off regularization is softer than that of dimensional. 0.3

energy momentum tensor Energy densityPressure P (GeV 4 ) cut-off dim D=2.24 dim D=

15 The relation of pressure P and energy density (EoS) P (GeV 4 ) cut-off dim D= Since the shapes of these phase transitions and are different, two slopes become different.

high dense star gravity pressure Tolman-Oppenheimer-Volkoff equation Inside the star, gravitational force is balanced with the pressure of the matter. The mass and radius relation of the star is expressed by the TOV equation. Our interest is in 4 dimensional space-time. Here we regard “D” as a parameter of our effective model and solve the TOV equation in 4 space-time dimensions under the EoS obtained by dimensional regularization.

17 D=2.20 D=2.24 D=2.28 R (km) M (M ◎ ) The mass M and radius R relation of the star M (M ◎ ) R (km) cut-off dim D= ~ GeV 4 Since the slopes of P- relations are different, M-R relations become different. In the dim. regularization, D=2.20, 2.24, 2.28 o

18 Summary of section 2 2. Extended NJL model in the dimensional regularization ・ It was found that the mass-radius relations of the star have the large dependence of the regularization methods. These results come from the shapes of the phase transitions and. Future works ・ We evaluate the extended NJL model at D~2.3 in the dimensional regularization. ・ We want to adopt the neutrality conditions for electric and color charge, the effect of strange quark. Adopting these effect in cut-off, one can not explain the experimental data of high dense star because it is still small radius. But I think we can explain it using the dimensional regularization. PRD.69,045011, Ruster, Rischke