Phase diagram at finite temperature and quark density in the strong coupling limit of lattice QCD for color SU(3)

Transcription

1 Phase diagram at finite temperature and quark density in the strong coupling limit of lattice QCD for color SU(3) Akira Ohnishi in Collaboration with N. Kawamoto, K.Miura, T.Ohnuma Hokkaido University, Sapporo, Japan Eprint hep-lat/ A. Ohnishi, 2006/11/15 1

2 Quark and Hadronic Matter Phase Diagram Dense quark & hadronic matter contains rich physics, but Lattice QCD simulation is not yet reliable. Model/Approximate approaches are necessary! Monte-Carlo calc. of Lattice QCD: Improved ReWeighting Method (Fodor-Katz) Taylor Expansion in μ (Bielefeld U.) Analytic Continuation (de Forcrand-Philipssen) Model / Phen. Approaches: NJL, QMC, RMF,... Strong Coupling Limit of Lattice QCD z A. Ohnishi, QM2006@Shanghai, 2006/11/15 2

3 Strong Coupling Limit of Lattice QCD Chiral Restoration at μ=0. Damgaard, Kawamoto, Shigemoto, PRL53(1984),2211 Phase Diagram with Nc=3 Nishida, PRD69, (2004) A. Ohnishi, 2006/11/15 3

4 Previous Works in Strong Coupling Limit LQCD Strong Coupling Limit Lattice QCD re-attracts interests Ref T μ Nc Baryon CSC Nf Damgaard-Kawamoto-Shigemoto('84) Finite 0 U(Nc) X X 1 Damgaard-Hochberg-Kawamoto('85) 0 Finite 3 Yes X 1 Bilic-Karsch-Redlich('92) Finite Finite 3 X X 1 ~ 3 Azcoiti-Di Carlo-Galante-Laliena('03) 0 Finite 3 Yes Yes 1 Nishida-Fukushima-Hatsuda('04) Finite Finite 2 Yes (*) Yes (*) 1 Nishida('04) Finite Finite 3 X X 1~2 Kawamoto-Miura-AO-Ohnuma('05) Finite Finite 3 Yes Yes (+) 1 *: bosonic baryon=diquark in SU(2) +: analytically included, but ignored in numerical calc. Baryons should be important at High Baryon Densities, but they have been ignored in finite T treatments! This work: Baryonic effects at Finite T (and μ) for SU c (3) A. Ohnishi, QM2006@Shanghai, 2006/11/15 4

5 Strong Coupling Limit Lattice QCD QCD Lattice Action Z D[,,U ]exp [ S G S F s S F t m 0 M ] S G = 1 g 2 x S F s = 1 2 x, j + [TrU TrU ] j x x U j x x j U + x j j x x U ν + U μ + U μ U ν U j U j + χ U μ χ χ Uμ + χ (U j ) 3 M= χ χ S F t = 1 2 x e x U 0 x x 0 e x 0 U + 0 x x Strong Coupling Limit: g We can ignore S G and perform one-link integral after 1/d expansion. M(x) M(x+j) B= /6 B= /6 du U ab U + cd = 1 N ad bc c du U ab U cd U ef = 1 6 ace bdf S F s 1 2 M V M M B V B B = 1 4 N c x, j 0 M x M x j x, j 0 j 8 [ B x B x j B x j B x ] A. Ohnishi, QM2006@Shanghai, 2006/11/15 5

6 SCL-LQCD w/o Baryons Lattice Action (staggered fermion) in SCL Z D[,,U ]exp [ S F s S F t m 0 S G ] Spatial Link Integral D[,,U 0 ]exp[ 1 2 M,V M M B,V B B G 0 ] Strong Coupling 1/d Expansion (1/ d) Bosonization (Hubburd-Stratonovich transformation) Quark and U 0 Integral Damgaad-Kawamoto-Shigemoto 1984, Faldt-Petersson 1986, Bilic-Karsch-Redlich 1992, Nishida 2004,... D[,,U 0, ]exp[ 1 2,V M,V M M G 0 ] G exp N 3 S N [ 1 2 a 2 T log G U ] =exp N 3 S F eff /T Local Bi-linear action in quarks Effective Free Energy A. Ohnishi, QM2006@Shanghai, 2006/11/15 6

7 SCL-LQCD with Baryons Effective Action up to O(1/ d) Z D[,,U 0 ]exp[ 1 2 M,V M M B,V B B G 0 ] M = a a B= abc a b c /6 = D[,,U 0, b,b]exp[ 1 2 M V M M bv B 1 b b, B B,b G 0 ] Decomposition of bb by using diquark condensate (Azcoiti et al., 2004) exp[ b, B B,b ]=exp [ 1 6 b, 1 6, b ] = [ D[ a, * a ]exp * * 2 b 3 2 b ] 3 exp M 2 /2 M b b/9 2 Decomposition of Mbb using baryon potential field ω exp M b b/9 2 = D [ ]exp [ M b b 9 2 note: with one species of staggered fermion! b b 2 =0 2] 2 2 M A. Ohnishi, QM2006@Shanghai, 2006/11/15 7

8 Effective Free Energy with Baryon Effects Effective Action in local bilinear form of quarks S F = 1 2 M V M M 1 2, b, V B 1 g b, M G 0 = N 3 s N Bosonization + MFA +No diquark cond. quark & gluon int. * * D D + 2 a 2 2 a, M G 0 b, V B 1 g b b int. F eff, = 1 2 a F q eff a F b eff g = 1 2 a 2 1 O(ω 2 ) O(ω 4 ) 2 a 2 F q eff a F b eff g Linear Approx. (ω ~ ασ/a ω ) F eff = 1 2 b 2 F q eff b F b eff g A. Ohnishi, QM2006@Shanghai, 2006/11/15 8

9 Effective Free Energy with Baryon Effects F eff = 1 2 b 2 F q eff b ;T, F b eff g is analytically derived based on many previous works, including Strong Coupling Limit (Kawamoto-Smit, 1981) 1/d expansion (Kluberg-Stern-Morel-Petersson, 1983) Lattice chemical potential (Hasenfratz-Karsch, 1983) Quark and time-like gluon analytic integral (Damgaad-Kawamoto-Shigemoto, 1984, Faldt-Petersson, 1986) F q eff ;T, = T log C C 1 4 C 3 Decomposition of baryon-3 quark coupling (Azcoiti-Di Carlo-Galante-Laliena, 2003) and auxiliary baryon potential and baryon integral (Kawamoto-Miura-AO-Ohnuma, hep-lat/ ) C =cosh sinh 1 /T C 3 =cosh 3 /T A. Ohnishi, QM2006@Shanghai, 2006/11/15 9

10 Phase diagram in SCL-LQCD with Baryons (Kawamoto-Miura-AO-Ohnuma, hep-lat/ ) What is the baryonic composite effect on phase diagram? Auxiliary baryon integral & diquarks generate terms M 2 modifies the energy scale! Compare the phase diagram scaled with T c. Baryon Effects Baryons Gain Free Energy Extention of Hadron Phase to Larger μ! A. Ohnishi, QM2006@Shanghai, 2006/11/15 10

11 Small Critical μ : Common in SCL-LQCD? Finite T SCL-LQCD No B: μ c (0)/T c (0) ~ ( ) (Nishida2004, Bilic et al. 1992(Bielefeld),...) Present: μ c (0)/T c (0) < 0.44 (Parameter dep.) Monte-Carlo: μ c (0)/T c (0) > 1 Bilic et al. Fodor-Katz (Improved Reweighting) Bielefeld (Taylor expansion), de Forcrand-Philipsen (AC),... Real World: μ c (0)/T c (0) > 3 T c (0) ~ 170 MeV, μ c (0) > 330 MeV Present A. Ohnishi, QM2006@Shanghai, 2006/11/15 11

12 Towards Realistic Understanding Reality Axis: 1/g 2, n f, m 0,... would enhance μ c /T c ratio Example: 1/g 2 correction enhances μ c /T c by a factor ~(2-3). exp[ 1 g TrU 0j x ] ~exp [ V 2 x V + x j /4 N 2 c g 2 ] exp[ 2 2 V x V + x j /16 N 2 c g 2 ] S F t = 1 2 e V x e V x+ 2 exp V x exp V x+ V x = x U 0 x x 0 Time-like plaquetts can modify effective chemical potential U 0 U μ U 0 + χ Uμ + χ U μ + Preliminary! A. Ohnishi, QM2006@Shanghai, 2006/11/15 12 χ U μ U U χ

13 Summary We obtain an analytical expression of effective free energy at finite T and finite μ with baryonic composite effects in the strong coupling limit of lattice QCD for color SU(3). MFA, QG integral, 1/d expansion (NLO, O(1/ d)), bosonization with diquarks and baryon potential field using bb 2 =0, Linear approx., zero diquark cond.(color Angle Average), variational parameter choice Baryonic action is found to result in Free Energy Gain and Extension of Hadron Phase to Larger μ by around 30 %. Problem: Too small μ c /T c in the Strong Coupling Limit. 1/g 2 correction and other may help. Strong Coupling Limit is useful to understand Dense Matter RG evolution exp(-c/g 2 ) deps., 1/g 2 correction seems to work well Application to chiral RMF (K. Tsubakihara, AO, nucl-th/ ) A. Ohnishi, QM2006@Shanghai, 2006/11/15 13

14 Color Angle Average Problem: Diquark Condensates induce quark-baryon coupling, and Baryon integral becomes difficult. Solution: Color Angle Average Integral of Color Angle Variables D= 2 b 3 Three-Quark and Baryon Coupling is ReBorn! Solve Self-Consistent Equaton A. Ohnishi, QM2006@Shanghai, 2006/11/15 14

15 Effective Free Energy with Diquark Condensate Bosonization of M k b b Introduce k bosons exp M k b b= d k exp[ 1 2 k k M 1/ k M k 1 b b 2 M k bb] = d k exp[ k 2 /2 k M 1/ k M k 1 bb k 2 M 2 /2] Effective Free Energy The same F eff is obtained at v=0. Diquark Effects in interaction start from v 4. c.f. Ipp, Yamamoto A. Ohnishi, QM2006@Shanghai, 2006/11/15 15

16 Backups A. Ohnishi, 2006/11/15 16

17 Parameter Choice In bosonization, two parameters (γ and α) are introduced through identities. Major effects Modify the energy scale Minor effects Controls the higher order potential terms We have fixed them to minimize F eff /T c at vacuum A. Ohnishi, QM2006@Shanghai, 2006/11/15 17

18 1/g correction 2 Gluons tend to break hadrons, then 1/g 2 correction is expected to reduce T c. (Bilic-Cleymans 1995) Naive extapolation of 1/g2 correction seems to give μ c /T c ~ 6/g 2 =5 A. Ohnishi, QM2006@Shanghai, 2006/11/15 18

19 Baryon Integral Baryon integral can be evaluated in an almost analytic way! A. Ohnishi, 2006/11/15 19

20 Energy surface Figures Validity of Linear Approx. A. Ohnishi, 2006/11/15 20

21 RMF with σ Self Energy from SCL-LQCD σ Self Energy from simple SCL-LQCD S 1 2 M V M M 1 2 V M V M U 1 2 b 2 N c log 2 Chiral RMF with logarithmic σ potential (Tsubakihara-AO, nucl-th/ ) A. Ohnishi, QM2006@Shanghai, 2006/11/15 21

22 Chirally Symmetric Relativistic Mean Field and Its Application K. Tsubakihara and AO, nucl-th/ A. Ohnishi, 2006/11/15 22

23 RMF with Chiral Symmetry Good Sym. in QCD, and Spontaneous breaking generates hadron masses. Schematic model: Linear σ model L= c N i Problems and Prescriptions N g N i 5 N χ Sym. is restored at a very small density. σω Coupling stabilizes normal vacuum, but gives too stiff EOS. (Boguta PLB120,34, Ogawa et al. PTP111(2004)75) Loop effects (N.K. Gledenning, NPA480,597; M. Prakash and T. L. Ai nsworth, PRC36, 346; Tamenaga et al.) Higher order terms (Hatsuda-Prakashi 1989, Sahu-Ohnishi, 2000) Dielectric Field (Papazoglou et al. (Frankfurt), 1998) Different Chiral partner assignment (Kunihiro et al., Hatsuda et al. Har ada et al.) A. Ohnishi, QM2006@Shanghai, 2006/11/15 23

24 Problems in RMF with Chiral Symmetry Sudden Change of <σ> σ ω Coupling L = 1 4 F F 1 2 C 2 2 g N N =g B /C 2 V = g 2 B 2 2C 2 Stiff EOS A. Ohnishi, QM2006@Shanghai, 2006/11/15 24

25 Linear σ Model Chiral restor. Below ρ 0. Baryon Loop & Sahu-Ohnishig models Unstable at large σ Boguta model Too Stiff EOS Instability in Chiral Models A. Ohnishi, 2006/11/15 25

26 RMF with σ Self Energy from SCL-LQCD σ Self Energy from simple Strong Coupling Limit LQCD RMF Lagrangian Non-Analytic Type σ Self Energy σ is shifted by f π, and small explicit χ breaking term is added. U =2a f / f, f x = 1 2[ ] x2 log 1 x x 2, a= f 2 2 m 2 2 m A. Ohnishi, QM2006@Shanghai, 2006/11/15 26

27 Nuclear Matter and Finite Nuclei Nuclear Matter: By tuning λ, g ωn, m σ, EOS can be Soft! Finite Nuclei: By tuning g ρn, Global behavior of B.E. is reproduced, except for j-j closed nuclei (C, Si, Ni). A. Ohnishi, QM2006@Shanghai, 2006/11/15 27

28 Free Energy Surface and Phase Diagram At μ 0, quark can gain Free Energy even at σ =0 Two Min. Structure First Order A A B B C C α =1/2 A. Ohnishi, QM2006@Shanghai, 2006/11/15 28