Effects of fluctuations for QCD phase diagram with isospin chemical potential

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1 Effects of fluctuations for QCD phase diagram with isospin chemical potential Kazuhio Kamiado (Yuawa Institute, Kyoto Univ) woring with Nils Strodthoff, Lorenz von Smeal and Jochen Wambach (TU Darmstadt)

2 QCD phase diagram µ I Fuushima and Hatsuda(21) Rich structure is expected. The sign problem exists for finite baryon-chemical potential thus Lattice calculation is not available.

3 Isospin chemical potential Three-dimensional phase diagram Temperature [T] Quar-chemical potential [µ] Isospin-chemical potential [µi] µ u = µ + µ I µ d = µ µ I Quar Determinant is real (µ=). The important sampling method is available. M. Alford, A. Kapustin and F. Wilcze, Phys. Rev. D59, 5452 (1999).

4 Property at T = (Silver blaze) a typical lattice result order parameter h i LSM NJL + µ I m m.5 π/σ σ/σ CPT.1 m π µ I (GeV) µ I [GeV] G. 16. The ratio p/ɛ (upper panel) and the ch I = P µ I W. Detmold, K. Orginos and Z. Shi, arxiv: L. He, M. Jin, and P. Zhuang, Phys. Rev. D 71 (25) At T =, nothing happens before µi reaches to the mass of the lightest charged particle (charged pion). T. D. Cohen, Phys. Rev. Lett. 91, 2221 (23) Silver blaze Arthur Conan Doyle

5 Fluctuations beyond mean-field U = 1 2 µ = = 1 2 µ 2 = a + b 2 + c 3 + d 4 U b Neglect Φ 3 and Φ 4 terms (mean-field approximation.) breas down in a critical region (b ~ ) Include the effects of c or d by solving functional-rg quar-meson model with µi = + = L = [i/@ + g( + i 5 ~ ~ )+µ I 3 ] (@ +2µ I )( + + i )(@ 2µ I )( + i ) + U( 2 + ~ 2 ) c

6 Functional Renormalization Group (FRG) [ ]= Tr [ ]: effective potential at scale Γ = R F R F + (2,) [ ] Tr R B R B + (,2) [ ] { C. Wetterich, Phys. Lett. B31, 9 (1993) = [ ]=S[ ] classical =[ ] = [ ] quantum Γ (,2) = Γ (2,1) Γ (2,1) 2 + Γ (2,2) local potential approximation LP A = Kinetic part + U ( 2 + 2, ) c Γ (,3) Γ (,3) 1 2 Γ (,4) Γ (,3) abi 3 Γ LPA φ i φ a φ b, Γ (,4) abi j 4 Γ LPA φ i φ j φ a φ b, Γ (2,1) i 3 Γ LPA, Γ (2,2) ij 4 Γ LPA. ψ ψ φ i ψ ψ φ i φ j e quar sector we find

7 Pion masses Real part of pion 2-point function 35 3 scr 2 (p ) 1/2 25 (-i ) 1/2 [MeV] p 2 2 U 2 5 pole mass [MeV] p [MeV] :onset of pion condensation curvature mass Pion pole mass and pion curvature mass are difference at 2%. The pole mass well agree the onset of pion condensation (the difference is just at 3%).

8 Isospin density (T=, µ=) the lattice data is from W. Detmold, K. Orginos and Z. Shi, arxiv: [hep-lat] χpt calculation is from D. T. Son and M. A. Stephanov, Phys. Rev. Lett. 86, 592 (21) 2µ I m m χpt C FRG B A A B C FRG M [MeV] analytic form for linear sigma model I(x, y) =2f 2 m x y 2 3 y x y 2 1 x2 x =2µ I /m, y = m /m 1 2 I Both QM and χpt models reproduce the charge density of the LQCD near the onset of the pion condensation. The difference comes from the mass of sigma.

9 Quars in pion condensation phase E ± () = g ( 2 + g 2 2 ± µ I ) 2 gh i >µ I gh i <µ I E + E- E + E- E E E E M = p g 2 h + i 2 +(gh i µ I ) 2 gh + i quar dispersion relation in pion condensation phase up and down quars are mixed by the charged pion condensation.

10 µ-µi phase diagram Results (T = ) << f B = µ = p g 2 h + i 2 +(gh i µ I ) 2 or gh + i µ 25 2 h + i = 15 h + i6= 1 5 SB region pion cond. 1st order sigma second 1st order CEP µ I /m µi c = 1/2 Mπ is satisfied. Another 1st-order transition appear. Baryon-density jumps at the boundary.

11 Phase diagram Mean-field FRG T[MeV] 2nd-order h + i6= µ[mev] µ[mev ] 1st-order 1st pion cond 2nd pion cond TCP 1st order sigma SB region h + i = µ I [m ] µ I /m T µ[mev ] h + i = µ[mev] nd-order.2 pion cond. 1st order sigma SB region second 1st order CEP h + i6= µ I [m ] µ I /m Meson fluctuation hide TCP line. Ordinary chiral 1st-order phase boundary (red surface) is shrunen by the fluctuations.

12 Summary Silver Blaze relation is satisfied by the pion pole mass. The result of QM model agree with the Lattice QCD calculation. The result for higher µi depends on the sigma meson mass. We need to a light sigma mode. At low T, We have found the extra 1st-order phase transition at which quar density jumps. Meson fluctuations hide the TCP line which exists on the pion condensation surface.