QCD ( ), ( ) 8 23 Typeset by FoilTEX

QCD ( ), ( ) 8 23 Typeset by FoilTEX

Introduction:QCD Phase diagram of QCD system Critical end point Quark Gluon Plasma T c Temperature 1st order phase transition line Hadron 0 CSC 0 Chemical potential summer school 2008 1/20

Introduction: G = SU R (N f ) SU L (N f ) G SU v (N f ) at T = 0( ) T > T c ( ) Q. Banks-Casher Dirac Dirac chrmt QCD summer school 2008 2/20

Table of contents 1. Introduction 2. Dirac Dirac 3. Dirac 4. 5. summer school 2008 3/20

2.1 QCD Dirac Z QCD (m) = = D ψdψda µ e R d 4 x ψ(d+m)ψ(x) S YM DA µ det(d + m f )e S YM det(d + m f ) f f YM QCD Dirac D = γ µ ( µ + ia µ ) Yang- Mills D iλ n det(d + m f ) = n (iλ n + m f ) summer school 2008 4/20

2.2 Dirac 1. D = D iλ (λ R) 2. {D, γ 5 } = 0 ( ) ±iλ (..Ḋφ = iλφ γ 5Dφ = Dγ 5 φ = iλγ 5 φ Dγ 5 φ = iλγ 5 φ) D D = [ 0 iw iw 0 ] summer school 2008 5/20

2.3 Banks-Casher Dirac (Dirac )ρ(λ): ρ(λ) = δ(λ λ n ) n QCD Banks-Casher : < A.Casher,1980) ψψ = πρ(0) V λ ρ(λ) (cf. ρ free (λ) λ 3 0 (λ 0)) ψψ > ρ(λ) (T.Banks & summer school 2008 6/20

3.1 (E.V.Shuryak& J.J.M.Verbaarschot,1993) Z RM ν=0(m) = DW f det(d + m f )e nσ2 tr(w W ) D = [ 0 iw iw 0 ] D n n W W (chiral Gaussian ensemble) W (Random distribution) summer school 2008 7/20

3.2 chrmt (1) chrmt Dirac (J.J.M.Verbaarschot & I. Zahed,1993; K. Takahashi,1998) (chgue, ν = 0, m 0) Z RM = dλ 1 dλ n p(λ 1,, λ n ) p(λ 1,, λ n ) λ 2 j λ 2 l 2 λ 2N f +1 j e nσ2 λ 2 j j<l j ρ(λ) dλ 2 dλ n p(λ, λ 2,, λ n ) summer school 2008 8/20

3.3 chrmt (2) ρ(λ)/v = λσ 2 (nσ 2 λ 2 ) N fe nσ2 λ 2 n! (n + N f 1)! [ L N f n (nσ 2 λ 2 ) 2 L N f +1 n (nσ 2 λ 2 )L N f 1 n (nσ 2 λ 2 ) ] Σ2 2π 4 Σ 2 λ2 (V ( )) (L a n(x):laguerre ) Banks-Casher ψψ = πρ(0)/v = Σ> 0 chrmt summer school 2008 9/20

3.4 chrmt Dirac (N f = 1) Ρ Λ V 0.35 0.30 0.25 0.20 0.15 n 5 n 10 n 0.10 0.05 0.5 1.0 1.5 2.0 2.5 Λ V (= 2n) ( ) summer school 2008 10/20

3.5 QCD chrmt ( ) QCD Dirac chrmt Dirac Q:chRMT A1 QCD (cf. NJL model) schematic model chrmt (cf. (A.D.Jackson& J.J.M.Verbaarschot,1996; M.A.Stephanov,1996)etc.) A2 chrmt QCD (Universality) summer school 2008 11/20

3.6 Universality ρ(λ) λ = 0 ( ) Microscopic spectrumρ s (z) Universal (E.V.Shuryak& J.J.M.Verbaarschot,1993) ρ s (z) lim V 1 V Σ ρ( z ) (z = V Σλ) V Σ = z 2 [J 2 N f (z) J 2 N f +1(z)J 2 N f 1(z)] (cf. Leutwyler- Smilga sum rule(h.leutwyler& A.Smilga,1992), Gaussian distribution stability etc.) summer school 2008 12/20

4.1 QCD Dirac Dirac chrmt Dirac :Staggered Fermion SU(2) (β = 4/g 2 0) V = 6 4, 1000 configurations V = 8 4, 1081 configurations summer school 2008 13/20

4.2 :global spectrum 0.25 0.2 8 4 global spectrum 6 4 global spectrum )/V ( 0.15 0.1 0.05 0 0 0.5 1 1.5 2 2.5 3 ρ(0)/v = 0.21(L = 6), 0.20(L = 8) > 0(lattice unit) summer school 2008 14/20

4.3 :microscopic spectrum 0.6 0.5 chrmt prediction 0.4 s (z) 0.3 0.2 0.1 0 0 5 10 15 20 z ρ s (z) = 2z 2 1 0 duu2 1 0 dv[j 0(2uvz)J 1 (2uz) vj 0 (2uz)J 1 (2uvz)] (T.Nagao & P.J.Forrester,1995). summer school 2008 15/20

4.3 :microscopic spectrum(l = 6) 0.6 0.5 microscopic spectrum chrmt prediction 0.4 s (z) 0.3 0.2 0.1 0 0 5 10 15 20 z ρ s (z) = 2z 2 1 0 duu2 1 0 dv[j 0(2uvz)J 1 (2uz) vj 0 (2uz)J 1 (2uvz)] (T.Nagao & P.J.Forrester,1995). summer school 2008 16/20

4.3 :microscopic spectrum(l = 8) 0.6 0.5 microscopic spectrum chrmt prediction 0.4 s (z) 0.3 0.2 0.1 0 0 5 10 15 20 z ρ s (z) = 2z 2 1 0 duu2 1 0 dv[j 0(2uvz)J 1 (2uz) vj 0 (2uz)J 1 (2uvz)] (T.Nagao & P.J.Forrester,1995). summer school 2008 17/20

4.4 :microscopic spectrum 0.6 0.5 0.4 8 4 microscopic spectrum 6 4 microscopic spectrum chrmt prediction s (z) 0.3 0.2 0.1 0 0 5 10 15 20 z summer school 2008 18/20

4.5 Dirac ρ s (z) quenched SU(2) (M.E.Berbenni-Bitsch, S.Meyer, A.Schäfer, J.J.M.Verbaarschot & T.Wettig,1998) 2-flavor QCD (H.Fukaya, S.Aoki, T.W.Chiu, S.Hashimoto, T.Kaneko, H.Matsufuru, J.Noaki, K.Ogawa, T.Onogi, & N. Yamada, 2007) summer school 2008 19/20

5 chrmt chrmt schematic model ρ s (z) Universality Fermion model chrmt: summer school 2008 20/20

A.1 Banks-Casher (N f = 1) ψψ = lim m 0 = lim m 0 = lim m 0 = lim m 0 lim V lim V lim V 1 V m ln ZQCD = lim 1 1 V iλ n n + m QCD 1 2m V λ 2 n + m 2 dλ ρ(λ) V λ n >0 2m λ 2 + m 2 = m 0 QCD lim V dλ ρ(λ) V 1 V 1 Z QCD m det(d + m) YM (... ±λ ) πδ(λ) = πρ(0) V summer school 2008 21/20