Chiral Random Matrix Model as a simple model for QCD

Transcription

1 Chiral Random Matrix Model as a simple model for QCD Hirotsugu FUJII (University of Tokyo, Komaba), with Takashi Sano T. Sano, HF, M. Ohtani, Phys. Rev. D 80, (2009) HF, T. Sano, Phys. Rev. D 81, ; D 83, , and in progress

2 QCD phase diagram One of the fundamental challenges in modern physics Ø Needs non-perturbative analyses Ø Ø Ø Lattice QCD at small µ; model studies w/ (P)NJL, etc. Beam Energy Scan programs are underway at RHIC/SPS

3 Chiral Random Matrix (ChRM) model and UA(1) anomaly Before we started our project, Hope: The simplest model for dynamical breaking of chiral symmetry should reveal the most common features of chiral phase transition Problem: It was unknown how to implement UA(1) breaking term, and then no flavor dependence in ChRM models

4 Outline Motivation Chiral Random Matrix (ChRM) Incorporating the UA(1) anomaly term Meson masses Phase diagram Columbia plot (Meson condensation at finite mu at T=0) Outlook & Summary

5 1. Chiral Random Matrix

6 QCD & Chiral Random Matrix Theory Review: Verbaarschot-Wettig QCD partition function Chiral symmetry in pair or λn=0; n right-, m left-handed modes hermitian with W is n x m complex matrix D has ν=n - m exact zero modes (index theorem) Topological sectors Fluctuation of ν = susceptibility w.r.t. θ

7 QCD & Chiral Random Matrix Theory Symmetry breaking: Banks-Casher rel Free theory: = k, ( ) ~ 3 JLQCD χsb = accumulation of low-lying Dirac modes by non-perturbative effects

8 QCD & Chiral Random Matrix Theory Restrict only to low-lying (constant) modes represented by a 2Nx2N matrix with Gaussian random elements (C.f. Nuclear Structure) QCD partition func. ChRM theory Equivalent to QCD in the ε regime, mπ<< 1/L << mρ, where constant pion fluctuations dominate in the partition function Application: (1) Universal spectral correlation (2) QCD-like model at N infinity

9 Model of QCD: sigma model representation Shuryak & Verbaarschot (1993) Integration over W Bosonization In thermodynamic limit & equal mass case Chiral symmetry is broken Nf is factorized (angular fluctuation of S is equiv to Nonlin sigma model) Broken phase

10 Finite Temperature extension Jackson & Verbaarschot (1996) Stephanov (1996) Introduce a deterministic field t respecting symmetry Symmetry restoration at finite T 2nd order for any Nf (Landau mean-field theory)

11 Extension to Finite T & µ Halasz et al. (1998) t : respects symmetry µ : breaks hermiticity Consistent with Landau-Ginzburg analysis T T& µ enter by symmetry consideration Independent of Nf m µ

12 2. Implementing the UA(1) anomaly term

13 Index theorem in ChRM model Index theorem: N+ : #(eigenmodes), ν : topological # of a gauge config Instantons UA(1) breaking Total partition fn is obtained by summing over ν (w/angle θ) In ChRM model, D of N x (N +ν) matrix W has ν exact-zero eigenvalues (P(ν): gauge field weight)

14 Extension of Zero-mode Space Janik, Nowak & Zahed (1997) Sano, HF, Ohtani (2009) Idea: Divide low-lying modes into two categories N+, N- : Topological (instanton-) zero modes and fluctuating 2N : Near-zero modes near-zero mode part Last term gives the phase e2inf ν θ when S S e2iθ

15 Complete partition fn. Sum over ν (Ι) Janik, Nowak & Zahed (1997) Poisson dist for instantons : KMT-type UA(1) breaking term appears! Potential is unbound φ3 term wins at large φ PPo dist modified by quark d.o.f. 't Hooft (1986)

16 Complete partition fn. Sum over ν (ΙΙ) T. Sano, HF, M. Ohtani (2009) cells Total number of modes must be finite N~ V Binomial dist 1-p p p: occupation prob Finite d.o.f. Z is a polynomial (except for Gauss weight) KMT int. appears under the log. in Ω Stable ground state

17 Nf Dependent Thermal Phase Transition Chiral condensate Nf=2 Σ=1, α=0.3, γ=2 Nf=3

18 Topological susceptibility at finite T

19 Topological susceptibility at finite T Our model satisfies the UA(1) identity! consistent with symmetries of QCD Top.suscept. follows the chiral condensate for small m Σ=1, α=0.3, γ=2

20 Topological susceptibility at finite T Σ=1, α=0.3, γ=2 follows the chiral condensate

21 3. meson masses

22 Meson curvature masses Σ=1, α=0.3, γ=2 Singlet pseudo-scalar meson is massive by anomaly

23 Meson curvature masses Σ=1, α=0.3, γ=2 Singlet pseudo-scalar meson is massive by anomaly All the masses degenerate in symm phase in spite of UA(1) breaking See Hatsuda-Lee, also Jido's lecture

24 Singularity at the critical point mud=0.01 & ms=0.2; α=0.5, γ=1 Only σ becomes massless Note that, at CP, σ mixes with density and heat fluctuations; all susceptibilities χmm, χµµ, χtt diverge

25 4. Columbia Plot

26 2+1 flavor phase diagram: µ=0 plane TCP crossover 1st order The stronger KMT term makes the 1st order region wider Boundary curve is consistent with mean-field prediction

27 Critical Surface 1st order region expands as µ increases Familiar situation with constant KMT coupling O(4) criticality α=0.5 & γ=1

28 5. Meson condensation at T=0 at finite µ

29 Meson condensation at T=0, finite µi Chemical pontentials, and condensates HF, T. Sano (2010), Cf. B. Klein, et al. (2003) α = 0.5 & γ =1 Gap eq for ρ (m=0) In vacuum, chiral&meson condensed phases degenerate if m=0 At larger µi, a pion condensed phase appears At small µi, a single chiral transition along µq due to anomaly

30 Meson condensation at T=0, finite µi & µy HF, T. Sano (2010), Cf. Araki, Yoshinaga, (2008) Kaon condensation appear in the diagram Chiral restoration&meson conds compete with each other Regarding CSC phase, see Sano-Yamazaki (2011)

31 6. Outlook Complex Langevin simulation

32 ChRM to study Complex Langevin simulation D is non(anti)hermitian at finite µ; the same sign problem as QCD invalidates importance sampling Langevin eq makes a system distributed around a minimum When S becoms complex at some config, originally real vars become complex after evolution (average must be real, though) Known old problems in Complex Langevin simulations: Convergence: avoided using adoptive step size! Correctness of equilib dist: trial&error situation

33 ChRM to study Complex Langevin simulation Converging, but wrong sign problem or other reason? N=2; finite system Sano-HF-Kikukawa, in progress N=infinity Han-Stephanov (2008)

34 Summary ChRM model with UA(1) anomaly is constructed consistent with symmetries of QCD Fluctuations of #(zero modes) N+ result in physical behavior of top. susceptibility meson masses and T-µ phase diagram are qualitatively the same as those in other models Meson condensation is studied as a response to chemical potentials Chiral Random Matrix model is a useful toy model for QCD e.g., investigation of the sign problem

35 Introduction: Chiral Random Matrix Theory Reviewed in Verbaarschot & Wettig (2000 Chiral random matrix theory 1. Exact description for QCD in ε regime 2. A schematic model with chiral symmetry In-mediun Models Jackson & Verbaarschot (1996) Chiral restoration at finitewettig, T Schaefer & Weidenmueler(1996) Halasz et. al. (1998) Phase diagram in T-µ Han & Stephanov (2008) Sign problem, etc Bloch, & Wettig(2008) U(1) problem & resolution (vacuum) Janik, Nowak, Papp, & Zahed (1997) Known problems at finite T 1. Phase transition is 2nd-order irrespective of Nf 2. Topological susceptibility behaves unphysically Ohtani, Lehner, Wettig & Hatsuda (2008)