Chiral Random Matrix Model as a simple model for QCD Hirotsugu - - PowerPoint PPT Presentation

Chiral Random Matrix Model as a simple model for QCD

Hirotsugu FUJII (University of T

• kyo, Komaba),

with T akashi Sano

• T. Sano, HF, M. Ohtani, Phys. Rev. D 80, 034007 (2009)
• HF, T. Sano, Phys. Rev. D 81, 037502; D 83, 014005, and in progress

QCD phase diagram

Ø One of the fundamental challenges in modern physics Ø Needs non-perturbative analyses

Ø Lattice QCD at small µ; model studies w/ (P)NJL, etc. Ø Beam Energy Scan programs are underway at RHIC/SPS

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

• f chiral phase transition

Problem:

• It was unknown how to implement U

A(1) breaking

term, and then no flavor dependence in ChRM models

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

• 1. Chiral Random Matrix

QCD & Chiral Random Matrix Theory

• QCD partition function
• Chiral symmetry

Review: Verbaarschot-Wettig 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)

• T
• pological sectors

Fluctuation of ν = susceptibility w.r.t. θ

QCD & Chiral Random Matrix Theory

• Symmetry breaking: Banks-Casher rel
• χSB = accumulation of low-lying Dirac modes by

non-perturbative effects

Free theory: λ = k, ρ(λ) ~ λ3 JLQCD

QCD & Chiral Random Matrix Theory

QCD partition func. → ChRM theory

Restrict only to low-lying (constant) modes represented by a 2Nx2N matrix with Gaussian random elements (C.f. Nuclear Structure)

Equivalent to QCD in the ε regime, m

π<< 1/L << m ρ , where

constant pion fluctuations dominate in the partition function Application: (1) Universal spectral correlation (2) QCD-like model at N → infinity

Model of QCD: sigma model representation

In thermodynamic limit & equal mass case Integration over W →

Bosonization → Broken phase

• Chiral symmetry is broken
• Nf is factorized

(angular fluctuation of S is equiv to Nonlin sigma model)

Shuryak & Verbaarschot (1993)

Finite T emperature extensionJackson & 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)

Extension to Finite T & µ

Halasz et al. (1998)

T m µ Independent of Nf

t : respects symmetry µ : breaks hermiticity

Consistent with Landau-Ginzburg analysis T& µ enter by symmetry consideration

• 2. Implementing the UA(1) anomaly term

Index theorem in ChRM model

• Index theorem:
• N+ – : #(eigenmodes), ν : topological # of a gauge config
• Instantons → UA(1) breaking
• T
• tal 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)

Extension of Zero-mode Space Janik, Nowak & Zahed (1997)

Sano, HF, Ohtani (2009)

• N+, N- : T
• pological (instanton-) zero modes and fluctuating
• 2N : Near-zero modes

Idea: Divide low-lying modes into two categories

near-zero mode part

Last term gives the phase e2iNf ν θ when S → S e2iθ

Complete partition fn. – Sum over ν (Ι)

't Hooft (1986)

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.

Complete partition fn. – Sum over ν (ΙΙ)

cells

p 1-p

• T. Sano, HF, M. Ohtani (2009)

p: occupation prob T

• tal number of modes must be finite N~ V

Binomial dist

Finite d.o.f. → Z is a polynomial (except

for Gauss weight)

KMT int. appears under the log. in Ω Stable ground state

Nf Dependent Thermal Phase Transition

Nf=3

Σ=1, α=0.3, γ=2

Nf=2

Chiral condensate

T

• pological susceptibility at finite T

T

• pological susceptibility at finite T

Σ=1, α=0.3, γ=2

Our model satisfies the UA(1) identity ! → consistent with symmetries of QCD

Top.suscept. follows the chiral condensate for small m

T

• pological susceptibility at finite T

Σ=1, α=0.3, γ=2

follows the chiral condensate

• 3. meson masses

Meson curvature masses

Σ=1, α=0.3, γ=2

Singlet pseudo-scalar meson is massive by anomaly

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

Singularity at the critical point

Only σ becomes “massless” Note that, at CP, σ mixes with density and heat fluctuations; all susceptibilities χ

mm, χ µµ, χ TT diverge

mud=0.01 & ms=0.2; α=0.5, γ=1

• 4. Columbia Plot

2+1 flavor phase diagram: µ=0 plane

The stronger KMT term makes the 1

st order region wider

Boundary curve is consistent with mean-field prediction

crossover

1st order TCP

Critical Surface

1

st order region expands as µ increases

Familiar situation with constant KMT coupling α=0.5 & γ=1

O(4) criticality

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

Meson condensation at T=0, finite µI

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

α = 0.5 & γ =1

HF, T. Sano (2010),

• Cf. B. Klein, et al. (2003)

Gap eq for ρ (m=0) Chemical pontentials, and condensates

Meson condensation at T=0, finite µI & µY

Kaon condensation appear in the diagram Chiral restoration&meson conds compete with each other

Regarding CSC phase, see Sano-Yamazaki (2011) HF, T. Sano (2010),

• Cf. Araki, Yoshinaga, (2008)

• 6. Outlook

Complex Langevin simulation

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

ChRM to study Complex Langevin simulation

Converging, but wrong – sign problem or other reason?

Han-Stephanov (2008) N=2; finite system Sano-HF-Kikukawa, in progress N=infinity

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

Introduction: Chiral Random Matrix Theory

• Chiral random matrix theory
• 1. Exact description for QCD in ε regime
• 2. A schematic model with chiral symmetry

Reviewed in Verbaarschot & Wettig (2000)

• In-mediun Models
• Chiral restoration at finite T
• Phase diagram in T-µ
• Sign problem, etc…
• U(1) problem & resolution (vacuum)

Jackson & Verbaarschot (1996) Halasz et. al. (1998) Han & Stephanov (2008) Janik, Nowak, Papp, & Zahed (1997) Wettig, Schaefer & Weidenmueler(1996) Bloch, & Wettig(2008)

Known problems at finite T

• 1. Phase transition is 2nd-order irrespective of Nf
• 2. Topological susceptibility behaves unphysically

Ohtani, Lehner, Wettig & Hatsuda (2008)