A Chiral Random Matrix Model for 2+1 Flavor QCD at Finite Temperature and

Density

Takashi Sano (University of Tokyo, Komaba),with H. Fujii, and M. Ohtani

• UA(1) breaking and phase transition in chiral random matrix model arXiv:0904.1860v2 [hep-ph](to appear in PRD) TS, H. Fujii & M. Ohtani • Work in progress , H. Fujii & TS

Outline

1. Introduction2. Chiral Random Matrix Models 3. ChRM Models with Determinant Interaction4. 2 & 3 Equal-mass Flavor Cases5. Extension to Finite T & m with 2+1 Flavors 6. Conclusions & Further Studies

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１． Introduction

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Introduction: Chiral Random Matrix Theory

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•Chiral random matrix theory1. Exact description for finite volume

QCD2. A schematic model with chiral

symmetry

Reviewed in Verbaarschot & Wettig (2000)

• In-mediun Models• Chiral restoration at finite T• Phase diagram in T-m• 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 T1. Phase transition is 2nd-order irrespective of Nf2. Topological susceptibility behaves unphysically

Ohtani, Lehner, Wettig & Hatsuda (2008)

2. Chiral Random Matrix Models

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Chiral Random Matrix Models

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• Dirac operator:

: : Complex random matrix

• Topological charge:• Thermodynamic limit: # of (quasi-)zero

modes

Shuryak & Verbaarschot (1993)

: Chiral Symmetry

•Model definition(Vacuum)• Partition function

Gaussian

Effective Potential W in the Vacuum

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3 2 1 1 2 3

4

6

8

10

φ

ΩThe effective potential

Gaussian integral over W

Hubbard-Stratonovitch transformation

Broken phase

Finite Temperature ChRM Model

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•Temperature effect: periodicity in imaginary time

effective potential

Jackson & Verbaarschot(1996)• Deterministic external field t

4 2 2 4

5

10

15

t=1

t=0.2

Ω

φ

t=4

Chiral symmetry is restored at finite T • 2nd-order for any number of Nf• Inadequate as an effective model for QCD

Determinant interaction should be incorporated

Unphysical Suppression of ctop

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T/Tc

as N

Adopted from Ohtani’s slide (2007)

B. Alles, M. D‘Elia & A. Di Giacomo NPB483(2000)139

Ohtani, Lehner, Wettig & Hatsuda (2008)

ChRM model lattice

Our model describes physical & Nf-dependent phase transition

3. ChRM models with determinant interaction

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Extension of Zero-mode Space

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Janik, Nowak & Zahed (1997)

•N+, N- : Topological (quasi) zero modes = instanton origin (localized)•2N : near zero modes temperature effects

• N+=N-=0 reduced to conventional model with n=0

Sum over Instanton Distribution

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Example: Poisson dist. (free instantons)

‘t Hooft int.

2 1 1 2 3

3

2

1

1

2 W

f

Nf=3Unbound potential

•f ^3 terms dominate (Nf=3)

Unfortunately…

‘ Hooft (1986)

Binomial Distribution for N+, N-

13 . With this distribution, the effective potential become

cells

p1-p

: unit cell size

TS, H. Fujii & M. Ohtani (2009)

p: single instanton existence probabilityWith binomial summation formula,

Regularized distribution

‘t Hooft int. appears under the log.3 2 1 1 2 3

3

2

1

1

2

3

Poisson

Binomial

f

W

Stable ground state

4. 2 & 3 Equal-mass Flavors

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Nf Dependent Phase Transition

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1st2ndNf=2 Nf=3

•2nd-order for Nf=2, 1st-order for Nf=3 in the chiral limit

S=1, a=0.3, g=2

Topological Susceptibility

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• No unphysical suppressionNf=2 Nf=3

• correct q dependence: Axial Ward identity ：

Mesonic Masses

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Anomaly makes h heavy Consistent with Lee & Hatsuda (1996)

Nf=2 Nf=3

h(ps0)

s(s0)

d(s)

p(ps)

h(ps0)

d(s)

p(ps)

s(s0)

m=0.10 m=0.10

5. Extension to Finite T & m with 2+1 Flavors

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Conventional Model at Finite T & m

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W

f

equal-massm=T=0

• m-m symmetry

Halasz et. al. (1998)

Tm

m

Independent of Nf

Proposed Model at Finite T & m

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• S can be absorbed: S=1 • a & g : “anomaly effects”

W

f

equal-mass Nf=3m=T=0

near-zero mode

m=0 Plane

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Critical line on mud-ms plane TCP on ms axis

crossover

g=1

Critical Surface

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Positive curvature for all m

a=0.5,& g=1

Tri-critical line

Equal-mass Nf=3 Limit

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g=1g

aA. Curvature at m=0 seems positive for whole parameters

Q. How does the curvature depend on a & g?

m-dependent a

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2g=1, a0=0.5, & m0=0.2

• Negative curvature can be generated

Conclusions & Further Studies

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We have constructed the ChRM model with U(1) breaking determinant term Stable ground state solution binomial distribution 1st order phase transition for Nf=3 at finite T Physical topological susceptibility & Axial Ward

identity We apply the model to the 2+1 flavor case at

finite T & m Critical surface: Positive curvature for constant

parameters Outlook More on the 2+1 flavor case (in progress) Isospin & strangeness chemical potential Color superconductivity …etc

cf. Vanderheyden,& Jackson (2000)

Thank you

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