a chiral random matrix model for 2+1 flavor qcd at finite temperature and density
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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
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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
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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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