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: v2 [hep-ph](to appear in PRD) TS, H. Fujii & M. Ohtani Work in progress, H. Fujii & TS

Outline 1. Introduction 2. Chiral Random Matrix Models 3. ChRM Models with Determinant Interaction 4. 2 & 3 Equal-mass Flavor Cases 5. Extension to Finite T &  with 2+1 Flavors 6. Conclusions & Further Studies 2

１． Introduction 3

Introduction: Chiral Random Matrix Theory 4 Chiral random matrix theory 1.Exact description for finite volume QCD 2.A schematic model with chiral symmetry Reviewed in Verbaarschot & Wettig (2000) In-mediun Models Chiral restoration at finite T Phase diagram in T-  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)

2. Chiral Random Matrix Models 5

Chiral Random Matrix Models 6 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  in the Vacuum 7 φ Ω The effective potential Gaussian integral over W  Hubbard-Stratonovitch transformation  Broken phase

Finite Temperature ChRM Model 8 Temperature effect : periodicity in imaginary time effective potential Jackson & Verbaarschot(1996) Deterministic external field t t=1 t=0.2 Ω φ t=4  Chiral symmetry is restored at finite T How to include the determinant interaction? 2nd-order for any number of Nf Inadequate as an effective model for QCD Determinant interaction should be incorporated

Unphysical Suppression of  top 9  T/TcT/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 The effective potential is not analytic at =0 (includes | | term)

3. ChRM models with determinant interaction 10

Extension of Zero-mode Space 11 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 =0 Lehner, Ohtani, Verbaarschot, & Wettig (2009)

Sum over Instanton Distribution 12 Example: Poisson dist. (free instantons) ‘t Hooft int.   Nf=3 Unbound potential  ^3 terms dominate (Nf=3) Unfortunately… ‘ Hooft (1986)

Binomial Distribution for N+, N- 13 .. γ, p: parameters With this distribution, the effective potential become  The potential is bounded.  Anomalous U A (1) breaking is included. cells p 1-p : unit cell size TS, H. Fujii & M. Ohtani (2009) p: single instanton existence probability With binomial summation formula,  Regularized distribution ‘t Hooft int. appears under the log. Poisson Binomial    Stable ground state

4. 2 & 3 Equal-mass Flavors 14

Nf Dependent Phase Transition 15 1st 2nd Nf=2 Nf=3 2nd-order for Nf=2, 1st-order for Nf=3 in the chiral limit 

Topological Susceptibility 16 No unphysical suppression Nf=2 Nf=3 correct  dependence:  Axial Ward identity ：

Mesonic Masses 17  Anomaly makes  heavy  Consistent with Lee & Hatsuda (1996) Nf=2Nf=3  (ps0)  (s0)  (s)  (ps)  (ps0)  (s)  (ps)  (s0) m=0.10

5. Extension to Finite T &  with 2+1 Flavors 18

Conventional Model at Finite T &  19   equal-mass m=T=0   -  symmetry Halasz et. al. (1998) T m  Independent of Nf

Proposed Model at Finite T &  20  can be absorbed:  =1  “anomaly effects”   equal-mass Nf=3 m=T=0 near-zero mode

 =0 Plane 21  Critical line on m ud -m s plane  TCP on m s axis crossover 

Critical Surface 22  Positive curvature for all m  Tri-critical line

Equal-mass Nf=3 Limit 23    A. Curvature at  =0 seems positive for whole parameters Q. How does the curvature depend on  ?

 -dependent  24 2      Negative curvature can be generated

Conclusions & Further Studies 25  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 &   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 26

Remarks on Binomial Distribution 27 At  Stable ground state Poisson Binomial φ Ω

More on Binomial Distribution 28 Free instantons: Poisson distribution with fixed limit leads Poisson dist. with Ω Po Binomial φ Ω Excluding instantons to occupy the same cells Not free, but repulsive No stable solutions with Poisson distribution

Extended Structure of the Dirac Operator 29 We assume that D is N+ + N- + 2N dimensional matrix and that the near zero mode number 2N does not vary when a volume V is given. We set 2N/V=1 for convenience. The thermodynamic limit is taken by N to infinity. Near zero modes 2N Topological zero modes N+ and N- Bosonization ・ topological zero modes : instanton-origin ・ near-zero modes : matter effects Janik, Nowak & Zahed (1997)

N+, N- Distribution 30 .. γ, p: parameters With this distribution, the effective potential become  The potential is bounded.  Anomalous U A (1) breaking is included.  N cells p 1-p : unit cell size with TS, H. Fujii & M. Ohtani (2009) p: single instanton existence probability Conventional model + U(1) breaking term

QCD chiral phase structure 31  Several effective models(NJL, σ -model, ladder QCD etc…) indicate: the existence of CP depends on the strange quark mass and it exist if the transition is cross over at μ =0.  A recent lattice QCD calculation(de Forcrand & Philipsen JHEP 0701, 077(2007)) suggests completely different mass- dependence of the QCD phase structure: CP does not exist (at least in small μ region) if the transition is cross over at μ =0.  Complete contradiction Does QCD critical point(CP) exist on the temperature(T)- chemical potential( μ ) plane? Effective models(NJL, σ -model, ladder QCD etc…): Nf=2: TCP(ch. limit), CP(m>0) Nf=2+1: ms dependent Lattice QCD calc. Nf=2+1: No critical points de Forcrand & Philipsen JHEP 0701, 077(2007) T μ μ T 2nd Asakawa &Yazaki NPA504(1989)

Chiral Phase Transition of 2 & 3 flavor QCD 32 Finite temperature chiral phase transition Breaking pattern SSB Phase transition (restoration) Landau-Ginzburg approach (mean field theory) Remnant of UA(1)Effective free energy# of flavors

Universality argument: summary 33 Order of phase transition (in the chiral limit) Second order for First order for The difference is due to anomaly term (mean filed level) Correct even if fluctuation is introduced (Pisarski-Wilczek 1984) Effective models must show the same dependence (p)NLJ model, sigma model… How about chiral random matrix model

Instanton distribution 34  Statistical factor for free instanton ensemble:  Poisson distribution With the effective potential (We set S=phi x Identity)The above procedure is, indeed, constructed to reproduce this effective potential. reasonable form, but… ‘t Hooft Phys. Rept.142(1986) The total partition function

The problem: No Stable Solutions for Nf=3 35  The determinant interaction produces the φ ^Nf terms which is unbound for Nf=3 at large φ.  In NJL models, the kinetic term-origin momentum integrals serve higher order of φ which make the potentials bounded.  The ChRM model, however, does not have a kinetic term.  For the gaussian ChRM model, the effective potential should be bounded by the φ ^2 term. Is this problem unavoidable? Ω φ ( )Assuming term dominates for large Unbound Nf=3, m=0 How to avoid the problem?

2nd-order Phase Transition 36  Unsuitable for the Nf=3 & Nf=2+1 case.  Cannot produce Columbia plot.  The effective potential is a summation of that of 1 flavor  Flavor-mixing term is needed Critical point(t=1) Broken phase(t=0.2) Ω φ Symmetric phase(t=4) At the chiral limit Irrespective of the # of flavors How to include the determinant interaction?

Binomial to Poisson 37 with fixed limit leads Poisson dist. with cells large cells Excludes instantons to occupy the same cells Size of the cells may be the typical repulsive interaction range Poisson distribution is the infinitely fine limit cf. ``vacuum’’ of Poisonian model phi=∞ corresponds to a state of infinite # of instantons

Topological Susceptibility 38 Janik, Nowak, Papp, & Zahed (1997) Anomaly effect at T=0 Gaussian distribution of Non-zero eta mass: resolution to U(1) problem Quenched limit with Screening of the topological susceptibility with light quarks At non-zero T Unphysical suppression of Modified models are proposed. Ohtani, Lehner, Wettig & Hatsuda (2008) Lehner, Ohtani, Verbaarschot & Wettig (2009)

2+1 flavors at finite T &  39 T m   can be absorbed:  =1  “anomaly effects” scale: not yet Halasz et. al. (1998)     -  symmetry Nf=3 m=T=0

2+1 flavors at finite T &  40  Phase structure under 4 parameters : m ud, m s, T &   (Tri-)CP exists or not?  parameters:    can be absorbed:  =1   “anomaly effects”  =0  T m 