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

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.

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

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  1. A Chiral Random Matrix Model for 2+1 Flavor QCD at Finite Temperature and Density TakashiSano (UniversityofTokyo,Komaba), with H. Fujii, and M. Ohtani • UA(1)breakingandphasetransitioninchiralrandommatrixmodelarXiv:0904.1860v2 [hep-ph](toappear inPRD)TS,H.Fujii&M.Ohtani • Work in progress , H. Fujii & TS

  2. Outline • Introduction • Chiral Random Matrix Models • ChRMModelswithDeterminantInteraction • 2 & 3 Equal-mass Flavor Cases • Extension to Finite T & m with 2+1 Flavors • Conclusions&FurtherStudies

  3. １． Introduction

  4. Introduction: Chiral Random Matrix Theory ReviewedinVerbaarschot&Wettig(2000) • Chiral random matrix theory • Exact description for finite volume QCD • A schematic model with chiral symmetry • In-mediun Models • ChiralrestorationatfiniteT • Phasediagram in T-m • Signproblem,etc… • U(1)problem & resolution(vacuum) Jackson&Verbaarschot(1996) Wettig, Schaefer & Weidenmueler(1996) Halaszet.al.(1998) Han & Stephanov (2008) Bloch, & Wettig(2008) Janik, Nowak, Papp, & Zahed (1997) Known problemsatfiniteT Phasetransitionis2nd-orderirrespectiveofNf Topologicalsusceptibility behavesunphysically Ohtani,Lehner,Wettig&Hatsuda(2008)

  5. 2.Chiral Random MatrixModels

  6. Chiral Random Matrix Models • Model definition(Vacuum) • Partition function Shuryak & Verbaarschot (1993) Gaussian • Dirac operator: :Chiral Symmetry : :Complex random matrix • Topologicalcharge: • Thermodynamic limit: #of(quasi-)zeromodes

  7. Effective Potential W in the Vacuum Gaussian integral over W  Hubbard-Stratonovitch transformation  Ω Theeffectivepotential φ Broken phase

  8. FiniteTemperatureChRMModel Ω • Deterministicexternalfield t Jackson&Verbaarschot(1996) • Chiral symmetry is restored at finite T t=4 t=1 t=0.2 effective potential φ • 2nd-orderfor any number ofNf • Inadequate as an effective model for QCD Determinant interaction should be incorporated • Temperatureeffect:periodicityinimaginarytime Howtoincludethedeterminantinteraction?

  9. as N   T/Tc Unphysical Suppression of ctop Adopted from Ohtani’sslide(2007) ChRMmodel lattice Ohtani,Lehner,Wettig&Hatsuda(2008) B. Alles, M. D‘Elia&A. Di GiacomoNPB483(2000)139 Our model describes physical & Nf-dependent phase transition Theeffectivepotentialisnotanalyticatn=0(includes|n|term)

  10. 3.ChRMmodelswithdeterminant interaction

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

  12. Sum overInstantonDistribution Example: Poisson dist. (free instantons) ‘ Hooft (1986) ‘t Hooft int. Nf=3 W Unfortunately… f Unbound potential • f^3 termsdominate (Nf=3)

  13. Binomial Distribution forN+, N- TS,H.Fujii&M.Ohtani(2009) cells • .  Regularized distribution W Binomial f 1-p p p:singleinstantonexistenceprobability Poisson Withbinomialsummationformula, :unitcellsize ‘t Hooft int. appears under the log. Stablegroundstate With this distribution, the effective potential become • The potential is bounded. • Anomalous UA(1) breaking is included. γ,p:parameters

  14. 4. 2 & 3 Equal-mass Flavors

  15. Nf DependentPhase Transition S=1, a=0.3, g=2 Nf=2 Nf=3 1st 2nd • 2nd-order for Nf=2, 1st-order for Nf=3 in the chirallimit

  16. Topological Susceptibility Nf=2 Nf=3 • Nounphysicalsuppression • correct q dependence: Axial Ward identity：

  17. Mesonic Masses • Anomaly makes hheavy • Consistent with Lee & Hatsuda (1996) Nf=2 Nf=3 d(s) m=0.10 m=0.10 d(s) s(s0) s(s0) h(ps0) h(ps0) p(ps) p(ps)

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

  19. Conventional Model at Finite T & m Halaszet.al.(1998) W equal-mass m=T=0 f • m-m symmetry T m Independent of Nf m

  20. Proposed Model atFiniteT&m W equal-mass Nf=3 m=T=0 near-zero mode f • S can be absorbed: S=1 • a & g : “anomaly effects”

  21. m=0 Plane • Criticallineonmud-msplane • TCPonmsaxis g=1 crossover

  22. CriticalSurface • Positivecurvatureforallm a=0.5,& g=1 Tri-critical line

  23. Equal-mass Nf=3Limit Q. How does the curvature depend on a & g? g=1 g A. Curvatureatm=0seemspositiveforwholeparameters a

  24. m-dependent a 2 g=1, a0=0.5, & m0=0.2 • Negative curvature can be generated

  25. Conclusions&FurtherStudies • We have constructed the ChRM model with U(1) breaking determinant term • Stable ground state solution  binomial distribution • 1storderphasetransition for Nf=3 at finite T • Physicaltopologicalsusceptibility &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+1flavorcase(inprogress) • Isospin&strangeness chemical potential • Color superconductivity • …etc cf.Vanderheyden,&Jackson(2000)

  26. Thankyou

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