Axial U(1) symmetry in 2-flavor QCD at finite temperature Sinya - PowerPoint PPT Presentation

Axial U(1) symmetry in 2-flavor QCD at finite temperature Sinya AOKI for JLQCD Collaboration Center for Gravitational Physics, Yukawa Institute for Theoretical Physics, Kyoto University YITP workshop MIN16 - Meson in Nucleus 2016 - 31 July - 2 Aug., Panasonic Hall, YITP, Kyoto University

1. Introduction low T high T Chiral symmetry of QCD restoration of chiral symmetry phase transition 2. Eigenvalue distribution of Dirac operator 1. Recovery of U(1)_A symmetry at high T ? relation ? Theoretical questions U (1) B ⊗ SU ( N f ) L ⊗ SU ( N f ) R U (1) B ⊗ SU ( N f ) V ρ ( λ ) λ : eigenvalue of Dirac operator

Eigenvalue density Banks-Casher relation (See later.) gap 0 What are general consequences ? (This talk) if chiral symmetry is restored. λ n � ρ ( λ ) = ρ n n ! n m → 0 � ¯ lim ψψ � = πρ (0) ρ (0) = ρ 0 = 0 If ρ ( λ ) has a gap Anomalous U (1) A symmetry is fully restored. ρ ( λ ) λ

Susceptibility Γ = 1 � χ A d 4 x � M A Γ ( x ) M A N f = 2 Γ (0) � V Γ ( x ) = ¯ α ( Γ ⊗ T A ) fg αβ ψ a ( x ) g M A ψ a ( x ) f β chiral SU (2) σ meson: 1 ⊗ 1 π meson: γ 5 ⊗ τ a U (1) A U (1) A chiral SU (2) η meson: γ 5 ⊗ 1 δ meson: 1 ⊗ τ a U (1) A susceptibilities χ π − η ≡ χ π − χ η χ σ − η ≡ χ σ − χ η χ π − δ ≡ χ π − χ δ If U (1) A is recovered, χ σ − η = χ π − δ = χ π − η = 0.

2. Previous Theoretical Investigation S.A, H. Fukaya, Y. Taniguchi, “Chiral symmetry restoration, eigenvalue density of Dirac operator and axial U(1) anomaly at finite temperature”, Phys. Rev D86(2012)114512.

Set up Lattice regularization with Overlap fermion, 2-flavors Eigenvalue spectrum complex pair Exact “chiral” symmetry but explicit U(1)_A anomaly form Ginsparg-Wilson relation doublers(chiral) zero modes(chiral) D γ 5 + γ 5 D = aDR γ 5 D γ 5 D γ 5 = D † λ A λ A λ A n λ A n + ¯ n = aR ¯ A : gauge configuration n λ 1/Ra 0 y 2/Ra 1/Ra − 1/Ra x n = ¯ D ( A ) φ A n = λ A n φ A D ( A ) γ 5 φ A λ A n γ 5 φ A n n

Some assumptions Note that this does not hold if the chiral symmetry is spontaneously broken. topological charge non-singlet chiral symmetry is restored. Ex. (Too strong. We should loosen this condition.) Assumption 2 Assumption 1 if O ( A ) is m -independent A : gauge configuration � O ( A ) � m = f ( m 2 ) f ( x ) is analytic at x = 0 1 V � Q ( A ) 2 � m = m Σ + O ( m 2 ) lim N f V →∞

Non-singlet chiral Ward-Takahashi identities eigenvalues density Results ∞ λ n 1 � � � � ¯ ρ A � ρ A ( λ ) ≡ lim = λ A n λ A δ λ − n n n ! V V →∞ n n =0 | λ | 3 m → 0 � ρ A ( λ ) � m = lim m → 0 � ρ A 3! + O ( λ 4 ) lim 3 � m No constraints to higher � ρ A n � m lim m → 0 h ρ A 3 i m 6 = 0 even for ”free” theory.

topological charge total number of zero modes � ρ A 0 � m = 0 1 1 V k � ( N A R + L ) k � m = 0 , V k � Q ( A ) 2 k � m = 0 lim lim V →∞ V →∞ N A R + L = N A R + N A L Q ( A ) = N A R − N A L N A R a number of right-handed zero modes N A L a number of left-handed zero modes

Consequences Singlet susceptibility at high T This, however, does not mean U(1)_A symmetry is recovered at high T. is necessary but NOT “sufficient” for the recovery of U(1)_A . 1st or 2nd ? What is the order of chiral phase transition in 2-flavor QCD ? Effective symmetry at hight T N 2 V → 0 χ π − η = lim f m 2 V � Q ( A ) 2 � m = 0 lim m → 0 lim V →∞ m → 0 χ π − η = 0 ” m π = m η ” lim full U (1) A is not recovered. SU (2) L ⊗ SU (2) R ⊗ Z 4 not SU (2) L ⊗ SU (2) R ⊗ U (1) A

Order of phase transition at N f =2 U(1) A is still broken at T > T c U(1) A is restored at T > T c 2nd order 1st order ? ? phase diagram of 2+1 flavor QCD SU (2) L ⊗ SU (2) R SU (2) L ⊗ SU (2) R ⊗ U (1) SU (2) L ⊗ SU (2) R ⊗ Z 4

Remarks Important conditions Large volume limit chiral limit lattice chiral symmetry Ginsparg-Wilson relation Fractional power for the eigenvalue density non-singlet chiral symmetry is recovered. consistent with the integer case (n > 2) Universal treatment ? (future investigations) V → ∞ m → 0 D γ 5 + γ 5 D = aDR γ 5 D ρ A ( λ ) � c A λ γ , γ > 0 γ ≤ 2 is excluded. γ > 2

3. Recent Numerical Results A. Tomiya et al. (JLQCD), Lat2015 G. Cossu et al. (JLQCD), Lat2015

Gap seems to open at Eigenvalue densities smaller quark mass. 1 Cossu et al. (JLQCD), Overlap � ρ ( λ ) = lim δ ( λ − λ n ) Phys. Rev. D87 (2013) 114514 V V →∞ n T c � 180 MeV

temperature Small eigenvalues appear. Gap seems to close at or above critical Buchoff et al. (LLNL/RBC), DomainWall, Phys. Rev. D89 (2014) 5,054514 0.02 0.02 149MeV 159MeV 0.015 0.015 ρ ( λ ) (GeV 3 ) ρ ( λ ) (GeV 3 ) 0.01 0.01 16 3 × 8 16 3 × 8 0.005 0.005 Min( λ 100 ) Min( λ 100 ) � � m l m l � � m s m s 0 0 0 0.05 0.1 0.15 0.2 0.25 0 0.05 0.1 0.15 0.2 0.25 λ (GeV) λ (GeV) 0.02 0.02 168MeV 177MeV 0.015 0.015 ρ ( λ ) (GeV 3 ) ρ ( λ ) (GeV 3 ) 0.01 0.01 16 3 × 8 16 3 × 8 0.005 0.005 Min( λ 100 ) Min( λ 100 ) � � m l m l � � m s m s 0 0 0 0.05 0.1 0.15 0.2 0.25 0 0.05 0.1 0.15 0.2 0.25 λ (GeV) λ (GeV) 0.02 0.02 T c � 180 MeV 186MeV 195MeV 0.015 0.015 16 3 × 8 16 3 × 8 ρ ( λ ) (GeV 3 ) ρ ( λ ) (GeV 3 ) Min( λ 100 ) Min( λ 100 ) � � 0.01 0.01 m l m l � � m s m s 0.005 0.005 0 0 0 0.05 0.1 0.15 0.2 0.25 0 0.05 0.1 0.15 0.2 0.25 λ (GeV) λ (GeV) 0.003 0.003 0.003 16 3 × 8 16 3 × 8 16 3 × 8 177MeV 186MeV 195MeV � � � m l m l m l ρ ( Λ ) (GeV 3 ) 0.002 ρ ( Λ ) (GeV 3 ) 0.002 ρ ( Λ ) (GeV 3 ) 0.002 0.001 0.001 0.001 0 0 0 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 Λ (GeV) Λ (GeV) Λ (GeV)

8 No Group Hot QCD (2013,2014) (Möbius)- Domain-wall (W/ ov) 2, 4, Viktor Dick et 3 fm al (2015) OV on HISQ sea No Akio Tomiya(Osaka Univ.) Why ? Fermion(Chiral sym.), Volumes or Topology ? Restored LLNL/RBC, domain-wall Overlap Fermion Size Gap in the spectrum U A (1) Correlator U(1) A JLQCD (2013) (Top. fixed) 2 fm Gap Degenerate Restored TWQCD (2013) Optimal Summary of recent results from chiral fermions. Even DW-type quarks do not agree... T & Tc No gap Degenerate degeneracy Violated 11 fm No gap degeneracy Violated 3, 4 fm No gap

What causes this difference ? (1)calculate eigenvalue distribution of overlap operator on these configurations (0)calculate eigenvalue distribution of DW operator on these configurations Preliminary full Overlap the full eigenvalue distribution (2)reweighting factor from the improved DW to Overlap is introduced to obtain partially quenched very small violation of GW relation volume ? quark mass ? lattice chiral symmetry ? generate gauge configurations with an improved DomainWall quarks Recent study by A. Tomiya et al. for JLQCD collaboration LLNL/RBC collaborations JLQCD collaboration DomainWall: approximated GW relation Overlap: exact GW relation original

T=190 MeV for L=3 fm, T=1.05 T c Domain-wall Preliminary may destroy the theoretically expected relation. An exact lattice chiral symmetry is essential. A tiny violation of the chiral symmetry open in full Overlap. After the reweighting, small eigenvalues in PQ disappear, and the gap seems to Akio Tomiya(Osaka Univ.) 17 (finer lattice) (reweighted)Overlap sea (partially quenched) Overlap on domain-wall peak Unphysical T= 190 MeV for L=3fm,T=1.05 Tc A. Tomiya et al. (JLQCD), Lat2015 L=32 x12 Domain-wall � =4.24 (T=195 MeV) 0.015 am=0.0025 0.01 � ( � ) (GeV 3 ) 0.005 0 0 20 40 60 80 100 120 140 � (MeV) L=32 x12 Partially quenched Overlap � =4.24 (T=195 MeV) � 0.015 0.015 am=0.0025 am=0.0025 0.01 0.01 � ( � ) (GeV 3 ) � � 0.005 0.005 0 0 0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 140 � (MeV) � (MeV)

G. Cossu et al. (JLQCD), Lat2015 U(1)_A susceptibility reweighted Overlap DomainWall zero-modes ∆ := χ π − χ δ 2 m 2 ∆ = 2 N R + L � + V ( λ 2 + m 2 ) 2 V m 2 λ � =0 Before After Topology from mode counting Topology from smeared conf. ∆ ∆

G. Cossu et al. (JLQCD), Lat2015 Partially Quenched If the gap opens, the effective symmetry is REWEIGHTING IS CRUCIAL ∆ ∆ Point: Reweighting is crucial Partially quenched results show accumulation of unphysical near zero modes SU(2) L ⊗ SU(2) R ⊗ U(1) A

S. Sharma, V. Dick, F. Karsch, E. Laermann, S. Mukherjee, Lattice2015 Eigenvalues density of Overlap on DomainWall (partially quenched !) From Sharma’s talk@Lat2015, This is an artifact due to PQ ! General features: Near zero mode peak +bulk A � λ 2 + A + B λ γ We fit to the ansatz: ρ ( λ ) = Bulk rises linearly as λ ,no gap seen. No gap even when quark mass reduced! 2 m π =135 MeV m π =200 MeV 1.8 1.08 T c 1.6 1.4 1.2 ρ ( λ )/T 3 1 0.8 γ =1 0.6 0.4 0.2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 λ /T

4. Conclusion