U(1) axial symmetry at high temperature Sinya AOKI for JLQCD - PowerPoint PPT Presentation

U(1) axial symmetry at high temperature Sinya AOKI for JLQCD Collaboration Yukawa Institute for Theoretical Physics, Kyoto University 研究会「有限温度密度系の物理と格子QCDシミュレーション」 2015 September 5, CCS, University of Tsukuba, Tsukuba

1. Introduction Symmetries of QCD at high temperature Restoration of non-singlet chiral symmetry Theoretical questions 2. Eigenvalue distribution of Dirac operator 1. U(1)_A symmetry at high T ? relation ? U (1) B ⊗ SU ( N f ) L ⊗ SU ( N f ) R ρ ( λ ) λ : eigenvalue of Dirac operator

Eigenvalue density Banks-Casher relation gap 0 (See later.) 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 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-flavor Exact “chiral” symmetry but explicit U(1)_A anomaly form Ginsparg-Wilson relation doublers(chiral) zero modes(chiral) complex pair Eigenvalue spectrum D γ 5 + γ 5 D = aDR γ 5 D λ A λ A λ A n λ A n + ¯ n = aR ¯ A : gauge configuration n � 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

Propagator Measure # of zero modes N_f=2 in this talk. # of doublers zero modes(chiral) doublers(chiral) bulk modes(non-chiral) N R + L N D � φ n ( x ) φ † f m λ n − m + γ 5 φ n ( x ) φ † n ( y ) n ( y ) γ 5 � 1 Ra � � m φ k ( x ) φ † � 2 φ K ( x ) φ † S ( x, y ) = k ( y ) + K ( y ) f m ¯ − λ n − m n k =1 K =1 f m = 1 + Rma 2 � 2 � N f N A D P m ( A ) = e � S Y M ( A ) ( − m ) N f N A m ¯ � Z 2 n + m 2 � λ A n λ A � R + L Ra � λ A n > 0 m = 1 − ( ma ) 2 R 2 Z 2 4 positive definite and even function of m � = 0 for even N f

scalar Chiral symmetry is restored pseudo-scalar chiral rotation at N_f=2 m → 0 � δ a O n 1 ,n 2 ,n 3 ,n 4 � m = 0 lim � � S a = P a = d 4 x S a ( x ) , d 4 x P a ( x ) O n 1 ,n 2 ,n 3 ,n 4 = ( P a ) n 1 ( S a ) n 2 ( P 0 ) n 3 ( S 0 ) n 4 δ a S b = 2 δ ab P 0 , δ a P b = − 2 δ ab S 0 δ a S 0 = 2 P a , δ a P 0 = − 2 S a

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 →∞

forms in future investigations.) (We should remove this assumption and use more general At finite lattice spacing, integrals over all eigenvalues are convergent, since expanded at the origin are “measure zero” in the configuration space. More precisely, configurations whose eigenvalue density can not be Assumption 3 eigenvalues density can be expanded as ∞ λ n 1 � � � � ¯ ρ A � ρ A ( λ ) ≡ lim = at � = 0 ( � < � ) λ A n λ A δ λ − n n n ! V V →∞ n n =0 | λ | ≤ 2 Ra

Analysis (some examples) source of m singularity � S 0 � m m → 0 � ¯ ψψ � m = 0 lim m → 0 lim lim = 0 V V →∞ F (¯ + N A � � V = − 1 F ( λ n ) λ n ) S 0 � R + L f m λ n − m + f m ¯ V V m λ n − m n V → ∞ � Λ R 1 2 m R d λ ρ A ( λ ) g 0 ( λ 2 ) = πρ A I 1 = 0 + O ( m ) λ 2 + m 2 Z m 0 R � ρ A 0 � m = O ( m 2 ) � N R + L � = O ( m 2 ) lim V V →∞ m

from others topological charge repeat these analysis for higher susceptibilities. =0 χ η − δ = 1 m → 0 χ η − δ = 0 lim V � P 2 0 � S 2 a � � I 1 � m 2 V { 2 N R + L − N f Q ( A ) 2 } + 1 1 �� χ η − δ = N f + I 2 Q ( A ) = N A R − N A L Z m m R m � Λ R d λ ρ A ( λ ) m 2 R − λ 2 g 0 ( λ 2 ) g m 1 + m 2 2 1 � � I 2 = g m = , ( λ 2 + m 2 Z 2 2 Λ 2 R ) 2 Z m 0 m R � π m m + 2 � I 1 + I 2 = ρ A + 2 ρ A 1 + O ( m ) , 0 m R Λ R � ρ A 0 � m = O ( m 2 ) N 2 f � Q ( A ) 2 � m m → 0 � ρ A lim = 2 lim 1 � m m 2 V m → 0

Final Results | 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 � ρ A 3 � m � = 0 even for ”free” theory. � 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 →∞

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 . 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 lim ” m π = m η ”

More general Singlet WT identities anomaly(measure) singlet rotation We can show for Breaking of U(1)_A symmetry is invisible for these “bulk quantities”. � J 0 O + δ 0 O � m = O ( m ) O n 1 ,n 2 ,n 3 ,n 4 = ( P a ) n 1 ( S a ) n 2 ( P 0 ) n 3 ( S 0 ) n 4 O = � Q ( A ) 2 1 � V k � J 0 O � m = lim � O ( V 0 ) lim = 0 mV V →∞ V →∞ m where k is the smallest integer which makes the V → ∞ limit finite. S 0 ∼ O ( V ) , P a , S a , P 0 ∼ O ( V 1 / 2 ) 1 V k � δ 0 O � m = 0 m → 0 lim lim V →∞ 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