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Chiral Symmetry Restoration, Eigenvalue Density of Dirac Operator and axial U(1) anomaly at Finite Temperature Sinya AOKI University of Tsukuba with H. Fukaya and Y. Taniguchi for JLQCD Collaboration GGI workshop, New Frontiers in Lattice


  1. Chiral Symmetry Restoration, Eigenvalue Density of Dirac Operator and axial U(1) anomaly at Finite Temperature Sinya AOKI University of Tsukuba with H. Fukaya and Y. Taniguchi for JLQCD Collaboration GGI workshop, “New Frontiers in Lattice Gauge Theory”, GGI, Firenze, Italy, September 5, 2012

  2. 1. Introduction low T high T Chiral symmetry of QCD restoration of chiral symmetry phase transition Some questions 1. Eigenvalue distribution of Dirac operator 2. Recovery of U(1)_A symmetry at high T ? related ? U (1) B ⊗ S ( N f ) L ⊗ SU ( N f ) R U (1) B ⊗ S ( N f ) V

  3. 1 Previous studies on 1 T=177,192MeV T=172MeV T=209MeV � Lin (HotQCD11), DW ρ ( λ ) = lim δ ( λ − λ n ) V V →∞ n 0.08 0.08 m l + m res m l + m res 150 MeV 160 MeV 0.07 0.07 Cossu et al. (JLQCD11), Overlap m s + m res m s + m res Δ < Δ < ψψ > / π ψψ > / π 0.06 0.06 < ψψ > / π < ψψ > / π 0.05 0.05 ��� ρ ( Λ ) ρ ( Λ ) �������������������������� � ������� � ��� 0.04 0.04 0.03 0.03 ��� ������ ��� 0.02 0.02 ������������� ��� 0.01 0.01 0 0 0 0.05 0.1 0.15 0.2 0 0.05 0.1 0.15 0.2 �������������� ��� Λ Λ 0.08 0.08 m l + m res m l + m res � 170 MeV 180 MeV 0.07 0.07 ��� m s + m res m s + m res Δ < ψψ > / π Δ < ψψ > / π 0.06 0.06 < ψψ > / π < ψψ > / π ��� ������ ��� 0.05 0.05 ρ ( Λ ) ρ ( Λ ) 0.04 0.04 ������������� ��� 0.03 0.03 �������������� 0.02 0.02 ��������������� ��� �������������� 0.01 0.01 0 0 � 0 0.05 0.1 0.15 0.2 0 0.05 0.1 0.15 0.2 ��� Λ Λ 0.08 0.08 ��� ������ ��� m l + m res m l + m res 190 MeV 200 MeV 0.07 0.07 m s + m res m s + m res 0.002 0.002 Δ < ψψ > / π Δ < ψψ > / π ������������� 0.06 0.06 ��� < ψψ > / π < ψψ > / π 0.05 0.05 �������������� ρ ( Λ ) ρ ( Λ ) ��������������� ��� 0.04 0.04 �������������� 0.03 0.03 0 0 0 0.015 0 0.015 � 0.02 0.02 � ��� ��� ��� ��� ��� ��� ���������� 0.01 0.01 0 0 0 0.05 0.1 0.15 0.2 0 0.05 0.1 0.15 0.2 Λ Λ 0.018 1.0e-01 m l /m s = 1/20 m l /m s = 1/20 T = 173.0 MeV T = 173.0 MeV T = 177.7 MeV T = 177.7 MeV 0.016 T = 188.7 MeV T = 188.7 MeV Ohno et al. (11), HISQ 0.014 T = 210.6 MeV T = 210.6 MeV T = 239.7 MeV T = 239.7 MeV 1.0e-02 0.012 T = 275.9 MeV T = 275.9 MeV T = 331.6 MeV 0.01 ρ ( λ ) ρ ( λ ) 0.008 1.0e-03 0.006 0.004 Is small λ suppressed ? 0.002 1.0e-04 0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0 0.002 0.004 0.006 0.008 λ a λ a

  4. � Previous studies on 2 No ! Chiral symmetry is restored. No ! No ! Lattice results zero mode contributions are important. U(1) A is NOT. Yes ! d 4 x � σ ( x ) σ (0) � δ ( x ) δ (0) � χ U (1) A = χ U (1) A /V = 0 , ( m → 0) Cohen(96), Theory Lee-Hatsuda(96), Theory χ U (1) A = O ( m 2 ) + ∆ ∆ = O (1) at N f = 2: contributions from Q = ± 1 Bernard, et al. (96), KS Chandrasekharan et al. , (98), KS 5 0.14 χ P 0.12 4 omega 0.1 chi-bar-chi chi_P 3 − < χχ > − < χχ > 0.08 ω 0.06 2 0.04 1 0.02 0 0 0 0.005 0.01 0.015 0.02 0.025 0.03 m

  5. Yes ?! Recent lattice results No ?! meson correlators 200 Hegde (HotQCD11), DW 160 χ disc /T 2 ���� �� ������ �� ��������������������������� χ 5,disc /T 2 ��������������������������� 120 ( χ π - χ δ )/T 2 ���� �� ����������� ���� �� ����������� ���������� χ U (1) A = 0 or not ? ��������� ���� �� 80 ������ �� ���� �� 40 ���� �� �� �� T[MeV] 0 140 150 160 170 180 190 200 ������ �� ���������������� � ����������� ��������������� Cossu et al. (JLQCD11), Overlap ��������� �� �� ����� �� � � � � � �� �� �� �� � � � � �� �� �� �� �������� �������� ������ �� ���� �� ��������������������������� ���������������������������� ���� �� ������ �� ������ �� �� �� �� �� ���� �� ���� �� ���������������� ���������������� ����������� ����������� ��������������� ��������������� ��������� �������� � � � � � � � �� �� �� �� � � � � � �� �� �� �� �������� ��������

  6. Our work give constraints on eigenvalue densities of 2-flavor overlap fermions, if chiral symmetry in QCD is restored at finite temperature. discuss a behavior of singlet susceptibility using the constraints. 1. Introduction 2. Overlap fermions 3. Constraints on eigenvalue densities 4. Discussions: singlet susceptibility Content

  7. 2. Overlap fermions Action zero modes(chiral) Eigenvalue spectrum doublers(chiral) Ginsparg-Wilson relation F ( D ) = 1 − Ra S = ¯ ψ [ D − mF ( D )] ψ , 2 D D γ 5 + γ 5 D = aDR γ 5 D λ A λ A λ A n λ A n + ¯ n = aR ¯ 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

  8. 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

  9. Ward-Takahashi identities under “chiral” rotation scalar explicit from WT identities If the chiral symmetry is restored, Integrated operators pseudo-scalar chiral rotation at N_f=2 θ a ( x ) δ a i θ a ( x ) T a γ 5 (1 − RaD ) x ψ ( x ) = x ¯ i ¯ θ a ( x ) δ a ψ ( x ) θ a ( x ) T a γ 5 , ψ ( x ) = ¯ S a ( x ) ψ ( x ) T a F ( D ) ψ ( x ) , � � = S a = P a = d 4 x S a ( x ) , d 4 x P a ( x ) ¯ P a ( x ) ψ ( x ) T a i γ 5 F ( D ) ψ ( x ) , = δ a S b = 2 δ ab P 0 , δ a P b = − 2 δ ab S 0 m → 0 � δ a O n 1 ,n 2 ,n 3 ,n 4 � m = 0 lim � 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 N = n i , n 1 + n 2 = odd , n 1 + n 3 = odd i δ a 2 O n 1 ,n 2 ,n 3 ,n 4 = − n 1 O n 1 − 1 ,n 2 ,n 3 ,n 4 +1 + n 2 O n 1 ,n 2 − 1 ,n 3 +1 ,n 4 − n 3 O n 1 ,n 2 +1 ,n 3 − 1 ,n 4 + n 4 O n 1 +1 ,n 2 ,n 3 ,n 4 − 1

  10. 3. Constraints on eigenvalue densities Note that this does not hold if the chiral symmetry is spontaneously broken. non-singlet chiral symmetry is restored: Ex. Assumption 1 Assumption 2 O m → 0 lim lim V →∞ � δ a O � m = 0 (for a � = 0) , � O ( A ) � m = 1 � � D A P m ( A ) O ( A ) , Z = D A P m ( A ) . Z P m ( A ): even in m if O ( A ) is m -independent � 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 →∞

  11. Assumption 3 consequence finite if O ( A ) is m -independent and positive, and satisfies 1 lim m 2 k � O ( A ) � m = 0 m → 0 � D A ˆ � O ( A ) � m = m 2( k +1) P ( m 2 , A ) O ( A ) ˆ P (0 , A ) � = 0 for ∃ A for ∀ l integer � P ( m 2 , A ) O ( A ) l = O ( m 2( k +1) ) D A ˆ � O ( A ) l � m = m 2( k +1) since O ( A ) and O ( A ) l are both positive and share the same support.

  12. are “measure zero” in the configuration space. More precisely, configurations which can not be expanded at the origin eigenvalues density can be expanded as Assumption 4 ∞ 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

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