SU (4)-Ward identities for QCD with restored chiral symmetry Vasily - PowerPoint PPT Presentation

SU (4)-Ward identities for QCD with restored chiral symmetry Vasily Sazonov LPT Orsay, University of Paris Sud 11

Low mode truncation According to the Banks-Casher relation: m → 0 � ψψ � = πρ (0) lim When chiral symmetry is restored (for instance at T > T c ) � ψψ � = 0 = ⇒ ρ (0) = 0 What if we artificially enforce ρ (0) = 0, by removing zero and near-zero modes of the Dirac operator from computations? How one can do it? [C.B. Lang, Mario Schr¨ ock, Phys. Rev. D 84 (2011) 087704 ]: k � 1 S = S Full − | λ i �� λ i | λ i i =1 The correlators of the hadron interpolators � t →∞ � O ( t ) O † (0) � = � 0 | O | n �� n | O † | 0 � e − tE n lim n are expressed as convolutions of S .

Expected degeneracies in the spectrum Each meson is denoted as ( I , J PC ), with I isospin, J total spin, P parity and C charge conjugation. The left column represents the irreducible representations of the parity-chiral group, ( I R , I L ). Below each state its generating current is given. The scheme is adopted from [L. Ya. Glozman, M. Pak, Phys. Rev. D 92, 016001 (2015)]

Evolution of hadron masses under the low-mode truncation [M.Denissenya, L.Ya.G., C.B.Lang, PRD 89(2014)077502]: J = 1 k is the number of removed lowest eigenmodes and σ is the corresponding energy gap (100 gauge field configurations, generated with N F = 2 dynamical overlap fermions on a 16 3 × 32 lattice with the spacing a ∼ 0 . 12 fm)

Observed degeneracies in the spectrum Each meson is denoted as ( I , J PC ), with I isospin, J total spin, P parity and C charge conjugation. The left column represents the irreducible representations of the parity-chiral group, ( I R , I L ). Below each state its generating current is given. The scheme is adopted from [L. Ya. Glozman, M. Pak, Phys. Rev. D 92, 016001 (2015)]

Development of the topic ◮ J = 2 mesons, [M. Denissenya, L.Ya.Glozman, M.Pak, PRD 91(2015)114512] ◮ J = 1 / 2 baryons, [M. Denissenya, L.Ya.G, M.Pak, PRD 92 (2015) 074508] ◮ SU (4) symmetry, [L.Ya.Glozman, EPJA 51(2015)27], [L.Ya.Glozman, M. Pak, PRD 92(2015)016001] ◮ Prediction of the SU (4) restoration at T > T c , relation to confinement, [L.Ya.Glozman, arXiv 1512.06703] ◮ Approximate SU (4) degeneracy, observed at T > T c , [C. Rohrhofer, Y. Aoki, G. Cossu, H. Fukaya, L. Ya. Glozman, S. Hashimoto, C. B. Lang, S. Prelovsek, Phys. Rev. D 96, 094501 (2017)]

SU (4)-symmetry transformations The quark QCD Lagrangian with N F = 2 flavors in the Euclidean space is L = Ψ( 1 F ⊗ ( γ µ D µ + m ))Ψ , µ = 1 .. 4 , The transformations of the SU (4) group are completely determined by generators T l , satisfying the su (4) algebra commutation relations. T l are given by 15 matrices { ( τ a ⊗ 1 D ) , ( 1 F ⊗ Σ i ) , ( τ a ⊗ Σ i ) } , a , i = 1 , 2 , 3 . τ a are the generators of SU (2) L × SU (2) R chiral symmetry.

The matrices Σ i are Σ i = { γ 4 , i γ 5 γ 4 , − γ 5 } and satisfy the su (2) algebra commutation relations. The generators Σ i define chiral spin SU (2) CS group with transformations acting only in the Dirac space Ψ → Ψ ′ = e i ǫ ( x ) · ( 1 F ⊗ Σ) Ψ . The transformations of the full SU (4) group   u L   u R Ψ → Ψ ′ = e i ǫ ( x ) · T Ψ ,   Ψ =   d L d R mix both quark flavors and left-/right-handed components.

In the Euclidean space spinors Ψ and Ψ are completely independent. The action of SU (4) on the spinor Ψ is defined to be equivalent of the SU (4) action on the � Ψ = − i Ψ † γ 4 . Generators δ Ψ δ Ψ 1 F ⊗ γ 4 i ǫ ( x ) γ 4 Ψ − i ǫ ( x )Ψ γ 4 1 F ⊗ i γ 5 γ 4 − ǫ ( x ) γ 5 γ 4 Ψ − ǫ ( x )Ψ γ 5 γ 4 1 F ⊗ ( − γ 5 ) − i ǫ ( x ) γ 5 Ψ − i ǫ ( x )Ψ γ 5 τ a ⊗ 1 D i ǫ ( x ) τ a Ψ − i ǫ ( x )Ψ τ a τ a ⊗ γ 4 i ǫ ( x ) τ a γ 4 Ψ − i ǫ ( x )Ψ γ 4 τ a τ a ⊗ i γ 5 γ 4 − ǫ ( x ) τ a γ 5 γ 4 Ψ − ǫ ( x )Ψ γ 5 γ 4 τ a τ a ⊗ ( − γ 5 ) − i ǫ ( x ) τ a γ 5 Ψ − i ǫ ( x )Ψ γ 5 τ a Table: Infinitesimal variations of spinors Ψ and Ψ under the SU (4) transformations, the tensorial product of flavor and Dirac spaces is omitted to shorter notations.

Lagrangian transformations The quark Lagrangian is not invariant under the SU (4) transformations. Generators δ L = δ Ψ D Ψ + Ψ D δ Ψ 1 F ⊗ γ 4 i ( ∂ µ ǫ ( x ))Ψ γ µ γ 4 Ψ − 2 i ǫ ( x )Ψ γ 4 γ i [ ∂ i − igA i ]Ψ 1 F ⊗ i γ 5 γ 4 − ( ∂ µ ǫ ( x ))Ψ γ µ γ 5 γ 4 Ψ − 2 m ǫ ( x )Ψ γ 5 γ 4 Ψ − 2 ǫ ( x )Ψ γ i γ 5 γ 4 [ ∂ i − igA i ]Ψ 1 F ⊗ ( − γ 5 ) − i ( ∂ µ ǫ ( x ))Ψ γ µ γ 5 Ψ − 2 i ǫ ( x ) m Ψ γ 5 Ψ τ a ⊗ 1 D i ( ∂ µ ǫ ( x ))Ψ γ µ τ a Ψ τ a ⊗ γ 4 i ( ∂ µ ǫ ( x ))Ψ γ µ γ 4 τ a Ψ − 2 i ǫ ( x )Ψ γ 4 γ i τ a [ ∂ i − igA i ]Ψ τ a ⊗ i γ 5 γ 4 − ( ∂ µ ǫ ( x ))Ψ γ µ γ 5 γ 4 τ a Ψ − 2 m ǫ ( x )Ψ γ 5 γ 4 τ a Ψ − 2 ǫ ( x )Ψ γ i γ 5 γ 4 τ a [ ∂ i − igA i ]Ψ τ a ⊗ ( − γ 5 ) ( ∂ µ ǫ ( x ))Ψ γ µ γ 5 τ a Ψ − 2 i ǫ ( x ) m Ψ γ 5 τ a Ψ Table: Variations of the quark Lagrangian under the SU (4) transformations. Here D = ( 1 F ⊗ ( γ µ D µ + m )) and in the right part of the table the tensorial product of flavor and Dirac structures is omitted to shorter notations.

Anomalous violation of the SU (4) symmetry Consider QCD in a finite volume with appropriate boundary conditions. Then, the anomaly term of the action is given by the sum � � � � φ † d 4 x ǫ ( x ) − i Tr F [ g F ] k ( x ) Σ + Σ φ k ( x ) , k where ′ = Ψ e i ǫ ( x ) · T , Ψ → Ψ T = ( g F ⊗ Σ) , g F = { 1 F , τ a } and an appropriate regularization is assumed (see later). When g F = 1 F , Tr F [ g F ] = N F , alternatively, one has Tr F [ g F = τ a ] = 0

Identities We treat terms containing ∂ µ ǫ ( x ) employing periodic boundary conditions for ǫ ( x ) and periodic/anti-periodic boundary conditions for quark fields. The bilinear forms containing the mixing of quark flavors performed by τ a vanish after the integration over the quark fields and one ends up with the three following identities ∂ µ � Ψ γ µ γ 4 Ψ � A + 2 � Ψ γ 4 γ i [ ∂ i − igA i ]Ψ � A = 0 , ∂ µ � Ψ γ µ γ 5 γ 4 Ψ � A − 2 m � Ψ γ 5 γ 4 Ψ � A − 2 � Ψ γ i γ 5 γ 4 [ ∂ i − igA i ]Ψ � A � φ † k ( x ) γ 5 γ 4 e − ( λ k / M ) 2 φ k ( x ) = 0 , +4 i lim M →∞ k 1 8 π 2 Tr ∗ F µν F µν = 0 ∂ µ � Ψ γ µ γ 5 Ψ � A + 2 im � Ψ γ 5 Ψ � A +

Spectral properties of SU (4)-Ward identities Classical part of identities Consider the terms, related to the non-invariance of the classical action utilizing the spectral representation � 1 φ † � Ψ( x ) O Ψ( x ) � A = n ( x ) O φ n ( x ) m − i λ n n The summation over n is ill-defined and has to be regularized � 1 n ( x ) Oe − ( λ n / M ) 2 φ n ( x ) φ † � Ψ( x ) O Ψ( x ) � A = lim m − i λ n M →∞ n Then, we split the sum over n as � � � + + λ n > 0 λ n < 0 λ n =0

The anti-commutation of γ 5 with matrices γ µ ensures that i γ µ D µ ( γ 5 φ n ) = − λ n ( γ 5 φ n ) ≡ λ − n φ − n and φ † − n = φ † φ − n = γ 5 φ n ⇒ n γ 5 Now we join the sums over the positive and negative eigenvalues � � n ( x ) O φ n ( x ) 1 φ † � Ψ( x ) O Ψ( x ) � A = lim m + M →∞ λ n =0 � � 1 n ( x ) Oe − ( λ n / M ) 2 φ n ( x ) φ † m − i λ n λ n > 0 �� 1 n ( x ) γ 5 Oe − ( λ n / M ) 2 γ 5 φ n ( x ) + φ † m + i λ n

Operators from Ward identities O = { γ 4 γ i [ ∂ i − igA i ] , γ i γ 5 γ 4 [ ∂ i − igA i ] } commute with γ 5 [ O , γ 5 ] = 0 Then, � � n ( x ) O φ n ( x ) 1 φ † � Ψ( x ) O Ψ( x ) � A = lim m + M →∞ λ n =0 � � 2 m n ( x ) Oe − ( λ n / M ) 2 φ n ( x ) φ † λ 2 n + m 2 λ n > 0 Therefore, in the zero mass limit the dominant contributions to the expectation values � Ψ( x ) O Ψ( x ) � A come from zero modes of the Dirac operator.

Anomalous part ◮ Axial anomaly = n + − n − , consequently vanishes in a presence of a spectral gap ◮ ( 1 F ⊗ ( i γ 5 γ 4 )) - related anomaly � � φ † k ( x ) γ 5 γ 4 e − ( λ k / M ) 2 φ k ( x ) = φ † k ( x ) γ 5 γ 4 e − ( λ k / M ) 2 φ k ( x ) + k k =0 � � � φ † k ( x ) γ 5 γ 4 e − ( λ k / M ) 2 φ k ( x ) + φ † − k ( x ) γ 5 γ 4 e − ( λ k / M ) 2 φ − k ( x ) = k > 0 � � k ( x ) γ 5 γ 4 e − ( λ k / M ) 2 φ k ( x ) + k ( x ) { γ 5 , γ 4 } e − ( λ k / M ) 2 φ k ( x ) = φ † φ † k =0 k > 0 � k ( x ) γ 5 γ 4 e − ( λ k / M ) 2 φ k ( x ) φ † k =0 hence, it is also inconsistent with the spectral gap

Banks-Casher relation, T > T C , SU (4) m → 0 � ψψ � = πρ (0) . lim Consequently, above T c , when � ψψ � = 0 ◮ ρ (0) = 0 ◮ ρ ( λ ) = | λ | α , α > 2, [S. Aoki, H. Fukaya, Y. Taniguchi, Phys. Rev. D 86, 114512 (2012)] ◮ or ρ ( λ ) = 0, for λ < λ c ◮ The latter means that the pion, sigma, delta, and eta(-prime) meson correlators are all identical, realizing the U (1) A symmetry, [T. D. Cohen, Phys. Rev. D 54, 1867 (1996)].