Z ¯ q ( x ) t $^{a}$γ$_{µ}$ γ$_{5}$q ( x ). The renormalization constant Z needed for the local currents to match their continuum counterpart is non-perturbatively determined as Z = 1 . 39360(48) [15]. The lattice calculation of 〈 J a, cv µ ( x ) J b, loc ν ( y ) 〉 is standard except for the complicated form of J a, cv µ ( x ). Namely, we calculate the quark propagator originating from a fixed space-time point y = 0 and construct the two-point function with the conserved current J a, cv µ ( x ) located at arbitrary space-time point x . We then apply the Fourier transform in all four dimensions to obtain the two-point function in the momentum space. Because of the current conservation of J a, cv µ , we may derive the WT identities for the two-point functions where a ˆ Q$_{µ}$ = sin( aQ$_{µ}$ ) are a momentum definition corresponding to the backward derivative operator ∂ x ∗ µ . We use a convention that the two-point function after the Fourier transfor- mation, such as 〈 J a $_{µ}$J b $_{ν}$〉 ( Q ), is a function of aQ$_{µ}$ = 2 πn$_{µ}$/L$_{µ}$ with L$_{µ}$$_{=1}$$_{∼}$$_{4}$ the extent of the lattice in the µ -th direction. The second term in (9) represents the correlation function of the pseudo-scalar density operator P $^{a}$( x ) = ¯ q ( x ) t $^{a}$γ$_{5}$ (1 − D$_{ov}$/m$_{0}$ ) q ( x ) and the local axial- vector current A b, loc ν ( y ). A possible term arising from the axial transformation of J b, loc ν ( y ) ( J = V or A ) vanishes when we take the vacuum expectation value, since the vacuum has axis-interchange symmetry while the index ν remains in J b, loc ( y ). The vector and axial-vector VPFs are now given by Here, Π (0) V ( Q ) vanishes because of the conservation of V a, cv µ , while Π (0) A ( Q ) represents a remnant due to PCAC: (Repeated indices a ’s are not summed.) The transverse part Π (1) J ( Q ) can be extracted as (repeated indices µ ’s are not summed) if one ignores the additional term Δ J $_{µν}$( Q ), which reflects the violation of the current conservation of the local current J a, loc ν . Since the current 6 Π (1) J ( Q ) = 〈 J a, cv µ J a, loc µ 〉 ( Q ) / ( ˆ Q 2 − ˆ Q$_{µ}$ ˆ Q$_{µ}$ ) , (12) Π (0) A ( Q ) = − 2 m$_{q}$ 〈 P $^{a}$Aa, loc ν 〉 ( Q ) / ( ˆ Q 2 ˆ Q$_{ν}$ ) . (11) 〈 J a, cv µ J b, loc ν 〉 ( Q ) = δ ab [ ( δ$_{µν}$ ˆ Q 2 − ˆ Q$_{µ}$ ˆ Q$_{ν}$ )Π (1) J ( Q ) − ˆ Q$_{µ}$ ˆ Q$_{ν}$ Π (0) J ( Q ) + Δ J $_{µν}$( Q ) ] . (10) ∑ µ ˆ Q$_{µ}$ 〈 V a, cv µ V b, loc ν 〉 ( Q ) = 0 , (8) µ ˆ Q$_{µ}$ 〈 A a, cv µ A b, loc ν 〉 ( Q ) − 2 m$_{q}$ 〈 P $^{a}$Ab, loc ν 〉 ( Q ) = 0 , (9)