content_image/359849.jpg content_image/359850.jpg content_image/359848.jpg content_image/359847.jpg content_image/359846.jpg
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 

<img src='content_image/359849.jpg'>

<img src='content_image/359850.jpg'>

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 ν ( y ). 

The vector and axial-vector VPFs are now given by 

<img src='content_image/359848.jpg'>

Here, Π V ( Q ) vanishes because of the conservation of V µ , while Π A ( Q ) represents a remnant due to PCAC: 

<img src='content_image/359847.jpg'>

(Repeated indices a ’s are not summed.) The transverse part Π (1) J ( Q ) can be extracted as 

<img src='content_image/359846.jpg'>

(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 

