mation with α ( x ) a local parameter [10]. The derivative ∂ x ∗ µ denotes a backward derivative operator ∂ x ∗ µ V$_{µ}$ ( x ) ≡ V$_{µ}$ ( x ) − V$_{µ}$ ( x − a ˆ µ ) on the lattice. Similarly, for flavor non-singlet transforma- tions, we can derive flavor non-singlet conserved vector and axial-vector currents as where t a denotes the generator of SU ( N$_{f}$ ) normalized as Tr t $^{a}$tb = δ $^{ab}$/ 2. For flavor SU(2), t a = τ $^{a}$/ 2 with τ 2 the Pauli matrix. In the following, we consider the flavor non-singlet currents. In practical implementation for numerical calculations, we approximate the sign function in (2) by a rational function with Zolotarev’s optimized coefficients. In our setup, the sign function is approximated to the level of 10 − (7 − 8) with the number of pole N ≃ 10. (For details, see [7] for instance.) Accordingly, the kernel K$_{µ}$ ( w, z | x ) is constructed as with and h$_{W}$ ( w, z ) = H$_{W}$ ( w, z ; − m$_{0}$ ) /λ$_{min}$ , where λ$_{min}$ is a lower limit of the eigenvalue of | H$_{W}$ | to be approximated by the rational function. (In (6) the site indices of W$_{µ}$ and h$_{W}$ are omitted, but they are multiplied as matrices.) The Zolotarev’s coefficients b$_{l}$ , c$_{l}$ , d$_{0}$ are given in [7]. In this study, we consider the two-point functions of flavor non-singlet conserved and local currents 〈 J a, cv µ ( x ) J b, loc ν (0) 〉 , where J a, loc µ ( x ) is either V a, loc µ ( x ) = Z ¯ q ( x ) t $^{a}$γ$_{µ}$ q ( x ) or A a, loc µ ( x ) = 5 W$_{µ}$ ( z, w | x ) = − 1 2 γ$_{5}$ { (1 − γ$_{µ}$ ) U$_{µ}$ ( z ) δ$_{x}$$_{+ˆ}$ $_{µ,w}$δ$_{x,z}$ − (1 + γ$_{µ}$ ) U † $_{µ}$( z − ˆ µ ) δ$_{x,w}$δ$_{z}$$_{−}$$_{ˆ}$ µ,x } , (7) K$_{µ}$ ( w, z | x ) = m$_{0}$ ( 1 − m 2 m$_{0}$ ) γ$_{5}$ [ d$_{0}$ λ$_{min}$ W$_{µ}$ ( h 2 W + c$_{2}$$_{n}$ ) N ∑ l =1 b$_{l}$ h 2 W + c$_{2}$$_{l}$$_{−}$$_{1}$ + d$_{0}$ λ$_{min}$ h$_{W}$ N ∑ l =1 ( c$_{2}$$_{l}$$_{−}$$_{1}$ − c$_{2}$$_{n}$ ) b$_{l}$ h 2 W + c$_{2}$$_{l}$$_{−}$$_{1}$ ( W$_{µ}$h$_{W}$ + h$_{W}$ W$_{µ}$ ) 1 h 2 W + c$_{2}$$_{l}$$_{−}$$_{1}$ ] , (6) A a, cv µ ( x ) = w,z ¯ q ( w ) t $^{a}$K$_{µ}$ ( w, z | x )[ˆ γ$_{5}$q ]( z ) , (5) V a, cv µ ( x ) = ∑ w,z ¯ q ( w ) t $^{a}$K$_{µ}$ ( w, z | x ) q ( z ) , (4) δS$_{ov}$ = ∑ x,y ¯ q ( x ) [ − α ( x ) D$_{ov}$ ( x, y ) + D$_{ov}$ ( x, y ) α ( y )] q ( y ) = x,y,z ¯ q ( x ) α ( z ) ∂ z ∗ µ K$_{µ}$ ( x, y | z ) q ( y ) (3)