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mation 

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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 

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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 

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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]. 

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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 ) = 

