0 0.002 0.004 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 (aQ) 2 0 0.002 0.004 m$_{s}$=0.08 m$_{s}$=0.10 V $^{diag}$- V offd FIG. 6: Difference between Π diag V ( Q ) and Π offd V ( Q ) at all valence quark masses m$_{q}$ = 0 . 015 (circles), 0 . 025 (squares), 0 . 035 (diamonds), and 0 . 050 (triangles). Top panel is the result at m$_{s}$ = 0 . 08 and the bottom is in m$_{s}$ = 0 . 10. suppressed by an additional factor of α$_{s}$ and thus well below ± 0 . 0001. The discretization effect may also come from the non-conserved local current J loc ν in (13), which is represented by the term Δ J $_{µν}$( Q ) in (10). We estimate its leading contribution Δ J $_{µν}$( Q ) = ( δ$_{µν}$ ˆ Q $^{2}$− ˆ Q$_{µ}$ ˆ Q$_{ν}$ ) Q 2 $_{ν}$F J $_{11}$( Q ) in the expansion (13) in terms of small aQ$_{µ}$ , by solving linear equations (10) for different sets of µ and ν . We find that the maximum magnitude of ( δ$_{µν}$ ˆ Q 2 − ˆ Q$_{µ}$ ˆ Q$_{ν}$ ) Q 2 $_{ν}$F V + A 11 ( Q ) is less than 1% of Π$_{V}$$_{+}$$_{A}$( Q ) in the fit range 0 . 463 ≤ ( aQ ) 2 ≤ 0 . 994. The contribution of Δ J $_{µν}$( Q ) may also be estimated by looking at a difference between VPFs obtained with µ = ν (diag) and with µ = ν (offd) components, i.e. respectively. Figure 6 shows Π diag V ( Q ) − Π offd V ( Q ) as a function of ( aQ ) $^{2}$. The maximum magnitude of the difference in the range 0 . 463 ≤ ( aQ ) 2 ≤ 0 . 994 is 0.003, which is the same Π diag J ( Q ) = 〈 J cv µ J loc µ 〉 ( Q ) / ( ˆ Q 2 − ˆ Q$_{µ}$ ˆ Q$_{µ}$ ) , (15) Π offd J ( Q ) = 〈 J cv µ J loc ν 〉 ( Q ) / ( − ˆ Q$_{µ}$ ˆ Q$_{ν}$ ) , (16) 15 C. Non-conserved current