conservation is recovered in the continuum limit, this term can be expanded in terms of small aQ$_{µ}$ as where F$_{mn}$ denotes the scalar function depends on the index m,n and momentum Q . It satisfies the condition ∑ µ ˆ Q$_{µ}$ Δ J $_{µν}$( Q ) = 0 coming from the WT identity for J a, cv µ . In this work, we confirmed that this term is numerically negligible in the range ( aQ ) 2 < 1, and ignore its contribution as we discuss later. Defining Π$_{J}$( Q ) = Π (0) J ( Q ) + Π (1) J ( Q ), the Operator Product Expansion (OPE) of VPF, Π$_{V}$$_{+}$$_{A}$( Q ) = Π$_{V}$ ( Q ) + Π$_{A}$( Q ), is given by for large Q $^{2}$. The perturbative expansion of the coefficients C ( V + A ) X ( X = 0, ¯ qq and GG ) is known up to two- to four-loop order in the continuum renormalization scheme, i.e. the MS scheme, depending on the terms. The first term c in (14) is a scheme-dependent constant, divergent in the limit of infinite ultraviolet cutoff. For the Adler function D ( Q $^{2}$) = − Q $^{2}$d Π( Q $^{2}$) /dQ $^{2}$, which is a physical observable, the first term disappears and the contributions from other terms remain finite. The coefficients in the second and third terms are perturbatively calculated to four-loop order in the MS scheme [16–18]; the expression explicitly contains α (3) s ( Q ) defined in the MS scheme. (The superscript (3) stands for the number of flavors.) The third term contains the running mass ¯ m ( Q ) whose anomalous dimension is known to three-loop order [19, 20]. The fourth and fifth terms represent higher order effects in OPE containing dimension-four operators. Their Wilson coefficients are calculated at three-loop order [20]. 7 Π$_{V}$$_{+}$$_{A}$ |$_{OPE}$ ( Q $^{2}$,α$_{s}$ ) = c + C$_{0}$ ( Q $^{2}$,µ$^{2}$,α$_{s}$ ) + C V + A m ( Q $^{2}$,µ$^{2}$,α$_{s}$ ) ¯ m $^{2}$( Q ) Q 2 + ∑ q = u,d,s C V + A ¯ qq ( Q $^{2}$,α$_{s}$ ) 〈 m$_{q}$ ¯ qq 〉 Q 4 + C$_{GG}$ ( Q $^{2}$,α$_{s}$ ) 〈 ( α$_{s}$/π ) GG 〉 Q 4 + O ( Q − $^{6}$) (14) Δ J $_{µν}$( Q ) = ∑ m,n =1 ( δ$_{µν}$ ∑ ρ ˆ Q 2 m ρ − ˆ Q 2( m − 1) µ ˆ Q$_{µ}$ ˆ Q$_{ν}$ ) Q 2 n ν F$_{mn}$ ( ˆ Q ) . (13) III. FIT WITH THE PERTURBATIVE FORMULA