The rest of the paper is organized as follows. In Section II, we describe the details of the lattice calculation of VPF, including the definition of the overlap fermion used in this work and a construction of the conserved current for the overlap fermion. Section III discusses the fit of VPF to the perturbative formula. The possible systematic errors are discussed in Section IV, followed by our final results. Our conclusions are given in Section V. In the continuum theory, transverse (Π (1) J ( Q )) and longitudinal (Π (0) J ( Q )) parts of VPF are defined through two-point functions 〈 J a $_{µ}$( x ) J b $_{ν}$(0) 〉 of either vector ( J$_{µ}$ = V$_{µ}$ ) or axial-vector ( J$_{µ}$ = A$_{µ}$ ) currents with a , b the flavor indices. Namely, after a Fourier transformation to the momentum space, the two-point functions are parametrized as where the momentum Q$_{µ}$ is space-like as we work on an Euclidean space-time lattice. Be- cause of the WT identities, the longitudinal part of the vector channel vanishes, Π (0) V ( Q ) = 0, while the axial-vector channel is proportional to the quark mass. for a quark mass m . Here, m$_{0}$ is a parameter to define the overlap kernel H$_{W}$ ( x, y ; − m$_{0}$ ) = γ$_{5}$ ( D$_{W}$ ( x, y ) − m$_{0}$ ) with D$_{W}$ ( x, y ) the conventional Wilson-Dirac operator. In this study we take m$_{0}$ = 1 . 6. (Here and in the following we set a = 1, unless otherwise stated.) The overlap action has an exact symmetry under a chiral rotation defined with the modified chirality operator ˆ γ$_{5}$ ( x, y ) ≡ γ$_{5}$ ( δ$_{x,y}$ − D$_{ov}$ ( x, y ) /m$_{0}$ ), so that the continuum-like axial WT identities are hold on the lattice at finite lattice spacings. The conserved vector current for this action has a complicated form, which can be written in a general form V 0 , cv µ ( x ) = ∑ w,z ¯ q ( w ) K$_{µ}$ ( w, z | x ) q ( z ) with a non-local kernel K$_{µ}$ ( w, z | x ). K$_{µ}$ ( w, z | x ) is determined such that it forms a Noether current under a local vector transfor- D$_{ov}$ ( x, y ) = ( m$_{0}$ + m 2 ) + ( m$_{0}$ − m 2 ) γ$_{5}$ sgn[ H$_{W}$ ( x, y ; − m$_{0}$ )] (2) 〈 J a $_{µ}$J b $_{ν}$〉 ( Q ) = δ ab [ ( δ$_{µν}$Q 2 − Q$_{µ}$Q$_{ν}$ )Π (1) J ( Q ) − Q$_{µ}$Q$_{ν}$ Π (0) J ( Q ) ] , (1) On the lattice, we employ the overlap fermion formulation [11, 12], whose action S$_{ov}$ = ∑ x,y ¯ q ( x ) D$_{ov}$ ( x, y ) q ( y ) is specified by the massive overlap-Dirac operator 4 II. LATTICE CALCULATION OF VACUUM POLARIZATION FUNCTIONS