Lattice study of the vacuum polarization function and determination of the strong coupling constant

We calculate the vacuum polarization functions on the lattice using the overlap fermion formulation. By matching the lattice data at large momentum scales with the perturbative expansion supplemented by the operator product expansion (OPE), we extract the strong coupling constant ${\alpha }_s(\mu )$ in two-flavor QCD as ${\Lambda }_{\overline{MS}}^{(2)}=0.234(9)(\genfrac{}{}{0ex}{}{+16}{-0})\mathrm{GeV}$, where the errors are statistical and systematic, respectively. In addition, from the analysis of the difference between the vector and axial-vector channels, we obtain some of the four-quark condensates.

Figure 1

Momentum dependence of $B_1^J(Q)$, $B_2^J(Q)/a^2$, and $C_{11}^J(Q)/a^2$ at $m_q=0.015$. Circles (crosses) show the vector (axial-vector) channel. The solid curves represent a polynomial fit and the dashed curves show the one-loop results.

Figure 2

Momentum dependence of $B_1^J(Q)$, $C_{11}^J(Q)/a^2$, and $B_2^J(Q)/a^2$ calculated in perturbation theory.

Figure 3

${\Pi }_V^{(0+1)}(Q)$ from off-diagonal $\mu \ne \nu$ and diagonal $\mu =\nu$ correlators with (lower panel) and without (upper panel) the subtraction of $B_{1,2}^J(Q)$ and $C_{11}^J(Q)$.

Figure 4

Fit range dependence of ${\Lambda }_{\overline{MS}}^{(2)}$, $⟨({\alpha }_s/\pi )GG⟩$, and the constant term $c$. The maximum momentum squared $(aQ{)}_{\max }^2$ is 1.324.

Figure 5

${\Pi }_{V+A}^{(0+1)}(Q)$ as a function of $(aQ{)}^2$. The lattice data at different quark masses are shown by open symbols. Fit curves for each quark mass and in the chiral limit are drawn. The full result in the chiral limit (dashed-dots curves are at the finite masses, and solid curve is in the chiral limit), as well as that without $⟨{\alpha }_s{G}^2⟩/Q^4$ term (dashed curve), are shown.

Figure 6

${\Lambda }_{\overline{MS}}^{(2)}$ from the data at each quark mass.

Figure 7

One-loop calculation of ${\Pi }_V(Q)$ (upper panel) at representative values of $(aQ{)}^2$ (circles) and a fit with the log-plus-polynomial form. The dashed curve shows the purely logarithmic contribution. The term that represents the lattice artifact $\Delta {\Pi }_V(Q)$ is shown in the lower panel.

Figure 8

Fit range dependence of $a_6(\mu )$, $b_6(\mu )$, and $a_8$. The horizontal axis denotes the minimum momentum squared $(aQ{)}_{\min }^2$.

Figure 9

$Q^6{\Pi }_{V-A}^{(0+1)}(Q)$ as a function of $(aQ{)}^2$. The lattice data at different quark masses are shown by open symbols. Fit curves for each quark mass and in the chiral limit are drawn.