QCDSparameter from inhomogeneous Bethe-Salpeter equation

We calculate the low-energy parameters S in QCD, which is also known as ${\mathit{L}}_{10}$, and the pion decay constant ${\mathit{f}}_{\mathrm{\pi }}$ using the inhomogeneous Bethe-Salpeter equation in the improved ladder approximation. To extract these quantities we calculate the ‘‘V-A’’ two-point function ${\mathrm{\Pi }}_{\mathit{V}\mathit{V}}$(${\mathit{q}}^2$)-${\mathrm{\Pi }}_{\mathit{A}\mathit{A}(}$ ${\mathit{q}}^2$) in the spacelike region. We obtain S=0.43–0.48, which is about 30% larger than the experimental value. The calculated ${\mathit{f}}_{\mathrm{\pi }}$ is quite consistent with the result by solving the homogeneous Bethe-Salpeter equation for the pion. We also evaluate the S parameter in SU(3) gauge theory with ${\mathit{N}}_{\mathit{D}}$ doublets of fermions in connection with the walking technicolor model, and find that the value of S/${\mathit{N}}_{\mathit{D}}$ hardly depends on ${\mathit{N}}_{\mathit{D}}$.