Fourth-order quantum-chromodynamic contributions to thee+−e−annihilation cross section

We compute analytically the logarithmic corrections to the photon propagator to order $g^4$ in massless quantum chromodynamics. With ${\alpha }_c=\frac{g^2}{4{\pi }^2}$ defined by momentum-space subtraction, we find that $\frac{\sigma (e^{+}{e}^{-}\to \mathrm{hadrons})}{\sigma (e^{+}{e}^{-}\to {\mu }^{+}{\mu }^{-})}={\Sigma }_q{{\epsilon }_q}^2(1+{\alpha }_c+K{{\alpha }_c}^2+\cdots )$, where $K=\frac{463}{48}+\left(\frac{85}{36}\sqrt{3}\right){\mathrm{Cl}}_2\left(\frac{\pi }{3}\right)-11\mathrm{}\zeta \mathrm{}\left(3\right)+\left[\right(\frac{2}{3}\left)\zeta \mathrm{}\right(3)-\frac{23}{36}]n_f=-2.193+0.162\mathrm{}{n}_f$ for $n_f$ flavors of quark. The computation is done in momentum space using a novel generalization to $4-\epsilon$ dimensions of the usual Chebyshev-polynomial expansion of Feynman propagators.