Hadronic light-by-light corrections to the muong−2:The pion-pole contribution

Abstract

The correction to the muon anomalous magnetic moment from the pion-pole contribution to the hadronic light-by-light scattering is considered using a description of the ${\pi }^0-{\gamma }^{*}-{\gamma }^{*}$ transition form factor based on the large-$N_C$ and short-distance properties of QCD. The resulting two-loop integrals are treated by first performing the angular integration analytically, using the method of Gegenbauer polynomials, followed by a numerical evaluation of the remaining two-dimensional integration over the moduli of the Euclidean loop momenta. The value obtained, $a_{\mu }^{\mathrm{LbyL};{\pi }^0}=+5.8\mathrm{}\left(1.0\right)\mathrm{}\times {10}^{-10},$ disagrees with other recent calculations. In the case of the vector meson dominance form factor, the result obtained by following the same procedure reads $a_{\mu }^{\mathrm{LbyL};{\pi }^0}{|}_{\mathrm{VMD}}=+5.6\mathrm{}\times {10}^{-10},$ and differs only by its overall sign from the value obtained by previous authors. The inclusion of the $\eta$ and ${\eta }^{\prime }$ poles gives a total value $a_{\mu }^{\mathrm{L}\mathrm{b}\mathrm{y}\mathrm{L};\mathrm{P}\mathrm{S}}=+8.3\mathrm{}\left(1.2\right)\mathrm{}\times {10}^{-10}$ for the three pseudoscalar states. This result substantially reduces the difference between the experimental value of $a_{\mu }$ and its theoretical counterpart in the standard model.