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 ${\ensuremath{\pi}}^{0}\ensuremath{-}{\ensuremath{\gamma}}^{*}\ensuremath{-}{\ensuremath{\gamma}}^{*}$ 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, ${a}_{\ensuremath{\mu}}^{\mathrm{LbyL};{\ensuremath{\pi}}^{0}}=+5.8(1.0)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}10},$ disagrees with other recent calculations. In the case of the vector meson dominance form factor, the result obtained by following the same procedure reads ${a}_{\ensuremath{\mu}}^{\mathrm{LbyL};{\ensuremath{\pi}}^{0}}{|}_{\mathrm{VMD}}=+5.6\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}10},$ and differs only by its overall sign from the value obtained by previous authors. The inclusion of the $\ensuremath{\eta}$ and ${\ensuremath{\eta}}^{\ensuremath{'}}$ poles gives a total value ${a}_{\ensuremath{\mu}}^{\mathrm{L}\mathrm{b}\mathrm{y}\mathrm{L};\mathrm{P}\mathrm{S}}=+8.3(1.2)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}10}$ for the three pseudoscalar states. This result substantially reduces the difference between the experimental value of ${a}_{\ensuremath{\mu}}$ and its theoretical counterpart in the standard model.