Hadronic light-by-light scattering in the muong−2: A new short-distance constraint on pion exchange

Abstract

Recently it was pointed out that for the evaluation of the numerically dominant pion-exchange contribution to the hadronic light-by-light scattering correction in the muon $g\ensuremath{-}2$, a fully off-shell pion-photon-photon form factor should be used. Following this proposal, we first derive a new short-distance constraint on the off-shell form factor which enters at the external vertex for the muon $g\ensuremath{-}2$ and show that it is related to the quark condensate magnetic susceptibility in QCD. We then evaluate the pion-exchange contribution in the framework of large-${N}_{C}$ QCD using an off-shell form factor which fulfills all short-distance constraints. With a value for the magnetic susceptibility as estimated in the same large-${N}_{C}$ framework, we obtain the result ${a}_{\ensuremath{\mu}}^{\mathrm{LbyL};{\ensuremath{\pi}}^{0}}=(72\ifmmode\pm\else\textpm\fi{}12)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}11}$. Updating our earlier results for the contributions from the exchanges of the $\ensuremath{\eta}$ and ${\ensuremath{\eta}}^{\ensuremath{'}}$ using simple vector-meson dominance form factors, we obtain ${a}_{\ensuremath{\mu}}^{\mathrm{LbyL};\mathrm{PS}}=(99\ifmmode\pm\else\textpm\fi{}16)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}11}$ for the sum of all light pseudoscalars. Combined with available evaluations for the other contributions to hadronic light-by-light scattering this leads to the new estimate ${a}_{\ensuremath{\mu}}^{\mathrm{LbyL};\mathrm{had}}=(116\ifmmode\pm\else\textpm\fi{}40)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}11}$.