<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>η</mml:mi><mml:mo stretchy="false">→</mml:mo><mml:msup><mml:mrow><mml:mi>γ</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>γ</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup></mml:mrow></mml:math> transition form factor and the hadronic light-by-light <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>η</mml:mi></mml:math>-pole contribution to the muon <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>g</mml:mi><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:math> from lattice QCD
Constantia Alexandrou, Simone Bacchio, Sebastian Burri, Jacob Finkenrath, Andrew Gasbarro, Kyriakos Hadjiyiannakou, Karl Jansen, Gurtej Kanwar, Bartosz Kostrzewa, Konstantin Ottnad, Marcus Petschlies, Ferenc Pittler, Carsten Urbach, Urs Wenger
Abstract
We calculate the double-virtual $\ensuremath{\eta}\ensuremath{\rightarrow}{\ensuremath{\gamma}}^{*}{\ensuremath{\gamma}}^{*}$ transition form factor ${\mathcal{F}}_{\ensuremath{\eta}\ensuremath{\rightarrow}{\ensuremath{\gamma}}^{*}{\ensuremath{\gamma}}^{*}}({q}_{1}^{2},{q}_{2}^{2})$ from first principles using a lattice QCD simulation with ${N}_{f}=2+1+1$ quark flavors at the physical pion mass and at one lattice spacing and volume. The kinematic range covered by our calculation is complementary to the one accessible from experiment and is relevant for the $\ensuremath{\eta}$-pole contribution to the hadronic light-by-light scattering in the anomalous magnetic moment ${a}_{\ensuremath{\mu}}=(g\ensuremath{-}2)/2$ of the muon. From the form factor calculation we extract the partial decay width $\mathrm{\ensuremath{\Gamma}}(\ensuremath{\eta}\ensuremath{\rightarrow}\ensuremath{\gamma}\ensuremath{\gamma})=338(87{)}_{\mathrm{stat}}(17{)}_{\mathrm{syst}}\text{ }\text{ }\mathrm{eV}$ and the slope parameter ${b}_{\ensuremath{\eta}}=1.34(28{)}_{\mathrm{stat}}(14{)}_{\mathrm{syst}}\text{ }\text{ }{\mathrm{GeV}}^{\ensuremath{-}2}$. For the $\ensuremath{\eta}$-pole contribution to ${a}_{\ensuremath{\mu}}$ we obtain ${a}_{\ensuremath{\mu}}^{\ensuremath{\eta}\text{-pole}}=13.8(5.2{)}_{\mathrm{stat}}(1.5{)}_{\mathrm{syst}}\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}11}$.