Lattice calculation of the pion transition form factor with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msub><mml:mrow><mml:mi>N</mml:mi></mml:mrow><mml:mrow><mml:mi>f</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>2</mml:mn><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math> Wilson quarks

Abstract

We present a lattice QCD calculation of the double-virtual neutral pion transition form factor, with the goal to cover the kinematic range relevant to hadronic light-by-light scattering in the muon $g\ensuremath{-}2$. Several improvements have been made compared to our previous work. First, we take into account the effects of the strange quark by using the ${N}_{f}=2+1$ coordinated lattice simulation gauge ensembles. Second, we have implemented the on-shell $\mathcal{O}(a)$ improvement of the vector current to reduce the discretization effects associated with Wilson quarks. Finally, in order to have access to a wider range of photon virtualities, we have computed the transition form factor in a moving frame as well as in the pion rest frame. After extrapolating the form factor to the continuum and to physical quark masses, we compare our results with phenomenology. We extract the normalization of the form factor with a precision of 3.5% and confirm within our uncertainty previous somewhat conflicting estimates for a low-energy constant that appears in chiral perturbation theory for the decay ${\ensuremath{\pi}}^{0}\ensuremath{\rightarrow}\ensuremath{\gamma}\ensuremath{\gamma}$ at the next-to-leading order. With additional input from experiment and theory, we reproduce recent estimates for the decay width $\mathrm{\ensuremath{\Gamma}}({\ensuremath{\pi}}^{0}\ensuremath{\rightarrow}\ensuremath{\gamma}\ensuremath{\gamma})$. We also study the asymptotic large-${Q}^{2}$ behavior of the transition form factor in the double-virtual case. Finally, we provide as our main result a more precise model-independent lattice estimate of the pion-pole contribution to hadronic light-by-light scattering in the muon $g\ensuremath{-}2$: ${a}_{\ensuremath{\mu}}^{\text{HLbL};{\ensuremath{\pi}}^{0}}=(59.7\ifmmode\pm\else\textpm\fi{}3.6)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}11}$. Using in addition the normalization of the form factor obtained by the PrimEx experiment, we get the lattice and data-driven estimate ${a}_{\ensuremath{\mu}}^{\text{HLbL};{\ensuremath{\pi}}^{0}}=(62.3\ifmmode\pm\else\textpm\fi{}2.3)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}11}$.