Lattice calculation of the pion transition form factor with $N_f=2+1$ 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 $g-2$. Several improvements have been made compared to our previous work. First, we take into account the effects of the strange quark by using the $N_f=2+1$ CLS gauge ensembles. Secondly, 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 ${\pi }^0\to \gamma \gamma$ at NLO. With additional input from experiment and theory, we reproduce recent estimates for the decay width $\Gamma ({\pi }^0\to \gamma \gamma )$. 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 $g-2$: $a_{\mu }^{\mathrm{HLbL};{\pi }^0}=(59.7\pm 3.6)\times 10^{-11}$. Using in addition the normalization of the form factor obtained by the PrimEx experiment, we get the lattice and data-driven estimate $a_{\mu }^{\mathrm{HLbL};{\pi }^0}=(62.3\pm 2.3)\times 10^{-11}$.

29 pages, 14 figures. v2: minor corrections to match the published version. A file with the transition form factor data at the physical pion mass and in the continuum is included in the submission

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