Rho resonance, timelike pion form factor, and implications for lattice studies of the hadronic vacuum polarization
Felix Erben, Jeremy Green, Daniel Mohler, Hartmut Wittig
Abstract
We study isospin-1 $P$-wave $\ensuremath{\pi}\ensuremath{\pi}$ scattering in lattice QCD with two flavors of $\mathrm{O}(a)$ improved Wilson fermions. For pion masses ranging from ${m}_{\ensuremath{\pi}}=265\text{ }\text{ }\mathrm{MeV}$ to ${m}_{\ensuremath{\pi}}=437\text{ }\text{ }\mathrm{MeV}$, we determine the energy spectrum in the center-of-mass frame and in three moving frames. We obtain the scattering phase shifts using L\"uscher's finite-volume quantization condition. Fitting the dependence of the phase shifts on the scattering momentum to a Breit-Wigner form allows us to determine the corresponding $\ensuremath{\rho}$ mass ${m}_{\ensuremath{\rho}}$ and ${g}_{\ensuremath{\rho}\ensuremath{\pi}\ensuremath{\pi}}$ coupling. By combining the scattering phase shifts with the decay matrix element of the vector current, we calculate the timelike pion form factor, ${F}_{\ensuremath{\pi}}$, and compare the results to the Gounaris-Sakurai representation of the form factor in terms of the resonance parameters. In addition, we fit our data for the form factor to the functional form suggested by the Omn\`es representation, which allows for the extraction of the charge radius of the pion. As a further application, we discuss the long-distance behavior of the vector correlator, which is dominated by the two-pion channel. We reconstruct the long-distance part in two ways: one based on the finite-volume energies and matrix elements, and the other based on ${F}_{\ensuremath{\pi}}$. It is shown that this part can be accurately constrained using the reconstructions, which has important consequences for lattice calculations of the hadronic vacuum polarization contribution to the muon anomalous magnetic moment.