Rho resonance, timelike pion form factor, and implications for lattice studies of the hadronic vacuum polarisation.

We study isospin-1 P-wave $\pi\pi$ scattering in lattice QCDwith two flavours of O($a$) improved Wilson fermions. For pionmasses ranging from $m_\pi=265$ MeV to $m_\pi=437$ MeV, we determine the energy spectrum in the centre-of-mass frame and in three moving frames. We obtain the scattering phase shifts using Luscher's finite-volume quantisation condition. Fitting the dependence of the phase shifts on the scattering momentum to a Breit-Wigner form allows us to determine the resonance parameters $m_\rho$ and $g_{\rho\pi\pi}$. By combining the scattering phase shifts with the decay matrix element of the vector current, we calculate the timelike pionform factor, $F_\pi$, 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 Omnes representation, which allows for the extraction of the charge radiusof the pion. As a further application, we discuss the long-distance behaviour of the vector correlator, which is dominated by the two- pionchannel. We reconstruct the long-distance part in two ways: one based on the finite-volume energies and matrix elements and the other based on $F_\pi$. It is shown that this part can be accurately constrained using the reconstructions, which has important consequences for lattice calculations of the hadronic vacuum polarisation contribution to the muon anomalous magnetic moment.