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We study isospin-1 P-wave $\pi\pi$ scattering in lattice QCD with twoflavours of O($a$) improved Wilson fermions. For pion masses ranging from$m_\pi=265$ MeV to $m_\pi=437$ MeV, we determine the energy spectrum in thecentre-of-mass frame and in three moving frames. We obtain the scattering phaseshifts using L\"uscher's finite-volume quantisation condition. Fitting thedependence of the phase shifts on the scattering momentum to a Breit-Wignerform allows us to determine the resonance parameters $m_\rho$ and$g_{\rho\pi\pi}$. By combining the scattering phase shifts with the decaymatrix element of the vector current, we calculate the timelike pion formfactor, $F_\pi$, and compare the results to the Gounaris-Sakurai representationof the form factor in terms of the resonance parameters. In addition, we fitour data for the form factor to the functional form suggested by the Omn\`esrepresentation, which allows for the extraction of the charge radius of thepion. As a further application, we discuss the long-distance behaviour of thevector correlator, which is dominated by the two-pion channel. We reconstructthe long-distance part in two ways: one based on the finite-volume energies andmatrix elements and the other based on $F_\pi$. It is shown that this part canbe accurately constrained using the reconstructions, which has importantconsequences for lattice calculations of the hadronic vacuum polarisationcontribution to the muon anomalous magnetic moment.

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