A Lattice QCD study of the $\rho$ resonance

We present a lattice QCD study of the $\rho$ resonance with $N_f=2+1$ clover fermions at a pion mass of approximately $320$ MeV and lattice size $3.6$ fm. We consider two processes involving the $\rho$. The first process is elastic scattering of two pions in P-wave with isospin $1$. Using the L\"uscher method we determine the scattering phase shift, from which we obtain the $\rho$ resonance mass and decay width $\Gamma(\rho\to\pi\pi)$. The second process is the radiative transition $\pi\gamma\to\pi\pi$, where we follow the Brice\~no-Hansen-Walker-Loud approach to determine the transition amplitude in the invariant mass region near the $\rho$ resonance and for both space- and time-like photon momentum. This allows us to determine the coupling between the $\rho$, the pion and the photon, and the resulting $\rho$ radiative decay width.