$P$-wave $\pi\pi$ scattering and the $\rho$ resonance from lattice QCD

We calculate the parameters describing elastic $I=1$, $P$-wave $\pi\pi$ scattering using lattice QCD with $2+1$ flavors of clover fermions. Our calculation is performed with a pion mass of $m_\pi \approx 320\:\:{\rm MeV}$ and a lattice size of $L\approx 3.6$ fm. We construct the two-point correlation matrices with both quark-antiquark and two-hadron interpolating fields using a combination of smeared forward, sequential and stochastic propagators. The spectra in all relevant irreducible representations for total momenta $|\vec{P}| \leq \sqrt{3} \frac{2\pi}{L}$ are extracted with two alternative methods: a variational analysis as well as multi-exponential matrix fits. We perform an analysis using L\"uscher's formalism for the energies below the inelastic thresholds, and investigate several phase shift models, including possible nonresonant contributions. We find that our data are well described by the minimal Breit-Wigner form, with no statistically significant nonresonant component. In determining the $\rho$ resonance mass and coupling we compare two different approaches: fitting the individually extracted phase shifts versus fitting the $t$-matrix model directly to the energy spectrum. We find that both methods give consistent results, and at a pion mass of $am_{\pi}=0.18295(36)_{stat}$ obtain $g_{\rho\pi\pi} = 5.69(13)_{stat}(16)_{sys}$, $am_\rho = 0.4609(16)_{stat}(14)_{sys}$, and $am_{\rho}/am_{N} = 0.7476(38)_{stat}(23)_{sys}$, where the first uncertainty is statistical and the second is the systematic uncertainty due to the choice of fit ranges.