P-wave nucleon-pion scattering amplitude in the $\mathrm{\Delta }(1232)$ channel from lattice QCD

We determine the $\mathrm{\Delta }(1232)$ resonance parameters using lattice QCD and the L"uscher method. The resonance occurs in elastic pion-nucleon scattering with $J^P=3\mathrm{/}{2}^{+}$ in the isospin $I=3\mathrm{/}2$, $P$-wave channel. Our calculation is performed with $N_f=2+1$ flavors of clover fermions on a lattice with $L\approx 2.8$fm\mathrm{.}Thepionandnucleonmassesare$m_\pi =255.4(1.6)$MeVand$m_N=1073(5)$MeV,andthestrongdecaychannel$\Delta \rightarrow \pi N$ is found to be above the threshold. To thoroughly map out the energy-dependence of the nucleon-pion scattering amplitude, we compute the spectra in all relevant irreducible representations of the lattice symmetry groups for total momenta up to $\vec{P}=\frac{2\pi }{L}(1,1,1)$, including irreps that mix $S$ and $P$ waves. We perform global fits of the amplitude parameters to up to 21 energy levels, using a Breit-Wigner model for the $P$-wave phase shift and the effective-range expansion for the $S$-wave phase shift. From the location of the pole in the $P$-wave scattering amplitude, we obtain the resonance mass $m_{\mathrm{\Delta }}=1378(7)(9)$ MeV and the coupling $g_{\Delta$\text{-}$\pi N}=23.8(2.7)(0.9)$.