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

TL;DR

This study uses lattice QCD and the Lüscher method to precisely determine the properties of the $
$ resonance in pion-nucleon scattering, including its mass and coupling, by analyzing energy spectra in various lattice symmetry channels.

Contribution

The paper presents a comprehensive lattice QCD calculation of the $
$ resonance parameters, including a detailed analysis of scattering amplitudes and phase shifts in multiple irreducible representations.

Findings

Resonance mass $m_
=1378(7)(9)$ MeV obtained.

Coupling constant $g_{
-
} ext{to} \u0001 N$ determined as 23.8(2.7)(0.9).

Effective-range and Breit-Wigner models successfully fit the scattering data.

Abstract

We determine the $\Delta(1232)$ resonance parameters using lattice QCD and the L\"uscher method. The resonance occurs in elastic pion-nucleon scattering with $J^P=3/2^+$ in the isospin $I = 3/2$, $P$-wave channel. Our calculation is performed with $N_f=2+1$ flavors of clover fermions on a lattice with $L\approx 2.8$ fm. The pion and nucleon masses are $m_\pi =255.4(1.6)$ MeV and $m_N=1073(5)$ MeV, and the strong decay channel $\Delta \rightarrow \pi N$ 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 $m_\Delta=1378(7)(9)$ MeV and the coupling $g_{\Delta\text{-}\pi N}=23.8(2.7)(0.9)$.