# Towards the P-wave nucleon-pion scattering amplitude in the $\Delta (1232)$ channel

## TL;DR

This study employs lattice QCD and the Lüscher method to analyze pion-nucleon scattering in the $ ext{I} = 3/2$ channel, focusing on the $ ext{Δ}(1232)$ resonance near the decay threshold, and extracts resonance parameters.

## Contribution

It introduces a lattice QCD approach to determine the P-wave nucleon-pion scattering amplitude and resonance parameters in the $ ext{Δ}(1232)$ channel at near-threshold pion mass.

## Key findings

- Preliminary energy spectra in various moving frames obtained.
- Scattering phase shifts extracted from lattice data.
- Resonance mass and coupling determined via Breit-Wigner fit.

## Abstract

We use lattice QCD and the L\"uscher method to study elastic pion-nucleon scattering in the isospin $I = 3/2$ channel, which couples to the $\Delta(1232)$ resonance. Our $N_f=2+1$ flavor lattice setup features a pion mass of $m_\pi \approx 250$ MeV, such that the strong decay channel $\Delta \rightarrow \pi N$ is close to the threshold. We present our method for constructing the required lattice correlation functions from single- and two-hadron interpolating fields and their projection to irreducible representations of the relevant symmetry group of the lattice. We show preliminary results for the energy spectra in selected moving frames and irreducible representations, and extract the scattering phase shifts. Using a Breit-Wigner fit, we also determine the resonance mass $m_\Delta$ and the $g_{\Delta-\pi N}$ coupling.