Pion-nucleon scattering in the Roper channel from lattice QCD

Abstract

We present a lattice QCD study of $N\ensuremath{\pi}$ scattering in the positive-parity nucleon channel, where the puzzling Roper resonance ${N}^{*}(1440)$ resides in experiment. The study is based on the PACS-CS ensemble of gauge configurations with ${N}_{f}=2+1$ Wilson-clover dynamical fermions, ${m}_{\ensuremath{\pi}}\ensuremath{\simeq}156\text{ }\text{ }\mathrm{MeV}$ and $L\ensuremath{\simeq}2.9\text{ }\text{ }\mathrm{fm}$. In addition to a number of $qqq$ interpolating fields, we implement operators for $N\ensuremath{\pi}$ in $p$-wave and $N\ensuremath{\sigma}$ in $s$-wave. In the center-of-momentum frame we find three eigenstates below 1.65 GeV. They are dominated by $N(0)$, $N(0)\ensuremath{\pi}(0)\ensuremath{\pi}(0)$ [mixed with $N(0)\ensuremath{\sigma}(0)$] and $N(p)\ensuremath{\pi}(\ensuremath{-}p)$ with $p\ensuremath{\simeq}2\ensuremath{\pi}/L$, where momenta are given in parentheses. This is the first simulation where the expected multi-hadron states are found in this channel. The experimental $N\ensuremath{\pi}$ phase shift would---in the approximation of purely elastic $N\ensuremath{\pi}$ scattering---imply an additional eigenstate near the Roper mass ${m}_{R}\ensuremath{\simeq}1.43\text{ }\text{ }\mathrm{GeV}$ for our lattice size. We do not observe any such additional eigenstate, which indicates that $N\ensuremath{\pi}$ elastic scattering alone does not render a low-lying Roper. Coupling with other channels, most notably with $N\ensuremath{\pi}\ensuremath{\pi}$, seems to be important for generating the Roper resonance, reinforcing the notion that this state could be a dynamically generated resonance. Our results are in line with most of the previous lattice studies based just on $qqq$ interpolators, which did not find a Roper eigenstate below 1.65 GeV. The study of the coupled-channel scattering including a three-particle decay $N\ensuremath{\pi}\ensuremath{\pi}$ remains a challenge.