Pion-nucleon scattering in the Roper channel from lattice QCD

Abstract

We present a lattice QCD study of $N\pi$ 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, $m_\pi \simeq 156~$MeV and $L\simeq 2.9~$fm. In addition to a number of $qqq$ interpolating fields, we implement operators for $N\pi$ in $p$-wave and $N\sigma$ in $s$-wave. In the center-of-momentum frame we find three eigenstates below 1.65 GeV. They are dominated by $N(0)$, $N(0)\pi(0)\pi(0)$ (mixed with $N(0)\sigma(0)$) and $N(p)\pi(-p)$ with $p\simeq 2\pi/L$, where momenta are given in parentheses. This is the first simulation where the expected multi-hadron states are found in this channel. The experimental $N\pi$ phase-shift would -- in the approximation of purely elastic $N\pi$ scattering -- imply an additional eigenstate near the Roper mass $m_R\simeq 1.43~$GeV for our lattice size. We do not observe any such additional eigenstate, which indicates that $N\pi$ elastic scattering alone does not render a low-lying Roper. Coupling with other channels, most notably with $N\pi\pi$, 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 previous lattice studies based just on $qqq$ interpolators, that did not find a Roper eigenstate below $1.65~$GeV. The study of the coupled-channel scattering including a three-particle decay $N\pi\pi$ remains a challenge.