Elastic $I=3/2$ $p$-wave nucleon-pion scattering amplitude and the $\mathrm{\Delta }(1232)$ resonance from $N_f=2+1$ lattice QCD

We present the first direct determination of meson-baryon resonance parameters from a scattering amplitude calculated using lattice QCD. In particular, we calculate the elastic $I=3/2$, $p$-wave nucleon-pion amplitude on a single ensemble of $N_{\mathrm{f}}=2+1$ Wilson-clover fermions with $m_{\pi }=280\mathrm{MeV}$ and $m_K=460\mathrm{MeV}$. At these quark masses, the $\mathrm{\Delta }(1232)$ resonance pole is found close to the $N-\pi$ threshold and a Breit-Wigner fit to the amplitude gives $g_{\mathrm{\Delta }N\pi }^{\mathrm{BW}}=19.0(4.7)$ in agreement with phenomenological determinations.

Figure 1

Variation of $t_{\min }$, the lower end of the range for single-exponential fits to the correlator ratio in Eq. (1). The vertical axis is the dimensionless center-of-mass momentum $u_{\mathrm{cm}}^2=L^2{q}_{\mathrm{cm}}^2/(2\pi {)}^2$. Shown here are the ground state energies from each of the five irreps $({\mathbf{P}}^2,\mathrm{\Lambda })=\{(0,H_g),(1,G_2),(3,F_1),(3,F_2),(4,G_2)\}$ and a single first excited state from the $(3,F_2)$ irrep. For each fit $t_{\max }=25a$ while the dashed lines indicate the chosen fit.

Figure 2

The real part of the inverse scattering amplitude for $I=3/2$, $p$-wave elastic $N-\pi$ scattering. Different colors indicate different energy levels, for each of which a colored circle is plotted for each of the central 68% of bootstrap samples and a black square indicates the mean value. The solid and dotted lines denote the mean values and bootstrap errors, respectively, for the Breit-Wigner fit described in the text. The lower panel focusses on the resonance region shown in the upper panel.