Lattice QCD studies of the $\mathrm{\Delta }$ baryon resonance and the $K_0^{\ast }(700)$ and $a_0(980)$ meson resonances: the role of exotic operators in determining the finite-volume spectrum

Studies of the $\mathrm{\Delta }$ baryon resonance and the $K_0^{\ast }(700)$ and $a_0(980)$ meson resonances using $N_f=2+1$ lattice QCD for pion masses near 200 MeV are presented. The $s$-wave scattering lengths for both the $I=1\mathrm{/}2$ $N\pi$ and $I=3\mathrm{/}2$ $N\pi$ channels and properties of the $\mathrm{\Delta }$ resonance are identified from the finite-volume energy levels of the lattice simulation. The importance of a three-quark $\mathrm{\Delta }$-operator in the $N\pi$ system and tetraquark operators in the mesonic systems is investigated.