Lattice QCD studies of the $\Delta$ 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 $\Delta$ 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 $I=1/2$ $N \pi$ and $I=3/2$ $N \pi$ channels and properties of the $\Delta$ resonance are identified from the finite-volume energy levels of the lattice simulation. The importance of a three-quark $\Delta$-operator in the $N\pi$ system and tetraquark operators in the mesonic systems is investigated.