Roy equation analyses of $\pi\pi$ scatterings at unphysical pion masses

An extended Roy equation including a bound state pole is used to study $\pi\pi$ scatterings at unphysical large pion masses when $\sigma$ becomes a bound state in one situation and stays as a broad resonance in the other case. The coupled integral equations at large pion masses are solved by taking the lattice driving terms and the Regge amplitudes as inputs. Relying on the solutions of Roy equations that respect unitarity, analyticity and crossing symmetry, we give predictions to the phase shifts with $IJ=00,11,20$ in the elastic energy region. We then perform analytic continuation into the complex $s$ plane to search for various poles, all of which are inside the validity domain of the Roy equation. This is the first time that lattice data at unphysical large pion masses are analyzed within the rigorous Roy equation method.