Massless S-matrix Bootstraps and RG Flows

We find complete solutions of S-matrix bootstrap equations for the scatterings of massless particles in (1+1) dimensions with A_n symmetries. We show that only three types (minimal, diagonal, and saturated cases) of these S-matrices can generate RG flows that lead to UV-complete theories, and other RG flows predicted in [11] are inconsistent with the bootstrap equations. Using these S-matrices, we derived the TBA equations and corresponding Y-systems that generate the RG flows between the IR and UV CFTs.