TBA equations for excited states in the O(3) and O(4) nonlinear $sigma$-model

TBA integral equations are proposed for 1-particle states in the sausage- and SS-models and their $sigma$-model limits. Combined with the ground state TBA equations the exact mass gap is computed in the O(3) and O(4) nonlinear $sigma$-model and the results are compared to 3-loop perturbation theory and Monte Carlo data.