Exact Mass-Coupling Relation for the Homogeneous Sine-Gordon Model

We derive the exact mass-coupling relation of the simplest multi-scale quantum integrable model, i.e., the homogeneous sine-Gordon model with two mass scales. The relation is obtained by comparing the perturbed conformal field theory description of the model valid at short distances to the large distance bootstrap description based on the model's integrability. In particular, we find a differential equation for the relation by constructing conserved tensor currents which satisfy a generalization of the $\Theta$ sum rule Ward identity. The mass-coupling relation is written in terms of hypergeometric functions.