  We determine exactly the mass-coupling relation for the simplest multi-scale
quantum integrable model, the homogenous sine-Gordon model with two independent
mass-scales. We first reformulate its perturbed coset CFT description in terms
of the perturbation of a projected product of minimal models. This
representation enables us to identify conserved tensor currents on the UV side.
These UV operators are then mapped via form factor perturbation theory to
operators on the IR side, which are characterized by their form factors. The
relation between the UV and IR operators is given in terms of the sought-for
mass-coupling relation. By generalizing the $\Theta$ sum rule Ward identity we
are able to derive differential equations for the mass-coupling relation, which
we solve in terms of hypergeometric functions. We check these results against
the data obtained by numerically solving the thermodynamic Bethe Ansatz
equations, and find a complete agreement.
