coupling. Furthermore, for theories that are also supersymmetric we find exact agreement.

Integrable bosonic theories {#s31ibt}
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Let us first discuss on a class of generalized sine-Gordon models [@Hollowood:1994vx; @Bakas:1995bm]. These theories are defined by a gauged WZW model for a coset $G/H$ plus a potential and their classical integrability can be demonstrated through the existence of a Lax connection. Here we will consider the coset $G/H = \text{SO}(n+1)/\text{SO}(n)$, in which case the asymptotic excitations are a free $\text{SO}(n)$ vector with unit mass. Therefore, the S-matrix will again have the structure . This class includes the sine-Gordon and complex sine-Gordon models for $n=1$ and $2$ respectively, for which the exact S-matrices are known [@Zamolodchikov:1978xm; @Dorey:1994mg] and agree with perturbation theory.

Starting from the gauged WZW formulation of these models, the gauge symmetry can be fixed and at the classical level the unphysical modes integrated out [@Hoare:2010fb] giving a Lagrangian for just the physical excitations. To quartic order this is given by with $\text{c}_1 = 0$ and $\text{c}_2 h = \tfrac{\pi}{2k}$, where $k$ is the coupling. The functions parametrizing the tree-level S-matrix are therefore $$\begin{aligned}
T^{(0)}(\theta) = \frac{i\pi}{k}{\operatorname{csch}}\theta \ ,\qquad  R^{(0)}(\theta) = \frac{i\pi}{k}\coth \theta\ .\label{s2tlbos} \end{aligned}$$ For general $n$ this Lagrangian 