understood as just a shift of the coupling $k \to k + n - 1$. This is analogous to the shift mentioned in , but it is not completely equivalent as here we are not comparing to standard perturbation theory, but rather to integrable S-matrix found from the gauged WZW model. It is important to observe that a consequence of this is that we do not need to introduce an additional coupling, and as such this can still be interpreted as a difference in the regularization scheme.

For the sine-Gordon model ($n=1$) the S-matrix we are considering describes the scattering of a single particle type (the Lagrangian-field excitation) and therefore for the $2\to 2$ scattering is given by $T(\theta) + R(i\pi- \theta) + R(\theta)$. In this case the two results agree exactly. This should be expected as using field redefinitions the interaction piece of the sine-Gordon Lagrangian can be written in a form without derivatives. For $n\geq 2$ the shift in the coupling is by the dual Coxeter number of the group $G=\text{SO}(n)$. This structure appears regularly in the quantization of WZW and gauged WZW models, where $k$ is the quantized level [@Witten:1983ar; @Knizhnik:1984nr; @Leutwyler:1991tv; @Tseytlin:1992ri; @Tseytlin:1993my; @deWit:1993qv].

In summary, 