is only valid for the computation of the tree-level S-matrix – the procedure of integrating out the unphysical fields picks up a one-loop correction [@Hoare:2010fb] $$\Delta \mathcal{L} = -\frac{\pi}{2k^2} (X\cdot \partial_+ X)(X\cdot \partial_-X) - \frac{\pi(n-2)}{2k^2} (X\cdot X)(\partial_+ X \cdot \partial_- X) \ .\label{s2lagcon}$$ Furthermore, the corresponding contributions to the one-loop S-matrix restore various properties of integrability. Indeed these counterterms were first studied from this perspective for the complex sine-Gordon model in [@deVega:1981ka].

Including these contributions the resulting functions parametrizing the one-loop S-matrix following from perturbation theory with dimensional regularization are given by [^6] $$\begin{aligned}
T^{(1)}(\theta) &= \frac{i\pi}{2k^2}  \big(i\pi  (1+\cosh^2\theta) {\operatorname{csch}}^2\theta - (n-2){\operatorname{csch}}\theta\big)  \ ,\no
\\
R^{(1)}(\theta) &=
\frac{i\pi}{2k^2}  \big(2\pi i\coth\theta {\operatorname{csch}}\theta-2 (n-2) \coth \theta + \theta  (n-2) \coth^2 \theta \big)
 \ .  \label{s2olbos}\end{aligned}$$

Inputting the tree-level S-matrix into the candidate formula for the one-loop S-matrix following from the unitarity techniques described in Section \[sec:general\] we find the following expressions for the parametrizing functions $$\begin{aligned}
T^{(1)}_{\text{u.c.}}(\theta) &= \frac{i\pi}{2k^2}  \big(i\pi  (1+\cosh^2\theta) {\operatorname{csch}}^2\theta + n{\operatorname{csch}}\theta\big)  \ ,\no
\\
R^{(1)}_{\text{u.c.}}(\theta) &= 
\frac{i\pi}{2k^2}  \big(2\pi i\coth\theta {\operatorname{csch}}\theta+2 \coth \theta + \theta  (n-2) \coth^2 \theta \big)\ . \label{s2olbosuc}\end{aligned}$$ In this case these results agree with up to a contribution proportional to tree-level S-matrix , which can be 