\partial_+ X)(X \cdot \partial_- X)$ using field redefinitions. $\text{c}_{1,2}$ are arbitrary constants, while $h$ is the small parameter that we use to do perturbation theory. We assume that the theory is renormalizable and the ellipses denote any higher-order interaction terms required therefor.

The requirements of $\text{SO}(n)$ invariance and crossing symmetry imply that the S-matrix can be parametrized in terms of two functions, $T(\theta)$ and $R(\theta)$, as follows $$\begin{aligned}
& S_{ij}^{kl} (\theta) = \delta_{ij}\delta^{kl} \, R (i\pi - \theta) + \delta_i^k \delta_j^l \ T(\theta) + \delta_i^l \delta_j^k \ R(\theta) \ .\label{s2son}\end{aligned}$$ where $i,j,\ldots=1,...,n$ are $\text{SO}(n)$ vector indices. Using standard perturbation theory with dimensional regularization [^5] to one loop we find the following parametrizing functions $$\begin{aligned}
T(\theta) &= 1 + 2 i h (\text{c}_1 + \text{c}_2) {\operatorname{csch}}\theta \no   \vphantom{\frac{2i h^2}{\pi}}
\\ & \quad \ \ \, + \frac{2ih^2}{\pi} \big(2 \text{c}_1 \text{c}_2 (i\pi - 2 \theta ) \coth \theta{\operatorname{csch}}\theta + i \pi  (2 \text{c}_1^2+2 \text{c}_1 \text{c}_2+\text{c}_2^2 (1+\cosh^2 \theta))  {\operatorname{csch}}^2\theta \no
\\ & \qquad \qquad \quad +((\text{c}_1 + \text{c}_2) (4 \text{c}_1+n (\text{c}_1+\text{c}_2))-2 \text{c}_2^2)  {\operatorname{csch}}\theta \big)\ ,  \vphantom{\frac{2i h^2}{\pi}}\no
\\
R(\theta) &= 2 i h (\text{c}_1 + \text{c}_2 \cosh\theta) {\operatorname{csch}}\theta  \vphantom{\frac{2i h^2}{\pi}}\label{s2com1}
\\ & \quad \ \ \, + \frac{2i h^2}{\pi}\big(\theta \, (\text{c}_1+\text{c}_2 \cosh \theta 