in terms of rapidities $${\text p}_i = \sinh \vartheta_i \ , \qquad \epsilon_{{\text p}_i} = \cosh \vartheta_i \ ,$$ where the mass has been set to unity. Lorentz invariance then implies that the $2\to 2$ scattering S-matrix should depend solely on the difference of the two rapidities associated to asymptotic states $$\theta = \vartheta_1 - \vartheta_2 \ .$$

The candidate expression for the one-loop S-matrix given in equation  is then given by $$\begin{aligned}
 {S^{(1)}}{}_{{M}{N}}^{{P}{Q}}(\theta)=\frac{1}{8 \sinh\theta}\,\Big[{\tilde S^{(0)}}{}_{{M}{N}}^{{R}{S}}(\theta){\tilde S^{(0)}}{}_{{R}{S}}^{{P}{Q}}(\theta)I(i\pi-\theta)\nonumber  &+ {\tilde S^{(0)}}{}_{{M}{R}}^{{S}{P}}(0){\tilde S^{(0)}}{}_{{S}{N}}^{{R}{Q}}(\theta)I(0) \\ &
 +{\tilde S^{(0)}}{}_{{M}{R}}^{{S}{Q}}(\theta){\tilde S^{(0)}}{}_{{S}{N}}^{{P}{R}}(\theta)I(\theta)\,\Big]
 \label{eqn:final_rel}~,\end{aligned}$$ where $\tilde S{}^{(0)} (\theta)= 4 \sinh \theta \ S^{(0)}(\theta)$ and the one-loop integrals are given by $$\label{olrel}
I(\theta) = \frac{\theta\,{\operatorname{csch}}\theta}{4\pi i } \ .$$ The consistency condition now reads \[s2const\] [S\^[(0)]{}]{}\_[[M]{}[R]{}]{}\^[[S]{}[P]{}]{}(0)[S\^[(0)]{}]{}\^[[R]{}[Q]{}]{}\_[[S]{}[N]{}]{}()=[S\^[(0)]{}]{}\^[[P]{}[S]{}]{}\_[[M]{}[R]{}]{}()[S\^[(0)]{}]{}\^[[Q]{}[R]{}]{}\_[[S]{}[N]{}]{}(0) .

As a starting point let us consider a general $\text{SO}(n)$-invariant theory with quartic interactions up to second order in derivatives for a single $\text{SO}(n)$ vector of unit mass $X$ (the light-cone derivatives $\partial_\pm$ are defined as $\partial_\tau \pm \partial_\sigma$) $$\label{s2laggen}
\mathcal{L} = \frac12 \partial_+ X \cdot \partial_- X -\frac12 X\cdot X + h \, \text{c}_1 (X\cdot X)^2 + h \, \text{c}_2 (X\cdot X)(\partial_+ X \cdot \partial_- X) + \ldots \ ,$$ where we have eliminated the other allowed term $(X \cdot 