I have a system where the two-particle scattering matrix $S_{12}(p_1,p_2)$ depends on the momentum difference $p_1-p_2$, and also on the total momentum $P=p_1+p_2$ in some non-trivial way. One can use parametrization of energies and momenta $E=m\cosh(\alpha)$ and $p=m\sinh(\alpha)$ to express everything in terms of the rapidities $\alpha$; then I obtain $S_{12}(\alpha,\beta)$ as a function of $\alpha-\beta$ and $\alpha+\beta$. It is know that if $S_{12}$ satisfies the Yang-Baxter equation, then the $N$-body $S$ matrix factorizes as a product of 2-body $S$ matrices.

But if $S$ depends on the combination of $\alpha+\beta$, not only on $\alpha-\beta$, can we already rule out the possibility for such $S$ matrix to satisfy the YBE? Or is it possible that even depending on $\alpha+\beta$ (and $\alpha-\beta$), the YBE can still be satisfied?

The quantity $p_1+p_2$ can always be expressed in terms of te invariant $s^{2}=(E_1+E_2)^{2}-(p_1+p_2)^{2}$, but I don't know if that is of any help in such cases.

I have a system where the two-particle scattering matrix $S_{12}(p_1,p_2)$ depends on the momentum difference $p_1-p_2$, and also on the total momentum $P=p_1+p_2$ in some non-trivial way. One can use parametrization of energies and momenta $E=m\cosh(\alpha)$ and $p=m\sinh(\alpha)$ to express everything in terms of the rapidities $\alpha$. Then, I obtain $S_{12}(\alpha,\beta)$ as a function of $\alpha-\beta$ and $\alpha+\beta$. It is known that if $S_{12}$ satisfies the Yang-Baxter equation (TBE), then the $N$-body $S$ matrix factorizes as a product of 2-body $S$ matrices.

But if $S$ depends on the combination of $\alpha+\beta$, not only on $\alpha-\beta$, can we already rule out the possibility for such $S$ matrix to satisfy the YBE? Or, is it possible that even depending on $\alpha+\beta$ (and $\alpha-\beta$), the YBE can still be satisfied?

The quantity $p_1+p_2$ can always be expressed in terms of the invariant:

$$s^{2}=(E_1+E_2)^{2}-(p_1+p_2)^{2}$$

But I don't know if that is of any help in such cases.