Fock-state evolution through splittings only (Sec. \[sec2\]). A unitary $S$-matrix was constructed from multiple scatterings in the center-of-mass frame with linearly-evolving states (Sec. \[sec3A\]). However, we know that theoretically, this procedure has a limited validity, that we have estimated as $y<2y_F$. In order to compare the results for scattering obtained in that framework to what would be obtained in the case of saturated states, we need to introduce a saturation mechanism.

We can imagine different ways of enforcing saturation. The simplest way consists in forbidding new splittings to a site on which the number of particles has reached the value $1/\alpha^2$. We will call this method “veto”. The resulting model is close to other models that have been studied before, e.g. in Ref. [@EGBM].

However, in order to get an approximately boost-invariant $S$-matrix[^3], one needs instead to replace the splitting probability $\frac12$ at site $i$ by $\frac12 e^{-\alpha^2 n_i}$, in such a way that for small $n_i$ compared to $1/\alpha^2$, the splitting probability tends to the free splitting rate $\frac12$, and for large $n_i$, splittings are frozen. This prescription was proposed by Mueller and Salam [@MS] long ago and revisited more recently [@ISAST]. We will check numerically the approximate boost 