invariance. This method will be called “saturation”. It is this one that makes sense in a field-theory framework where of course observables have to be independent of the frame in which the scattering is observed. Note that with this choice for the saturation mechanism, the resulting model is close to the one that was extensively studied in Ref. [@ISAST], except for the discretized rapidity.

To our knowledge, the uniqueness of the prescription that leads to boost-invariant saturation has not been formally proven in models with spatial dimensions such as the one that we are building. It is also not known whether the prescription could be slightly modified in a way that approximately preserves boost invariance, the characteristics of the amplitude being at the same time significantly changed.

Finally, we note that the consequences of boost invariance (or rather of “projectile-target duality”) has been thoroughly investigated very recently, also at a quite formal and rigorous level [@KL1; @KL2]: The transformation that corresponds to exchanging the scattering particles was translated into an operation on the effective action of the corresponding model. So we could check the (approximate) boost invariance of our model with the technology developed in Refs. [@K1; @K2]. (We have 