1-\sigma$ after the average over the realizations), the scattering occurs in the lab frame. In this case, the above formula gives the solution to the Balitsky-Kovchegov equation if the states evolve linearly: $S=S_\text{BK}$, where $S_\text{BK}$ was discussed in Sec. \[sec2B\]. When $\sigma=\frac12$ instead, it means that the scattering takes place in the center-of-mass frame.

By definition, $S$ is unitary, whether or not there is a limit on the growth of the number of partons that interact. So a priori there is no violation of unitarity, and saturation is not necessary to prevent $S$ from becoming negative. However, we know that the discussion is more subtle, and that the $S$-matrix defined like that violates boost invariance [@Mueller:2004sea]. Nevertheless, for small enough values of $y$, there should be no need to put saturation effects in the evolution of the states.

Let us discuss quantitatively the expected range in which saturation effects may be left out. In these considerations, we follow the discussion given by Mueller [@M2]. To simplify, let us first go to the center-of-mass frame, that is, we set $\sigma=\frac12$. Imagine that we start from a situation in which both objects are at rest. When the two systems are gradually boosted 