Non-linear equation: Energy conservation and impact parameter dependence

In this paper we address two questions: how energy conservation affects the solution to the non-linear equation, and how impact parameter dependence influences the inclusive production. Answering the first question we solve the modified BK equation which takes into account energy conservation. In spite of the fact that we used the simplified kernel, we believe that the main result of the paper: the small (≤40%) suppression of the inclusive production due to energy conservation, reflects a general feature. This result leads us to believe that the small value of the nuclear modification factor is of a non-perturbative nature. In the solution a new scale appears Q_fr=Q_sexp(-1/(2α_S)) and the production of dipoles with the size larger than 2/Q_fr is suppressed. Therefore, we can expect that the typical temperature for hadron production is about Q_fr (T≈Q_fr). The simplified equation allows us to obtain a solution to Balitsky-Kovchegov equation taking into account the impact parameter dependence. We show that the impact parameter (b) dependence can be absorbed into the non-perturbative b dependence of the saturation scale. The solution of the BK equation, as well as of the modified BK equation without b dependence, is only accurate up to ±25%.