Non-linear equation: energy conservation and impact parameter dependence

TL;DR

This paper investigates how energy conservation and impact parameter dependence influence solutions to the non-linear BK equation, revealing a small suppression in inclusive production and introducing a new scale related to hadron production temperature.

Contribution

It provides a modified BK equation incorporating energy conservation and impact parameter dependence, highlighting a new scale and its implications for hadron production.

Findings

Energy conservation causes up to 40% suppression of inclusive production.

A new scale, Q_{fr}, emerges, affecting dipole production and hadron temperature.

Impact parameter dependence can be incorporated into the saturation scale's non-perturbative b dependence.

Abstract

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 ($\leq 40%$) suppression of the inclusive productiondue 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 $Q_{fr} = Q_s \exp(-1/(2 \bas))$ 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 \approx 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 $\pm 25%$.