Integrating Heisenberg uncertainty and maximum entropy principles verses statistical approach to nucleon structure

Abstract

There are many methods to calculate the parton distributions. In this work, we are trying to determine the unknown parameters of valence quark and gluon densities, taking into account some constraints simultaneously. For this purpose, we consider two different approaches. At first approach, a parameterized form for the valence densities are assumed. Using the constraints imposed by the Heisenberg uncertainty and maximum entropy principles and also considering the quark number and momentum sum rules, the unknown parameters of the parton distributions are determined. In second approach, we attribute a statistical distributions to quark, anti-quarks and gluon densities which are including temperature T, volume V and chemical potential, $\mu$, as thermodynamic parameters. In this case, using only the maximum entropy principle and the quark number together with momentum sum rules, the parton distributions in terms of thermodynamic parameters are extracted. The results for valence quark densities in both approaches and additionally the gluon density in second approach are in {satisfactory match} with results from some parametrization models. By evolving these densities to the desired energy scales, we are able to calculate the nucleon $xF_3$ structure function and also the ratio of valence quark densities.