A phenomenological investigation of the beauty content of a proton in the framework of $k_t$-factorization using KMR and MRW unintegrated parton distributions

Abstract

In this paper, we address the reduced beauty cross section (${\sigma_{red}}^{b\overline{b}}(x, Q^2)$) and the beauty structure function (${F_2}^{b\overline{b}}(x, Q^2)$), to study the beauty content of a proton. We calculate ${\sigma_{red}}^{b\overline{b}}$ and ${F_2}^{b\overline{b}}$ in the $k_t$-factorization formalism by using the integral form of the Kimber-Martin-Ryskin and Martin-Ryskin-Watt unintegrated parton distribution function with the angular ordering constraint ($AOC$) and the MMHT2014 PDF set as the input. Recently Guiot and van Hameren demonstrated that the upper limit, $K_{max}$, of the transverse-momentum integration performed in the $k_t$-factorization formalism should be almost equal to Q, where Q is the hard scale, otherwise it leads to an overestimation of the proton structure function ($F_2(x, Q^2)$). In the present work, we show that $k_{max}$ cannot be equal Q at low and moderate energy region, and also by considering the gluon and quark contributions to the same perturbative order and a physical gauge for the gluon, i. e., $A^\mu q^{\prime}_\mu=0$ in the calculation of ${F_2}^{b\overline{b}}$ in the $k_t$-factorization formalism, we do not encounter any overestimation of the theoretical predictions due to different choices of $K_{max}>Q$. Finally, the resulted ${\sigma_{red}}^{b\overline{b}}$ and ${F_2}^{b\overline{b}}$ are compared to the experimental data and the theoretical predictions. In general, the extracted ${\sigma_{red}}^{b\overline{b}}$ and ${F_2}^{b\overline{b}}$ based on the KMR and MRW approaches are in perfect agreement with the experimental data and theoretical predictions at high energies, but at low and moderate energies, the one developed from the KMR approach has better consistency than that of MRW approach.