In this paper, we address the reduced beauty cross section [σb¯b red ðx;Q2Þ] and the beauty structure function [Fb¯b 2 ðx;Q2Þ], to study the beauty content of a proton. We calculate σb¯b red and Fb¯b 2 in the ktfactorization formalism by using the integral form of the Kimber-Martin-Ryskin and Martin-Ryskin-Watt unintegrated parton distribution function (KMR and MRW-UPDF) with the angular ordering constraint (AOC) and the MMHT2014 PDF set as the input. Recently Guiot and van Hameren demonstrated that the upper limit, kmax, of the transverse-momentum integration performed in the kt-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 [F2 ðx;Q2Þ]. In the present work, we show that kmax cannot be equal to 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., Aμq0 μ ¼ 0 in the calculation of Fb¯b 2 in the kt-factorization formalism, we do not encounter any overestimation of the theoretical predictions due to different choices of kmax > Q. Finally, the resulted σb¯b red and Fb¯b 2 are compared to the experimental data and the theoretical predictions. In general, the extracted σb¯b red and Fb¯b 2 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 the MRW approach.
