  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.
