In this paper, we address the reduced beauty cross section [<a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msubsup><a:mi>σ</a:mi><a:mi>red</a:mi><a:mrow><a:mi>b</a:mi><a:mover accent="true"><a:mi>b</a:mi><a:mo stretchy="false">¯</a:mo></a:mover></a:mrow></a:msubsup><a:mo stretchy="false">(</a:mo><a:mi>x</a:mi><a:mo>,</a:mo><a:msup><a:mi>Q</a:mi><a:mn>2</a:mn></a:msup><a:mo stretchy="false">)</a:mo></a:math>] and the beauty structure function [<g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:msubsup><g:mi>F</g:mi><g:mn>2</g:mn><g:mrow><g:mi>b</g:mi><g:mover accent="true"><g:mi>b</g:mi><g:mo stretchy="false">¯</g:mo></g:mover></g:mrow></g:msubsup><g:mo stretchy="false">(</g:mo><g:mi>x</g:mi><g:mo>,</g:mo><g:msup><g:mi>Q</g:mi><g:mn>2</g:mn></g:msup><g:mo stretchy="false">)</g:mo></g:math>], to study the beauty content of a proton. We calculate <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:msubsup><m:mi>σ</m:mi><m:mi>red</m:mi><m:mrow><m:mi>b</m:mi><m:mover accent="true"><m:mi>b</m:mi><m:mo stretchy="false">¯</m:mo></m:mover></m:mrow></m:msubsup></m:math> and <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:msubsup><q:mi>F</q:mi><q:mn>2</q:mn><q:mrow><q:mi>b</q:mi><q:mover accent="true"><q:mi>b</q:mi><q:mo stretchy="false">¯</q:mo></q:mover></q:mrow></q:msubsup></q:math> in the <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:msub><u:mi>k</u:mi><u:mi>t</u:mi></u:msub></u:math>-factorization 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 <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"><w:mi>M</w:mi><w:mi>M</w:mi><w:mi>H</w:mi><w:mi>T</w:mi><w:mn>2014</w:mn></w:math> <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"><y:mi>P</y:mi><y:mi>D</y:mi><y:mi>F</y:mi></y:math> set as the input. Recently Guiot and van Hameren demonstrated that the upper limit, <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"><ab:msub><ab:mi>k</ab:mi><ab:mi>max</ab:mi></ab:msub></ab:math>, of the transverse-momentum integration performed in the <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"><cb:msub><cb:mi>k</cb:mi><cb:mi>t</cb:mi></cb:msub></cb:math>-factorization formalism should be almost equal to <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML" display="inline"><eb:mi>Q</eb:mi></eb:math>, where <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML" display="inline"><gb:mi>Q</gb:mi></gb:math> is the hard scale, otherwise it leads to an overestimation of the proton structure function [<ib:math xmlns:ib="http://www.w3.org/1998/Math/MathML" display="inline"><ib:msub><ib:mi>F</ib:mi><ib:mn>2</ib:mn></ib:msub><ib:mo stretchy="false">(</ib:mo><ib:mi>x</ib:mi><ib:mo>,</ib:mo><ib:msup><ib:mi>Q</ib:mi><ib:mn>2</ib:mn></ib:msup><ib:mo stretchy="false">)</ib:mo></ib:math>]. In the present work, we show that <mb:math xmlns:mb="http://www.w3.org/1998/Math/MathML" display="inline"><mb:msub><mb:mi>k</mb:mi><mb:mi>max</mb:mi></mb:msub></mb:math> cannot be equal to <ob:math xmlns:ob="http://www.w3.org/1998/Math/MathML" display="inline"><ob:mi>Q</ob:mi></ob:math> 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., <qb:math xmlns:qb="http://www.w3.org/1998/Math/MathML" display="inline"><qb:msup><qb:mi>A</qb:mi><qb:mi>μ</qb:mi></qb:msup><qb:msubsup><qb:mi>q</qb:mi><qb:mi>μ</qb:mi><qb:mo>′</qb:mo></qb:msubsup><qb:mo>=</qb:mo><qb:mn>0</qb:mn></qb:math> in the calculation of <sb:math xmlns:sb="http://www.w3.org/1998/Math/MathML" display="inline"><sb:msubsup><sb:mi>F</sb:mi><sb:mn>2</sb:mn><sb:mrow><sb:mi>b</sb:mi><sb:mover accent="true"><sb:mi>b</sb:mi><sb:mo stretchy="false">¯</sb:mo></sb:mover></sb:mrow></sb:msubsup></sb:math> in the <wb:math xmlns:wb="http://www.w3.org/1998/Math/MathML" display="inline"><wb:msub><wb:mi>k</wb:mi><wb:mi>t</wb:mi></wb:msub></wb:math>-factorization formalism, we do not encounter any overestimation of the theoretical predictions due to different choices of <yb:math xmlns:yb="http://www.w3.org/1998/Math/MathML" display="inline"><yb:msub><yb:mi>k</yb:mi><yb:mi>max</yb:mi></yb:msub><yb:mo>></yb:mo><yb:mi>Q</yb:mi></yb:math>. Finally, the resulted <ac:math xmlns:ac="http://www.w3.org/1998/Math/MathML" display="inline"><ac:msubsup><ac:mi>σ</ac:mi><ac:mi>red</ac:mi><ac:mrow><ac:mi>b</ac:mi><ac:mover accent="true"><ac:mi>b</ac:mi><ac:mo stretchy="false">¯</ac:mo></ac:mover></ac:mrow></ac:msubsup></ac:math> and <ec:math xmlns:ec="http://www.w3.org/1998/Math/MathML" display="inline"><ec:msubsup><ec:mi>F</ec:mi><ec:mn>2</ec:mn><ec:mrow><ec:mi>b</ec:mi><ec:mover accent="true"><ec:mi>b</ec:mi><ec:mo stretchy="false">¯</ec:mo></ec:mover></ec:mrow></ec:msubsup></ec:math> are compared to the experimental data and the theoretical predictions. In general, the extracted <ic:math xmlns:ic="http://www.w3.org/1998/Math/MathML" display="inline"><ic:msubsup><ic:mi>σ</ic:mi><ic:mi>red</ic:mi><ic:mrow><ic:mi>b</ic:mi><ic:mover accent="true"><ic:mi>b</ic:mi><ic:mo stretchy="false">¯</ic:mo></ic:mover></ic:mrow></ic:msubsup></ic:math> and <mc:math xmlns:mc="http://www.w3.org/1998/Math/MathML" display="inline"><mc:msubsup><mc:mi>F</mc:mi><mc:mn>2</mc:mn><mc:mrow><mc:mi>b</mc:mi><mc:mover accent="true"><mc:mi>b</mc:mi><mc:mo stretchy="false">¯</mc:mo></mc:mover></mc:mrow></mc:msubsup></mc:math> 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. Published by the American Physical Society 2024
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N. Olanj (2024) studied this question.
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