Analysis of the fragmentation function of gluon at next-to-leading order approximation

We are investigating the behavior of the fragmentation function of a gluon, denoted as <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msub><a:mi>D</a:mi><a:mi>g</a:mi></a:msub><a:mo stretchy="false">(</a:mo><a:mi>x</a:mi><a:mo>,</a:mo><a:msup><a:mi>μ</a:mi><a:mn>2</a:mn></a:msup><a:mo stretchy="false">)</a:mo></a:math>, where <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>μ</e:mi></e:math> represents the observable scale. This function is derived from the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi evolution equations. Our objective is to evolve the fragmentation function of a gluon for heavy-quark-antiquark bound states with large transverse momentum using a Laplace transform technique. This method enables us to calculate numerical solutions for two and four quarkonium states based on the known initial fragmentation function of gluons. We examine both leading-order and higher-order approximations for the fragmentation function of a gluon, <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mo stretchy="false">[</g:mo><g:mi>g</g:mi><g:mo stretchy="false">→</g:mo><g:msub><g:mi>T</g:mi><g:mrow><g:mi>n</g:mi><g:mi>Q</g:mi></g:mrow></g:msub><g:mo stretchy="false">]</g:mo></g:math>, by integrating the evolved fragmentation function of the gluon at the initial scale. In our computations, we utilize the initial scales for <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"><l:mrow><l:msubsup><l:mrow><l:mi>T</l:mi></l:mrow><l:mrow><l:mi>g</l:mi></l:mrow><l:mrow><l:mn>2</l:mn><l:mi>c</l:mi></l:mrow></l:msubsup></l:mrow></l:math> from Braaten and Yuan [] and for <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"><n:msubsup><n:mi>T</n:mi><n:mi>g</n:mi><n:mrow><n:mn>4</n:mn><n:mi>c</n:mi></n:mrow></n:msubsup></n:math> and <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"><p:msubsup><p:mi>T</p:mi><p:mi>g</p:mi><p:mrow><p:mn>4</p:mn><p:mi>b</p:mi></p:mrow></p:msubsup></p:math> from Celiberto [] and Celiberto and Gatto [], respectively. Through comparing our predictions with existing literature results, we can accurately determine the evolution of the fragmentation function of a gluon at scale <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" display="inline"><r:mi>μ</r:mi></r:math>.