Single parton fragmentation functions of heavy quarkonium in soft gluon factorization

We study the single parton fragmentation functions (FFs) at the input factorization scale <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msub><a:mi>μ</a:mi><a:mn>0</a:mn></a:msub><a:mo>≳</a:mo><a:mn>2</a:mn><a:msub><a:mi>m</a:mi><a:mi>Q</a:mi></a:msub></a:math>, with heavy quark mass <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:msub><c:mi>m</c:mi><c:mi>Q</c:mi></c:msub></c:math>, in the soft gluon factorization (SGF) approach. We express the FFs in terms of perturbatively calculable short distance hard parts for producing a heavy quark-antiquark pair in all possible states, convoluted with corresponding soft gluon distribution for the hadronization of the pair to a heavy quarkonium. We compute the perturbative short distance hard parts for producing a heavy quark pair in all possible <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>S</e:mi></e:math>-wave and <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>P</g:mi></g:math>-wave states up to <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi>O</i:mi><i:mo stretchy="false">(</i:mo><i:msubsup><i:mi>α</i:mi><i:mi>s</i:mi><i:mn>2</i:mn></i:msubsup><i:mo stretchy="false">)</i:mo></i:math>. With our results, the SGF can be further used to study the heavy quarkonium production at the hadron colliders and heavy quarkonium production within a jet. Published by the American Physical Society 2024