One-loop matching factors for singlet quasiparton distribution functions in the hybrid-ratio scheme

The one loop matching kernels between parton distribution functions (PDFs) for parton <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>i</a:mi><a:mo>=</a:mo><a:mi>u</a:mi></a:math>, <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mrow><c:mi>d</c:mi></c:mrow></c:math>, <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mrow><e:mi>s</e:mi></e:mrow></e:math>, <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mrow><g:mi>g</g:mi></g:mrow></g:math> and their corresponding quasi-PDFs are computed at one loop in the hybrid-ratio scheme. We found that, in addition to the conservation of the quasiquark number for each flavor, the second moment <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mo stretchy="false">⟨</i:mo><i:mi>x</i:mi><i:msub><i:mo stretchy="false">⟩</i:mo><i:mover accent="true"><i:mi>i</i:mi><i:mo stretchy="false">˜</i:mo></i:mover></i:msub><i:mo>=</i:mo><i:mo stretchy="false">⟨</i:mo><i:mi>x</i:mi><i:msub><i:mo stretchy="false">⟩</i:mo><i:mi>i</i:mi></i:msub></i:math> of quasi-PDF of parton <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:mi>i</q:mi></q:math> (denoted as <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:mover accent="true"><s:mi>i</s:mi><s:mo stretchy="false">˜</s:mo></s:mover></s:math>) and PDF of parton <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"><w:mi>i</w:mi></w:math> is the same in our approach. This is demonstrated numerically using the CTEQ14 global analysis as input.