Decoupling of the structure functions in momentum space based on the Laplace transformation

Using Laplace transform techniques, we describe the determination of the longitudinal structure function <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msub><a:mi>F</a:mi><a:mi>L</a:mi></a:msub><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>, at the leading-order approximation in momentum space, from the structure function <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:msub><e:mi>F</e:mi><e:mn>2</e:mn></e:msub><e:mo stretchy="false">(</e:mo><e:mi>x</e:mi><e:mo>,</e:mo><e:msup><e:mi>Q</e:mi><e:mn>2</e:mn></e:msup><e:mo stretchy="false">)</e:mo></e:math> and its derivative with respect to <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi>ln</i:mi><i:msup><i:mi>Q</i:mi><i:mn>2</i:mn></i:msup></i:math> in a kinematical region of low values of the Bjorken variable <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mi>x</k:mi></k:math>. Since the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mi>x</m:mi></m:math> dependence of <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:msub><o:mi>F</o:mi><o:mn>2</o:mn></o:msub><o:mo stretchy="false">(</o:mo><o:mi>x</o:mi><o:mo>,</o:mo><o:msup><o:mi>Q</o:mi><o:mn>2</o:mn></o:msup><o:mo stretchy="false">)</o:mo></o:math> and its evolution with <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:msup><s:mi>Q</s:mi><s:mn>2</s:mn></s:msup></s:math> are determined much better by the data than <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:msub><u:mi>F</u:mi><u:mi>L</u:mi></u:msub><u:mo stretchy="false">(</u:mo><u:mi>x</u:mi><u:mo>,</u:mo><u:msup><u:mi>Q</u:mi><u:mn>2</u:mn></u:msup><u:mo stretchy="false">)</u:mo></u:math>, this method provides both a direct check on <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"><y:msub><y:mi>F</y:mi><y:mi>L</y:mi></y:msub><y:mo stretchy="false">(</y:mo><y:mi>x</y:mi><y:mo>,</y:mo><y:msup><y:mi>Q</y:mi><y:mn>2</y:mn></y:msup><y:mo stretchy="false">)</y:mo></y:math> where measured, and a way of extending <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"><cb:msub><cb:mi>F</cb:mi><cb:mi>L</cb:mi></cb:msub><cb:mo stretchy="false">(</cb:mo><cb:mi>x</cb:mi><cb:mo>,</cb:mo><cb:msup><cb:mi>Q</cb:mi><cb:mn>2</cb:mn></cb:msup><cb:mo stretchy="false">)</cb:mo></cb:math> into regions of <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML" display="inline"><gb:mi>x</gb:mi></gb:math> and <ib:math xmlns:ib="http://www.w3.org/1998/Math/MathML" display="inline"><ib:msup><ib:mi>Q</ib:mi><ib:mn>2</ib:mn></ib:msup></ib:math> where there are currently no data. In our calculations, we ultilize the Block-Durand-Ha parametrization for the structure function <kb:math xmlns:kb="http://www.w3.org/1998/Math/MathML" display="inline"><kb:msub><kb:mi>F</kb:mi><kb:mn>2</kb:mn></kb:msub><kb:mo stretchy="false">(</kb:mo><kb:mi>x</kb:mi><kb:mo>,</kb:mo><kb:msup><kb:mi>Q</kb:mi><kb:mn>2</kb:mn></kb:msup><kb:mo stretchy="false">)</kb:mo></kb:math> [M. M. Block , .]. We find that the Laplace transform method in momentum space provides correct behaviors of the extracted longitudinal structure function <ob:math xmlns:ob="http://www.w3.org/1998/Math/MathML" display="inline"><ob:msub><ob:mi>F</ob:mi><ob:mi>L</ob:mi></ob:msub><ob:mo stretchy="false">(</ob:mo><ob:mi>x</ob:mi><ob:mo>,</ob:mo><ob:msup><ob:mi>Q</ob:mi><ob:mn>2</ob:mn></ob:msup><ob:mo stretchy="false">)</ob:mo></ob:math> and that our obtained results are in line with data from the H1 Collaboration and other results for <sb:math xmlns:sb="http://www.w3.org/1998/Math/MathML" display="inline"><sb:msub><sb:mi>F</sb:mi><sb:mi>L</sb:mi></sb:msub><sb:mo stretchy="false">(</sb:mo><sb:mi>x</sb:mi><sb:mo>,</sb:mo><sb:msup><sb:mi>Q</sb:mi><sb:mn>2</sb:mn></sb:msup><sb:mo stretchy="false">)</sb:mo></sb:math> obtained using Mellin transform method. Published by the American Physical Society 2024