Unraveling the collisional energy loss of a heavy quark in a quark-gluon plasma

At leading order in the QCD coupling constant, we compute the energy loss per traveling distance of a heavy quark <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>d</a:mi><a:mi>E</a:mi><a:mo>/</a:mo><a:mi>d</a:mi><a:mi>z</a:mi></a:math> from elastic scattering off thermal quarks and gluons at a temperature <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>T</c:mi></c:math>, including the thermal perturbative description of soft scatterings (<e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mo>−</e:mo><e:mi>t</e:mi><e:mo><</e:mo><e:mo>−</e:mo><e:msup><e:mi>t</e:mi><e:mo>*</e:mo></e:msup></e:math>) and a perturbative QCD-based calculation for hard collisions (<g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mo>−</g:mo><g:mi>t</g:mi><g:mo>></g:mo><g:mo>−</g:mo><g:msup><g:mi>t</g:mi><g:mo>*</g:mo></g:msup></g:math>). Within this soft-hard factorization model, we find that the full results of <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi>d</i:mi><i:mi>E</i:mi><i:mo>/</i:mo><i:mi>d</i:mi><i:mi>z</i:mi></i:math> behaves a mild sensitivity to the intermediate cutoff <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:msup><k:mi>t</k:mi><k:mo>*</k:mo></k:msup></k:math>, supporting the validity of the soft-hard approach within the temperature region of interest. We rederive the analytic formula for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mi>d</m:mi><m:mi>E</m:mi><m:mo>/</m:mo><m:mi>d</m:mi><m:mi>z</m:mi></m:math> in the high-energy approximation, <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:msub><o:mi>E</o:mi><o:mn>1</o:mn></o:msub><o:mo>≫</o:mo><o:msubsup><o:mi>m</o:mi><o:mn>1</o:mn><o:mn>2</o:mn></o:msubsup><o:mo>/</o:mo><o:mi>T</o:mi></o:math>, where <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:msub><q:mi>E</q:mi><q:mn>1</q:mn></q:msub></q:math> is the injected heavy quark energy and <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:msub><s:mi>m</s:mi><s:mn>1</s:mn></s:msub></s:math> is its mass. It is realized that the soft logarithmic contribution, <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:mi>d</u:mi><u:mi>E</u:mi><u:mo>/</u:mo><u:mi>d</u:mi><u:mi>z</u:mi><u:mo>∝</u:mo><u:mi>ln</u:mi><u:mo stretchy="false">(</u:mo><u:mo>−</u:mo><u:msup><u:mi>t</u:mi><u:mo>*</u:mo></u:msup><u:mo>/</u:mo><u:msubsup><u:mi>m</u:mi><u:mi>D</u:mi><u:mn>2</u:mn></u:msubsup><u:mo stretchy="false">)</u:mo></u:math>, arises from the <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"><y:mi>t</y:mi></y:math>-channel scattering off thermal partons, while the hard logarithmic term, <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"><ab:mi>d</ab:mi><ab:mi>E</ab:mi><ab:mo>/</ab:mo><ab:mi>d</ab:mi><ab:mi>z</ab:mi><ab:mo>∝</ab:mo><ab:mi>ln</ab:mi><ab:mo stretchy="false">[</ab:mo><ab:msub><ab:mi>E</ab:mi><ab:mn>1</ab:mn></ab:msub><ab:mi>T</ab:mi><ab:mo>/</ab:mo><ab:mo stretchy="false">(</ab:mo><ab:mo>−</ab:mo><ab:msup><ab:mi>t</ab:mi><ab:mo>*</ab:mo></ab:msup><ab:mo stretchy="false">)</ab:mo><ab:mo stretchy="false">]</ab:mo></ab:math>, stems from the <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML" display="inline"><gb:mi>t</gb:mi></gb:math>-channel scattering off thermal partons, and the one <ib:math xmlns:ib="http://www.w3.org/1998/Math/MathML" display="inline"><ib:mi>d</ib:mi><ib:mi>E</ib:mi><ib:mo>/</ib:mo><ib:mi>d</ib:mi><ib:mi>z</ib:mi><ib:mo>∝</ib:mo><ib:mi>ln</ib:mi><ib:mo stretchy="false">(</ib:mo><ib:msub><ib:mi>E</ib:mi><ib:mn>1</ib:mn></ib:msub><ib:mi>T</ib:mi><ib:mo>/</ib:mo><ib:msubsup><ib:mi>m</ib:mi><ib:mn>1</ib:mn><ib:mn>2</ib:mn></ib:msubsup><ib:mo stretchy="false">)</ib:mo></ib:math> comes from the <mb:math xmlns:mb="http://www.w3.org/1998/Math/MathML" display="inline"><mb:mi>s</mb:mi></mb:math>- and <ob:math xmlns:ob="http://www.w3.org/1998/Math/MathML" display="inline"><ob:mi>u</ob:mi></ob:math>-channel scattering off gluons. The sum of these contributions cancels the <qb:math xmlns:qb="http://www.w3.org/1998/Math/MathML" display="inline"><qb:msup><qb:mi>t</qb:mi><qb:mo>*</qb:mo></qb:msup></qb:math>-dependence as observed in the full result. The mass hierarchy is observed <sb:math xmlns:sb="http://www.w3.org/1998/Math/MathML" display="inline"><sb:mrow><sb:mi>d</sb:mi><sb:mi>E</sb:mi><sb:mo>/</sb:mo><sb:mi>d</sb:mi><sb:mi>z</sb:mi><sb:mo stretchy="false">(</sb:mo><sb:mtext>charm</sb:mtext><sb:mo stretchy="false">)</sb:mo><sb:mo>></sb:mo><sb:mi>d</sb:mi><sb:mi>E</sb:mi><sb:mo>/</sb:mo><sb:mi>d</sb:mi><sb:mi>z</sb:mi><sb:mo stretchy="false">(</sb:mo><sb:mtext>bottom</sb:mtext><sb:mo stretchy="false">)</sb:mo></sb:mrow></sb:math>. Our full results are crucial for a better description of heavy quark transport in QCD medium, in particular at low and moderate energy. We also calculate the energy loss by imposing the Einstein’s relationship. The related results appear to be systematically larger than that without imposing the Einstein’s relationship. Published by the American Physical Society 2024