Soft-hard factorization of heavy-quark transport in QCD matter at finite chemical potential

We calculate the collisional energy loss and momentum diffusion coefficients of heavy quarks traversing a hot and dense QCD medium at finite quark chemical potential, <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mrow> <a:mi>μ</a:mi> <a:mo>≠</a:mo> <a:mn>0</a:mn> </a:mrow> </a:math> . The analysis is performed within an extended soft-hard factorization model that consistently incorporates the <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mi>μ</c:mi> </c:math> -dependence of the Debye screening mass <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:msub> <e:mi>M</e:mi> <e:mi>D</e:mi> </e:msub> <e:mo stretchy="false">(</e:mo> <e:mi>μ</e:mi> <e:mo stretchy="false">)</e:mo> </e:math> and of the fermionic thermal distribution functions. Both the energy loss and the diffusion coefficients are found to increase with <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:mi>μ</i:mi> </i:math> , with the enhancement being most pronounced at low temperatures where the chemical potential effects dominate the medium response. To elucidate the origin of this dependence, we derive analytic high-energy approximations in which the leading <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:mi>μ</k:mi> </k:math> -corrections appear as logarithmic terms: a soft logarithm <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mo>∼</m:mo> <m:msup> <m:mi>μ</m:mi> <m:mn>2</m:mn> </m:msup> <m:mi>ln</m:mi> <m:mo stretchy="false">(</m:mo> <m:mo stretchy="false">|</m:mo> <m:msup> <m:mi>t</m:mi> <m:mo>*</m:mo> </m:msup> <m:mo stretchy="false">|</m:mo> <m:mo>/</m:mo> <m:msubsup> <m:mi>M</m:mi> <m:mi>D</m:mi> <m:mn>2</m:mn> </m:msubsup> <m:mo stretchy="false">)</m:mo> </m:math> from <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"> <s:mi>t</s:mi> </s:math> -channel scattering off thermal gluonic excitations, and a hard logarithm <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"> <u:mo>∼</u:mo> <u:msup> <u:mi>μ</u:mi> <u:mn>2</u:mn> </u:msup> <u:mi>ln</u:mi> <u:mo stretchy="false">(</u:mo> <u:msub> <u:mi>E</u:mi> <u:mn>1</u:mn> </u:msub> <u:mi>T</u:mi> <u:mo>/</u:mo> <u:mo stretchy="false">|</u:mo> <u:msup> <u:mi>t</u:mi> <u:mo>*</u:mo> </u:msup> <u:mo stretchy="false">|</u:mo> <u:mo stretchy="false">)</u:mo> </u:math> from scattering off thermal quarks. In the complete result the dependence on the intermediate separation scale <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"> <ab:msup> <ab:mi>t</ab:mi> <ab:mo>*</ab:mo> </ab:msup> </ab:math> cancels, as required. We also confirm the expected mass hierarchy <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"> <cb:mrow> <cb:mo>−</cb:mo> <cb:mi>d</cb:mi> <cb:mi>E</cb:mi> <cb:mo>/</cb:mo> <cb:mi>d</cb:mi> <cb:mi>z</cb:mi> <cb:mo stretchy="false">(</cb:mo> <cb:mi>charm</cb:mi> <cb:mo stretchy="false">)</cb:mo> <cb:mo><</cb:mo> <cb:mo>−</cb:mo> <cb:mi>d</cb:mi> <cb:mi>E</cb:mi> <cb:mo>/</cb:mo> <cb:mi>d</cb:mi> <cb:mi>z</cb:mi> <cb:mo stretchy="false">(</cb:mo> <cb:mi>bottom</cb:mi> <cb:mo stretchy="false">)</cb:mo> </cb:mrow> </cb:math> at fixed velocity. Our findings demonstrate that finite chemical potential plays a significant role in heavy-quark transport and must be included in theoretical descriptions of heavy-flavor dynamics in baryon-rich environments, such as those probed in the RHIC Beam Energy Scan, and at FAIR and NICA.