Glauber quark and gluon contributions to quark energy loss at next-to-leading order and next-to-leading twist

The higher-twist formalism is used at <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mrow> <a:mi>O</a:mi> <a:mo>(</a:mo> <a:msubsup> <a:mi>α</a:mi> <a:mi>s</a:mi> <a:mn>2</a:mn> </a:msubsup> <a:mo>)</a:mo> </a:mrow> </a:math> to compute all possible medium-induced single-scattering emission kernels for an incoming highly energetic and virtual quark traversing the nuclear environment. The effects of the heavy-quark mass scale are taken into account [] both in the initial state as well as in the final state, along with interactions involving both in-medium Glauber gluons and quarks [], as well as coherence effects []. As this study is a continuation of our work on medium-induced photon production [], the general factorization procedure for <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mrow> <b:mi>e</b:mi> <b:mtext>−</b:mtext> <b:mi>A</b:mi> </b:mrow> </b:math> deep-inelastic scattering is still used. An incoming quark energy loss in the nuclear medium yields four possible scattering kernels <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:msub> <c:mi mathvariant="script">K</c:mi> <c:mi>i</c:mi> </c:msub> </c:math> with the following final states: (i) <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"> <e:mrow> <e:mi>q</e:mi> <e:mo>+</e:mo> <e:mi>g</e:mi> </e:mrow> </e:math> ; (ii) <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"> <f:mrow> <f:mi>g</f:mi> <f:mo>+</f:mo> <f:mi>g</f:mi> </f:mrow> </f:math> ; and (iii) <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"> <g:mrow> <g:mi>q</g:mi> <g:mo>+</g:mo> <g:msup> <g:mover accent="true"> <g:mi>q</g:mi> <g:mo>¯</g:mo> </g:mover> <g:mo>′</g:mo> </g:msup> </g:mrow> </g:math> , where the quark <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"> <i:mi>q</i:mi> </i:math> may have a flavor different from the antiquark <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"> <j:msup> <j:mover accent="true"> <j:mi>q</j:mi> <j:mo>¯</j:mo> </j:mover> <j:mo>′</j:mo> </j:msup> </j:math> ; and (iv) <l:math xmlns:l="http://www.w3.org/1998/Math/MathML"> <l:mrow> <l:mi>q</l:mi> <l:mo>+</l:mo> <l:msup> <l:mi>q</l:mi> <l:mo>′</l:mo> </l:msup> </l:mrow> </l:math> , where, again, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>q</m:mi> </m:math> may have a flavor different from <n:math xmlns:n="http://www.w3.org/1998/Math/MathML"> <n:msup> <n:mi>q</n:mi> <n:mo>′</n:mo> </n:msup> </n:math> . The collisional kernels include full phase factors from all nonvanishing diagrams and complete first-order derivative in the longitudinal direction ( <o:math xmlns:o="http://www.w3.org/1998/Math/MathML"> <o:msup> <o:mi>k</o:mi> <o:mo>−</o:mo> </o:msup> </o:math> ) as well as second-order derivative in the transverse momentum ( <p:math xmlns:p="http://www.w3.org/1998/Math/MathML"> <p:msub> <p:mi>k</p:mi> <p:mo>⊥</p:mo> </p:msub> </p:math> ) gradient expansion. Furthermore, in-medium parton distribution functions and the related jet transport coefficients have a hard transverse-momentum dependence (of the emitted quark or gluon) present within the phase factor.