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 $O(\alpha^2_s)$ 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 [Phys. Rev. C 94, 054902 (2016)] both in the initial state as well as in the final state, along with interactions involving both in-medium Glauber gluons and quarks [Nucl. Phys. A 793, 128 (2007)], as well as coherence effects [Phys. Rev. C 105, 024908 (2022)]. As this study is a continuation of our work on medium-induced photon production [Phys. Rev. C 112, 025204 (2025)], the general factorization procedure for $e$-$A$ deep-inelastic scattering is still used. An incoming quark energy loss in the nuclear medium yields four possible scattering kernels $K_i$ with the following final states: (i) $q+g$, (ii) $g+g$, (iii) $q+\bar{q}'$, where the quark $q$ may have a flavor different from the antiquark $\bar{q}'$, and (iv) $q+q'$, where, again, $q$ may have a flavor different from $q'$. The collisional kernels include full phase factors from all non-vanishing diagrams and complete first-order derivative in the longitudinal direction ($k^-$) as well as second-order derivative in the transverse momentum ($k_{\perp}$) 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.