Theoretical formalism of radiative jet energy loss in a finite size dynamical QCD medium

The computation of radiative energy loss in a finite-size QCD medium with dynamical constituents is a key ingredient for obtaining reliable predictions for jet quenching in ultrarelativistic heavy-ion collisions. We here present a theoretical formalism for the calculation of the first order in opacity radiative energy loss of a quark jet traveling through a finite-size dynamical QCD medium. We show that, while each individual contribution to the energy loss is infrared divergent, the divergence is naturally regulated once all diagrams are taken into account. Finite-size effects are shown to induce a nonlinear path-length dependence of the energy loss, recovering both the incoherent Gunion-Bertsch limit, as well as destructive Landau-Pomeanchuk-Migdal limit. Finally, our results suggest a remarkably simple general mapping between energy-loss expressions for static and dynamical QCD media.

Figure 1

Feynman diagrams $M_{1,0,2,C}$, $M_{1,0,2,L}$, and $M_{1,0,2,R}$ contribute to the radiative energy loss to the first order in the opacity. On each panel, the left (right) gray ellipse represents the source $J$, which at time $x_0$ ($x_5$) produces an on-shell jet with momentum $p_2$ ($p_5$). The large dashed circles (“blob”) represent effective HTL gluon propagators [14]. A cut gluon propagator with momentum $k$ and color $c$ corresponds to the radiated gluon. A cut gluon propagator with momentum $q$ and color $a$ corresponds to a collisional interaction with a parton in the medium. Specific time points are represented by $x_i$. The diagrams are calculated in light cone coordinate system, and $x_2^{+}-x_0^{+}<2L$ on the first and the third panels, as well as $x_3^{+}-x_5^{+}<2L$ on the first and the second panels, represent the condition that the distance between collisional interaction and jet production has to be smaller than the size $L$ of the medium. (Left) Middle and right panels present three possible cuts (central, left, and right, respectively) of the same 2-HTL Feynman diagram, all of which contribute to the first order in opacity radiative energy loss.

Figure 2

Feynman diagram $M_{1,0,1,C}$, contributing to the radiative energy loss to first order in opacity, labeled the same way as in Fig. 1.

Figure 3

Feynman diagrams $M_{1,0,2,C}$, $M_{1,0,2,L}$, and $M_{1,0,2,R}$ contributing to the radiative energy loss to first order in opacity, labeled the same way as in Fig. 1. The same figure is also presented as Fig. 1 in the main text and is repeated here for completeness.

Figure 4

Feynman diagrams $M_{1,0,3,C}$ and $M_{1,0,4,C}$ contributing to the radiative energy loss to first order in opacity, labeled the same way as in Fig. 1.

Figure 5

Feynman diagrams $M_{1,0,3,R}$ and $M_{1,0,4,L}$ contributing to the radiative energy loss to first order in opacity, labeled the same way as in Fig. 1.

Figure 6

Feynman diagrams $M_{1,0,5,R}$ and $M_{1,0,6,L}$ contributing to the radiative energy loss to first order in opacity, labeled the same way as in Fig. 1.

Figure 7

Feynman diagrams $M_{1,1,1,C}$ and $M_{1,1,2,C}$ contributing to the radiative energy loss to first order in opacity, labeled the same way as in Fig. 1.

Figure 8

Feynman diagrams $M_{1,1,1,L}$ and $M_{1,1,2,R}$ contributing to the radiative energy loss to first order in opacity, labeled the same way as in Fig. 1.

Figure 9

Feynman diagrams $M_{1,1,3,C}$ and $M_{1,1,4,C}$ contributing to the radiative energy loss to first order in opacity, labeled the same way as in Fig. 1.

Figure 10

Feynman diagrams $M_{1,1,3,R}$, $M_{1,1,3,L}$, $M_{1,1,4,R}$, and $M_{1,1,4,L}$ contributing to the radiative energy loss to first order in opacity, labeled the same way as in Fig. 1.

Figure 11

Feynman diagram $M_{1,2}$ contributing to the radiative energy loss to first order in opacity, labeled the same way as in Fig. 1.

Figure 12

Feynman diagram $M_{1,2}$ contributing to the radiative energy loss to first order in opacity, labeled the same way as in Fig. 1.

Figure 13

Tadpole Feynman diagrams $M_{t,R}$ and $M_{t,L}$, labeled the same way as in Fig. 1.

Figure 14

Ratio $\frac{J_{t,\max }({\overrightarrow{\mathbf{q}}}^2)}{v({\overrightarrow{\mathbf{q}}}^2)}$ is shown as a function of momentum $\left|\overrightarrow{\mathbf{q}}\right|$ for Debye mass $\mu =0.5$.