# Jet thermalization in QCD kinetic theory

Yacine Mehtar-Tani<EMAIL_ADDRESS>c Soeren Schlichting
<EMAIL_ADDRESS>c,d Ismail Soudi<EMAIL_ADDRESS>Physics Department, Brookhaven National Laboratory, Upton, NY 11973, USARIKEN
BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973,
USAFakultät für Physik, Universität Bielefeld, D-33615 Bielefeld,
GermanyDepartment of Physics and Astronomy, Wayne State University, Detroit,
MI 48201.

###### Abstract

We perform numerical studies in QCD kinetic theory to investigate the energy
and angular profiles of a high energy parton - as a proxy for a jet produced
heavy ion collisions - passing through a Quark-Gluon Plasma (QGP). We find
that the fast parton loses energy to the plasma mainly via a radiative
turbulent gluon cascade that transport energy locally from the jet down to the
temperature scale where dissipation takes place. In this first stage, the
angular structure of the turbulent cascade is found to be relatively
collimated. However, when the lost energy reaches the plasma temperature is it
rapidly transported to large angles w.r.t. the jet axis and thermalizes. We
investigate the contribution of the soft jet constituents to the total jet
energy. We show that for jet opening angles of about 0.3 rad or smaller the
effect is negligible. Conversely, larger opening angles become more and more
sensitive to the thermal component of the jet and thus to medium response. Our
result showcase the importance of the jet cone size in mitigating or enhancing
the details of dissipation in jet quenching observables.

###### Keywords:

Perturbative QCD, Jet quenching, Kinetic theory

##  1 Introduction

Understanding how QCD jets produced in heavy ion collisions degrade their
energy in the presence of the Quark-Gluon Plasma (QGP) due to strong final
state interactions is one of the key theoretical and experimental challenges
of the heavy ion programs at RHIC and LHC. Not only do they constitute unique
probes of the transport properties of the quark gluon plasma, but they also
provide valuable information on non-equilibrium dynamics and the approach to
thermal equilibrium of non-Abelian plasmas.

Jet evolution in a QCD medium has been extensively studied in the past decade
to extend the picture of leading radiative parton energy loss to jets as
multi-partonic systems. The theoretical foundations of the former were laid in
the late 1990’s, by Zakharov, Baier, Dokshitzer, Mueller, Peigné, Schiff,
Baier:1996kr ; Zakharov:1996fv ; Baier:1996sk ; Zakharov:1997uu ;
Zakharov:1998sv ; Baier:2000mf ; Baier:2001yt  who showed that not only can an
energetic parton passing through dense QCD matter lose energy elastically but
also by medium-induced gluon radiation triggered coherently by multiple
scatterings during the quantum mechanical formation time of the fluctuation.
The resulting radiative spectrum is suppressed at high gluon frequency due to
the so-called Landau-Pomeranchuk-Migdal effect Migdal:1956tc  as compared to
incoherent Bethe-Heitler radiation Bethe:1934za . As a result, the average
energy loss was shown to scale quadratically with medium size in contrast with
the linear growth of elastic energy loss and therefore becomes the dominant
energy loss mechanisms for large QCD media. Because the jet spectra are
steeply falling as a function of the transverse momentum $p_{T}$ of the jet,
jet suppression as measured by the nuclear modification factor $R_{AA}$ is
sensitive to multiple soft gluon radiation that are produced copiously over
the extent of the medium length. Hence, multiple radiation can be assumed to
be independent in a first approximation, allowing for a Markovian description
of the primary gluon radiation Baier:1996kr ; Blaizot:2013vha . More
generally, the full medium-induced parton cascade can be studied in the
framework of kinetic theory, which enables an end-to-end description of jet
thermalization in the QGP Schenke:2009gb ; Kutak:2018dim ; Mehtar-Tani:2018zba
; Adhya:2019qse ; Blanco:2020uzy ; Schlichting:2020lef ; Blanco:2021usa .

Clearly, the main limitation of the kinetic description for the medium-induced
parton cascade is the fact that it does not account for the collinear QCD
shower that forms as a result of the high virtuality of the initial parton
associated with the hard scattering process at the origin of the jet
production. The probability for such a branching process is enhanced
logarithmically near the hard vertex and thus, most of the perturbative part
of the collinear QCD cascade occurs early before final state interactions take
place. The junction between these parton cascades of two kinds was poorly
understood up until recently. Many models and Monte Carlo event generators
available today use ad hoc prescriptions in order to combine the collinear
virtual cascade with the medium induced one, introducing substantial theory
uncertainties. The difficulty lies in the fact that the branching process is
of quantum nature, and therefore one may reasonably expect quantum
interference to play a crucial role in the build up of the parton shower. Some
progress has been made in this direction recently in a series of works that
investigated the radiative pattern of a system of two color correlated
partons, a.k.a., a parton antenna Mehtar-Tani:2011vlz ; Mehtar-Tani:2011lic ;
Armesto:2011kh ; Mehtar-Tani:2012mfa ; Calvo:2014cba ; Moldes:2015zya ;
Mehtar-Tani:2021fud . Instead of propagating independently in the plasma, if
the two final state parton system is collimated enough such that its typical
transverse size inside the plasma is smaller than the medium resolution scale
the di-parton system remains in a color coherent state and thus behaves
effectively as a single color charge. As a result, the two-parton antenna
loses energy as a single parton that carries the total charge of the system.
The generalization to an arbitrary number of partons in the collinear shower
was worked out in the so-called leading logarithmic approximation, where color
coherence constrains the phase space for independent partons in the jet
Mehtar-Tani:2017web ; Mehtar-Tani:2017ypq ; Caucal:2018dla .

Given these complications, the goal of this work is not to provide a
comprehensive study of all aspects of jet evolution in the plasma in view of
data comparison. Therefore, we won’t touch on the initial collinear cascade
and instead focus on dissecting the medium induced cascade initiated by a
single parton to gain more insight onto this phase of jet evolution and
provide guidance for future phenomenological studies.

Our motivation is twofold. In addition to studying in detail the 3D structure
of the medium-induced cascade, which includes energy and angular dependence of
the jet constituents, we aim at gauging the sensitivity of jet evolution in
the plasma to the soft medium scale in particular to the underlying
dissipative processes, whereby dissipation we mean the irreversible process of
depositing energy and momentum to the soft thermal bath. Indeed, jet quenching
observables are typically computed assuming that the medium induced cascade is
perturbative and that medium properties are fully encoded in transport
coefficients such as $\hat{q}$, which are then fitted to the data. However, it
is not a given that this is a pertinent approximation at RHIC and LHC
energies, where the details of energy dissipation may have a quantitative
impact on seemingly perturatively computable observables. This question is
reflected to a large extent by the ongoing discussion on the impact of medium
response or back reaction on jet quenching observables Casalderrey-
Solana:2004fdk ; Renk:2006mv ; Qin:2009uh ; Li:2010ts ; Tachibana:2014lja ;
Tachibana:2017syd ; Casalderrey-Solana:2020rsj ; Schlichting:2020lef ;
Yang:2022nei . It is therefore necessary to identify the domain of validity of
systematic perturbative computations for jet observables for precision
phenomenology in the future.

The in-medium parton cascade that resums to all orders of medium induced
branchings was analyzed analytically under certain simplifying assumptions and
approximations Blaizot:2012fh ; Blaizot:2013hx ; Blaizot:2015jea . A
remarkable feature of this cascade is the emergent turbulent nature of energy
transport from the leading parton energy to the infrared. In a wide range of
scales connecting the jet energy typically of the order of 100-500 GeV at LHC
and the plasma temperature $T\simeq 0.5$ GeV, known as the inertial window,
energy is transported with constant flux without any accumulation. This
feature is the hallmark of turbulence, whether it is due to quantum mechanical
splittings or related to classical nonlinear wave systems. In the inertial
range of frequencies, this process is characterized by self-similar dynamics
encoded in universal exponents; it is local in frequency space, that is, it
does not depend on the injection scale $p_{T}$ nor the dissipation scale $T$.
This represents an efficient mechanism for energy transport from large to low
frequency.

In this work we shall go beyond previous works by investigating the fate of
the jet four-momentum across the turbulent phase as well as the thermalization
processes in the framework of kinetic theory. By including the angular
dimension of the turbulent cascade into the effective kinetic description, we
will in particular address the question, how energy is efficiently transferred
from small to large angles w.r.t. to the jet axis.

The paper is organized as follows. In Sec. 2, we present the linearized
kinetic description and define the evolution equations which will be used to
study energy loss and thermalization of high-energy partons. Sec. 3 provides
an overview of our results for the energy loss and equilibration of a high
energy parton in a thermal medium. Subsequently, in Sec. 4 and 5, we
investigate the sensitivity of out-of-cone energy loss to the equilibration of
soft fragments, and the dependence on the energy scale of the hard parton. We
illustrate in Sec. 6, how our results in QCD kinetic theory can be employed in
phenomenological studies of parton and jet quenching. We summarize our most
important findings and conclude in Sec. 7.

##  2 Setup – Linearized kinetic equations

Figure 1: Schematic representation of the in-medium cascade of a hard parton
produced by a hard scattering, which subsequently interacts with the QGP. We
consider an initial hard parton with momentum along the $z$ direction which
loses its energy due to elastic scatterings with the medium and collinear
radiation. The energy distribution $D(x,\theta)$ represents the average energy
and momentum distributions of the fragments originating for the initial hard
parton. In this work, we investigate the average energy lost out of a jet cone
of size $R$ as a result of the cascading process induced by interactions with
the plasma constituents, depicted in red. Due to energy-momentum conservation,
recoil effects may also contribute to the in-cone energy of the jet.

We are interested in the problem of energy deposition of high energy partons
in a thermal QCD medium. We will neglect collinear branchings that may be
triggered by a nascent parton with large virtuality, hence, we only consider
initially on-shell quarks or gluons. Of course, a complete treatment of jet
evolution in a heavy ion collisions must include the latter parton cascade
that forms in vacuum as well. We leave this formidable problem for a future
work and focus here on the problem of energy loss and medium-induced
fragmentation of a single high energy parton passing through the Quark-Gluon
plasma (QGP), as illustrated schematically in Fig.1.

Owing to the fact that in a typical event the elementary elastic and inelastic
processes are local, i.e., their characteristic time scales are much shorter
than the system size, we may use kinetic theory to describe the evolution of
highly energetic partons traversing the QGP. Within this study, we will
describe the QGP as a weakly coupled system in thermal equilibrium at a
constant temperature $T$ (independent of spacial coordinates or time). Both
the thermal QGP as well as the hard partons are then described in terms of on-
shell particles by the phase-space distribution $f_{a}({\bm{p}},\mathbf{x},t)$
where $a=g,q_{f},\bar{q}_{f}$ labels gluons ($g$) and quarks/anti-quarks
($q_{f}/\bar{q}_{f}$ of a given flavor $f$)111We consider $N_{f}=3$ degenerate
flavors of massless quarks.. In this framework, the evolution of the systems
is described by the leading order QCD Boltzmann equation for the phase-space
distributions $f_{a}$, which includes both the number conserving
$2\leftrightarrow 2$ processes as well as effective collinear
$1\leftrightarrow 2$ processes, and reads Arnold:2002zm ; Arnold:2002ja ;
Ghiglieri:2015ala ; Du:2020dvp

$\left(\partial_{t}+\frac{{\bm{p}}}{|{\bm{p}}|}\cdot\mathbf{\nabla}_{x}\right)f_{a}({\bm{p}},\mathbf{x},t)=-C_{a}^{2\leftrightarrow
2}[\\{f_{i}\\}]-C_{a}^{1\leftrightarrow 2}[\\{f_{i}\\}]\;.$ (1)

In Eq. (1), the elastic collision rates at leading order in the QCD coupling
constant are given by Arnold:2002zm ; Arnold:2002ja ; Ghiglieri:2015ala ;
Du:2020dvp

$\displaystyle C_{a}^{2\leftrightarrow
2}[\\{f_{i}\\}]=\frac{1}{4|{\bm{p}}_{1}|\nu_{a}}\sum_{bcd}\int_{{\bm{p}}_{2}{\bm{p}}_{3}{\bm{p}}_{4}}\left|{\cal
M}^{ab}_{cd}({\bm{p}},{\bm{p}}_{2};{\bm{p}}_{3},{\bm{p}}_{4})\right|^{2}(2\pi)^{4}\delta^{(4)}(P+P_{2}-P_{3}-P_{4})$
$\displaystyle\times\big{\\{}f_{a}({\bm{p}})f_{b}({\bm{p}}_{2})(1\pm
f_{c}({\bm{p}}_{3})(1\pm
f_{d}({\bm{p}}_{4}))-f_{c}({\bm{p}}_{3})f_{d}({\bm{p}}_{4})(1\pm
f_{a}({\bm{p}}))(1\pm f_{b}({\bm{p}}_{2}))\big{\\}}\;,$ (2)

where ${\cal
M}^{ab}_{cd}({\bm{p}}_{1},{\bm{p}}_{2};{\bm{p}}_{3},{\bm{p}}_{4})$ stand for
the relevant spin and color averaged two-to-two tree-level matrix elements
(see. e.g. Arnold:2002zm ), while $\nu_{g}=2d_{A}=2(N_{c}^{2}-1)$ and
$\nu_{q_{f}}=\nu_{\bar{q}_{f}}=2d_{F}=2N_{c}$ denote the color and spin
degeneracy factors of gluons and quarks. We note that in a thermal QGP the
infrared divergences of the matrix elements ${\cal
M}^{ab}_{cd}({\bm{p}}_{1},{\bm{p}}_{2};{\bm{p}}_{3},{\bm{p}}_{4})$ associated
with small momentum exchange in the $t$ and $u$ channels at tree-level are
regularized by screening effects that can be accounted for by using HTL
propagators as discussed in Refs. Arnold:2002zm ; Arnold:2002ja ;
Arnold:2003zc ; Ghiglieri:2015ala . We will follow the approach of Ref.
Arnold:2003zc  and refer to App. B for additional details.

Since each gluon radiation comes at the expense of an additional power of the
coupling constant, inelastic processes are naively suppressed in plain
perturbation theory. However, this suppression is compensated by an
enhancement of (nearly) collinear splittings, such that even at leading order
these processes cannot be neglected. Since nearly collinear splittings can be
induced by coherent multiple splittings over the course of the formation time,
in fact it becomes necessary to resum their effects into an effective
collinear rate $\frac{d\Gamma^{a}_{bc}({\bm{p}},z)}{dz}$, which is integrated
over the typically small transverse momentum transfer generated during the
splitting, and most importantly accounts for the Landau, Pomeranchuk and
Migdal (LPM) effect Landau:1953um ; Migdal:1956tc ; Baier:1996kr ;
Baier:1996sk ; Zakharov:1996fv ; Zakharov:1997uu ; Zakharov:1998sv . By
following the approach by Arnold, Moore and Yaffe (AMY) Arnold:2002zm ;
Arnold:2002ja , to perform the re-summation of said multiple scatterings, one
obtains the collision integral for such effective $1\leftrightarrow 2$
collinear processes which reads Arnold:2002zm ; Arnold:2002ja ; Du:2020dvp

$\displaystyle C_{a}^{1\leftrightarrow 2}[\\{f_{i}\\}]=$
$\displaystyle\sum_{bc}\int_{0}^{1}dz\Bigg{\\{}\frac{1}{2}\frac{d\Gamma^{a}_{bc}({\bm{p}},z)}{dz}\Big{[}f_{a}({\bm{p}})(1\pm
f_{b}(z{\bm{p}}))(1\pm
f_{c}(\bar{z}{\bm{p}}))-f_{b}(z{\bm{p}})f_{c}(\bar{z}{\bm{p}})(1\pm
f_{a}({\bm{p}}))\Big{]}-\frac{\nu_{b}}{\nu_{a}}$
$\displaystyle\times\frac{1}{z^{3}}\frac{d\Gamma^{b}_{ac}(\frac{{\bm{p}}}{z}\;,z)}{dz}\Big{[}f_{b}\left(\frac{{\bm{p}}}{z}\right)(1\pm
f_{a}({\bm{p}}))\left(1\pm
f_{c}\left(\frac{\bar{z}}{z}{\bm{p}}\right)\right)-f_{a}({\bm{p}})f_{c}\left(\frac{\bar{z}}{z}{\bm{p}}\right)\left(1\pm
f_{b}\left(\frac{{\bm{p}}}{z}\right)\right)\Big{]}\Bigg{\\}}\;.$ (3)

Here the first term in the second line describes the $1\to 2$ collinear
splitting of a parent parton of momentum $p$ into two offspring of momenta
$zp$ and $\bar{z}p=(1-z)p$, while the second term in the first line describes
the reverse $2\to 1$ process. Similarly, the second term in the third line
describes the merging of two partons with momenta $p$ and $\bar{z}p/z$ into a
parton of momentum $p/z$, while the first term in the third line describes the
reverse splitting process.

Note that, since we are interested in the evolution of the hard parton in a
large medium, we shall neglect finite size effects in the calculation of the
medium induced splitting rates $\frac{d\Gamma^{a}_{bc}({\bm{p}},z)}{dz}$.
Stated differently, we will work in the limit of a constant or saturated
radiation rate $\frac{d\Gamma^{a}_{bc}({\bm{p}},z)}{dz}$ as computed for an
infinite medium Arnold:2002zm ; Arnold:2002ja ; Ghiglieri:2015ala , and we
follow the procedure outlined in App. A of Schlichting:2020lef  to calculate
the associated rates. Clearly, this is a good approximation, so long as the
quantum mechanical formation time $t_{f}$ of the splitting is much smaller
than the medium length $L$. Specifically, for the splitting of a highly
energetic parton $p\gg T$, this condition takes the form

$\displaystyle t_{f}(\omega)\simeq\sqrt{\frac{\omega}{\hat{q}}}\ll L\,,$ (4)

where $\omega=z(1-z)p$ and $\hat{q}$ is the quenching parameter (see the
following reviews Mehtar-Tani:2013pia ; Blaizot:2015lma  and references
therein for more details). Hence, this treatment is justified whenever

$\displaystyle\omega\ll\omega_{c}\equiv\hat{q}L^{2}\,,$ (5)

i.e. for very asymmetric splittings and/or for sufficiently small parent
momentum $p\ll\hat{q}L^{2}$ in a large medium where $\hat{q}L^{2}\gg T$. What
we call the quenching parameter $\hat{q}$, can be related to the mean
transverse momentum squared a parton acquires due to subsequent scatterings
with a medium of length $L$ as

$\displaystyle\hat{q}=\frac{\langle k_{\bot}^{2}\rangle}{L}\;.$ (6)

Specifically, at leading order of the perturbative expansion the coefficient
is given by Arnold:2002zm ; Ghiglieri:2015ala

$\displaystyle\hat{q}=C_{R}\hat{\bar{q}}=\frac{g^{2}TC_{R}}{2\pi}m_{D}^{2}\ln\frac{\mu_{q}}{m_{D}}\;,$
(7)

where we introduce the notation $\hat{\bar{q}}$ as the quenching parameter
stripped of its color factor $C_{R}$ of the hard parton, while
$m_{D}^{2}=g^{2}T^{2}(N_{c}/3+N_{f}/6)$ is the leading order Debye mass and
$\mu_{q}$ is used as an ultraviolet regulator, usually taken as the geometric
mean between the hard parton energy $E$ and the temperature $T$, i.e.,
$\mu_{q}\sim\sqrt{ET}$. While for a realistic medium of length $L=5{\rm fm}$
and a quenching factor $\hat{q}=1GeV/fm^{2}$, we find a typical scale
$\omega\ll 25GeV$, throughout this manuscript will extend beyond this phase-
space for simplicity.

Evidently, the QCD Boltzmann equation (1) is a non-linear kinetic equation
that, in its general form, can and has been used to study the thermalization
of far-from equilibrium QCD plasmas Kurkela:2018oqw ; Du:2020dvp . Since, we
are interested in the relaxation of hard partons that are characterized by
very small occupation numbers $\delta f_{a}({\bm{p}},\mathbf{x},t)\ll 1$, the
problem is actually simpler, as we may treat the disturbance due to the
presence of the hard parton as a perturbation on top of thermal QGP. By
treating the thermal QGP as static and homogeneous for simplicity, we can then
decompose the phase space distribution as

$\displaystyle f_{a}({\bm{p}},\mathbf{x},t)=$ $\displaystyle n_{a}(p)+\delta
f_{a}({\bm{p}},\mathbf{x},t)\;,$ (8)

where $n_{a}(p)$ stands for the equilibrium Bose-Einstein distribution
$\left(n_{g}(p)=\frac{1}{e^{p/T}-1}\right)$ for gluons and respectively the
Fermi-Dirac distribution $\left(n_{q}(p)=\frac{1}{e^{p/T}+1}\right)$ for
quarks and anti-quarks. We then linearize Eq. (1) w.r.t. the perturbation
$\delta f_{a}({\bm{p}},\mathbf{x},t)\ll n_{a}(p)$ induced by the initial
presence of a hard parton. Since the explicit form of the linearized kinetic
equations turns out to be rather lengthy, we refrain from reproducing them in
this section and instead refer the interested reader to App. B for the
implementation of the elastic collision integral, while the inelastic
radiation collision integrals can readily be found in Schlichting:2020lef .

While previous works Schlichting:2020lef  have employed essentially the same
QCD kinetic setup222Note that in contrast to the present study, the elastic
processes in Schlichting:2020lef  were treated in the small-angle
approximation. to study the degradation of the energy of hard partons, we will
go beyond that and as illustrated in Fig. 1 study the fully differential
momentum distribution of jet fragments in energy ($xE$) and polar angle
($\theta$) w.r.t. the initial parton momentum. We emphasize, that the
extension of the previous study to the angular distribution is crucial in the
context of jet physics since by definition a jet is defined by a collection of
particles that fall within a cone defined by an opening angle $R$. By
explicitly keeping track of the inclusive momentum distribution of jet
fragments, the problem of jet energy loss then reduces to investigating the
mechanism by which energy carried by fragments “leaks” out of the jet cone as
illustrated in Fig. 1, which is the central objective of this study.

Before we discuss this further, we wish to make contact with standard
notations for the distribution of jet fragments in terms of the momentum
fraction $z$ and the angle $\theta$ formed by the initial parton momentum
${\bm{p}}_{\rm in}$ and the measured final parton ${\bm{p}}$. By following
previous works Blaizot:2013vha ; Blaizot:2014ula ; Blaizot:2014rla , the in-
medium evolution of the jet shower is then conveniently studied in terms of
the differential energy distribution $D_{a/jet}$ of particles $a$ inside a jet

$\displaystyle D_{a/\rm jet}(x,\theta,t)\equiv$ $\displaystyle x\frac{{\rm
d}N_{a}}{{\rm d}x{\rm d}\cos\theta}$ $\displaystyle=$
$\displaystyle\nu_{a}\int
d^{3}\mathbf{x}\int\frac{d^{3}p}{(2\pi)^{3}}~{}\frac{|{\bm{p}}|}{E}~{}\delta\Big{(}\frac{|{\bm{p}}|}{E}-x\Big{)}\delta\Big{(}\frac{{\bm{p}}\cdot\hat{e}_{z}}{xE}-\cos\theta\Big{)}\delta
f_{a}({\bm{p}},\mathbf{x},t)\;,$ (9)

where $E$ is the total jet energy and $\hat{e}_{z}=\frac{{\bm{p}}_{\rm
in}}{|{\bm{p}}_{\rm in}|}$ is the unit vector along the initial parton
direction, and we have integrated out the azimuth angle $\phi$ as well as the
spatial coordinate $\mathbf{x}$. By integrating over polar angle
$\cos(\theta)$, we then recover the energy distribution studied in earlier
works Kutak:2018dim ; Mehtar-Tani:2018zba ; Adhya:2019qse ; Blanco:2020uzy ;
Schlichting:2020lef ; Blanco:2021usa

$\displaystyle D_{a/\rm jet}(x,t)=\int_{-1}^{1}{\rm d}\cos(\theta)~{}D_{a/\rm
jet}(x,\theta,t)\,,$ (10)

which as a result of energy conservation is normalized to unity, i.e.

$\displaystyle\sum_{a}\int_{0}^{\infty}{\rm d}x\int_{-1}^{1}{\rm
d}\cos\theta~{}D_{a/\rm jet}(x,\theta,t)=\sum_{a}\int_{0}^{\infty}{\rm
d}x~{}D_{a/\rm jet}(x,t)=1\,.$ (11)

at any time.

##  3 Energy loss and equilibration

Figure 2:  Evolution of the energy distribution
$D(x,\theta)=\sum_{a}D_{a/g-\rm jet}(x,\theta)$ of all species for a gluon jet
with initial energy $E=500~{}T$ at different times $t=1.1,4.1,9.8$ fm/c as a
function of momentum fraction $x=\frac{p}{E}$ and angle $\theta$. The bottom
right panel shows the asymptotic equilibrium distribution. White lines along
the angle axis represent constant angles from small to large
$\theta=0.16,0.32,0.62$, while the one along the momentum fraction axis
represent constant momentum fraction of $x=2\pi T/E$.

We are now equipped with the linearized kinetic equations that are necessary
to study the problem of energy loss and equilibration of hard partons or jets
inside a thermal QGP. Starting from an initial condition that represents a
single hard quark or gluon moving in the $+z$ direction

$\displaystyle D^{(0)}_{a/b-\rm jet}(x,\theta)\equiv D_{a/b-\rm
jet}(x,\theta,t=0)=\delta_{ab}\delta(1-x)\,\delta(1-\cos\theta)\,,$ (12)

we aim at a quantitative understanding of the non-equilibrium evolution of the
system from the initial condition in Eq. (12) all the way towards thermal
equilibrium, where at asymptotically late times the only information retained
from the jet is associated with the excess of the conserved energy $\delta
E=E$, longitudinal momentum $\delta P_{z}=E$ and valence charge $\delta
N_{f}=0$ for a gluon jet and $\delta N_{f}=1$ for a quark jet.

We will consider two types of initial conditions, corresponding to a highly
energetic gluon or quark, which we will refer to as $g-\rm jet$ and $q-\rm
jet$ respectively. In practice, in our numerical simulations we will model the
delta functions in the initial condition by a narrow Gaussian of width
$\sigma=10^{-3}/\sqrt{2}$, such that e.g. for a gluon jet

$\displaystyle D_{g/g-\rm
jet}(x,\theta)=\frac{1}{N}\exp\left\\{-\frac{(1-x)^{2}+x^{2}\sin^{2}\theta}{2\sigma^{2}}\right\\}\;,$
(13) $\displaystyle D_{q/g-\rm jet}(x,\theta)=0\;,\qquad D_{\bar{q}/g-\rm
jet}(x,\theta)=0\;,$ (14)

where the normalization factor is given by

$N=\int_{-1}^{1}{\rm
d}{\cos\theta}\int_{0}^{\infty}dx~{}e^{-\frac{(xE-E)^{2}+x^{2}E^{2}(1-{\cos\theta}^{2})}{2\sigma^{2}}}\;\;,$
(15)

such that the normalization condition in Eq. (11) remains valid. When
considering quark jets, we will for simplicity assume an equal abundance of
$u,d,s$ light flavor jets, such that the initial conditions for the evolution
are also degenerate in flavor, as discussed in more detail in App. B. If not
stated otherwise, throughout this section, we will present results for the
evolution of gluon jets with energy $E=500~{}T$ and consider $g=2$ which
corresponds to a realistic coupling strength
$\alpha_{s}=\frac{g^{2}}{4\pi}\simeq 0.3$ at RHIC an LHC energies.

When studying the time evolution of the system, it is useful to recall that
the medium induced cascade, is characterized by a single time scale
Blaizot:2012fh ; Blaizot:2013hx ; Blaizot:2015jea

$t_{\rm
th}\sim\frac{1}{\alpha_{s}}t_{f}(E)\sim\frac{1}{\alpha_{s}}\sqrt{\frac{E}{\hat{q}}}\,.$
(16)

Its physical meaning is that of the time it takes for a hard parton to undergo
a radiative break-up and thermalize in the plasma, and it is also sometimes
referred to as the stopping time. We may use this characteristic time scale to
adopt the dimensionless time variable

$\tau\equiv\frac{t}{t_{\rm th}}=g^{4}T\sqrt{\frac{T}{E}}t\;,$ (17)

where we have used the fact that for a thermal plasma $\hat{q}\propto
g^{4}T^{3}$. Alternatively, to facilitate a connection with heavy ion
phenomenology, our results will also be presented in terms of the physical
time in units $\rm fm/c$ by using a thermal medium with temperature $T=200$
MeV to set the physical scale. However, we shall refrain from a detailed
phenomenological study since our evolution does not include important effects
such as finite size effects for the medium-induced splitting rates, the space-
time evolution of the medium and the collinear vacuum cascade.

Before delving into the detailed discussion of our numerical results, let us
first highlight and summarize the generic features of the time evolution of
the energy and angular distribution of the jet fragments in a 3D plot. For
this purpose, the evolution of the total energy distribution of all species

$\displaystyle D(x,\theta,t)=\sum_{a}D_{a/\rm jet}(x,\theta,t)$ (18)

is presented in Fig. 2 as a function of momentum fraction $x=p/E$ and polar
angle $\theta$ at three different evolution times $t=1.1,4.1,9.8{\rm fm/c}$
and compared to the equilibrium distribution (asymptotic behavior) in the
bottom right panel of Fig. 2.333Note that we use an unconventional plotting
scheme for the $z$-axis representing the distribution $D(x,\theta)$, with a
linear scale from the values $D(x,\theta)=-10$ to $D(x,\theta)=10$ and a
logarithmic scaling beyond for $|D(x,\theta)|>10$. The negative values of the
distribution are represented as a dip in the $z$-axis below $z=0$, as opposed
to conventional logarithmic scales.

Starting at early times in the top left panel, the initial hard parton peak is
clearly visible at momentum fraction $x\sim 1$ and small angles $\theta\sim
0$, however collinear radiation leads to a sizable population of the
intermediate to soft sector $x<1$ in the collinear region with no noticeable
broadening to large angles.

While the hard momentum sector $x\sim 1$ continues to be depleted by collinear
radiation, the soft fragments undergo a successive broadening by which the
energy deposited in the soft sector $x\lesssim 2\pi T/E$ at small angles is
transported to larger angles out of the jet cone $R<1$. By intermediate times
$t=4.1$fm/c the angular distribution in the soft sector thus extends all the
way to large angles $\theta\sim 1$. Conversely, only a minimal broadening of
the hard partons is achieved, such that only the broadening near the medium
scales $x\lesssim 2\pi T/E$ contributes to the energy at large angles.

Eventually, at late times $t=9.8$fm/c, we observe that the hard parton peak is
mostly depleted and only the remnants of the energy at intermediate scales
continues to equilibrate with the medium, via the same thermalization
mechanism. Due to the continuous influx of energy from these semi-hard
fragments, the soft sector $x\lesssim 2\pi T/E$ also remains out of
equilibrium, featuring an enhancement of the energy distribution at narrow
angles. Nevertheless, by this time the peak in the soft sector has broadened
considerably to angles $\theta\sim 1$, as energy deposited into the soft
sector at an earlier stage has had sufficient amount of time to undergo
kinetic equilibration. Notably, at very large angles $\theta\gtrsim 1.3$,
there is also a negative contribution to the energy distribution, which can be
associated with recoiling thermal particles that due to their interaction with
the nearly collinear constituents get dragged along the jet axis. While
initially, this is a rather small effect, this negative contribution becomes
increasingly important as more and more nearly collinear fragments $\theta\ll
1$ drag along the constituents of the thermal QGP. Stated differently, the
nearly collinear cascade deposits an almost equal amount of energy $\delta E$
and longitudinal momentum $\delta P^{z}$ in the soft sector, which over the
course of the thermalization of the soft sector is then redistributed among
the soft fragments, leading to an increase of the number of particles moving
parallel and a decrease of the number of particles moving anti-parallel to the
jet axis. Such a depletion of the away side is also referred to as the
diffusion wake valley and has been subject to ongoing extensive investigation
using the hydrodynamic response to jet energy loss Casalderrey-Solana:2004fdk
; Renk:2006mv ; Qin:2009uh ; Li:2010ts ; Tachibana:2014lja ; Tachibana:2017syd
; Casalderrey-Solana:2020rsj ; Yang:2022nei  (see Cao:2020wlm  for a review).

Eventually, at asymptotically late times all hard fragments have thermalized
inside the medium, and the only leftover signal is due to the change of the
conserved energy $\delta E$, momentum along the jet axis $\delta P_{z}$ and
valence charge $\delta N_{f}$ of the jet, which lead to a change of the
temperature $T$, the flow velocity $u^{z}$ and chemical potential $\mu_{f}$.
By matching the changes in the thermodynamic variables as detailed in App. C,
the corresponding equilibrium energy distribution for each species can be
determined as

$\displaystyle
D_{a/jet}^{(eq)}(x,\theta)=\frac{\nu_{a}}{(2\pi)^{2}}(xE)^{3}\left[\frac{1}{4}\frac{\delta
E}{e(T)}T\partial_{T}+\frac{3}{4}\frac{\delta
P^{z}}{e(T)}\partial_{u^{z}}+\frac{\delta
N_{f}}{\chi(T)T}\partial_{\frac{\mu_{f}}{T}}\right]\left.n_{a}\Big{(}p_{\mu}u^{\mu}\Big{)}\right|_{u^{z}=0}$
(19)

and the equilibrium energy distribution of all species is given by

$\displaystyle
D^{(eq)}_{a}(x,\theta)=\sum_{a}\frac{\nu_{a}}{(2\pi)^{2}}\frac{(xE)^{4}}{4e(T)}~{}\frac{E}{T}~{}\left[1+3\cos(\theta)\right]n_{a}(xE)(1\pm
n_{a}(xE))$ (20)

where $\chi(T)$ and $e(T)$ are the quark flavor susceptibility and energy
density as a function of temperature, and we assumed the equation of state of
an ultra-relativistic ideal gas. By following the evolution up to very late
times, we find that the above equilibrium distribution, which features an
increase in temperature and a boost along the jet axis, is correctly recovered
in our numerical simulations as can be seen in the bottom right panel of Fig.
2.

Summarizing the results in Fig. 2, we find that the re-distribution of energy
outside a jet cone is governed by a two-step scenario, where

  * 1)

Hard partons with $x\gg 2\pi T/E$ remain collinear but undergo a radiative
break-up due to multiple successive splittings, that deposits energy and
momentum into the soft sector $x\lesssim 2\pi T/E$ via a turbulent cascade

  * 2)

Soft fragments with $x\lesssim 2\pi T/E$ thermalize, thereby transporting the
continuous influx of energy and momentum due to the collinear cascade out to
large angles $\theta\sim 1$. However, due to the continuous influx of energy,
the soft sector remains out of equilibrium throughout the entire evolution.

Below, we further scrutinize the evolution of the energy and angular spectra
during this process and explore the consequences of this dynamics for parton
and jet energy loss.

###  3.1 Momentum structure of the cascade

Figure 3: Evolution of the energy distribution as a function of the momentum
fractions $x=\frac{p}{E}$ with a decomposition into different angular regions
as described in Eq. (21). The shaded regions help to visualize the change in
the energy distribution as function of the jet cone size $R$.

Now that we have established the general picture, we will further investigate
the energy spectra. In order to make explicit contact with QCD jets, we shall
explore the structure of the energy distribution by considering the angular
integrated energy distribution that measures the energy that remains inside a
cone of opening angle $R$,

$D(x,R,t)=\sum_{a}\int_{\cos R}^{1}{\rm d}{\cos\theta}~{}D_{a/\rm
jet}(x,\theta,t)\;,$ (21)

where we integrate the energy distribution up to a cone of size $R$ w.r.t. the
momentum of the initial hard parton. We show in Fig. 3 the full distribution
$D(x,t)\equiv D(x,R=\pi,t)$ integrated over all angles, together with the
distributions $D(x,R,t)$ for different angular cones of size
$R=0.16,0.32,0.62$ at three different times $t=1.1,4.1,9.8$ of the evolution
and at asymptotically late times.

Before we turn to the discussion of our numerical results, we note that the
fully integrated distribution $D(x,t)$, has been investigated in earlier
kinetic evolution studies by including in-medium collinear radiation only
Jeon:2003gi ; Blaizot:2013hx ; Blaizot:2015jea ; Mehtar-Tani:2018zba  or by
also including elastic scattering with the medium Iancu:2015uja ;
Schlichting:2020lef . One of the most important conclusions of these studies
is that the energy loss is driven by the radiative break-up of hard fragments
due to successive in-medium splittings. Since the radiative break-up due to
multiple (quasi-democratic) splittings is a self-similar process, it leads to
a so-called _inverse energy cascade_ , i.e. a turbulent cascade that
transports energy all the way from very high $(\sim E)$ to very low $(\sim T)$
momentum scales, which can be studied using the theory of weak wave turbulence
Blaizot:2013hx ; Blaizot:2015jea ; Mehtar-Tani:2018zba ; Schlichting:2020lef .
By focusing on the inertial range of intermediate momentum fractions $T/E\ll
x\ll 1$ where radiative processes dominate, the evolution equations can be
approximated as Mehtar-Tani:2018zba ; Schlichting:2020lef

$\displaystyle\partial_{t}D_{a/\rm jet}(x,\theta,t)\simeq$
$\displaystyle\sum_{bc}\int_{0}^{1}dz~{}\left[\frac{d\Gamma^{b}_{ac}}{dz}\Big{(}\frac{xE}{z},z\Big{)}D_{b/\rm
jet}\Big{(}\frac{x}{z},\theta,t\Big{)}-\frac{1}{2}\frac{d\Gamma^{a}_{bc}}{dz}\Big{(}xE,z\Big{)}D_{a/\rm
jet}\Big{(}x,\theta,t\Big{)}\right]\;,$ (22)

where we dropped all sub-leading elastic and inelastic processes, as well as
all Bose enhancement and Fermi suppression factors. By employing the so called
harmonic-oscillator approximation for the high-energy rates Arnold:2008zu ;
Arnold:2008iy ; Schlichting:2021idr

$\displaystyle\frac{d\Gamma^{a}_{bc}}{dz}\Big{(}xE,z\Big{)}\simeq\frac{\alpha_{s}}{2\pi}P_{ba}(z)~{}\sqrt{\frac{\hat{\bar{q}}}{z(1-z)xE}}~{}\sqrt{zC_{c}^{R}+(1-z)C_{b}^{R}-z(1-z)C_{a}^{R}}$
(23)

where $P_{ba}(z)$ denote the leading order QCD splitting functions and
$C_{a/b/c}^{R}$ denote the Casimir of the representation of particles $a,b,c$,
i.e. $C_{a}^{R}=C_{F}=\frac{N_{c}^{2}-1}{2Nc}$ for quarks
$a=q_{f},\bar{q}_{f}$ and $C_{a}^{R}=C_{A}=N_{c}$ for gluons $a=g$, the
evolution equations reduce to the ones in Mehtar-Tani:2018zba  and the
turbulent power spectrum can be determined as $D_{a/jet}(x)\propto 1/\sqrt{x}$
within the inertial range of momentum fractions $T/E\ll x\ll 1$ Blaizot:2013hx
; Blaizot:2015jea ; Mehtar-Tani:2018zba . Since the in-medium splittings are
concentrated in the collinear region, we can expect to recover this turbulent
spectrum at intermediate momentum fractions $T/E\ll x\ll 1$ in the small angle
region $\theta\ll 1$. However, it is also clear that for smaller momentum
fractions $x\lesssim T/E$ the effective description in Eq. (3.1) breaks down,
as all kinds of elastic and inelastic processes become equally relevant for
the kinetic evolution and ultimately lead to the thermalization of the soft
sector.

Starting at early times ($t=1.1{\rm fm}/c$) shown in the top left panel of
Fig. 3, the energy distribution is strongly peaked around the energy of the
original hard parton ($x\sim 1$), and at intermediate momentum fractions
$T/E\ll x\ll 1$ shows the expected turbulent $\sim 1/\sqrt{x}$ spectrum due to
medium-induced radiation. Since the radiative emission is nearly collinear,
the hard $(x\sim 1)$ and intermediate $(1\gg x\gg T/E)$ fragments largely
remain confined within a narrow angular cone $(\theta\leq 0.16)$. Even at the
later stages of the evolution ($t=4.1,9.8{\rm fm}/c$) when the hard ($x\sim
1$) peak has significantly depleted, the hard and intermediate partons largely
remain within such a narrow cone, except for rare large angle scatterings that
will be discussed further below.

Energy lost by the hard fragments accumulates in the soft sector starting at
momentum fractions $x\lesssim 4\pi T/E$, where the energy distribution shows a
pronounced peak. Since elastic scattering are most efficient at these soft
scales, they lead to sizable broadening out to cone-sizes $R\gtrsim 0.3$
already at early times. Over the course of the evolution, broadening of the
soft sector continues to drive energy out of the collinear region to larger
angles. By the time $t=9.8{\rm fm}/c$, most of the energy resides in the soft
sector ($x\lesssim 4\pi T/E$) and has been transported to large angles between
$R=0.3$ and $R=0.6$ cone size. Eventually, once the residual hard fragments
have lost all of their energy, the evolution reaches the equilibrium
distribution in Eq. (20), whose angular structure will be further discussed
below.

###  3.2 Angular structure of the cascade

Figure 4: Evolution of the angular distribution in different momentum regions,
as described by Eq. (24).

We will now investigate the angular structure of the energy cascade and study
the redistribution of energy out to large angles. Our results are compactly
summarized in Fig. 4 where we analyze the angular structure of the energy
distribution by dividing the spectrum into three energy bins corresponding to
the large momentum sector ($xE\geq 90~{}T$) in the left panel, intermediate
scales ($10~{}T\leq xE\leq 90~{}T$) in the middle and the soft sector ($xE\leq
10~{}T$) in the right. Each panel shows the integrated energy distribution as
follows

$D(\theta,t)|_{x_{\rm min}}^{x_{\rm max}}=\sum_{a}\int_{x_{\rm min}}^{x_{\rm
max}}{\rm d}x~{}D_{a/\rm jet}(x,\theta,t)\;.$ (24)

Starting with the hard sector, we observe that the peak is only marginally
broadened as it is depleted from early to late times and energy is transferred
to intermediate and lower momentum scales $(x\leq 90T/E)$. Similar effects are
seen at intermediate scales, where the distribution displays a rapid fall off
at large angles while the peak is depleted from the collinear region. Due to
the constant influx of energy at small angles $\theta\ll 1$, the energy
distribution in the soft sector also remains peaked at small angles;
nevertheless, over the course of the evolution more and more of the energy in
the soft sector is redistributed out to large angles $\theta\sim 1$ due to
elastic interactions. One also observes, that throughout the entire evolution
the energy distribution in the soft sector features a negative away side peak,
which accounts for the depletion of thermal QGP particles moving in the
opposite direction of the jet that is necessary to account for momentum
conservation.

Summarizing the results in Fig. 4, we observe that for typical cone-sizes
$\theta\sim 0.3$ the angular broadening at high and intermediate scales is a
sub-leading effect. Instead, the re-distribution of energy to large angles
proceeds predominantly via the collinear radiative break-up of the hard
parton, with subsequent broadening of the soft sector where the elastic
processes are more efficient in redistributing energy out to large angles.

Figure 5: Evolution of the angular distribution for hard particles with
momentum fraction $x>30~{}T/E$. The gray line that represents the expected
Coulomb tail $1/\theta^{4}$ at large angles is shown for reference.

Despite the fact that the broadening of high momentum particles provides a
subleading effect for re-distribution of energy to large angles, it is
interesting to investigate this phenomenon more closely, as is done in Fig. 5,
where we show the angular distribution $D(\theta,t)$ of high-energy fragments
$x>30T/E$ at various times of the evolution. Starting from early times where
the distribution is extremely narrow, one observes a simultaneous broadening
and depletion of the peak along with the emergence of a power-law tail at
intermediate angles, which can be attributed to rare large angle scatterings
Kurkela:2014tla  – as originally discussed by Molière theory for QED
Moliere:1947zz . Specifically for high energy fragments $x\gg T/E$, the
effects of elastic collisions on the evolution of the energy distribution
$D_{a/\rm jet}(x,{\bf q}_{\bot},t)$ as a function of energy fraction $x$ and
transverse momentum ${\bf q}_{\bot}$

$\displaystyle D_{a/\rm jet}(x,{\bf q}_{\bot},t)=\nu_{a}\int
d^{3}\mathbf{x}\int\frac{d^{3}p}{(2\pi)^{3}}~{}\frac{|{\bm{p}}|}{E}~{}\delta\Big{(}\frac{|{\bm{p}}|}{E}-x\Big{)}\delta^{(2)}\Big{(}{\bf
q}_{\bot}-{\bf p}_{\bot}\Big{)}\delta\bar{f}_{a}({\bm{p}},\mathbf{x},t)\;.$
(25)

Can be approximately described by Blaizot:2014rla ; Mehtar-Tani:2018zba ;
Schlichting:2020lef

$\displaystyle\partial_{t}D_{a/jet}(x,{\bf p}_{\bot},t)\simeq\int_{{\bf
q}_{\bot}}C_{a}({\bf q}_{\bot})\left[D_{a/jet}(x,{\bf p}_{\bot}-{\bf
q}_{\bot},t)-D_{a/jet}(x,{\bf p}_{\bot},t)\right]$
$\displaystyle+\sum_{bc}\int_{0}^{1}dz~{}\left[\frac{1}{z^{2}}\frac{d\Gamma^{b}_{ac}}{dz}\Big{(}\frac{xE}{z},z\Big{)}D_{b/jet}\Big{(}\frac{x}{z},\frac{{\bf
p}_{\bot}}{z},t\Big{)}-\frac{1}{2}\frac{d\Gamma^{a}_{bc}}{dz}\Big{(}xE,z\Big{)}D_{a/jet}\Big{(}x,{\bf
p}_{\bot},t\Big{)}\right]\;,$ (26)

where as in Eq. (3.2), the first line describes the effects of radiative
emissions. The second line in Eq. (3.2) describes transverse momentum
broadening due to elastic interactions (while neglecting long momentum and
energy transfer as well as species changing processes), as described by the
collisional broadening kernel $C_{a}({\bf q}_{\bot})$, which has been
determined in Arnold:2008vd ; Aurenche:2002pd  for leading order QCD kinetic
theory. Within this framework, the rare large angle scatterings correspond to
a $C_{a}({\bf q}_{\bot})\sim 1/{\bf q}_{\bot}^{4}$ power-law tail of the
collisional broadening kernel, for momentum transfers $|{\bf q}_{\bot}|\gg
m_{D}$. When expressed in terms of angular variables, the momentum transfer is
given by $|{\bf q}_{\bot}|=xE~{}\sin\theta\simeq xE~{}\theta$, such that the
angular distribution $D(x,\theta)$ can be expected to follow a $\sim
1/\theta^{4}$ behavior at intermediate angles where large angle scatterings
dominate. While small angle scatterings lead to a diffusive broadening at
small angles, we observe in Fig. 5 that the $\sim 1/\theta^{4}$ power law
indicated by the gray dashed line dominates the large angle region up to
angles $\theta\sim 0.4$, beyond which the energy distribution drops sharply as
energy and momentum conservation further restrict the available phase space
for large angle scatterings.

##  4 Sensitivity to soft fragments

Based on our analysis, we noted that only the soft fragments of the medium
induced shower escape to large angles, while partons with momenta larger than
the temperature stay mostly collinear throughout the evolution. In this
section, we will further explore the consequences of this behavior by
comparing the energy carried by soft and hard fragments inside the cone. This
question is of both conceptual and phenomenological relevance. Conceptually,
in the standard approach to jet quenching the degradation of energy due to the
radiative break-up cascade can be discussed without an explicit microscopic
description of the dissipation of energy in the thermal medium. This is due to
the so-called inviscid property of turbulence that states that the turbulent
cascade transports energy from high to low momentum scales, regardless of the
detailed properties of the hard source and the dissipation of energy in the
thermal medium. We recall that the effect of dissipation is to stop the
turbulent cascade by causing the accumulation of energy around scales $\sim
2\pi T$ and its subsequent thermalization. Hence, when a proper microscopic
description of the dynamics of the soft sector is included, the turbulent
cascade may form only well above the temperature scale.

We may therefore wonder to what extent the details of the kinetic
thermalization process and the resulting deformation of the spectrum affect
energy loss, as compared to a purely high-momentum description that ignores
detailed balance at scales $\lesssim 2\pi T$. This question is crucial as far
as the predictive power of the theory is concerned. While the QCD coupling
constant can be treated as small for the radiative break-up of high-energy
fragments, such that perturbation theory can be justified and systemically
improved, the dynamics near the temperature scale may in general turn out to
be non-perturbative in which case our kinetic description can at best be
viewed as a perturbative model of the soft sector.

Beyond the conceptual importance of clarifying the role of low scale
thermalization in jet quenching, this question is also of considerable
phenomenological importance. In particular, it would be important to quantify
the sensitivity of jet observables to the dynamics of soft thermalization, in
order to devise jet observables that can be accurately calculated within a
perturbative QCD approach on the one hand and on the other hand explore means
to study QCD thermalization via jet quenching in heavy-ion collisions.

While our kinetic framework is in principle well suited to provide an
understanding of the energy recovery due to medium response and recoil, which
constitutes a highly active area of research in QCD be quenching Casalderrey-
Solana:2004fdk ; Renk:2006mv ; Qin:2009uh ; Li:2010ts ; Tachibana:2014lja ;
Tachibana:2017syd ; Casalderrey-Solana:2020rsj ; Yang:2022nei , a detailed
phenomenological analysis is well beyond the scope of the present work.
Nevertheless, we will attempt to provide some new insights into this question
by comparing the in-cone energy with and without the soft modes that fall
below $2\pi T$, which can be viewed as an ad hoc  energy cutoff of the
radiative break-up cascade.

Figure 6: Evolution of the energy inside the cone ($\theta\leq R$) for gluon
(left) and quark (right) initial jets with $E=100T$. Full line-points
represent the full momentum region as in Eq. (27), while the dashed lines
describe only the high momentum fraction region $xE\geq 2\pi T$ as in Eq.
(28).

The energy remaining inside the jet cone of size $R$ is computed by
integrating the energy distribution up to an angle $R$

$E(R,\tau)=E\sum_{a}\int{\rm d}x~{}\int_{\cos R}^{1}{\rm
d}{\cos\theta}~{}D_{a/\rm jet}(x,\theta,\tau)\;.$ (27)

Now, for the energy carried by hard partons in the cascade only the
integration over the momentum fraction $x$ is cutoff at the low momentum
$(x\geq 2\pi T/E)$

$E_{\rm hard}(R,\tau)=E\sum_{a}\int_{2\pi T/E}^{\infty}{\rm d}x~{}\int_{\cos
R}^{1}{\rm d}{\cos\theta}~{}D_{a/\rm jet}(x,\theta,\tau)\;.$ (28)

Our results are shown in Fig. 6 for three cone-sizes $R=(0.16,0.32,0.62)$
(green, red, blue) for an initial gluon jet in the left panel and an initial
quark jet in the right panel. Solid lines with points at the upper edge of the
band represent the full energy inside the cone in Eq. (27), while the dashed
lines at the lower edge of the band represent only the contribution from the
hard momentum region in Eq. (28), such that the bands showcase the discrepancy
between the two.

We find that for narrow cones $(R\leq 0.3)$ the dynamics of the soft sector
does not play a major role as the energy loss is very similar in both momentum
regions. However, for larger cone sizes, a transition occurs where the initial
energy loss becomes slower, and we observe a significant discrepancy between
the two curves, which highlights the sensitivity to the dynamics of the soft
sector for cone sizes $R\gtrsim 0.3$. Clearly, the physical origin of this
discrepancy can be easily understood, as the energy loss in the hard sector is
driven by the depletion due to the in-medium splitting, while the energy loss
in the soft sector is due to elastic scatterings which re-distribute the
energy to large angles outside the cone. While frequent small angle elastic
scatterings induce a sufficient amount of broadening for narrow cone sizes,
with increasing cone size the energy transport to large out-of-cone angles
takes an increasing amount of time. We can therefore anticipate, that for
larger cone sizes, the out-of-cone energy loss shows a larger sensitivity to
the detailed dynamics of the thermalization of the soft sector, which is
confirmed by our analysis in App. A where we compare numerical results with
full HTL elastic scattering to the corresponding ones obtained in the small-
angle/diffusion approximation Schlichting:2020lef .

Eventually at late times, where the jet has nearly equilibrated, the energy
inside the cone slowly approaches a constant value, which depends only on the
cone size $R$. Indeed, the amount of energy that remains inside the jet cone
can be estimated from the equilibrium distribution as

$\displaystyle E_{\rm eq}(R)=$ $\displaystyle E\sum_{a}\int{\rm
d}x~{}\int_{\cos R}^{1}{\rm d}{\cos\theta}~{}D_{a}^{\rm
eq}(x,\theta)=E\left[\frac{5}{2}+\frac{3}{2}\cos(R)\right]\sin^{2}\Big{(}\frac{R}{2}\Big{)}\;,$
(29)

such that for small cone sizes $R\ll 1$ one has $E_{\rm eq}(R)\sim ER^{2}$.
While Eq. (29) provides the cone size dependence of the equilibrium energy
distribution, we observe from Fig. 6 that irrespective of the cone size, the
soft fragments carry a substantial fraction of this energy $(\gtrsim 60\%)$,
at late times when the jet has nearly equilibrated.

Figure 7: Evolution of the energy stored inside the cone $R$ as a function of
the cone size $R$. (left) Gluon jet, (right) Quark jet. Linespoints represent
the energy carried by all partons while dashed lines represent the energy
carried by hard partons with momentum fraction $x\geq 2\pi T/E$.

Next, in order to showcase the relevant angular scale where the separation
between soft and hard partons is relevant for the out-of-cone energy loss, we
display in Fig. 7, the cumulative energy $E(R,\tau)$ inside a cone as a
function of the cone size $R$ for different times $t=(2.1,3.1,5,6.9)$fm/c with
an initial gluon jet in the left panel and an initial quark jet in the right
panel. Solid lines with points represent the full energy inside the cone,
while dashed lines represent the energy carried by hard partons with momentum
fraction $x\geq 2\pi T/E$ only. Note that the angular scale is logarithmic.
Starting from early times, one observes a strong depletion of the energy
inside narrow cones, along with a slight broadening of the core structure,
which is better visible for the gluon jet that experiences more broadening.
Eventually, as the energy is depleted from the initial hard parton at $x\sim
1$ and $R\sim 0$, we observe the emergence of an angular scale, where the
energy inside narrower cones is mostly carried by hard partons while for
larger cone sizes the soft sector gradually starts to contribute a substantial
fraction of the energy. Over the course of the evolution, this scale evolves
from $R\sim 0.03$ at $t=2.1{\rm fm}/c$ to around $R\sim 0.1$ at $t=6.9{\rm
fm}/c$. Since the contribution of hard fragments saturates at large angles, we
can also confirm that the hard partons mostly stay collinear throughout the
evolution, and except for very narrow cones, the out-of-cone energy loss
occurs primarily due to the re-distribution of soft fragments out to large
angles. While the soft sector always dominates the energy distribution at
large angles, the precise scale where this transition happens depends on the
evolution time, i.e. on the amount of quenching that the jet has suffered
inside the medium. Beyond cone size of $R\sim 1$, the distribution overshoots
unity and decreases again due to the negative contributions of recoiling
partons, before eventually all energy is recovered at $\theta=\pi$ where
$E(\pi,\tau)=E$ at all times.

##  5 Stopping time scaling of energy loss

Figure 8: Evolution of the fraction of the initial energy carried by particles
inside cones $R=0.16,0.32,0.62$ with different initial energies $E=100-1000T$.
The left panels represent the evolution of the total energy
$E(R,\tau)/E=\sum_{a}E_{a}/E$, starting with a gluon initial jet (top) or
quark initial jet (bottom). The bottom right panel represents the valence
energy, defined as the difference of energy carried by quarks and anti-quarks
$E_{V}(R,\tau)/E=(E_{q}-E_{\bar{q}})/E$.

So far, we have analyzed the in-medium evolution for a fixed initial energy
$E=500T$ of the hard parton. Based on our observation that the out-of-cone
energy loss proceeds via radiative break-up of hard fragments, followed by
angular broadening of soft fragments, we can anticipate that the evolution of
the out-of-cone energy loss is governed by the splitting time $t_{\rm
th}\sim\frac{1}{\alpha_{s}}\sqrt{\frac{E}{\hat{q}}}$ of the initial hard
parton in Eq. (16) that governs the energy deposition into the soft sector.
Verifying to what extent a simple scaling with $t_{\rm th}$ holds in our
numerical simulations is thus not only important for phenomenological
applications, but also provides an important verification of the underlying
energy loss mechanism.

Our results for the energy dependence of the out-of-cone energy loss are
compactly summarized in Fig. 8, where we display the energy remaining inside a
cone of size $R=0.16,0.32,0.62$. By scaling the evolution time in terms of the
typical splitting time $t_{\rm th}$ as in Eq. (17), we account for the leading
energy dependence when comparing different initial parton energies
$E=(100,300,500,700,1000)T$ for an initial gluon jet in the top panel and
initial quark jet in the bottom left panel.

We also compare the evolution with the analytical estimates computed for the
energy deposition in the soft sector due to the turbulent cascade, which leads
to the following linear energy loss rate Mehtar-Tani:2018zba ;
Schlichting:2020lef

$\displaystyle\frac{1}{E}\frac{dE}{d\tau}=\frac{C_{A}^{1/2}C_{R}}{8\pi^{2}}\sqrt{\frac{\hat{\bar{q}}(E)}{g^{4}T^{3}}}\left(\tilde{\gamma}_{g}+\frac{S}{G}\tilde{\gamma}_{q}\right)\tau\;,$
(30)

where $C_{R}=C_{A}$ for gluon and $C_{R}=C_{F}$ for quarks jets,
$S/G=0.07\times 2N_{f}$ is the quark to gluon ratio in the turbulent cascade
Schlichting:2020lef , and

$\displaystyle\tilde{\gamma}_{g}=$ $\displaystyle\int_{0}^{1}{\rm
d}z~{}\sqrt{\frac{xE}{g^{8}T^{3}}}\Big{[}\frac{d\Gamma^{g}_{gg}}{dz}\Big{(}xE,z\Big{)}+2N_{f}\frac{d\Gamma^{g}_{q\bar{q}}}{dz}\Big{(}xE,z\Big{)}\Big{]}~{}z\log(z)$
$\displaystyle=$
$\displaystyle\frac{1}{8\pi^{2}}\sqrt{\frac{\hat{\bar{q}}(\sqrt{TE})}{g^{4}T^{3}}}(25.78+2N_{f}0.177)\;,$
(31) $\displaystyle\tilde{\gamma}_{q}=$ $\displaystyle\int_{0}^{1}{\rm
d}z~{}\sqrt{\frac{xE}{g^{8}T^{3}}}\Big{[}\frac{d\Gamma^{q}_{gq}}{dz}\Big{(}xE,z\Big{)}+\frac{d\Gamma^{q}_{qg}}{dz}\Big{(}xE,z\Big{)}\Big{]}~{}2z\log(z)=\frac{1}{8\pi^{2}}\sqrt{\frac{\hat{\bar{q}}(\sqrt{TE})}{g^{4}T^{3}}}(11.595)\;,$
(32)

are flux constants governing the energy flux along the turbulent cascade, as
discussed in detail in Schlichting:2020lef .444Note that to evaluate the
integrals, one employs the leading logarithmic approximation of the splitting
rates Arnold:2008zu  where
$\sqrt{\frac{xE}{g^{8}T^{3}}}\frac{d\Gamma^{a}_{bc}}{dz}\Big{(}xE,z\Big{)}$
becomes approximately independent of the coupling constants $g$ and the energy
$xE$ of the emitter, e.g.
$\sqrt{\frac{xE}{g^{8}T^{3}}}\frac{d\Gamma^{g}_{gg}}{dz}\Big{(}xE,z\Big{)}\simeq\frac{1}{8\pi^{2}}P_{gg}(z)\sqrt{\frac{\hat{\bar{q}}(xE)}{g^{4}T^{3}}}~{}\sqrt{\frac{(1-z)C_{A}+z^{2}C_{A}}{z(1-z)}}\;.$
We evaluate this at the scale $xE=\sqrt{TE}$ which is the geometric mean of
the hard $E$ and thermal scale $x=T$. In practice, when we present our
numerical estimate we extract
$\sqrt{\frac{\hat{\bar{q}}(E)}{g^{4}T^{3}}}\overset{E=1000T}{\simeq}1.39$ and
$\sqrt{\frac{\hat{\bar{q}}(\sqrt{TE})}{g^{4}T^{3}}}\overset{E=\sqrt{1000}T}{\simeq}1.20$
from fits to the numerical results for the in-medium splitting rates, as
discussed in App. A of Schlichting:2020lef . Collecting everything this yields
$E(\tau)=E\big{(}1-0.014C_{R}C_{R}\frac{\tau^{2}}{2}\big{)}$ as the estimate
for the energy loss, which is depicted in Fig. 8.  By assuming that the energy
in the soft sector escapes the jet cone on a time scale that is much smaller
than the energy deposition due to the turbulent cascade, the linear energy
loss rate in Eq. (30), then leads to a quadratic decrease of the energy inside
the cone, which is shown as a dashed line in Fig. 8.

When considering the energy inside a cone, we observe that the scaling works
extremely well for narrow cone sizes $R\leq 0.16$, where the energy deposited
in the soft sector escapes the cone quasi-instantaneously on the time scale of
the energy loss of the hard partons. In this regime, the out-of-cone energy
loss is also remarkably well described using the analytical estimate in Eq.
(30). Conversely, for larger cone sizes the scaling only holds approximately
and gradually starts to break down as the time scale for the re-distribution
of energy to large angles becomes significant, leading to a breakdown of the
simple scaling with the time scale for in-medium splittings. Energy loss to
large angles is then also sensitive to the thermalization of the soft sector,
and can no longer be captured by simple estimates as in Eq. (30).

We also show the valence energy by taking the difference of the energy carried
by quarks and anti-quarks in the bottom right panel. Since, the energy
difference of quarks and anti-quarks is not linked to a conserved quantity,
there is an actual loss of energy in the valence channel, primarily due to in-
medium splittings of hard valence quarks. We observe that the valence energy
exhibits virtually no dependence on the cone-size, as the momentum broadening
of the leading valence parton is negligible. Since the energy loss in the
valence channel is governed by in-medium splittings, the results for different
initial energies scale with the splitting time to very good accuracy.

##  6 Modeling medium quenching

Figure 9: (left) Leading parton quenching factor as a function of initial
parton energy. We take the medium temperature $T=200$MeV.

Our investigations of energy loss in the previous sections have been of rather
theoretical nature, and we would now like to illustrate how our results can be
used in phenomenological studies of jets in heavy-ion collisions. Such studies
rely on the assumption that the hard partonic process that initiates the jet
is independent from the subsequent final state interactions with the QGP. This
is rooted in the principle of factorization of short distance/time physics,
that take place over scales of the order $1/p_{T}$, and thus is independent of
the QGP dynamics so long as the in-medium mean free path is larger than
$1/p_{T}$, and long distance physics where the interactions of the jet with
the plasma constituents take place, which lead to jet energy loss and its
substructure modifications Mehtar-Tani:2017web ; Qiu:2019sfj . By comparing
with proton-proton collisions where no final state interactions occur, one can
infer quantitatively the effects of medium modifications.

Since the results of our kinetic evolution can be seen as a Green’s function
for the evolution of a (resolved) hard parton inside the medium, we can
compute the suppression of hard partons by folding our results of energy loss
with hard partonic spectrum from proton-proton collisions. However, our
effective kinetic evolution lacks important physics due to vacuum like
emissions and path length dependence of the in-medium radiation rates, as well
as a proper space-time evolution of the medium. While these aspects are known
to be important for phenomenological studies, we will nevertheless employ our
results to obtain estimates for quenching of spectra yields in order to
illustrate their use and gain qualitative insights.

###  6.1 Leading parton quenching

Firstly, we consider the yield of the inclusive parton spectrum after passing
through the medium which can be obtained as a convolution of the energy
distribution for a given starting energy $p^{in}_{T}$ with the initial parton
spectrum555We assume parton-hadron duality, i.e., the fragmentation does not
substantially alter the leading parton such that each leading parton leads to
a high-$p_{T}$ hadron in the final state. Baier:2001yt ; Salgado:2003gb ;
Mehtar-Tani:2014yea ; Adhya:2019qse

$\displaystyle\frac{d^{2}\sigma_{AA}}{dp^{2}_{T}}(p_{T})=$
$\displaystyle\int_{0}^{\infty}{\rm d}^{2}p^{in}_{T}\int_{0}^{1}\frac{{\rm
d}x}{x}\int_{-1}^{1}{\rm d}{\cos\theta}~{}\delta^{(2)}(p_{T}-xp^{in}_{T})$
$\displaystyle D\left(x,\theta,\tau\equiv
g^{4}T\sqrt{T/p_{T}^{in}}t\right)\frac{{\rm d}^{2}\sigma_{\rm vac}}{{\rm
d}^{2}p^{in}_{T}}(p^{in}_{T})\;,$ (33)

where we make use of the rescaled time $\tau=g^{4}T\sqrt{\frac{T}{E}}t$ from
Eq. (17) Adhya:2019qse  in order to account for the initial parton energy
(c.f. Fig. 8 for the energy dependence). Dividing by the vacuum spectrum
approximated by a power-law $\frac{d^{2}\sigma_{\rm
vac}}{dp^{2}_{T}}(p^{in}_{T})\propto p_{T}^{-1-n}$, we define the inclusive
leading parton quenching factor

$\displaystyle Q^{\rm p}_{AA}(p_{T})=$
$\displaystyle\frac{\frac{d^{2}\sigma_{AA}}{{\rm d}p^{2}_{T}}}{\frac{{\rm
d}^{2}\sigma_{\rm vac}}{{\rm d}p^{2}_{T}}}=\int_{0}^{1}{\rm
d}x\int_{-1}^{1}{\rm
d}\cos\theta~{}D\left(x,\theta,g^{4}T\sqrt{xT/p_{T}}t\right)x^{n-2}\;.$ (34)

We follow Spousta:2015fca  and take $n_{g}=5.66$ and $n_{q}=4.19$ for gluon
and quark jets, respectively. The resulting leading parton quenching factors
are shown in Fig. 9, for different times $t=1,2,3,5$fm/c and for both gluon
and quark jets. We will not directly compare with experimental data, still we
observe a similar qualitative behavior to the suppression $R_{AA}$ of charged
hadrons STAR:2002ggv ; PHENIX:2005zfm ; CMS:2012aa ; ATLAS:2018bvp ;
STAR:2016jdz ; PHENIX:2020alr . The quenching factor display a suppression
($Q<1$) in the full range of energy $p_{T}=100-1000T$, where higher energies
are less suppressed as expected. However, since we treat medium-induced
splitting in the infinite medium limit which tends to overestimate the rates,
the quenching factor we obtain at comparable times of heavy-ion collisions
($t\sim 5$fm/c) is much more suppressed than experimental results. Hence, we
conclude that other effects such as the path length dependence of radiative
emission rates are important to reproduce experimental observables. As
expected, we find that gluon initiated fragmentation leads to more suppression
of final state particles than quark initiated fragmentation, which is
explained by the fact that gluons lose their energy much faster than quarks
due to larger color factors in the splitting matrix elements. Additionally,
since the exponent of the momentum fraction $x$ in Eq. (34) is positive
($n-2>0$) the factor $x^{n-2}$ suppresses the soft modes and the leading
parton quenching factor is mostly sensitive to the hard sector constituents
which sit at small angle and large momentum fraction.

###  6.2 Jet quenching

Figure 10: (left) Energy remaining inside the cone for different cone sizes
$R=0.16,0.32,0.62$ considering all fragments in full lines or only the hard
partons with momentum fraction $x\geq 2\pi T/E$ in dashed lines. (right) Jet
quenching factor for a quark jet with cone sizes $R=0.16,0.32,0.62$ after
$L=5$fm/c evolution in the medium. Different curves of the same color show the
results, including all fragments $x\geq 0$ (solid) or hard fragments only
$x\geq 2\pi T/E$ (dashed), such that bands highlight the sensitivity to soft
fragments.

Jets are collimated bunches of particles produced in high energy collisions,
which in a first approximation reflect the elementary hard partonic process.
The nascent energetic partons created at time scales of order $1/Q$ with large
virtuality tend to shower into multi partonic system. This strongly collimated
parton cascade is a prediction of pertubative QCD. Of course, beyond these
rather generic considerations the precise definition of a jet, from an
operational view point, requires the introduction of the notion of the jet
cone size $R$ in addition to the use of a jet reconstruction algorithm.

In the context of heavy ion collisions, it has been recently shown that the
collinear in-vacuum parton cascade factorizes from in-medium jet evolution
responsible for parton energy loss Mehtar-Tani:2017web ; Mehtar-Tani:2017ypq ;
Caucal:2018dla ; Caucal:2018ofz ; Caucal:2019uvr . The former forms typically
over much shorter time scales than medium generated scales. If we consider a
single parton splitting, it takes place between $Q^{-1}\equiv(Rp_{T})^{-1}\ll
t_{f}\ll\sqrt{z(1-z)P_{T}/\hat{q}}$ where $z$ is the longitudinal momentum
fraction carried by the offspring partons.

In this leading-log picture of jet evolution in the QGP, after the individual
color charges that are viewed as single partons by the medium are resolved by
it, the in-medium parton cascade discussed in the present manuscript develops.
In order to compute the suppression of the inclusive jet spectrum in $AA$
compared to $pp$, one then needs to first compute the probability for a single
collinear parton to lose energy out of a jet cone of size $R$.

A full-fledged study of jet quenching withing our kinetic approach is beyond
the scope of this work, but we aim in this section to achieve a first step
towards providing more computationally friendly tools for jet quenching
phenomenology that do not rely on computationally intensive approaches such as
MC event generators or transport models.

Our main goal in this section is therefore to focus on the case of single
collinear parton, that is, neglecting in-vacuum jet evolution to gain some
insight on how kinetic approaches can be implemented into jet quenching
calculations. This will also enable us to address the importance of the medium
response of soft sector for inclusive jet spectra.

In the presence of a QGP, the jet spectrum factorizes as follows,

$\displaystyle\frac{{\rm d}\sigma}{{\rm d}p_{T}}=\int_{0}^{\infty}{\rm
d}~{}\epsilon P(\epsilon,R)\frac{{\rm d}\sigma_{\rm vac}}{{\rm
d}p_{T}^{in}}(p_{T}^{in}\equiv p_{T}+\epsilon)\,,$ (35)

where, as before, $p_{T}^{in}$ denotes the energy of the original (unquenched)
parton and $\epsilon$ is the energy loss. Despite the fact that the formula is
quite general since the energy loss probability $P(\epsilon,R)$ may apply for
a multiple partonic system Mehtar-Tani:2017web  we will restrict our analysis
to a single parton.

Following the seminal work by Baier et al. Baier:2001yt  (see also Mehtar-
Tani:2017web ), because $P(\epsilon,R)$ is dominated by short lived soft gluon
radiation that are not measured if they are not considered part of the jet, it
takes a Poisson-like form

$\displaystyle
P(\epsilon,R)=\sum_{n=0}^{\infty}\frac{1}{n!}\left[\prod_{i=1}^{n}\omega_{i}\frac{{\rm
d}I}{\omega_{i}}{\rm
d}\omega_{i}\right]\delta\left(\epsilon-\sum_{i=1}^{n}\omega_{i}\right)\exp\left[-\int_{0}^{\infty}{\rm
d}\omega\frac{{\rm d}I}{{\rm d}\omega}\right]$ (36)

Interestingly, in Laplace space the convolution in $\omega$ space turns into a
direct product and hence leads to the exponentiation of the single gluon
radiative spectrum

$\displaystyle{\cal L}[P(\epsilon,R)](\nu)=\int{\rm d}\epsilon~{}{\rm
e}^{-\epsilon\nu}P(\epsilon,R)=\exp\left[-\int{\rm d}\omega\frac{{\rm
d}I}{{\rm d}\omega}\left(1-{\rm e}^{-\omega\nu}\right)\right]$ (37)

This observation will be used hereafter in evaluating the inclusive jet
spectrum. To do so, we need to make an additional approximation. Using the
fact that the power jet spectrum is steeply falling, that is,

$\displaystyle\frac{{\rm d}\sigma_{\rm vac}}{{\rm
d}p_{T}}\sim\frac{1}{p_{T}^{n}}\quad\text{with}\quad n\gg 1\,,$ (38)

one may thus expand the shifted power spectrum $(P_{T}+\epsilon)^{-n}$ around
small $\epsilon\ll p_{T}$ while keeping $n\epsilon$ finite. This yields,

$\displaystyle\frac{1}{(p_{T}+\epsilon)^{n}}=\frac{1}{p_{T}^{n}}\exp\left[-\frac{n\epsilon}{p_{T}}\right]\left(1+{\cal
O}(n\epsilon^{2})\right)$ (39)

This is the exponential we need to connect with the Laplace transform. Thus,
we have

$\displaystyle\frac{{\rm d}\sigma}{{\rm d}p_{T}}=Q(p_{T},R)\frac{{\rm
d}\sigma_{\rm vac}}{{\rm d}p_{T}}\,,$ (40)

where the quenching weight $Q$ is simply the Laplace transform of the energy
loss probability evaluated at $\nu=n/p_{T}$,

$\displaystyle Q(p_{T},R)\approx{\cal
L}[P(\epsilon,R)](n/p_{T})=\exp\left[-\int{\rm d}\omega\frac{{\rm d}I}{{\rm
d}\omega}\left(1-{\rm e}^{-n\omega/p_{T}}\right)\right]$ (41)

Since Eq. (36) accounts for multiple emissions in a probabilistic fashion,
this formula has the advantage of accounting for the fluctuations of energy
loss. However, it neglects the possibility that part of the radiated energy
may remain inside the jet cone, hence contributing to the total jet energy. On
the other hand, the kinetic equation, while it allows to treat the full
dynamics of energy loss down to the dissipation scale, cannot be directly used
to compute jet quenching as it allows to compute the inclusive average parton
distribution and hence, does not handle fluctuations in and out of the jet
cone. However, we may combine the strengths of these two approaches by
accounting for fluctuations of energy loss via the Poisson like distribution
of the primary gluon emissions and by assuming that after being radiated the
primary gluons will rapidly cascade in to a large number of gluons allowing
for an event by event averaging of energy in and out of cone.

By replacing

$\displaystyle\omega\to\omega-E(\omega|R,\tau_{L})$ (42)

in Eq. (41), this assumption amounts to shifting the frequency of the primary
radiated gluon $\omega$ by the average energy $E(\omega|R,\tau_{L})$ remaining
inside a cone of size $R$ after evolving in the medium for a time $\tau_{L}$,
which we compute using kinetic theory as in Eq. (27). This idea was previously
explored in a simplified fashion in Mehtar-Tani:2021fud .

Now following this approach, let us consider an initial hard quark with energy
$p_{T}$, radiating the first gluon emission with energy $\omega$ along the
$z$-axis following the finite-size radiation rate in the soft radiation
$\omega$ limit written as Arnold:2008iy

$\displaystyle\frac{{\rm d}\rm I^{q}_{gq}}{{\rm d}\omega{\rm
d}t}\equiv\frac{{\rm d}\Gamma^{q}_{gq}}{{\rm
d}\omega}(p_{T},\omega)=\frac{\alpha_{s}}{2\pi}C_{F}\sqrt{\frac{C_{A}\hat{\bar{q}}(p_{T})}{\omega^{3}}}\;,$
(43)

and we will employ $\hat{\bar{q}}(p_{T})=787\,{\rm MeV/fm}^{2}$ in the
following numerical estimates. Once it is emitted after a time $t$, the gluon
traverses the medium and loses its energy following the kinetic evolution,
leaving the medium after time $L$. Its energy is then suppressed by the medium
during a time $(L-t)$, therefore the energy remaining in the cone of size $R$
is given by the energy fraction computed in Sec. 3 evaluated at $(L-t)$, which
leads to the slightly modified function
$E\left(\omega|R,\tau=\frac{L-t}{t_{\rm th}}\right)$ from Eq. (27) where we
include a variable $\omega$ to account for the starting energy of the gluon in
the kinetic evolution. We have computed a kinetic evolution for the energies
$\omega=(1,3,10,30,100,300,500,700,1000)T$ which we interpolate between to
obtain the energy remaining inside the cone as a smooth function of the
initial gluon energy $\omega$, making use of the time scaling with the
stopping time $t_{\rm th}$ which works fairly well for small cone sizes. Using
the energy remaining inside the cone after a time $(L-t)$, the quenching
weight in Eq. (41) is then written as

$\displaystyle Q(p_{T})=\exp\left[\int_{0}^{L}{\rm d}t\int{\rm
d}\omega\frac{{\rm d}\Gamma}{{\rm d}\omega}\left(1-{\rm
e}^{-n\frac{\omega}{p_{T}}\left[1-\frac{E\left(\omega|R,\tau=\frac{L-t}{t_{\rm
th}}\right)}{\omega}\right]}\right)\right]\;.$ (44)

In order to investigate the sensitivity to the soft fragments, we also employ
the modified energy inside the cone $E_{\rm
hard}\left(\omega|R,\tau=\frac{L-t}{t_{\rm th}}\right)$ from Eq. (28) which
account for only the hard fragments with momentum fraction $x\geq 2\pi T/E$.
In the left panel of Fig. 10, we display the remaining energy inside a cone of
size $R$ as a function of the initial parton momentum $p_{T}$ at $L=5$fm/c for
the different cone-sizes $R=0.16,0.32,0.62$. Full line-points represent the
full energy $E\left(\omega|R,\tau=\frac{L}{t_{\rm th}}\right)$, while dashed
lines represent $E_{\rm hard}\left(\omega|R,\tau=\frac{L}{t_{\rm th}}\right)$,
i.e., only the energy stored in the hard partons with momentum fraction $x\geq
2\pi T/E$. The right panel of Fig. 10 shows the jet quenching factor for a
single prong quark jet, i.e. a quark initiated jet loosing energy as one
initial hard quark, using the same color coding. We observe that by passing
through a medium of length $L=5{\rm fm}/c$, single prong quark jets with
energies $p_{T}\leq 25$GeV have lost a significant fraction of their energy
out to large cones $R\geq 0.62$. While the jet quenching factor shows a large
suppression of at low $p_{T}$, depending on the cone size the suppression
becomes milder for $p_{T}\geq 200-400$T. When comparing the results for the
different cone-sizes, we find that while for large cone-sizes a clear
sensitivity to the soft fragments is apparent, for a narrow cone, the
contribution of soft fragments to the energy inside the cone is negligible
leading to negligible difference in the quenching. We obtain a clear
indication that jet cone sizes larger $\gtrsim 0.3$ are more sensitive to the
soft fragments which heavily depend on the details of the thermalization of
the soft sector, thus requiring more refined studies of near-equilibrium
physics and recoils of the jet onto the medium.

##  7 Conclusion

In this paper, we investigated the dynamics of energy loss and thermalization
of a hard parton as it passes through a hot deconfined QCD plasma. As the
highly energetic parton traverses a quark-gluon plasma, it undergoes
interactions with the medium, leading to a transfer of energy from the hard
modes $(\sim E\ll T)$ all the way to near-equilibrium $(\sim T)$ soft modes.
While the hard modes can be treated perturbatively, this treatment starts to
break down once the energy becomes soft and is deposited in the medium,
similarly for the response of the medium due to energy loss. Whereas,
typically, studies of the thermalization transform the energy deposited into a
source term for the hydrodynamic simulation Casalderrey-Solana:2004fdk ;
Renk:2006mv ; Qin:2009uh ; Li:2010ts ; Tachibana:2014lja ; Tachibana:2017syd ;
Casalderrey-Solana:2020rsj ; Yang:2022nei , during this study we instead used
an effective kinetic description to bridge the apparent gap in the theoretical
description of jet quenching. By using a common framework for the hard parton
and the soft fragments, we are able to follow the hard parton’s energy as it
is carried out to large angles and thermalizes with the medium.

We employed a leading order effective kinetic description of the interactions
of a hard parton with a thermal medium, which takes into account both the
elastic interactions with the medium and the collinear medium-induced
radiation. Several studies have shown that the medium cascade is governed by
the successive splitting driving an inverse energy cascade Blaizot:2013hx ;
Blaizot:2015jea ; Mehtar-Tani:2018zba ; Schlichting:2020lef . Since the
inelastic radiation is enhanced in the collinear region, this energy cascade
is collinear to the leading hard parton. Conversely, the elastic interactions
are suppressed at large energies, such that the broadening of the hard parton
is marginal. We observe only a small diffusion of the energy, which dies as a
fourth power of the angle due to the Molière tail. However, the elastic
interactions play a more important role for the soft scales, where the
scattering rate becomes comparable to the inelastic interactions. Therefore,
once the energy is transferred to the low energy modes, elastic scatterings
thermalizes the partons with the medium and push the distribution out to large
angles.

While our study has been centered around the energy loss of a hard parton once
it has already been resolved by the medium, we also illustrated how kinetic
theory can be used in phenomenological studies of parton/jet quenching. In the
future, it would be interesting to further refine the effective kinetic
description, by considering more realistic initial conditions that emerge from
the early vaccuum-like emissions and lifting some assumptions that simplify
the dynamics. By considering the parton to spend a sufficiently long time
inside the medium, we employed an infinite medium calculation to obtain a
simpler form of the effective medium-induced rates that does not explicitly
depend on time. However, for realistic medium sizes, this simplification also
leads to a significant over-quenching of partons and jets. Clearly, a more
realistic description can readily be obtained by employing a time dependent
splitting rate that captures the interplay of the formation time of the
radiation and the medium length CaronHuot:2010bp ; Andres:2020vxs ;
Schlichting:2021idr . When one eventually implements these finite length
effects, the simulation can also move away from the simple description of the
medium background as a static equilibrated medium and allow for a hydrodynamic
evolution of the background QGP medium.

##  8 Acknowledgements

S.S. and I.S. are supported by the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) through the CRC-TR 211 ’Strong-interaction matter
under extreme conditions’– project number 315477589 – TRR 211 and the German
Bundesministerium für Bildung und Forschung (BMBF) through Grant No.
05P21PBCAA. I.S. is supported in part by the U.S. Department of Energy (DOE)
under grant number DE-SC0013460 and in part by the National Science Foundation
(NSF) under grant number OAC-2004571 within the framework of the JETSCAPE
collaboration. The authors also gratefully acknowledge computing time provided
by the Paderborn Center for Parallel Computing (PC2). This research used
resources of the National Energy Research Scientific Computing Center, a DOE
Office of Science User Facility supported by the Office of Science of the U.S.
Department of Energy under Contract No. DE-AC02-05CH11231. Y. M.-T. is
supported by the U.S. Department of Energy, Office of Science, Office of
Nuclear Physics, under contract No. DE- SC0012704 an acknowledges support from
the RHIC Physics Fellow Program of the RIKEN BNL Research Center.

##  Appendix A Comparison with the small angle approximation

Figure 11: Evolution of the energy inside the cone ($\theta\leq R$) as given
by Eq. (27), for gluon (left) and quark (right) initial jets with $E=100T$. We
compare with the ‘Diffusion’ approximation, taking the momentum broadening
coefficient to be either $\hat{q}$ or $\frac{\hat{q}}{4}$.

Our investigation of the out-of-cone energy loss in the main text, are based
on a leading order QCD kinetic description involving medium induced
$1\leftarrow 2$ collinear splittings/mergings and $2\leftarrow 2$ elastic
interactions described by leading order perturbative matrix elements with HTL
screening. While inelastic interactions are responsible for the radiative
break-up of hard partons, elastic interactions provide momentum broadening
outside the jet cone. By considering small momentum exchange for elastic
interactions the momentum broadening can effectively be described using a
Fokker-Planck diffusion equation Blaizot:2014jna ; Ghiglieri:2015ala ;
Schlichting:2020lef , where the energy dependence is then accounted for in the
Coulomb logarithm of the momentum diffusion coefficient $\hat{q}$. Below we
investigate to what extent this ‘Diffusion’ approximation can describe the
effects of momentum broadening, both on a qualitative and quantitative level.
We chose the equilibrium momentum broadening coefficient $\hat{q}$ for the
diffusion evolution where the logarithmic scale dependence is taken to be
$\sim 1$,

$\hat{q}=\frac{g^{4}T^{3}C_{R}}{2\pi}\left(\frac{N_{c}}{3}+\frac{N_{f}}{6}\right)\;.$
(45)

We then proceed as before, and compute the fraction of energy remaining inside
the cone ($\theta\leq R$) from Eq. (27) within the full leading order QCD
kinetic evolution, and in ‘Diffusion’ approximation. The logarithmic
dependence of the momentum broadening coefficient, which we do not consider,
affects the momentum broadening and consequently out-of-cone energy loss. In
the full matrix element evolution this effect is scale dependent, however, in
our ‘Diffusion’ simulation we do not set up a scale dependent $\hat{q}$, but
instead compare our results with two different values for $\hat{q}$, either
taking the equilibrium value in Eq. (45) or taking a quarter
($\hat{q}\to\frac{\hat{q}}{4}$). We find that these two values provide a large
enough separation to describe the range of cone dependence of the energy loss
we focus on in Fig. 11. We display in Fig. 11 the energy remaining inside the
cone $R=(0.16,0.32,0.62)$ from Eq. (27) for the full matrix element evolution
using linespoints as well as for the ‘Diffusion’ approximation with long
dashed lines for $\hat{q}$ and long dashed lines for $\frac{\hat{q}}{4}$ where
on the left we have a Gluon initial jet and on the right a Quark initial jet
with initial energy $E=100T$. First, we see that, as expected, the evolution
using $\frac{\hat{q}}{4}$ loses energy slower than for $\hat{q}$. Likewise,
how quickly energy is lost also depends on the cone size, where narrower cone
sizes lose energy more quickly. Comparing with the full matrix element
evolution, we observe how the hard particles with narrow cones $R\sim 0.16$
are better described by $\hat{q}$ while for larger cone sizes when softer
particles are more relevant the evolution is closer to $\frac{\hat{q}}{4}$,
confirming our earlier conclusion that, for large cone sizes, the out-of-cone
energy loss is sensitive to the detailed dynamics of the soft sector, such
that a single scale independent $\hat{q}$ cannot simultaneously describe the
broadening at all energy and angular scales. We note also that the late time
limits of the different curves display the energy inside each cone size in the
equilibrium distribution, which is the same independently of the evolution
used.

##  Appendix B Elastic scatterings

$ab\longleftrightarrow cd$ | $\qquad\left|{\cal M}^{ab}_{cd}\right|^{2}$
---|---
$qq\leftrightarrow qq$ | $\displaystyle 8N_{f}g^{4}\,\frac{d_{\rm F}^{2}\,C_{\rm F}^{2}}{d_{\rm A}}\left(\frac{s^{2}+u^{2}}{t^{2}}+\frac{s^{2}+t^{2}}{u^{2}}\right)+16g^{4}\,d_{\rm F}\,C_{\rm F}\left(C_{\rm F}{-}\frac{C_{\rm A}}{2}\right)\frac{s^{2}}{tu}$
$q\bar{q}\longleftrightarrow q\bar{q}$ | $\displaystyle 8N_{f}g^{4}\,\frac{d_{\rm F}^{2}\,C_{\rm F}^{2}}{d_{\rm A}}\left(\frac{s^{2}+u^{2}}{t^{2}}+\frac{t^{2}+u^{2}}{s^{2}}\right)+16g^{4}\,d_{\rm F}\,C_{\rm F}\left(C_{\rm F}{-}\frac{C_{\rm A}}{2}\right)\frac{u^{2}}{st}$
$q\bar{q}\longleftrightarrow g\,g$ | $\displaystyle 8g^{4}\,d_{\rm F}\,C_{\rm F}^{2}\left(\frac{u}{t}+\frac{t}{u}\right)-8g^{4}\,d_{\rm F}\,C_{\rm F}\,C_{\rm A}\left(\frac{t^{2}+u^{2}}{s^{2}}\right)$
$\begin{array}[]{c}q\,g\longleftrightarrow q\,g\,,\\\ \bar{q}\,g\longleftrightarrow\bar{q}\,g\end{array}$ | $\displaystyle-8g^{4}\,d_{\rm F}\,C_{\rm F}^{2}\left(\frac{u}{s}+\frac{s}{u}\right)+8g^{4}\,d_{\rm F}\,C_{\rm F}\,C_{\rm A}\left(\frac{s^{2}+u^{2}}{t^{2}}\right)$
$g\,g\leftrightarrow g\,g$ | $\displaystyle 16g^{4}\,d_{\rm A}\,C_{\rm A}^{2}\left(3-\frac{su}{t^{2}}-\frac{st}{u^{2}}-\frac{tu}{s^{2}}\right)$
Table 1: Different $2\leftrightarrow 2$ processes in QCD and their squared
matrix elements Arnold:2002zm  written in terms of the Mandelstam variables
$s=(P_{1}+P_{2})^{2}$, $t=(P_{1}-P_{3})^{2}$ and $u=(P_{1}-P_{4})^{2}$. For
QCD theory with $SU(N_{c})$ gauge symmetry, the fundamental representation has
dimension $d_{F}=N_{c}$ and quadratic Casimir
$C_{F}=\tfrac{N_{c}^{2}-1}{2N_{c}}$, while in the adjoint representation the
dimension is $d_{A}=N_{c}^{2}-1$ and the quadratic Casimir is $C_{A}=N_{c}$.

During this section, we will briefly compile the elastic collision integrals
we use during this study. Since following the evolution for each quark flavor
separately is numerically costly, we will instead sum over the different
flavor by introducing the distribution,

$\displaystyle D_{Q}=\sum_{i}D_{q_{i}}\;,\qquad
D_{\bar{Q}}=\sum_{i}D_{\bar{q}_{i}}\;.$ (46)

We present the elastic scattering matrix element squared in Tab. 1 in this
convention, where processes that can evolve different flavors (e.g., the
t-channel of quark-quark scattering) acquire a factor $N_{f}$. The collision
kernel of the elastic scattering is written

$\displaystyle C_{a}^{2\leftrightarrow 2}[\\{f_{i}\\}]=$
$\displaystyle\frac{1}{2|p_{1}|\nu_{a}}\sum_{bcd}\int
d\Omega^{2\leftrightarrow 2}\left|{\cal
M}^{ab}_{cd}({\bm{p}}_{1},{\bm{p}}_{2};{\bm{p}}_{3},{\bm{p}}_{4})\right|^{2}\delta\mathcal{F}({\bm{p}}_{1},{\bm{p}}_{2};{\bm{p}}_{3},{\bm{p}}_{4})\;,$
(47)

where the statistical factor
$\delta\mathcal{F}({\bm{p}}_{1},{\bm{p}}_{2},{\bm{p}}_{3},{\bm{p}}_{4})$ is
now

$\displaystyle\delta\mathcal{F}({\bm{p}}_{1},{\bm{p}}_{2},{\bm{p}}_{3},{\bm{p}}_{4})=$
$\displaystyle~{}~{}~{}\delta
f_{a}({\bm{p}}_{1})\left[\pm_{a}n_{c}(p_{3})n_{d}(p_{4})-n_{b}(p_{2})(1\pm
n_{c}(p_{3})\pm n_{d}(p_{4}))\right]$ $\displaystyle+\delta
f_{b}({\bm{p}}_{2})\left[\pm_{b}n_{c}(p_{3})n_{d}(p_{4})-n_{b}(p_{1})(1\pm
n_{c}(p_{3})\pm n_{d}(p_{4}))\right]$ $\displaystyle-\delta
f_{c}({\bm{p}}_{3})\left[\pm_{c}n_{a}(p_{1})n_{b}(p_{2})-n_{b}(p_{4})(1\pm
n_{a}(p_{1})\pm n_{b}(p_{2}))\right]$ $\displaystyle-\delta
f_{d}({\bm{p}}_{4})\left[\pm_{d}n_{a}(p_{1})n_{b}(p_{2})-n_{b}(p_{3})(1\pm
n_{a}(p_{1})\pm n_{b}(p_{2}))\right]\;,$ (48)

while $\pm_{i}$ is plus if particle $i$ is a boson and minus for a fermion.
Since we consider the phase-space distribution to be isotropic in the azimuth
angle $\phi_{p}$ of momentum ${\bf p}$, we can express the phase-space
distributions of gluons and quarks as

$\delta f_{g}(x_{1},\theta_{1})=\int\frac{d\phi_{p}}{2\pi}~{}\delta
f_{a}({\bm{p}})=(2\pi)^{2}\frac{D_{g}(x_{1},\theta_{1})}{\nu_{g}x_{1}^{3}E^{2}}\,,\qquad\delta
f_{q}(x_{1},\theta_{1})=(2\pi)^{2}\frac{D_{Q}(x_{1},\theta_{1})}{N_{f}\nu_{q}x_{1}^{3}E^{2}}$
(49)

and similarly for anti-quarks. Expressing the contributions to the collision
kernels in terms of the energy distribution $D_{a}$, the contributions to the
gluon collision integral from different processes are given by

$\displaystyle C_{g}^{qg\leftrightarrow qg}=$
$\displaystyle\frac{1}{2|p_{1}|\nu_{a}}\int d\Omega^{2\leftrightarrow
2}\left|{\cal M}^{gq}_{gq}\right|^{2}$
$\displaystyle\Bigg{[}N_{f}\frac{D_{g}(x_{1},\theta_{1})}{\nu_{g}x_{1}^{3}E^{2}}\left[n_{F}(p_{3})n_{B}(p_{4})-n_{b}(p_{2})(1-n_{F}(p_{3})+n_{B}(p_{4}))\right]$
$\displaystyle+\frac{D_{Q}(x_{2},\theta_{2})}{\nu_{q}x_{2}^{3}E^{2}}\left[-n_{F}(p_{3})n_{B}(p_{4})-n_{F}(p_{1})(1-n_{F}(p_{3})+n_{B}(p_{4}))\right]$
$\displaystyle-\frac{D_{Q}(x_{3},\theta_{3})}{\nu_{q}x_{3}^{3}E^{2}}\left[-n_{B}(p_{1})n_{F}(p_{2})-n_{F}(p_{4})(1+n_{B}(p_{1})-n_{F}(p_{2}))\right]$
$\displaystyle-
N_{f}\frac{D_{g}(x_{4},\theta_{4})}{\nu_{g}x_{4}^{3}E^{2}}\left[n_{B}(p_{1})n_{F}(p_{2})-n_{F}(p_{3})(1+n_{B}(p_{1})-n_{F}(p_{2}))\right]\Bigg{]}$
$\displaystyle+\frac{1}{2|p_{1}|\nu_{a}}\int d\Omega^{2\leftrightarrow
2}\left|{\cal M}^{gq}_{qg}\right|^{2}$
$\displaystyle\Bigg{[}N_{f}\frac{D_{g}(x_{1},\theta_{1})}{\nu_{g}x_{1}^{3}E^{2}}\left[n_{B}(p_{3})n_{F}(p_{4})-n_{b}(p_{2})(1+n_{B}(p_{3})-n_{F}(p_{4}))\right]$
$\displaystyle+\frac{D_{Q}(x_{2},\theta_{2})}{\nu_{q}x_{2}^{3}E^{2}}\left[-n_{B}(p_{3})n_{F}(p_{4})-n_{b}(p_{1})(1+n_{B}(p_{3})-n_{F}(p_{4}))\right]$
$\displaystyle-
N_{f}\frac{D_{g}(x_{3},\theta_{3})}{\nu_{g}x_{3}^{3}E^{2}}\left[n_{B}(p_{1})n_{F}(p_{2})-n_{F}(p_{4})(1+n_{B}(p_{1})-n_{F}(p_{2}))\right]$
$\displaystyle-\frac{D_{Q}(x_{4},\theta_{4})}{\nu_{q}x_{4}^{3}E^{2}}\left[-n_{B}(p_{1})n_{F}(p_{2})-n_{B}(p_{3})(1+n_{B}(p_{1})-n_{F}(p_{2}))\right]\Bigg{]}\;,$
(50) $\displaystyle C_{g}^{gg\leftrightarrow q\bar{q}}=$
$\displaystyle\frac{1}{2|p_{1}|\nu_{a}}\int d\Omega^{2\leftrightarrow
2}\left|{\cal M}^{q\bar{q}}_{gg}\right|^{2}$
$\displaystyle\Bigg{[}N_{f}\frac{D_{g}(x_{1},\theta_{1})}{\nu_{q}x_{1}^{3}E^{2}}\left[-n_{F}(p_{3})n_{B}(p_{4})-n_{F}(p_{2})(1-n_{F}(p_{3})+n_{B}(p_{4}))\right]$
$\displaystyle+N_{f}\frac{D_{g}(x_{2},\theta_{2})}{\nu_{q}x_{2}^{3}E^{2}}\left[n_{F}(p_{3})n_{B}(p_{4})-n_{B}(p_{1})(1-n_{F}(p_{3})+n_{B}(p_{4}))\right]$
$\displaystyle-\frac{D_{Q}(x_{3},\theta_{3})}{\nu_{q}x_{3}^{3}E^{2}}\left[-n_{F}(p_{1})n_{B}(p_{2})-n_{F}(p_{4})(1-n_{F}(p_{1})+n_{B}(p_{2}))\right]$
$\displaystyle-\frac{D_{\bar{Q}}(x_{4},\theta_{4})}{\nu_{q}x_{4}^{3}E^{2}}\left[n_{F}(p_{1})n_{B}(p_{2})-n_{F}(p_{3})(1-n_{F}(p_{1})-n_{B}(p_{2}))\right]\Bigg{]}\;,$
$\displaystyle+\frac{1}{2|p_{1}|\nu_{a}}\int d\Omega^{2\leftrightarrow
2}\left|{\cal M}^{q\bar{q}}_{gg}\right|^{2}$
$\displaystyle\Bigg{[}N_{f}\frac{D_{g}(x_{1},\theta_{1})}{\nu_{q}x_{1}^{3}E^{2}}\left[-n_{F}(p_{3})n_{B}(p_{4})-n_{F}(p_{2})(1-n_{F}(p_{3})+n_{B}(p_{4}))\right]$
$\displaystyle+N_{f}\frac{D_{g}(x_{2},\theta_{2})}{\nu_{q}x_{2}^{3}E^{2}}\left[n_{F}(p_{3})n_{B}(p_{4})-n_{B}(p_{1})(1-n_{F}(p_{3})+n_{B}(p_{4}))\right]$
$\displaystyle-\frac{D_{\bar{Q}}(x_{3},\theta_{3})}{\nu_{q}x_{3}^{3}E^{2}}\left[-n_{F}(p_{1})n_{B}(p_{2})-n_{F}(p_{4})(1-n_{F}(p_{1})+n_{B}(p_{2}))\right]$
$\displaystyle-\frac{D_{Q}(x_{4},\theta_{4})}{\nu_{q}x_{4}^{3}E^{2}}\left[n_{F}(p_{1})n_{B}(p_{2})-n_{F}(p_{3})(1-n_{F}(p_{1})-n_{B}(p_{2}))\right]\Bigg{]}\;,$

where the sum over the quark flavors cancels with the factor $1/N_{f}$ for the
quark energy distribution. Similarly, the contribution of quark-gluon
scattering to the quark collision integral, is obtained by performing a sum
over the $N_{f}$ degenerate flavors as follows

$\displaystyle C_{Q}^{qg\leftrightarrow qg}=$
$\displaystyle\sum_{f}C_{q_{f}}=\frac{1}{2|p_{1}|\nu_{a}}\int
d\Omega^{2\leftrightarrow 2}\left|{\cal M}^{gq}_{gq}\right|^{2}$
$\displaystyle\Bigg{[}\frac{D_{Q}(x_{1},\theta_{1})}{\nu_{q}x_{1}^{3}E^{2}}\left[-n_{F}(p_{3})n_{B}(p_{4})-n_{F}(p_{2})(1-n_{F}(p_{3})+n_{B}(p_{4}))\right]$
$\displaystyle+N_{f}\frac{D_{g}(x_{2},\theta_{2})}{\nu_{g}x_{2}^{3}E^{2}}\left[n_{F}(p_{3})n_{B}(p_{4})-n_{B}(p_{1})(1-n_{F}(p_{3})+n_{B}(p_{4}))\right]$
$\displaystyle-\frac{D_{Q}(x_{3},\theta_{3})}{\nu_{q}x_{3}^{3}E^{2}}\left[-n_{F}(p_{1})n_{B}(p_{2})-n_{F}(p_{4})(1-n_{F}(p_{1})+n_{B}(p_{2}))\right]$
$\displaystyle-
N_{f}\frac{D_{g}(x_{4},\theta_{4})}{\nu_{g}x_{4}^{3}E^{2}}\left[n_{F}(p_{1})n_{B}(p_{2})-n_{F}(p_{3})(1-n_{F}(p_{1})-n_{B}(p_{2}))\right]\Bigg{]}$
$\displaystyle+\frac{1}{2|p_{1}|\nu_{a}}\int d\Omega^{2\leftrightarrow
2}N_{f}\left|{\cal M}^{gq}_{qg}\right|^{2}$
$\displaystyle\Bigg{[}\frac{D_{Q}(x_{1},\theta_{1})}{\nu_{q}x_{1}^{3}E^{2}}\left[n_{B}(p_{3})n_{F}(p_{4})-n_{B}(p_{2})(1+n_{B}(p_{3})-n_{F}(p_{4}))\right]$
$\displaystyle+N_{f}\frac{D_{g}(x_{2},\theta_{2})}{\nu_{g}x_{2}^{3}E^{2}}\left[-n_{B}(p_{3})n_{F}(p_{4})-n_{F}(p_{1})(1+n_{B}(p_{3})-n_{F}(p_{4}))\right]$
$\displaystyle-
N_{f}\frac{D_{g}(x_{3},\theta_{3})}{\nu_{g}x_{3}^{3}E^{2}}\left[n_{F}(p_{1})n_{B}(p_{2})-n_{F}(p_{4})(1-n_{F}(p_{1})+n_{B}(p_{2}))\right]$
$\displaystyle-\frac{D_{Q}(x_{4},\theta_{4})}{\nu_{q}x_{4}^{3}E^{2}}\left[-n_{B}(p_{1})n_{b}(p_{2})-n_{B}(p_{3})(1-n_{F}(p_{1})+n_{B}(p_{2}))\right]\Bigg{]}\;.$
(52)

Next, the contributions of elastic scattering of quarks is given by

$\displaystyle C_{Q}^{qq\leftrightarrow qq}=$
$\displaystyle\sum_{i}C_{q_{i}}=\frac{1}{2|p_{1}|\nu_{a}}\int
d\Omega^{2\leftrightarrow 2}\left|{\cal M}^{qq}_{qq}\right|^{2}$
$\displaystyle\Bigg{[}\frac{D_{Q}(x_{1},\theta_{1})}{\nu_{q}x_{1}^{3}E^{2}}\left[-n_{F}(p_{3})n_{B}(p_{4})-n_{F}(p_{2})(1-n_{F}(p_{3})+n_{B}(p_{4}))\right]$
$\displaystyle+\frac{D_{Q}(x_{2},\theta_{2})}{\nu_{q}x_{2}^{3}E^{2}}\left[n_{F}(p_{3})n_{B}(p_{4})-n_{B}(p_{1})(1-n_{F}(p_{3})+n_{B}(p_{4}))\right]$
$\displaystyle-\frac{D_{Q}(x_{3},\theta_{3})}{\nu_{q}x_{3}^{3}E^{2}}\left[-n_{F}(p_{1})n_{B}(p_{2})-n_{F}(p_{4})(1-n_{F}(p_{1})+n_{B}(p_{2}))\right]$
$\displaystyle-\frac{D_{Q}(x_{4},\theta_{4})}{\nu_{q}x_{4}^{3}E^{2}}\left[n_{F}(p_{1})n_{B}(p_{2})-n_{F}(p_{3})(1-n_{F}(p_{1})-n_{B}(p_{2}))\right]\Bigg{]}\;,$
(53)

where we combined the two processes $q_{i}q_{f}\to q_{i}q_{f}$ and
$q_{i}q_{f}\to q_{f}q_{i}$ by writing the second processes as u-channel matrix
element, and added the process $q_{i}q_{i}\to q_{i}q_{i}$ in order to obtain
the full matrix element in Tab. 1.

Eventually, the remaining quark anti-quark scattering and annihilation
processes are written as

$\displaystyle C_{Q}^{q\bar{q}\leftrightarrow q\bar{q}}=$
$\displaystyle\sum_{i}C_{q_{i}}=\frac{1}{2|p_{1}|\nu_{a}}\int
d\Omega^{2\leftrightarrow 2}\left|{\cal M}^{q\bar{q}}_{q\bar{q}}\right|^{2}$
$\displaystyle\Bigg{[}\frac{D_{Q}(x_{1},\theta_{1})}{\nu_{q}x_{1}^{3}E^{2}}\left[-n_{F}(p_{3})n_{B}(p_{4})-n_{F}(p_{2})(1-n_{F}(p_{3})+n_{B}(p_{4}))\right]$
$\displaystyle+\frac{D_{\bar{Q}}(x_{2},\theta_{2})}{\nu_{q}x_{2}^{3}E^{2}}\left[n_{F}(p_{3})n_{B}(p_{4})-n_{B}(p_{1})(1-n_{F}(p_{3})+n_{B}(p_{4}))\right]$
$\displaystyle-\frac{D_{Q}(x_{3},\theta_{3})}{\nu_{q}x_{3}^{3}E^{2}}\left[-n_{F}(p_{1})n_{B}(p_{2})-n_{F}(p_{4})(1-n_{F}(p_{1})+n_{B}(p_{2}))\right]$
$\displaystyle-\frac{D_{\bar{Q}}(x_{4},\theta_{4})}{\nu_{q}x_{4}^{3}E^{2}}\left[n_{F}(p_{1})n_{B}(p_{2})-n_{F}(p_{3})(1-n_{F}(p_{1})-n_{B}(p_{2}))\right]\Bigg{]}\;,$
$\displaystyle+\frac{1}{2|p_{1}|\nu_{a}}\int d\Omega^{2\leftrightarrow
2}\left|{\cal M}^{q\bar{q}}_{\bar{q}q}\right|^{2}$
$\displaystyle\Bigg{[}\frac{D_{Q}(x_{1},\theta_{1})}{\nu_{q}x_{1}^{3}E^{2}}\left[-n_{F}(p_{3})n_{B}(p_{4})-n_{F}(p_{2})(1-n_{F}(p_{3})+n_{B}(p_{4}))\right]$
$\displaystyle+\frac{D_{\bar{Q}}(x_{2},\theta_{2})}{\nu_{q}x_{2}^{3}E^{2}}\left[n_{F}(p_{3})n_{B}(p_{4})-n_{B}(p_{1})(1-n_{F}(p_{3})+n_{B}(p_{4}))\right]$
$\displaystyle-\frac{D_{\bar{Q}}(x_{3},\theta_{3})}{\nu_{q}x_{3}^{3}E^{2}}\left[-n_{F}(p_{1})n_{B}(p_{2})-n_{F}(p_{4})(1-n_{F}(p_{1})+n_{B}(p_{2}))\right]$
$\displaystyle-\frac{D_{Q}(x_{4},\theta_{4})}{\nu_{q}x_{4}^{3}E^{2}}\left[n_{F}(p_{1})n_{B}(p_{2})-n_{F}(p_{3})(1-n_{F}(p_{1})-n_{B}(p_{2}))\right]\Bigg{]}\;,$
(54) $\displaystyle C_{Q}^{q\bar{q}\leftrightarrow gg}=$
$\displaystyle\sum_{i}C_{q_{i}}=\frac{1}{2|p_{1}|\nu_{a}}\int
d\Omega^{2\leftrightarrow 2}\left|{\cal M}^{q\bar{q}}_{gg}\right|^{2}$
$\displaystyle\Bigg{[}\frac{D_{Q}(x_{1},\theta_{1})}{\nu_{q}x_{1}^{3}E^{2}}\left[-n_{F}(p_{3})n_{B}(p_{4})-n_{F}(p_{2})(1-n_{F}(p_{3})+n_{B}(p_{4}))\right]$
$\displaystyle+\frac{D_{\bar{Q}}(x_{2},\theta_{2})}{\nu_{q}x_{2}^{3}E^{2}}\left[n_{F}(p_{3})n_{B}(p_{4})-n_{B}(p_{1})(1-n_{F}(p_{3})+n_{B}(p_{4}))\right]$
$\displaystyle-
N_{f}\frac{D_{g}(x_{3},\theta_{3})}{\nu_{q}x_{3}^{3}E^{2}}\left[-n_{F}(p_{1})n_{B}(p_{2})-n_{F}(p_{4})(1-n_{F}(p_{1})+n_{B}(p_{2}))\right]$
$\displaystyle-
N_{f}\frac{D_{g}(x_{4},\theta_{4})}{\nu_{q}x_{4}^{3}E^{2}}\left[n_{F}(p_{1})n_{B}(p_{2})-n_{F}(p_{3})(1-n_{F}(p_{1})-n_{B}(p_{2}))\right]\Bigg{]}\;,$

where, we combined the two processes $q_{i}\bar{q}_{i}\to q_{i}\bar{q}_{i}$
and $q_{i}\bar{q}_{i}\to\bar{q}_{f}q_{f}$. Analogously, one can obtain the
remaining processes involving anti-quark particles. In the following sections,
we will outline our treatment of the matrix element using Hard thermal loop
(HTL) propagators.

###  B.1 Hard thermal loop matrix element

The matrix elements in Tab. 1 are for particles in vacuum, which leads to an
infrared divergence in the momentum exchange in the t- and u-channels666The
mixed channels $s^{2}/(tu)$ and $u^{2}/(st)$ also generate divergences, but
these cancel between the gain and loss term Arnold:2002zm .. For a proper
treatment using thermal propagators, the divergences are regulated by the
self-energies. However, as the authors of Arnold:2002zm  argue, medium-
dependent effects are only important for small angle scatterings corresponding
to the regions where $-t$ or $-u$ are of the order of the thermal mass squared
and the modification of the other terms can be neglected. In this section, we
will follow the AMY approach Arnold:2002zm ; Arnold:2002ja ; Arnold:2003zc  in
order to re-write the divergent terms using the retarded self-energy, which
cuts off the divergence.

#### Gluon exchange:

We apply the rewriting for the following t- and u-channel gluon exchange
processes: $gg\leftrightarrow gg$, $q^{i}q^{j}\leftrightarrow q^{i}q^{j}$,
$q^{i}\bar{q}^{j}\leftrightarrow q^{i}\bar{q}^{j}$,
$\bar{q}^{i}\bar{q}^{j}\leftrightarrow\bar{q}^{i}\bar{q}^{j}$ and
$q^{i}g\leftrightarrow q^{i}g$. The matrix elements in terms of Mandelstam
variables can be rewritten as

$\displaystyle\frac{s^{2}+u^{2}}{t^{2}}=\frac{1}{2}+\frac{1}{2}\frac{(s-u)^{2}}{t^{2}}\;,$
$\displaystyle\qquad\frac{su}{t^{2}}=\frac{1}{4}-\frac{1}{4}\frac{(s-u)^{2}}{t^{2}}\;,$
(56)
$\displaystyle\frac{s^{2}+t^{2}}{u^{2}}=\frac{1}{2}+\frac{1}{2}\frac{(s-t)^{2}}{u^{2}}\;,$
$\displaystyle\qquad\frac{st}{u^{2}}=\frac{1}{4}-\frac{1}{4}\frac{(s-t)^{2}}{u^{2}}\;.$
(57)

Using thermal propagators amounts to the replacement Arnold:2003zc

$\displaystyle\frac{(s-u)^{2}}{t^{2}}\longrightarrow|G_{\mu\nu}(P_{1}-P_{3})(P_{1}+P_{3})^{\mu}(P_{2}+P_{4})^{\nu}|^{2}\;,$
(58)
$\displaystyle\frac{(s-t)^{2}}{u^{2}}\longrightarrow|G_{\mu\nu}(P_{1}-P_{4})(P_{1}+P_{4})^{\mu}(P_{2}+P_{3})^{\nu}|^{2}\;,$
(59)

where $G_{\mu\nu}(P_{1}-P_{3})$ is the retarded thermal gluon propagator,
computed in the HTL approximation. In the Coulomb gauge, it is given by

$\displaystyle G_{00}(\omega,{\bm{q}})=$
$\displaystyle\frac{-1}{q^{2}+\Pi_{00}(\omega,q)}\;,\qquad
G_{ij}(\omega,{\bm{q}})=\frac{\delta_{ij}-\tfrac{{\bm{q}}_{i}{\bm{q}}_{j}}{q^{2}}}{q^{2}-\omega^{2}+\Pi_{T}(\omega,q)}\;,$
(60) $\displaystyle G_{i0}(\omega,{\bm{q}})=$ $\displaystyle
G_{0i}(\omega,{\bm{q}})=0\;.$ (61)

The transverse and longitudinal gluon self-energies are given by

$\displaystyle\Pi_{00}(\omega,q)=$ $\displaystyle
m_{D}^{2}\left[1-\frac{\omega}{2q}\left(\ln\left(\frac{q+\omega}{q-\omega}\right)-i\pi\right)\right]\;,$
(62) $\displaystyle\Pi_{T}(\omega,q)=$ $\displaystyle
m_{D}^{2}\left[\frac{\omega^{2}}{q^{2}}+\frac{\omega(q^{2}-\omega^{2})}{4q^{3}}\left(\ln\left(\frac{q+\omega}{q-\omega}\right)-i\pi\right)\right]\;,$
(63)

in the $(+---)$ metric convention, the propagator with 4-momentum
$Q=(\omega,{\bm{q}})$ is space-like (i.e., $|\omega|<q$) for the t- and
u-channels and the logarithm is well-behaved.

#### Quark exchange:

The other t- and u-channel processes that require a rewriting are the quark
exchange processes: $q^{i}\bar{q}^{i}\leftrightarrow gg$ and
$qg\leftrightarrow qg$. The matrix elements are computed using the following
replacement for the four-momentum exchange,

$Q^{\mu}\to\mathcal{Q}^{\mu}\equiv Q^{\mu}-\Sigma^{\mu}(Q)\;,$ (64)

where the quark self-energy in the Coulomb gauge is given by

$\displaystyle\Sigma^{0}(Q)=$
$\displaystyle\frac{m_{F}^{2}}{2q}\left[\ln\left(\frac{q+\omega}{q-\omega}\right)-i\pi\right]\;,$
(65) $\displaystyle\bm{\Sigma}(Q)=$
$\displaystyle-{\bm{q}}\frac{m_{F}^{2}}{q^{2}}\left[1-\frac{\omega}{2q}\left(\ln\left(\frac{q+\omega}{q-\omega}\right)-i\pi\right)\right]\;.$
(66)

The relevant matrix elements become

$\displaystyle\frac{u}{t}\longrightarrow$ $\displaystyle\left.\frac{4{\rm
Re}[(P_{1}\cdot\mathcal{Q})(P_{2}\cdot\mathcal{Q}^{*})]-s\mathcal{Q}\cdot\mathcal{Q}^{*}}{|\mathcal{Q}\cdot\mathcal{Q}|^{2}}\right|_{\mathcal{Q}^{\mu}=P_{1}^{\mu}-P_{3}^{\mu}-\Sigma^{\mu}(P_{1}-P_{3})}\;,$
(67) $\displaystyle\frac{t}{u}\longrightarrow$ $\displaystyle\left.\frac{4{\rm
Re}[(P_{1}\cdot\mathcal{Q})(P_{2}\cdot\mathcal{Q}^{*})]-s\mathcal{Q}\cdot\mathcal{Q}^{*}}{|\mathcal{Q}\cdot\mathcal{Q}|^{2}}\right|_{\mathcal{Q}^{\mu}=P_{1}^{\mu}-P_{4}^{\mu}-\Sigma^{\mu}(P_{1}-P_{4})}\;,$
(68) $\displaystyle\frac{s}{u}\longrightarrow$
$\displaystyle\left.\frac{-4{\rm
Re}[(P_{1}\cdot\mathcal{Q})(P_{3}\cdot\mathcal{Q}^{*})]-t\mathcal{Q}\cdot\mathcal{Q}^{*}}{|\mathcal{Q}\cdot\mathcal{Q}|^{2}}\right|_{\mathcal{Q}^{\mu}=P_{1}^{\mu}-P_{4}^{\mu}-\Sigma^{\mu}(P_{1}-P_{4})}\;.$
(69)

##  Appendix C Equilibrium distributions

Below we derive the equilibrium distributions for the energy distribution
$D_{a}(x,\theta)$. We first consider the change of the volume integrated
phase-space distribution

$\displaystyle\delta\bar{f}^{(eq)}_{a}({\bf p})=\int_{{\bf
x}}\delta\bar{f}^{(eq)}_{a}({\bf x},{\bf p})$ (70)

which can be represented as a linear superposition of the changes of the
temperature $T$, the rest-frame four velocity $u^{z}$ and the chemical
potential $\frac{\mu_{f}}{T}$ according to

$\displaystyle\delta\bar{f}^{(eq)}_{a}({\bf p})=V\left[\frac{\delta
T}{T}T\partial_{T}+\delta
u^{z}\partial_{u^{z}}+\delta\Big{(}\frac{\mu_{f}}{T}\Big{)}\partial_{\frac{\mu_{f}}{T}}\right]n_{a}({\bf
p})\;.$ (71)

We note that in the above expression the factor of the volume $V$ has to
appear on dimensional grounds, however this will cancel against the amplitudes
$\delta T/T$, $\delta u^{z}$ and $\delta\Big{(}\frac{\mu_{f}}{T}\Big{)}$ all
of which are inversely proportional to the volume. Specifically, by performing
the thermodynamic matching for an ideal gas of massless partons, these
amplitudes can be expressed as

$\displaystyle\frac{\delta T}{T}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\frac{\delta E}{E_{\rm tot}}=\frac{1}{4}\frac{\delta
E}{Ve(T)}\;,$ (72) $\displaystyle\delta u^{z}$ $\displaystyle=$
$\displaystyle\frac{\delta P^{z}}{E_{\rm tot}+P_{\rm
tot}}=\frac{3}{4}\frac{\delta P^{z}}{Ve(T)}\;,$ (73)
$\displaystyle\delta\Big{(}\frac{\mu_{f}}{T}\Big{)}$ $\displaystyle=$
$\displaystyle\frac{\delta N_{f}}{V\chi(T)T},$ (74)

where $E_{\rm tot}=Ve(T)$ and $P_{\rm tot}=E_{\rm tot}/3$ are the total energy
and total pressure,
$e(T)=\left(\frac{\pi^{2}}{30}\nu_{g}+\frac{7\pi^{2}}{120}\nu_{q}N_{f}\right)T^{4}$
is the energy density, and $\chi(T)=\frac{\nu_{q}}{6}T^{2}$ is the charge
susceptibility. By inserting Eqns. (71) and (72) into the definition of the
energy distribution in Eq. (2) and employing $\delta E=\delta P_{z}$ equal to
the energy $E$ of the jet, the equilibrium energy distribution then takes the
form

$\displaystyle
D_{a}^{(eq)}(x,\theta)=\frac{\nu_{a}}{(2\pi)^{2}}(xE)^{3}\left[\frac{1}{4}\frac{E}{e(T)}T\partial_{T}+\frac{3}{4}\frac{P^{z}}{e(T)}\partial_{u^{z}}+\frac{\delta
N_{f}}{\chi(T)T}\partial_{\frac{\mu_{f}}{T}}\right]\left.n_{a}\Big{(}p_{\mu}u^{\mu}\Big{)}\right|_{u^{z}=0}$
(75)

which is to be evaluated at $|\mathbf{p}|=xE$, with $\delta N_{f}=0$ for gluon
jets and $\delta N_{f}=\pm\delta_{fi}$ for quark/anti-quark jets of a given
flavor $i$. Since the change due to the chemical potential contributes with
opposite signs for particles and anti-particles, it cancels in the total
energy distribution, which is then given by

$\displaystyle
D^{(eq)}(x,\theta)=\sum_{a}\frac{\nu_{a}}{(2\pi)^{2}}\frac{(xE)^{4}}{4e(T)}~{}\frac{E}{T}~{}\left[1+3\cos(\theta)\right]n_{a}(xE)(1\pm
n_{a}(xE))$ (76)

where in the last step we used the identity

$\displaystyle
T\partial_{T}n_{a}(p)=-p\partial_{p}n_{a}(p)=\frac{p}{T}n_{a}(p)\Big{(}1\pm
n_{a}(p)\Big{)}\;.$ (77)

By use of this identity it is also straightforward to verify that the
equilibrium distribution in Eq. (76) satisfies the normalization condition in
Eq. (11).

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