aainstitutetext: Institute of Particle and Nuclear Physics, Faculty of
Mathematics and Physics, Charles University, V Holešovičkách 2, 180 00 Prague
8, Czech Republicbbinstitutetext: Instituto Galego de Física de Altas Enerxías
IGFAE, Universidade de Santiago de Compostela, E-15782 Galicia-
Spainccinstitutetext: Department of Physics and Technology, University of
Bergen, 5007 Bergen, Norway

#  Multi-partonic medium induced cascades in expanding media

Souvik Priyam Adhya b Carlos A. Salgado a Martin Spousta c Konrad Tywoniuk
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Going beyond the simplified gluonic cascades, we introduce both gluon and
quark degrees of freedom for partonic cascades inside the medium. We then
solve the set of coupled evolution equations numerically with splitting
kernels calculated for static, exponential, and Bjorken expanding media to
arrive at medium-modified parton spectra for quark and gluon initiated jets.
Using these, we calculate the inclusive jet $R_{\mathrm{AA}}$ where the
phenomenologically driven combinations of quark and gluon jet fractions are
included. Then, the rapidity dependence of the jet $R_{\mathrm{AA}}$ is
examined. We also study the path-length dependence of jet quenching for
different types of expanding media by calculating the jet $v_{2}$.
Additionally, we qualitatively study the sensitivity of observables on the
time of the onset of the quenching for the case of Bjorken media. All
calculations are compared with recently measured data.

##  1 Introduction

One of the most striking phenomena that was predicted Bjorken:1982tu ;
Gyulassy:1990ye ; Wang:1994fx  to occur in relativistic collisions of heavy
nuclei and that was confirmed by numerous experiments at RHIC Adcox:2001jp ;
Adler:2002xw ; Adler:2002tq  and the LHC CMS:2012aa ; Aad:2012vca ;
Aad:2015wga ; Aamodt:2010jd ; Aad:2014bxa ; Adam:2015ewa ; Khachatryan:2016jfl
is the suppression of high-$p_{T}$ hadron and jet production, generally
referred to as jet “quenching”, for reviews see dEnterria:2009xfs ;
Majumder:2010qh ; Mehtar-Tani:2013pia ; Blaizot:2015lma . This phenomenon is
generally understood as a consequence of energy loss of partons traversing the
hot and dense QCD medium consisting of the soft degrees of freedom released in
the collisions. Over the last two decades, the theoretical description of jet
quenching evolved from single-parton energy loss to include multi-parton
coherence effects and a deeper understanding of thermalization effects, for
reviews see CasalderreySolana:2007zz ; Mehtar-Tani:2013pia ; Blaizot:2015lma ;
Cao:2020wlm , paralleled by a rapid development of Monte Carlo models
Lokhtin:2005px ; Zapp:2008gi ; Armesto:2009fj ; Renk:2010zx ; Young:2011ug ;
Casalderrey-Solana:2014bpa ; He:2015pra ; Cao:2017zih ; Putschke:2019yrg ;
Caucal:2019uvr ; Ke:2020clc .

Not all observables have the same sensitivity to the degree of complexity
implemented in the theoretical description and therefore not the same
sensitivity to the underlying physics which is modeled. Consequently, it is
important to study the impact of individual aspects of complex modeling on
individual observables. In this paper, we focus mainly on the canonical jet
spectrum in lead-lead collisions at the LHC and the related nuclear
modification factor, $R_{\mathrm{AA}}$. These observables are mainly sensitive
to the energy lost by the leading particles in the jet, due to the bias from
the steeply falling hard cross-section of jet production Baier:2001yt . To
quantify in better detail the dependence on path length of jets, we also focus
on the azimuthal asymmetry of jet production quantified in the second harmonic
coefficient of the jet multiplicity distribution, $v_{2}$. All these
observables have been analyzed in various frameworks before. However, in this
paper we present a novel implementation of the effects of expanding medium
which allows to quantify the sensitivity to the plasma evolution (for related
work, see also Caucal:2020uic ), and we present the calculation of the
inclusive jet suppression evaluated differentially in the full set of basic
kinematic quantities: transverse momentum, rapidity, and the azimuth.

In a recent work Adhya:2019qse , we have presented a study of the distribution
of medium-induced gluons and the jet calculated using the evaluation of in-
medium evolution with splitting kernels derived from the gluon emission
spectra for expanding profiles. Scaling behavior of splitting kernels was
derived for low-$x$ and high-$x$ regimes in the asymptote of large times and
its impact on the resulting jet $R_{\mathrm{AA}}$ was discussed.111Since, at
that point, we were including only gluons, we were referring to the
phenomenological jet suppression factor as $Q_{AA}$. For the full phase space
of the radiation, the scaling of jet $R_{\mathrm{AA}}$ with an effective
quenching parameter was introduced.

In this paper, the main aim is to study the impact of the medium expansion on
observed jet suppression within more realistic description of the jet
production. The description is improved by: 1) by including both gluon and
quark degrees of freedom for partonic cascades into the evolution equations;
2) by improving on the description of the initial parton spectra. This should
provide more realistic calculation of nuclear modification factor and more
realistic values of quenching parameter, $\hat{q}$, to be extracted from the
data. Within this description we evaluate the rapidity dependence of
$R_{\mathrm{AA}}$ and path-length dependence of the energy loss (via jet
$v_{2}$) and study if measurements of inclusive jet suppression differential
in rapidity and azimuthal angle are sensitive to the way how the medium
expands. Furthermore, we study the sensitivity of these more differential jet
observables on the time of the onset of the quenching for the case of Bjorken
expansion.

The paper is organized as follows. In Sec. 2 we summarize the single parton
emission spectra and the in-medium splitting rates for possible flavor
combinations at the leading order. Next, we introduce the coupled evolution
equations to arrive at the medium evolved parton spectra with the in-medium
splitting functions as the kernels of the evolution. In Sec. 3, we first
calculate jet $R_{\mathrm{AA}}$ from moments of medium evolved partonic
spectra for quark and gluon initiated jets. Then, in the next four subsection
we analyze the transverse momentum dependence, initial-time dependence,
rapidity dependence, and path-length dependence of the jet suppression. We
summarize and conclude in Sec. 4.

##  2 In-medium emissions for different media profiles

###  2.1 Coupled parton cascade evolution equations

Medium-induced radiation was traditionally considered in the context of soft
gluon emissions Baier:1996kr ; Baier:1996sk ; Zakharov:1996fv ;
Zakharov:1997uu . Naturally, the turbulent nature of the cascade was first
discussed in the context of gluon fragmentation Blaizot:2013hx ;
Blaizot:2015jea . Nevertheless, because of the important role played by quasi-
democratic splittings, it is natural to consider the interplay of quark and
gluon degrees of freedom in the cascade Mehtar-Tani:2018zba ;
Schlichting:2020lef . From our point of view, this is also a natural demand
when considering quenching effects for a realistic initial hard spectrum of
initial partons, which is dominated by gluons at low- and intermediate-$p_{T}$
and by quarks at high-$p_{T}$. For consistency, one should also consider the
feedback from gluon to quark-antiquark splittings.

The main quantity of interest is the single-inclusive in-medium parton
distributions, defined as

$\displaystyle D_{i}(x,\tau)\equiv x\frac{{\rm d}N_{i}}{{\rm d}x},$ (1)

where $x=\omega/E$ denotes the energy fraction carried by a parton of
frequency $\omega$ with respect to the initial energy $E$ of the original
parton222Here, we concretely refer to the light-cone momentum fraction
$E\equiv p^{+}=(p^{0}+p^{3})/2$ and $p^{-}=p^{0}-p^{3}$. and $i$ refers to the
parton flavor. This is measured at the dimensionless “time”
$\tau\equiv\sqrt{\hat{q}_{0}/E}L$, where $L$ is the in-medium path-
length.333Sometimes the evolution variable is defined with an additional
$\alpha_{s}$, so that $\tau=L/t_{\rm stop}$ , where $t_{\rm
stop}=\bar{\alpha}^{-1}\sqrt{E/\hat{q}}$ is the stopping time of a jet with
energy $E$. Because we are including finite-size effects, we have instead
chosen to keep the coupling constant explicitly in the splitting functions.
Here, $\hat{q}_{0}$ is the value of the jet quenching coefficient $\hat{q}$ at
an initial reference time. Hence, the distribution $D(x,\tau)$ also
intrinsically depends on this time scale. For a medium with constant or
exponentially decaying density the natural initial time is simply $t=0$ and
this dependence is trivial. However, this is not the case for power-like
decaying medium profiles, which are associated with a finite starting time.
This will be discussed in more detail below.

Concretely, we focus on the gluon $D_{g}(x)$ and quark singlet distributions
$D_{S}(x)\equiv\sum_{f}\big{[}D_{q_{f}}(x)+D_{\bar{q}_{f}}(x)\big{]}$, where
the sum runs over all active flavors, which describes the energy distribution
of quarks inside a jet. The evolution equations for the inclusive in-medium
parton distribution can be written as Mehtar-Tani:2018zba

$\displaystyle\frac{\partial}{\partial\tau}D_{\rm g}\left(x,\tau\right)$
$\displaystyle=\int_{0}^{1}dz\,{\cal K}_{\rm
gg}(z)\left[\sqrt{\frac{z}{x}}D_{\rm
g}\left(\frac{x}{z},\tau\right)\Theta(z-x)-\frac{z}{\sqrt{x}}D_{\rm
g}(x,\tau)\right]$ $\displaystyle-\int_{0}^{1}{\rm d}z\,{\cal K}_{\rm
qg}(z)\frac{z}{\sqrt{x}}\,D_{\rm g}\left(x,\tau\right)+\int_{0}^{1}{\rm
d}z\,{\cal K}_{\rm gq}(z)\,\sqrt{\frac{z}{x}}\,D_{\rm
S}\left(\frac{x}{z},\tau\right)\,,$ (2)
$\displaystyle\frac{\partial}{\partial\tau}D_{\rm S}\left(x,\tau\right)$
$\displaystyle=\int_{0}^{1}dz\,{\cal K}_{\rm
qq}(z)\left[\sqrt{\frac{z}{x}}D_{\rm
S}\left(\frac{x}{z},\tau\right)\Theta(z-x)-\frac{1}{\sqrt{x}}D_{\rm
S}(x,\tau)\right]$ $\displaystyle+\int_{0}^{1}dz\,{\cal K}_{\rm
qg}(z)\,\sqrt{\frac{z}{x}}D_{\rm g}\left(\frac{x}{z},\tau\right)\,.$ (3)

Each equation involves a positive gain term and a negative loss term which
describe the production of a parton with energy fraction $x$ from the
splitting of a parent parton with energy fraction $x/z$ and the decay of a
parton with energy fraction $x$ into softer fragments with fractions $zx$ and
$(1-z)x$, respectively.

The splitting kernels ${\cal K}_{ij}$ quantify the rate of splittings per unit
evolution time $\tau$. They can be expressed as

$\displaystyle{\cal K}_{ij}(z,\tau)\equiv\frac{{\rm d}I_{ij}}{{\rm d}z\,{\rm
d}\tau}\,,$ (4)

where ${\rm d}I_{ij}/{\rm d}z$ is the medium-induced emission of parton $i$
from parton $j$, carrying a longitudinal momentum fraction $z$. The main
purpose of this work is to explore the evolution of the cascades for expanding
media. Below, we focus on three main scenarios:

Static medium:

In a static medium, $\hat{q}(t)=\hat{q}_{0}\Theta(L-t)$, where $L$ is the
medium length. The splitting rate reads

$\displaystyle\mathcal{K}^{\rm
stat}_{ij}(z,\tau)=\frac{\alpha_{s}}{2\pi}P_{ij}(z)\kappa_{ij}(z)\,\text{Re}\left[(i-1)\tan\left(\frac{1-i}{2}\kappa_{ij}(z)\tau\right)\right]\,.$
(5)

For the case of soft gluon emissions, i.e. $1-z,z\ll 1$, in a static medium,
the splitting function is given as,

$\displaystyle\mathcal{K}^{\rm
stat(soft)}_{ij}(z,\tau)\simeq\frac{\alpha_{s}}{2\pi}P_{ij}(z)\kappa_{ij}(z)\,.$
(6)

Exponentially decaying medium:

For exponentially decaying media the profile of the jet quenching parameter is
given by $\hat{q}(t)=\hat{q}_{0}{\rm e}^{-t/L}$. The splitting rate reads,

$\displaystyle\mathcal{K}^{\rm
exp}_{ij}(z,\tau)=\frac{\alpha_{s}}{\pi}P_{ij}(z)\kappa_{ij}(z)\,\text{Re}\left[(i-1)\frac{J_{1}\big{(}(1-i)\kappa_{ij}(z)\tau\big{)}}{J_{0}\big{(}(1-i)\kappa_{ij}(z)\tau\big{)}}\right]\,.$
(7)

Power-law decaying medium:

For power-law decaying media, the jet quenching parameter is defined as

$\displaystyle\hat{q}(t)=\begin{cases}0&{\rm for}\quad t<t_{0}\,,\\\
\hat{q}_{0}(t_{0}/t)^{\alpha}&{\rm for}\quad t_{0}<t<L+t_{0}\,,\\\ 0&{\rm
for}\quad L+t_{0}<t\,,\end{cases}$ (8)

with $\alpha=1$ corresponding to the Bjorken expansion Baier:1998yf . The
splitting rate reads in this case,

$\displaystyle\mathcal{K}^{\rm pow}_{ij}(z,\tau,\tau_{0})$
$\displaystyle=\frac{\alpha_{s}}{2\pi}P_{ij}(z)\kappa_{ij}(z)\sqrt{\frac{\tau_{0}}{\tau+\tau_{0}}}$
$\displaystyle\times\text{Re}\left[(1-i)\frac{J_{1}(z_{L})Y_{1}(z_{0})-J_{1}(z_{0})Y_{1}(z_{L})}{J_{1}(z_{0})Y_{0}(z_{L})-J_{0}(z_{L})Y_{1}(z_{0})}\right],$
(9)

where

$\displaystyle z_{0}$ $\displaystyle=(1-i)\kappa_{ij}(z)\tau_{0}\,,$ (10)
$\displaystyle z_{L}$
$\displaystyle=(1-i)\kappa_{ij}(z)\sqrt{\tau_{0}(\tau+\tau_{0})}\,,$ (11)

with $\tau_{0}=\sqrt{\hat{q}_{0}/E}t_{0}$. Due to this additional dependence
on $\tau_{0}$ in the splitting function, the resulting distribution naturally
also depends on it, i.e. $D(x,\tau)\to D(x,\tau,\tau_{0})$.

The splitting rates have been written in terms of the color-kinematical
factors $\kappa_{ij}(z)$ and (unregularised) Altarelli-Parisi splitting
functions $P_{ij}(z)$ and can be found explicitly in Appendix A.

###  2.2 Numerical results for medium-induced parton spectra

We use the coupled evolution equations (2.1) and (2.1) to numerically
calculate the evolution of the quark and gluon spectra for both quark-
initiated and gluon-initiated jets as a function of energy fraction $x$ for
different evolution time $\tau$. The rates and solution for the pure gluon
cascade was discussed in Adhya:2019qse . In the current work, we solve the
coupled evolution equations using finite difference methods. For future
reference, the resulting distributions are evaluated for two sets of initial
conditions, as follows:

  * •

gluon-initiated jets: $D_{g}(x,0)=\delta(1-x)$, $D_{S}(x,0)=0$.

  * •

quark-initiated jets: $D_{g}(x,0)=0$, $D_{S}(x,0)=\delta(1-x)$.

Figure 1 shows the medium evolved spectra for the gluon ($D_{\rm g}$) and the
quark ($D_{\rm S}$) distributions separately for gluon-initiated (Fig. 1(a))
and quark-initiated (Fig. 1(b)) jets as a function of $x$ evolved up to
$\tau=1.0$. The evolution has been performed in media with different density
profiles, i.e. performing the evolution with the different kernels in Eqs.
(6), (7) and (Power-law decaying medium:), see the figure legends for details.
For the Bjorken case, we have chosen $\tau_{0}=0.5$.

One can see that gluon spectra follow the $1/\sqrt{x}$ turbulent behavior for
both quark and gluon initiated jets _and_ for both the static medium and the
expanding medium (the static case was already discussed in Mehtar-Tani:2018zba
). However, for the quark spectra this scaling is only approximate for all the
medium profiles including the static profile. This is mainly because the
stationary behavior, i.e. $D_{g}(x)\sim c_{1}/\sqrt{x}$ and $D_{S}\sim
c_{2}/\sqrt{x}$ (where $c_{1}$ and $c_{2}$ are approximately constants), has
not yet had sufficient time to establish itself Mehtar-Tani:2018zba . Note,
that the speed of the evolution is slower when considering the splitting rates
for finite media, as done above Adhya:2019qse .

Outside the region $x\sim 1$, the exponential spectra have a greater amplitude
than the full static spectra and similar amplitude as the static soft spectra.
Spectra for the Bjorken expanding media have the least amplitude due to a
smaller rate of splittings compared to the other profiles. We have shown
previously Adhya:2019qse  that the “singular” parts of gluon spectra
($z\rightarrow 0$, $z\rightarrow 1$) for different medium profiles scale with
the static one when using an effective quenching parameter, defined as
$4\hat{q}_{0}$ and $4\hat{q}_{0}t_{0}/L$ for the exponential and Bjorken
profiles, respectively. This scaling between different medium profiles holds
for individual contributions discussed here as well. However, such a scaling
does not work when employing the full evolution kernels as obtained in Ref.
Adhya:2019qse  for gluon-only cascades.

In general, the $D_{\rm S}$ and $D_{\rm g}$ dominate the high-$x$ part of the
spectra for quark-initiated and gluon-initiated jets, respectively. For gluon-
initiated jets, the probability of finding a gluon in the parton cascade at
high as well as small $x$ is significantly larger than a quark for all the
cases of expanding media. For quark-initiated jets, the probability to find a
gluon is smaller than a quark at high $x$ and larger at small $x$. This is a
generalization of similar observations done in Ref. Mehtar-Tani:2018zba  for
static media.

(a) Gluon-initiated jets.

(b) Quark-initiated jets.

Figure 1: Evolution of the quark singlet spectra $(D_{S})$ and the gluon
spectra $(D_{g})$ for the ”gluon initiated” (left panels) and the ”quark
initiated” jets (right panels) for a particular evolution time $\tau=1.0$ for
different medium profiles.

##  3 Inclusive jet suppression

The main aim of this paper is to continue advancing studies of the inclusive
jet suppression in heavy-ion collisions. In this work we will treat the jet as
consisting of a single partonic color charge, and postpone issues of possible
radiative corrections to this picture Mehtar-Tani:2017web  to future efforts.
Although we consider scenarios with expanding media, we assume a fixed in-
medium path length, $L=5$ fm.

###  3.1 Transverse momentum dependence of jet suppression

To quantify the suppression factor of jets in heavy-ion collisions one usually
assumes a separation between the short-distance production of hard partons and
their subsequent propagation through the medium. We define medium evolved
spectra of quark and gluon cascades for quark-initiated jets (denoted by
“$\\{q\\}$”) and gluon-initiated jets (denoted by “$\\{g\\}$”) as

$\displaystyle D(x,\tau;\\{q\\})$
$\displaystyle=\big{[}D_{g}(x,\tau)+D_{S}(x,\tau)\big{]}_{D_{g}(x,0)=0,\,D_{S}(x,0)=x\delta(1-x)}\,,$
(12) $\displaystyle D(x,\tau;\\{g\\})$
$\displaystyle=\big{[}D_{g}(x,\tau)+D_{S}(x,\tau)\big{]}_{D_{g}(x,0)=x\delta(1-x),\,D_{S}(x,0)=0}\,.$
(13)

For high-$p_{T}$ jets, where $\tau\lesssim 1$, one is naturally dominated by
$D(x,\tau;\\{g\\})\approx D_{g}(x,\tau)$ and $D(x,\tau;\\{q\\})\approx
D_{S}(x,\tau)$, as discussed above. For the moment, we also neglect possible
early vacuum-like evolution that can take place prior to when medium effects
become dominant Mehtar-Tani:2017web ; Mehtar-Tani:2021fud ; Takacs:2021bpv .

One of the key observables quantifying inclusive jet suppression is the jet
nuclear modification factor, measured by the LHC experiments Aad:2014bxa ;
Chatrchyan:2012raa ; Aaboud:2018twu ; Adam:2015ewa ; Acharya:2019jyg . In our
approach, the yield for the inclusive jet suppression can be obtained as a
convolution of the $D_{i}(x,\tau)$ distribution with the initial parton
spectra Baier:2001yt ; Salgado:2003gb ; Mehtar-Tani:2014yea ; Adhya:2019qse .
For a given parton species $i$ that is produced in the hard cross section
${\rm d}\sigma^{0}_{i}/({\rm d}p_{\scriptscriptstyle T}\,{\rm d}y)$, we write

$\displaystyle\frac{{\rm d}\sigma^{\rm AA}_{i}}{{\rm d}p_{\scriptscriptstyle
T}\,{\rm d}y}=\int_{0}^{1}\frac{{\rm
d}x}{x}\,D\left(x,\sqrt{x}\tau;\\{i\\}\right)\left.\frac{{\rm
d}\sigma^{0}_{i}}{{\rm d}p_{\scriptscriptstyle T}^{\prime}\,{\rm
d}y}\right|_{p_{\scriptscriptstyle T}^{\prime}=p_{\scriptscriptstyle T}/x}\,,$
(14)

for $i=q,g$ and where the evolution time
$\tau=\sqrt{\hat{q}_{0}/p_{\scriptscriptstyle T}}L$ is written in terms of the
final, _measured_ transverse momentum $p_{\scriptscriptstyle T}$. Here, ${\rm
d}\sigma^{0}/{\rm d}p_{\scriptscriptstyle T}$ denotes the hard spectrum for
producing a parton $i$ in proton-proton collisions, or in “vacuum”. For the
Bjorken case, where the distribution has an explicit dependence on the initial
time $\tau_{0}$, we approximate $D(x,\sqrt{x}\tau,\sqrt{x}\tau_{0})\approx
D(x,\sqrt{x}\tau,\tau_{0})$ Adhya:2019qse  since $\tau_{0}$ is already very
small and the integral (14) is dominated by $x\sim 1$. The quenching factor
for parton species $i$, ${\cal Q}_{i}(p_{\scriptscriptstyle T})$ , defined as
the ratio of partonic medium and vacuum spectra, is then

$\displaystyle{\cal Q}_{i}(p_{\scriptscriptstyle T})\equiv\frac{{\rm
d}\sigma^{\rm AA}_{i}\big{/}({\rm d}p_{\scriptscriptstyle T}\,{\rm d}y)}{{\rm
d}\sigma^{0}_{i}\big{/}({\rm d}p_{\scriptscriptstyle T}\,{\rm d}y)}\,,$ (15)

measured at the same final $p_{\scriptscriptstyle T}$ and the same rapidity
$y$ (or rapidity range). If we approximate the initial parton spectrum by a
power law ${\rm d}\sigma^{0}/{\rm d}p_{\scriptscriptstyle T}\propto
p_{\scriptscriptstyle T}^{-n}$, with fixed power index $n=$const., the
quenching factor is then a $(n-1)^{st}$ moment of the in-medium parton
distribution, i.e. ${\cal Q}_{i}=\int_{0}^{1}{\rm
d}x\,x^{n-1}D(x,\sqrt{x}\tau;\\{i\\})$.

In this study, we go beyond this simple approximation and extend the power law
into a more general form

$\displaystyle\frac{{\rm d}\sigma^{0}_{i}}{{\rm d}p_{\scriptscriptstyle
T}\,{\rm d}y}=A_{0}\left(\frac{p_{\scriptscriptstyle
T,0}}{p_{\scriptscriptstyle T}}\right)^{-n_{i}(p_{\scriptscriptstyle
T},y)}\,,$ (16)

where $n_{i}(p_{\scriptscriptstyle
T},y)=n_{i,0}-\beta_{i}\log(p_{\scriptscriptstyle T}/p_{\scriptscriptstyle
T,0})-\gamma_{i}\log^{2}(p_{\scriptscriptstyle T}/p_{\scriptscriptstyle
T,0})-\delta_{i}\log^{3}(p_{\scriptscriptstyle T}/p_{\scriptscriptstyle T,0})$
which provides a good characterization of the initial parton spectra
Spousta:2015fca . Here $\beta_{i}$, $\gamma_{i}$, and $\delta_{i}$ are
additional free parameters to be fitted separately for quark- and gluon-
initiated jets and $p_{\scriptscriptstyle T,0}$ is fixed to 40 GeV. Note that
$A_{0}$ scales out in any ratio observable we are interested in, and the
rapidity dependence is parameterized via the coefficients $\beta_{i}$,
$\gamma_{i}$ and $\delta_{i}$ that are extracted for each rapidity selection.
We will keep the rapidity dependence implicit as a rule in the formulas below.

In this case, the quenching factor for a given parton species $i$, is written
as

$\displaystyle{\cal Q}_{i}(p_{\scriptscriptstyle T})=\int_{0}^{1}{\rm
d}x\,(p_{\scriptscriptstyle
T})^{V_{1}}x^{n-1+V_{2}}D(x,\sqrt{x}\tau;\\{i\\})\,,$ (17)

where,

$\displaystyle V_{1}$ $\displaystyle=$
$\displaystyle\beta_{i}\log(x)-\gamma_{i}\log^{2}(x)+2\gamma_{i}\log\Big{(}\frac{p_{\scriptscriptstyle
T}}{p_{\scriptscriptstyle T,0}}\Big{)}\log(x)$ (18) $\displaystyle+$
$\displaystyle 3\delta_{i}\log^{2}\Big{(}\frac{p_{\scriptscriptstyle
T}}{p_{\scriptscriptstyle
T,0}}\Big{)}\log(x)-3\delta_{i}\log\Big{(}\frac{p_{\scriptscriptstyle
T}}{p_{\scriptscriptstyle T,0}}\Big{)}\log^{2}(x)+\delta_{i}\log^{3}(x)$
$\displaystyle V_{2}$ $\displaystyle=$
$\displaystyle\beta_{i}\log\Big{(}\frac{p_{\scriptscriptstyle
T}}{xp_{\scriptscriptstyle
T,0}}\Big{)}+\gamma_{i}\log^{2}\Big{(}\frac{p_{\scriptscriptstyle
T}}{xp_{\scriptscriptstyle
T,0}}\Big{)}+\delta_{i}\log^{3}\Big{(}\frac{p_{\scriptscriptstyle
T}}{xp_{\scriptscriptstyle T,0}}\Big{)}.$ (19)

This parameterization can be extended to a larger number of terms in
$n(p_{\scriptscriptstyle T})$, but in was previously found that this provides
a good description of the jet spectrum in proton-proton collisions over a
large $p_{\scriptscriptstyle T}$ range, $p_{\scriptscriptstyle T}<1000$ GeV
Spousta:2015fca .

The full nuclear modification factor contains the contributions from both
quarks and gluons to the final jet spectra, and reads

$\displaystyle R_{\mathrm{AA}}(p_{\scriptscriptstyle
T})=\frac{\sum_{i=q,g}\frac{{\rm d}\sigma^{\rm AA}_{i}}{{\rm
d}p_{\scriptscriptstyle T}}}{\sum_{i=q,g}\frac{{\rm d}\sigma^{0}_{i}}{{\rm
d}p_{\scriptscriptstyle T}}}\,.$ (20)

For the time being, we will neglect nuclear effects on PDFs, since they are
small in the relevant $100\text{ GeV}<p_{\scriptscriptstyle T}<400\text{ GeV}$
range we are mainly interested in. The nuclear modification factors calculated
for quark-initiated and gluon-initiated cascades are then combined using the
fraction of quark-initiated jets,

$\displaystyle f_{q}(p_{\scriptscriptstyle
T})=\left[1+\left(\frac{1-f_{q,0}}{f_{q,0}}\right)\left(\frac{p_{\scriptscriptstyle
T,0}}{p_{\scriptscriptstyle T}}\right)^{V_{3}}\right]^{-1}$ (21)

where $f_{q,0}$ is fraction of quark-initiated jets at $p_{\scriptscriptstyle
T,0}$ and

$\displaystyle V_{3}$ $\displaystyle=$
$\displaystyle{n_{g}-n_{q}}+(\beta_{g}-\beta_{q})\log\bigg{(}\frac{p_{\scriptscriptstyle
T}}{p_{\scriptscriptstyle
T,0}}\bigg{)}+(\gamma_{g}-\gamma_{q})\log^{2}\bigg{(}\frac{p_{\scriptscriptstyle
T}}{p_{\scriptscriptstyle T,0}}\bigg{)}$ (22) $\displaystyle+$
$\displaystyle(\delta_{g}-\delta_{q})\log^{3}\bigg{(}\frac{p_{\scriptscriptstyle
T}}{p_{\scriptscriptstyle T,0}}\bigg{)}.$

Finally, the combined nuclear modification factor $R_{\mathrm{AA}}$ reads

$\displaystyle R_{\mathrm{AA}}=f_{q}(p_{\scriptscriptstyle T})\,{\cal
Q}_{q}(p_{\scriptscriptstyle T})+\big{[}1-f_{q}(p_{\scriptscriptstyle
T})\big{]}\,{\cal Q}_{g}(p_{\scriptscriptstyle T})\,.$ (23)

The parameter $f_{q,0}$ and parameters of the extended power law description
of input parton spectra were obtain from fits of jet spectra from PYTHIA8
Sjostrand:2007gs  and they are summarized in Tab. 1. Jet spectra were obtained
by running the anti-$k_{t}$ algorithm Cacciari:2008gp  with resolution
parameter $R=0.4$ for the center of mass energy of
$\sqrt{s_{\mathrm{NN}}}=5.02$ TeV. Jet spectra were reconstructed separately
for several rapidity selections specified in Tab. 1 and used later in the
study. Distributions calculated inclusively in rapidity were obtained for
rapidity selection of $|y|<2.8$. The jets were matched to one of two outgoing
partons from the leading order hard-scattering process by choosing the parton
with the smallest angular distance, $R=\sqrt{\Delta\phi^{2}+\Delta\eta^{2}}$,
and the flavor of the jet was assigned to be that of the matched parton.
Furthermore, for the numerical results presented below we set
$\alpha_{s}=0.1$.

parameters | $n_{g}$ | $n_{q}$ | $\beta_{g}$ | $\beta_{q}$ | $\gamma_{g}$ | $\gamma_{q}$ | $\delta_{g}$ | $\delta_{q}$ | $f_{q,0}$
---|---|---|---|---|---|---|---|---|---
$|y|<0.3$ | 3.21 | 2.52 | 0.99 | 0.93 | 0 | 0 | 0 | 0 | 0.16
$|y|<2.8$ | 3.63 | 3.14 | 1.00 | 0.92 | 0 | 0 | 0 | 0 | 0.27
$2.1<|y|<2.8$ | 5.63 | 5.00 | 0.35 | 0.30 | $-1.5\cdot 10^{-10}$ | -0.19 | 0.24 | 0.27 | 0.42

Table 1: List of parameters from the extended power law characterization of
input parton spectra. Figure 2: The jet $R_{\mathrm{AA}}$ for four medium
profiles: static soft (black), static (green), exponential (blue), and Bjorken
with $t_{0}=0.1$ fm (red) The $\hat{q}_{0}$ values were optimized for each
profile to reproduce the data (in gray) Aaboud:2018twu .

The nuclear modification factor $R_{\mathrm{AA}}$ is plotted in Fig. 2 for
four types of medium expansion. We also include experimental data on measured
anti-$k_{T}$ jets with $R=0.4$ Aaboud:2018twu . For each medium profile, we
use value of $\hat{q}_{0}$ obtained from the $\chi^{2}$ minimization of the
difference in the nuclear modification factor between the data and the theory.
We have fixed the parameter $L=5$ fm for all the medium profiles. The obtained
$\hat{q}_{0}$ values for realistic quark and gluon initial spectra are
tabulated together with the values obtained for a gluon-only calculation with
a simple power-law spectrum discussed in Ref. Adhya:2019qse  in Table 2. One
can see that the extracted value of the jet quenching parameter differs
significantly between the simple configuration which uses gluon-only cascades
and power-law input parton spectra and the more realistic treatment which uses
both the quark and gluon degrees of freedom and improved description of input
parton spectra. This illustrates a simple fact that the value of jet quenching
parameter has an important dependence on the level of completeness of the jet
energy loss description and approximations used. Since the $R_{\mathrm{AA}}$
distributions do not scale when using the above quoted $\hat{q}_{0}$ values,
we confirm the breakdown of the “effective” scaling laws for
$R_{\mathrm{AA}}$, which was discussed in Ref. Adhya:2019qse , also for more
realistic treatment presented in this paper.

The results in Fig. 2 shows that the single-parton description of the jet
$R_{\mathrm{AA}}$ used here describes the shape of the experimental data in
the $p_{\scriptscriptstyle T}$ region below $\sim 250$ GeV. On the contrary,
at $p_{\scriptscriptstyle T}\gtrsim 250$ GeV the calculated $R_{\mathrm{AA}}$
is steeper compared to the data. This was already observed in Ref.
Adhya:2019qse  for the gluon-only calculation and the more complete
description of jet energy loss used here does not lead to an improvement in
the description of the data. This behavior signals the onset of the quenching
of multiple resolved partons in a jet CasalderreySolana:2012ef . These appear
due to hard vacuum-like splittings inside the medium Mehtar-Tani:2017web , see
also Caucal:2018dla . A resummation of such fluctuations leads to stronger
suppression at high-$p_{\scriptscriptstyle T}$ due to enhanced energy loss
Mehtar-Tani:2021fud ; Takacs:2021bpv .

The inability to describe the full $p_{\scriptscriptstyle T}$ region of
measured $R_{\mathrm{AA}}$ is in contrast with parametric modeling presented
in Ref. Spousta:2015fca , where the use of improved description of input
parton spectra and use of appropriate quark and gluon mixing was shown to lead
to a good description of the $R_{\mathrm{AA}}$ over a large
$p_{\scriptscriptstyle T}$ range under the assumption of
$\sqrt{p_{\scriptscriptstyle T}}$-dependence of the lost energy. Within the
BDMPS the $\sqrt{p_{\scriptscriptstyle T}}$-dependence of the lost energy
holds at low-$p_{\scriptscriptstyle T}$, but does not hold at
high-$p_{\scriptscriptstyle T}$ Baier:2001yt , where the single-scattering
contribution to the medium-induced energy loss dominates in calculations
Wiedemann:2000za , see also Mehtar-Tani:2021fud ; Takacs:2021bpv . Moreover,
both here an in Ref. Spousta:2015fca  the nPDF effects are missing which can
also contribute to the behavior of the $R_{\mathrm{AA}}$ at
high-$p_{\scriptscriptstyle T}$ Mehtar-Tani:2021fud ; Takacs:2021bpv .

Quenching parameter ($\hat{q}$) | Static (soft) | Static | Expo | Bjorken
---|---|---|---|---
$t_{0}=0.1$ fm
$\hat{q}_{0}$ (full) [GeV3] | 0.15 | 0.2 | 0.077 | 1.8
$\hat{q}_{0}$ (gluon-only) [GeV3] | 0.20 | 0.2 | 0.090 | 2.6
Table 2: The jet quenching parameter for different media profiles obtained
from fitting the calculation to the data from Ref. Aaboud:2018twu . Full
description of the input parton spectra (first line) can be compared with
results using gluon-only cascades and power-law description of input parton
spectra (second line), published in Ref. Adhya:2019qse .

###  3.2 Initial-time dependence of jet suppression

The value of $t_{0}$ in the Bjorken medium quantifies the time during which no
interaction with the medium occurs. This reflects a situation with a finite
time needed for the onset of the quenching interactions in the medium. In this
section, we discuss the impact of the choice of $t_{0}$ on the observable jet
suppression by comparing two different starting times of the quenching in the
Bjorken medium, namely $t_{0}=0.1$ fm and $t_{0}=1.0$ fm. Left panel of Fig. 3
shows a comparison of the $\langle\hat{q}\rangle$ for the Bjorken media with
these two choices of $t_{0}$ evaluated as a function of $L$ calculated as

$\displaystyle\langle\hat{q}\rangle=\frac{2}{(L-t_{0})^{2}}\int_{t_{0}}^{L}{\rm
d}t\,(t-t_{0})\hat{q}(t)\,,$ (24)

where $\hat{q}(t)=\hat{q}_{0}t_{0}/t$ for the Bjorken scenario. One can see
that delaying the onset of quenching leads to a larger average value of
$\hat{q}$ at $L>t_{0}$ compared to an earlier time of the quenching. In
general, the later the quenching sets in, the larger the
$\langle\hat{q}\rangle$ is, which makes the medium more opaque at the later
stages of the evolution.

Figure 3: Left: The $\hat{q}$ as a function of the length of medium, $L$, for
Bjorken medium for two values of initial time of quenching, $t_{0}=0.1$ fm and
$t_{0}=1.0$ fm. Right: Jet $R_{\mathrm{AA}}$ as a function of
$p_{\scriptscriptstyle T}$ for Bjorken medium with $L=5$ fm and with
$\hat{q}_{0}$ values optimized to reproduce the data (in gray) Aaboud:2018twu
.

We used again the $\chi^{2}$ minimization procedure to find the $\hat{q}_{0}$
value which minimizes the difference in the $R_{\mathrm{AA}}$ between the data
and the theory prediction for the two $t_{0}$ values with medium length fixed
to $L=5$ fm. The results along with the data are shown in the right panel of
Fig. 3. For $t_{0}=0.1$ and $1.0$ fm, the $\hat{q}_{0}$ is found to be
approximately 1.8 GeV3 and 1.0 GeV3, respectively. That is for ten times
larger initial time, the $\hat{q}_{0}$ has to be roughly two times smaller to
obtain the same $R_{\mathrm{AA}}$. While the choice of $t_{0}$ can be somehow
absorbed into an adjusted value of $\hat{q}_{0}$, we will see below that it
affects other observables. In the next two sections, we use these optimized
$\hat{q}_{0}$ values, and their respective $t_{0}$ parameters, to calculate
more differential observables derived from $R_{\mathrm{AA}}$ which
characterize the inclusive jet suppression and quantify how these settings
affect them.

###  3.3 Rapidity dependence of jet suppression

Following the measurement done in Ref. Aaboud:2018twu , we study the rapidity
dependence of the jet suppression. The optimized $\hat{q}_{0}$ values from
Secs. 3.1 and 3.2 along with parameters from Tab. 1 are used to calculate the
rapidity dependent $R_{\mathrm{AA}}$. The results are presented in Fig. 4 in
terms of the ratio of the $R_{\mathrm{AA}}$ in a given rapidity region to the
$R_{\mathrm{AA}}$ in central rapidity region ($|y|<0.3$). In detail, Fig. 4(a)
shows the ratio of $R_{\mathrm{AA}}$ in $|y|<2.8$ and $|y|<0.3$ as a function
of $p_{\scriptscriptstyle T}$ for four medium profiles: static soft, static,
exponential, and Bjorken with $t_{0}=0.1$ fm. The distributions for different
profiles differ by less than 10% for all $p_{\scriptscriptstyle T}$ values.
One can also see that the ratio decreases with the increasing
$p_{\scriptscriptstyle T}$. Particularly significant is the enhancement at
$p_{\scriptscriptstyle T}\lesssim 100$ GeV which may be tested by experiments
and which was not yet measured. For $p_{\scriptscriptstyle T}>100$ GeV the
calculation is consistent with measured data.

The middle and right panels, Fig 4(b), detail the rapidity dependence for a
fixed $p_{\scriptscriptstyle T}$, namely 316 GeV $<p_{\scriptscriptstyle T}<$
562 GeV. The middle panel of Fig. 4 shows the ratio as a function of rapidity
for one particular $p_{\scriptscriptstyle T}$ bin and for the same selection
of medium profiles as in Fig. 4(a). One can see that with increasing rapidity
the ratio decreases in the same way for all the profiles, but somewhat less
strongly than in the data. The right panel of Fig. 4 shows the ratio as a
function of rapidity for Bjorken media with $t_{0}=0.1$ fm and $t_{0}=1.0$ fm.
One can see a sizable difference in the magnitude of the double ratio between
these two configurations – ten times larger $t_{0}$ leads to almost 2 times
smaller values of the double ratio in the most forward rapidity bin. The
values of the double ratio for larger $t_{0}$ are in general better matching
the data for this particular observable.

The fact that the rapidity dependence of the inclusive jet suppression is very
similar for static, exponential, and Bjorken medium with small $t_{0}$ implies
that the origin of the rapidity dependence is somewhat universal. To check
this, we have calculated the ratio of $R_{\mathrm{AA}}$ for gluon-only
evolution. When evaluated as a function of $|y|$ we see the same trends as in
the default case that includes both quark and gluon fragmentation. This
implies that the rapidity dependence is primarily a consequence of the change
in the steepness of input parton spectra. However, besides that, the timing of
the onset of the jet quenching has clearly also an impact on this observable
quantity.

(a) $p_{\scriptscriptstyle T}$ dependence.

(b) $y$ dependence.

Figure 4: Left: The ratio of $R_{\mathrm{AA}}$ in $2.1<|y|<2.8$ and
$R_{\mathrm{AA}}$ in $|y|<0.3$ as a function of jet $p_{\scriptscriptstyle
T}$. Middle: The ratio of $R_{\mathrm{AA}}$ in different $|y|$ bins and
$R_{\mathrm{AA}}$ in $|y|<0.3$ for $p_{\scriptscriptstyle T}=316-512$ GeV.
Black, green, blue, and red color encodes static soft, static, exponential,
and Bjorken medium profiles, respectively. Right: The ratio of
$R_{\mathrm{AA}}$ in different $|y|$ bins and $R_{\mathrm{AA}}$ in $|y|<0.3$
for $p_{\scriptscriptstyle T}=316-512$ GeV for Bjorken medium profile with
$t=0.1$ fm (red) and $t=1.0$ fm (blue). The ATLAS data taken from Ref.
Aaboud:2018twu .

###  3.4 Path-length dependence of jet suppression and jet $v_{2}$

The path-length dependence of jet suppression can be studied using the jet
$v_{2}$ evaluated in a simple approximation Zigic:2018smz  as

$v_{2}=\frac{1}{2}\frac{R_{\mathrm{AA}}(L^{in})-R_{\mathrm{AA}}(L^{out})}{R_{\mathrm{AA}}(L^{in})+R_{\mathrm{AA}}(L^{out})},$
(25)

where $L^{in}$ and $L^{out}$ are path lengths in the direction of the event
plane and in the direction perpendicular to the event plane, respectively.
Values of $L^{in}$ and $L^{out}$ were obtained from a hard sphere model where
vertices of hard processes and directions of back-to-back initial partons in
the azimuthal plane were randomly generated in the overlapping region of two
spheres. The mapping between the impact parameter and $N_{coll}$ values is
done using a Glauber model Loizides:2017ack . It should be stressed that this
approach represents a tool for basic investigation of the relation between the
medium expansion and the jet $v_{2}$, rather than a modeling with a full
description of the problem. The values $L^{in}$ and $L^{out}$ for different
centralities are listed in Tab. 3.

The upper left panel of Fig. 5 shows the jet $v_{2}$ evaluated as a function
of $p_{\scriptscriptstyle T}$ for central collisions ($N_{coll}=1580$) along
with the result of measurements from Refs. Aad:2013sla ; ATLAS:2020qxc . One
can see that the magnitude and the trends of the jet $v_{2}$ distribution
obtained from the calculation match those seen in the data. The upper right
panel of Fig. 5 shows the jet $v_{2}$ evaluated as a function of $N_{coll}$.
Both $p_{\scriptscriptstyle T}$ and $N_{coll}$ dependence of jet $v_{2}$ show
rather small differences among static, static soft, and exponential medium.
The results for Bjorken medium with $t_{0}=0.1$ fm differ significantly from
results for other medium profiles. Even larger difference can be seen for the
Bjorken medium with two different choices of $t_{0}$ shown in lower panels of
Fig. 5. These observations have similar implications as those discussed in the
case of the rapidity dependence of $R_{\mathrm{AA}}$: the impact of the medium
expansion can be largely scaled out by a suitable choice of $\hat{q}_{0}$,
which is however not the case for the starting time of the quenching; the jet
$v_{2}$ remains sensitive to choice of $t_{0}$. This is in agreement with
findings of the sensitivity of $v_{2}$ on $t_{0}$ published in Ref.
Andres:2019eus  which was done in more complex modeling of the collision
geometry, but less complex modeling of the medium induced showering than in
this paper.

$N_{coll}$ | 1910 | 1800 | 1580 | 1300 | 975 | 660 | 380 | 170 | 70
---|---|---|---|---|---|---|---|---|---
$L_{in}$ | 5.6 | 4.8 | 4.3 | 3.6 | 3.0 | 2.4 | 1.8 | 1.1 | 0.5
$L_{out}$ | 5.6 | 5.5 | 5.2 | 5.0 | 4.7 | 4.3 | 3.7 | 2.7 | 2.1
Table 3: The length parameters $L_{in}$ and $L_{out}$ from the hard sphere
model for ten values of number of binary collisions, $N_{coll}$.

(a) $p_{\scriptscriptstyle T}$ dependence.

(b) Centrality dependence.

Figure 5:  The jet $v_{2}$ elliptic flow for the static and expanding profiles
with optimized $\hat{q_{0}}$ as a function of impact parameter
$p_{\scriptscriptstyle T}$ (left) and $N_{coll}$ right. Upper panels show the
jet $v_{2}$ for static medium, exponential medium, and Bjorken medium with
$t_{0}=0.1$ fm. Lower panels show results for Bjorken media for $t_{0}=0.1$ fm
and $t_{0}=1.0$ fm.

##  4 Summary and Conclusions

Both quark and gluon degrees of freedom were used to calculate the evolution
of partonic cascades in expanding media (static, exponential, and Bjorken).
This together with an improved description of input parton spectra makes the
modeling of medium induced radiation in the limit of multiple soft scattering
more realistic compared to gluon-only cascades and power-law modeling of input
parton spectra.

Using numerical estimates of medium-induced parton spectra, the nuclear
modification factor $R_{\mathrm{AA}}$ of jets at a fixed cone-size $R=0.4$ was
calculated and jet quenching parameter, $\hat{q}$, was extracted by comparing
the calculations with the data. The differences between extracted $\hat{q}$ in
the improved description and in the simple, gluon-only scenario Adhya:2019qse
, were found to be substantial (e.g. for Bjorken medium with $t_{0}=0.1$ fm
the $\hat{q}_{0}$ value is by $30\%$ smaller in the improved description
compared the simple gluon-only case). At the same time, the shape of the
$R_{\mathrm{AA}}$ is not modified significantly in the improved description
compared to the simple scenario. Especially the lack of flattening of the
$R_{\mathrm{AA}}$ at high-$p_{\scriptscriptstyle T}$ is noticeable, and will
be addressed in more detail in future efforts.

The rapidity dependence of the jet suppression is studied and it is found that
the decrease of the $R_{\mathrm{AA}}$ seen in the forward region by ATLAS
Aaboud:2018twu  is primarily a consequence of the change in the steepness of
input parton spectra. When comparing results for different medium profiles we
conclude that the rapidity dependence of the inclusive jet suppression is not
very sensitive to the way how the medium expands, however it is sensitive to
the choice of the time of the onset of jet quenching. In the most forward
rapidity bin the difference in the double ratio of $R_{\mathrm{AA}}$ is
achieving almost a factor of two between Bjorken scenario with $t_{0}=0.1$ fm
and $t_{0}=1.0$ fm. Further, the jet $v_{2}$ is studied in an approximation of
hard-sphere geometry. It is found that the differences in the jet $v_{2}$
between full calculation of static medium, static soft, and exponential medium
are within 15%. On the contrary, jet $v_{2}$ differs in maximum roughly by a
factor of two between Bjorken scenario with $t_{0}=0.1$ fm and $t_{0}=1.0$ fm.

It is noticeable that the trends of the evolution with $t_{0}$ are opposite
for jet $v_{2}$ and the rapidity dependence of $R_{\mathrm{AA}}$: while the
jet $v_{2}$ increases with increasing $t_{0}$ the double ratio of
$R_{\mathrm{AA}}$ decreases with increasing $t_{0}$. This indicates that the
simultaneous study of jet $v_{2}$ and rapidity dependence of the jet
production may help revealing the details of the medium expansion.

The presented study, which quantifies the inclusive jet suppression
differentially in the full set of basic kinematic quantities – transverse
momentum, rapidity, and the azimuth, represents one more step towards
realistic modeling of the parton energy loss. It should contribute to the
effort of disentangling the impact of different effects on measured quantities
characterizing inclusive jet suppression.

###### Acknowledgements.

KT is supported by a Starting Grant from Trond Mohn Foundation (BFS2018REK01)
and the University of Bergen. CAS receives financial support from European
Union’s Horizon 2020 research and innovation program under the grant agreement
No. 82409; from Xunta de Galicia (Centro de investigacion de Galicia
accreditation 2019-2022); from the European Union ERDF; from the Spanish
Research State Agency by “Maria de Maeztu” Units of Excellence program
MDM-2016-0692 and project FPA2017-83814-P and from the European Research
Council project ERC-2018-ADG-835105 YoctoLHC. SPA and MS are supported by
Grant Agency of the Czech Republic under grant 18-12859Y, by the Ministry of
Education, Youth and Sports of the Czech Republic under grant LTT 17018, and
by Charles University grant UNCE/SCI/013.

##  Appendix A Kinematical factors in splitting rates

The kinematical factors $\kappa_{ij}(z)$, appearing in the splitting rates,
read

$\displaystyle\kappa_{\rm gg}(z)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{(1-z)C_{A}+z^{2}C_{A}}{z(1-z)}}\;,$ (26)
$\displaystyle\kappa_{\rm qg}(z)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{C_{F}-z(1-z)C_{A}}{z(1-z)}}\;,$ (27)
$\displaystyle\kappa_{\rm gq}(z)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{(1-z)C_{A}+z^{2}C_{F}}{z(1-z)}}\;,$ (28)
$\displaystyle\kappa_{\rm qq}(z)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{zC_{A}+(1-z)^{2}C_{F}}{z(1-z)}}\;.$ (29)

Finally, the un-regularized Altarelli-Parisi splitting functions at leading
order are given by

$\displaystyle P_{\rm gg}(z)$ $\displaystyle=$
$\displaystyle\frac{1}{2}2C_{A}\frac{[1-z(1-z)]^{2}}{z(1-z)}\;,$ (30)
$\displaystyle P_{\rm qg}(z)$ $\displaystyle=$
$\displaystyle\frac{1}{2}2n_{f}T_{R}\Big{(}z^{2}+(1-z)^{2}\Big{)}\;,$ (31)
$\displaystyle P_{\rm gq}(z)$ $\displaystyle=$
$\displaystyle\frac{1}{2}C_{F}\frac{1+(1-z)^{2}}{z}\;,$ (32) $\displaystyle
P_{\rm qq}(z)$ $\displaystyle=$
$\displaystyle\frac{1}{2}C_{F}\frac{1+z^{2}}{(1-z)}\;.$ (33)

## References

  * (1) J. D. Bjorken, _Energy Loss of Energetic Partons in Quark - Gluon Plasma: Possible Extinction of High p(t) Jets in Hadron - Hadron Collisions_ , .
  * (2) M. Gyulassy and M. Plumer, _Jet Quenching in Dense Matter_ , _Phys.Lett._ B243 (1990) 432.
  * (3) X.-N. Wang, M. Gyulassy and M. Plumer, _The LPM effect in QCD and radiative energy loss in a quark gluon plasma_ , _Phys.Rev._ D51 (1995) 3436 [hep-ph/9408344].
  * (4) PHENIX collaboration, _Suppression of hadrons with large transverse momentum in central Au+Au collisions at $\sqrt{s_{NN}}$ = 130-GeV_, _Phys. Rev. Lett._ 88 (2002) 022301 [nucl-ex/0109003].
  * (5) STAR collaboration, _Centrality dependence of high $p_{T}$ hadron suppression in Au+Au collisions at $\sqrt{s}_{NN}$ = 130-GeV_, _Phys. Rev. Lett._ 89 (2002) 202301 [nucl-ex/0206011].
  * (6) STAR collaboration, _Disappearance of back-to-back high $p_{T}$ hadron correlations in central Au+Au collisions at $\sqrt{s_{NN}}$ = 200-GeV_, _Phys. Rev. Lett._ 90 (2003) 082302 [nucl-ex/0210033].
  * (7) CMS collaboration, _Study of high-pT charged particle suppression in PbPb compared to $pp$ collisions at $\sqrt{s_{NN}}=2.76$ TeV_, _Eur. Phys. J._ C72 (2012) 1945 [1202.2554].
  * (8) ATLAS collaboration, _Measurement of the jet radius and transverse momentum dependence of inclusive jet suppression in lead-lead collisions at $\sqrt{s_{NN}}$= 2.76 TeV with the ATLAS detector_, _Phys. Lett. B_ 719 (2013) 220 [1208.1967].
  * (9) ATLAS collaboration, _Measurement of charged-particle spectra in Pb+Pb collisions at $\sqrt{{s}_{\mathsf{{NN}}}}=2.76$ TeV with the ATLAS detector at the LHC_, _JHEP_ 09 (2015) 050 [1504.04337].
  * (10) ALICE collaboration, _Suppression of Charged Particle Production at Large Transverse Momentum in Central Pb-Pb Collisions at $\sqrt{s_{NN}}=$ 2.76 TeV_, _Phys. Lett._ B696 (2011) 30 [1012.1004].
  * (11) ATLAS collaboration, _Measurements of the Nuclear Modification Factor for Jets in Pb+Pb Collisions at $\sqrt{s_{\mathrm{NN}}}=2.76$ TeV with the ATLAS Detector_, _Phys. Rev. Lett._ 114 (2015) 072302 [1411.2357].
  * (12) ALICE collaboration, _Measurement of jet suppression in central Pb-Pb collisions at $\sqrt{s_{\rm NN}}$ = 2.76 TeV_, _Phys. Lett. B_ 746 (2015) 1 [1502.01689].
  * (13) CMS collaboration, _Measurement of inclusive jet cross sections in $pp$ and PbPb collisions at $\sqrt{s_{NN}}=$ 2.76 TeV_, _Phys. Rev._ C96 (2017) 015202 [1609.05383].
  * (14) D. d’Enterria, _Jet quenching_ , _Landolt-Bornstein_ 23 (2010) 471 [0902.2011].
  * (15) A. Majumder and M. Van Leeuwen, _The Theory and Phenomenology of Perturbative QCD Based Jet Quenching_ , _Prog.Part.Nucl.Phys._ A66 (2011) 41 [1002.2206].
  * (16) Y. Mehtar-Tani, J. G. Milhano and K. Tywoniuk, _Jet physics in heavy-ion collisions_ , _Int. J. Mod. Phys. A_ 28 (2013) 1340013 [1302.2579].
  * (17) J.-P. Blaizot and Y. Mehtar-Tani, _Jet Structure in Heavy Ion Collisions_ , _Int. J. Mod. Phys._ E24 (2015) 1530012 [1503.05958].
  * (18) J. Casalderrey-Solana and C. A. Salgado, _Introductory lectures on jet quenching in heavy ion collisions_ , _Acta Phys. Polon._ B38 (2007) 3731 [0712.3443].
  * (19) S. Cao and X.-N. Wang, _Jet quenching and medium response in high-energy heavy-ion collisions: a review_ , _Rept. Prog. Phys._ 84 (2021) 024301 [2002.04028].
  * (20) I. P. Lokhtin and A. M. Snigirev, _A Model of jet quenching in ultrarelativistic heavy ion collisions and high-p(T) hadron spectra at RHIC_ , _Eur. Phys. J. C_ 45 (2006) 211 [hep-ph/0506189].
  * (21) K. Zapp, G. Ingelman, J. Rathsman, J. Stachel and U. A. Wiedemann, _A Monte Carlo Model for ’Jet Quenching’_ , _Eur. Phys. J. C_ 60 (2009) 617 [0804.3568].
  * (22) N. Armesto, L. Cunqueiro and C. A. Salgado, _Q-PYTHIA: A Medium-modified implementation of final state radiation_ , _Eur.Phys.J._ C63 (2009) 679 [0907.1014].
  * (23) T. Renk, _YaJEM: a Monte Carlo code for in-medium shower evolution_ , _Int. J. Mod. Phys. E_ 20 (2011) 1594 [1009.3740].
  * (24) C. Young, B. Schenke, S. Jeon and C. Gale, _MARTINI event generator for heavy quarks: Initialization, parton evolution, and hadronization_ , _Phys. Rev._ C86 (2012) 034905 [1111.0647].
  * (25) J. Casalderrey-Solana, D. C. Gulhan, J. G. Milhano, D. Pablos and K. Rajagopal, _A Hybrid Strong/Weak Coupling Approach to Jet Quenching_ , _JHEP_ 10 (2014) 019 [1405.3864].
  * (26) Y. He, T. Luo, X.-N. Wang and Y. Zhu, _Linear Boltzmann Transport for Jet Propagation in the Quark-Gluon Plasma: Elastic Processes and Medium Recoil_ , _Phys. Rev. C_ 91 (2015) 054908 [1503.03313].
  * (27) JETSCAPE collaboration, _Multistage Monte-Carlo simulation of jet modification in a static medium_ , _Phys. Rev. C_ 96 (2017) 024909 [1705.00050].
  * (28) J. H. Putschke et al., _The JETSCAPE framework_ , 1903.07706.
  * (29) P. Caucal, E. Iancu and G. Soyez, _Deciphering the $z_{g}$ distribution in ultrarelativistic heavy ion collisions_, _JHEP_ 10 (2019) 273 [1907.04866].
  * (30) W. Ke and X.-N. Wang, _QGP modification to single inclusive jets in a calibrated transport model_ , _JHEP_ 05 (2021) 041 [2010.13680].
  * (31) R. Baier, Y. L. Dokshitzer, A. H. Mueller and D. Schiff, _Quenching of hadron spectra in media_ , _JHEP_ 0109 (2001) 033 [hep-ph/0106347].
  * (32) P. Caucal, E. Iancu and G. Soyez, _Jet radiation in a longitudinally expanding medium_ , 2012.01457.
  * (33) S. P. Adhya, C. A. Salgado, M. Spousta and K. Tywoniuk, _Medium-induced cascade in expanding media_ , _JHEP_ 07 (2020) 150 [1911.12193].
  * (34) R. Baier, Y. L. Dokshitzer, A. H. Mueller, S. Peigne and D. Schiff, _Radiative energy loss of high-energy quarks and gluons in a finite volume quark - gluon plasma_ , _Nucl.Phys._ B483 (1997) 291 [hep-ph/9607355].
  * (35) R. Baier, Y. L. Dokshitzer, A. H. Mueller, S. Peigne and D. Schiff, _Radiative energy loss and p(T) broadening of high-energy partons in nuclei_ , _Nucl.Phys._ B484 (1997) 265.
  * (36) B. Zakharov, _Fully quantum treatment of the Landau-Pomeranchuk-Migdal effect in QED and QCD_ , _JETP Lett._ 63 (1996) 952.
  * (37) B. Zakharov, _Radiative energy loss of high-energy quarks in finite size nuclear matter and quark - gluon plasma_ , _JETP Lett._ 65 (1997) 615 [hep-ph/9704255].
  * (38) J.-P. Blaizot, E. Iancu and Y. Mehtar-Tani, _Medium-induced QCD cascade: democratic branching and wave turbulence_ , _Phys. Rev. Lett._ 111 (2013) 052001 [1301.6102].
  * (39) J.-P. Blaizot and Y. Mehtar-Tani, _Energy flow along the medium-induced parton cascade_ , _Annals Phys._ 368 (2016) 148 [1501.03443].
  * (40) Y. Mehtar-Tani and S. Schlichting, _Universal quark to gluon ratio in medium-induced parton cascade_ , _JHEP_ 09 (2018) 144 [1807.06181].
  * (41) S. Schlichting and I. Soudi, _Fragmentation and equilibration of jets in a QCD plasma_ , 2008.04928.
  * (42) R. Baier, Y. L. Dokshitzer, A. H. Mueller and D. Schiff, _Radiative energy loss of high-energy partons traversing an expanding QCD plasma_ , _Phys. Rev._ C58 (1998) 1706 [hep-ph/9803473].
  * (43) Y. Mehtar-Tani and K. Tywoniuk, _Sudakov suppression of jets in QCD media_ , _Phys. Rev._ D98 (2018) 051501 [1707.07361].
  * (44) Y. Mehtar-Tani, D. Pablos and K. Tywoniuk, _Cone size dependence of jet suppression in heavy-ion collisions_ , 2101.01742.
  * (45) A. Takacs and K. Tywoniuk, _Quenching effects in the cumulative jet spectrum_ , 2103.14676.
  * (46) CMS Collaboration collaboration, “Nuclear modification factor of high transverse momentum jets in pbpb collisions at $\sqrt{s_{NN}}=2.76$ tev.” 2012.
  * (47) ATLAS collaboration, _Measurement of the nuclear modification factor for inclusive jets in Pb+Pb collisions at $\sqrt{s_{\mathrm{NN}}}=5.02$ TeV with the ATLAS detector_, _Phys. Lett._ B790 (2019) 108 [1805.05635].
  * (48) ALICE collaboration, _Measurements of inclusive jet spectra in pp and central Pb-Pb collisions at $\sqrt{s_{\rm{NN}}}$ = 5.02 TeV_, 1909.09718.
  * (49) C. A. Salgado and U. A. Wiedemann, _Calculating quenching weights_ , _Phys.Rev._ D68 (2003) 014008 [hep-ph/0302184].
  * (50) Y. Mehtar-Tani and K. Tywoniuk, _Jet (de)coherence in PbPb collisions at the LHC_ , _Phys. Lett._ B744 (2015) 284 [1401.8293].
  * (51) M. Spousta and B. Cole, _Interpreting single jet measurements in Pb $+$ Pb collisions at the LHC_, _Eur. Phys. J._ C76 (2016) 50 [1504.05169].
  * (52) T. Sjostrand, S. Mrenna and P. Z. Skands, _A Brief Introduction to PYTHIA 8.1_ , _Comput.Phys.Commun._ 178 (2008) 852.
  * (53) M. Cacciari, G. P. Salam and G. Soyez, _The Anti-k(t) jet clustering algorithm_ , _JHEP_ 0804 (2008) 063 [0802.1189].
  * (54) J. Casalderrey-Solana, Y. Mehtar-Tani, C. A. Salgado and K. Tywoniuk, _New picture of jet quenching dictated by color coherence_ , _Phys. Lett._ B725 (2013) 357 [1210.7765].
  * (55) P. Caucal, E. Iancu, A. H. Mueller and G. Soyez, _Vacuum-like jet fragmentation in a dense QCD medium_ , _Phys. Rev. Lett._ 120 (2018) 232001 [1801.09703].
  * (56) U. A. Wiedemann, _Gluon radiation off hard quarks in a nuclear environment: Opacity expansion_ , _Nucl. Phys._ B588 (2000) 303 [hep-ph/0005129].
  * (57) D. Zigic, I. Salom, J. Auvinen, M. Djordjevic and M. Djordjevic, _DREENA-C framework: joint $R_{AA}$ and $v_{2}$ predictions and implications to QGP tomography_, _J. Phys. G_ 46 (2019) 085101 [1805.03494].
  * (58) C. Loizides, J. Kamin and D. d’Enterria, _Improved Monte Carlo Glauber predictions at present and future nuclear colliders_ , _Phys. Rev. C_ 97 (2018) 054910 [1710.07098].
  * (59) ATLAS collaboration, _Measurement of the Azimuthal Angle Dependence of Inclusive Jet Yields in Pb+Pb Collisions at $\sqrt{s_{NN}}=$ 2.76 TeV with the ATLAS detector_, _Phys. Rev. Lett._ 111 (2013) 152301 [1306.6469].
  * (60) ATLAS collaboration, _Measurements of Jet Azimuthal Anisotropies in Pb+Pb collisions at $\sqrt{s_{NN}}$ = 5.02 TeV_, .
  * (61) C. Andres, N. Armesto, H. Niemi, R. Paatelainen and C. A. Salgado, _Jet quenching as a probe of the initial stages in heavy-ion collisions_ , _Phys. Lett. B_ 803 (2020) 135318 [1902.03231].