# Pushing forward jet substructure measurements in heavy-ion collisions

Daniel Pablos<EMAIL_ADDRESS>INFN, Sezione di Torino, via
Pietro Giuria 1, I-10125 Torino, Italy  Alba Soto-Ontoso
<EMAIL_ADDRESS>Université Paris-Saclay, CNRS, CEA, Institut de
physique théorique, 91191, Gif-sur-Yvette, France  Theoretical Physics
Department, CERN, 1211 Geneva 23, Switzerland

###### Abstract

Energetic jets that traverse the quark-gluon plasma created in heavy-ion
collisions serve as excellent probes to study this new state of deconfined QCD
matter. Presently, however, our ability to achieve a crisp theoretical
interpretation of the crescent number of jet observables measured in
experiments is hampered by the presence of selection biases. The aim of this
work is to minimise those selection biases associated to the modification of
the quark- vs. gluon-initiated jet fraction in order to assess the presence of
other medium-induced effects, namely color decoherence, by exploring the
rapidity dependence of jet substructure observables. So far, all jet
substructure measurements at mid-rapidity have shown that heavy-ion jets are
narrower than vacuum jets. We show both analytically and with Monte Carlo
simulations that if the narrowing effect persists at forward rapidities, where
the quark-initiated jet fraction is greatly increased, this could serve as an
unambiguous experimental observation of color decoherence dynamics in heavy-
ion collisions.

††preprint: CERN-TH-2022-164

##  I Introduction

Ultra-relativistic heavy-ion collisions have succeeded in recreating the
extreme temperature and pressure conditions that our Universe experienced
during the first microseconds after the Big Bang. Unraveling the microscopic
properties of the medium that permeated our Universe during this epoch, namely
the quark-gluon plasma (QGP), is one of the long-standing questions of
particle physics Baym (2016); Busza  _et al._ (2018). A widely used approach
to this challenge is to study the modification of high-momentum particles, or
jets, when traversing the QGP, very much like in the Rutherford experiment.
Data recorded during the last two decades at both RHIC Adcox  _et al._ (2002);
Adler  _et al._ (2002); Adam  _et al._ (2020) and the LHC Adam  _et al._
(2015); Khachatryan  _et al._ (2017); Aaboud  _et al._ (2019a); Sirunyan  _et
al._ (2021) has confirmed that the interaction between jets and the QGP leads
to an overall depletion of the jet yield at high-$p_{t}$. Theoretically, this
suppression, commonly known as jet quenching d’Enterria (2010); Majumder and
Van Leeuwen (2011); Mehtar-Tani  _et al._ (2013), is understood as a result of
the wide-angle nature of medium-induced emissions which end up being radiated
outside of the jet cone and thus lead to a net energy loss.

Aiming at a more detailed picture of the multi-scale evolution of jets in the
presence of a thermal medium, experimental measurements in the last five years
have explored jet substructure observables such as the momentum sharing
fraction or the opening angle of a pair of hard subjets Sirunyan  _et al._
(2018a); Acharya  _et al._ (2020, 2022). We refer the reader to Cunqueiro and
Sickles (2022) for a comprehensive review of the latest jet measurements.
These observables can be designed such that the perturbative part of the
radiation phase-space dominates and are, consequently, under better
theoretical control than global ones. Up to now, a varied set of jet
substructure measurements has revealed an overall narrowing of the jet core
with respect to the vacuum baseline Acharya  _et al._ (2022). However, the
current experimental precision is not high enough to discriminate between
disparate theoretical models. In this paper, we develop a strategy that shall
allow future measurements to identify the actual physical mechanism behind the
observed narrowing effect.

The first model, introduced in Ref. Spousta and Cole (2016), argues that the
experimental trend is driven by a larger number of quark-initiated jets, known
to be more collimated, after the $p_{t}$ selection cut in the $Pb+Pb$ sample
with respect to $p+p$. An enhanced quark fraction in $Pb+Pb$ could originate
from a combination of the color charge dependence of jet quenching and the jet
spectrum. That is, since gluon jets radiate more, they will lose more energy
and, as a consequence of the steeply falling spectrum, will not pass the jet
$p_{t}$ selection cut. A critical feature of this model is that if one fixes
the color charge of the jet initiator, no modifications are expected with
respect to vacuum jet evolution (modulo any potential $p_{t}$ dependence of
the observable itself). The natural question is how different the $q/g$
fractions need to be in order to quantitatively describe the data, or
equivalently, how much stronger is the quenching that gluon jets experience.
The authors of Ref. Ringer  _et al._ (2020) achieved a quantitative
description of jet substructure observables with a factor of 4 more quark jets
in $Pb+Pb$ with respect to $p+p$. Since this number was extracted via a global
fit to jet spectrum data Qiu  _et al._ (2019), it is agnostic to the dynamics
of energy loss.111This quark fraction is in tension with the one extracted in
Ref. Brewer  _et al._ (2021) that corresponds to $1.5$ increase (see Fig. 1,
panels (c) and (f)), although an apples-to-apples comparison is not possible
due to the different jet selections used in those studies. Experimentally, no
sizeable modification of the quark and gluon fractions has been observed
Sirunyan  _et al._ (2020); Li and Vitev (2020). Thus, the physical mechanism
that would lead to such a dramatic quenching of gluon jets, far larger than
that expected from Casimir scaling 222Casimir scaling is violated, even in
vacuum, beyond leading order as was shown in Ref. Apolinário  _et al._
(2020)., remains to be settled. Throughout the rest of this paper, we will
refer to this hypothesis as ‘modified $q/g$ fraction model’ and we will
elaborate more on it in Secs. III.1.1 and III.3.

An alternative explanation to the narrowing effect relies on the existence of
a critical resolution angle of the QGP. This angular scale, defined as
$\theta_{c}=2/\sqrt{\hat{q}L^{3}}$ with $\hat{q}$ being the quenching
parameter and $L$ the medium length, naturally emerges when considering the
soft radiation pattern of an antenna in the multiple soft scattering
approximation Mehtar-Tani  _et al._ (2011, 2012a); Casalderrey-Solana and
Iancu (2011); Mehtar-Tani and Tywoniuk (2013); Casalderrey-Solana  _et al._
(2013); Apolinário  _et al._ (2015), and splits the radiation phase-space into
resolved and unresolved emissions.333A resolution scale also appears when
calculating the gluon emission pattern of an antenna using the opacity
expansion formalism Wiedemann (2000); Gyulassy  _et al._ (2000); Sievert and
Vitev (2018), particularly relevant for thin media Mehtar-Tani  _et al._
(2012b); Casalderrey-Solana  _et al._ (2016a). In short, if the opening angle
of a vacuum-like splitting is larger than $\theta_{c}$, its two prongs behave
as independent emitters of medium-induced gluons. On the contrary, collinear
branchings with $\theta<\theta_{c}$ are not resolved by the medium and thus
lose energy coherently as an individual color charge. Therefore, jets with
$\theta>\theta_{c}$ are more quenched, leading to an overall narrowing of the
jet sample. Two jet quenching Monte Carlos that incorporate some notion of
color coherence, namely JetMed Caucal  _et al._ (2018, 2019) and the Hybrid
Strong/Weak Coupling Model Casalderrey-Solana  _et al._ (2014); Hulcher  _et
al._ (2018); Casalderrey-Solana  _et al._ (2020), are able to quantitatively
describe jet substructure data. Naturally, these two models are also sensitive
to the different degree of quenching of quark and gluon jets. However, a
purely coherent description of energy loss in these models is not sufficient
to match the experimental data and thus a resolution criterion in terms of a
critical angle/length is required.

This paper addresses the question on how to experimentally disentangle between
the ‘modified $q/g$ fraction’ and ‘color decoherence’ models. Our strategy is
to explore the rapidity dependence of jet substructure observables. For
simplicity, we focus on the $k_{t}$-distribution of the hardest splitting in a
jet, but our conclusions apply to any jet substructure measurement.444Since
$k_{t}=z\theta$ and the $z$-distribution is barely modified in heavy-ion
collisions, the observed narrowing in terms of $\theta$ is directly translated
into a shift towards smaller $k_{t}$. Preliminary experimental results on
$k_{t}$ can be found in Ref. Bossi (2022). The idea is based on a simple
observation: increasing the jet rapidity enhances the fraction of quark-
initiated jets. For a fixed color charge of the jet initiator, the two models
under study lead to dramatically different predictions for the
$k_{t}$-distribution. On the one hand, as we have already anticipated, the
‘modified $q/g$ fraction’ heavy-ion result would approach the vacuum one when
moving to forward rapidities, since the ensemble is dominated by quark-
initiated jets in both collision systems. In contrast, if the medium is able
to resolve the substructure fluctuations developed during the DGLAP evolution
of the jet, the $k_{t}$-distribution would differ from the $p+p$ baseline for
every rapidity bin, as both quark- and gluon-initiated jets feature wide
($\theta>\theta_{c}$) and narrow ($\theta<\theta_{c}$) configurations.

In this work, we study the rapidity dependence of the leading-$k_{t}$
distribution up to $|y|=4.5$. Experimentally, jet substructure measurements at
such forward rapidities are rather challenging and require dedicated
instrumentation. For this reason, we deem necessary to address the feasibility
of the proposed measurements, both with the current data and in the upcoming
high-luminosity phase of the LHC (HL-LHC). The ATLAS collaboration has
pioneered the study of the rapidity dependence of jet quenching by measuring
both the inclusive jet spectrum Aaboud  _et al._ (2019a) and the fragmentation
function Aaboud  _et al._ (2018) at $|y|<2.8$ and $|y|<2.1$, respectively. CMS
has explored even more forward kinematics and measured the jet cross-section
in the pseudorapidity region $-6.6<\eta<-5.2$ using CASTOR Sirunyan  _et al._
(2019); Khachatryan  _et al._ (2021), but only in $p+Pb$ collisions.
Similarly, the LHCb detector allows for the reconstruction of charged and
neutral particles in the very forward rapidity region ($2<|y|<4.5$) but, to
this date, heavy-ion measurements were limited to peripheral collisions, where
medium effects are expected to be reduced. It is then possible to measure the
leading-$k_{t}$ distribution in both $p+p$ and $Pb+Pb$ collisions with the
current technology of both ATLAS and CMS using charged, high-$p_{t}$ jets up
to $|y|=2.5$ with data recorded during Runs 2 and 3.

As we show below, this rapidity interval could be enough, for jets at
sufficiently high $p_{t}$, to disentangle between the theoretical models
studied in this paper. However, more stringent constraints on the two
confronting pictures can be obtained by pushing the measurements to even
larger rapidities. This can be achieved in the future HL-LHC, where an
integrated luminosity of about $\mathcal{L}=$10 $\rm nb^{-1}$ in $Pb+Pb$
collisions at a centre-of-mass of energy of 5.5 TeV per nucleon pair will be
delivered Citron  _et al._ (2019). A further increase in the integrated
luminosity is expected in case the heavy-ion program is extended up to the end
of Run 6, as recently proposed by the ALICE3 experiment ALICE (2022a). In
fact, this detector would be ideal for this study since it is designed to
cover 8 units in rapidity with very good tracking and $p_{t}$-resolution of
charged particles ALICE (2022a). Upgrades in both ATLAS and CMS will also
extend their rapidity coverage CMS (2017); ATLAS (2021) and LHCb plans to
extend its centrality coverage to semi-central collisions in Run 3 and central
collisions in Run 4 Citron  _et al._ (2019). To sum up, we consider two
experimental settings: (i) ‘current LHC’, which includes measurements up to
$|y|<2.5$, and (ii) ‘future LHC’, where we study jet rapidities as high as
$|y|=4.5$.

Figure 1: Left: Quark-initiated jet fraction as a function of the jet $p_{t}$
for different intervals of jet rapidity $y$. Right: Same as left panel but for
the inclusive jet spectrum.

The rest of the paper is organised as follows. In Section II we consider dijet
events in $p+p$ collisions and study the rapidity dependence of (i) quark and
gluon fractions, (ii) the jet $p_{t}$ spectrum, and (iii) the leading-$k_{t}$
distribution using Pythia Bierlich  _et al._ (2022). We also provide an
analytic estimate of the vacuum $k_{t}$ distribution. Next, in Sec. III we
explore the modification of all these observables when including medium
effects, both analytically (Sec. III.1) and with Monte Carlo simulations
(Secs. III.2, III.3). We calculate the medium-modified $k_{t}$ distribution
within two different semi-analytic approaches. One model is based on the
modified $q/g$ fraction picture and the results are displayed in Sec. III.1.1.
The second model, presented in Sec. III.1.2, incorporates the dynamics of
color decoherence, on a similar fashion as Ref. Caucal  _et al._ (2022). For
the numerical results we pursue two paths. In Sec. III.2, we use the Hybrid
Monte Carlo model to study the rapidity dependence of jet suppression and to
predict the rapidity dependence of the leading-$k_{t}$ distribution. Next, in
Sec. III.3, we present an implementation of the modified $q/g$ fraction model
in which Pythia is used to obtain the vacuum distributions that are then
combined within the logic of a fully coherent energy loss model. We conclude
with a summary of the main results of this manuscript in Sec. IV.

##  II Vacuum baseline

###  II.1 Engineering the quark-initiated jet fraction

A key quantity in this paper is the quark-initiated jet fraction, $q$-fraction
in short. We begin by analysing its rapidity dependence on a dijet sample in
$p+p$ collisions at LHC energies ($\sqrt{s}=5.02$ TeV) using the Pythia
Bierlich  _et al._ (2022) event generator. This quantity is defined as
follows. First, we reconstruct anti-$k_{t}$ Cacciari  _et al._ (2008) jets of
radius $R=0.4$ with FastJet Cacciari  _et al._ (2012). The quark- or gluon-
initiated tag is assigned according to a coincidence procedure, introduced in
Ref. Spousta and Cole (2016), and valid at leading order (LO): for a given
jet, we find the parton produced by the hard-scattering matrix element whose
angular distance, $\Delta R\equiv\sqrt{\Delta\phi^{2}+\Delta\eta^{2}}$, with
respect to the jet axis is minimized. We then assign the quark- or gluon-
initiated jet tag depending on the identity of the selected hard parton,
considered to be the shower-initiator. By selecting different jet momenta and
rapidities one effectively changes the value of Bjorken-$x$ explored in the
incoming protons, i.e. $x=2\,p_{t}\,\textrm{cosh}(y)/\sqrt{s}$, and thus the
flavour of the parton that participates in the hard scattering. The quark-
initiated jet fraction as a function of both the jet rapidity $y$ and
transverse momentum $p_{t}$ is shown in the left panel of Fig. 1. We observe
that the $q$-fraction is enhanced both at forward rapidities and at high jet
$p_{t}$. This is expected since at large values of $x$ the parton distribution
function (PDF) of valence quarks dominate. Thus, by selecting jets within a
given $p_{t}$ range and different bins in rapidity, we can engineer the
$q$-fraction of a given jet ensemble.

Number of jets
---
|  $q$-fraction
Jet $p_{t}$ | $\lesssim 0.3$ | $\lesssim 0.6$ | $\lesssim 0.9$
$20<p_{t}<80$ GeV | $3.2\times 10^{8}$ | $2.7\times 10^{8}$ | $1\times 10^{7}$
$100<p_{t}<150$ GeV | $8\times 10^{5}$ | $4\times 10^{5}$ | $5.4\times 10^{4}$
$225<p_{t}<300$ GeV | $1.6\times 10^{4}$ | $3.4\times 10^{4}$ | $1.5\times 10^{3}$
Table 1: Estimated number of jets, $N_{\rm jets}$, to be measured in heavy-ion
collisions in the high-luminosity phase of the LHC for a given $q$-fraction
range depending on the jet $p_{t}$. Guiding us with the left panel of Fig. 1,
we select the following rapidity windows. For $q$-fraction $\lesssim 0.3$ all
jet $p_{t}$ bins use the window $|y|<0.3$. For $q$-fraction $\lesssim 0.6$,
low-$p_{t}$ uses $2.5<|y|<3$, mid-$p_{t}$ uses $2.1<|y|<2.5$ and high-$p_{t}$
uses $1.2<|y|<2.1$. Finally, for $q$-fraction $\lesssim 0.9$, low-$p_{t}$ uses
$4<|y|<4.5$, mid-$p_{t}$ uses $3<|y|<4$ and high-$p_{t}$ uses $2.1<|y|<2.5$.

As one moves towards larger rapidities, thereby approaching the kinematic
limit determined by the centre-of-mass energy of the hadronic collision, the
jet $p_{t}$ spectrum becomes increasingly steeper, as shown in the right panel
of Fig. 1. These two rapidity-dependent properties of jets, namely the
evolution of the $q$-fraction and the evolution of the power index of the
spectrum, $n$, are the two determining factors to understand jet suppression,
or $R_{\rm AA}$, as a function of rapidity, as will be discussed in Section
III.2.

Using the spectra from the right panel of Fig. 1 we can estimate the expected
number of jets in the HL-LHC corresponding to a range of $q$-fraction values
for different windows in jet $p_{t}$ and $y$.555The actual centre-of-mass
energies will be $\sqrt{s}=5.5$ ATeV, so we expect a slightly larger number of
jets than the ones we estimate by using the spectrum at $\sqrt{s}=5.02$ ATeV.
The number of jets is simply given by $N_{\rm jets}(\Delta p_{t},\Delta
y)\approx\mathcal{L}N_{\rm coll}\int_{\Delta p_{t},\Delta y}d\sigma/dp_{t}dy$,
and we set $\mathcal{L}=10\,\text{nb}^{-1}$ and $N_{\rm coll}\approx 2000$ for
central collisions. The results are shown in Table 1. As we will explicitly
see when computing medium modifications in Secs. III.2, III.3 these numbers
guarantee enough statistics to disentangle the physical mechanisms under
consideration in the present work.

###  II.2 Analytic results at double-logarithmic accuracy

Figure 2: Leading-$k_{t}$ distribution as a function of rapidity in vacuum at
DLA for jets with $p_{t}=120$ GeV and $R=0.4$. The bottom panel display the
ratio to the mid-rapidity result.

Next, we focus on the $k_{t}$-distribution of the hardest splitting in the jet
clustering sequence, as defined by the Dynamical Grooming procedure Mehtar-
Tani  _et al._ (2020) with $a=1$.666Since we are measuring the $k_{t}$
distribution of the splitting with the highest $k_{t}$, the groomed
distribution is equivalent to the plain one. We use the small-angle
approximation and define $k_{t}=z\theta$, where $z$ and $\theta$ are the
momentum sharing fraction and opening angle of the splitting, respectively.
This observable was calculated at next-to-next-to double logarithmic accuracy
and compared to ALICE data in Refs. Caucal  _et al._ (2021a); ALICE (2022b).
In the present analytic study, we do not aim at providing precise predictions
but rather focus on qualitative features of the distribution. As such, we
consider all emissions to be soft and collinear. At this double-logarithmic
accuracy (DLA) the self-normalised $k_{t}$-distribution is given by

$\displaystyle\frac{1}{\sigma}\frac{{\rm d}\sigma}{{\rm
d}k_{t}}\Big{|}_{p_{t},y}$
$\displaystyle=\sum_{i\in\\{q,g\\}}f_{i}\int_{0}^{1}{\rm d}z\int_{0}^{R}{\rm
d}\theta P^{\rm vac}(z,\theta)\delta(k_{t}-z\theta)$ $\displaystyle\times
e^{-\int{\rm d}z^{\prime}\int{\rm d}\theta^{\prime}P^{\rm
vac}(z^{\prime},\theta^{\prime})\Theta(z^{\prime}\theta^{\prime}-k_{t})}$
$\displaystyle\stackrel{{\scriptstyle\text{DLA}}}{{=}}\sum_{i\in\\{q,g\\}}f_{i}\frac{2\bar{\alpha}}{k_{t}}\ln\frac{R}{k_{t}}e^{-\bar{\alpha}\ln^{2}\frac{R}{k_{t}}},$
(1)

where we fixed the strong coupling to
$\bar{\alpha}\equiv\alpha_{s}(p_{t}R)C_{i}/\pi$ with $C_{i}$ the color factor,
$f_{i}$ corresponds to the quark (or gluon) fraction and we have used that the
branching kernel reduces to $P^{\rm vac}=\bar{\alpha}/(z\theta)$ in the soft-
and-collinear limit. In Fig. 2 we plot Eq. (II.2) for different rapidities.
777Our analytic results, both in vacuum and in the medium, rely in general on
the properties of the species- and rapidity-dependent initial jet spectra,
such as their power index $n$ and $q$-fraction. We have used Pythia results to
perform a fit of the jet $p_{t}$-spectra, for jet-initiator species $k$ within
a rapidity window $y$, as ${\rm d}\hat{\sigma}^{(k,y)}/{\rm
d}p_{t}=\sigma^{(k,y)}_{0}\,(p^{(k,y)}_{t,0}/p_{t})^{n^{(k,y)}(p_{t})}$ and
$n^{(k,y)}(p_{t})=\sum_{i=0}^{2}c^{(k,y)}_{i}\ln^{i}(p^{(k,y)}_{t,0}/p_{t})$.
We observe that the low-$k_{t}$ regime is enhanced with increasing rapidity
due to the more collimated DGLAP evolution of $q$-initiated jets. Note that,
at this level of accuracy, the maximum of the distribution is dictated by the
strong coupling constant, i.e. $k_{t,{\rm max}}\propto e^{-1/(2\bar{\alpha})}$
Caucal  _et al._ (2021a). We cut the plot at $z\theta/R=10^{-3}$ to downplay
the region where non-perturbative corrections would dominate.

###  II.3 Pythia results

Since the previous calculation misses several ingredients of a realistic
parton shower, we also calculate the leading-$k_{t}$ distribution using Pythia
simulations. The results are shown in Fig. 3 for quark- and gluon-initiated
jets, as well as the total result. We reconstruct the particles of an
anti-$k_{t}$ jet with $R=0.4$ using the Cambridge/Aachen algorithm Dokshitzer
_et al._ (1997), and look for the highest $k_{t}$ throughout its clustering
history, starting from the last clustering and following the leading branch.
Note that for this particular figure we use a definition of $k_{t}$ with
dimensions of energy, $k_{t}\equiv zp_{t}^{\rm parent}\sin\theta$, where
$p_{t}^{\rm parent}$ is the momentum of the parent branch. In consistency with
what is shown in Fig. 1, the inclusive results for leading-$k_{t}$ almost
coincide with the gluon results at mid-rapidity, while they are dominated by
the quark results when increasing the jet $p_{t}$ and/or selecting larger
rapidities. As expected from the analytics (see Fig. 2), the
$k_{t}$-distribution for quark-initiated jets is shifted towards smaller
values compared to that of gluon-initiated jets.

Figure 3: Leading-$k_{t}$ distribution for quark-initiated and gluon-initiated
jets, as well as the total result, where each panel selects jets with
different rapidity cuts that tag different $q$-fractions, as specified by
Table 1. From top to bottom the jet $p_{t}$ increases. The $p_{t}$ and
rapidity cuts that are accessible with the current LHC technology are marked
in teal, while those of the future LHC runs are highlighted in purple.

##  III Medium results

In the previous section we gained intuition on the shape of the
leading-$k_{t}$ distribution for quark- and gluon-initiated jets, and how we
can engineer jet samples with different $q$-fractions by selecting jets that
belong to specific $p_{t}$ ranges and rapidity windows. We now move to the
description of the medium effects on this observable. Our main goal is to
study how the different physical descriptions of the jet-medium interaction
(‘modified $q/g$ fraction’ and ‘color decoherence’) provide distinctive
results as we study the medium modifications of jet ensembles with different
$q$-fractions.

###  III.1 Analytic calculation in the multiple, soft scattering approximation

Let us begin by generalising Eq. (II.2) to include medium modifications
following the approximations introduced in Ref. Caucal  _et al._ (2022). The
medium-modified leading-$k_{t}$ distribution is given by

$\displaystyle\frac{1}{\sigma}\left.\frac{{\rm d}\sigma}{{\rm
d}k_{t}}\right|_{p_{t},y}$
$\displaystyle=\frac{1}{\mathcal{N}}\sum_{i\in\\{q,g\\}}f^{\rm
n}_{i}\int_{0}^{1}{\rm d}z\int_{0}^{R}{\rm d}\theta P_{i}^{\rm med}(z,\theta)$
$\displaystyle\times{\rm e}^{-\int{\rm d}z^{\prime}\int{\rm
d}\theta^{\prime}P^{\rm
med}(z^{\prime},\theta^{\prime})\Theta(z^{\prime}\theta^{\prime}-k_{t})}\delta(k_{t}-z\theta)$
$\displaystyle\times\int_{0}^{\infty}{\rm
d}\varepsilon\mathcal{E}_{i}(\varepsilon|z,\theta){\rm
e}^{-\frac{n\varepsilon}{p_{t}}},$ (2)

where $\mathcal{N}$ is a normalisation factor, $f^{\rm n}_{i}$ is the
quark/gluon fraction computed using nuclear PDFs (nPDFs) and $n$ is the
spectral index of the $p_{t}$ spectrum.888Similarly to the vacuum case, we
have performed a fit to the jet spectra ${\rm d}\sigma^{(k,y)}/{\rm d}p_{t}$,
with the only difference being the use of the central set of the nPDFs
provided by EPPS16 Eskola  _et al._ (2017). The two main novelties of Eq.
(III.1) are the in-medium branching kernel, $P^{\rm med}$, and the energy loss
probability distribution, $\mathcal{E}$. The former accounts for the fact that
the tagged emission can be either vacuum-like or medium-induced, and may
include constraints on the radiation phase-space. The latter represents the
probability for an $i$-initiated splitting to radiate energy $\varepsilon$ out
of the jet cone via a medium-induced cascade.

In what follows we will specify the exact form of $P^{\rm med}$ and
$\mathcal{E}$ in two different models: the modified $q/g$ modification model
and another based on color decoherence. We would like to emphasize that we
make numerous approximations to simplify the models as much as possible, while
maintaining their main physical ingredients. Consequently, the following
results serve an illustrative purpose and are not meant to be quantitative
predictions. For the color decoherence-based model we follow Ref. Caucal  _et
al._ (2022) and consider the QGP to be a brick of length $L=4$ fm
characterised by the transport coefficient $\hat{q}=0.3$ GeV3 and a coupling
constant for medium-induced emissions of $\alpha^{\rm med}_{s}=0.24$. These
values lead to a reasonably good description of the jet yield depletion
defined as

$R_{\rm AA}=\frac{{\rm d}\sigma^{AA}/{\rm d}p_{t}}{{\rm d}\sigma^{pp}/{\rm
d}p_{t}},$ (3)

in heavy-ion collisions, i.e. $R_{\rm{AA}}\sim 0.47$ at $p_{t}=120$ GeV and
$R=0.4$. Keeping the same parameters for the modified $q/g$ fraction model
would lead to a different value of $R_{{\rm AA}}$, since the energy loss model
is different. Imposing that $R_{{\rm AA}}$ coincides in both models at
$p_{t}=120$ GeV leads to $L=4$ fm, $\hat{q}=0.7$ GeV3 and $\alpha^{\rm
med}_{s}=0.28$.999Many other combinations of ($L,\hat{q},\alpha^{\rm
med}_{s}$) would yield the same $R_{{\rm AA}}$.

We would like to remark that we keep $L$ fixed for all rapidity values. To
understand why this is correct, let us consider a particle moving
perpendicular to the beam axis that travels a length $x$. If instead the
particle moves with some longitudinal momentum, it will traverse, in the
centre-of-mass frame, a length $x^{\prime}=x/\cos\theta$, where $\theta$ is
the angle with respect to the direction perpendicular to the beam. Using the
definition of rapidity, $y\equiv\operatorname{arctanh}{(p_{z}/p)}$, with
$p_{z}$ and $p$ being the longitudinal momentum and total momentum of the
particle, respectively, one sees that $\cos\theta=1/\cosh y$, so we can
express the length travelled by the particle with a finite rapidity in the
centre-of-mass frame as $x^{\prime}=x\cosh y$. Now, the quantity that matters
for energy loss is the distanced travelled by a particle in the local fluid
rest frame (LFRF). In general, the collision centre-of-mass frame will not
coincide with the LFRF, and so a Lorentz transformation needs to be performed
to translate $x^{\prime}$ into its LFRF analogue, that we denote
$x^{\prime}_{F}$. For the case in which a parton is moving with rapidity $y$
and some transverse velocity through a Bjorken-flow, one gets that
$x^{\prime}_{F}(y)=x^{\prime}/\cosh y$ (see, e.g., Ref. Casalderrey-Solana
_et al._ (2016b)). Then, after taking this factor into account we obtain that
$x^{\prime}_{F}=x$. That is, regardless of the rapidity of the particle, it
will always traverse a length $x$ in the LFRF, as if it was moving
transversely to the beam axis. This justifies the use of a rapidity-
independent medium length $L$.

####  III.1.1 Modified $q/g$ fraction model

We consider a description of the in-medium jet evolution that resembles the
one presented in Refs. Spousta and Cole (2016); Ringer  _et al._ (2020). To
start with, the branching kernel is taken to be the vacuum one, i.e. the
leading-$k_{t}$ condition is always met by a vacuum emission and therefore we
set $P^{\rm med}\to\bar{\alpha}/(z\theta)$ in Eq. (III.1). Regarding the
energy loss distribution, we assume that the intrajet activity is irrelevant
and thus every jet loses energy as if it was a single color charge. That is,
we follow the quenching weights paradigm Baier  _et al._ (2001) and calculate
the probability for a single parton of flavor $i$ to lose energy $\varepsilon$
as

$\displaystyle\mathcal{Q}_{i}(p_{t},R)$
$\displaystyle\equiv\int_{0}^{\infty}{\rm
d}\varepsilon\mathcal{E}_{i}(\varepsilon|z,\theta){\rm
e}^{-\frac{n\varepsilon}{p_{t}}}$ (4)
$\displaystyle=\exp\left[\int_{R}^{\infty}{\rm d}\theta\int_{0}^{1}{\rm
d}zP_{i}^{\rm mie}(z,\theta)\left({\rm e}^{-nz}-1\right)\right],$

where we adopt the multiple, soft scattering approximation to describe the
spectrum of medium-induced emissions, i.e.

$\displaystyle P^{\rm mie}(z,\theta)$ $\displaystyle=\bar{\alpha}_{s,\rm
med}\sqrt{\frac{2\omega_{c}}{z^{3}p_{t}}}\Theta(\omega_{c}-zp_{t})$ (5)
$\displaystyle\times
2\theta\frac{z^{2}p^{2}_{t}}{Q^{2}_{s}}\Gamma\left(0,\frac{z^{2}p^{2}_{t}\theta^{2}}{Q^{2}_{s}}\right)\,,$

with $\omega_{c}=\hat{q}L^{2}/2$ being the maximum frequency that a medium-
induced emission can acquire and $Q_{s}=\sqrt{\hat{q}L}$ its typical
transverse momentum. The first line in Eq. (5) corresponds to the energy
spectrum while the second line describes transverse momentum broadening. Note
that Eq. (5) is an approximation of the fully differential medium-induced
spectrum that is only valid in the $k_{t}\ll Q_{s}$ and $\omega\ll\omega_{c}$
limit, i.e. in the soft-and-collinear limit.

Figure 4: Quark fraction as a function of rapidity for jets with $p_{t}=120$
GeV calculated in the modified q/g fraction model. The bottom panel display
the ratio to the $p+p$ result.

Having specified the ingredients of the model, an important remark is in
order. The shape of the energy loss distribution impacts the $q/g$ fraction of
the jet sample after quenching.101010The $q$-fraction can be obtained by
fixing the flavour in Eq. (III.1), integrating over $k_{t}$ and dividing by
the total jet cross section $\mathcal{N}$. Our specific choice, given by Eq.
(4), is comparable in spirit to that of Ref. Spousta and Cole (2016), but it
is expected to differ from Ref. Ringer  _et al._ (2020). The $q$-fractions
obtained with our toy model are displayed in Fig. 4. At mid-rapidity, our
$q$-fraction, driven by a Casimir scaling of energy loss, is similar to that
of Ref. Brewer  _et al._ (2021) but substantially lower than that of Ref.
Ringer  _et al._ (2020). This implies that our results for the $k_{t}$
distribution correspond to a conservative version of the modified $q/g$
fraction model. We will explore $q$-fraction values similar to those of Ref.
Ringer  _et al._ (2020) using Monte Carlo simulations in Sec. III.3. The
evolution with rapidity of the ratio of the $q$-fraction between $Pb+Pb$ and
$p+p$ is displayed in the lower panel of Fig. 4. This marked evolution,
getting close to no modification at all at forward rapidities, compactly
elucidates the motivations of the proposed rapidity scan.

In Fig. 5 we show results for the coherent modification of the leading-$k_{t}$
distribution as a function of the jet rapidity and take the ratio with respect
to the vacuum baseline. Attending to the message provided by Fig. 4, the
interpretation of these results is quite transparent: the visible narrowing of
the $k_{t}$ distribution at mid-rapidities is greatly reduced when the
$q$-fraction is so high that the depletion of gluon jets in the measured
$Pb+Pb$ ensemble becomes irrelevant. Naturally, this model predicts a ratio of
the $k_{t}$-distribution with respect to $p+p$ close to one at large
rapidities. Note that, if following Ref. Ringer  _et al._ (2020), one started
with a larger value of the $q$-fraction in $Pb+Pb$ at mid-rapidity, the
approach of the $k_{t}$-distribution towards the vacuum result with increasing
rapidities would be more abrupt.

Figure 5: Leading-$k_{t}$ distribution in the medium using the modified $q/g$
model for jets with $p_{t}=120$ GeV. The bottom panel displays the ratio to
the $p+p$ result.

####  III.1.2 Color decoherence model

We now turn to a theoretical description of the leading-$k_{t}$ distribution
based on Ref. Caucal  _et al._ (2022). Here we give a brief summary of the
main ingredients of the model, while a more detailed description can be found
in Ref. Caucal  _et al._ (2022). This calculation, grounded in perturbative
QCD, accounts for the factorization in time between vacuum-like and medium-
induced processes in the double-logarithmic approximation. In this case, the
medium branching kernel reads

$P^{\rm med}(z,\theta)=P^{\rm vac}(z,\theta)\Theta_{\notin\rm
veto}(z,\theta)+P^{\rm mie}(z,\theta),$ (6)

where $\Theta_{\notin\rm veto}$ constraints the phase-space for vacuum like
emissions to be Caucal  _et al._ (2018)

$\Theta_{\notin\rm
veto}(z,\theta)=1-\Theta(\theta-\theta_{c})\Theta(k^{2}_{t}-k^{2}_{t,\rm
med})\Theta(L-t_{f}),$ (7)

with the formation time of an emission given by $t_{f}=2/(k_{t}\theta)$ and
$k^{2}_{t,\rm med}=\hat{q}t_{f}$ is the minimum transverse momentum acquired
by the emission via multiple soft collisions during its formation. In a
nutshell, Eq. (7) imposes that vacuum emissions inside the medium are only
allowed for sufficiently high values of $k_{t}$ or, equivalently, short
formation times such that they are not affected by medium dynamics. This
separation has been rigorously proven at DLA in Ref. Caucal  _et al._ (2018).

Note that compared to the model presented in Sec. III.1.1, Eq. (6) includes
two new ingredients: (i) the possibility that a medium-induced emission is the
one with the highest-$k_{t}$, and (ii) a restriction in the phase-space for
vacuum emissions. Regarding the first point, since we describe the
interactions between the hard propagating parton and the medium in the
multiple, soft scattering approximation, we use Eq. (5) to describe the
medium-induced branching probability. This is an important aspect of the
calculation since the leading-$k_{t}$ distribution has been proposed as a
potential observable to search for QCD Molière scattering in the medium
D’Eramo  _et al._ (2019); Ehlers (2021); Hulcher  _et al._ (2022).
Importantly, our analytic estimates provide the multiple, soft scattering
baseline for future studies which will account for rare, hard scatterings.
Note that the transverse diffusion term in Eq. (5) can lead to a broadening of
the $k_{t}$ distribution.

The calculation of the energy loss probability distribution in this model
reduces the whole jet to just the tagged splitting. Then, depending on whether
the angle of the splitting is smaller or larger than the QGP resolution angle
$\theta_{c}$ the jet will lose less or more energy. That is, we replace the
last line of Eq. (III.1) with

$\displaystyle\int_{0}^{\infty}{\rm
d}\varepsilon\mathcal{E}_{i}(\varepsilon|z,\theta){\rm
e}^{-\frac{n\varepsilon}{p_{t}}}$ $\displaystyle=(1-\Theta_{\rm
res})\mathcal{Q}_{i}(p_{t},R)$ (8) $\displaystyle+\Theta_{\rm
res}\mathcal{Q}_{g}(p_{t},R)\mathcal{Q}_{i}(p_{t},R),$

where the resolution criterion reads

$\Theta_{\rm res}(z,\theta)=\Theta(\theta-\theta_{c})\Theta(k_{t}-k_{t,\rm
med}).$ (9)

The physical meaning of Eq. (8) is rather simple. The first term accounts for
the case in which the splitting is unresolved and thus the jet loses energy as
a global color charge, just like in the modified $q/g$ fraction model. In
turn, the second term in Eq. (8) describes the energy loss of a splitting
resolved by the medium as the product of the quenching weight of both prongs.
We would like to emphasize that this model is a rather crude simplification of
the dynamics of color coherence. We plan to explore the use of a resummed
quenching weight Mehtar-Tani and Tywoniuk (2018) for jet substructure
calculations together with a determination of the phase-space for vacuum-like
emissions beyond the double-logarithmic approximation in a separate
publication Barata  _et al._ (2022).

Figure 6: Leading $k_{t}$-distribution in the medium using the color
decoherence model for jets with $p_{t}=120$ GeV. The bottom panel displays the
ratio to the $p+p$ result. We have highlighted two particularly relevant
scales that are further discussed in the main text.

The leading-$k_{t}$ distribution for this model is displayed in Fig. 6. We
observe a drastically different result compared to Fig. 5: medium
modifications on the $k_{t}$-distribution are enhanced at forward rapidities.
In fact, we observe, for all rapidities, an enhancement of $k_{t}$ values
around the scale $\theta_{c}/(2R)$. That is, when the energy is shared
democratically between the two prongs and the opening angle is identical to
the critical one. The only scale in this model is $\theta_{c}$ and it is thus
natural that the $k_{t}$-distribution exhibits a great sensitivity to it, as
it was also the case in the study performed in Ref. Caucal  _et al._ (2022)
for the opening angle of the splitting. We also observe that this enhancement
at $k_{t}=\theta_{c}/(2R)$ is more pronounced at forward rapidities. In this
plot, we have highlighted as well the typical scale for medium induced-
emissions, i.e. $k_{t}=Q_{s}/p_{t}$.

Let us now focus on the rapidity dependence of this model. At asymptotically
large rapidities, quarks dominate the sample and the quenching weight
corresponding to the emitter in Eq. (8) always reduces to that of a quark. All
those splittings with $\theta>\theta_{c}$ are resolved and, therefore,
suppressed, leading to an enhancement of the relative yield of jets featuring
small-$k_{t}$ (equivalently small-angle) splittings. This filtering mechanism
becomes less effective at mid-rapidity since there one has an admixture of
quarks and gluons. The fact that the sample is not pure is important since
energy loss depends both on the color factor and the opening angle of the
splitting. Due to their vacuum-like evolution, gluon-initiated jets are on
average broader than quark-initiated jets, so it is more likely that they can
be resolved by the medium. However, gluon-initiated jets with unresolved,
small-$k_{t}$ splittings can be more quenched than quark-initiated ones, since
$\mathcal{Q}_{g}\propto\mathcal{Q}_{q}^{C_{A}/C_{F}}$. The competition between
these two effects, i.e. the color charge and the opening angle of the
splitting, yields an overall milder narrowing at mid-rapidity than in the
forward regime.

To sum up this analytic section, we have identified an experimental
measurement that could pin down the origin of the narrowing effect observed
for mid-rapidity jets. These analytic estimates have allowed us to highlight
the core ideas underlying the potential of jet substructure measurements at
forward rapidities to disentangle between different jet quenching dynamics. In
what follows, we turn to a more quantitative approach and present predictions
both with the Hybrid Monte Carlo model and with a modified $q/g$ fraction
model that uses Pythia as its vacuum baseline.

###  III.2 The Hybrid Strong/Weak Coupling Model

The hybrid strong/weak coupling model Casalderrey-Solana  _et al._ (2014,
2016b) combines a perturbative high-$Q^{2}$ evolution of the parton shower
together with a non-perturbative description of the dynamics between the jet
partons and the strongly coupled QGP. The amount of hydrodynamized energy per
unit length has been computed in a strongly coupled $\mathcal{N}=4$ SYM plasma
at large-$N_{c}$ and infinite coupling Chesler and Rajagopal (2014, 2016):

$\left.\frac{{\rm d}E}{{\rm d}x}\right|_{\rm strongly\leavevmode\nobreak\
coupled}=-\frac{4}{\pi}E_{\rm in}\frac{x^{2}}{x_{\rm
stop}^{2}}\frac{1}{\sqrt{x_{\rm stop}^{2}-x^{2}}}\quad,$ (10)

where $x_{\rm stop}\equiv E_{\rm in}^{1/3}/(2{T}^{4/3}\kappa_{\rm sc})$ is the
distance an energetic parton with initial energy $E_{\rm in}$ will travel
within the strongly-coupled medium before completely hydrodynamizing.
$\kappa_{\rm sc}$ is an $\mathcal{O}(1)$ parameter fitted to hadron and jet
heavy-ion data measured at the LHC Casalderrey-Solana  _et al._ (2019). The
energy and momentum lost by the colored charges, cf. Eq. (10), hydrodynamizes
and excites a wake in the flowing plasma, described hydrodynamically, which
later decays into soft hadrons at the freeze-out hypersurface. We estimate the
distributions of those hadrons by applying the Cooper-Frye prescription to the
jet-induced perturbations, where we assume that the background fluid is
characterized by Bjorken-flow and that the perturbations stay close in
rapidity around the jet Casalderrey-Solana  _et al._ (2017). Improvements to
this approximated description of medium response are in development
Casalderrey-Solana  _et al._ (2021a).

All results in this work will include the wake, and comparisons against the
results without the wake will not be presented. Even though the wake has an
important effect for jets with larger cones Tachibana  _et al._ (2017); Pablos
(2020, 2021); Mehtar-Tani  _et al._ (2021), its impact on moderate $R\sim 0.4$
cones is milder, and does not play an important role for the kind of jet
ensembles here considered. The effect of the soft hadrons from the wake on jet
substructure, for inclusive jet ensembles at high enough jet $p_{t}$, is also
subleading, due to a selection bias effect Casalderrey-Solana  _et al._ (2020,
2021b) in which the narrower, less quenched jets, which naturally have
associated a smaller wake, represent a large fraction of the measured jet
population after imposing $p_{t}$-cuts Rajagopal  _et al._ (2016).111111The
effect of the wake on jet substructure observables can be enhanced by
selecting very quenched jets using boson-jet samples Brewer  _et al._ (2022)
or machine learning techniques Du  _et al._ (2020). Note that alternative
descriptions of medium response in which the recoils from the elastic
scattering processes stay relatively close to the jet axis can present more
sizeable effects on, e.g., groomed observables of inclusive jet samples
Milhano  _et al._ (2018).

Figure 7: $R_{\rm AA}$ ratio between results at a given rapidity $|y|$ and
those at mid-rapidity ($|y|<0.3$). Jets are reconstructed with anti-$k_{t}$
and $R=0.4$. Results are shown for different jet $p_{t}$ bins.

Finite resolution effects are incorporated in the hybrid model Hulcher  _et
al._ (2018) in analogy to the notion of screening: the resolution length
$L_{\rm res}$ determines the minimal distance between two color charges such
that they engage with the QGP independently. We will explore two limiting
regimes: (i) $L_{\rm res}=0$, when partons are resolved the instant after they
are formed, and (ii) $L_{\rm res}=\infty$, that corresponds to the fully
unresolved scenario in which the QGP is sensitive to the global color charge
of the jet only. Compared to our analytic results from the previous section,
$L_{\rm res}=\infty$ resembles the coherent energy loss implemented for the
modified $q/g$ fraction model, while a finite value of $L_{\rm res}$ (not
explored here) would play the role of $\theta_{c}$ in the color decoherence
model.

In view of the goals of the present study, the distinctive color-charge
dependence of energy loss in this model deserves some comment. Within an
holographic energy loss scenario, the different quenching between quarks and
gluons scales like $\mathcal{C}\equiv(C_{A}/C_{F})^{1/3}\simeq 2^{1/3}$ at
large-$N_{c}$ Gubser  _et al._ (2008), in contrast to the linear $C_{A}/C_{F}$
scaling expected from perturbative arguments. Since the energy loss rate Eq.
(10) was derived for the dual of a parton in the fundamental representation, a
quark, this means that $\kappa_{\rm sc}^{G}=\kappa_{\rm sc}\,\mathcal{C}$.
Therefore, the pure $q/g$ fraction effect is by construction milder than in
pQCD-inspired energy loss scenarios. However, when the traversed distance is
much smaller than the stopping distance, i.e. $\epsilon\equiv x/x_{\rm
stop}\ll 1$, the energy loss rate Eq. (10) can be expanded in powers of
$\epsilon$ and, to leading order, the dependence on the Casimirs becomes
linear, as in pQCD. Deviations from the full expression start to become
significant ($\mathcal{O}(10\%)$) at around $\epsilon\approx 0.7$, which for a
$T=0.25$ GeV and a traversed length of $x=5$ fm means that for partons with
$E_{\rm in}>100$ GeV, the color charge dependence is well described by a
linear Casimir scaling.

Before moving on to analyze the modification of the leading-$k_{t}$
distribution with the Hybrid model, and specially its rapidity dependence, it
is important to understand whether the rapidity dependence of jet suppression
itself is reasonably described.

####  III.2.1 Rapidity dependence of jet suppression

The energy loss rate in Eq. (10) was actually derived for a parton moving
through a fluid in the local fluid rest frame. Following the discussion by the
end of the introduction to Sec. III.1, given that the QGP is well approximated
as a Bjorken-flow, energy loss of a given single parton using Eq. (10) is in
practice very mildly dependent on rapidity. Therefore, any rapidity dependence
of jet suppression found in observables has to have its origin elsewhere.

In Fig. 7 we show results for jet suppression, for different cuts in rapidity
using $L_{\rm res}=0$ (using $L_{\rm res}=\infty$ yields a very similar
picture). We present the results as ratios to the $|y|<0.3$ result for
different jet $p_{t}$ bins and compare against ATLAS data Aaboud  _et al._
(2019a). In addition, we calculate this observable with and without nuclear
PDFs using the EPPS16 Eskola  _et al._ (2017) set. We discuss the results
without nPDF first. As the quark fraction increases with rapidity (cf. left
panel of Fig. 1), so does $R_{\rm AA}$, meaning less suppression, as quark
jets tend to be less quenched. On the other hand, by increasing rapidity, the
initial jet spectrum becomes steeper, specially at higher jet $p_{t}$ (cf.
right panel of Fig. 1), which translates into a reduction of $R_{\rm AA}$.
These competing effects yield an evolution of $R_{\rm AA}$ that is almost
independent of rapidity within $|y|\leq 2$ (as was also observed in Refs.
Spousta and Cole (2016); He  _et al._ (2019)). For very large rapidities,
where one is approaching the kinematic limit at increasingly lower jet
$p_{t}$, the steepness of the spectrum is the dominant effect, notably
reducing $R_{\rm AA}$. Overall, there is good agreement between the Hybrid
model without nPDF and the experimental data.

In turn, the results of the Hybrid Model with the inclusion of nPDF (using
only the central set for computing time reasons) are quite different. The
nPDF-induced modification of the initial jet spectrum at moderate rapidities
yields a reduction of $R_{\rm AA}$ with respect to mid-rapidity that is
notable for a wide range in jet $p_{t}$. To highlight the sizeable effect of
nPDF on this observable, we show in a dotted bisque-colored line the results
that use nPDF, but without quenching. For this ‘no quenching’ baseline we have
also included the uncertainties associated to the 40 error sets, as prescribed
in Ref. Eskola  _et al._ (2017). We can appreciate how the deviation from
unity of the full results with nPDF are actually dominated by the nPDF effect
itself, yielding these inconsistent with data in the first two $p_{t}$ bins
displayed in the top panels of Fig. 7. While the results in the second to
highest jet $p_{t}$ bin, in the bottom left panel, are the least sensitive to
the effects of the nPDF, the highest jet $p_{t}$ bin, in the bottom right
panel, indicates that data seems to slightly prefer those that include nPDF.
For this jet $p_{t}$ bin, both nPDF and quenching effects are needed to
achieve the best description of experimental data. The results of Fig. 7
provide strong motivation to revisit the impact of nPDF in other jet quenching
models. Along with the previous observations on the sizeable effects of nPDF
in jet $R_{\rm AA}$ at high-$p_{t}$ Pablos (2020); Caucal  _et al._ (2021b),
which questions the direct interpretation of $R_{\rm AA}\simeq 1$ as a signal
of the recovery of lost energy for large-$R$ jets Pablos (2020), these results
contribute to build a case for a thorough consideration of the role of nPDF in
jet quenching physics, specially in view of the future precision studies
programmed at RHIC and the LHC.

####  III.2.2 Predictions for a rapidity scan of the leading-$k_{t}$
distribution

Figure 8: Ratio of the $q$-fraction between $Pb+Pb$ and $p+p$ for different
jet $p_{t}$ intervals and rapidity windows, comparing $L_{\rm res}=0$ ($L_{\rm
res}=\infty$) in solid (grid pattern) bands.

In the previous section, we have demonstrated that the rapidity dependence of
jet $R_{\rm AA}$ can be reasonably described by the Hybrid Model. We are now
in a position to perform predictions on the rapidity dependence of jet
substructure observables, such as the leading-$k_{t}$ distribution. Before
presenting our results for this observable, it is relevant to discuss the
modification of the $q$-fraction due to medium effects in the Hybrid Model. We
show in Fig. 8 the ratio between the $q$-fraction in $Pb+Pb$ and in $p+p$ both
for the fully resolved case, $L_{\rm res}=0$, in solid bands, and for the
fully unresolved case, $L_{\rm res}=\infty$, in grid pattern bands. In
general, this ratio tends to decrease with increasing rapidities, as we saw in
Fig. 4, because the value of the $p+p$ baseline in the denominator steadily
increases.121212Discrepancies between the results in Figs. 4 and 8 are
expected since the energy loss model differs. There is an exception to this
trend, observed in the different behaviour for the results of the lowest jet
$p_{t}$ bin, in black. For jets with $20<p_{t}<80$ GeV, the ratio of the
$q$-fraction can be somewhat larger for rapidity values around $|y|\gtrsim
2.5$ than the one at mid-rapidity. The reason for this is the strong jet
$p_{t}$ dependence in the value of the $q$-fraction for jets with
$p_{t}\lesssim 100$ GeV at forward rapidities, as seen in the left panel of
Fig. 1. Shifting the jet $p_{t}$ has a notable effect on the value of the
$q$-fraction at $|y|\approx 2.5$, while the impact is much smaller at mid-
rapidity. Therefore, due to medium-induced energy loss, jet $p_{t}$ bin
migration leads to a sizeable increase in the number of quark-initiated jets
for those kinematic ranges in which the $q$-fraction features such a strong
dependence with $p_{t}$.

In Fig. 8 we also observe that the scenario with $L_{\rm res}=\infty$ produces
larger differences in the $q$-fraction between $Pb+Pb$ and $p+p$. The
differences with respect to the case with $L_{\rm res}=0$ are approximately
independent of the rapidity window, and decrease in magnitude as jet $p_{t}$
increases. We can understand this based on simple arguments. The ratio of the
$q$-fraction can be expressed in terms of jet suppression as $f_{q}^{\rm
ratio}\equiv R_{\rm AA}^{q}/R_{\rm AA}^{\rm total}$, this is, the suppression
of quark-initiated jets divided by the total jet suppression. One can
approximate $R_{\rm AA}\approx 1-\varepsilon n/p_{t}$, see Eq. (4), where
$\varepsilon$ is the average energy lost per jet, $n$ is the power index of
the spectrum and $p_{t}$ is the jet transverse momentum. Then,

$\displaystyle R_{\rm AA}^{q}\approx 1-\varepsilon_{q}\frac{n}{p_{t}},$ (11)
$\displaystyle R_{\rm AA}^{\rm total}\approx
1-[\varepsilon_{q}f_{q}+\varepsilon_{g}(1-f_{q})]\frac{n}{p_{t}},$ (12)

where $f_{q}$ represents the vacuum $q$-fraction, and where we have assumed
that quark and gluon jets have similar values of $n$. To leading power in
$\varepsilon_{i}n/p_{t}$, we write

$f_{q}^{\rm ratio}\approx
1+(1-f_{q})(\varepsilon_{g}-\varepsilon_{q})\frac{n}{p_{t}}+\mathcal{O}((\varepsilon
n/p_{t})^{2}).$ (13)

In order to express the energy loss for a jet with a given flavour, we make
use of the resummed quenching weights formalism in the small energy loss
approximation Mehtar-Tani and Tywoniuk (2018). In this limit, we can simply
write

$\displaystyle\varepsilon_{q}\sim
C_{F}\hat{\varepsilon}[1+C_{A}\mathcal{A}(p_{t},R)],$ (14)
$\displaystyle\varepsilon_{g}\sim
C_{A}\hat{\varepsilon}[1+C_{A}\mathcal{A}(p_{t},R)]\,,$ (15)

where $\mathcal{A}(p_{t},R)$ refers to the phase-space of extra energy loss
sources (assumed to be all vacuum-like gluons) resolved by the medium, and
$\hat{\varepsilon}$ is now the average energy lost _per parton_ (stripped of
the color charge dependence). In the $L_{\rm res}=\infty$ case, the medium
cannot resolve the phase-space by definition and so $\mathcal{A}_{\infty}\to
0$. Instead, when $L_{\rm res}=0$, the medium resolves all emissions that are
formed within the medium. At fixed coupling, $\mathcal{A}_{0}$ is thus given
by

$\displaystyle\mathcal{A}_{0}$
$\displaystyle=\frac{\alpha_{s}}{\pi}\int\frac{{\rm d}z}{z}\int\frac{{\rm
d}\theta}{\theta}\Theta(t_{f}<L)\Theta(\theta<R)$ (16)
$\displaystyle=\frac{\alpha_{s}}{4\pi}\ln^{2}\left(\frac{p_{t}R^{2}L}{2}\right),$

where we have used $t_{f}=2/(z\theta^{2}p_{t})$. Finally, we can obtain the
difference in the $q$-fraction ratio between these two scenarios as

$\displaystyle f^{\rm ratio}_{q,\infty}-f^{\rm ratio}_{q,0}\approx$
$\displaystyle(1-f_{q})\frac{n}{p_{t}}(C_{A}-C_{F})$ (17)
$\displaystyle\times$
$\displaystyle\,\hat{\varepsilon}_{\infty}[1-\frac{\hat{\varepsilon}_{0}}{\hat{\varepsilon}_{\infty}}(1+C_{A}\mathcal{A}_{0})]\,.$

Similarly to the fitting procedure adopted within the Hybrid Model, we impose
a comparable $R_{\rm AA}\approx 0.5$ for both scenarios for jets with
$p_{t}\approx 150$ GeV and $R=0.4$. Using $L=4$ fm, $f_{q}=0.5$,
$\alpha_{s}=0.2$ and $n=5$ we get that $\hat{\varepsilon}_{\infty}\approx 7$
GeV and $\hat{\varepsilon}_{0}\approx 2.8$ GeV. By numerically evaluating Eq.
(17), we obtain $\mathcal{O}(10\%)$ in the difference of $f_{q}^{\rm ratio}$
at $p_{t}\approx 80$ GeV, decreasing down to less than 2% at high jet $p_{t}$,
around $p_{t}\approx 250$ GeV. These results are in qualitative agreement with
what is observed in Fig. 8.

We discuss next the results of the medium-modified $k_{t}$ distribution. In
Fig. 9, we show the ratio between $Pb+Pb$ and $p+p$ for the leading-$k_{t}$
distribution, using the same cuts as in Fig. 3. For each of the individual
plots, the main message is clearly the presence of a sizeable narrowing of the
leading-$k_{t}$ distribution when the jet substructure can be resolved by the
QGP, represented in this model by $L_{\rm res}=0$. Note that the observed
narrowing is notably stronger than the one obtained analytically in Fig. 6.
This is most likely due to the fact that, as expressed in Eq. (8), quenching
of a resolved jet simply consists in the energy loss from the two tagged
prongs; there is no additional quenching for each of the possible subsequent
emissions off these partons, as we do in the present Monte Carlo analysis. If
one were to include the resummation of the quenching weights Mehtar-Tani and
Tywoniuk (2018); Mehtar-Tani  _et al._ (2021); Takacs and Tywoniuk (2021) in
the analytic model, the quenching of the wider structures would be stronger,
and the narrowing more pronounced. Another important remark is that the
narrowing in the $L_{\rm res}=0$ case persists for all rapidities and that the
differences between $L_{\rm res}=\infty$ and $L_{\rm res}=0$ are sufficiently
large at the kinematic regions that already are experimentally accessible with
current detector capabilities at the LHC.

Focusing now on the $L_{\rm res}=\infty$ curves in Fig. 9, we can observe that
the ratio becomes increasingly flat as we increase the vacuum $q$-fraction by
moving towards larger rapidities. This is fully consistent with the picture
obtained in Fig. 5. In addition, the evolution of this flattening becomes
milder with decreasing jet $p_{t}$ since the $q$-fraction ratio does not
evolve as strongly with rapidity, as we saw in Fig. 8. In the lowest jet
$p_{t}$ bin there are a few interesting features that we would like to comment
on. In this case, since the ratio of the $q$-fraction can actually increase
(cf. Fig. 8) we find that the leading-$k_{t}$ distribution is actually
narrower for the $2.5<|y|<3$ bin than for the $|y|<0.3$ bin. In fact, the mid-
rapidity result is even slightly wider than in vacuum. We have checked that
this last point is an effect of the wake, as the soft particles at large
angles produced via jet-energy thermalisation can contribute to generate, or
enhance, structures sitting at the jet boundary.131313This effect is
irrelevant at higher jet $p_{t}$ since they are not affected by the relatively
small energy injection from the thermal soft particles.

Figure 9: Leading-$k_{t}$ distribution calculated with the Hybrid model for
different rapidity and transverse momentum intervals. Note that we use a
dimensionless expression for $k_{t}$, defined as $k_{t}\equiv z(p_{t}^{\rm
parent}/p_{t}^{\rm jet})\sin\theta/R$.

###  III.3 Modified $q/g$ fraction model with Pythia templates

Figure 10: Leading-$k_{t}$ distribution calculated with the toy model for
modified $q/g$ fraction for different rapidity and transverse momentum
intervals. Same definition of $k_{t}$ as in Fig. 9 is used.

The goal of this section is to extend the analytic results presented in Sec.
III.1.1 by using as a vacuum baseline templates from Pythia with the estimated
statistics of the future HL-LHC run. We do so as to realistically assess the
feasibility of future experiments to pin down the differences between the
results from a modified $q/g$ fraction hypothesis and those obtained when the
medium is capable of resolving the fluctuating substructure of each individual
jet, i.e. $L_{\rm res}=0$ results of the Hybrid model presented in the
previous section.

Our starting point consists in expressing $R_{\rm AA}$ in terms of the
suppression experienced by quark- and gluon-initiated jets. The definition of
jet suppression, Eq. (3), can be recast into

$R_{\rm AA}=f_{q}\mathcal{Q}_{q}+(1-f_{q})\mathcal{Q}_{g}\,,$ (18)

where $f_{q}$ is the quark-initiated jet fraction in vacuum, and
$\mathcal{Q}_{q}$ and $\mathcal{Q}_{g}$ are the (bare) quenching weights of
quark-initiated and gluon-initiated jets, respectively. These bare quenching
weights can be computed as in Eq. (4) within the BDMPS-Z approach. We can
further use the fact that $\mathcal{Q}_{i}$ scales with the Casimir of the
color charge exponentially Baier  _et al._ (2001), so that
$\mathcal{Q}_{g}=\mathcal{Q}_{q}^{C_{A}/C_{F}}$. For the purposes of the
present section, it is convenient to express Eq. (18) in terms of the relative
suppression of gluon versus quark jets, $f_{\rm rel}$, as

$\displaystyle R_{\rm AA}$
$\displaystyle=\mathcal{Q}_{q}[f_{q}+(1-f_{q})\mathcal{Q}_{q}^{C_{A}/C_{F}-1}]$
$\displaystyle=\mathcal{Q}_{q}[f_{q}+(1-f_{q})f_{\rm rel}]\,.$ (19)

Let us discuss a few meaningful values of $f_{\rm rel}$. To start with,
choosing $f_{\rm rel}=\mathcal{Q}_{q}^{C_{A}/C_{F}-1}$ corresponds to the
relative suppression motivated by the BDMPS-Z physics encapsulated in Eq. (6),
which for a reasonable value of $\mathcal{Q}_{q}=0.6$ leads to $f_{\rm
rel}\approx 0.5$, i.e. a factor of two increase in the relative number of
quark jets over gluon jets, due to medium effects, when compared to that same
number in vacuum. As mentioned in the introduction, the value $f_{\rm
rel}=0.25$ coincides with the one extracted via global fits to the jet
fragmentation function Qiu  _et al._ (2019), and used in further quenching
studies of the modification of the groomed radius Ringer  _et al._ (2020);
Acharya  _et al._ (2022). Finally, we also explore $f_{\rm rel}=0.1$ to test
the maximal effects from the modified $q/g$ fraction hypothesis.

The medium-modified leading-$k_{t}$ distribution, in close analogy to the
formulas presented in Sec. III.1.1, then reads

$\frac{1}{\sigma}\frac{{\rm d}\sigma}{{\rm
d}k_{t}}\Big{|}_{AA}=\mathcal{N}^{-1}\left[f_{q}\frac{{\rm d}\sigma_{q}}{{\rm
d}k_{t}}\Big{|}_{pp}+f_{\rm rel}(1-f_{q})\frac{{\rm d}\sigma_{g}}{{\rm
d}k_{t}}\Big{|}_{pp}\right],$ (20)

where $\mathcal{N}=f_{q}+f_{\rm rel}(1-f_{q})$. Note that the individual
distributions are assumed to be unmodified with respect to vacuum. That is,
they correspond to templates obtained using Pythia and shown in Fig. 3. An
important caveat of Eq. (20) is that we keep $f_{\rm rel}$ fixed for all
values of jet $p_{t}$ and rapidity. As a consequence, this approach ignores
the strong $p_{t}$-dependence of the $q$-fraction for lower jet $p_{t}$ at
higher rapidities, as highlighted when discussing Fig. 8. This
$p_{t}$-dependence becomes important when spectra shifts, due to energy loss,
modify the average jet $p_{t}$ of a given jet ensemble. Keeping $f_{\rm rel}$
fixed also neglects any intrinsic $p_{t}$-dependence of the substructure
observable in consideration. Accounting for all these effects would require
$f_{\rm rel}\to f_{\rm rel}(p_{t},y)$.

In Fig. 10 we show results for the modification of the leading-$k_{t}$
distribution using the toy model encapsulated in Eq. (20). They are presented
as ratios with respect to the vacuum result, using the same kinematic cuts as
in Figs. 3 and 9. The presence of the grey band responds to a sanity check
that Eq. (20) reproduces the total vacuum result, $f_{\rm rel}=1$, given the
correct $f_{q}$. The values of $f_{q}$ are estimated from Fig. 1, and then
more finely tuned to get an exact ratio of 1. The statistics used for the
$Pb+Pb$ samples approximately correspond to those estimated in Table (1). In
turn, the $p+p$ reference has a factor $\sim 10$ higher statistics.

The first observation concerning Fig. 10 is that this toy model is able to
generate a visible narrowing of the leading-$k_{t}$ distribution at mid-
rapidity for all $p_{t}$ intervals, as it was the case in the comparison to
the groomed jet radius data in Ref. Acharya  _et al._ (2022). In agreement
with the analytic results of Sec. III.1.1, the ratio becomes close to one as
one evolves towards more forward rapidities. Note that for the low-$p_{t}$ bin
the narrowing persists even at the most forward rapidity bin since the value
of $f_{q}$ is smaller than for the other $p_{t}$ bins. The key outcome of this
exercise is that we can now quantify for which values of jet $p_{t}$ and
rapidity can one discriminate between our two predictions: the one based on
the modified $q/g$ fraction, and that of the Hybrid model with $L_{\rm
res}=0$. By comparing Figs. 9 and 10 we identify two jet selection cuts that
achieve such goal: (i) $2.1<|y|<2.5$ and $225<p_{t}<300$ GeV, and (ii)
$3<|y|<4$ and $100<p_{t}<150$ GeV. The former is potentially accessible with
the current technology of ATLAS and CMS experiments. The latter requires
detector upgrades and higher statistics, so it can be targeted by all four LHC
experiments beyond Run 3. This is the main result of this paper.

##  IV Conclusions

Jet substructure measurements at LHC energies have revealed that the hard core
of the jet is narrower in heavy-ion collisions than in $p+p$. A lack of
consensus exists in the theoretical interpretation of this narrowing effect.
On the one hand, fully coherent energy loss models propose that a relative
increase in the number of quark-initiated jets, known to possess a narrower
fragmentation, with respect to vacuum suffices to explain current
measurements. This increase in the number of quark-initiated jets would be
based on the fact that they lose less energy than gluon-initiated jets due to
their different color charge, and are more likely to pass the jet $p_{t}$ cut
required to enter the inclusive jet ensemble. It is noteworthy that in order
to quantitatively describe the data, such a model requires a quark fraction 4
times larger than in $p+p$, a number far beyond any estimate based on the
Casimir scaling of energy loss. The second category of models argue that the
jet sample is mainly dominated by unresolved splittings. That is, only
sufficiently wide splittings $\theta>\theta_{c}$, either quark- or gluon-
initiated, will be resolved by the medium, lose more energy and make the
energy of the jet they belong to fall below the $p_{t}$-selection cut.

This paper proposes a rapidity scan of jet substructure measurements to pin
down the origin of the narrowing effect. We show that a gradual increase of
the jet rapidity enhances the fraction of quark-initiated jets in $p+p$. Then,
in a fully coherent picture of energy loss the narrowing effect would vanish
at asymptotically forward rapidites, where the flavour of the jet-initiator is
fixed. In turn, the resolution scale of the medium $\theta_{c}$ is independent
of the jet rapidity and, thus, if the narrowing effect is driven by this
phenomenon it should persist for all rapidity values. To make these statements
more quantitative we have studied the relative transverse momentum
distribution of the hardest splitting in a jet via analytic toy models and
Monte Carlo simulations, the latter using the expected statistics of the high-
luminosity run of the LHC. We demonstrate that the the current rapidity reach
of LHC experiments at high-$p_{t}$, i.e. $225<p_{t}<300$ GeV, would already
have strong discriminating power between these two theoretical interpretations
of the narrowing effect. Even stronger constraints can be achieved with
detector upgrades capable of measuring jet substructure at $|y|>3$ for
moderate jet $p_{t}$, i.e. $100<p_{t}<150$ GeV. This kinematic regime is well
within the expected acceptance of all LHC experiments in Run 4 and beyond.

Complementary constraints on the interaction of quark-initiated and gluon-
initiated jets with the QGP can be obtained by the comparison of the
suppression patterns of inclusive jets and (i) heavy-flavour tagged jets
Chatrchyan  _et al._ (2014); ALICE (2022c); Li and Vitev (2019), which are
expected to be initiated by either a charm or a bottom quark, or (ii)
$Z/\gamma$+jet events Gallicchio and Schwartz (2011); Wang and Zhu (2013);
Aaboud  _et al._ (2019b); Sirunyan  _et al._ (2018b); Kang  _et al._ (2017);
Yang  _et al._ (2021); Brewer  _et al._ (2022); Takacs and Tywoniuk (2021),
where the quark fraction is also enhanced due to the Born level matrix
element. In our opinion, measurements at forward rapidities are cleaner than
these two alternative options since the latter are subject to multiple sources
of contamination. For example, a heavy-flavour tagged jet can originate from a
$g\to q\bar{q}$ splitting, and so a non-vanishing fraction of $g$-initiated
jets makes its way into the sample Attems  _et al._ (2022). Equivalently,
sophisticated isolation cuts are required in $\gamma$+jet events to guarantee
that the reconstructed photon was generated in the hard scattering process and
not in the aftermath of the collision via electroweak radiation – and even
after applying these cuts a sizeable number of these secondary photons still
persist in the final sample. Finally, $Z$-tagged events suffer from low
statistics Sirunyan  _et al._ (2022). It would be interesting to perform a
more systematic comparison of the pros and cons of these alternative methods
with respect to the measurements at forward rapidities proposed in this work.
We leave this task for future studies.

###### Acknowledgements.

We are grateful to Benjamin Audurier, Cristian Baldenegro, Gian Michele
Innocenti, Dennis Perepelitsa, Martin Rybar and Ricardo Vázquez for fruitful
exchanges on the experimental feasibility of rapidity-dependent jet
substructure measurements. We also thank Carlota Andrés for guidance on the
use of the nPDF error sets. The authors would like to thank the organizers of
the XXIXth International Conference on Ultra-relativistic Nucleus-Nucleus
Collisions, Quark Matter 2022, which took place in Krakow, Poland, where this
collaboration started. DP has received funding from the European Union’s
Horizon 2020 research and innovation program under the Marie Skłodowska-Curie
grant agreement No. 754496. A.S.O.’s work was supported by the European
Research Council (ERC) under the European Union’s Horizon 2020 research and
innovation programme (grant agreement No. 788223, PanScales). She would also
like to thank the Munich Institute for Astro-, Particle and BioPhysics
(MIAPbP), funded by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) under Germany´s Excellence Strategy – EXC-2094 – 390783311, for
hospitality while this publication was finalised.

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