# Isolating perturbative QCD splittings in heavy-ion collisions

Leticia Cunqueiro  Sapienza Università di Roma & INFN, Piazzale Aldo Moro 5,
Roma 00185, Italy  Daniel Pablos  INFN, Sezione di Torino, via Pietro Giuria
1, I-10125 Torino, Italy  Departamento de Física, Universidad de Oviedo, Avda.
Federico García Lorca 18, 33007 Oviedo, Spain  Instituto Universitario de
Ciencias y Tecnologías Espaciales de Asturias (ICTEA), Calle de la
Independencia 13, 33004 Oviedo, Spain  Alba Soto-Ontoso  CERN, Theoretical
Physics Department, CH-1211 Geneva 23, Switzerland  Martin Spousta  Institute
of Particle and Nuclear Physics, Faculty of Mathematics and Physics, Charles
University, V Holešovičkách 2, 180 00 Prague 8, Czech Republic  Adam Takacs
Department of Physics and Technology, University of Bergen, Allegaten 55, 5007
Bergen, Norway  Institute for Theoretical Physics, Heidelberg University,
Philosophenweg 16, 69120 Heidelberg, Germany  Marta Verweij  Institute for
Gravitational and Subatomic Physics (GRASP), Utrecht University, Utrecht,
Netherlands  Nikhef, National institute for subatomic physics, Amsterdam,
Netherlands

###### Abstract

We define a new strategy to scan jet substructure in heavy-ion collisions. The
scope is multifold: (i) test the dominance of vacuum jet dynamics at early
times, (ii) capture the transition from coherent to incoherent jet energy
loss, and (iii) study elastic scatterings in the medium, which are either hard
and perturbative or soft and responsible for jet thermalisation. To achieve
that, we analyse the angular distribution of the hardest splitting,
$\theta_{\rm hard}$, above a transverse momentum scale, $k_{t}^{\rm min}$, in
high-$p_{t}$ jets. Sufficiently high values of $k_{t}^{\rm min}$ target the
regime in which the observable is uniquely determined by vacuum-like
splittings and energy loss, leaving the jet substructure unmodified compared
to proton-proton collisions. Decreasing $k_{t}^{\rm min}$ enhances the
sensitivity to the relation between energy loss and the intra-jet structure
and, in particular, to observe signatures of colour decoherence at small
angles. At wider angles it also becomes sensitive to hard elastic scatterings
with the medium and, therefore, the perturbative regime of medium response.
Choosing $k_{t}^{\rm min}\approx 0$ leads to order one effects of non-
perturbative origin such as hadronisation and, potentially, soft scatterings
responsible for jet thermalisation. We perform a comprehensive analysis of
this observable with three state-of-the-art jet-quenching Monte Carlo event
generators. Our study paves the way for defining jet observables in heavy-ion
collisions dominated by perturbative QCD and thus calculable from first
principles.

##  I Introduction

Collisions of heavy ions at ultra-relativistic energies produce a myriad of
particles ($\mathcal{O}(10^{3})$). The presence of a quark-gluon plasma (QGP)
phase during the system evolution leaves an imprint on the characteristic
pattern of final state multi-particle correlations. The ultimate goal of the
heavy-ion program is to characterise the QGP in terms of its transport
properties and microscopic structure. One of the main challenges is to design
and measure observables that are both QGP standard candles and amenable to
theoretical computations. A successful approach to this problem is the use of
jet observables to measure their modification compared to jets in proton-
proton collisions. Experimental overviews on jet modification are provided in
Refs. [1, 2, 3, 4, 5], while for theory see Refs. [6, 7, 8]. While early
measurements used jets as monolithic objects, most recent measurements focus
on the internal structure of jets (see Refs. [2, 3, 4] for more details). The
main model-agnostic conclusion that can be drawn from these measurements is
that in-medium jets lose around $10\%$ of their energy, i.e. are quenched, by
means of copious out-of-cone radiation induced by the QGP and that this energy
degradation is sensitive to the jet substructure.

The theoretical description of some of the key processes in jet formation in
heavy-ion collisions is in the realm of perturbative QCD and is thus
calculable. Effects that pertain to this category are, e.g. nuclear-modified
parton distribution functions [9, 10, 11], the initial hard-scattering (prior
to QGP formation), and early vacuum-like QCD radiation [12, 13]. Medium-
induced emissions, triggered by the interaction between the jet particles and
the QGP colour fields [14, 15, 16, 17], in most cases, are assumed to be
perturbative. Their description requires phenomenological input, namely a
model of the QGP. A pQCD description of jet evolution breaks down when the
partonic cascade reaches energy scales around the QGP temperature, that is,
$\mathcal{O}(\Lambda_{\rm QCD})$. Then, other non-perturbative effects such as
thermalisation and hadronisation become relevant. Thermalisation can be
studied with QCD kinetic theory [18, 19, 20] (using an artificially big
coupling), while the formation of hadrons rely on phenomenological modelling
[21, 22, 23]. Alternatively, strongly coupled descriptions of parton energy
loss have also been studied using the gauge/gravity duality [24, 25, 26, 27,
28]. While they offer a natural scenario for jet hydrodynamisation in a
strongly coupled plasma, the underlying QFT is not QCD, but $\mathbf{N}=4$
SYM, implying the presence of large theoretical uncertainties in the
extrapolation of the results to the system actually produced in heavy-ion
collisions.

Due to the intricate interplay among the aforementioned effects and the multi-
scale nature of the process, an end-to-end analytic approach to in-medium jet
evolution that matches the experimental precision is currently beyond reach.
As a consequence, the theoretical interpretation of jet observables relies
almost exclusively on phenomenological modelling by means of Monte Carlo event
generators. Several implementations of in-medium parton showers have been
proposed in the literature [29, 30, 31, 32, 33, 34, 35, 36, 13, 37, 38]. They
differ not only in the precision in which they describe the individual
ingredients of jet evolution but also in the way they assemble them. A
paradigmatic example, that will be relevant in this paper, is the interleaving
of vacuum and medium-induced emissions. While some approaches implement either
a partial or an exact factorisation between vacuum and medium-induced
emissions [13, 39], others include them on equal footing in their evolution
equations [40, 31, 41]. Experimental measurements have not yet been able to
pin down which is the correct approach. Consolidating the theoretical
description of in-medium jet evolution thus requires experimental guidance by
means of more differential and/or precise measurements.

This paper aims to disentangle different stages of jet evolution by combining
two jet substructure observables. Our proposal strongly relies on the use of
high-$p_{t}$ jets to ensure a clear separation between perturbative and
medium-related scales. A key point in our study is the introduction of an
auxiliary, intermediate energy scale that allows us to sweep through different
phases of jet evolution in a controlled fashion. By means of state-of-the-art
Monte Carlo simulations, we quantitatively address the following fundamental
questions:

  * •

Is there a regime of pure vacuum evolution in the in-medium development of a
parton shower?

  * •

Does energy loss depend on the opening angle of the splitting? If so, at which
energy scale does this effect become relevant?

  * •

Are elastic scatterings with the medium visible in jet substructure
observables?

The underlying philosophy and technical details of the proposed observable
together with its connection to previous measurements are explained in Section
II. The proton-proton baseline is studied in Sec. III including a comparison
between state-of-the-art pQCD and the vacuum prediction of jet quenching event
generators. Quantitative results showing the discriminating power of the
proposed observable in heavy-ion collisions can be found in Sec. IV. We end up
with a brief summary of our results in Sec. V. The experimental feasibility of
this measurement with the upcoming Runs 3 and 4 of the LHC, some analytic
considerations, the impact of energy loss prescriptions and medium response
are studied in Appendices A, B, C, and D, respectively.

##  II Analysis strategy

Figure 1: Sketch depicting the different regimes of in-medium jet evolution in
a Lund-plane style representation. The gray bands around $\theta=\theta_{c}$
and $t_{f}=L$ indicate that these two scales fluctuate on an event-by-event
basis.

Our goal is to design and study an observable to probe certain corners of the
radiation phase-space of an in-medium jet, as sketched in a Lund-plane fashion
in Fig. 1. Before entering into the precise definition of the observable, let
us briefly comment on the most relevant features of this radiation phase
space.

###  II.1 Brief reminder of in-medium jet evolution

For a given jet with transverse momentum $p_{t}$ and cone size $R$, we
characterise its branchings by their opening angle $\theta$ and their relative
transverse momentum $k_{t}$.111We follow the standard Lund jet plane
definitions, $\theta=\sqrt{(y_{1}-y_{2})^{2}+(\phi_{1}-\phi_{2})^{2}}$,
$k_{t}=\min(p_{t1},p_{t2})\theta$, where the two prongs of the splitting are
denoted with subscripts 1 and 2 [42]. The maximum relative transverse momentum
is $k_{t}<p_{t}R$ ($\mathcal{O}(10^{2})$ GeV in this study). Another important
kinematic variable is the formation time of the splitting
$t_{f}=2/(k_{t}\theta)$, which describes the quantum mechanical extent of a
branching [43].

Jet constituents interact elastically and inelastically with the QGP colour
fields. The former results in transverse momentum broadening, while the latter
induces additional radiation. Therefore, splittings in an in-medium parton
shower can be either vacuum-like (i.e. described by the Altarelli-Parisi
splitting function [44]) or medium-induced, with a branching probability that
depends on the medium properties. For example, modelling the jet-QGP
interaction in the multiple soft scattering approximation [14, 15, 45] results
into Gaussian broadening with accumulated transverse momentum $\langle
k_{t}^{2}\rangle\sim\hat{q}L$, with $\hat{q}$ the so-called quenching
parameter and $L$ the propagation length. Since the typical value of $\hat{q}$
is around $1$ to $3$ GeV2/fm [46, 47, 48], elastic and inelastic medium
processes mainly contribute in the soft regime ($k_{t}\sim\mathcal{O}(1)$ GeV)
of Fig. 1. Another important feature is that these medium-induced emissions
typically occur at wide angles, drifting energy out of the jet cone.
Alternatively, hard collisions with the medium can occur above $\langle
k_{t}^{2}\rangle>\hat{q}L$ and induce emissions with a Coulomb-like transverse
momentum distribution, i.e. $\propto 1/k_{t}^{4}$ tail [16, 17]. These
Molière/higher-twist scatterings [49, 50] compete with vacuum-like radiation
in the high-$k_{t}$ regime, although their occurrence is relatively rare for
dense media.

Another medium-related scale highlighted in Fig. 1 is the de-coherence angle,
$\theta_{c}\sim(\sqrt{\hat{q}L^{3}})^{-1}$ [51, 52]. If the opening angle of
the splitting is small, the medium cannot resolve the two daughters through
soft scatterings, and emissions are coherently induced on this composite,
$2$-parton object. Above $\theta_{c}$, the daughters are resolved, and both
source medium-induced radiation.

To illustrate the importance of $\theta_{c}$, consider the extreme case in
which the first splitting of an angular-ordered parton shower has
$\theta<\theta_{c}$. All emissions inside the jet will be unresolved by the
medium, its substructure remains unmodified, and the jet loses energy as a
single object (just the initiator is quenched). In turn, when a splitting
satisfies $\theta>\theta_{c}$, both prongs act as independent radiators of
medium-induced emissions, resulting in a larger energy loss. This angular
dependence of energy loss translates into a sizeable modification of jet
substructure observables by means of a selection bias, or survivor bias
effect. Due to the presence of a steeply falling jet $p_{t}$ spectrum,
inclusive jet ensembles at any given $p_{t}$ in heavy-ion collisions will be
dominated by those jets that on average lost the least amount of energy.
Therefore, the number of jets with wide-angle substructure,
$\theta>\theta_{c}$, is suppressed compared to proton-proton collisions in any
inclusive jet observable where unavoidably minimum jet $p_{t}$ thresholds are
imposed [53, 54, 55, 56, 57, 58].222Selection biases are also present even if
energy loss is independent of the jet substructure. For example, gluon jets
are more suppressed since, on average, they lose more energy than quark jets
due to their larger colour factor. As gluon jets are typically wider, this
bias would also cause some narrowing of the final quenched jet ensemble. This
is sometimes referred to as a bias in the $q/g$-fraction [59, 60, 61].
Additionally, the selection bias mixed with colour resolution (i.e. small
angle splittings do not source energy loss) gives even stronger narrowing and
it is necessary to explain jet measurements [62, 58, 39, 63].

An important observation is that varying the jet reconstruction parameters,
($p_{t},R$), enables the separation of medium dynamics from vacuum physics.
For example, raising the jet $p_{t}$ increases the phase space for high-energy
emissions (i.e. the upper diagonal of the emission phase space in Fig. 1 moves
to higher values). Since $\hat{q}$ depends only mildly on energy [64, 65], the
scale at which medium-induced emissions dominate ($k^{2}_{t}\sim\hat{q}L$)
will remain the same and the new phase-space region should be mainly populated
by vacuum emissions. Similarly, increasing the jet radius $R$ further
separates the $\theta=R$ boundary from the $\theta_{c}$ scale (again almost
independent of the jet energy). We will exploit these facts in the definition
of our observable.

###  II.2 Observable definition

We cluster events with the anti-$k_{t}$ algorithm [66, 67] using a jet radius
of $R=0.2$ and select all jets that satisfy $p_{t}>400$ GeV, and $|\eta|<2.8$.
This subset of jets is then reclustered with the Cambridge/Aachen algorithm
[68, 69], effectively reordering the branching history in decreasing angles.
For each of these jets, we find the hardest splitting, i.e. the splitting with
maximum $k_{t}=\min(p_{t1},p_{t2})\theta$. If the splitting satisfies
$k_{t}>k_{t}^{\rm min}$, we record the opening angle of this hardest
branching, that we denote $\theta_{\rm hard}$.

Let us now justify each of the choices that we have made in the previous
paragraph and comment on their experimental feasibility.

  * •

$R=0.2$: there are certain advantages of using jets with small cone sizes. The
physics associated with vacuum-like emissions and coherence that we are
interested in the present work typically appear at small angles ($<0.1$). The
reduced jet area decreases the contamination from the uncorrelated fluctuating
background and notably improves experimental systematic uncertainties.
Moreover, smaller jet cones also reduce the contribution from medium response,
which typically populates the larger angle region, thus further improving the
signal.

  * •

$p_{t}>400$ GeV: the larger the jet $p_{t}$ is, the more pQCD dominated the
observable is. This guarantees not only calculability but also a large
separation of scales, breaking up the problem of in-medium jet evolution into
simpler pieces. Going to higher momenta, however, significantly reduces the
statistics. The expected number of jets in PbPb collisions during Runs 3 and 4
at the LHC is discussed in Appendix A.

  * •

$k^{\rm min}_{t}$: this auxiliary variable is central to our approach. Its
role is to slice the radiation phase space, separating different regimes of
jet evolution. We explore three values: $20$, $5$, and $1$ GeV. The highest
$k^{\rm min}_{t}$ selects splittings at the top of the emission phase space
(Fig. 1), with formation times less than $0.4$ fm, and addresses the question
of whether vacuum splittings indeed dominate the early evolution of jets. In
the multiple soft scatterings approximation, medium-induced emissions will
typically appear at $k_{t}^{2}\approx\hat{q}L\approx 2^{2}$ GeV2 and
$\theta>\theta_{c}$. Therefore, if the value of the cut is large enough
$k_{t}^{\min}\gg\sqrt{\hat{q}L}$, one expects a vacuum-like substructure even
in the medium. We refer to this as the factorisation (or separation) of
vacuum-like and medium physics. Lowering $k^{\rm min}_{t}$ (but still keeping
it far from non-perturbative scales $\mathcal{O}(1)$ GeV) opens up the phase
space for perturbative collinear splittings below $\theta_{c}$, and the
observable becomes sensitive to splittings being resolved or unresolved by the
medium. In addition, it could potentially be affected by perturbative elastic
scatterings and medium-induced emissions, whose modelling changes in different
jet quenching models. Finally, the lowest value of $k^{\rm min}_{t}$ maps the
region dominated by non-perturbative physics.

  * •

$\theta_{\rm hard}$: the angular distribution of the hardest splitting is
known to be sensitive to the substructure-dependence of energy loss [62], e.g.
as caused by $\theta_{c}$ [51, 70]. It is also motivated by the simplified
theoretical picture in which jet energy loss depends on an angular resolution
scale, namely $\theta_{c}$ [51, 70]. From the point of view of distinguishing
vacuum from medium-induced emissions, this choice is also reasonable since the
latter typically occur at large angles as a result of transverse momentum
broadening.

In the landscape of other jet substructure studies, the proposed algorithm is
unique in the sense that it first selects the hardest splitting above the
$k_{t}^{\rm min}$ cut and then studies its angular distribution. The main
difference with respect to dynamical grooming observables with $a=1$ [71] is
precisely $k_{t}^{\rm min}$. Predictions for dynamically groomed observables
in heavy-ion collisions have been put forward in Refs. [62, 72, 73],
demonstrating its sensitivity to colour coherence. Given our interest in
studying the earliest stages of the shower, selecting the splitting with the
shortest formation time might a priori be appealing [71, 74, 75].
Nevertheless, these splittings can have an arbitrarily low $k_{t}$, and
therefore the observable is polluted with non-perturbative corrections.
Including a transverse momentum cut was also explored in the Late-$k_{t}$
approach [76], where instead of the hardest, the most collinear splitting
above a certain $k_{t}^{\rm min}$ is selected. This option is not well suited
for our purposes since we want to explore the full angular distribution.
Furthermore, SoftDrop [77] with $\beta=1$ and $z_{\rm
cut}=k_{t}^{\min}/(p_{t}R)$ would probe the same phase space as we do, but it
also biases the angular information by selecting the widest instead of the
hardest splitting. So far, the SoftDrop setup mostly explored in heavy-ion
collisions considers $\beta=0$ [78, 79, 80, 81, 82, 83, 63, 73, 84]. Setting
$\beta=0$ does not restrict the $k_{t}$ of branchings and, therefore, non-
perturbative effects might still be sizeable. Instead of selecting the hardest
splitting per jet, one could choose all primary splittings with
$k_{t}>k_{t}^{\rm min}$, corresponding to a slice of the primary jet Lund
plane [42, 85]. This option would explore all the physics mechanisms we are
interested in but, for the sake of simplicity, we stick to the one-splitting-
per-jet case. Finally, resolution effects on energy-correlators are also
currently under study [86, 87].

##  III Proton-proton baseline

Figure 2: The angular distribution of the hardest-$k_{t}$ splitting inside
inclusive, pp jets for different $k^{\rm min}_{t}$ values in NLO
Powheg+Pythia8. The bottom panels show: (i) the impact of NLO matching, and
(ii) the impact of hadronisation on the pure shower samples, i.e. not matched
to NLO. In all cases, the band represents the statistical uncertainty.

Figure 3: The pp baseline prediction for $\theta_{\rm hard}$ by different jet
quenching Monte Carlos. From right to left the value of $k^{\rm min}_{t}$
decreases. The bottom panel displays the ratio to the state-of-the-art
prediction, namely Powheg+Pythia8.

We begin our analysis by exploring the behaviour of the observable in proton-
proton collisions at $\sqrt{s}=5.02$ TeV. In Fig. 2, we show the self-
normalised $\theta_{\rm hard}$-distribution for different values of $k^{\rm
min}_{t}$ as obtained by matching, in a Powheg approach [88], the exact $pp\to
jj$ next-to-leading order (NLO) matrix element to the Pythia8 parton shower
[89]. We observe a clear shift towards smaller angles when decreasing $k^{\rm
min}_{t}$. This feature can be easily understood from Fig. 1, since phase-
space in the collinear regime opens up when lowering the $k^{\rm min}_{t}$
constraint. The lowest accessible angle is given by $\theta^{\rm min}_{\rm
hard}\propto k^{\rm min}_{t}/p_{t}$.333Notice that our jet selection is
$p_{t}>400$ GeV, and therefore there is no hard cutoff on the smallest angle.
We anticipate that the net angular reach for a given $k^{\rm min}_{t}$ will be
an important aspect when including medium effects. On the wide-angle side, we
note that angles larger than $R$ are possible due to the C/A reclustering, but
they are highly unlikely. The error bands correspond to statistical
uncertainties only (combined in the ratio plots).

In the lower panels of Fig. 2 we quantify the impact of NLO corrections and
hadronisation.444We also performed NLO studies with MadGraph5_aMC@NLO+Herwig
[90, 91], resulting in similar curves, although with a slower statistical
convergence.,555Multi-parton interaction effects were also studied and they
are negligible. The former is expected to be more relevant for the largest
$k^{\rm min}_{t}$ since, in that regime, the true matrix element notably
differs from the soft-and-collinear approximation used by the parton shower.
Indeed, we observe that for $k^{\rm min}_{t}=1,5$ GeV, NLO corrections are
practically irrelevant. In turn, we find a moderate effect of about $20\%$ for
$k^{\rm min}_{t}=20$ GeV.

Hadronisation effects have the opposite $k^{\rm min}_{t}$-dependence compared
to NLO corrections. The deeper we go into the infrared regime by lowering
$k^{\rm min}_{t}$, the bigger the corrections are. For $k^{\rm min}_{t}=20$
GeV we find that hadron and parton curves agree within statistical uncertainty
(except in the $\theta_{\rm hard}>R$ region coming from hadrons on the
boundary of the jet cone), thus demonstrating the pQCD purity of this region
of phase-space. The intermediate value of $k^{\rm min}_{t}$ receives less than
$20\%$ hadronisation corrections in a wide angular region, while they reach up
to $50\%$ for $k^{\rm min}_{t}=1$ GeV.

Next, we compare these state-of-the-art pQCD results with the pp (i.e. vacuum)
baseline of the jet quenching Monte Carlo codes that we use in this paper:
Hybrid [36], JetMed [13, 92], and Jewel [30, 41]. This comparison is relevant
since none of the current jet-quenching MCs implement NLO corrections. It
could be argued that deficiencies in vacuum modelling are irrelevant since
they would potentially cancel in a medium-to-vacuum ratio. However, this
statement is exact only when vacuum and medium physics are entirely decoupled.
This is actually not the case in most jet-quenching Monte Carlo event
generators. A deficient description of the pp baseline can lead to a
misleading interpretation of the medium-modified distributions.

Figure 4: 2D distribution of the hardest splitting ($\theta_{\rm
hard},k_{t,{\rm hard}}$), using different jet-quenching models (similar to the
sketch in Fig. 1). The ratio to pp illustrates the modification of jets inside
the quark-gluon plasma. The approximated resolution phase spaces are denoted
with black lines. Note that the ratio extends in the upper diagonal because of
our $p_{t}>400$ GeV selection.

The models under comparison have different vacuum showers: Hybrid is built on
the default tune of Pythia8.3, while Jewel implements a virtuality-ordered
Pythia6-like shower [93]. They both include initial-state radiation and
hadronisation, in contrast to JetMed, which relies on a pure final-state
shower at parton level and thus misses many ingredients that might be
particularly relevant at low $p_{t}$ and large jet radii. The three models are
compared in Fig. 3 for all values of $k^{\rm min}_{t}$. For the highest
$k_{t}$-cut we observe quite a good agreement ($<10\%$ differences) among the
three models. Remarkably, we also find the ratios to the state-of-the-art
vacuum baseline to be compatible with unity in the whole angular range,
although statistical uncertainties blow up in the edge bins. The fact that the
parton-level curve obtained with JetMed agrees with the rest in the bulk of
the distribution indicates that the observable is indeed dominated by pQCD
physics. This well-defined vacuum baseline is an essential requirement to
unambiguously identify medium modifications. For the $k^{\rm min}_{t}=5$ GeV
setup, JetMed and Jewel deviate up to $25\%$ in the $\theta_{\rm hard}<0.05$
region from the Powheg+Pythia8 baseline. It is important to keep in mind that
this regime of very small angles will play a role in the medium studies. This
consensus among models breaks down when $k_{t,\rm min}=1$ GeV. The $50\%$
disagreement in the JetMed case is mainly caused by the lack of hadronisation,
as we show in Fig. 2. Naturally, the Hybrid model ratio to NLO+Pythia8 is
compatible with $1$ since they share the same shower and, in this regime, NLO
corrections are negligible. The Pythia6 vs. Pythia8.3 differences result in an
up to $20\%$ deviation in the Jewel case.

After this detailed analysis of the pp baseline in which we have found a
remarkably good agreement between Monte Carlo codes for the highest $k^{\rm
min}_{t}$ values, we proceed to study medium modifications.

##  IV Heavy-ion results

In this section we apply our analysis technique for jets in heavy-ion
collisions, using some of the most popular jet quenching models. We
demonstrate the advantage of slicing the jet substructure by isolating
different effects in the medium one by one. These MC studies employ a high
number of jets to better illustrate the physics under study. A discussion on
the projected experimental statistics and its uncertainties can be found in
App. A.

###  IV.1 Radiation phase-space for MCs under study

To gain intuition on the medium modifications to the $\theta_{\rm
hard}$-observable shown below, we first simulate the analog of Fig. 1 with the
three jet-quenching Monte Carlo codes introduced in the previous section:
JetMed, Hybrid, and Jewel. That is, we plot in Fig. 4 the PbPb-to-pp ratio of
the ($k_{t}$, $\theta_{\rm hard}$) self-normalised density of the hardest
splitting within our jet selection. Let us discuss each of the three plots
individually and, in particular, the relevant scales therein:

  * •

JetMed [13, 92]. The description of the QGP in this model is rather
simplified, i.e. it can either run with a Björken expanding media [94] or a
brick. In this paper, we use the former with $L=4$ fm and $\hat{q}=1.5$
GeV2/fm. The parton-medium interaction is calculated in the multiple soft
scatterings approximation. An explicit realisation of colour coherence
dynamics is implemented by means of $\theta_{c}$ (with a constant value event-
by-event). The typical relative transverse momentum of induced emissions is
$\langle k_{t}\rangle=\sqrt{\hat{q}L}\approx 2.4$ GeV, and they appear above
the critical angle $\theta>\theta_{c}=2/\sqrt{\hat{q}L^{3}}=0.04$. These
induced emissions are responsible for the small enhancement below the
$t_{f}=t_{d}=[4/(\hat{q}\theta^{2})]^{1/3}$ line. 666The decoherence time
$t_{d}$ is the time at which a colour-connected dipole gets resolved by the
medium, by destroying its colour correlation via multiple colour rotations.
The most prominent feature in Fig. 4 is the depletion of emissions in the
polygon delimited by the $t_{f}<t_{d}$ and $\theta>\theta_{c}$ boundaries.
Vacuum-like emissions in this corner of phase space are resolved by the medium
and source more energy loss. Due to the selection bias (described in Sec. II)
these wide-angle substructure jets are thus suppressed. Another relevant
feature of this model is that vacuum-like emissions are forbidden (vetoed) in
the $t_{d}<t_{f}<L$ region since this is the regime dominated by broadening
dynamics and not enhanced by the jet energy. In this context, vacuum and
medium-induced cascades do factorise exactly, with the former acting as
sources for the latter. Finally, a third stage of vacuum-like showering is
also implemented with an extended angular region, i.e., the first emission
outside the medium can be emitted at any angle due to the color randomisation
suffered by the emitting dipoles during their evolution throughout the medium
[70]. This third stage of the shower fills the lowermost part of the phase
space, down to $1$ GeV. The hard-coded value of $\theta_{c}$ helps to quantify
the sensitivity of the $\theta_{\rm hard}$ observable to coherent vs.
incoherent energy loss. Finally, no medium response and hadronisation are
implemented in JetMed.

  * •

Hybrid [36, 95]. This model runs a perturbative vacuum shower down to the
hadronisation scale and then rewinds through the branching history so as to
incorporate an energy loss rate computed at strong coupling [28, 96]. No
medium-induced emissions are considered. The total amount of energy lost by
splittings depends on their propagation distance within the QGP, as determined
by embedding the Pythia8 event in a realistic heavy-ion simulation using 2+1D
viscous hydrodynamics [97]. The jet production points are sampled from two
overlapped nuclear density profiles using the Glauber model [98]. Emissions
whose formation time is smaller than the hydrodynamic initialization time
$t_{0}=0.6$ fm are not quenched. Another feature of the Hybrid model is that
only those partons that are “separated enough”, i.e. resolved by the medium,
lose energy independently. The medium-resolution power is controlled by the
$L_{\rm res}$ parameter, which depends on medium properties such as the local
temperature [99]. For two colour-connected legs to lose energy independently,
their transverse distance, $r_{\perp,\rm dip}\approx\theta(L-t_{f})$, has to
be larger than the resolution length $L_{\rm res}$. In order to illustrate the
resolved phase-space, we pick a representative value of the medium length in
central PbPb collisions, $L\sim 6$ fm, and $L_{\rm res}=2/(\pi\cdot
0.25{\rm[GeV]})\approx 0.5$ fm. All emissions above the $r_{\perp,\rm
dip}=(L-t_{f})\theta>L_{\rm res}$ (black curve in Fig. 4) are resolved by the
medium, lose more energy, and are therefore suppressed by the selection bias.
Here, the phase space boundary is not as sharp as for JetMed because both $L$
and $L_{\rm res}(T)$ fluctuate. In what follows, we choose $L_{\rm res}=2/(\pi
T)$ and explore other values of $L_{\rm res}$ in App. C. The lost energy is
then hydrodynamised, producing jet-induced wakes that decay into hadrons at
the freeze-out hypersurface of the flowing medium [100]. These wake particles
get clustered with the jet resulting in an excess of soft, wide-angle
splittings (lower left corner in Fig. 4). All results shown below include
medium response and a detailed study of its impact is presented in App. D.

  * •

Jewel [30, 41]. At each evolution step in the parton shower, the propagating
parton can either emit vacuum radiation or undergo elastic ($2$-$2$)
scatterings with medium partons. The process with the shorter “time”
(formation time, or virtuality, compared to light-cone time) is realized.
Elastic scatterings result in transverse momentum broadening, drifting energy
out of the jet cone, and causing energy loss. Scatterings also reset the
shower scale and induce collinear emissions without changing the jet energy.
To estimate at which time ($t_{\rm el}$) elastic scatterings become dominant,
we compare the scattering probability $p_{\rm el}\approx(t_{\rm
el}-t_{0})/l_{\rm mfp}$ with the radiation one $p_{\rm rad}\approx\int^{t_{\rm
el}}_{t_{0}}dt\int_{0}^{1}dzP_{1\to 2}(t,z)$. Here, $l_{\rm mfp}$ is the
average distance between scatterings, $t_{0}$ is the initial formation time,
and the splitting probability (expressed with formation time) is $P_{1\to
2}\approx\alpha_{s}P(z)/(\pi t)$, where we used the Altarelli–Parisi splitting
function $P(z)\approx 2C_{i}/z$. When $t\gg t_{\rm el}$, elastic scatterings
interrupt vacuum radiation and induce extra collinear emissions (small angle,
hard enhancement, $z\sim 1$ in Fig. 4). Additionally, elastic scatterings
broaden soft splittings to wider angles (wide angle enhancement below $1$ GeV
in Fig. 4).777In Fig. 4, induced collinear emissions and broadening overlap.
In our numerical tests, we however identified the separation of these two
regions. In contrast to the previous two models, an additional notion of
colour coherence is not implemented. The medium in Jewel is modelled as a
thermal, longitudinally expanding parton gas. The medium response is accounted
for by keeping track of the recoiled medium partons after an elastic
scattering takes place. These recoilers however free stream without further
scatterings or radiation. In what follows we will disregard recoil particles
and explore their impact in App. D. We do this as our current setup of Jewel
would overestimate all medium response effects since, after the scattering,
their evolution is frozen, i.e. they are non-dynamical.

###  IV.2 Early times/vacuum-dominated regime

We first study jets in PbPb collisions whose hardest splitting satisfies
$k_{t}>20$ GeV. This region corresponds to very short formation times, i.e.
parametrically $t_{f}<2p_{t}/(k^{\rm min}_{t})^{2}\approx 0.4$ fm, smaller
than the mean free path $l_{\rm{mfp}}=(\rho\sigma_{\rm{el}})^{-1}$ (where
$\rho$ is the density of scattering centers and $\sigma_{\rm el}$ the total
elastic cross section). The $\theta_{\rm hard}$-distribution is displayed in
Fig. 5. We observe that all models agree within statistical uncertainties and
that the medium-to-vacuum ratio is remarkably close to unity.888Note that
these distributions are self-normalised and thus the overall suppression of
the jet spectrum is factored out. This choice is justified since our main
interest is a potential shape modification. This result suggests that, in this
upper corner of phase space, vacuum-like splittings are likely to be tagged
and that its prongs lose the same amount of energy, independently of their
opening angle. We show this independence analytically in App. B. We refer to
this as the factorisation (or separation) of vacuum and medium-induced
physics. This separation is very general in the sense that it is independent
of the medium modelling when one goes to asymptotically high jet-$p_{t}$ and
$k_{t}^{\rm min}$. In the JetMed context, these results are easily interpreted
since the cut excludes the possibility of tagging medium-induced emissions
($k^{\rm min}_{t}\gg\sqrt{\hat{q}L}$) and $\theta_{\rm hard}$ is always bigger
than $\theta_{c}$. We would like to note that Hybrid wake effects at $k^{\rm
min}=20$ GeV are completely negligible, and recoil particles in Jewel are
strongly suppressed (see in App. D).

We would like to highlight that not all jet-quenching models predict a flat
ratio. For example, a potential source of substructure modifications at such
high energy scales is the single-hit (or higher-twist) corrections induced by
rare, hard interactions with the medium. These effects are included in the
SCETg formalism [101] and in MATTER+LBT Monte Carlo generator [34, 35].
However, the latest implementation of MATTER makes use of a virtuality-
dependent $\hat{q}$ [102, 39] that effectively reduces the impact of such
corrections in this high-$k_{t}$ regime.

The experimental measurement of the proposed observable has the potential to
unambiguously establish whether vacuum physics dominates the early stages of
jets in heavy-ion collisions. Knowing that there is a region of phase-space
dominated by vacuum splittings implies that all jet quenching models should
agree not only among them, but also with the pp baseline, within that given
region. This would represent a major step forward in our understanding of jet
quenching and is one of the main results of this paper.

Figure 5: The angular distribution of the hardest-$k_{t}$ splitting inside
inclusive jets in heavy-ion collisions with $k_{t}^{\rm min}=20$ GeV for the
jet quenching Monte Carlo event generators described in Fig. 4. The bottom
panel displays the medium-to-vacuum ratio.

###  IV.3 Substructure-dependent energy loss

Figure 6: Same as Fig. 5 but for $k^{\rm min}_{t}=5$ GeV.

When lowering the cut to $k_{t}^{\rm min}=5$ GeV, we enter a domain where one
can test the substructure dependence of energy loss. Here, emissions with much
longer formation times $\lambda_{\rm{mfp}}\ll t_{f}<6.4$ fm appear, while soft
medium-induced splittings are still suppressed $k_{t}^{\rm
min}>\sqrt{\hat{q}L}\approx 2.4$ GeV. In the colour coherence picture sketched
in Fig. 1, we expect splittings with $\theta<\theta_{c}$ to appear, which are
unresolved by the medium. As we have discussed, jets with wider splittings
$\theta>\theta_{c}$ loose more energy and are suppressed due to selection
bias. Consequently, we expect the suppression of $\theta_{\rm
hard}>\theta_{c}$ (or equivalently an enhancement of $\theta_{\rm
hard}<\theta_{c}$). Most modern jet quenching models implement this coherence
(or resolution) effect, but they differ in the shape of the phase space
boundary as discussed in Sec. IV.1 (see in Fig. 4). Theoretically, this
boundary is not known beyond the soft-and-collinear limit, and models
implement it differently. An experimental measurement in this regime,
therefore, could put tighter constraints on the modelling of the resolution
phase space.

In Fig. 6, we show the angular distribution of the hardest splitting
$\theta_{\rm hard}$ for $k_{t}^{\rm min}=5$ GeV. The PbPb-to-pp ratio in
JetMed displays a sharp enhancement at $\theta_{c}=0.04$, a clear consequence
of colour resolution and selection bias. Our expectation is that, on both
sides of $\theta_{c}$, the jet substructure does not change much, it is still
vacuum-like (the ratio is flat). For the Hybrid model, the resolution boundary
is not as sharp as it gets washed out due to fluctuations in the propagation
length, temperature, and parton energy. The overall picture is however similar
to JetMed: wide-angle splittings are suppressed. Results for the Hybrid model
with different $L_{\rm res}$ values are shown in App. C. We would like to
highlight that the angular distribution in the $L_{\rm res}=\infty$ fully
coherent limit of the Hybrid model (only the initiator loses energy
independent of the substructure) is barely modified as compared to vacuum, as
expected. Even though the wake is included in Fig. 6, the $5$ GeV cut is still
large enough to completely remove its effects (see it in more detail in App.
D).

The Jewel model is somewhat distinct from the other two models since it lacks
an explicit implementation of colour coherence. The competition between
elastic scattering and vacuum radiation still outlines a boundary that we
referred to as $t_{\rm el}$. However, this boundary is beyond the phase space
limited by $k_{t}^{\min}=5$ GeV (see Fig. 4) and therefore the medium-to-
vacuum ratio is compatible with unity. Jets in Jewel of course lose energy,
but their internal structure is vacuum-like even at $k_{t}\sim 5$ GeV. This
ratio mildly changes when keeping track of the recoiling partons as we present
in App. D.

Let us re-emphasise that the proposed measurement, after identifying the
vacuum-like dominated region of phase-space, has the potential to pin down the
angular dependence of energy loss. In particular, it can quantify the angular
resolution power of the QGP, tightly connected with its microscopic, transport
properties, while putting tighter constraints on the actual size of the
resolved phase space.

Furthermore, at wider angles ($\theta_{\rm hard}\gg\theta_{c}$) our observable
with $k_{t}^{\rm min}=5$ GeV is sensitive to the presence of hard recoils
produced in perturbative elastic scatterings (also referred to as Molière
scatterings), while remaining insensitive to the presence of the soft hadrons
from the jet-induced wake (see the detailed study in App. D). This exemplifies
the discriminating power of future experimental measurements of this
observable to distinguish between different physical assumptions on the
thermalisation dynamics of jets in the QGP.

###  IV.4 Non-perturbative regime

Figure 7: Same as Fig. 5 but for $k^{\rm min}_{t}=1$ GeV.

As a last step, we set a lower selection cutoff $k_{t}^{\rm min}=1$ GeV. By
doing this, we extend the low-angle reach of the distributions and we also
enter a region where contributions from non-perturbative effects are expected.
In addition, as discussed in App. A, fake splittings might hamper the
unfolding of experimental results. Even if it might not be possible to produce
particle-level measurements in experiments, we think it is important to show
when and how the perturbative picture studied above breaks down. The results
are shown in Fig. 7. With the extended low-angle reach, the threshold
behaviour of the JetMed distribution at $\theta_{\rm hard}=\theta_{c}$ becomes
more apparent. The ratio on both sides of $\theta_{c}$ is flat, showing the
dominance of vacuum-like substructure. The contribution of medium-induced
emissions seems to be subleading (recall that $k_{t}^{\rm
min}<\sqrt{\hat{q}L}=2.4$ GeV). An important caveat is that hadronisation
corrections can reach up to $50$% in this regime (see Fig. 3) and thus JetMed
might not be accurate in this regime.999It is often argued that hadronisation
effects cancel in the PbPb-to-pp ratio. This is, however, a model-dependent
statement that we have explicitly checked by computing the double ratio
between the PbPb and pp distributions at hadron and parton levels. In the
Hybrid model, this ratio is consistent with unity for this value of $k^{\rm
min}_{t}$, while in Jewel $\mathcal{O}(1)$ hadronisation effects are observed
for $\theta<0.05$. This difference originates from the fact that only Jewel
changes the number of partons that undergo string fragmentation, while Hybrid
either reduces their energy or removes them completely.

For the Hybrid model, the narrowing of the distribution persists, which
indicates that wake particles, typically manifesting as a bump at large angles
(see App. D), are suppressed by this observable with this set of fiducial
cuts, mainly due to the small $R=0.2$ choice. We note that the fact that
JetMed and Hybrid ratio curves cross unity at the same angle is coincidental.
In fact, while the JetMed curves switch from depletion to enhancement at
$\theta_{\rm hard}\sim\theta_{c}$ for $k^{\rm min}_{t}=1\to 5$ GeV, the
crossing point in the Hybrid case changes from $\theta_{\rm hard}\sim 0.1$ to
$0.05$ when lowering $k^{\rm min}_{t}$ from $5\to 1$ GeV. Thus, with a high
enough angular resolution, the experimental measurement of the crossing point
as a function of $k^{\rm min}_{t}$ could potentially disentangle between
selection bias effects. Differentiating in centrality and jet $p_{t}$ would
further prove the existence of a critical resolution angle.

In the case of Jewel, we also observe an enhancement of narrow structures in
PbPb versus pp. As we previously discussed, we attribute this to elastic
scatterings, which reset the vacuum shower scale inducing collinear emissions.
Further elastic scatterings did not broaden these emissions enough towards
wider angles. All three models indicate that, even in this low domain of
$k^{\rm min}_{t}$, the prevailing medium modification of the $\theta_{\rm
hard}$ distribution is the intra-jet structure dependence of energy loss.

It is worth noting that this distribution is significantly altered when
introducing recoil effects in Jewel (see Fig. 11). At the same time, it turns
out to be remarkably resilient to wake particles. One can still study the wake
particles and therefore the thermalisation of jets by opening the jet cone to
$R=0.4$ (and keeping $k_{t}^{\rm min}=1$ GeV), see in App. D.

##  V Summary

This study proposes a novel analysis strategy aiming at experimentally
accessing different stages of jet evolution in heavy-ion collisions. It
explores the angular distribution of the hardest splitting, $\theta_{\rm
hard}$, above a transverse momentum threshold $k_{t}^{\rm min}$. This momentum
scale can be tuned so as to enhance or suppress certain features of jet
dynamics. For example, selecting hard enough splittings ($k_{t}^{\rm min}\sim
20$ GeV) the observable is dominated by perturbative physics and, in
particular, by the vacuum evolution of an in-medium jet. This perturbative
region extends to even lower cuts ($\sim 5$ GeV) with the additional
sensitivity to the colour resolution power of the medium and hard elastic
scatterings. In contrast, when the hardest splitting becomes commensurate to
the QGP and confinement scales ($k_{t}^{\rm min}\sim\Lambda_{\rm QCD},T$), it
becomes sensitive to thermalisation and hadronisation effects in the medium. A
key observation is that this strategy is optimal when the jet radiation phase-
space is so large ($p^{\rm jet}_{t}=\mathcal{O}(100)$ GeV) that it admits
well-separated regions between vacuum- and medium-dominated dynamics. Although
the outlined strategy is completely general/data-driven, we illustrate the
potential of this observable by using three of the most popular jet quenching
Monte Carlo event generators: Hybrid, JetMed, and Jewel.

First, we performed a systematic study of the proton-proton baseline using
state-of-the-art techniques (Powheg+Pythia8). We compared this baseline to the
vacuum prediction of jet-quenching event generators. At sufficiently high
$k_{t}^{\rm min}$ cut, we find remarkably good agreement among all vacuum
models, and therefore any potential modifications due to medium effects arise
from a well-controlled baseline. Differences in the vacuum mode of the jet-
quenching models gradually appear at low $k_{t}$, where hadronisation
corrections become sizeable (up to 50% for $k_{t}^{\rm min}=1$ GeV).

On the heavy-ion side, we first address the question of whether the early
stages of in-medium evolution are dominated by vacuum dynamics. To that end,
we selected $k_{t}^{\rm min}=20$ GeV, which is high enough to ensure
perturbativity. The three models under study predict a PbPb-to-pp ratio
compatible with unity across all $\theta_{\rm hard}$ values. We interpret this
as the result of a vacuum-like splitting being tagged, followed by energy loss
that is largely independent of the substructure. We show analytically why the
amount of lost energy is independent of the opening angle of the splitting in
this hard regime. An experimental confirmation of this high $k_{t}^{\rm min}$
result will show, for the first time, a vacuum-dominated observable in a
heavy-ion environment.

Next, we extend the angular coverage of the observable by lowering to
$k_{t}^{\rm min}=5$ GeV. This intermediate value opens up the possibility of
understanding the small-angle dependence of energy loss while staying in the
perturbative regime. At small angles, a reduction of energy loss is expected
due to colour resolution effects. This angular dependence is then enhanced by
selection bias effects, where the jet ensemble contains fewer jets with wider
substructure, as they typically lose more energy than narrower ones.101010See
footnote 2 for an extended discussion on selection bias effects. We indeed
observe a clear narrowing in the $\theta_{\rm hard}$-distribution in those
models where some sort of resolution scale is implemented. In regards to wide-
angle enhancement, we observe the imprints of hard elastic scatterings,
especially for wider $R=0.4$ jets, in Jewel. For the Hybrid model, there is no
signal from the medium response as the implemented thermal wake is removed by
the $k_{t}^{\min}=5$ GeV cut. The experimental test of this intermediate
$k_{t}^{\rm min}$ result could reveal the colour resolution scale in the
medium, a fundamental property of the QGP. Furthermore, this region also gives
experimental access to explore hard (Molière-like) elastic scatterings in the
medium.

Decoupling selection bias and colour coherence effects might be possible with
high enough experimental precision or with complementary measurements in
$\gamma$/Z+jet events [103, 56], at forward rapidities [58] or by means of a
centrality scan [104]. In fact, recent experimental results from CMS in
$\gamma$+jet events have shown that the selection bias can be suppressed by
reducing $x_{J}=p_{t,{\rm jet}}/p_{t,\gamma}$ [105]. We leave the study of the
$\theta_{\rm hard}$ observable in boson-tagged events for future work.

Finally, we explore the scenario in which $k_{t}^{\rm min}=1$ GeV, which lies
in the non-perturbative regime. We find that in JetMed the $\theta_{\rm
hard}$-distribution is narrowed in PbPb due to the colour resolution with
selection bias effects and only a few medium-induced emissions get tagged for
narrow jets ($R=0.2$). This spoils the possibility of measuring the medium-
induced emission kernel with this observable and endorses the idea of doing so
by exploring a regime in which vacuum-like emissions are strongly suppressed,
such as the dead-cone [106, 76, 107]. Another effect that this value of
$k_{t}^{\rm min}$ targets is the soft medium response, i.e. jet
thermalisation. Wake particles in the Hybrid model leave a weak imprint on
this observable mainly due to the use of relatively narrow $R=0.2$ jets. In
the Jewel case, recoil medium particles create a broadening of the
$\theta_{\rm hard}$-distribution around the jet boundary. This regime,
although experimentally challenging to unfold, has the strongest potential to
help us understand jet thermalisation and constrain the modeling of medium
response in jet-quenching Monte Carlo generators.

As future steps, we will work on the analytic calculation of this observable
in proton-proton collisions at high-logarithmic accuracy by extending the
results presented in Ref. [108] to the case of an additional $k_{t}^{\rm min}$
selection. In regards to the heavy-ion scenario, the present Monte Carlo study
has informed us about the feasibility to perturbatively describe the
high-$k_{t}$ regime, which motivates carrying out analytic calculations,
including medium-induced effects, that can be sensibly confronted with
experimental data.

We are confident that an experimental realisation of the proposed strategy
will provide new key inputs to improve our understanding of the rich interplay
between the vacuum and medium scales that govern jet evolution in the quark-
gluon plasma.

## Acknowledgements

The authors would like to thank the organisers of the “QCD challenges from pp
to AA workshop” which took place in Padova, Italy, where this collaboration
started. We also thank Jack Holguin for participating in the early stages of
this project and his feedback on the manuscript. We are grateful to Paul
Caucal for providing the JetMed samples and comments on the manuscript, Luca
Rottoli for providing the Powheg samples, and Silvia Ferrario-Ravasio, Aidin
Masouminia, and Simon Plätzer for helping with the Herwig results. The work of
LCM was supported by the European Research Council project
ERC-2020-COG-101002207 QCDHighDensityCMS. The work of DP has received funding
from the European Union’s Horizon 2020 research and innovation program under
the Marie Sklodowska-Curie grant agreement No. 754496. The work of ASO has
been funded by the European Research Council (ERC) under the European Union’s
Horizon 2020 research and innovation program (grant agreement No 788223). The
work of MS has been supported by the Ministry of Education, Youth and Sports
of the Czech Republic under grant ERC-CZ LL2327. The work of AT is supported
by the Starting Grant from Trond Mohn Foundation (BFS2018REK01), the
University of Bergen, and by DFG – Project number 496831614. The work of MV
was supported by the Dutch Research Council (NWO) - Project number
STU.019.019.

Figure 8: Left: Fraction of fake splittings obtained by embedding Pythia8 pp
events in a thermal background [85, 109] for various minimum selections of
$k_{t}$. Right: the relative statistical uncertainty of the $\theta_{\rm
hard}$ distribution by considering the experimentally accessible number of
jets in runs 3 and 4 of the LHC.

##  Appendix A Experimental considerations

Here we address the experimental feasibility of the proposed observable. To
start with, this would be the first triple-differential measurement in heavy-
ion collisions, i.e. the measurement will require the simultaneous unfolding
of background and detector effects in three quantities,
$p_{t}^{\mathrm{jet}}$, $k_{t}^{\rm min}$ and $\theta_{\rm hard}$. However,
since only thresholds on jet $p_{t}$ and $k_{t}$ are imposed, the unfolding
problem is simpler than when performing the triple differential measurement
over many bins.

An important aspect when correcting the measurement to particle level is the
fraction of fake splittings (splittings originating entirely from upward
fluctuations of the underlying event). We have estimated the fraction of fake-
splittings in our observable by embedding Pythia8 events into a thermal
background mimicking the underlying event of a heavy-ion collision. The
results are shown in the left panel of Fig. 8 after applying full event
constituent subtraction [110] to the embedded jets. Jet substructure
measurements have been reported for fake fractions smaller than $20\%$. This
threshold is met by our observable as long as $k_{t}^{\rm min}\gtrsim 2$ GeV.
ALICE studies of the dynamically groomed $k_{t}$ have shown that for $R=0.2$
jets in central collisions, a detector-level cutoff of $k_{t}>1.5$ GeV renders
the observable unfoldable. Due to the strong migrations in $k_{T}$ between
detector and true levels, the fully corrected results can be reported at
$p_{t}=60$ GeV and $k_{t}>3$ GeV in central collisions [111]. The higher jet
$p_{t}$ required in our study implies higher purities and thus we conclude
that unfolding this triple-differential observable should not be accompanied
by large systematic uncertainties.

Other challenges to realize experimentally the scan of the emission phase-
space described in the previous Sections concern the ability to measure small-
angle splittings. Track-based measurements of the Lund plane by ATLAS [112],
CMS [113], and ALICE [114] have demonstrated the ability to reconstruct
splittings down to angles $\theta\approx 0.005$ and should therefore provide
sufficient sensibility to colour coherence effects.

In this study, jets were required to have $p_{t}>400$ GeV. For LHC Run 3 and
Run 4, assuming total collected luminosity of 13 nb-1 [115], we estimated the
total number of $R=0.2$ jets with $p_{t}>400$ GeV within $|\eta|<2.8$ to
exceed $1.5\cdot 10^{4}$ in $0-10$% central PbPb collisions. The estimate is
based on the data published in Ref. [116]. The right panel of Fig. 8 shows the
expected statistical uncertainty of the $\theta_{\rm hard}$-distribution with
this projected statistics obtained with Monte Carlo simulations. We would like
to remark that reducing the jet $p_{t}$ selection down to $200$ GeV can
increase the statistics by a factor of $50$. The exact jet $p_{t}$ selection
is thus to be decided by experiments based on the available luminosity and
effects seen in the data.

##  Appendix B The reduction of bias effects at high $k_{t}^{\rm min}$

Figure 9: Size of resolved phase-space for jets with $R=0.2$, $p_{t}>400$ GeV
as a function of $\theta_{\rm hard}$ for different $k_{t}^{\rm min}$ values
(corresponding to splittings with $\theta>\theta_{c})$ in the soft-and-
collinear limit, see Eq. (2).

Figure 10: Sensitivity of Hybrid results to the value of the resolution length
$L_{\rm res}$. From left to right the value of $k_{t}^{\rm min}$ decreases.

The absence of selection bias at high-$k_{t}^{\rm min}$ values can be
understood based on simple considerations. The object of interest is the
energy lost by a jet featuring a splitting with $k_{t,\rm hard}>k^{\rm
min}_{t}$ as a function of its opening angle $\theta_{\rm hard}$, that we
denote $W(\theta_{\rm hard},k^{\rm min}_{t})$. The amount of energy lost is
directly proportional to the size of the resolved phase space [117], that at
$\mathcal{O}(\alpha_{s})$ reads

$\displaystyle\Omega(k_{t,\rm{hard}})$
$\displaystyle=\int_{0}^{R}d\theta\int_{0}^{k_{t,\rm
hard}}dk_{t}\frac{2C_{i}\alpha_{s}(k_{t})}{\pi}\frac{d\sigma}{dk_{t}d\theta}$
$\displaystyle\times\Theta(t_{f}<L)\Theta(t_{f}<t_{d}),$ (1)

where $d^{2}\sigma/dk_{t}d\theta$ is the double-differential cross section for
producing a vacuum-like splitting (stripped off colour factor and coupling)
and the second line of the equation contains all phase-space constraints
required for an emission to be part of the resolved phase-space. In order to
estimate the amount of energy loss differential in angle, this quenched phase-
space has to be weighted by the cross-section for producing the tagged
splitting, i.e.

$\displaystyle\frac{dW}{d\theta_{\rm hard}}\Big{|}_{k_{t}^{\rm
min}}=\frac{\displaystyle\int_{k_{t}^{\rm min}}^{p_{t}\theta_{\rm
hard}}dk_{t,\rm hard}\frac{1}{\sigma}\frac{d\sigma}{dk_{t,\rm
hard}d\theta_{\rm hard}}\Omega(k_{t,\rm hard})}{\displaystyle\int_{k_{t}^{\rm
min}}^{p_{t}\theta_{\rm hard}}dk_{t,\rm
hard}\frac{1}{\sigma}\frac{d\sigma}{dk_{t,\rm hard}d\theta_{\rm hard}}}\,.$
(2)

In the soft-and-collinear approximation and at fixed-coupling, Eq. 2 can be
solved analytically and the result is plotted in Fig. 9 for a quark-jet. We
observe that for $k_{t}^{\rm min}=20$ GeV, the size of the quenched phase
space is largely independent of $\theta_{\rm hard}$ thus confirming the
picture drawn by the Monte Carlo generators in Fig. 5. We note that only
$k_{t}^{\rm min}$ values for which $\theta^{\rm min}>\theta_{c}$ are shown in
the figure. This is so due to the toy model not being applicable when both
resolved and unresolved splittings contribute.

##  Appendix C $L_{\rm res}$ dependence of Hybrid results

The Hybrid model allows for smoothly changing the size of the resolved phase-
space by varying the $L_{\rm res}$ parameter. In Fig. 10 we show the
$\theta_{\rm hard}$-distribution for all $k^{\rm min}_{t}$ values in three
different scenarios: only the jet initiator is quenched, $L_{\rm res}=\infty$
(fully coherent energy loss), all particles are quenched inside the medium,
$L_{\rm res}=0$ (fully incoherent energy loss), and the intermediate scenario
present in the main text in which splittings whose transverse size satisfy
$r_{\perp,\rm dip}>L_{\rm res}=2/(\pi T)$ are resolved and lose energy. For
the highest value of $k^{\rm min}_{t}$ we observe that the resulting
$\theta_{\rm hard}$-distribution is barely dependent on $L_{\rm res}$ thus
confirming that, in this corner of phase-space, energy loss details are not
relevant and that the ingredient at test is the vacuum splitting
evolution.111111There is still some ordering among different $L_{\rm res}$,
which disappears for $R=0.4$ jets, and thus has its origin in statistical
fluctuations occurring for $R=0.2$. By lowering $k^{\rm min}_{t}$ the
observable becomes sensitive to the correlation between energy loss and jet
substructure. In fact, $L_{\rm res}=\infty$ yields no in-medium modification
as it is independent of the jet substructure. A finite value of $L_{\rm res}$
results in a narrowing of the distribution since more collimated jets tend to
lose less energy. This result endorses the potential of the proposed
observable to test the presence of an angular scale that controls the degree
of quenching, i.e. the medium resolution power. Setting $L_{\rm res}=0$
(corresponding to all splittings with $t_{f}<L$ losing energy) leads to an
even stronger narrowing of the $\theta_{\rm hard}$ distribution. In this case,
the resolution boundary does not fluctuate with temperature and/or parton
energy, resulting in a sharper enhancement.

##  Appendix D Medium response in Jewel and Hybrid

Figure 11: Same as the ratio plots in Figs. 5,6, and 7. The upper panels are
Hybrid events with and without the wake for different $k_{t}^{\rm min}$. The
lower panels are Jewel events with and without keeping track of recoilers for
different $k_{t}^{\rm min}$. The left and right side are jets with different
cone sizes $R=0.2$ and $R=0.4$.

In this section, we study the sensitivity of the proposed observable to medium
response effects in the Hybrid and Jewel models. We remind the reader that
JetMed does not account for this effect. In the main text figures we included
the Hybrid wake, while discarded Jewel recoils.

In Fig. 11 we show the PbPb/pp ratio of the $\theta_{\rm hard}$ distributions
with and without medium response for both Hybrid and Jewel models (similar to
the ratio plots in Figs. 5,6, and 7). The results of the Hybrid model (upper
left panel in Fig. 11) suggest that the medium response effects remain below
$10$% across all $k^{\rm min}_{t}$ choices for $R=0.2$ jets. The reason is
that in the Hybrid model all lost energy is assumed to thermalize instantly
and contributes as a source term in the hydrodynamic equations of motion.
Perturbative, hard elastic scatterings are neglected in the present version of
Hybrid, and the resulting medium response is a soft, wide-angle hadron
distribution from the wake. To further illustrate wake effects in Hybrid, we
repeat the previous plot for wider, $R=0.4$ jets (upper right panel in Fig.
11). Here, the wake becomes substantially more important at lower $k_{t}^{\rm
min}$ cuts, implying that the wake appears at wide angles ($\theta>0.2$). We
would like to remind the reader that the overall decreasing trend of the
PbPb/pp ratio with increasing angle is caused by the substructure-dependent
energy loss and selection bias effects as we discussed in the main text. The
effect from medium response sits on top of this effect, typically at larger
angles.

Jewel implements medium response in a radically different manner. The medium
response dynamics can affect the $\theta_{\rm hard}$ distribution even for
moderate $k^{\rm min}_{t}$ value (lower left panel in Fig. 11). The size of
this effect can reach up to $50$% for $k^{\rm min}_{t}=1$ GeV for $R=0.2$.
These recoils are thermal particles that become part of correlated background
after undergoing elastic $2\to 2$ scatterings with the jet particles. These
scatterings can be perturbative or not depending on the exchanged
momentum.121212$2\to 2$ pQCD scatterings are modified in the QGP due to
thermal screening effects typically at small momenta [118, 119]. We refer to
these soft scatterings as non-perturbative as the plasma temperature is
relatively low $T_{\rm QGP}\sim\Lambda_{\rm QCD}$.Therefore, the medium
response in Jewel produces harder particles at smaller angles than the wake in
Hybrid. Two technical details on the treatment of these recoils are relevant:
we have used the method presented in Ref. [120] to ensure energy-momentum
conservation (subtracting “holes”). Also, in our runs (default in Jewel),
further rescatterings and splittings of recoil particles are neglected, and
they free stream. In the most recent version of Jewel, recoil rescatterings
can be activated. Due to the associated computational cost, we did not
consider it here. Recoil in Jewel therefore implements the other extreme
scenario, a completely weakly coupled medium response. To further illustrate
the impact of recoil in Jewel, we repeated the analysis for wider $R=0.4$
jets. In this case, even the highest $k_{t}^{\rm min}=30$ GeV cut is sensitive
to recoil particles, evidencing the stark consequences of the free streaming
setup adopted in the present Jewel study. This is the reason why we did not
include the Jewel recoils in the main text. Indeed, it would be very
instructive to test more realistic modelling of medium response, where re-
scatterings can occur, for instance using kinetic theory [35, 121].
Differences in the PbPb/pp ratio between Hybrid and Jewel in the absence of
medium response are due to the different energy loss mechanisms adopted by
each model, where in particular one observes a flatter ratio in Jewel than in
Hybrid.

To sum up, our observable not only separates perturbative from non-
perturbative effects in terms of radiation and energy loss, but it is also
sensitive to the physics of perturbative scatterings and the thermalisation
process of jets. The choice of using jets with high-enough $p_{t}$ is in fact
crucial, as it allows enough phase space for large-momentum transfer, and thus
perturbative, scatterings with the thermal QGP constituents to occur.
Measuring this observable in experiments will in this way serve to guide the
modelling of medium response in Monte Carlo event generators.

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