# The Fox-Wolfram Moment of jet production in relativistic heavy ion
collisions

Wei-Xi Kong  Key Laboratory of Quark & Lepton Physics (MOE) and Institute of
Particle Physics, Central China Normal University, Wuhan 430079, China  Ben-
Wei Zhang<EMAIL_ADDRESS>Key Laboratory of Quark & Lepton Physics
(MOE) and Institute of Particle Physics, Central China Normal University,
Wuhan 430079, China

( )

###### Abstract

We present the first theoretical investigation of Fox-Wolfram moments (FWMs)
for multi-jet production in relativistic heavy ion collisions. In this work,
jet productions in p+p collisions are computed with a Monte Carlo event
generator SHERPA, while the Linear Boltzmann Transport model is utilized to
simulate the multiple scattering of energetic partons in the hot and dense QCD
matter. The event-normalized distributions of the lower-order FWM, $H_{1}^{T}$
in p+p and Pb+Pb collisions are calculated. It is found that for events with
jet number $n_{\text{jet}}=2$ the $H_{1}^{T}$ distribution in Pb+Pb is
suppressed at small $H_{1}^{T}$ while enhanced at large $H_{1}^{T}$ region as
compared to p+p. For events with $n_{\text{jet}}>2$, the jet number reduction
effect due to jet quenching in the QGP decreases the $H_{1}^{T}$ distribution
at large $H_{1}^{T}$ in Pb+Pb relative to p+p. The medium modification of the
Fox-Wolfram moment $H_{1}^{T}$ for events with $n_{\text{jet}}\geq 2$ are also
presented, which resemble those of events with $n_{\text{jet}}=2$. Its reason
is revealed through the relative contribution fractions of events with
different final-state jet numbers to $H_{1}^{T}$.

###### pacs:

13.87.-a; 12.38.Mh; 25.75.-q

##  I introduction

The quark-gluon plasma (QGP), a new state of QCD matter composed of de-
confined quarks and gluons, has been a principal focus of contemporary
research in high-energy nuclear physics for the last thirty years Gyulassy and
McLerran (2005); Muller  _et al._ (2012); Wang (2016); Busza  _et al._ (2018).
Highly energetic partons generated in the initial collision traverse the QGP
medium, experiencing energy loss through interactions with constituent partons
of the medium. This phenomenon, recognized as jet quenching, serves as a
potent investigative tool for probing the characteristics of the QGP Bjorken
(1982); Gyulassy and Plumer (1990); Wang and Gyulassy (1992); Baier  _et al._
(1997); Gyulassy  _et al._ (2004); Qin and Wang (2015); Wicks  _et al._
(2007); Bass  _et al._ (2009); Cao and Wang (2021); Baier  _et al._ (2001);
Kumar  _et al._ (2021); Pablos  _et al._ (2022); Luo  _et al._ (2024); Xie
_et al._ (2023, 2024). Various observables have emerged to quantitatively
investigate the phenomena of jet quenching. Initially, investigations focused
on how the medium modifies hadron spectra and hadron correlations, such as the
suppression of leading hadron spectra, modifications of di-hadron
correlations, and other hadronic observables studies Brenner  _et al._ (1982);
Adler  _et al._ (2003); Albacete and Marquet (2010); Aamodt  _et al._ (2011,
2012); Aggarwal  _et al._ (2010); Xing  _et al._ (2020); Dai  _et al._ (2015);
Xing  _et al._ (2024); Li  _et al._ (2024a); Zhang  _et al._ (2023). With the
sustained running of the Large Hadron Collider (LHC) and the development of
theory, various observables at jet level have been developed. These
observables respectively concentrate on the production or correlation of
inclusive jets Aad  _et al._ (2013a); Adam  _et al._ (2015); Adamczyk  _et
al._ (2015); Abelev  _et al._ (2008); Kang  _et al._ (2016a); Adamczyk  _et
al._ (2017); Khachatryan  _et al._ (2017); Acharya  _et al._ (2020); Aad  _et
al._ (2013b); Dasgupta  _et al._ (2016); He  _et al._ (2019); Kumar  _et al._
(2023), dijets Magestro (2006); Aad  _et al._ (2010); He  _et al._ (2012);
Milhano and Zapp (2016); Dai  _et al._ (2020); Li  _et al._ (2024b), and
tagged jets Wang  _et al._ (1996); Adare  _et al._ (2009); Abelev  _et al._
(2010); Neufeld  _et al._ (2011); Dai  _et al._ (2013); Sirunyan  _et al._
(2017); Zhang  _et al._ (2018); Wang  _et al._ (2019); Li  _et al._ (2023);
Acharya  _et al._ (2022); Dai  _et al._ (2022); Wang  _et al._ (2021); Yang
_et al._ (2021); Xiao  _et al._ (2024); Liu  _et al._ (2022); Gao  _et al._
(2023); Sirimanna  _et al._ (2024), with some also scrutinizing the structure
and substructure of aforementioned categories of jet events Aad  _et al._
(2013c); Chatrchyan  _et al._ (2014); Cal  _et al._ (2019); Chatrchyan  _et
al._ (2012); Aad  _et al._ (2014); Aaboud  _et al._ (2018); Kang  _et al._
(2016b); Lesser (2023); Larkoski  _et al._ (2020); Kogler  _et al._ (2019);
Milhano  _et al._ (2018); Feige  _et al._ (2012); Li and Vitev (2019); Wang
_et al._ (2023); Kang  _et al._ (2023a); Yang  _et al._ (2024, 2023a, 2023b);
Tachibana  _et al._ (2023).

In addition to these typical experimental observables under sustained
attention, there exists a category of measurements capturing the geometric
shape of the entire jet event, known as jet event shape observables Banfi  _et
al._ (2010); Khachatryan  _et al._ (2011); Aad  _et al._ (2012); Chatrchyan
_et al._ (2013). The early studies of event shape observables were primarily
conducted in electron-positron collisions, deep inelastic scattering (DIS) and
proton-antiproton collisions Catani  _et al._ (1993); Abbiendi  _et al._
(2005); Bauer  _et al._ (2008); Ellis  _et al._ (1980); Dasgupta and Salam
(2002); Adloff  _et al._ (2000); Abdallah  _et al._ (2003); Abreu  _et al._
(1997); Abe  _et al._ (1991); Catani  _et al._ (1992). With extensive research
in high-energy nuclear physics ongoing at the LHC, some theoretical
explorations involving jet event shape observables in heavy ion collisions
have emerged in recent years Chen  _et al._ (2020); Mallick  _et al._ (2020,
2021); Prasad  _et al._ (2022); Kang  _et al._ (2023b). These studies hold
significant importance in understanding the properties of nuclear medium Qiu
and Heinz (2011); Berger  _et al._ (2003); Acharya  _et al._ (2021); Adam  _et
al._ (2016); Acharya  _et al._ (2019) and elucidating the dynamics of extreme
relativistic collisions Schukraft  _et al._ (2013); Jia (2014).

In this work we extend the studying of medium modifications of jet shapes in
heavy-ion collisions to Fox-Wolfram moments (FWMs), a set of rather unique
event shape measurements characterized by expansions in terms of spherical
harmonics Fox and Wolfram (1978). The application of FWMs has been extensive
in collider physics, particularly in high-energy physics domains, including
studies in Higgs physics Bernaciak  _et al._ (2013, 2014), top physics Field
_et al._ (1997), and general investigations in $e^{+}e^{-}$ collisions Nagy
and Trocsanyi (1998); Li  _et al._ (2020) as well as hadron collisions Spiller
(2016). We make the first theoretical attempt to calculate the medium
modifications of the event-normalized distribution of the lower-order FWM,
$\text{FWM}_{\text{1st}}^{\text{T}}$ (denoted as $H_{1}^{T}$ in the subsequent
text) for events with $2$ jets and with $>$ 2 jets, by comparing $H_{1}^{T}$
distributions in p+p and Pb+Pb collisions at $\sqrt{s}=5.02$ TeV. For events
with $2$ jets, the results reveal a suppression in the small $H_{1}^{T}$
region and an enhancement in the large $H_{1}^{T}$ region for the distribution
in Pb+Pb collisions compared to p+p collisions. Meanwhile, for events with $>$
2 jets, there is a suppression of the Pb+Pb collisions relative to the p+p
collisions in the large $H_{1}^{T}$ region. Furthermore, when the jet
multiplicity is unspecified, i.e., events with $\geq$ 2 jets, the
modifications resemble that of events with 2 jets. And computations regarding
the relative contributions of the two types of events with respect to
$H_{1}^{T}$ elucidate the underlying reasons.

The subsequent organization of the content is as follows. In the Sec. II, we
introduce FWMs, from which we extract a specific observable of our focus. And
we discuss and elucidate its physical interpretation in p+p collisions. In
Sec. III, the results and discussions in p+p and Pb+Pb collisions are
presented, followed by the summary in Sec. IV.

##  II FWMs analysis and $\text{FWM}_{\text{1st}}^{\text{T}}$ in p+p
collisions

Giving by a superposition of spherical harmonics $Y_{\ell}^{m}(\Psi_{i})$,
FWMs is a set of such observables Fox and Wolfram (1978)

$\displaystyle
H_{\ell}^{o}=\frac{4\pi}{2\ell+1}\sum_{m=-\ell}^{\ell}\;\left|\sum_{i=1}^{n_{\text{jet}}}W_{i}^{o}\;Y_{\ell}^{m}(\Psi_{i})\right|^{2}\;$
(1)

FWMs are sensitive to the number of jets, angular correlation of jets, and
energy distribution of jets. They systematically describe geometric
correlations in trems of spherical harmonics. The inner sum is over the final
state objects (here are jets). There are various schemes for the construction
of weight factors $W_{i}^{o}$ as shown in the literature Ref. Bernaciak  _et
al._ (2013); Spiller (2016). The ‘o’ here serves as a marker for a different
scheme. It represents the individual components of momentum or the vector sum
of different components, such as $p_{x}$, $p_{y}$, $p_{z}$, $p_{T}$, and
$|\vec{p}|$. The typical schemes are as follows:

$\displaystyle
W_{i}^{o}=\frac{p^{o}_{i}}{\sum_{k=1}^{n_{\text{jet}}}p^{o}_{k}}$ (2)

The angular $\Psi_{i}$ being contingent upon the axis, we reformulate Eq.(1)
to achieve a formulation independent of axis, facilitated by the utilization
of the inherent relationship between spherical harmonics and Legendre
polynomials.

$\displaystyle
H^{o}_{\ell}=\sum_{i,j=1}^{n_{\text{jet}}}\;W_{ij}^{o}\;P_{\ell}(\cos\Delta\Psi_{ij})\text{,}\quad
W_{ij}^{o}=\frac{p^{o}_{i}p^{o}_{j}}{(\sum_{k=1}^{n_{\text{jet}}}p^{o}_{k})^{2}}$
(3)

According to different weight construction schemes, we can choose different
angle schemes $\Delta\Psi_{ij}$ to measure the distance between jets. Here,
$\Delta\Psi_{ij}$ can take on values of $\Delta\phi_{ij}$,
$\Delta\theta_{ij}$, and $\Delta\Omega_{ij}$, which adhere to the spherical
coordinates equation
$\cos\Delta\Omega_{ij}=\cos\theta_{i}\cos\theta_{j}+\sin\theta_{i}\sin\theta_{j}\cos(\phi_{i}-\phi_{j})$.
The angle $\phi_{i}$ and $\theta_{i}$ represent the azimuth angle of jet and
angle between jet and beam direction, among which the latter satisfies the
equation with jet rapidity $\eta_{i}=-\ln(\tan\frac{\theta_{i}}{2})$. So, it
is straightforward to ascertain that $\Delta\phi_{ij}$ and $\Delta\theta_{ij}$
represent the azimuthal separation and the difference in rapidity between jet
i and j. And $\Delta\Omega_{ij}$ represents the distance between them in
three-dimensional momentum space.

Given our emphasis on jet physics, the transverse momentum and azimuthal
separation $\Delta\phi_{ij}$ of jets are employed in the construction of the
FWMs. The values of $i,j(1,2,3,...)$ represent jets ordered by descending
transverse momentum. And the marker ‘o’ changes to the ‘T’:

$\displaystyle H_{\ell}^{T}$
$\displaystyle=\sum_{i,j=1}^{n_{\text{jet}}}\;W_{ij}^{T}\;P_{\ell}(\cos\Delta\phi_{ij})$
(4)
$\displaystyle=\sum_{i,j=1}^{n_{\text{jet}}}\;\frac{p_{Ti}p_{Tj}}{\left(\sum_{k=1}^{n_{\text{jet}}}p_{Tk}\right)^{2}}\;P_{\ell}(\cos\Delta\phi_{ij})$

Let’s first analyze the toy model given by Eq.(4) when $n_{\text{jet}}$ is
equal to 2 as shown in FIG. 1. For simplicity, we denote $p_{T2}=x\cdot
p_{T1}$ which simplify the functional form of the weights $W_{ij}^{T}$.
Expanding the sum in the definition Eq.(4) with weights expressed by
transverse momentum yields

$\displaystyle
H_{\ell}^{T}(n_{\text{jet}}=2)=\frac{1+2xP_{\ell}(\cos\Delta\phi_{12})+x^{2}}{1+2x+x^{2}}$
(5)

If $\ell=0$, then the Legendre polynomial $P_{1}=1$, resulting in
$H_{1}^{T}=1$, which is not meaningful for our study. Due to the properties of
the Legendre polynomials $P_{\ell}$, the FWMs $H_{\ell}^{T}$ exhibit strongly
oscillatory behavior, as shown in FIG. 1. The larger value of $\ell$, the
stronger oscillatory. This can be viewed as a change in the resolution with
which the FWMs probe the structure of the QCD jet geometry Bernaciak  _et al._
(2013). For this reason, the study of FWMs of different orders was used to
provide sufficient resolution to discriminate signal and backgrounds in Higgs
and t$\bar{\mathrm{t}}$ production researches Bernaciak  _et al._ (2013);
Field  _et al._ (1997).

As $\Delta\phi_{12}$ approaches $\pi$, $H_{\ell}^{T}$ approaches 1 (even
moments) or 0 (odd moments). And all weights $W_{ij}^{T}$ ensure that the
range for the moments is $0\leq H_{\ell}^{T}\leq 1$ and preserve the different
shapes of $H_{\ell}^{T}$ for even or odd moments. For simplicity, we focus our
study on the lower-order $H_{1}^{T}$. The definition is as follows:

$\displaystyle H_{1}^{T}$
$\displaystyle=\sum_{i,j=1}^{n_{\text{jet}}}\;\frac{p_{Ti}p_{Tj}}{\left(\sum_{k=1}^{n_{\text{jet}}}p_{Tk}\right)^{2}}\;P_{1}(\cos\Delta\phi_{ij})$
(6)
$\displaystyle=\sum_{i,j=1}^{n_{\text{jet}}}\;\frac{p_{Ti}p_{Tj}}{\left(\sum_{k=1}^{n_{\text{jet}}}p_{Tk}\right)^{2}}\;\cos\Delta\phi_{ij}$

Figure 1: The analytic result of the FWMs $H_{\ell}^{T}$ ($n_{\text{jet}}$ =
2) for $\ell=1,2,3,4,5,6$.

Figure 2: In the transverse plane, schematic diagrams illustrating the spatial
configurations of $n_{\text{jet}}=2$ and $n_{\text{jet}}=3$ events with polar
coordinates across three consecutive $H_{1}^{T}$ intervals.

To study event shape observables a satisfactory theoretical description needs
both the fix-order perturbative calculations and the inclusion of the large
contributions beyond fix-order when the event shape observables are very large
or very small Banfi  _et al._ (2010). In our study, jet productions in p+p
collisions are simulated within a Monte Carlo event generator SHERPA 2.2.11
Gleisberg  _et al._ (2009), which perform NLO Matrix Element calculations
matched to the Parton Shower with several merging schemes. AMEGIC++ Krauss
_et al._ (2002) and Comix Gleisberg and Hoeche (2008) are inbuilt matrix-
element generators of SHERPA, which automatically calculate and integrate
tree-level amplitudes, cooperating with their phase-space generator Phasic
Krauss  _et al._ (2002). OpenLoops Cascioli  _et al._ (2012) programs is
interfaced with SHERPA to provide virtual corrections. An method of MC@NLO-
type is performed to match a fixed-order next-to-leading order calculation
with the resummation of the parton shower Hoeche  _et al._ (2012). The parton
distribution function (PDF) set ‘NNPDF30$\\_$nnlo’ is loaded for beams Ball
_et al._ (2009a, b).

The spatial structure of $H_{1}^{T}$ is depicted in the top-left corner of
FIG. 1. To gain further insights into the configuration structure of
$H_{1}^{T}$, we calculate the average azimuthal angle correlations for events
with $n_{\text{jet}}=2$ and $n_{\text{jet}}>2$ in p+p collisions at a center-
of-mass energy of $\sqrt{s}$ = 5.02 TeV, considering three logarithmic
intervals of $H_{1}^{T}$, as presented in TABLE 1. The kinetic sets
corresponding to this calculation are described in Sec. III. From TABLE 1, we
observe that the azimuthal angle distance $\langle\Delta\phi_{12}\rangle$ of
$n_{\text{jet}}=2$ events decreases with increasing $H_{1}^{T}$. Additionally,
for $n_{\text{jet}}>2$ events, both $\langle\Delta\phi_{12}\rangle$ and
$\langle\Delta\phi_{13}\rangle$ decrease while $\langle\Delta\phi_{23}\rangle$
increases with increasing $H_{1}^{T}$. This indicates that the jet event shape
becomes broader as $H_{1}^{T}$ increases for both $n_{\text{jet}}=2$ and
$n_{\text{jet}}>2$ events. To present the geometric illustration more vividly,
we provide visual schematic diagrams obtained through calculations in FIG. 2.

$\ln(H_{1}^{T})$ | $n_{\text{jet}}=2$ events |  $n_{\text{jet}}>2$ events
---|---|---
$\langle\Delta\phi_{12}\rangle$ | $\langle\Delta\phi_{12}\rangle$ | $\langle\Delta\phi_{13}\rangle$ | $\langle\Delta\phi_{23}\rangle$
$\left[-10.0,-7.0\right]$ | 3.115 | 2.880 | 2.602 | 0.786
$\left[-7.0,-3.0\right]$ | 3.029 | 2.840 | 2.536 | 0.864
$\left[-3.0,0.0\right]$ | 2.573 | 2.541 | 2.055 | 1.245
Table 1: The averaged azimuthal angle correlations are calculated for
$n_{\text{jet}}=2$ and $n_{\text{jet}}>2$ events in p+p collisions at
$\sqrt{s}=5.02$ TeV at three $\ln(H_{1}^{T})$ intervals.

As shown in FIG. 2, schematic diagrams illustrating the spatial configurations
of $n_{\text{jet}}=2$ and $n_{\text{jet}}=3$ events across three consecutive
$H_{1}^{T}$ intervals are plotted based on TABLE 1. This schematic employs a
scheme in which the direction of the leading jet is the polar axis of the
polar coordinates, and in order to better schematize the geometric
configuration of the jet events, we rescale the transverse momentum of jets in
a dimensionless interval from 0 to 1, preserving the relative proportionality
between their transverse momentum magnitudes. These diagrams are
visualizations that as $H_{1}^{T}$ increases, both $n_{\text{jet}}=2$ and
$n_{\text{jet}}=3$ event shapes become broader.

##  III $\text{FWM}_{\text{1st}}^{\text{T}}$ in Pb+Pb collisions

In this section, we use p+p data generated by the SHERPA event generator as a
baseline to explore how the distribution of $H_{1}^{T}$ will be modified after
undergoing multiple scatterings with nuclear medium partons. In our study, The
Linear Boltzmann Transport (LBT) model Wang and Zhu (2013); He  _et al._
(2015); Cao  _et al._ (2016) is used to simulate the physical process of
multiple scattering between jet and medium partons in QGP. Additionally, the
information of the bulk partons is provided by (3+1)D CLVisc hydrodynamical
model Pang  _et al._ (2012, 2018, 2015); Wu  _et al._ (2022); Jiang  _et al._
(2023) with initial conditions simulated from A Multi-Phase Transport (AMPT)
model Lin  _et al._ (2005). Both elastic and inelastic scattering processes
are considered for the initial jet shower partons and the thermal recoil
partons.

In the LBT model, the $2\rightarrow 2$ elastic scattering process is simulated
by the linear Boltzmann transport equation,

$\displaystyle p_{1}\cdot\partial
f_{a}(p_{1})=-\int\frac{d^{3}p_{2}}{(2\pi)^{3}2E_{2}}\int\frac{d^{3}p_{3}}{(2\pi)^{3}2E_{3}}\int\frac{d^{3}p_{4}}{(2\pi)^{3}2E_{4}}$
$\displaystyle\frac{1}{2}\sum_{b(c,d)}[f_{a}(p_{1})f_{b}(p_{2})-f_{c}(p_{3})f_{d}(p_{4})]|M_{ab\rightarrow
cd}|^{2}$ $\displaystyle\times
S_{2}(s,t,u)(2\pi)^{4}\delta^{4}(p_{1}+p_{2}-p_{3}-p_{4})$ (7)

The inelastic scattering, what is called the medium-induced gluon radiation in
the LBT model, is described by the higher twist formalism Guo and Wang (2000);
Zhang  _et al._ (2004); Zhang and Wang (2003); Majumder (2012),

$\frac{dN_{g}}{dxdk_{\perp}^{2}dt}=\frac{2\alpha_{s}C_{A}P(x)\hat{q}}{\pi
k_{\perp}^{4}}\left(\frac{k_{\perp}^{2}}{k_{\perp}^{2}+x^{2}M^{2}}\right)^{2}\sin^{2}\left(\frac{t-t_{i}}{2\tau_{f}}\right)$
(8)

We refer to Wang and Zhu (2013); He  _et al._ (2015); Cao  _et al._ (2016) for
the entensive discussions on the interpretation of various symbols and
implementations of the elastic and inelastic scattering of fast partons with
the medium in Eq. (7) and Eq. (8).

Figure 3: The event normalized $H_{1}^{T}$ distributions for
$n_{\text{jet}}=2$ (left panel) and $n_{\text{jet}}>2$ (right panel) events in
p+p and Pb+Pb collisions at $\sqrt{s}=5.02$ TeV.

By generating p+p events with Sherpa + OpenLoops program packages and
simulating the production of jets in Pb+Pb collisions at $\sqrt{s_{NN}}=5.02$
TeV with LBT model, we present our results and analyze them in this section.
The anti-$k_{\perp}$ algorithm is used to reconstruct jets with a cone size of
R = 0.4 using the FastJet package Cacciari  _et al._ (2012). The jet events
are required to have at least two in our study. The kinematic sets are as
follows: the transverse momentum of leading jet $p_{T}^{\text{leading}}>110$
GeV/c, the threshold $p_{T}^{\text{jet}}>30$ GeV/c and rapidity $|\eta|\leq$
2.5 for all jets.

We start the investigation by studying the nuclear modifications of the
$H_{1}^{T}$ distributions for events with a fixed number of jets. The jet
events are classified into two types, events with $n_{\text{jet}}=2$ and
$n_{\text{jet}}>2$.

Let us first focus on the event normalized spectrum of the $H_{1}^{T}$ for the
$n_{\text{jet}}=2$ events and its nuclear modification, as shown in the left
panel of FIG. 3. The left panel of FIG. 3 shows the results of the p+p (black
line) and Pb+Pb (red line) collisions at the $\sqrt{s}=5.02$ TeV, and the
ratio of these two, also known as the nuclear modification factor, is shown in
the bottom panel. The distribution is apparently suppressed and enhanced
separately in the small and large regions, taking approximately
$H_{1}^{T}=-4.0$ as a split. This medium modification means that the
proportion of back-to-back jets, which are events with narrow shapes, is
reduced, and therefore the proportion of $n_{\text{jet}}=2$ events is enhanced
in regions characterized by large $H_{1}^{T}$, where the event shapes are
broader. The jet quenching effect indicates that the jet primarily loses
energy through medium-induced gluon radiation. This leads to a more diffusely
distributed total energy after passing through the medium, resulting in the
broadening of $n_{\text{jet}}=2$ events.

$n_{\text{jet}}=2$ events |  $n_{\text{jet}}>2$ events
---|---
p+p | Pb+Pb | p+p | Pb+Pb
$\langle\Delta\phi_{12}\rangle$ | $\langle\Delta\phi_{12}\rangle$ | $\langle\Delta\phi_{12}\rangle$ | $\langle\Delta\phi_{13}\rangle$ | $\langle\Delta\phi_{23}\rangle$ | $\langle\Delta\phi_{12}\rangle$ | $\langle\Delta\phi_{13}\rangle$ | $\langle\Delta\phi_{23}\rangle$
2.946 | 2.922 | 2.830 | 2.521 | 0.872 | 2.804 | 2.464 | 0.958

Table 2: The averaged azimuthal angle correlations are calculated for
$n_{\text{jet}}=2$ and $n_{\text{jet}}>2$ events in p+p and Pb+Pb collisions
at $\sqrt{s}=5.02$ TeV, across the entire $\ln(H_{1}^{T})$ statistical region.

Subsequent to the preceding paragraph, the event normalized $H_{1}^{T}$
spectrum of $n_{\text{jet}}>2$ events in p+p collisions (black line) and Pb+Pb
collisions (red line) and their nuclear modifications are shown in the right
panel of FIG. 3. The distributions exhibit an extreme enhancement in the bins
where $\ln(H_{1}^{T})\in$ [-5, -1.75] by the medium, with suppressions
observed in the other regions. The reason for the medium modification in the
bins $\ln(H_{1}^{T})\in$ [-10, -5] and [-5, -1.75] is the same as that for
$n_{\text{jet}}=2$, as mentioned above. There is another thing that should be
considered, which is the jet number reduction effect Apolinario  _et al._
(2013) when we focus on $n_{\text{jet}}>2$ events (see FIG. 4). It leads to
the proportion of $n_{\text{jet}}>2$ events reduce significantly in the large
region where $\ln(H_{1}^{T})\in$ [-1.75, 0]. This phenomenon may result in
$n_{\text{jet}}>2$ events having a more broader shape.

To more intuitively reflect the medium modification of the event shape, we
present the average azimuthal angle distance between jets for events with
$n_{\text{jet}}=2$ and $n_{\text{jet}}>2$ over the entire $\ln(H_{1}^{T})$
statistical region in TABLE 2. It should be noted that we only consider the
top three jets ranked by transverse momentum for $n_{\text{jet}}>2$ events. As
seen in TABLE 2, the shape of $n_{\text{jet}}=2$ events broadens, with the
average azimuthal angle distance $\langle\Delta\phi_{12}\rangle$ decreasing
due to medium modification. For $n_{\text{jet}}>2$ events, both
$\langle\Delta\phi_{12}\rangle$ and $\langle\Delta\phi_{13}\rangle$ decrease,
while $\langle\Delta\phi_{23}\rangle$ increases, indicating that the event
shape is also broadened by medium modification.

Figure 4: Histograms of the fraction of different numbers of jets in all jet
events in p+p and Pb+Pb collisions at $\sqrt{s}$ = 5.02 TeV.

The histograms showing fractions of different numbers of jets in total jet
events for p+p and Pb+Pb collisions at $\sqrt{s}=5.02$ TeV are presented in
FIG. 4. In Pb+Pb collisions, the change in the number of jets as a fraction of
total jet events is significant compared to p+p collisions. The fraction of
events with $n_{\text{jet}}=2$, increases from 0.55 in p+p collisions to 0.62
in Pb+Pb collisions, and correspondingly, the total fraction of other events,
i.e., $n_{\text{jet}}>2$ events, decreases from 0.45 to 0.38. The decrease in
the number of events is known as the jet number reduction effect. The reason
for this is that some of the sub-leading jets experience energy loss in the
medium, causing their transverse momentum magnitude to fall below our
observational threshold of 30 GeV/c. As a result, the fraction of observed
$n_{\text{jet}}=2$ events increases in Pb+Pb collisions compared to p+p
collisions.

Figure 5: Top: event normalized $H_{1}^{T}$ distribution of
$n_{\text{jet}}\geq 2$ events in p+p and Pb+Pb collisions at $\sqrt{s}=5.02$
TeV; Bottom: medium modification factor of event normalized $H_{1}^{T}$
distribution of $n_{\text{jet}}\geq 2$ events.
Figure 6: The relative contribution fractions of $n_{\text{jet}}=2$ and
$n_{\text{jet}}>2$ events from the distribution of $H_{1}^{T}$ normalized by
the total number of events in p+p and Pb+Pb collisions at $\sqrt{s}=$ 5.02
TeV.

Now we consider the $H_{1}^{T}$ distribution for the combination of
$n_{\text{jet}}=2$ and $n_{\text{jet}}>2$ events in proportion, as shown in
FIG. 5. The nuclear modification factor exhibits a similar pattern to that of
$n_{\text{jet}}=2$ events, characterized by suppression at low $H_{1}^{T}$
values and enhancement at high $H_{1}^{T}$ values. To understand why the
nuclear medium modification for total ($n_{\text{jet}}\geq 2$) events is as
shown in FIG. 5, we draw FIG. 6, which illustrates the relative contributions
of $n_{\text{jet}}=2$ and $n_{\text{jet}}>2$ events to $H_{1}^{T}$ in both p+p
and Pb+Pb collisions at $\sqrt{s}=5.02$ TeV. In the low $H_{1}^{T}$ region,
$n_{\text{jet}}>2$ events constitute a slightly larger proportion than
$n_{\text{jet}}=2$ events. As $H_{1}^{T}$ increases, the percentage of
$n_{\text{jet}}=2$ ($n_{\text{jet}}>2$) events gradually rises from
approximately 44% (56%) in low the $H_{1}^{T}$ region to around 91% (9%) in
the high $H_{1}^{T}$ region in p+p collisions. Similarly, the corresponding
percentages transition from approximately 47% (53%) to around 93% (7%) in
Pb+Pb collisions. The transition in the relative contributions between
$n_{\text{jet}}=2$ and $n_{\text{jet}}>2$ events occurs at around
$H_{1}^{T}=-5$. The nuclear modification remains suppressed in the low
$H_{1}^{T}$ region, where both $n_{\text{jet}}=2$ and $n_{\text{jet}}>2$
events have similar fractions. In contrast, an enhancement effect is observed
in the high $H_{1}^{T}$ region, consistent with the behavior of
$n_{\text{jet}}=2$ events, which dominate in this region.

Figure 7: Distribution correlation between $H_{1}^{T}$ and x for
$n_{\text{jet}}=2$ event in p+p (left) and Pb+Pb (right) system at
$\sqrt{s}=5.02$ TeV.

##  IV summary

We utilize the Monte Carlo event generator SHERPA to simulate jet production
in p+p collisions. To incorporate virtual corrections, the OpenLoops programs
are seamlessly integrated into SHERPA environment. Additionally, our study
employs LBT model to replicate the intricate phenomenon of multiple
scatterings between jets and the medium.

We investigate $H_{1}^{T}$, which is the lower-order member of the set of FWMs
related to QCD geometric patterns, and apply it to the field of jet quenching
physics. This application allows us to examine how the medium modifies jet
event shape of varying multiplicities ($n_{\text{jet}}$). The results indicate
that $n_{\text{jet}}=2$ events undergo a pronounced broadening modification
due to the influence of nuclear matter. Meanwhile, the event shape for
$n_{\text{jet}}>2$ becomes broader due to medium modification, except in the
highest $\ln(H_{1}^{T})$ bin where the jet number effect is significant.
Additionally, we investigate the underlying reasons for the medium
modification of $H_{1}^{T}$ spectra for total jet events, without
distinguishing between different multiplicities. This is achieved by analyzing
the relative fraction of $H_{1}^{T}$ between the two types of events.

Acknowledgments: We extend our sincere gratitude to Jin-Wen Kang, Wei Dai,
Shi-Yong Chen, Meng-Quan Yang, Sa Wang, and Shan-Liang Zhang for their
invaluable discussions. We would also like to express our appreciation to the
open-source software packages Sherpa Gleisberg  _et al._ (2009), FastJet
Cacciari  _et al._ (2012), HepMC2 Dobbs and Hansen (2001) and ROOT Brun and
Rademakers (1997). This work has been supported by the Guangdong Major Project
of Basic and Applied Basic Research under Grant No. 2020B030103008, as well as
the Natural Science Foundation of China with Project Nos. 11935007 and
12035007.

##  Appendix A Formal simplification of the $H_{1}^{T}$ definition for
$n_{\text{jet}}=3$

The definition of $H_{1}^{T}$ is complex when $n_{\text{jet}}>2$ and is not
discussed in detail in the main text. Here, we discuss the case of
$n_{\text{jet}}=3$ as an example.

$\displaystyle H_{1}^{T}$
$\displaystyle=\sum_{i,j=1}^{3}\;\frac{p_{Ti}p_{Tj}}{\left(\sum_{k=1}^{3}p_{Tk}\right)^{2}}\;\cos\Delta\phi_{ij}$
(9)

Following Eq. (5), we denote $p_{T2}=y\cdot p_{T1}$ and $p_{T3}=z\cdot p_{T1}$
which simplify the functional form of Eq. (LABEL:eq:njet3). We get:

$\displaystyle H_{1}^{T}$
$\displaystyle=\frac{1+y^{2}+z^{2}+2y\cos\phi_{12}+2z\cos\phi_{13}+2yz\cos\phi_{23}}{1+y^{2}+z^{2}+2y+2z+2yz}$
(10)

The above simplification is intended to inform readers of a form similar to
Eq. (5); however, due to the increase in the number of variables, we have not
conducted an in-depth analysis. Nonetheless, as discussed above, the physical
interpretation of $H_{1}^{T}$ in events with $n_{\text{jet}}\geq 2$ is that
the event shape broadens as $H_{1}^{T}$ increases.

##  Appendix B Correlations between $H_{1}^{T}$ and x

To further explore the factors responsible for the medium modification of the
$H_{1}^{T}$ distribution, we select $n_{\text{jet}}=2$ events for correlation
studies. The transverse momentum asymmetry is defined as $x=p_{T2}/p_{T1}$. We
plot the distribution correlation between $H_{1}^{T}$ and x for
$n_{\text{jet}}=2$ events in both p+p and Pb+Pb systems at $\sqrt{s}$ = 5.02
TeV in FIG. 7. Remarkably pronounced medium modification phenomena can be
discerned from this result. It can be inferred that the medium modification of
$H_{1}^{T}$ exhibits moderate sensitivity to the transverse momentum imbalance
between jets, which accentuates the transverse momentum asymmetry.

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