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# Measurements of the groomed and ungroomed jet angularities in pp collisions at $$\sqrt{s}$$ = 5.02 TeV

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A bstract The jet angularities are a class of jet substructure observables which characterize the angular and momentum distribution of particles within jets. These observables are sensitive to momentum scales ranging from perturbative hard scatterings to nonperturbative fragmentation into final-state hadrons. We report measurements of several groomed and ungroomed jet angularities in pp collisions at $$\sqrt{s}$$ s = 5 . 02 TeV with the ALICE detector. Jets are reconstructed using charged particle tracks at midrapidity ( |η| < 0 . 9). The anti- k T algorithm is used with jet resolution parameters R = 0 . 2 and R = 0 . 4 for several transverse momentum $${p}_{\mathrm{T}}^{\mathrm{ch}}$$ p T ch jet intervals in the 20–100 GeV/ c range. Using the jet grooming algorithm Soft Drop, the sensitivity to softer, wide-angle processes, as well as the underlying event, can be reduced in a way which is well-controlled in theoretical calculations. We report the ungroomed jet angularities, λ α , and groomed jet angularities, λ α ,g , to investigate the interplay between perturbative and nonperturbative effects at low jet momenta. Various angular exponent parameters α = 1, 1.5, 2, and 3 are used to systematically vary the sensitivity of the observable to collinear and soft radiation. Results are compared to analytical predictions at next-to-leading-logarithmic accuracy, which provide a generally good description of the data in the perturbative regime but exhibit discrepancies in the nonperturbative regime. Moreover, these measurements serve as a baseline for future ones in heavy-ion collisions by providing new insight into the interplay between perturbative and nonperturbative effects in the angular and momentum substructure of jets. They supply crucial guidance on the selection of jet resolution parameter, jet transverse momentum, and angular scaling variable for jet quenching studies.
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JHEP05(2022)061
Published for SISSA by Springer
Revised:December 13, 2021
Accepted:April 8, 2022
Published:May 10, 2022
Measurements of the groomed and ungroomed jet
angularities in pp collisions at s= 5.02 TeV
The ALICE collaboration
E-mail: ALICE-publications@cern.ch
Abstract: The jet angularities are a class of jet substructure observables which characterize
the angular and momentum distribution of particles within jets. These observables are
sensitive to momentum scales ranging from perturbative hard scatterings to nonperturbative
fragmentation into ﬁnal-state hadrons. We report measurements of several groomed and
ungroomed jet angularities in pp collisions at
s
= 5
.
02 TeV with the ALICE detector.
Jets are reconstructed using charged particle tracks at midrapidity (
|η|<
0
.
9). The anti-
kT
algorithm is used with jet resolution parameters
R
= 0
.
2and
R
= 0
.
4for several transverse
momentum
pch jet
T
intervals in the 20–100 GeV/
c
range. Using the jet grooming algorithm
Soft Drop, the sensitivity to softer, wide-angle processes, as well as the underlying event,
can be reduced in a way which is well-controlled in theoretical calculations. We report
the ungroomed jet angularities,
λα
, and groomed jet angularities,
λα,g
, to investigate the
interplay between perturbative and nonperturbative eﬀects at low jet momenta. Various
angular exponent parameters
α
= 1, 1.5, 2, and 3 are used to systematically vary the
sensitivity of the observable to collinear and soft radiation. Results are compared to
analytical predictions at next-to-leading-logarithmic accuracy, which provide a generally
good description of the data in the perturbative regime but exhibit discrepancies in the
nonperturbative regime. Moreover, these measurements serve as a baseline for future ones
in heavy-ion collisions by providing new insight into the interplay between perturbative and
nonperturbative eﬀects in the angular and momentum substructure of jets. They supply
crucial guidance on the selection of jet resolution parameter, jet transverse momentum, and
angular scaling variable for jet quenching studies.
Keywords: Heavy Ion Experiments
ArXiv ePrint: 2107.11303
for the beneﬁt of the ALICE Collaboration.
Article funded by SCOAP3.
https://doi.org/10.1007/JHEP05(2022)061
JHEP05(2022)061
Contents
1 Introduction 1
2 Experimental setup and data sets 3
3 Analysis method 4
3.1 Jet reconstruction 4
3.2 Corrections 6
4 Systematic uncertainties 7
5 Results and discussion 9
5.1 Comparison to analytical calculations 14
5.1.2 Shape function based correction 18
5.2 Discussion 19
6 Conclusion 22
The ALICE collaboration 36
1 Introduction
In high-energy particle collisions, jet observables are sensitive to a variety of processes
in quantum chromodynamics (QCD), from the initial hard (high
Q2
) parton scattering
to a scale evolution culminating in hadronization near Λ
QCD
. Jets reconstructed with a
R
= 1 and with suﬃciently large transverse momentum
pjet
T
provide a proxy for the dynamics of the initial hard parton scattering, whereas those
reconstructed with smaller
R
or at lower
pjet
T
become sensitive to nonperturbative eﬀects. In
this article, jet substructure observables are deﬁned by clustering particles into a jet and then
constructing an observable from its constituents to characterize its internal radiation pattern.
Jet substructure techniques have provided one of the key tools to study rare event
topologies in pp collisions, for example by tagging boosted objects that decay into jets [
1
].
Moreover, measurements of jet substructure enable stringent tests of perturbative QCD
(pQCD) and facilitate studies of nonperturbative eﬀects which are not yet under satisfactory
theoretical control [
2
]. Jet substructure observables oﬀer both ﬂexibility and rigor: they can
be constructed to be theoretically calculable from ﬁrst-principles pQCD while simultaneously
maintaining sensitivity to jet radiation in speciﬁc regions of phase-space. Jet grooming
algorithms, such as Soft Drop [
3
5
], can additionally be used to remove soft, wide-angle
radiation via well-controlled approaches, reducing nonperturbative eﬀects. This deﬁnes two
1
JHEP05(2022)061
families of jet substructure observables: one that can be constructed from all jet constituents
and one based on a subset of jet constituents which remain after grooming procedures.
One such set of observables are the generalized jet angularities [
6
,
7
]. Expanding upon
the jet girth
g
(also known as the jet radial moment), the generalized jet angularities form
a class of jet substructure observables deﬁned by
λκ
αX
i
zκ
iθα
i,(1.1)
where the sum runs over the jet constituents
i
, and
κ
and
α
are continuous free parameters.
1
The ﬁrst factor zipT,i/pjet
Tdescribes the momentum fraction carried by the constituent,
and the second factor
θi
Ri/R
denotes the separation in rapidity (
y
) and azimuthal
angle (
ϕ
) of the constituent from the jet axis, where
Riqy2
i+ ϕ2
i
and
R
is the jet
resolution parameter. The jet angularities are infrared- and collinear- (IRC-)safe for
κ
= 1
and
α >
0[
8
,
9
]. We consider the ungroomed jet angularities, denoted as
λα
, as well as the
groomed jet angularities in which the sum runs only over the constituents of the groomed
jet, denoted as
λα,g
. These include the jet girth [
10
],
λ1
, and the jet thrust [
11
],
λ2
, which is
related to the jet mass
mjet
by
λ2
= (
mjet/pjet
T
)
2
+
O
(
λ2
2
);
λ2
, however, is more robust against
nonperturbative eﬀects than
mjet
since it does not depend explicitly on the hadron masses.
The IRC-safe jet angularities oﬀer the possibility to systematically vary the observable
deﬁnition in a way that is theoretically calculable and therefore provide a rich opportunity to
study both perturbative and nonperturbative QCD [
12
15
ities constructed from charged-particle jets. While charged-particle jets are IRC-unsafe [
16
],
comparisons to these theoretical predictions can nonetheless be carried out by following a non-
perturbative correction procedure, as outlined in section 5.1. Jet angularities were recently
calculated in pp collisions both in the ungroomed [
9
] and groomed [
17
] cases, as well as for jets
produced in association with a Z boson [
18
]. These calculations use all-order resummation of
large logarithms up to next-to-leading-logarithmic (NLL
0
) accuracy [
19
]. Measurements of
λα
and
λα,g
will serve to test these analytical predictions, in particular the role of resummation
eﬀects and power corrections. Moreover, by measuring multiple values of
α
, one can test the
predicted scaling of nonperturbative shape functions that are used to model hadronization,
which depend only on a single nonperturbative parameter for all values of α[20,21].
Several measurements of jet angularities have been performed in hadronic collisions.
The ungroomed jet angularity
λ1
has been measured in pp collisions by the ATLAS, CMS,
and ALICE Collaborations [
22
24
p
collisions by the CDF Collaboraiton [
25
].
The ungroomed jet angularity
λ2
has also been measured in pp collisions by the CMS
Collaboration [
24
]. The closely related ungroomed and groomed jet mass have been
extensively measured in pp collisions by the ATLAS and CMS Collaborations [
23
,
24
,
26
35
], and the ungroomed mass was also studied in p
p
collisions by the CDF Collaboration [
25
]
and in
p–Pb
collisions by the ALICE Collaboration [
36
]. Many of these measurements
have focused on using jet substructure for tagging objects at high
pT
, rather than for
1
The notation
λα
is employed to represent the jet angularities instead of the commonly-used notation
λβ
in order to avoid conﬂict with the letter
β
, which is also used to denote the angular parameter of the Soft
Drop grooming algorithm.
2
JHEP05(2022)061
fundamental studies of QCD, and with the exception of the jet mass there have not yet been
comparisons of jet angularities to analytical calculations, nor have any such comparisons
groomed jet angularities in pp collisions, and a systematic scan of the IRC-safe ungroomed
jet angularities. These measurements focus on low to moderate
pjet
T
, and small to moderate
R
. Moreover, the measurements are performed in pp collisions at a center-of-mass energy
s
= 5
.
02 TeV, the same center-of-mass energy at which ALICE recorded data in heavy-ion
collisions during LHC Run 2, and where no jet angularity measurements have been made.
These measurements serve as a baseline for future measurements of the jet angularities
in heavy-ion collisions, in which a deconﬁned state of strongly-interacting matter is pro-
duced [
37
40
]. Measurements of jets and jet substructure in heavy-ion collisions may provide
key insight into the physical properties of this deconﬁned state [
41
43
]. The jet angularities
are sensitive both to medium-induced broadening as well as jet collimation [
44
46
]; by
systematically varying the weight of collinear radiation, one may be able to eﬃciently
discriminate between jet quenching models. In
Pb–Pb
collisions,
λ1
has been measured for
R
= 0
.
2by the ALICE Collaboration [
22
], and the ungroomed and groomed jet mass have
been measured for
R
= 0
.
4by the ATLAS, CMS, and ALICE Collaborations [
30
,
34
,
36
].
The interpretation of previous measurements is unclear, with strong modiﬁcation being
observed in
Pb–Pb
collisions compared to pp collisions for the case when
α
= 1 and
R
= 0
.
2,
but little to no modiﬁcation seen for the
R
= 0
.
4jet mass. Future measurements over a
range of
R
and
α
oﬀer a compelling opportunity to disentangle the roles of medium-induced
broadening, jet collimation, and medium response in jet evolution. By measuring small to
moderate
R
jets in pp collisions, which are theoretically challenging and involve signiﬁcant
resummation eﬀects [
47
], the ability of pQCD to describe the small-radius jets that are
measured in heavy-ion collisions can be tested.
α
= 1,
1.5, 2, and 3 in pp collisions at
s
= 5
.
02 TeV. In addition to the standard jet girth (
α
= 1)
and jet mass (related to
α
= 2) parameters,
α
= 1
.
5and
α
= 3 are included to test
the universality of a nonperturbative shape function by varying eﬀects of soft, wide-angle
radiation, as discussed below in section 5.1.2, and to serve as a reference for future jet
quenching measurements in heavy-ion collisions. Grooming is performed according to the
Soft Drop grooming procedure with
zcut
= 0
.
2and
β
= 0 [
48
]. Charged particle jets
were reconstructed at midrapidity using the anti-
kT
parameters
R= 0.2
and
R= 0.4
in four equally-sized
pch jet
T
intervals from 20 to 100
GeV/c
.
The results are compared to NLL
0
pQCD predictions, as well as to the PYTHIA8 [
49
] and
Herwig7 [50,51] Monte Carlo generators.
2 Experimental setup and data sets
A description of the ALICE detector and its performance can be found in refs. [
52
,
53
]. The
pp data used in this analysis were collected in 2017 during LHC Run 2 at
s
= 5
.
02 TeV [
54
].
A minimum bias (MB) trigger was used; this requires a coincidence of hits in the V0
scintillator detectors, which provide full azimuthal coverage and cover the pseudorapidity
3
JHEP05(2022)061
ranges of 2
.
8
< η <
5
.
1and
3
.
7
< η <
1
.
7[
55
]. The event selection also requires the
location of the primary vertex to be within
±
10 cm from the nominal interaction point (IP)
along the beam direction and within 1 cm of the IP in the transverse plane. Beam-induced
background events were removed using two neutron Zero Degree Calorimeters located
at
±
112
.
5m along the beam axis from the center of the detector. Events with multiple
reconstructed vertices were rejected, and track quality selection criteria ensured that tracks
used in the analysis were from only one vertex. Events were acquired at instantaneous
luminosities between approximately 10
30
and 10
31 cm2
s
1
, corresponding to a low level of
pileup with approximately 0
.
004
<µ<
0
.
03 events per bunch crossing. The pp data sample
contains 870 million events and corresponds to an integrated luminosity of 18.0(4) nb
1
[
56
].
This analysis uses charged particle tracks reconstructed from clusters in both the Time
Projection Chamber (TPC) [
57
] and the Inner Tracking System (ITS) [
58
]. Two types of
tracks are deﬁned: global tracks and complementary tracks. Global tracks are required to
include at least one hit in the silicon pixel detector (SPD), comprising the ﬁrst two layers
of the ITS, and to satisfy a number of quality criteria [
59
], including having at least 70
out of a maximum of 159 TPC space points and at least 80% of the geometrically ﬁndable
space points in the TPC. Complementary tracks do not contain any hits in the SPD, but
otherwise satisfy the tracking criteria, and are reﬁt with a constraint to the primary vertex
of the event. Including this second class of tracks ensures approximately uniform azimuthal
acceptance, while preserving similar transverse momentum
pT
resolution to tracks with
SPD hits, as determined from the ﬁt quality. Tracks with
pT,track >
0
.
15
GeV/c
are accepted
over pseudorapidity
|η|<
0
.
9and azimuthal angle 0
< ϕ <
2
π
. All tracks are assigned a
mass equal to the π±mass.
The instrumental performance of the ALICE detector and its response to particles is
estimated with a GEANT3 [
60
] model. The tracking eﬃciency in pp collisions, as estimated
by propagating pp events from PYTHIA8 Monash 2013 [
49
] through the ALICE GEANT3
detector simulation, is approximately 67% at
pT,track
= 0
.
15
GeV/c
, rises to approximately
84% at
pT,track
= 1
GeV/c
, and remains above 75% at higher
pT
. The momentum resolution
σ
(
pT
)
/pT
is estimated from the covariance matrix of the track ﬁt [
53
] and is approximately
1% at
pT,track
= 1
GeV/c
. This increases with
pT,track
, reaching approximately 4% at
pT,track = 50 GeV/c.
3 Analysis method
3.1 Jet reconstruction
Jets are reconstructed from charged tracks with
pT>
150 MeV
/c
using the FastJet pack-
age [
61
]. The anti-
kT
algorithm is used with the
E
recombination scheme for resolution
parameters
R
= 0
.
2and 0
.
4[
62
]. All reconstructed charged-particle jets in the transverse
momentum range 5
< pch jet
T<
200 GeV
/c
are analyzed in order to maximize statistics
in the unfolding procedure (described below). Each jet axis is required to be within the
ﬁducial volume of the TPC,
|ηjet|<
0
.
9
R
. Jets containing a track with
pT>
100
GeV/c
are removed from the collected data sample, due to limited momentum resolution. In order
to make consistent comparisons between the data and the theoretical calculations, the
4
JHEP05(2022)061
R= 0.2R= 0.4
pch jet
T20 GeV/c 100 GeV/c 20 GeV/c 100 GeV/c
JES –12% –24% –13% –21%
JER 22% 21% 21% 21%
εreco 94% 100% 97% 100
Table 1
. Approximate values characterizing the jet reconstruction performance for
R
= 0
.
2and 0
.
4
in pp collisions.
JES
is the mean jet energy scale shift,
JER
is the jet energy resolution, and
reco
is the reconstruction eﬃciency.
background due to the underlying event is not subtracted from the data, and instead the
underlying event (along with other nonperturbative eﬀects) is included in model corrections,
as described in section 5.1.
The jet reconstruction performance is studied by comparing jets reconstructed from
PYTHIA8-generated events at “truth level” (before the particles undergo interactions with
the detector) to those at “detector level” (after the ALICE GEANT3 detector simulation).
Two collections of jets are constructed: pp truth level (PYTHIA truth) and pp detector
level (PYTHIA with detector simulation). The detector-level jets are then geometrically
matched with truth-level jets within
R <
0
.
6
R
match be unique. Table 1shows approximate values of the mean jet energy scale shift,
JES
=
Dpch jet
T,det pch jet
T,truth/pch jet
T,truthE
, the jet energy resolution,
JER
=
σpch jet
T,det.pch jet
T,truth
,
and the jet reconstruction eﬃciency,
εreco
, for both
R
= 0
.
2and
R
= 0
.
4, where
pch jet
T,det
is
the detector-level
pch jet
T
, and
pch jet
T,truth
is the truth-level
pch jet
T
. The jet energy scale shift is a
long-tailed asymmetric distribution due to tracking ineﬃciency [
63
] with a peak at
pch jet
T,det
=
pch jet
T,truth
, and
JES
should be understood only as a rough characterization of this distribution.
The ungroomed jet angularities are reconstructed using all of the charged-particle jet
constituents according to eq. (1.1). For the groomed jet angularities, Soft Drop grooming [
3
]
is performed, in which the constituents of each jet are reclustered with the Cambridge-
Aachen algorithm [
64
] with resolution parameter
R
, forming an angularly-ordered tree
data structure. Each node corresponds to a constituent track, and each edge is a branch
splitting deﬁned by
and
θR
Ry2+∆ϕ2
R
. The jet tree is then
traversed starting from the largest-angle splitting, and the Soft Drop condition,
z > zcut θβ
,
is recursively evaluated. Here,
z
pT
fraction deﬁned above, and
zcut
and
β
are tunable, free parameters of the grooming algorithm. For this analysis,
β
= 0 is
used to maximize the perturbative calculability [
17
], while
zcut
= 0
.
2is chosen (as opposed
to the more common
zcut
= 0
.
1) since higher-accuracy branch tagging can be achieved in
future heavy-ion collision analyses [
48
]. If the Soft Drop condition is not satisﬁed, then the
softer subleading branch is discarded and the next splitting in the harder branch is examined
in the same way. If, however, the condition is satisﬁed, then the grooming procedure is
concluded, with all remaining constituents deﬁning the groomed jet. The groomed jet
angularity is then deﬁned according to eq. (1.1) using the groomed jet constituents, but
still with the ungroomed
pch jet
T
and ungroomed jet axis to deﬁne
θi
, since the groomed
5
JHEP05(2022)061
jet observable is a property of the original (ungroomed) jet object. Note that while the
ungroomed
pch jet
T
is IRC-safe, the groomed
pch jet
T,g
is Sudakov safe [
65
]. If the jet does not
contain a splitting that passes the Soft Drop condition, then the groomed jet contains zero
constituents (“untagged”) and does not have a deﬁned groomed jet angularity.
3.2 Corrections
The reconstructed
pch jet
T
and
λα
diﬀer from their true values due to tracking ineﬃciency,
particle-material interactions, and track
pT
resolution. To account for these eﬀects,
PYTHIA8 Monash 2013 [
49
,
66
] and the ALICE GEANT3 detector simulation are used to
construct a 4D response matrix that describes the detector response mapping of
pch jet
T,truth
and
λα,truth
to
pch jet
T,det
and
λα,det
, where
pch jet
T,det
and
pch jet
T,truth
are as above, and
λα,det
and
λα,truth
are the analogous detector- and truth-level
λα
. The truth-level jet was constructed from
the charged primary particles of the PYTHIA event, deﬁned as all particles with a mean
proper lifetime larger than 1 cm/
c
, and excluding the decay products of these particles [
67
].
A 2D unfolding in
pch jet
T
and
λα
is then performed using the iterative Bayesian un-
folding algorithm [
68
,
69
] implemented in the RooUnfold package [
70
] to recover the true
jet spectrum at the charged-hadron level. This technique utilizes a “prior” distribution
(equivalent to the per-bin MC prediction) as a starting point, before iteratively updating
the distribution using Bayes’ theorem in conjunction with the calculated response matrix
and measured data (see refs. [
68
,
69
] for details). Since the jet yield in each reported
pch jet
T
interval varies widely, with higher-
pch jet
T
jets being less probable than lower-
pch jet
T
jets, and since the shape and mean value of the jet angularity distributions also changes
with
pch jet
T
, a separate 2D unfolding for each reported
pch jet
T
bin is performed in order to
optimize the observable binning at both truth and detector levels, thus ensuring suﬃcient
jet yield is included in the procedure for all distributions while simultaneously maximizing
the number of bins for regions of phase space where higher yield is available. The bin
migration in all cases is dominated by a strong diagonal mapping in the response matrix
coupled with a slight smearing along the
pch jet
T,truth
and
λα,truth
axes. The smearing in
λα
is
roughly symmetric about the diagonal, whereas the smearing in pch jet
Ttends to be skewed
towards lower values of pch jet
T,truth due to tracking eﬃciency eﬀects.
In the groomed case, the number of untagged jets in the unfolding procedure is included
λα
distributions. This is done so
that the unfolding procedure will correct for detector eﬀects on the groomed jet tagging
fraction as well as account for bin migration eﬀects for jets which are groomed away at
detector-level but not truth-level, or vice versa.
To validate the performance of the unfolding procedure, a set of refolding and closure
tests is performed, in which either the response matrix is multiplied by the unfolded data
and compared to the original detector-level spectrum, or in which the shape of the input
MC spectrum is modiﬁed to account for the fact that the actual distribution may be
diﬀerent than the MC input spectrum. The number of iterations, which sets the strength
of regularization, is chosen to be the minimal value such that all unfolding tests succeed.
This results in the number of iterations being equal to 3 for all distributions. In all cases,
closure is achieved compatible with statistical uncertainties.
6
JHEP05(2022)061
The distributions after unfolding are corrected for the kinematic eﬃciency, deﬁned as
the eﬃciency of reconstructing a truth-level jet at a particular
pch jet
T,truth
and
λα,truth
value
given a reconstructed jet
pch jet
T,det
and
λα,det
range. Kinematic ineﬃciency results from eﬀects
including smearing from the Soft Drop threshold and
pT
-smearing of the jet out of the
selected
pch jet
T,det
range. Any “missed” jets, those jets which exist at truth level but not at
detector level, are handled by this kinematic eﬃciency correction. In this analysis, minimal
detector-level cuts are applied, and the kinematic eﬃciency is therefore greater than 99%
in all cases. Since a wide
pch jet
T,truth
range is taken, the eﬀect of “fake” jets, those jets which
exist at detector level but not truth level, is taken to be negligible.
4 Systematic uncertainties
The systematic uncertainties in the unfolded results arise from uncertainties in the tracking
eﬃciency and unfolding procedure, as well as the model-dependence of the response matrix,
and the track mass assumption. Table 2summarizes the systematic uncertainty contributions.
Each of these sources of uncertainty dominate in certain regions of the measured observables,
with the exception of the track mass assumption which is small in all cases. The total
systematic uncertainty is taken as the sum in quadrature of the individual uncertainties
described below.
The tracking eﬃciency uncertainty is estimated to be 4% by varying track selection pa-
rameters and the ITS-TPC matching requirement. In order to assign a systematic uncertainty
to the nominal result, a response matrix is constructed using the same techniques as for the
ﬁnal result except that an additional 4% of tracks are randomly rejected before the jet ﬁnding.
This response matrix is then used to unfold the distribution in place of the nominal response
matrix, and the result is compared to the default result, with the diﬀerences in each bin taken
as a symmetric uncertainty. This uncertainty constitutes a smaller eﬀect in the groomed jet
angularities, where single-particle jets, being the most sensitive to the tracking eﬃciency,
are groomed away by the Soft Drop condition. The uncertainty on the track momentum
resolution is a sub-leading eﬀect to the tracking eﬃciency and is taken to be negligible.
Several variations of the unfolding procedure are performed in order to estimate the
systematic uncertainty arising from the unfolding regularization procedure:
1.
The number of iterations was varied by
±
2and the average diﬀerence with respect to
the nominal result is taken as the systematic uncertainty.
2.
The prior distribution is scaled by a power law in
pch jet
T
and a linear scaling in
λα
,
(
pch jet
T
)
±0.5×
[1
±
(
λα
0
.
5)]. The average diﬀerence between the result unfolded with
this prior and the original is taken as the systematic uncertainty.
3.
The binning in
λα
was varied to be slightly ﬁner and coarser than the nominal binning,
by combining (splitting) some adjacent bins with low (high) jet yield, or by shifting
the bin boundaries to be between the nominal boundaries.
4.
The lower and upper bounds in the
pch jet
T,det
range were increased to 10 and decreased
to 120
GeV/c
, respectively. These values are chosen as reasonable values to estimate
sensitivity to truncation eﬀects.
7
JHEP05(2022)061
Relative uncertainty
α R pch jet
T(GeV/c) Trk. eﬀ. Unfolding Generator Mass hypothesis Total
λα
1 0.4 60–80 1–15% 2–7% 1–5% 0–2% 7–16%
2 0.4 60–80 1–10% 1–8% 1–5% 1–3% 4–12%
3 0.4 60–80 1–10% 2–4% 1–4% 0–4% 4–11%
2 0.4 20–40 1–16% 1–4% 1–43% 0–5% 2–44%
2 0.2 60–80 2–12% 2–7% 1–9% 0–2% 3–12%
λα,g
1 0.4 60–80 1–7% 2–8% 1–6% 0–4% 2–13%
2 0.4 60–80 1–8% 2–9% 1–5% 0–4% 3–12%
3 0.4 60–80 1–6% 2–7% 1–11% 0–7% 4–16%
2 0.4 20–40 1–8% 2–5% 1–40% 0–3% 2–42%
2 0.2 60–80 1–7% 1–8% 1–12% 0–3% 1–15%
Table 2
. Summary of systematic uncertainties for a representative sample of
α
,
R
, and
pch jet
T
. A
moderately high 60
< pch jet
T<
80
GeV/c
with
R
= 0
.
4is chosen to show the variation with
α
, and
two additional rows show the trends with smaller pch jet
Tand R.
The total unfolding systematic uncertainty is then the standard deviation of the
variations,
qPN
i=1 σ2
i/N
, where
N
= 4 and
σi
is the systematic uncertainty due to a single
variation, since they each comprise independent measurements of the same underlying
systematic uncertainty in the regularization.
A systematic uncertainty associated with the model-dependent reliance on the Monte
Carlo generator which is used to unfold the spectra is included. We construct a fast
simulation to parameterize the tracking eﬃciency and track
pT
resolution, and build
response matrices using PYTHIA8 Monash 2013 and Herwig7 (default tune) as generators.
Even though a full detector simulation using PYTHIA8 has also been generated, a fast
simulation is used for this purpose so that there is complete parity between the two
generators in the calculation of this systematic uncertainty. This fast simulation provides
agreement within
±
10% of the full detector simulation for
R
= 0
.
2jets, with some larger
deviations seen in the tails of the jet angularity distributions for
R
= 0
.
4jets. These
two response matrices are then used to unfold the measured data, and the diﬀerences
between the two unfolded results in each interval are taken as a symmetric uncertainty.
This uncertainty is most signiﬁcant at lower pch jet
T.
In order to assess the uncertainty due to the track mass assumption, K
±
meson and
proton masses are randomly assigned to 13% and 5.5% of tracks, respectively, in both the
data and the response matrix. These numbers are chosen from the (approximate) inclusive
number of each respective particle measured at midrapidity in pp events by ALICE [
71
].
Neither the measurement inside the jets nor the
pch jet
T
-dependence are considered, so
these numbers are taken to constitute a reasonable maximum uncertainty. The bin-by-bin
diﬀerence of the unfolded result to the nominal result is taken as a symmetric uncertainty.
8
JHEP05(2022)061
5 Results and discussion
We report the
λα
and
λα,g
distributions for
α
= 1, 1.5, 2, and 3in four equally-sized
intervals of
pch jet
T
between 20 and 100 GeV/c. The distributions are reported as diﬀerential
cross sections:
1
σ
dσ
dλα1
Njets
dNjets
dλα
(ungroomed), or 1
σinc
dσ
dλα,g1
Ninc jets
dNgr jets
dλα,g
(groomed),
(5.1)
where
Njets
is the number of jets within a given
pch jet
T
range and
σ
is the corresponding
cross section. For the groomed case, some jets are removed by the grooming procedure,
and therefore two diﬀerent quantities are deﬁned: N
gr jets
, the number of jets which have
at least one splitting satisfying the Soft Drop condition, and N
inc jets
, the total number of
inclusive jets, with both N
gr jets
and N
inc jets
being within the given
pch jet
T
range.
σinc
is
the cross section corresponding to the latter inclusive quantity. For the ungroomed case,
N
inc jets
=N
jets
and
σ
=
σinc
, so the redundant labels are dropped. It is useful to normalize
the groomed diﬀerential cross section by the number of inclusive jets since the groomed jet
angularities are a property of the inclusively-measured jet population and are thus typically
normalized as such in theoretical calculations [17].
The ungroomed jet angularity distributions are shown in ﬁgure 1and ﬁgure 2for
R
= 0
.
4and
R
= 0
.
2, respectively. By the deﬁnitions given in eq. (5.1), these distributions
are all normalized to unity. As
α
increases, the distributions skew towards small
λα
, since
θi
is smaller than unity. For larger
R
, the distributions are narrower than for smaller
R
,
as expected due to the collinear nature of jet fragmentation. For small
R
and low
pch jet
T
there is a visible peak at
λα
= 0, which is due to single particle jets. These distributions
are compared to PYTHIA8 Monash 2013 [
49
,
66
] and Herwig7 (default tune) [
50
,
51
] from
truth-level projections of the respective response matrices, with jet reconstruction assigning
tracks the
π±
meson mass as in the measured data. These comparisons show deviations up
to approximately +50%(
30%). The largest deviations are for small values of
λα
, where
nonperturbative physics becomes signiﬁcant (see section 5.1 for discussion).
The groomed jet angularity distributions for
zcut
= 0
.
2and
β
= 0 are shown in ﬁgure 3
for
R
= 0
.
4and ﬁgure 4for
R
= 0
.
2. Note that these distributions are shown on a
logarithmic scale due to the distributions being more strongly peaked and falling faster
with
λα
as compared to the ungroomed distributions. The groomed jet angularities have
signiﬁcantly smaller values than the ungroomed jet angularities, due to the removal of soft
wide-angle radiation. The fraction of “untagged” jets, those that do not contain a splitting
which passes the Soft Drop condition, ranges from 10 to 12%. Unlike the ungroomed jet
angularities, which are normalized to unity, the groomed jet angularities are normalized
to the Soft Drop tagging fraction. Since the tagging rate is fairly large, the measured
distributions are therefore normalized close to unity. PYTHIA and Herwig describe the
groomed jet angularities slightly better than the ungroomed jet angularities, with most
deviations seen in the ungroomed distributions improving by 10–20% in the groomed case.
Comparing to the two MC generators, the data are in slightly better agreement with Herwig7
than with PYTHIA8, especially for R= 0.4.
9
JHEP05(2022)061
0 0.2 0.4 0.6
2
4
6
8
10
12
14
α
λ
d
σ
d
σ
1
= 1
α
= 1.5
α
= 2
α
0.5)× = 3 (
α
ALICE
= 5.02 TeVs
pp
T
kcharged jets anti-
| < 0.5
jet
η
= 0.4 |R
c < 40 GeV/
ch jet
T
p20 <
0 0.1 0.2 0.3 0.4 0.5 0.6
0.5
1
1.5
PYTHIA8
Data
0.5
1
1.5
Herwig7
Data
0 0.2 0.4 0.6
1
2
3
4
5
6
7
8
9
α
λ
d
σ
d
σ
1
= 1
α
0.7)× = 1.5 (
α
0.5)× = 2 (
α
0.3)× = 3 (
α
c < 80 GeV/
ch jet
T
p60 <
0 0.1 0.2 0.3 0.4 0.5 0.6
0.5
1
1.5
PYTHIA8
Data
0 0.1 0.2 0.3 0.4 0.5 0.6
α
λ
0
0.5
1
1.5
Herwig7
Data
0 0.2 0.4
2
4
6
8
10
12
14 = 1
α
= 1.5
α
= 2
α
0.5)× = 3 (
α
Syst. uncertainty
PYTHIA8 Monash 2013
Herwig7
c < 60 GeV/
ch jet
T
p40 <
0 0.1 0.2 0.3 0.4 0.5
0.5
1
1.5
0.5
1
1.5
0 0.2 0.4
1
2
3
4
5
6
7
8
9 = 1
α
0.7)× = 1.5 (
α
0.5)× = 2 (
α
0.3)× = 3 (
α
c < 100 GeV/
ch jet
T
p80 <
0 0.1 0.2 0.3 0.4 0.5
0.5
1
1.5
0 0.1 0.2 0.3 0.4 0.5
α
λ
0
0.5
1
1.5
Figure 1
. Comparison of ungroomed jet angularities
λα
in pp collisions for
R
= 0
.
4to MC predictions
using PYTHIA8 and Herwig7, as described in the text. Four equally-sized
pch jet
T
intervals are shown,
with edges ranging between 20 and 100 GeV/c. The distributions are normalized to unity.
10
JHEP05(2022)061
0 0.2 0.4 0.6
2
4
6
8
10
12
14
α
λ
d
σ
d
σ
1
= 1
α
= 1.5
α
= 2
α
0.5)× = 3 (
α
ALICE
= 5.02 TeVs
pp
T
kcharged jets anti-
| < 0.7
jet
η
= 0.2 |R
c < 40 GeV/
ch jet
T
p20 <
0 0.1 0.2 0.3 0.4 0.5 0.6
0.5
1
1.5
PYTHIA8
Data
0.5
1
1.5
Herwig7
Data
0 0.2 0.4 0.6
1
2
3
4
5
6
7
8
9
α
λ
d
σ
d
σ
1
= 1
α
0.7)× = 1.5 (
α
0.5)× = 2 (
α
0.3)× = 3 (
α
c < 80 GeV/
ch jet
T
p60 <
0 0.1 0.2 0.3 0.4 0.5 0.6
0.5
1
1.5
PYTHIA8
Data
0 0.1 0.2 0.3 0.4 0.5 0.6
α
λ
0
0.5
1
1.5
Herwig7
Data
0 0.2 0.4
2
4
6
8
10
12
14 = 1
α
= 1.5
α
= 2
α
0.5)× = 3 (
α
Syst. uncertainty
PYTHIA8 Monash 2013
Herwig7
c < 60 GeV/
ch jet
T
p40 <
0 0.1 0.2 0.3 0.4 0.5
0.5
1
1.5
0.5
1
1.5
0 0.2 0.4
1
2
3
4
5
6
7
8
9 = 1
α
0.7)× = 1.5 (
α
0.5)× = 2 (
α
0.3)× = 3 (
α
c < 100 GeV/
ch jet
T
p80 <
0 0.1 0.2 0.3 0.4 0.5
0.5
1
1.5
0 0.1 0.2 0.3 0.4 0.5
α
λ
0
0.5
1
1.5
Figure 2
. Comparison of ungroomed jet angularities
λα
in pp collisions for
R
= 0
.
2to MC predictions
using PYTHIA8 and Herwig7, as described in the text. Four equally-sized
pch jet
T
intervals are shown,
with edges ranging between 20 and 100 GeV/c. The distributions are normalized to unity.
11
JHEP05(2022)061
0 0.2 0.4 0.6
3
10
2
10
1
10
1
10
2
10
3
10
4
10
5
10
6
10
7
10
,g
α
λ
d
σ
d
inc
σ
1
= 1
α
0.5)× = 1.5 (
α
0.2)× = 2 (
α
0.03)× = 3 (
α
ALICE
= 5.02 TeVs
pp
T
kcharged jets anti-
| < 0.5
jet
η
= 0.4 |R
c < 40 GeV/
ch jet
T
p20 <
= 0β = 0.2
cut
zSoft Drop
0 0.1 0.2 0.3 0.4 0.5 0.6
0.5
1
1.5
PYTHIA8
Data
0.5
1
1.5
Herwig7
Data
0 0.2 0.4 0.6
2
10
1
10
1
10
,g
α
λ
d
σ
d
inc
σ
1
= 1
α
0.25)× = 1.5 (
α
0.1)× = 2 (
α
0.02)× = 3 (
α
c < 80 GeV/
ch jet
T
p60 <
0 0.1 0.2 0.3 0.4 0.5 0.6
0.5
1
1.5
PYTHIA8
Data
0 0.1 0.2 0.3 0.4 0.5 0.6
,g
α
λ
0
0.5
1
1.5
Herwig7
Data
0 0.2 0.4
3
2
1
1
10
2
3
4
5
6
7 = 1
α
0.5)× = 1.5 (
α
0.2)× = 2 (
α
0.03)× = 3 (
α
Syst. uncertainty
PYTHIA8 Monash 2013
Herwig7
c < 60 GeV/
ch jet
T
p40 <
0 0.1 0.2 0.3 0.4 0.5
0.5
1
1.5
0.5
1
1.5
0 0.2 0.4
1
1
10 = 1
α
0.25)× = 1.5 (
α
0.1)× = 2 (
α
0.02)× = 3 (
α
c < 100 GeV/
ch jet
T
p80 <
0 0.1 0.2 0.3 0.4 0.5
0.5
1
1.5
0 0.1 0.2 0.3 0.4 0.5
,g
α
λ
0
0.5
1
1.5
Figure 3
. Comparison of groomed jet angularities
λα,g
in pp collisions for
R
= 0
.
4to MC predictions
using PYTHIA8 and Herwig7, as described in the text. Four equally-sized
pch jet
T
intervals are shown
between 20 and 100 GeV/c. The distributions are normalized to the groomed jet tagging fraction.
12
JHEP05(2022)061
0 0.2 0.4 0.6
2
10
1
10
1
10
2
10
3
10
4
10
5
10
6
10
7
10
,g
α
λ
d
σ
d
inc
σ
1
= 1
α
0.5)× = 1.5 (
α
0.2)× = 2 (
α
0.03)× = 3 (
α
ALICE
= 5.02 TeVs
pp
T
kcharged jets anti-
| < 0.7
jet
η
= 0.2 |R
c < 40 GeV/
ch jet
T
p20 <
= 0β = 0.2
cut
zSoft Drop
0 0.1 0.2 0.3 0.4 0.5 0.6
0.5
1
1.5
PYTHIA8
Data
0.5
1
1.5
Herwig7
Data
0 0.2 0.4 0.6
3
10
2
10
1
10
1
10
2
10
,g
α
λ
d
σ
d
inc
σ
1
= 1
α
0.25)× = 1.5 (
α
0.1)× = 2 (
α
0.02)× = 3 (
α
c < 80 GeV/
ch jet
T
p60 <
0 0.1 0.2 0.3 0.4 0.5 0.6
0.5
1
1.5
PYTHIA8
Data
0 0.1 0.2 0.3 0.4 0.5 0.6
,g
α
λ
0
0.5
1
1.5
Herwig7
Data
0 0.2 0.4
2
1
1
10
2
3
4
5
6
7 = 1
α
0.5)× = 1.5 (
α
0.2)× = 2 (
α
0.03)× = 3 (
α
Syst. uncertainty
PYTHIA8 Monash 2013
Herwig7
c < 60 GeV/
ch jet
T
p40 <
0 0.1 0.2 0.3 0.4 0.5
0.5
1
1.5
0.5
1
1.5
0 0.2 0.4
1
1
10
= 1
α
0.25)× = 1.5 (
α
0.1)× = 2 (
α
0.02)× = 3 (
α
c < 100 GeV/
ch jet
T
p80 <
0 0.1 0.2 0.3 0.4 0.5
0.5
1
1.5
0 0.1 0.2 0.3 0.4 0.5
,g
α
λ
0
0.5
1
1.5
Figure 4
. Comparison of groomed jet angularities
λα,g
in pp collisions for
R
= 0
.
2to MC predictions
using PYTHIA8 and Herwig7, as described in the text. Four equally-sized
pch jet
T
intervals are shown
between 20 and 100 GeV/c. The distributions are normalized to the groomed jet tagging fraction.
13
JHEP05(2022)061
The data cover a wide range of
α
and multiple
R
down to low
pT
, and therefore are
subject to varying inﬂuence from nonperturbative eﬀects. Accordingly, these data can be
used to study nonpertubative eﬀects. The level and location of the disagreements with
PYTHIA and Herwig provide further constraints on nonperturbative eﬀects in MC event
generators. Moreover, the comparison of the groomed and the ungroomed jet angularities
with MC event generators allows direct sensitivity to radiation that was groomed away,
which is highly nonperturbative.
5.1 Comparison to analytical calculations
The measured ungroomed and groomed jet angularities are compared with analytical
calculations [
9
,
17
] which use all-order resummations of large logarithms to next-to-leading
logarithmic (NLL
0
) accuracy [
19
]. In particular, the calculations resum logarithms of
λα
,
R
, and
zcut
. In the case of the
λα
logarithms, the cumulant of the cross section
includes the complete set of terms of form
αn
slnkλα
for
k
= 2
n
,2
n
1, and 2
n
2. The
calculations are valid up to power corrections in
λα
,
R
, and
zcut
, and do not include
non-global logarithms [
72
]. These calculations are based on the framework of Soft Collinear
Eﬀective Theory (SCET) [
73
], in which the jet cross section is factorized into a “hard
function” corresponding to the initial scattering, and a “jet function” corresponding to
the fragmentation of a hard-scattered parton into a jet. For the calculation of the jet
angularities, the jet function is then further factorized into collinear and soft functions.
Systematic uncertainties on the analytical predictions are estimated by systematically
varying ﬁfteen combinations of scales that emerge in the calculation.
For the ungroomed jet angularities, the collinear-soft momentum scale for the factoriza-
tion formalism becomes nonperturbative for [9]
λα.Λ
pch jet
TR,(5.2)
where Λis the energy scale at which
αs
becomes nonperturbative, which is taken to be
approximately 1
GeV/c
. For the groomed jet angularities with
β
= 0, this soft factorization
scale becomes nonperturbative for [17]
λα,g.z1α
cut Λ
pch jet
TR!α
.(5.3)
Accordingly, the analytical predictions are expected to describe the data only at suﬃciently
large
λα
, which depends on
pch jet
T
,
R
, and
zcut
. On the other hand, for
λα
=
O
(1), power
corrections in
λα
become important, and are not included in the NLL
0
calculations. Note that
for λα,g> zcut, the groomed and ungroomed predictions are identical at the parton level.
For values of
λα
that are suﬃciently large to be described by SCET, corrections
for nonperturbative eﬀects must still be applied in order to compare these parton-level
calculations to our charged-hadron-level measurements. These nonperturbative eﬀects
include hadronization, the underlying event, and the selection of charged particle jets. Note
that track-based observables are IRC-unsafe. In general, nonperturbative track functions can
14
JHEP05(2022)061
be used to directly compare track-based measurements to analytical calculations [
16
,
74
,
75
];
however, such an approach has not yet been developed for jet angularities. Two techniques
are used, described in the following subsections, to apply the nonperturbative corrections.
The ﬁrst technique relies solely on MC generators to transform the parton-level calculations
into the ﬁnal predictions at the charged-hadron level. Two response matrices are constructed,
one using PYTHIA 8.244 and the other using Herwig7, which map the jet angularity
distributions from jets reconstructed at the ﬁnal-state parton level (after the parton
shower) to those from jets reconstructed at the charged-hadron level. This is done by
requiring a unique geometrical match between the parton and charged-hadron-level jets of
R < R/
2. The PYTHIA8 simulation uses the default Monash 2013 tune, which is tuned
to both e
+
e
and p
¯p
data [
66
], with the only change being that the minimum shower
pT
(
TimeShower:pTmin
) is set to 0.2
GeV/c
, one half of its default value, in order to better
match the NLL
0
predictions at parton level. Herwig7 is also run with the default tune [
76
].
The response matrix generated with both MC simulations is 4D, mapping
pparton jet
T
and
λparton jet
βto pch jet
T,truth and λα,truth.
Since the NLL
0
predictions are generated as normalized distributions, each
pch jet
T
interval
is ﬁrst scaled by a value corresponding to the inclusive
pjet
T
cross section, calculated at Next-
to-Leading Order (NLO) with NLL resummation of logarithms in the jet radius [
77
]. The
4D response matrix discussed above is then multiplied by these scaled 2D NLL
0
predictions
(in both
pjet
T
, ranging from 10 to 200
GeV/c
, and
λα
) to obtain the theoretical predictions
at charged-hadron level. To propagate the systematic uncertainty on the original NLL
0
calculations, this “folding” procedure is performed individually for each of ﬁfteen scale
variations, from which a total systematic uncertainty is constructed from the minimum
and maximum variation in each interval. Note that this procedure introduces a model-
dependence to the comparison, and in fact signiﬁcantly reduces the magnitude of the
systematic uncertainties compared to the parton level; the repetition of this procedure with
both PYTHIA8 and Herwig7 is meant to estimate the size of this model dependence.
Although the perturbative accuracy of the MC generators is not clear, by restricting
these comparisons to
pch jet
T>
60
GeV/c
, there is adequate matching between the analytical
calculations and the MC generators’ ﬁnal-state parton-level predictions to employ the
nonperturbative corrections via this mapping procedure. After the folding step, an additional
bin-by-bin correction is applied for multi-parton interactions in the underlying event using
the respective event generator. More speciﬁcally, a ratio is created between the 2D jet
angularity distributions generated with multi-parton interactions on versus oﬀ at the
charged-hadron level, which is then multiplied bin-by-bin by the folded distributions. In
all cases, the corrections performed with PYTHIA and those with Herwig are similar in
magnitude, indicating that this correction procedure is reasonable.
Figure 5shows comparisons of the measured ungroomed jet angularities to the folded
theoretical predictions for 60
< pch jet
T<
80
GeV/c
, for both
R
= 0
.
2(top) and
R
= 0
.
4
(bottom) and for
α
= 1
.
5(left), 2 (middle), and 3 (right). Figure 6shows the corresponding
comparisons for the groomed jet angularities. The comparisons for 80
< pch jet
T<
100
GeV/c
15
JHEP05(2022)061
2
4
6
8
10
12
14
16
18 Data
PYTHIA8NLL'
Herwig7NLL'
0.3)× = 2 (
α
0 0.1 0.2
α
λ
1
1
10
2
4
6
8
10
12
14
16
18 )R
ch jet
T
p / (Λ
NP
α
λ
Syst. uncertainty
0.12)× = 3 (
α
0 0.1 0.2
α
λ
1
1
10
2
4
6
8
10
12
14
16
18
20
22
24 Data
PYTHIA8NLL'
Herwig7NLL'
0.65)× = 2 (
α
0 0.1 0.2 0.3
α
λ
1
1
10
2
4
6
8
10
12
14
16
18
20
22
24 )R
ch jet
T
p / (Λ
NP
α
λ
Syst. uncertainty
0.27)× = 3 (
α
0 0.1 0.2
α
λ
1
1
10
Figure 5
. Comparison of ungroomed jet angularities
λα
in pp collisions for
R
= 0
.
2(top) and
R
= 0
.
4(bottom) to analytical NLL
0
predictions with MC hadronization corrections in the range
60
< pch jet
T<
80
GeV/c
. The distributions are normalized such that the integral of the perturbative
region deﬁned by
λα> λNP
α
(to the right of the dashed vertical line) is unity. Divided bins are
placed into the left (NP) region.
16
JHEP05(2022)061
0 0.1 0.2 0.3 0.4
1
10
2
10
3
10
,g
α
λ
d
,g
α
λ
d
σ
d
1
NP
,g
α
λ
,g
α
λ
d
σ
d
ALICE
= 5.02 TeVs
pp
T
kcharged jets anti-
| < 0.7
jet
η
= 0.2 |R
c < 80 GeV/
ch jet
T
p60 <
= 1.5
α
0 0.1 0.2 0.3
,g
α
λ
1
10
1
10
Theory
Data
1
10
2
3
Data
PYTHIA8NLL'
Herwig7NLL'
= 2
α
0 0.1 0.2 0.3
,g
α
λ
1
1
10
1
10
2
3
α
)
NP
α
λ
(
α
1-
cut
z
NP
,g
α
λ
Syst. uncertainty
= 3
α
= 0
β
= 0.2,
cut
zSoft Drop:
0 0.1 0.2
,g
α
λ
1
1
10
0 0.2 0.4
1
10
2
10
3
10
,g
α
λ
d
,g
α
λ
d
σ
d
1
NP
,g
α
λ
,g
α
λ
d
σ
d
ALICE
= 5.02 TeVs
pp
T
kcharged jets anti-
| < 0.5
jet
η
= 0.4 |R
c < 80 GeV/
ch jet
T
p60 <
= 1.5
α
0 0.2 0.4
,g
α
λ
1
10
1
10
Theory
Data
1
10
2
3
Data
PYTHIA8NLL'
Herwig7NLL'
= 2
α
0 0.1 0.2 0.3
,g
α
λ
1
1
10
1
10
2
3
α
)
NP
α
λ
(
α
1-
cut
z
NP
,g
α
λ
Syst. uncertainty
= 3
α
= 0
β
= 0.2,
cut
zSoft Drop:
0 0.1 0.2
,g
α
λ
1
1
10
Figure 6
. Comparison of groomed jet angularities
λα,g
in pp collisions for
R
= 0
.
2(top) and
R
= 0
.
4(bottom) to analytical NLL
0
predictions with MC hadronization corrections in the range
60
< pch jet
T<
80
GeV/c
. The distributions are normalized such that the integral of the perturbative
region deﬁned by
λα,g> λNP
α,g
(to the right of the dashed vertical line) is unity. Divided bins are
placed into the left (NP) region.
17
JHEP05(2022)061
are shown in appendix A. Predictions for the
α
= 1 distributions are not currently available
due to enhanced sensitivity to soft-recoil, which requires a diﬀerent factorization [22].
A dashed vertical line is drawn as a rough estimate for the division of perturbative-
and nonperturbative-dominated regions, via eq. (5.2) or eq. (5.3) with Λ=1
GeV/c
and
the mean
pch jet
T
for each interval. Note that the transition from values of
λα
which are
dominated by perturbative versus nonperturbative physics is actually smooth, and this
vertical line is merely intended as a visual guide. The nonperturbative-dominated region of
the jet angularities is denoted as λNP
α.
Since the integral for all of the distributions in ﬁgure 1through ﬁgure 4is ﬁxed at unity
by construction, it is important to note that disagreement in the nonperturbative-dominated
region induces disagreement in the perturbative-dominated region. Discrepancy in the
nonperturbative region is expected due to the divergence of
αs
and the corresponding
signiﬁcance of higher-order terms in the perturbative expansion and will necessarily
induce disagreement in the perturbative-dominated region. Accordingly, for these theoretical
comparisons, the distributions are normalized such that the integral above λNP
αis unity.
5.1.2 Shape function based correction
An alternate correction technique is also used, which employs a nonperturbative shape
function
F
(
k
)[
14
,
20
,
21
] to correct for the eﬀects caused by hadronization and the
underlying event. The shape function is deﬁned as
F(k) = 4k
2
α
exp 2k
α,(5.4)
where
k
is a momentum scale parameter of the shape function, and
α
is described by a
single parameter = O(1 GeV/c)obeying the scaling relation
α= /(α1),(5.5)
and expected to hold universally for hadronization corrections (but not necessarily for
underlying event corrections). To correct the parton-level calculations to the hadron
level, this shape function is convolved with the perturbative (parton level) jet angularity
distribution via numerical integration over argument k
dσ
dpjet
Tdλα
=ZF(k)dσpert
dpjet
Tdλαλαλshift
α(k)dk, (5.6)
where the shift term λshift
α(k)is either [17,21]:
λshift
α(k) = k
pjet
TR(ungroomed), or z1α
cut k
pjet
TR!α
(groomed, with β= 0). (5.7)
The limits of the integral are thus given by the values of
k
for which the argument
λαλshift
α(k)
is between 0 and 1. Since the nonperturbative parameter is not calculable
within perturbation theory, four values (0.2, 0.4, 0.8, and 2
GeV/c
) are chosen to observe
the diﬀerent shifting eﬀects. These distributions are then corrected once more using a
18
JHEP05(2022)061
similar PYTHIA8 folding procedure as described above to account for the eﬀects of only
reconstructing charged-particle jets. This correction is dominated by a shift and smearing
along the pjet
Taxis.
The comparisons to the ungroomed predictions are shown in ﬁgure 7, and the groomed
predictions are shown in ﬁgure 8. The shape function approach, speciﬁcally the scaling
given in eq. (5.5), is not fully justiﬁed in the groomed case [
78
,
79
]; nevertheless, reasonable
agreement is observed. Since this shape convolution does not require matching to MC at
the parton level, the comparisons are extended to the 40
< pch jet
T<
60
GeV/c
interval,
but below this the perturbative accuracy of the parton-level predictions is insuﬃcient for
rigorous comparisons. The comparisons for 40
< pch jet
T<
60
GeV/c
and 80
< pch jet
T<
100
GeV/c are shown in appendix A.
5.2 Discussion
The
λα
distributions are generally consistent with the calculations within uncertainties
when
λα
is suﬃciently large to be in the pQCD regime. This holds approximately inde-
pendent of
α
,
R
, and
pjet
T
, and whether or not the jets are groomed. In some distributions,
however, particularly for
R
= 0
.
4, modest disagreement is observed at large
λα
. This
disagreement cannot be unambiguously associated with a particular value of
λα
due to the
self-normalization of the observable, but rather demonstrates an overall inconsistency in the
shape of the distribution. This disagreement could be caused by the unaccounted power
corrections in
λα
, or other eﬀects and suggests a need for further theoretical investigation.
Nevertheless, the overall agreement with the perturbative calculations is striking, given the
low-to-moderate jet pTand Rconsidered.
For
α
= 1
.
5, the majority of the distributions can be described perturbatively, as
λNP
α
is
conﬁned towards the left-hand side of the distributions. As
α
increases to
α
= 3, the inﬂuence
of the
λNP
α
region grows, and the ungroomed distributions become strongly nonperturbative.
Similarly, as
R
increases from
R
= 0
.
2to
R
= 0
.
4, or as
pch jet
T
increases, the size of the
perturbative region increases. In the nonperturbative region
λα< λNP
α
, the
λα
distributions
diverge from the calculations. This is expected, since the perturbative approximations
break down for
λα< λNP
α
, and neither the MC or shape function corrections are necessarily
expected to fully correct for missing physics at higher orders or for nonperturbative coupling.
In some distributions, the shape-function-based correction is sometimes able to describe
the data partially into the nonperturbative regime for suitable values of .
While the overall level of agreement is comparable in both the ungroomed and groomed
cases, grooming widens the pQCD regime, as indicated by the location of the dashed blue
line in ﬁgures [58]. On the other hand, grooming shifts the distributions themselves to
signiﬁcantly smaller values of
λα
. Nevertheless, this highlights the potential beneﬁt of
grooming in heavy-ion collisions in order to retain a larger degree of perturbative control in
addition to controlling eﬀects of the underlying event.
The performance of the two nonperturbative correction methods based entirely
on MC generators, or on shape functions are comparable in the perturbative regime.
Comparing diﬀerent values of for the ungroomed case, where eq. (5.5) is justiﬁed, there
is in many cases only a small diﬀerence between the calculations with = 0
.
2, 0.4, and
19
JHEP05(2022)061
2
4
6
8
10
12
14
16
18
20 Data
PYTHIA8
=0.2
NP
FNLL'
PYTHIA8
=0.4
NP
FNLL'
PYTHIA8
=0.8
NP
FNLL'
PYTHIA8
=2.0
NP
FNLL'
0.3)× = 2 (
α
0 0.1 0.2
α
λ
1
1
10
2
4
6
8
10
12
14
16
18
20 )R
ch jet
T
p / (Λ
NP
α
λ
Syst. uncertainty
0.12)× = 3 (
α
0 0.1 0.2
α
λ
1
1
10
5
10
15
20
25
30 Data
PYTHIA8
=0.2
NP
FNLL'
PYTHIA8
=0.4
NP
FNLL'
PYTHIA8
=0.8
NP
FNLL'
PYTHIA8
=2.0
NP
FNLL'
0.65)× = 2 (
α
0 0.1 0.2 0.3
α
λ
1
1
10
5
10
15
20
25
30 )R
ch jet
T
p / (Λ
NP
α
λ
Syst. uncertainty
0.27)× = 3 (
α
0 0.1 0.2
α
λ
1
1
10
Figure 7
. Comparison of ungroomed jet angularities
λα
in pp collisions for
R
= 0
.
2(top) and
R
= 0
.
4(bottom) to analytical NLL
0
predictions using
F
(
k
)convolution in the range 60
< pch jet
T<
80
GeV/c
. The distributions are normalized such that the integral of the perturbative region deﬁned
by
λα> λNP
α
(to the right of the dashed vertical line) is unity. Divided bins are placed into the left
(NP) region.
20
JHEP05(2022)061
0 0.1 0.2 0.3 0.4
1
10
2
10
3
10
4
10
,g
α
λ
d
,g
α
λ
d
σ
d
1
NP
,g
α
λ
,g
α
λ
d
σ
d
ALICE
= 5.02 TeVs
pp
T
kcharged jets anti-
| < 0.7
jet
η
= 0.2 |R
c < 80 GeV/
ch jet
T
p60 <
= 1.5
α
0 0.1 0.2 0.3
,g
α
λ
1
10
1
10
Theory
Data
1
10
2
3
4Data
PYTHIA8
=0.2
NP
FNLL'
PYTHIA8
=0.4
NP
FNLL'
PYTHIA8
=0.8
NP
FNLL'
PYTHIA8
=2.0
NP
FNLL'
= 2
α
0 0.1 0.2 0.3
,g
α
λ
1
1
10
1
10
2
3
4
α
)
NP
α
λ
(
α
1-
cut
z
NP
,g
α
λ
Syst. uncertainty
= 3
α
= 0
β
= 0.2,
cut
zSoft Drop:
0 0.1 0.2
,g
α
λ
1
1
10
0 0.2 0.4
1
10
2
10
3
10
4
10
,g
α
λ
d
,g
α
λ
d
σ
d
1
NP
,g
α
λ
,g
α
λ
d
σ
d
ALICE
= 5.02 TeVs
pp
T
kcharged jets anti-
| < 0.5
jet
η
= 0.4 |R
c < 80 GeV/
ch jet
T
p60 <
= 1.5
α
0 0.2 0.4
,g
α
λ
1
10
1
10
Theory
Data
1
10
2
3
4
Data
PYTHIA8
=0.2
NP
FNLL'
PYTHIA8
=0.4
NP
FNLL'
PYTHIA8
=0.8
NP
FNLL'
PYTHIA8
=2.0
NP
FNLL'
= 2
α
0 0.1 0.2 0.3
,g
α
λ
1
1
10
1
10
2
3
4
α
)
NP
α
λ
(
α
1-
cut
z
NP
,g
α
λ
Syst. uncertainty
= 3
α
= 0
β
= 0.2,
cut
zSoft Drop:
0 0.1 0.2
,g
α
λ
1
1
10
Figure 8
. Comparison of groomed jet angularities
λα,g
in pp collisions for
R
= 0
.
2(top) and
R
= 0
.
4(bottom) to analytical NLL
0
predictions using
F
(
k
)convolution in the range 60
< pch jet
T<
80
GeV/c
. The distributions are normalized such that the integral of the perturbative region deﬁned
by
λα,g> λNP
α,g
(to the right of the dashed vertical line) is unity. Divided bins are placed into the
left (NP) region.
21
JHEP05(2022)061
0.8
GeV/c
. However, for
α
= 1
.
5and
α
= 2, larger values of ( = 2
GeV/c
) appear
to have more tension with the data in the perturbative regime than smaller values. For
α
= 3, the perturbative region is too small to make any clear statement. One must bear
in mind, however, that
λNP
α
is only a rough characterization of the regime of validity of the
perturbative calculation. Consequently, it is unknown whether this disagreement is due to
the value of or due to the breakdown of the perturbative calculation. For smaller values of
(e.g. Ω=0
.
2or 0
.
4
GeV/c
), the predicted scaling of eq. (5.5) is consistent with the data.
Note that the value of which describes the data is
O
corrections. These smaller values contrast with a previous result of Ω=3
.
5
GeV/c
for
the ungroomed mass of
R
= 0
.
4jets at 200
< pjet
T<
300
GeV/c
[
80
], suggesting that the
underlying event contribution to , which is not expected to obey the scaling of eq. (5.5),
may be modiﬁed by jets measured at diﬀerent
pjet
T
or by the choice to reconstruct jets using
only charged-particle tracks. No signiﬁcant
R
-dependence is observed in the scaling behavior
in this analysis, suggesting that any scaling-breaking underlying event contributions, when
also combined with hadronization corrections, are small for R= 0.2and 0.4.
6 Conclusion
The generalized jet angularities are reported both with and without Soft Drop grooming,
λα,g
and
λα
, respectively, for charged-particle jets in pp collisions at
s
= 5
.
02 TeV with
the ALICE detector. This measurement of both the ungroomed and, for the ﬁrst time,
the groomed jet angularities provides constraints on models and captures the interplay
between perturbative and nonperturbative eﬀects in QCD. Systematic variations of the
contributions from collinear and soft radiation of the shower, captured within a given
R
,
are provided by measuring the jet angularities for a selection of
α
parameters. These results
consequently provide rigorous tests of pQCD calculations.
The theoretical predictions at NLL
0
in SCET show an overall agreement with the
data for jets with values of
λα
in the perturbative regime delimited by a collinear-soft
momentum scale in the factorization framework of about 1
GeV/c
. The calculations, after
accounting for nonperturbative eﬀects by two diﬀerent methods, are compatible within
about 20% or better with the data in the perturbative region for all explored values of
R
and
α
. However, larger deviations of up to about 50% are observed in the tails of some
distributions, suggesting a need for further theoretical study. By making comparisons solely
in the perburbatively-dominated regime, consistency is seen with a predicted universal
scaling of the nonperturbative shape function parameter
α
with value
<
1. A clear
breakdown of the agreement is observed for small λ, where the perturbative calculation is
expected to fail. Such nonperturbative eﬀects include soft splittings and hadronization, and
these eﬀects dominate over signiﬁcant regions of the phase space of moderate and low-energy
jets. This is corroborated by the comparison of the measured groomed jet angularities to
the equivalent theoretical calculations, which demonstrate a wider range of agreement with
the perturbative calculations.
These comparisons provide critical guidance for measurements in high-energy heavy-ion
collisions where the internal structure of jets may undergo modiﬁcations via scatterings of jet
22
JHEP05(2022)061
fragments with the hot and dense QCD medium. Our measurements demonstrate that any
comparison to pQCD must also consider the regimes of
λα
and
λα,g
that are controlled by
perturbative processes as opposed to those that are dominated by nonperturbative processes.
This provides guidance for the selections of
α
,
R
, and
pch jet
T
, and indicates the importance
of capturing the complete spectrum of processes (perturbative and non-perturbative) in
theory calculations attempting to explain jet quenching.
These measurements further highlight that disagreement between theoretical predic-
tions and data in the nonperturbative regime will necessarily induce disagreement in the
perturbative regime, when in fact the perturbative accuracy of predictions should only be
scrutinized within the perturbative regime. In practice, these measurements give a clear
indication that careful inspection is needed when interpreting measurements of jet sub-
structure based on models of jet quenching in heavy-ion collisions for observables including
the jet angularity and the jet mass. Future measurements will beneﬁt from the provided
guidance demonstrating not only the agreement of jet angularities with pQCD calculations
in the perturbative regime but also on selecting on jet angularity diﬀerentially with
α
,
R
,
and
pch jet
T
in order to maximize theoretical control and interpretation of the perturbative
and nonpertubative regimes of jet substructure observables.
Acknowledgments
We gratefully acknowledge Kyle Lee and Felix Ringer for providing theoretical predic-
tions, and for valuable discussions regarding the comparison of these predictions to our
measurements.
The ALICE Collaboration would like to thank all its engineers and technicians for their
invaluable contributions to the construction of the experiment and the CERN accelerator
teams for the outstanding performance of the LHC complex. The ALICE Collaboration
gratefully acknowledges the resources and support provided by all Grid centres and the
Worldwide LHC Computing Grid (WLCG) collaboration. The ALICE Collaboration ac-
knowledges the following funding agencies for their support in building and running the
ALICE detector: A.I. Alikhanyan National Science Laboratory (Yerevan Physics Institute)
Foundation (ANSL), State Committee of Science and World Federation of Scientists (WFS),
Armenia; Austrian Academy of Sciences, Austrian Science Fund (FWF): [M 2467-N36]
and Nationalstiftung für Forschung, Technologie und Entwicklung, Austria; Ministry of
Communications and High Technologies, National Nuclear Research Center, Azerbaijan;
Conselho Nacional de Desenvolvimento Cientíﬁco e Tecnológico (CNPq), Financiadora
de Estudos e Projetos (Finep), Fundação de Amparo à Pesquisa do Estado de São Paulo
(FAPESP) and Universidade Federal do Rio Grande do Sul (UFRGS), Brazil; Ministry of
Education of China (MOEC), Ministry of Science & Technology of China (MSTC) and
National Natural Science Foundation of China (NSFC), China; Ministry of Science and
Education and Croatian Science Foundation, Croatia; Centro de Aplicaciones Tecnológi-
cas y Desarrollo Nuclear (CEADEN), Cubaenergía, Cuba; Ministry of Education, Youth
and Sports of the Czech Republic, Czech Republic; The Danish Council for Independent
Research | Natural Sciences, the VILLUM FONDEN and Danish National Research Foun-
23
JHEP05(2022)061
dation (DNRF), Denmark; Helsinki Institute of Physics (HIP), Finland; Commissariat à
l’Energie Atomique (CEA) and Institut National de Physique Nucléaire et de Physique