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

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 $\sqrt{s}=5.02$ TeV with the ALICE detector. Jets are reconstructed using charged particle tracks at midrapidity ($|\eta|<0.9$). The anti-$k_{\rm T}$ algorithm is used with jet resolution parameters $R=0.2$ and $R=0.4$ for several transverse momentum $p_{\rm T}^{\text{ch jet}}$ 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, $\lambda_{\alpha}$, and groomed jet angularities, $\lambda_{\alpha\text{,g}}$, to investigate the interplay between perturbative and nonperturbative effects at low jet momenta. Various angular exponent parameters $\alpha = 1$, 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.


Introduction
In high-energy particle collisions, jet observables are sensitive to a variety of processes in quantum chromodynamics (QCD), from the initial hard (high Q 2 ) parton scattering to a scale evolution culminating in hadronization near Λ QCD . Jets reconstructed with a radius (resolution) parameter near R = 1 and with sufficiently large transverse momentum p jet T provide a proxy for the dynamics of the initial hard parton scattering, whereas those reconstructed with smaller R or at lower p jet 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][4][5], can additionally be used to remove soft, wide-angle radiation via well-controlled approaches, reducing nonperturbative effects. This defines two 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 where the sum runs over the jet constituents i, and κ and α are continuous free parameters. 1 The first factor z i ≡ p T,i /p jet T describes the momentum fraction carried by the constituent, and the second factor θ i ≡ ∆R i /R denotes the separation in rapidity (y) and azimuthal angle (ϕ) of the constituent from the jet axis, where ∆R i ≡ ∆y 2 i + ∆ϕ 2 i and R is the jet resolution parameter. The jet angularities are infraredand collinear-(IRC-)safe for κ = 1 and α > 0 [8,9]. We consider the ungroomed jet angularities, denoted as λ α , as well as the groomed jet angularities in which the sum runs only over the constituents of the groomed jet, denoted as λ α,g . These include the jet girth [10], λ 1 , and the jet thrust [11], λ 2 , which is related to the jet mass m jet by λ 2 = (m jet /p jet T ) 2 + O(λ 2 2 ); λ 2 , however, is more robust against nonperturbative effects than m jet 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][13][14][15]. This article considers jet angularities 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 nonperturbative correction procedure, as outlined in Sec. 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 ) accuracy [19]. Measurements of λ α and λ α,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 Several measurements of jet angularities have been performed in hadronic collisions. The ungroomed jet angularity λ 1 has been measured in pp collisions by the ATLAS, CMS, and ALICE Collaborations [22][23][24] in addition to pp collisions by the CDF Collaboraiton [25]. The ungroomed jet angularity λ 2 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][27][28][29][30][31][32][33][34][35], and the ungroomed mass was also studied in pp 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 p T , rather than for 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 been made for charged-particle jets. In this article, we perform the first measurements of groomed jet angularities in pp collisions, and a systematic scan of the IRC-safe ungroomed jet angularities. These measurements focus on low to moderate p jet 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 produced [37][38][39][40]. Measurements of jets and jet substructure in heavy-ion collisions may provide key insight into the physical properties of this deconfined state [41][42][43]. The jet angularities are sensitive both to medium-induced broadening as well as jet collimation [44][45][46]; by systematically varying the weight of collinear radiation, one may be able to efficiently discriminate between jet quenching models. In Pb-Pb collisions, λ 1 has been measured for R = 0.2 by the ALICE Collaboration [22], and the ungroomed and groomed jet mass have been measured for R = 0.4 by 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 = 0.4 jet 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.
This article reports measurements of ungroomed and groomed jet angularities for 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 = 1.5 and 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 Sec. 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 z cut = 0.2 and β = 0 [48]. Charged particle jets were reconstructed at midrapidity using the anti-k T algorithm with jet resolution (radius) parameters R = 0.2 and R = 0.4 in four equally-sized p ch jet T intervals from 20 to 100 GeV/c. The results are compared to NLL pQCD predictions, as well as to the PYTHIA8 [49] and Herwig7 [50,51] Monte Carlo generators.

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 ranges of 2.8 < η < 5.1 and −3.7 < η < −1.7 [55]. The event selection also requires the location of the primary vertex to be within ±10 cm from the nom-Measurements of the jet angularities in pp collisions at √ s = 5.02 TeV ALICE Collaboration inal 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 ±112.5 m 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 cm −2 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 p T resolution to tracks with SPD hits, as determined from the fit quality. Tracks with p T,track > 0.15 GeV/c are accepted over pseudorapidity |η| < 0.9 and azimuthal angle 0 < ϕ < 2π. 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 p T,track = 0.15 GeV/c, rises to approximately 84% at p T,track = 1 GeV/c, and remains above 75% at higher p T . The momentum resolution σ (p T )/p T is estimated from the covariance matrix of the track fit [53] and is approximately 1% at p T,track = 1 GeV/c. This increases with p T,track , reaching approximately 4% at p T,track = 50 GeV/c.

Analysis method 3.1 Jet reconstruction
Jets are reconstructed from charged tracks with p T > 150 MeV/c using the FastJet package [61]. The anti-k T algorithm is used with the E recombination scheme for resolution parameters R = 0.2 and 0.4 [62]. All reconstructed charged-particle jets in the transverse momentum range 5 < p ch 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, η jet < 0.9 − R. Jets containing a track with p T > 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 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 Sec. 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 detectorlevel jets are then geometrically matched with truth-level jets within ∆R < 0.6 R while additionally requiring that each match be unique. Table 1   The ungroomed jet angularities are reconstructed using all of the charged-particle jet constituents according to Eq. (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 z ≡ p T,subleading p T,leading +p T,subleading and θ ≡ ∆R R ≡ √ ∆y 2 +∆ϕ 2 R . The jet tree is then traversed starting from the largest-angle splitting, and the Soft Drop condition, z > z cut θ β , is recursively evaluated. Here, z is the subleading branch p T fraction defined above, and z cut and β are tunable, free parameters of the grooming algorithm. For this analysis, β = 0 is used to maximize the perturbative calculability [17], while z cut = 0.2 is chosen (as opposed to the more common z cut = 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) using the groomed jet constituents, but still with the ungroomed p ch jet T and ungroomed jet axis to define θ i , since the groomed jet observable is a property of the original (ungroomed) jet object. Note that while the ungroomed p ch jet T is IRC-safe, the groomed p ch 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.

Corrections
The reconstructed p ch jet T and λ α differ from their true values due to tracking inefficiency, particlematerial interactions, and track p T 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 p ch jet T,truth and λ α,truth to p ch jet T,det and λ α,det , where p ch jet T,det and p ch jet T,truth are as above, and λ α,det and λ α,truth are the analogous detector-and truth-level λ α . 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 p ch jet T and λ α is then performed using the iterative Bayesian unfolding algorithm [68,69] implemented in the RooUnfold package [70] to recover the true jet spectrum at the chargedhadron 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 p ch jet T interval varies widely, with higher-p ch jet T jets being less probable than lower-p ch jet T jets, and since the shape and mean value of the jet angularity distributions also changes with p ch jet T , a separate 2D unfolding for each reported p ch jet T bin is performed in order to optimize the observable binning at both Measurements of the jet angularities in pp collisions at √ s = 5.02 TeV ALICE Collaboration 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 p ch jet T,truth and λ α,truth axes. The smearing in λ α is roughly symmetric about the diagonal, whereas the smearing in p ch jet T tends to be skewed towards lower values of p ch jet T,truth due to tracking efficiency effects. In the groomed case, the number of untagged jets in the unfolding procedure is included as an additional bin adjacent to the lower edge of the λ α 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.
The distributions after unfolding are corrected for the kinematic efficiency, defined as the efficiency of reconstructing a truth-level jet at a particular p ch jet T,truth and λ α,truth value given a reconstructed jet p ch jet T,det and λ α,det range. Kinematic inefficiency results from effects including smearing from the Soft Drop threshold and p T -smearing of the jet out of the selected p ch 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 p ch 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.

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 2 summarizes 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 parameters 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 ±2 and the average difference with respect to the nominal Measurements of the jet angularities in pp collisions at √ s = 5.02 TeV ALICE Collaboration result is taken as the systematic uncertainty.
2. The prior distribution is scaled by a power law in p ch jet T and a linear scaling in λ α , (p ch jet T ) ±0.5 × [1±(λ α −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 λ α 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 p ch 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.
The total unfolding systematic uncertainty is then the standard deviation of the variations, ∑ N i=1 σ 2 i /N, where N = 4 and σ i 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 p T 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 = 0.2 jets, with some larger deviations seen in the tails of the jet angularity distributions for R = 0.4 jets. 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 p ch 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 p ch 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.

Results and discussion
We report the λ α and λ α,g distributions for α = 1, 1.5, 2, and 3 in four equally-sized intervals of p ch jet T between 20 and 100 GeV/c. The distributions are reported as differential cross sections: where N jets is the number of jets within a given p ch 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 p ch jet T range. σ inc is the cross section corresponding to the latter inclusive quantity. For the ungroomed case, N inc jets = N jets and σ = σ inc , 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 Measurements of the jet angularities in pp collisions at √ s = 5.02 TeV ALICE Collaboration 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 Fig. 1 and Fig. 2 for R = 0.4 and R = 0.2, respectively. By the definitions given in Eq. 2, these distributions are all normalized to unity. As α increases, the distributions skew towards small λ α , since θ i 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 p ch jet T there is a visible peak at λ α = 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 λ α , where nonperturbative physics becomes significant (see Sec. 5.1 for discussion).
The groomed jet angularity distributions for z cut = 0.2 and β = 0 are shown in Fig. 3 for R = 0.4 and Fig. 4 for 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 λ α 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.
The data cover a wide range of α and multiple R down to low p T , 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                 Measurements of the jet angularities in pp collisions at √ s = 5.02 TeV ALICE Collaboration and the ungroomed jet angularities with MC event generators allows direct sensitivity to radiation that was groomed away, which is highly nonperturbative.

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 ) accuracy [19]. In particular, the calculations resum logarithms of λ α , R, and z cut . In the case of the λ α logarithms, the cumulant of the cross section includes the complete set of terms of form α n s ln k λ α for k = 2n, 2n − 1, and 2n − 2. The calculations are valid up to power corrections in λ α , R, and z cut , 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 factorization formalism becomes nonperturbative for [9] where Λ is the energy scale at which α s 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] Accordingly, the analytical predictions are expected to describe the data only at sufficiently large λ α , which depends on p ch jet T , R, and z cut . On the other hand, for λ α = O(1), power corrections in λ α become important, and are not included in the NLL calculations. Note that for λ α,g > z cut , the groomed and ungroomed predictions are identical at the parton level.
For values of λ α 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-hadronlevel 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 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.

MC-based hadronization correction
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 pp data [66], with the only change being that the minimum shower p T (TimeShower:pTmin) is set to 0.2 GeV/c, one half of its default value, in order to better match the NLL predictions at parton level. Herwig7 is also run with the default tune [76]. The response matrix generated with both MC simulations is 4D, mapping p Since the NLL predictions are generated as normalized distributions, each p ch jet T interval is first scaled by a value corresponding to the inclusive p jet 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 predictions (in both p jet T , ranging from 10 to 200 GeV/c, and λ α ) to obtain the theoretical predictions at charged-hadron level. To propagate the systematic uncertainty on the original NLL 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 p ch 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 5 shows comparisons of the measured ungroomed jet angularities to the folded theoretical predictions for 60 < p ch 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 6 shows the corresponding comparisons for the groomed jet angularities. The comparisons for 80 < p for each interval. Note that the transition from values of λ α which are dominated by perturbative versus nonperturbative physics is actually smooth, and this vertical line is merely intended as a visual guide. The nonperturbativedominated region of the jet angularities is denoted as λ NP α . Since the integral for all of the distributions in Fig. 1 through Fig. 4 is 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 α s 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 λ NP α is unity.

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 where k is a momentum scale parameter of the shape function, and Ω α is described by a single parameter Ω = O(1 GeV/c) obeying the scaling relation Measurements of the jet angularities in pp collisions at √ s = 5.02 TeV ALICE Collaboration   Measurements of the jet angularities in pp collisions at √ s = 5.02 TeV ALICE Collaboration 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 dp jet where the shift term λ shift α (k) is either [17, 21]: The limits of the integral are thus given by the values of k for which the argument λ α − λ shift α (k) is between 0 and 1. Since the nonperturbative parameter Ω 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 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 p jet T axis. The comparisons to the ungroomed predictions are shown in Fig. 7, and the groomed predictions are shown in Fig. 8. The shape function approach, specifically the scaling given in Eq. 6, 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 < p ch 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 < p ch jet T < 60 GeV/c and 80 < p ch jet T < 100 GeV/c are shown in Appendix A.

Discussion
The λ α distributions are generally consistent with the calculations within uncertainties when λ α is sufficiently large to be in the pQCD regime. This holds approximately independent of α, R, and p jet T , and whether or not the jets are groomed. In some distributions, however, particularly for R = 0.4, modest disagreement is observed at large λ α . This disagreement cannot be unambiguously associated with a particular value of λ α 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 λ α , 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 p T and R considered.
For α = 1.5, the majority of the distributions can be described perturbatively, as λ NP α is confined towards the left-hand side of the distributions. As α increases to α = 3, the influence of the λ NP α region grows, and the ungroomed distributions become strongly nonperturbative. Similarly, as R increases from R = 0.2 to R = 0.4, or as p ch jet T increases, the size of the perturbative region increases. In the nonperturbative region λ α < λ NP α , the λ α distributions diverge from the calculations. This is expected, since the perturbative approximations break down for λ α < λ NP 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 Ω.
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 [5][6][7][8]. On the other hand, grooming shifts the distributions themselves to significantly smaller values of λ α . 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.
Measurements of the jet angularities in pp collisions at √ s = 5.02 TeV ALICE Collaboration        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 Ω for the ungroomed case, where Eq. 6 is justified, there is in many cases only a small difference between the calculations with Ω = 0.2, 0.4, and 0.8 GeV/c. However, for α = 1.5 and α = 2, larger values of Ω (Ω = 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 λ NP 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 Ω or due to the breakdown of the perturbative calculation. For smaller values of Ω (e.g. Ω = 0.2 or 0.4 GeV/c), the predicted scaling of Eq. 6 is consistent with the data. Note that the value of Ω which describes the data is O(1) as expected for hadronization corrections. These smaller values contrast with a previous result of Ω = 3.5 GeV/c for the ungroomed mass of R = 0.4 jets at 200 < p jet T < 300 GeV/c [80], suggesting that the underlying event contribution to Ω, which is not expected to obey the scaling of Eq. 6, may be modified by jets measured at different p jet 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 = 0.2 and 0.4.

Conclusion
The generalized jet angularities are reported both with and without Soft Drop grooming, λ α,g and λ α , 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 in SCET show an overall agreement with the data for jets with values of λ α 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 Ω α with value Ω < 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 fragments with the hot and dense QCD medium. Our measurements demonstrate that any comparison to pQCD must also consider the regimes of λ α and λ α,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 p ch 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 predictions and data in the √ s = 5.02 TeV ALICE Collaboration 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 substructure 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 p ch jet T in order to maximize theoretical control and interpretation of the perturbative and nonpertubative regimes of jet substructure observables.

Acknowledgements
We gratefully acknowledge Kyle Lee and Felix Ringer for providing theoretical predictions, 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 acknowledges the following funding agencies for their support in building and running the ALICE detector:   [20] G. P. Korchemsky and G. Sterman, "Power corrections to event shapes and factorization", Nuclear Physics B 555 (Aug, 1999) 335-351, arXiv:hep-ph/9902341.              to analytical NLL predictions using F(k) convolution in the range 80 < p ch jet T < 100 GeV/c. The distributions are normalized such that the integral of the perturbative region defined by λ α,g > λ NP α,g (to the right of the dashed vertical line) is unity. Divided bins are placed into the left (NP) region.