Measurement of the angle between jet axes in pp collisions at $\sqrt{s} = 5.02$ TeV

This article reports measurements of the angle between differently defined jet axes in pp collisions at $\sqrt{s} = 5.02$ TeV carried out by the ALICE Collaboration. Charged particles at midrapidity are clustered into jets with resolution parameters $R=0.2$ and 0.4. The jet axis, before and after Soft Drop grooming, is compared to the jet axis from the Winner-Takes-All (WTA) recombination scheme. The angle between these axes, $\Delta R_{\mathrm{axis}}$, probes a wide phase space of the jet formation and evolution, ranging from the initial high-momentum-transfer scattering to the hadronization process. The $\Delta R_{\mathrm{axis}}$ observable is presented for $20<{p_{\mathrm{T}}^{\mathrm{ch\; jet}}}<100$ GeV/$c$, and compared to predictions from the PYTHIA 8 and Herwig 7 event generators. The distributions can also be calculated analytically with a leading hadronization correction related to the non-perturbative component of the Collins$-$Soper$-$Sterman (CSS) evolution kernel. Comparisons to analytical predictions at next-to-leading-logarithmic accuracy with leading hadronization correction implemented from experimental extractions of the CSS kernel in Drell$-$Yan measurements are presented. The analytical predictions describe the measured data within 20% in the perturbative regime, with surprising agreement in the non-perturbative regime as well. These results are compatible with the universality of the CSS kernel in the context of jet substructure.

< 100 GeV/c, and compared to predictions from the PYTHIA 8 and Herwig 7 event generators.The distributions can also be calculated analytically with a leading hadronization correction related to the non-perturbative component of the Collins-Soper-Sterman (CSS) evolution kernel.Comparisons to analytical predictions at next-to-leading-logarithmic accuracy with leading hadronization correction implemented from experimental extractions of the CSS kernel in Drell-Yan measurements are presented.The analytical predictions describe the measured data within 20% in the perturbative regime, with surprising agreement in the non-perturbative regime as well.These results are compatible with the universality of the CSS kernel in the context of jet substructure.

Introduction
Jets play a fundamental role in the study of quantum chromodynamics (QCD).Jets form when partons (quarks and gluons) scattered in high-momentum-transfer (hard) processes fragment into lower-energy (softer) partons.The fragmentation creates a shower of partons, until the average energy per particle falls below the scale at which color-neutral hadrons emerge.Jet substructure, which studies the radiation patterns inside jets, is a prolific field in both experiment [1] and theory [2].The large difference between the energy scale of the hard-scattered parton and the measured hadrons leaves a significant phase space for jet formation and evolution [3].Therefore, a multitude of jet-substructure measurements probing different regions of this phase space is needed to characterize the internal substructure of jets and advance our understanding of QCD.Numerous analyses have been carried out by the ALICE [4-12], ATLAS [13][14][15][16][17][18][19][20][21], CMS [22][23][24][25][26][27][28][29], and LHCb [30,31] Collaborations at the LHC, as well as at RHIC [32][33][34].
In this article, a novel jet-substructure observable proposed in Ref. 35 corresponding to the angle between two definitions of the axis of a jet: ∆R axis ≡ (y axis 1 − y axis 2 ) 2 + (ϕ axis 1 − ϕ axis 2 ) 2  (1) is studied.Here, a given axis corresponds to a set of coordinates in the rapidity (y) and azimuth (ϕ) plane.This is illustrated in Fig. 1.The "Standard" axis is determined by clustering the jet constituents with the anti-k T algorithm [36] and the E recombination scheme.Alternatively, the "Groomed" axis can be determined by first using a systematic procedure to remove the soft wide-angle radiation in the jet and then determining the axis of the anti-k T jet (with E recombination scheme), clustering only the constituents that remain after grooming.Given the removal of soft wide-angle radiation, this quantity is less sensitive to non-perturbative effects.The grooming procedure used in this analysis is reviewed below.
A third way to define the jet axis corresponds to reclustering the jet (initially clustered with the anti-k T algorithm and E recombination scheme) with the Cambridge-Aachen (C/A) algorithm [37,38] ensuring that the resulting jet includes all constituents from the original jet.The C/A algorithm clusters particles exclusively based on their spatial separation (in the y-ϕ plane) without taking into account their energies/momenta.Thus, particles closest in distance are clustered first, which results in an angular-ordered clustering sequence.Subsequently, the constituents are recombined with the Winner-Takes-All (WTA) transverse-momentum recombination scheme [39].This consists of going through the clustering his-

Collinear radiation Soft radiation
Groomed-away radiation WTA axis

Groomed axis
Standard axis

ΔR axis
Figure 1: Representation of different jet axes.The colored dashed lines represent particles comprising the jet in the initial sample constructed with the anti-k T algorithm and E recombination scheme.These define the "Standard" jet axis.The grooming procedure removes soft wide-angle radiation (represented by orange long-dashed lines), and the resulting axis ("Groomed (SD)") is defined by the remaining particles (red dashed and blue dotted lines).Finally, the "WTA" axis, which is determined with all the particles in the initial (ungroomed) jet, tends to be aligned with the most-energetic particle in the jet.The ∆R axis observable is determined from the angle between any pair of these axes.√ s = 5.02 TeV ALICE Collaboration tory and combining the pair of prongs in each 2 → 1 merging by assigning to the merged branch the direction of the harder of the two prongs and transverse momentum (p T ) corresponding to the sum of the two transverse momenta.The WTA scheme is infrared and collinear (IRC) safe and the resulting axis is insensitive to soft radiation at leading power of ∆R axis , which makes it amenable to perturbative calculations [35].
The ∆R axis observable is IRC safe and can be analytically calculated [35].The sensitivity of ∆R axis to non-perturbative effects can be controlled by changing the pair of axes that are used to determine the observable and also by varying the grooming and the jet resolution parameter.For instance, the difference between the Standard and Groomed axes specifically probes the influence of the groomed soft wide-angle radiation on the jet direction.Similarly, the WTA axis constitutes a robust reference against soft-radiation effects.The angle between the WTA and Standard/SD axes offers a proxy for studying how the fragmenting parton that initiated the jet gets distributed inside the jet cone.Consequently, this observable can be used both to test the performance of analytic predictions and to constrain nonperturbative models used in event generators.
The ∆R axis is sensitive to Transverse-Momentum-Dependent (TMD) physics [35].The leading hadronization correction for ∆R axis is related to the non-perturbative component of the Collins-Soper-Sterman (CSS) evolution kernel or rapidity anomalous dimension.Here, leading means logarithmically enhanced in the calculation of ∆R axis , in contrast to other nonperturbative components that are not logarithmically enhanced and thus expected to be small.The CSS kernel is a process-independent non-perturbative function constrained from fits to measured data that governs the non-perturbative evolution in rapidity and encodes information about soft-gluon exchanges between partons in vacuum [40].As such, extractions from diverse physics processes such as Drell-Yan and semi-inclusive deep-inelastic scattering [41][42][43][44] should adequately describe the hadronization corrections for the jet substructure observable presented in this study.This measurement can be included in future global fits to further constrain the CSS kernel.Furthermore, it has been proposed that the CSS kernel can be determined from lattice QCD [45,46], and such results can be benchmarked with our measurements.
The measurements presented here can also serve as a reference for measurements in heavy-ion collisions.
The internal structure of jets fragmenting in the strongly interacting, deconfined state of matter formed in heavy-ion collisions is modified relative to jet fragmentation in vacuum [47,48].Comparing to pp collisions allows us to study the medium properties.
In this analysis, the Soft Drop grooming procedure [49] is used to determine the Groomed axis (referred to as "SD" axis from here on).The jet is first reclustered with the C/A algorithm, with the same resolution parameter used in the original clustering of the anti-k T -based sample.This tree is then recursively declustered starting from the largest-distance splitting and at each 1 → 2 splitting, the condition is checked.Here, p T,1 and p T,2 are the transverse momenta of each prong, and ∆R 1,2 is the angular distance between them, calculated as in Eq. 1.The jet resolution parameter is denoted by R. The free parameters z cut and β determine how asymmetric and how wide the splitting can be, respectively.If the condition is not satisfied at a given splitting, the softer branch is removed from the tree and the procedure continues through the harder prong.When a splitting that satisfies the condition is found, the Soft Drop procedure stops and the branch parent to the splitting defines the groomed jet.If there is not a single splitting that satisfies the Soft Drop condition, then the grooming procedure fails.These jets are labeled "untagged" and their abundance is used in the normalization convention used in this analysis.A scan in Soft Drop parameters is carried out, both fixing z cut = 0.1 and varying β from 0 to 3 in increments of 1 and fixing β = 1 and varying z cut from 0.1 to 0.3 in increments of 0.1.
The jet reconstruction performance was estimated using PYTHIA 8 [60] with Monash 2013 tune [61] events ("truth" level) propagated through a GEANT 3 [62] model of the ALICE detector ("detector" level).In this sample, final-state particles are defined as those with a mean proper lifetime cτ > 1 cm [63].Particles (tracks) at truth (detector) level were individually clustered into jets, and the jets were matched between the two populations by requiring that the distance between their Standard axes in the y-ϕ plane satisfied ∆R < 0.6R and that the match was unique [54].  1 shows values characterizing the jet-reconstruction performance.In the case of the JES, this distribution is peaked at 0, but has an asymmetric tail that shifts the mean value, ∆ JES , due to tracking inefficiencies.The unfolding procedure described in the next section corrects for these JES and JER effects.

Corrections
To obtain distributions free of detector effects (truth level), the measured spectra (detector level) were unfolded using an iterative procedure based on Bayes' theorem [64,65], implemented in the RooUnfold package [66].This unfolding procedure accounts for effects such as track p T resolution, tracking inefficiencies, and particle interactions in the detector volume.The input to this procedure consisted of a 4-dimensional response matrix (RM) that maps the correspondence between detector and truth levels for p ch jet T and ∆R axis .The RM was created using the simulated jets described in the previous section.The p ch jet T,det axes were constructed in the range [10, 130] GeV/c to capture bin-migration effects.In the ∆R axis cases in which one of the two axes is the SD axis, the untagged jets were included in the unfolding procedure.The unfolding iterations were fixed at the number that minimizes the quadratic sum of the statistical and systematic uncertainties.In some cases, a few more iterations were carried out until a convergence within 5% between subsequent iterations was achieved.
The unfolding procedure was validated by performing a series of closure tests.A refolding test was performed to check that the detector-level spectrum can be recovered by reversing the unfolding procedure.In this test, the unfolded result was multiplied by the response matrix and the resulting spectrum was compared to the detector-level spectrum.Additionally, a statistical-closure test was performed to confirm that the unfolding procedure is robust against statistical fluctuations in the data.In this test, the simulated detector-level spectrum was smeared by an amount equal to the statistical uncertainty of the measured data.Subsequently, the smeared spectrum was unfolded, and the agreement between the resulting distribution and the truth-level simulated distribution was assessed.Finally, a shape-closure test was performed to account for the fact that the true detector-level distribution may be different than that from the generator, and to evaluate the robustness against this shape.In this test, the shapes of the simulated detector-and truth-level spectra were scaled by the prior-scaling function described in the systematic-uncertainty section.Subsequently, the scaled detector-level spectrum was unfolded, and the resulting spectrum was compared to the scaled truth-level distribution.In all these tests, closure within the statistical uncertainties was achieved.
Measurement of the angle between jet axes in pp collisions at √ s = 5.02 TeV ALICE Collaboration

Systematic uncertainties
Three sources of systematic uncertainties are considered in this analysis: those arising from the unfolding procedure, uncertainties in the tracking efficiency, and the choice of event generator used to produce the response matrix.
The unfolding uncertainty is estimated by performing four variations of the unfolding procedure: 1.The analysis is repeated with a number of iterations varied by ±2 around the nominal value and the average difference with respect to the nominal spectrum is taken as an uncertainty.
2. The analysis is repeated with a prior distribution multiplied by (p , where the parameters A and B are selected for each ∆R axis variation such that the endpoints of the spectrum are scaled by ≈ ±20%.The maximum difference between these two variations and the nominal result is taken as an uncertainty.4. The analysis is repeated with a different ∆R axis binning at detector level (slightly finer or coarser granularity) and the difference with respect to the main result is taken as an uncertainty.
Given that these four variations probe the same underlying source of uncertainty, the total unfolding uncertainty is defined as the standard deviation σ unfolding ≡ ∑ 4 i=1 σ 2 i /4, where σ i corresponds to the uncertainty from the individual variations.
The uncertainty on the ALICE tracking efficiency is 3-4%, determined by varying the track selection parameters and possible imperfections in the description of the ITS-TPC matching efficiency in the simulation.Consequently, the analysis was repeated with a RM populated with jets clustered over a track sample with 4% of all tracks randomly discarded, taking the more conservative value.Differences with respect to the nominal analysis were assigned as the systematic uncertainty due to the tracking efficiency.
To assess the model dependence of the analysis on the generated spectra used to compute the RM (based on PYTHIA 8 with Monash 2013 tune events propagated through GEANT 3), the analysis was repeated using RMs based on Herwig 7 (default tune) and PYTHIA 8 with Monash 2013 tune constructed using a fast simulation, and the difference between the two unfolded results was assigned as a systematic uncertainty.In a comparison based on PYTHIA 8 events, this fast simulation agrees with the GEANT 3based simulation within 10%.
The total systematic uncertainty is taken as the sum in quadrature of the contributions due to the unfolding, tracking efficiency, and choice of event generator.Table 2 summarizes the range of the systematic uncertainties in different ∆R axis intervals for jets of R = 0.2 and 0.4 in the transverse momentum range 40 < p ch jet T < 60 GeV/c.Most often, the dominant systematic uncertainty originates from the uncertainty on the tracking efficiency.

Results and discussion
The ∆R axis distributions are reported as normalized differential cross sections: Measurement of the angle between jet axes in pp collisions at √ s = 5.02 TeV ALICE Collaboration < 60 GeV/c.The unfolding, tracking efficiency, and generator systematic uncertainties can be found in the columns labeled Unf., Trk.Eff., and Gen., respectively.In the case of the groomed observables, the grooming parameters are specified as (z cut , β ).The displayed uncertainties correspond to the lowest and highest values for a given setting.
for jets of R = 0. ) is the number of inclusive jets in a given p ch jet T interval.In the groomed cases, the normalization factor N jets is obtained by including the number of untagged jets (i.e.without any splitting in the jet that satisfies the SD condition, so that the grooming process fails) in the unfolding procedure.Therefore, the number of jets appearing in the ∆R axis distributions differs from N jets by the number of jets that are untagged.Sections 5.1 and 5.2 present the experimental results as well as comparisons to predictions from MC generators and analytical calculations, respectively.).The bottom two panels in these figures correspond to the ratios of data to PYTHIA 8 and Herwig 7 distributions.The left panel of Fig. 2 shows the case in which z cut is fixed at 0.1 and β is varied from 0 to 3. In the right panel, β is fixed at 1 and z cut is varied from 0.1 to 0.3.These spectra are plotted with a logarithmic y-axis to better exhibit the entire distribution.Figure 3 shows the equivalent comparison for the case of the WTA-Standard and WTA-SD distributions.

Comparison to MC generators
Overall, the Standard-SD distributions are narrow and peaked at very small values.This implies that the Standard and SD axes are aligned and grooming does not significantly impact the jet direction.However, as the grooming becomes more aggressive (i.e. higher z cut or smaller β ), the alignment between the Standard and SD axes worsens somewhat.This trend is present for both R = 0.2 and R = 0.4.There is a maximum at ∆R Standard−SD axis = 0 that corresponds to jets for which the first splitting after reclustering the jet with the C/A algorithm already satisfies the SD condition.As a result, the difference with respect to the Standard axis is exactly 0.
The shape of the ∆R Standard−SD axis spectra is better described by Herwig 7 than by PYTHIA 8.In the          < 60 GeV/c range, bin-by-bin deviations with respect to PYTHIA 8 (Herwig 7) reach values up to ≈ 24% (≈ 7%).Given that this observable is particularly sensitive to soft-radiation effects, our data can be used to further constrain the hadronization models in these generators.
The ∆R axis distributions for the WTA-Standard and WTA-SD cases are broader (0 < ∆R axis ≲ R/2) and peak at larger ∆R axis , showing substantial deviation between the WTA and Standard/SD jet axes.Additionally, these distributions show very low sensitivity to the parameters chosen in the Soft Drop grooming procedure.for R = 0.2 (0.4).

Comparison to analytical calculations
The ∆R axis observable has been calculated in the Soft Collinear Effective Theory (SCET) framework [67].In this framework, the jet-production cross section is factorized into parton distribution functions (PDF), "hard", and "jet" contributions in order to separate physics processes at different scales.The PDFs encode the probability of finding a parton with a given flavor and momentum fraction from a proton and are non-perturbative objects extracted from global fits to measured data (see, e.g.Refs.68,69).The SCET calculations presented here use the CT14 NLO PDF set [70].The hard contribution encodes the shortdistance physics, i.e. the hard scattering of one parton from each colliding proton, and the distribution of the resulting partons.Finally, the jet function describes the evolution of a final-state parton from the Measurement of the angle between jet axes in pp collisions at √ s = 5.02 TeV ALICE Collaboration hard scattering into a collimated jet.Large logarithms are resummed to Next-to-Leading Logarithmic (NLL) accuracy for the Standard-SD ∆R axis , and NLL ′ for the WTA-Standard and WTA-SD ∆R axis , including the contribution from non-global logarithms [71].NLL ′ refers to the inclusion in the resummation of terms that formally only enter at Next-to-Next-to-Leading Logarithmic (NNLL) accuracy, but their inclusion in the NLL calculation improves the theoretical uncertainty.Specifically, logarithms of the grooming parameter z cut and the observable ∆R axis are resummed.Logarithms of the jet resolution parameter R are also resummed, which makes the calculations valid down to arbitrarily low values of R and allows for tests of such resummations.The resummed result is presented without matching to the fixed-order calculation in the high-∆R axis region.
Unlike other TMD-sensitive jet substructure observables, such as hadron-in-jet fragmentation [72,73], ∆R axis does not depend on collinear fragmentation functions.Thus, a significant source of uncertainty is removed from these calculations.
The leading hadronization correction for these observables includes terms of the form exp , where b ⊥ is the Fourier conjugate of k T , the projection of the transverse momentum of a jet axis transverse to the other axis in ∆R axis .The function g K (b ⊥ ; b max ⊥ ) is the non-perturbative component of the Collins-Soper-Sterman (CSS) evolution kernel or rapidity anomalous dimension.In the so-called b * prescription [74,75] it is often parametrized as In this work, the universality of the CSS kernel is tested by verifying that the ∆R axis observables are well described across two resolution parameters R, and several grooming settings and p 18 from a global data analysis to Drell-Yan lepton pair and Z 0 -boson production [76].
The analytic predictions are provided by the authors of Ref. 35 for the kinematics of our measurement at hadron level (i.e. after hadronization, in contrast to parton level) for full jets (i.e.including both charged and neutral hadrons), and without including multi-parton interaction (MPI) effects.To do a comparison to the measured distributions, the analytic predictions are corrected using data from Monte Carlo event generators.Final-state particles from pp events are clustered into full and charged-particle jets following the same procedure as in the data analysis (i.e. using the anti-k T algorithm with the E recombination scheme for a given R), and the resulting jets are required to satisfy |η jet | < 0.9 − R and p jet T > 5 GeV/c.The full and charged-particle jets are then geometrically matched following the matching procedure of the RM from the data analysis.This sample is used to construct a 4D response matrix that maps the p jet T and ∆R axis dependence from full-to charged-particle-jet levels for each observable.
The analytic predictions are provided as normalized densities (1/σ jet )(dσ jet /d∆R axis ) for the ∆R axis observable in different p jet T intervals (of 5 GeV/c width) for the nominal case, as well as for the results obtained by systematically varying the scales that appear in the calculation to account for theoretical uncertainties [35].The first step in the correction corresponds to multiplying the spectrum by the average value of the inclusive cross section in the considered p jet T interval to obtain the distribution for dσ jet /d∆R axis .The cross section used was calculated at Next-to-Leading Order (NLO) with resummation of logarithms of the jet radius at NLL [77].The scaled distributions are stored in 2D histograms in p T and ∆R axis and multiplied by the 4D response matrices described in the previous paragraph to obtain the analytic predictions for charged-particle jets.Subsequently, the resulting 2D histograms are corrected to account for MPI effects by multiplying bin-by-bin by the ratio of the simulated distribution without and with MPI effects included.The result is projected onto the observable axis for a range in p ch jet T equal to that of the measured distribution.The final theory prediction corresponds to the curve obtained from the nominal calculation.Additionally, the other results obtained from the systematic scale variations are equally corrected, and the envelope of all resulting distributions is taken as the theoretical uncertainty on Measurement of the angle between jet axes in pp collisions at √ s = 5.02 TeV ALICE Collaboration the calculations.
The use of a Monte Carlo event generator in these corrections introduces a model dependence, the significance of which is explored by applying the correction procedure with two different generators: PYTHIA 8 with the Monash 2013 tune [61] and Herwig 7 with the default tune [78].
Figure 5 shows a subset of the comparison between the measured distributions and the corrected analytic predictions.Equivalent comparisons for the rest of all available predictions are presented in Appendix B.
The black markers correspond to the distributions determined from measured data.These distributions are identical to those in Fig. 3 up to a normalization factor defined below.The vertical error bars correspond to the statistical uncertainties, and the rectangles correspond to the total systematic uncertainties.
The colored curves correspond to the SCET-based analytic predictions corrected for charge and MPI effects using two event generators (PYTHIA 8 and Herwig 7).
Differences between the SCET predictions corrected with either Monte Carlo event generator are very small.This is due to the fact that, since the input calculations are provided at hadron level, the most significant correction is done to the p T scale of the jet, and this correction is well modeled by both generators.Thus, the resulting distributions are not significantly model dependent.
The analytic calculations are only expected to describe the measured distributions in the perturbative region.The predictions presented here become non-perturbative approximately at ∆R axis ≲ ∆R NP axis = Λ/p ch jet T [35], where Λ corresponds to the scale at which the strong coupling constant becomes nonperturbative.The red vertical line in Fig. 5 corresponds to this value calculated using Λ = 1 GeV and p interval was determined by fitting the measured spectra with a power law and subsequently analytically calculating the mean value in the interval.The comparison is done by normalizing both the measured distributions and analytic predictions so that R/2 ∆R NP axis d∆R axis (dσ /d∆R axis ) = 1.The ∆R axis interval that contains the value ∆R NP axis is defined to belong to the non-perturbative region and excluded from the integral.In reality, there is no sharp boundary between the perturbative and non-perturbative regimes, but a continuous transition.This vertical line is therefore an indicative value defined for the sake of the comparison.
The Standard-SD variable is also IRC safe and therefore calculable in the SCET framework.However, these calculations are computationally expensive and the results are not available at the time of this article.These distributions are particularly sensitive to soft effects, and thus also to the hadronization corrections.

Discussion of analytic calculations
The analytic predictions describe the measured distributions in the perturbative regime within uncertainties for all variations of the observable (i.e.jet-resolution parameter, grooming setting and p ch jet T ) considered in the analysis.The agreement is excellent (largest deviations observed are within 10 to 20% in the lower p T bins, where the smaller systematic uncertainties allow us to draw more precise conclusions) given the low p ch jet T values from this measurement.Additionally, even though the distributions are normalized to their integral in the perturbative region, significant agreement is also found in the nonperturbative region, where the calculations are most sensitive to the non-perturbative correction.This agreement persists independently of jet-resolution parameter, grooming setting, and p ch jet T .There are larger shape differences between the data and analytic predictions for larger R.
The agreement at lower ∆R axis values is increasingly due to the inclusion of the non-perturbative correction from Eq. 4. The measured distributions are compatible with the universal behavior of the nonperturbative component of the CSS evolution kernel to the extent this hadronization correction is well modeled.This is the first time this has been verified in jet substructure observables.The measurements The theoretical uncertainties in the analytic predictions are significant, and reducing them can lead to a higher-precision test.Specifically, as shown in the ratio plots from Fig. 5, while the measured distributions and the calculations agree within uncertainties, there are systematic shape differences between them.For instance, there is a sudden rise in the interval at the highest ∆R axis which can be related to the lack of matching to a fixed-order calculation or power corrections in the observable in the SCET prediction.However, because the distributions are self-normalized, the disagreement cannot be pinned to a specific ∆R axis value.

Conclusions
The first measurement of the angle between the Standard, Soft Drop groomed, and Winner-Take-All charged-jet axes is reported.The measurement was carried out in pp collisions at √ s = 5.02 TeV with the ALICE detector for jets of R = 0.2 and 0.4.This analysis focused on jets below 100 GeV/c, where non-perturbative effects are more important than in highly energetic jets.Jet grooming does not substantially change the jet axis direction.The WTA and Standard jet axes differ substantially, though, showing that the Standard jet axis does not generally point in the direction of the highest-momentum jet fragment.However, the WTA and Standard or SD axes become more aligned in more energetic jets.The motivation to use the WTA axis arises from its insensitivity to soft radiation; it is amenable to perturbative calculations and is expected to be less modified in the medium created in ultrarelativistic nucleus-nucleus collisions.
The distributions from PYTHIA 8 and Herwig 7 show overall good agreement with the data; both groomed and ungroomed jet axis differences with WTA are generally described within the uncertainties.The Standard-SD jet axis difference is better described by Herwig 7 than by PYTHIA 8, suggesting that Herwig better reproduces the soft splittings which are groomed away by Soft Drop.There is also good agreement between the analytic predictions and the measured spectra, even in the region of ∆R axis where non-perturbative physics is important.Within the precision of the measurement and analytic calculations, our result is compatible with the universality of the non-perturbative component of the Collins-Soper-Sterman (CSS) TMD evolution kernel.This study constitutes the first experimental verification of such compatibility in the context of jet-substructure measurements.
These measurements shed light on the interplay between perturbative and non-perturbative effects.The majority of jets produced in pp collisions at this center-of-mass energy are initiated by gluons.It will be illuminating to compare these distributions to those from quark-initiated jets.This can be done by measuring ∆R axis in, for example, heavy-flavor jets or photon-jet coincidences.
The angle between different jet axes may also reflect medium modification of jet angular substructure in heavy-ion collisions.The results presented here will provide a valuable baseline for Pb-Pb studies.[8] ALICE Collaboration, S. Acharya

3 .
The analysis is repeated with the transverse-momentum range p ch jet T,det ∈ [5, 135] GeV/c and the difference with respect to the nominal analysis (p ch jet T,det ∈ [10, 130] GeV/c) is taken as an uncertainty.
2 and 0.4 in 20-GeV/c-wide p ch jet T intervals in the 20-100 GeV/c range.Here, N jets (p ch jet T

Figure 2
Figure 2 compares the measured Standard-SD distributions with predictions from Monte Carlo event generators, for jets of R = 0.4 (top) and 0.2 (bottom) in 40 < p ch jet T < 60 GeV/c.Results for different p ch jet T intervals are presented in Appendix A. The vertical error bars are statistical uncertainties, and the rectangles indicate the total systematic uncertainties.The solid (dashed) lines show results from PYTHIA 8 (Herwig 7).The bottom two panels in these figures correspond to the ratios of data to PYTHIA 8 and Herwig 7 distributions.The left panel of Fig.2shows the case in which z cut is fixed at 0.1 and β is varied from 0 to 3. In the right panel, β is fixed at 1 and z cut is varied from 0.1 to 0.3.These spectra are plotted with a logarithmic y-axis to better exhibit the entire distribution.Figure3shows the equivalent comparison for the case of the WTA-Standard and WTA-SD distributions.

Figure 2 :
Figure 2: Comparison between the ∆R axis Standard-SD measured distributions and Monte Carlo event generators for jets of R = 0.4 (top) and 0.2 (bottom) in 40 < p ch jet T < 60 GeV/c.Left: distributions with z cut = 0.1 and varying

Figure 3 :Figure 4 :
Figure 3: Comparison between the ∆R axis WTA-Standard and WTA-SD measured distributions and Monte Carlo event generators for jets of R = 0.4 (top) and 0.2 (bottom) in 40 < p ch jet T < 60 GeV/c.Left: distributions with z cut = 0.1 and varying β .Right: distributions with β = 1 and varying z cut .

The
∆R WTA−Standard axis and ∆R WTA−SD axis predictions from PYTHIA 8 and Herwig 7 generally agree with the data, deviating by up to ≈ 10% in the 40 < p ch jet T < 60 GeV/c interval and up to ≈ 40% in several other p ch jet Tintervals.The largest deviations are at low values of ∆R axis , corresponding to the region in which non-perturbative effects are significant.It should be noted, though, that the normalization convention implies that conclusions can only be made about the overall shape and not about a discrepancy in a specific ∆R axis interval.The axis differences with respect to WTA shift to lower ∆R axis values at higher p ch jet T , implying that the WTA and Standard or SD axes are more aligned in more energetic jets.This is summarized in Fig.4, where the most-probable value (MPV) of the ∆R WTA−Standard axis distribution normalized to the jet resolution parameter R is shown as a function of p ch jet T .The values in this figure are determined by repeatedly fitting a Landau distribution to the ∆R WTA−Standard axis spectra changing each time the fit range.A histogram is then filled with the peak-position values extracted from each fit.From this histogram, the mean and standard deviation are extracted as the central value and uncertainty, respectively.The values decrease from ≈ 11% (8.5%) at low p ch jet T to ≈ 5% (4%) at high p ch jet T jet-transverse-momentum value in the interval.The p ch jet T value for a given p ch jet T

Figure 5 :
Figure 5: Comparison between measured distributions and analytic predictions for the difference between the WTA axis and the Standard (left), SD (z cut = 0.2, β = 1) (center), and SD (z cut = 0.3, β = 1) (right) axes for jets of R = 0.4 (top) and 0.2 (bottom) in the transverse momentum range 40 < p ch jet T < 60 GeV/c.The black markers correspond to the distributions determined from measured data.The vertical error bars correspond to the statistical uncertainties, and the rectangles correspond to the total systematic uncertainties.The colored curves correspond to the SCET-based analytic predictions corrected for charge and MPI effects using two event generators (PYTHIA 8 and Herwig 7).The vertical dashed line defines the approximate boundary between the nonperturbative and perturbative regions.Both the measured and analytic predictions are normalized so that the integral R/2 ∆R NP axis

Figure A. 1 :Figure A. 2 :Figure A. 3 :Figure A. 4 :Figure A. 5 :
Figure A.1: Comparison between ∆R axis WTA-Standard and WTA-SD measured distributions Monte Carlo event generators for jets of R = 0.4 in 20 < p jet T < 40 GeV/c.Left: distributions with z cut = 0.1 and varying β .Right: distributions with β = 1 and varying z cut .

Table 1 :
Approximate values for the figures of merit characterizing the jet-reconstruction performance in this analysis.∆ JES is the mean value of the JES spectrum.See text for details.