Measurement of Groomed Jet Substructure Observables in \pp Collisions at $\sqrt{s} = 200$ GeV with STAR

In this letter, a comprehensive suite of jet substructure measurements via the SoftDrop algorithm, including the shared momentum fraction ($z_{\rm{g}}$) and the groomed jet radius ($R_{\rm{g}}$), are reported in \pp collisions at $\sqrt{s} = 200$ GeV collected by the STAR experiment. These substructure observables are differentially measured for jets of varying resolution parameters from $R = 0.2$ to $R = 0.6$ and transverse momentum range $15<p_{\rm{T, jet}}<60$ GeV$/c$. These studies show that, at RHIC kinematics with increasing jet resolution parameter and jet energy, the $z_{\rm{g}}$ distribution asymptotically converges to the DGLAP splitting kernel. The groomed jet radius measurements reflect a momentum-dependent narrowing of the jet structure for jets of a given resolution parameter, i.e., the larger the $p_{\rm{T, jet}}$, the narrower the first split. For the first time, these fully corrected measurements are compared to leading order Monte Carlo generators and to state-of-the-art theoretical calculations at next-to-leading-log accuracy. We observe that RHIC-tuned PYTHIA 6 is able to quantitatively reproduce data whereas the LHC-tuned event generators, PYTHIA 8 and HERWIG 7, are unable to provide a simultaneous description of both the $z_{\rm{g}}$ and $R_{\rm{g}}$, resulting in opportunities for fine parameter tuning of these models in \pp collisions at varying collision energies. We also find that the theoretical calculations without non-perturbative corrections are able to qualitatively describe the trend in data for jets of large resolution parameters at high $p_{\rm{T, jet}}$, but fail at small jet resolution parameters and low jet momenta.


Abstract
In this letter, a comprehensive suite of jet substructure measurements via the Soft-Drop algorithm, including the shared momentum fraction (z g ) and the groomed jet radius (R g ), are reported in p+p collisions at √ s = 200 GeV collected by the STAR experiment. These substructure observables are differentially measured for jets of varying resolution parameters from R = 0.2 to R = 0.6 and transverse momentum range 15 < p T,jet < 60 GeV/c. These studies show that, at RHIC kinematics with increasing jet resolution parameter and jet energy, the z g distribution asymptotically converges to the DGLAP splitting kernel. The groomed jet radius measurements reflect a momentum-dependent narrowing of the jet structure for jets of a given resolution parameter, i.e., the larger the p T,jet , the narrower the first split. For the first time, these fully corrected measurements are compared to leading order Monte Carlo generators and to state-of-the-art theoretical calculations at next-to-leading-log accuracy. We observe that RHIC-tuned PYTHIA 6 is able to quantitatively reproduce data whereas the LHC-tuned event generators, PYTHIA 8 and HERWIG 7, are unable to provide a simultaneous description of both the z g and R g , resulting in opportunities for fine parameter tuning of these models in p+p collisions at varying collision energies. We also find that the theoretical calculations without non-perturbative corrections are able to qualitatively describe the trend in data for jets of large resolution parameters at high p T,jet , but fail at small jet resolution parameters and low jet momenta.

Introduction
Jets are well-established signals of partons, i.e., quarks and gluons, created in the hard scatterings during high energy hadron collisions [1]. Jets have played a prominent role as an internal probe of partonic energy loss mechanisms in the quark-gluon plasma created in heavy-ion collisions. Refer to [2] and [3] for recent reviews of the experimental measurements and theoretical calculations on jet quenching. An important prerequisite of such studies is a quantitative understanding of jet properties related to its production, evolution and hadronization. The production of hard scattered partons is governed by 2 → 2 quantum chromodynamics (QCD) scattering at leading order (LO) and 2 → 3 at next-to-leading order (NLO), and is calculable using Parton Distribution Functions (PDFs) [4], which are extracted with fits to experimental measurements, including but not limited to jet cross-sections at various kinematics. Given a hard scattered parton, the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) splitting kernels [5,6,7] describe its evolution and fragmentation based on perturbative quantum chromodynamics (pQCD). At LO, the DGLAP splitting functions of a parton in vacuum are dependent on the momentum fraction of the radiated gluon and the corresponding angle of emission. The most efficient way for a highly virtual/off-shell parton to lose its virtuality is via consecutive radiation/splitting (for example q → q+g), resulting in a parton shower. Due to the double logarithmic structure of the splitting kernels and color coherence in the QCD, the evolution is expected to follow an angular or virtuality ordered shower. Such an ordering implies that the earliest splits are soft and wide in angle with the harder (referring to a high momentum radiated gluon) collinear splits happening later during jet evolution. Therefore, this process can be described by two natural scales: the split's momentum fraction and its angle with respect to the parton direction which, in turn, describe jet structure in vacuum. The primary focus of this letter is to study QCD and parton evolution in p+p collisions at RHIC. We establish a quantitative description of jet substructure that can serve as a reference for comparison to similar measurements in heavy-ion collisions where jet properties are expected to be modified due to jet quenching effects.
In this letter, we present fully corrected measurements of the SoftDrop groomed momentum fraction (z g ) and the groomed jet radius (R g ) in p+p collisions at centerof-mass energy √ s = 200 GeV. They allow a direct measurement of the DGLAP splitting functions during jet evolution. These measurements emerge as a "by-product" of the modified mass drop tagger or SoftDrop [8,9,10] grooming algorithm, used to remove soft, wide-angle radiation from sequentially clustered jets. This is achieved by recursively de-clustering the jet's angular-ordered branching history via the Cambridge/Aachen (C/A) clustering algorithm [11,12], which sequentially combines nearest constituents, i.e., those located closest in angle. Subjets are discarded until the transverse momenta, p T,1 and p T,2 , of the subjets from the current splitting fulfill the where R g is the groomed jet radius or a measure of the distance as defined in pseudorapidity-azimuthal angle (η − φ) space between the two surviving subjets and R is the jet resolution parameter. This analysis sets β = 0 and a momentum fraction cut of z cut = 0.1 [9] to determine if a subjet at a given clustering step survives the grooming procedure. The z cut parameter is set to reduce sensitivity to non-perturbative effects arising from the underlying event and hadronization [9,13]. It has been shown that for such a choice of z cut and β, along with the usage of the C/A algorithm for de-clustering, the distribution of the resulting z g converges to the vacuum DGLAP splitting functions for z > z cut in a "Sudakov-safe" manner [10], i.e., independent of the strong coupling constant (α s ) in the ultraviolet (UV) limit and under the fixed coupling approximation. Since the splitting kernels are defined to be independent of the momenta of initial partons, the UV limit corresponds to a jet of infinite momentum.
The SoftDrop z g was first measured by the CMS collaboration in p+p and Pb+Pb collisions at √ s NN = 5.02 TeV at the LHC for highly energetic jets with p T,jet > 140 GeV/c [14]. As the measurements are not corrected for smearing due to detector effects and resolution in Pb+Pb, the Monte Carlo (MC) generators, such as PYTHIA 6 [15], PYTHIA 8 [16] and HERWIG++ [17,18], are smeared instead to make meaningful comparisons. Due to the granularity of the CMS calorimeter, a R g > 0.1 threshold was enforced which consequently introduced a bias towards wider jets in the study [19]. It was shown that event generators at the LHC generally reproduce the trend in p+p collisions, but individually, neither PYTHIA 8 nor HERWIG 7 were able to quantitatively describe the measurements within systematic uncertainties. The large center-of-mass energies at the LHC increases NLO effects in jet production and fragmentation along with an increased sensitivity to multi-parton interactions and pileup. On the other hand, due to their large jet p T , the measurements are less sensitive to the hadronization process and higher-order power corrections [20,21] due to a small α s .
The p+p collisions at RHIC provide a complementary environment to study the jet structure and parton evolution. Due to the reduced center-of-mass energy (200 GeV as compared to 5.02 TeV), the study offers further insights regarding jet evolution by exploring different contributions of NLO effects and hadronization. For example, the higher-order effects in jet production at RHIC are suppressed compared to the LHC, while jets at RHIC are more susceptible to non-perturbative effects such as multi-parton interactions, the underlying event and hadronization effects by virtue of their kinematics at lower energies. Some of these effects are negated by the SoftDrop grooming procedure [20]. Jets used in this analysis are minimally biased since no additional selections are applied to the angular threshold. The measurements are fully corrected for detector response via a two-dimensional unfolding procedure. Thus in this letter, for the first time we present fully corrected jet substructure measurements at RHIC that are complementary to the LHC measurements. Additionally, they serve as a crucial baseline for tuning event generators, validating state-of-the-art theoretical calculations of jet functions, and for using similar measurements in heavy-ion collisions to extract medium-modified parton dynamics.

Experimental Setup and Jet Reconstruction
The data analyzed in this letter were collected by the STAR experiment [22] in jets that pass the SoftDrop criteria are then considered for the study.

Detector Simulation and Unfolding
In order to study the response of the STAR detector to jet substructure observables, p+p events at √ s = 200 GeV are generated using the PYTHIA 6.4.28 [15] event generator with the Perugia 2012 tune and CTEQ6L PDFs [26]. The PYTHIA 6 used in this analysis was further tuned to match the underlying event characteristics as measured by STAR in a recent publication [27]. These generated events are then passed through a GEANT3 [28] simulation of the STAR detector and embedded into zero-bias data from the same p+p run period. With the GEANT simulated PYTHIA 6 events, identical analysis procedures including event and jet selection criteria mentioned in Sect. 2 are implemented. Jets that are found from PYTHIA 6 simulations before and after the embedding procedures are hereafter referred to as particle-level and detector-level jets, respectively. The long-lived weak-decaying particles, which are not included in the jet finding at the particle level, are simulated in the event generation, and their decay prod- ucts are included in the detector-level jets as in real data analysis. The STAR detector response to a jet is estimated by comparing the properties of a PYTHIA 6 particle-level jet with its geometrically matched detector-level jet based on the following matching criterion, (∆η) 2 + (∆φ) 2 < R, where the ∆ refers to the difference between the detector-and particle-level jets in the same event and R is the jet resolution parameter.
With our jet quality selections, we have about 2% of detector-level jets with p det T,jet > 15 GeV/c that cannot be matched to particle-level jets. On the other hand, the jet finding efficiency for particle-level jets varies within 80-94% for 15 < p part T,jet < 60 GeV/c. The two dimensional p T,jet response matrix for R = 0.4 jets is shown in Fig. 1, in which the filled markers represent the average detector-level p det T,jet for a given particle-level p part T,jet . In comparison to the dashed diagonal line in Fig. 1, we find the mean p det T,jet to be smaller than the corresponding p part T,jet primarily due to tracking inefficiency. For the jet substructure observables, the detector response is shown in Fig. 2, plotted as the ratio of detector-level jet quantity to the matched particle-level jet quantity for a variety of p det for detector effects via a two-dimensional (e.g., p T,jet and z g ) unfolding procedure.  Fig. 3 for z g on the left and R g on the right. The bottom panels show the ratio of simulation to data where we observe a good agreement. In comparing the particle-level and detector-level PYTHIA 6 distributions, we see small but statistically significant differences due to the detector response which we correct for via an unfolding method described below.
The SoftDrop z g and R g distributions in this analysis are unfolded to the particle level to correct for detector effects including smearing and bin-by-bin migration. The fact that the detector response peaks at unity and is independent of p T,jet , as shown in Fig. 2, generates a more diagonal unfolding matrix in 4 dimensions (i.e., detector-and particle-level p T,jet and z g or p T,jet and R g ). Two-dimensional Bayesian unfolding [29] is done using the tools available in the RooUnfold package [30] with four iterations to take into account non-diagonal bin-to-bin migrations both in jet p T and SoftDrop observables. As a consequence of the detector simulation reproducing the uncorrected data as shown in Fig. 3, the unfolding procedure converges and is numerically stable.
The priors in the unfolding procedure are taken from the PYTHIA 6 simulation and their variations are studied as a source of systematic uncertainty.

Systematic uncertainties
There are two main categories of systematic uncertainties considered in this analysis. The first is related to the reconstruction performance of the STAR detector, including the uncertainty on the tower gain calibration (3.8%) and the absolute tracking efficiency (4%). The other source of systematic uncertainty is due to the analysis procedure, i.e., the use of hadronic correction (as described in Sec. 2) and the unfolding procedure. The correction to the tower energy, based on the matched tracks' momenta, is varied by subtracting half of the matched tracks' momenta from their corresponding tower E T . With regards to the unfolding procedure, the uncertainties include the variation of the iteration parameter from 2-6 with 4 as the nominal value, and a variation of the input prior shape for z g , R g and p T individually by using PYTHIA 8 and HERWIG 7. We estimated the effect of different sources on the final results by varying the detector simulation, following the same unfolding procedure and comparing to the nominal result. Since we are reporting self-normalized distributions, the luminosity uncertainty with respect to the data-taking is not considered. The total systematic uncertainties for the z g and R g measurements, calculated by adding individual sources in quadrature, are presented in Tab. 1 and 2 for R = 0.4 jets in 20 < p T,jet < 25 GeV/c range. For both measurements, the largest systematic uncertainty results from the unfolding procedure.
The total systematic uncertainties for these softdrop observables decrease slightly as the jet resolution parameter increases.

Results
The fully corrected z g and R g measurements are compared to leading order event generators, PYTHIA 6, PYTHIA 8 and HERWIG 7. Since our PYTHIA 6 events do not include weak decays at the particle level, we generate PYTHIA 8 and HERWIG 7 events with the same requirement. We note that for the observables discussed in this letter, we do not observe a significant effect due to weak decays. The parton shower  implementations are varied amongst the models, with PYTHIA 6 and PYTHIA 8 featuring virtuality ordered shower in contrast to HERWIG 7 with angular ordering. The description of the underlying event in PYTHIA 6 is based on the Perugia 2012 tune [31] and further tuned to match data from RHIC whereas PYTHIA 8 uses the Monash 2013 tune which was based on the LHC data [32]. The HERWIG 7 calculations use the EE4C underlying event tune [33] appropriately scaled for the collision energy at RHIC.
The fully corrected z g measurements for jets of varying p T,jet are compared to MC predictions as shown in Fig. 4. In addition, we show the symmetrized DGLAP splitting function at leading order for a quark emitting a gluon as the red dashed lines. The different panels represent jets with low p T,jet in the top middle and high p T,jet in the bottom right. We observe a more symmetric splitting (larger mean z g or, consequently, a flatter shape) function at lower p T,jet that gradually tends towards more asymmetric (smaller mean z g ) at higher p T,jet . The measurements also indicate a p T,jet -independent z g shape slightly steeper than the theoretical limit around p T,jet > 30 GeV/c within our kinematic range. With symmetric splitting functions, the probability to radiate a high-z gluon (where z is defined as the radiated object's energy fraction with respect to the original parton) is enhanced as opposed to an asymmetric splitting function dominated by low-z emissions. This evolution from a symmetric to asymmetric splitting function with increasing p T,jet is consistent with pQCD expectation wherein, a high-momentum parton has an enhanced probability to radiate a soft gluon. Such behavior is captured by both angular and virtuality ordered parton shower models. With default hadronization turned on, PYTHIA 6, PYTHIA 8 and HERWIG 7 describe the qualitative shape as observed in these measurements. To compare more quantitatively, the bottom panels show the ratio of the model calculations to data, and the shaded red region represents the total systematic uncertainty in data. Both PYTHIA versions are able to describe the z g measurements. However, HERWIG 7 seems to prefer more symmetric splits, especially at larger p T,jet .
The SoftDrop R g for R = 0.4 jets are presented in Fig. 5. The R g shows a momentumdependent narrowing of the jet structure as reflected in a shift to smaller values as the jet momentum increases. The measured R g distributions are qualitatively reproduced by all event generators. In contrast to the observations from the z g measurement, HER-  WIG 7 shows a slight tendency towards smaller R g , while PYTHIA 8 prefers a systematically wider R g distribution. For R = 0.4 jets, PYTHIA 6 is able to quantitatively describe data, whilst neither PYTHIA 8 nor HERWIG 7 is able to explain both z g and R g observables simultaneously within the experimental systematic uncertainties.
We further measured the splitting by varying the jet resolution parameter R as shown in Fig. 6 and Fig. 7 for the z g and R g , respectively. The left, middle and right panels represent R = 0.2, 0.4 and 0.6 jets. The top row is for jets with 15 < p T,jet < 20 GeV/c and the bottom row for jets with 30 < p T,jet < 40 GeV/c. Jets with smaller resolution parameters and at lower p T,jet display stronger z g shape modification with respect to the ideal DGLAP splitting and do not reproduce the characteristic 1/z shape seen at higher p T,jet . The narrowing of the R g with increasing p T,jet becomes more significant for jets of larger resolution parameters. The flattening of the z g shape for jets with R = 0.2 and low p T,jet are due to the stringent kinematic constraints on the phase space available. This observation is evident by the R g ranges seen in the top left panel in Fig. 7, for the splitting that is a direct consequence of virtuality/angular ordering.
The dashed black curve shows the z g and R g distribution from PYTHIA 8 events without hadronization (parton jets). We find that hadronization, as described in PYTHIA 8, tends to create softer z g or more asymmetric splits. In contrast, we observe the apparent robustness of the R g observable against hadronization effects.
Due to recent advances in theoretical calculations regarding jets of small resolution parameters and low momenta [34,35], we can now compare our fully corrected data to predictions at next-to-leading-log accuracy in Fig. 8 for z g (left panels) and R g (right panels). The systematic uncertainty in the theoretical calculations (gray shaded band) arises from QCD scale variations, including the p T -hard scale, the jet scale (p T,jet · R) and the scales associated with the substructure observables mentioned here [34]. We note that the systematic uncertainties for the calculations are large for the kinematic range studied in this measurement. These predictions are for jets at the parton level without non-perturbative corrections. This is one possible reason why the comparison to data at low jet momenta and small jet resolution parameter exhibits large deviations in the z g . On the other hand the predictions for the R g observable show large discrep-  ancies with the data for all of the jet resolution parameters and kinematics except the largest resolution parameter and highest p T,jet where the shape gets close to the data.
These comparisons highlight the need for more realistic calculations, including corrections arising from non-perturbative effects and higher-order corrections to further understand jet substructure more quantitatively.

Summary
In summary, we presented the first fully corrected SoftDrop z g and R g measurements of inclusive jets of varying resolution parameters with 15 < p T,jet < 60 GeV/c in p+p collisions at √ s = 200 GeV. The z g distribution converges towards an approximately p T,jet -independent shape above 30 GeV/c which is slightly more asymmetric than the idealized UV limit. On the other hand, the R g reflects a momentum-dependent narrowing of the jet structure. We observe that lower momentum jets are more likely to have a wider jet structure with more symmetric splitting within the jet. This behavior reverses for higher p T,jet jets wherein they are narrower and dominated by asymmetric splits. We also note that at small jet resolution parameters and low p T,jet , the z g is sensitive to hadronization effects resulting in a significant enhancement of asymmetric splitting, whereas for larger resolution parameters, 0.4 and 0.6, the effect is moderate and only results in a minor (shape) change towards more asymmetric splitting. The SoftDrop R g is observed to be less sensitive to hadronization. For both the measurements presented in this letter, we observe that the RHIC-tuned PYTHIA 6 is able to reproduce data whereas PYTHIA 8 and HERWIG 7 are unable to simultaneously describe both scales of the jet evolution. We also showed comparisons to theoretical calculations that extend the predictive power of pQCD at jet scales closer to the fundamental QCD scale, i.e., for jets with small momenta and resolution parameters. Such comparisons to data highlight the need for continued theoretical studies into the exact interplay between measured hadronic jet substructure observables and the underlying partonic splitting at RHIC energies. These studies offer a unique opportunity to further tune MC event generators and for understanding higher order effects on jet evolution at RHIC kinematics.