Measurement of jet fragmentation in Pb+Pb and $pp$ collisions at $\sqrt{s_{NN}} = 5.02$ TeV with the ATLAS detector

This paper presents a measurement of jet fragmentation functions in 0.49 nb$^{-1}$ of Pb+Pb collisions and 25 pb$^{-1}$ of $pp$ collisions at $\sqrt{s_{NN}} = 5.02$ TeV collected in 2015 with the ATLAS detector at the LHC. These measurements provide insight into the jet quenching process in the quark-gluon plasma created in the aftermath of ultra-relativistic collisions between two nuclei. The modifications to the jet fragmentation functions are quantified by dividing the measurements in Pb+Pb collisions by baseline measurements in $pp$ collisions. This ratio is studied as a function of the transverse momentum of the jet, the jet rapidity, and the centrality of the collision. In both collision systems, the jet fragmentation functions are measured for jets with transverse momentum between 126 GeV and 398 GeV and with an absolute value of jet rapidity less than 2.1. An enhancement of particles carrying a small fraction of the jet momentum is observed, which increases with centrality and with increasing jet transverse momentum. Yields of particles carrying a very large fraction of the jet momentum are also observed to be enhanced. Between these two enhancements of the fragmentation functions a suppression of particles carrying an intermediate fraction of the jet momentum is observed in Pb+Pb collisions. A small dependence of the modifications on jet rapidity is observed.


Introduction
Ultra-relativistic nuclear collisions at the Large Hadron Collider (LHC) produce hot dense matter called the quark-gluon plasma (QGP); recent reviews can be found in Refs. [1,2]. Hard-scattering processes occurring in these collisions produce jets which traverse and interact with the QGP. The study of modifications of jet rates and properties in heavy-ion collisions compared to pp collisions provides information about the properties of the QGP.
The rates of jet production are observed to be reduced by approximately a factor of two in lead-lead (Pb+Pb) collisions at LHC energies compared to expectations from the jet production cross-sections measured in pp interactions scaled by the nuclear overlap function of Pb+Pb collisions [3][4][5]. Similarly, back-to-back dijet [6-8] and photon-jet pairs [9] are observed to have unbalanced transverse momentum in Pb+Pb collisions compared to pp collisions. These observations imply that some of the energy of the parton showering process is transferred outside of the jet through its interaction with the QGP. This has been termed "jet quenching".
The distribution of particles within the jet are affected by this mechanism of energy loss. Several related observables sensitive to the properties of the medium can be constructed. Measurements of the jet shape [10] and the fragmentation functions were made in 2.76 TeV Pb+Pb collisions [11][12][13]. In Ref.
[13], jet fragmentation functions are measured as a function of both the charged-particle transverse momentum, p T , and the charged-particle longitudinal momentum fraction relative to the jet, The fragmentation functions are defined as: where p jet T is the transverse momentum of the jet, n ch is the number of charged particles in the jet, N jet is the number of jets under consideration, and ∆R = (∆η) 2 + (∆φ) 2 with ∆η and ∆φ defined as the differences between the jet axis and the charged-particle direction in pseudorapidity and azimuth,1 respectively. In order to quantify differences between Pb+Pb and pp collisions at the same collision energy, the ratios of the fragmentation functions are measured: and Relative to jets in pp collisions, it was found in Ref.
[13] that jets in Pb+Pb collisions have an excess of particles with transverse momentum below 4 GeV and an excess of particles carrying a large fraction of the jet transverse momentum. At intermediate charged-particle p T , there is a suppression of the chargedparticle yield. At the same time, an excess of low-p T particles is observed for particles in a wide region around the jet cone [14,15]. These observations may indicate that the energy lost by jets through the jet quenching process is being transferred to soft particles within and around the jet [16,17]; measurements of these soft particles have the potential to constrain the models describing such processes. A possible explanation for the enhancement of particles carrying a large fraction of the jet momentum is that it is related to the gluon-initiated jets losing more energy than quark-initiated jets. This leads to a higher quark-jet fraction in Pb+Pb collisions than in pp collisions. The change in flavor composition, combined with the different shapes of the quark and gluon fragmentation functions [18] then lead to the observed excess.
Proton-nucleus collisions, which do not generate a large amount of QGP, are used to differentiate between initial-and final-state effects due to the QGP formed in Pb+Pb collisions. Fragmentation functions in p+Pb collisions show no evidence of modification when compared with those in pp collisions [19]. Thus, any modifications observed in Pb+Pb collisions can be attributed to the presence of the QGP rather than to effects arising from the presence of the large nucleus.
The rapidity dependence of jet observables in Pb+Pb collisions is of great interest, in part because at fixed p jet T the fraction of quark jets increases with increasing |y jet | (see, for example, Refs. [18,20]). This makes the rapidity dependence of jet observables potentially sensitive to the different interactions of quarks and gluons with the QGP. Previous measurements of the rapidity dependence of jet fragmentation functions at √ s NN = 2.76 TeV in Pb+Pb collisions found a rapidity dependence of the fragmentation function modification with limited significance [13].
In this paper, the fragmentation functions and the R D(z) and R D(p T ) ratios are measured in Pb+Pb and pp collisions at 5.02 TeV using 0.49 nb −1 of Pb+Pb collisions and 25 pb −1 of pp collisions collected in 2015. 1 ATLAS uses a right-handed coordinate system with its origin at the nominal interaction point (IP) in the center of the detector and the z-axis along the beam pipe. The x-axis points from the IP to the center of the LHC ring, and the y-axis points upward. Cylindrical coordinates (r, φ) are used in the transverse plane, φ being the azimuthal angle around the beam pipe. The pseudorapidity is defined in terms of the polar angle θ as η = − ln tan(θ/2). The rapidity is defined as y = 0.5 ln[(E + p z )/(E − p z )] where E and p z are the energy and the component of the momentum along the beam direction.
Jets are measured over a rapidity range of |y jet | < 2.1 using the anti-k t reconstruction algorithm [21] with radius parameter R = 0.4 . The measurement is presented in intervals of p jet T , y jet , and collision centrality. These data extend the previous studies at √ s NN = 2.76 TeV in two ways. First, an increase in the peak energy density of the medium is expected. Second, the Pb+Pb integrated luminosity in the current dataset is 3.5 times the integrated luminosity available at 2.76 TeV, and the increase in the collision energy also increases the jet cross-sections. These two factors allow a measurement of the dependence of jet fragmentation functions on the transverse momentum of the jet over a wider range than was previously possible.

Experimental setup
The measurements presented in this paper were performed using the ATLAS inner detector, calorimeter, trigger and data acquisition systems [22]. The calorimeter system consists of a sampling liquid argon (LAr) electromagnetic (EM) calorimeter covering |η| < 3.2, a steel/scintillator sampling hadronic calorimeter covering |η| < 1.7, LAr hadronic calorimeters covering 1.5 < |η| < 3.2, and two LAr forward calorimeters (FCal) covering 3.1 < |η| < 4.9 [22]. The EM calorimeters are segmented longitudinally in shower depth into three layers with an additional presampler layer. They have segmentation in φ and η that varies with layer and pseudorapidity. The hadronic calorimeters have three sampling layers longitudinal in shower depth.
The inner detector measures charged particles within the pseudorapidity interval |η| < 2.5 using a combination of silicon pixel detectors, silicon microstrip detectors (SCT), and a straw-tube transition radiation tracker (TRT), all immersed in a 2 T axial magnetic field [22]. Each of the three detectors is composed of a barrel and two symmetric endcap sections. The pixel detector is composed of four layers: the "insertable B-layer" [23,24] and three layers with a pixel size of 50 µm × 400 µm. The SCT barrel section contains four layers of modules with 80 µm pitch sensors on both sides and each endcap consists of nine layers of double-sided modules with radial strips having a mean pitch of 80 µm. The two sides of each SCT layer in both the barrel and the endcaps have a relative stereo angle of 40 mrad. The TRT contains up to 73 (160) layers of staggered straws interleaved with fibres in the barrel (endcap).
The zero-degree calorimeters (ZDCs) are located symmetrically at z = ±140 m and cover |η| > 8. 3. They are constructed from tungsten absorber plates and Čerenkov light is transmitted via quartz fibers. In Pb+Pb collisions the ZDCs primarily measure "spectator" neutrons, i.e. neutrons that do not interact hadronically when the incident nuclei collide. A ZDC coincidence trigger is implemented by requiring the pulse height from each ZDC to be above a threshold set to accept the single-neutron peak.
A two-level trigger system is used to select the Pb+Pb and pp collisions. The first trigger level (L1) is hardware-based and implemented with custom electronics. The second level is the software-based highlevel trigger (HLT) and is used to further reduce the accepted event rate. Minimum-bias Pb+Pb events are recorded using a trigger defined by the logical OR of a L1 total energy trigger and the ZDC coincidence trigger. The total energy trigger required the total transverse energy measured in the calorimeter system to be greater than 50 GeV in Pb+Pb collisions. Jet events are selected by the HLT, after requiring the identification of a jet by the L1 jet trigger in pp collisions or the total energy trigger with a threshold of 50 GeV in Pb+Pb collisions. The L1 jet trigger utilized in pp collisions required a jet with transverse momentum greater than 20 GeV. The HLT jet trigger used a jet reconstruction algorithm similar to that used in the offline analysis (the offline jet reconstruction is discussed in Section 4). It selected events containing jets with transverse energy of at least 75 GeV in Pb+Pb collisions and at least 85 GeV in pp collisions. In pp collisions, the 85 GeV threshold jet trigger sampled the full delivered luminosity. The 75 GeV threshold jet trigger used in Pb+Pb collisions was prescaled2 in a small part of the Pb+Pb data-taking period; however, the trigger sampled more than 99% of the total integrated luminosity. The measurement is performed in the jet transverse momentum region where the triggers are fully efficient.

Data sets and event selection
The Pb+Pb and pp data used in this analysis were recorded in 2015. The data samples consist of 25 pb −1 of √ s = 5.02 TeV pp data and 0.49 nb −1 of √ s NN = 5.02 TeV Pb+Pb data. In Pb+Pb and pp collisions, events are required to have a reconstructed vertex within 150 mm of the nominal interaction point along the beam axis. Only events taken during stable beam conditions and satisfying detector and data-quality requirements, which include the calorimeters and inner tracking detectors being in nominal operation, are considered.
In Pb+Pb collisions, the event centrality reflects the overlap area of the two colliding nuclei and is characterized by ΣE FCal In addition to the jet-triggered sample, a separate Pb+Pb data sample was recorded with the minimum-bias trigger and two total transverse-energy triggers requiring 1.5 TeV and 6.5 TeV to enhance the rate of more central Pb+Pb events. This data sample is used to produce a Pb+Pb Monte Carlo (MC) events with conditions that match those registered while the data were recorded.
The performance of the detector and of the analysis procedure in Pb+Pb collisions is evaluated using 1.8 ×10 7 5.02 TeV MC events. These were produced from minimum-bias Pb+Pb data events overlaid with hard-scattering dijet pp events generated with P +P 8 [26, 27] using a set of tuned parameters called the A14 tune [28] and the NNPDF23LO parton distribution function (PDF) set [29]. The detector response was simulated using G 4 [30,31] and the simulated hits were combined with those from the data event. A weight is assigned to each MC event such that the event sample obtained from the minimum-bias trigger has the same centrality distribution as the sample collected by the jet trigger. A separate sample of 1.8 ×10 7 simulated 5.02 TeV P 8 pp hard-scattering events, generated with the same tune and PDFs as for the Pb+Pb MC sample, is used to evaluate the performance for measuring fragmentation functions in the pp data. The contribution from additional collisions in the same bunch crossing is not included in the MC simulation. A sample of Pb+Pb events generated with HIJING version 1.38b [32] is also used to evaluate the performance of the track reconstruction.

Jet and track selection
The jet reconstruction, underlying event (UE) determination, and subtraction procedures closely follow those used by ATLAS for jet measurements in pp and Pb+Pb collisions at √ s NN = 2.76 TeV [4]. The anti-k t algorithm is first run in four-momentum recombination mode, on ∆η × ∆φ = 0.1 × 0.1 calorimeter towers with the anti-k t radius parameter R = 0.2 and R = 0.4. The energies in the towers are obtained by summing the energies of calorimeter cells at the electromagnetic energy scale within the tower boundaries. Then, an iterative procedure is used to estimate the η-dependent UE transverse energy density on an eventby-event basis using the energy measurements in all calorimeter towers in the event while excluding the regions populated by jets. The resulting UE transverse energy density is modulated taking into account the presence of the azimuthal anisotropy of particle production [33]. The modulation includes contributions of the second-, third-, and fourth-order azimuthal anisotropy harmonics. Higher-order harmonics introduce negligible variation of the reconstructed jet energy. The UE transverse energy is subtracted from each calorimeter cell within the towers included in the reconstructed jet, and the fourmomentum of the jet is updated accordingly. Then, a jet ηand p T -dependent correction factor to the p jet T derived from the simulation samples is applied to correct for the calorimeter energy response [34]. An additional correction based on in situ studies of jets recoiling against photons, Z bosons, and jets in other regions of the calorimeter is applied [35,36]. The same jet reconstruction procedure without the azimuthal modulation of the UE is also applied to pp collisions.
Jets are required to have a rapidity within |y jet | < 2.1 so that all R = 0.4 jet cones are contained within the inner detector's acceptance. To prevent neighboring jets from distorting the measurement of the fragmentation functions, jets are rejected if there is another jet with higher p jet T anywhere within a distance ∆R < 1.0. A correction is applied to reduce the effects of the broadening of the jet direction measurement for R = 0.4 jets due to the UE. The correction uses jets reconstructed with a smaller distance parameter, R = 0.2 since their angular resolution evaluated in MC studies is found to be less affected by the UE fluctuations than that of larger-R jets. The jet direction is redefined as that of the closest R = 0.2 jet with p jet T > 35 GeV and matching the original jet direction within ∆R = 0.3 of the R = 0.4 jet, when such a matching jet is found. If no matching R = 0.2 jet is found the axis remains unchanged.
Charged-particle tracks are reconstructed from hits in the inner detector using the track reconstruction algorithm with settings optimized for the high hit density in heavy-ion collisions [37]. Tracks used in this analysis are required to have a total of at least 9 (11) hits in the silicon pixel and microstrip detectors for charged particles with pseudorapidity |η ch | ≤ 1.65 (|η ch | > 1.65). At least one hit is required in one of the two innermost pixel layers. If the track trajectory passed through an active module in the innermost layer, then a hit in this layer is required. Furthermore, a track must have no more than two holes in the Pixel and SCT detectors together, where a hole is defined by the absence of a hit predicted by the track trajectory. All charged-particle tracks used in this analysis are required to have reconstructed transverse momentum p ch T > 1 GeV. In order to suppress the contribution from secondary particles, the distance of closest approach of the track to the primary vertex in the transverse plane is required to be less than a value which varies from 0.45 mm at p ch T = 4 GeV to 0.2 mm at p ch T = 20 GeV, and at that point the track must be less than 1.0 mm from the primary vertex in the longitudinal direction.
The efficiency, ε(p truth T , p jet T , y jet ), for reconstructing charged particles within jets in Pb+Pb and pp collisions is evaluated from the matching of reconstructed tracks to generator-level primary particles3 using MC samples described above. The matching is based on contributions of generator-level particles to the hits in the detector layers. A reconstructed track is matched to a generator-level particle if it contains hits produced primarily by this particle [31]. The efficiency is evaluated separately in four |y jet | intervals and each interval of reconstructed p jet T used in the measurement. Furthermore, the efficiency is evaluated separately for each centrality interval in the case of Pb+Pb collisions. The charged-particle reconstruction efficiencies as a function of the generator-level primary particle transverse momentum, p truth T , are shown  Figure 1: Tracking efficiency, ε, smoothed using a third-order polynomial in log(p truth T ) as a function of p truth T in pp collisions in five different jet-p T intervals (left) and in Pb+Pb collisions (right) in two different jet-p T intervals and for 0-10% and 60-80% centrality intervals. In both plots the efficiency is evaluated for tracks within jets with |y jet | < 0.3.
in Figure 1 for jets with |y jet | < 0.3 in pp and Pb+Pb collisions. In order to remove fluctuations in the efficiency due to the limited MC sample size, the p truth T dependence of the efficiencies is parameterized and smoothed using a third-order polynomial in log(p truth T ) that gives a good description of the efficiency in the full range of p truth T . The efficiencies shown in Figure 1 exhibit only a modest variation with p truth T , centrality, and p jet T . A small almost continuous increase of the efficiency with the increasing p truth T is observed. The efficiency over the 20-100 GeV p truth T range is smaller for high p jet T compared to low p jet T by about 2% and 5% in pp and Pb+Pb collisions, respectively. This behavior is attributed to the higher probability to lose tracks in the dense core of high-p T jets than to lose tracks that are more isolated [38]. The efficiency is lower in more central Pb+Pb collisions due to the higher hit density. The efficiency exhibits only a small variation with y jet in the region |y jet | < 1.2, and it decreases by approximately 10% in the most forward y jet interval.
The contribution of reconstructed tracks which are not be matched to a generated primary particle in the MC samples of pp collision events produced without data overlay, along with the residual contribution of tracks matched to secondary particles, are together considered "fake" tracks. The fraction of fake tracks is less than 2% over the full kinematic range of this measurement. A possible degradation of the tracking performance at high occupancy is checked in the sample of Pb+Pb collision events simulated with the HIJING MC. No significant dependence of the rate of fake tracks on centrality is observed. The correction for the fake contribution is discussed in Section 5.

Analysis procedure
The analysis procedure closely follows the one used in the measurement of jet fragmentation at √ s NN = 2.76 TeV [13]. Reconstructed tracks are associated with a reconstructed jet if they fall within ∆R = 0.4 of the jet axis and for each of these particles the longitudinal momentum fraction, z, is calculated. The measured track yields, dn meas ch /dz or dn meas ch /dp ch T , are constructed as: and dn meas ch dp ch where the quantities ∆N ch (z) and ∆N ch (p ch T ) represent the number of associated tracks within the given z or p ch T range, respectively corrected for the track reconstruction efficiency. The efficiency correction is applied as a 1/ε(p ch T , p jet T , y jet ) weight on a track-by-track basis, assuming p ch T = p truth T . While that assumption is not strictly valid, the efficiency varies sufficiently slowly with p truth T that the error introduced by this assumption is less than 1%.
Tracks which are not correlated with the jet need to be subtracted from the measured distributions; these tracks come from both fake tracks and the UE. In Pb+Pb collisions, contributions to the fragmentation functions from the charged particles originating from the UE in Pb+Pb collisions are subtracted. This contribution is evaluated as a function of charge particle z or p ch T , y jet , p jet T , and the collision centrality. Additionally, the measured track yields in pp and Pb+Pb collisions are corrected for the presence of fake tracks.
The UE contribution is determined for each measured jet using a grid of R = 0.4 cones spanning the full coverage of the inner detector and following the method introduced in Ref. [11]. The method is applied to events containing jets included in the analysis. The cones have a fixed distance between their centers chosen such that the inner detector acceptance is uniformly covered while avoiding overlaps. Any cone having a charged particle with p ch T > 10 GeV or overlapping with a reconstructed jet with p jet T > 90 GeV is assumed to be associated with a hard process and is excluded from the UE estimation to avoid biasing it. The parameters defining the exclusion regions are evaluated in MC studies and are subjected to variations as part of the estimation of systematic uncertainties. The resulting UE charged-particle yields, dn UE ch /dz or dn UE ch /dp ch T , are evaluated over 1 < p ch T < 10 GeV according to: Here N cone is the number of background cones used in the UE determination of a given jet, ∆N cone ch represents the number of charged particles summed over all background cones, and ∆R represents the distance between the center of a cone and the direction of a given charged particle. The term ε(p ch T , η ch ) is the efficiency for reconstructing charged particles, estimated as a function of p ch T and η ch without requiring track-to-jet matching.
The estimated contribution from the UE in each cone is corrected for the difference in the average yield of UE charged particles at a given p ch T between the η position of the cone and η position of the jet. This correction is based on the centrality-, p ch T -, and η-dependent distribution of charged-particle yields in minimum-bias data events. An additional correction is applied to the charged-particle UE estimate to account for the difference in the azimuthal particle density, due to elliptic flow, between the φ angle of the cone and the φ angle of the jet. This utilizes a centrality-and p ch T -dependent parameterization of the measured elliptic flow coefficients [33].
The UE contribution is further corrected for the correlation between the actual UE charged particle yield underneath the jet and the jet energy resolution [11]; in regions where the UE has an upward fluctuation, the jet energy resolution is worse. The smearing due to jet energy resolution leads to a net migration of jets from lower p jet T to higher p jet T values. The effect of the migration causes the actual UE contribution underneath the jet to be larger than that estimated from the procedure described above. This effect is corrected for by applying multiplicative correction factors, depending on p ch T or z, y jet , p jet T , and collision centrality. The correction is estimated as a ratio of the UE charged particle yield evaluated by two different methods using the Pb+Pb MC samples. The first estimate uses the cone method discussed above. The second method calculates the UE contribution in the data overlay MC samples from tracks, within the area of a jet, that do not have an associated generated primary particle. The size of the correction is less than 2% at low z or p ch T where the UE has the largest impact, and has only a small dependence on p jet T . The contribution from fake tracks to the fragmentation functions is estimated from the MC samples without minimum-bias interactions overlaid. The fraction of these tracks is found to be below 2% of the tracks that pass the selection in all track and jet kinematic regions in this analysis.
The UE distributions corrected for the additive contribution of fake tracks, dñ UE+fake ch /dp ch T and dñ UE+fake ch /dz, are then subtracted from the measured distributions, and the subtracted charged-particle yields and fragmentation functions are evaluated: and dn sub ch dp ch where N meas jet is the total number of measured jets in a given p jet T interval. The signal-to-background ratio, n sub ch /n UE ch , strongly depends on the collision centrality and p ch T . Figure 2 shows the distributions prior to the UE and fake-track subtraction, dn meas ch dp ch T , divided by the distributions after the subtraction, dn sub ch dp ch T , as a function of p ch T for three centrality selections. In 0-10% central collisions, the distributions prior to subtraction are over ten times larger than the subtracted distributions for the most extreme case of 1 GeV charged particles. This ratio is reduced to approximately 2 in peripheral collisions at the same charged-particle p T . The fake-track contribution to the fragmentation functions is subtracted from the measured fragmentation functions in both the pp and Pb+Pb collisions; the UE subtraction is performed only for the Pb+Pb measurement as the UE contribution is negligible in the pp collisions (less than 2% over the entire kinematic range measured).
To remove the effects of bin migration due to the jet energy and track momentum resolution, the subtracted dn sub ch /dz and dn sub ch /dp ch T distributions are corrected by using a two-dimensional Bayesian unfolding procedure [39] in z or p T and p jet T as implemented in the RooUnfold package [40]. Two-dimensional unfolding is used because the calorimetric jet energy response depends on the fragmentation pattern of the jet [41]. Using MC samples, four-dimensional response matrices are created using the generator-level and reconstructed p jet T , and the generator-level and reconstructed charged-particle z or p T . Separate unfolding matrices are constructed for pp data and each centrality interval in Pb+Pb collisions. A separate onedimensional Bayesian unfolding is used to correct the measured p jet T spectra which are used to normalize the unfolded unnormalized fragmentation functions, dn unfolded ch /dp T and dn unfolded ch /dz. To achieve better agreement with the data, the MC jet spectra and fragmentation functions are reweighted to match the shapes in the reconstructed data. The Bayesian procedure requires a choice in the number of iterations. Additional iterations reduce the sensitivity to the choice of prior, but may amplify statistical fluctuations in the distributions. After four iterations for both the one-dimensional and two-dimensional unfoldings the fragmentation functions are stable for both the Pb+Pb and pp data. The final, particle-level corrected distributions are defined as: where N unfolded jet is the unfolded number of jets in a given p jet T interval. The performance of the analysis procedure is tested by dividing the MC events in half and using one half to generate response matrices with which the other half is unfolded and the ratio of unfolded to generator-level fragmentation functions4 is evaluated. This procedure tests all the analysis corrections and the unfolding procedure. Good recovery of the generator-level (truth) MC distributions is observed for the unfolded events. The deviations from the exact recovery of the generator-level MC distributions, the non-closure, are included in the systematic uncertainties. The ratios of D sub (z) and D sub (p ch T ) distributions 4 The generator-level fragmentation functions are constructed using generator-level jets and primary charged particles.   to the unfolded D(z) and D(p T ) distributions are shown in Figure 3 for pp collisions and 0-10% central Pb+Pb collisions. The magnitude of the unfolding effect varies as a function of p jet T , p ch T , and centrality. The effect of the unfolding is similar in pp and Pb+Pb collisions at low z and p T , but for higher-momentum particles within the jet, the effect of the unfolding in pp and Pb+Pb collisions differs by up to 25% between the two collision systems for 126 < p jet T < 158 GeV. This difference is due to UE fluctuations, which lead to poorer jet energy resolution in Pb+Pb collisions than in pp collisions. With increasing p jet T , the effect of UE fluctuations decreases; for 251 < p jet T < 316 GeV the effect of the unfolding is similar in Pb+Pb and pp collisions at all value of z and p T . The effect of the unfolding is larger at high z and p T due to the steepness of the fragmentation function near z = 1. The shaded boxes in Figure 3 show the size of systematic uncertainties associated with the unfolding which originate from the sensitivity of the unfolding to the shape of input MC distributions, as described in the next section.

Systematic uncertainties
The following sources of systematic uncertainty are considered: the jet energy scale (JES), the jet energy resolution (JER), the sensitivity of the unfolding to the prior, the residual non-closure of the analysis procedure, UE contribution, and tracking-related uncertainties. For each variation accounting for a source of systematic uncertainty, the fragmentation functions and ratios of D(z) and D(p T ) distributions in Pb+Pb and pp collisions are re-evaluated. The difference between the varied and nominal distributions is used as an estimate of the resulting uncertainty.
The systematic uncertainty due to the JES in Pb+Pb collisions is composed of two parts: a centralityindependent baseline component and a centrality-dependent component. Only the centrality-independent baseline component is used in pp collisions; it is determined from in situ studies of the calorimeter response [34, 42,43], and studies of the relative energy scale difference between the jet reconstruction procedure in heavy-ion collisions [42] and the procedure in pp collisions [34]. The centrality-dependent uncertainty reflects a modification of parton showers by the Pb+Pb environment. It is evaluated by comparing calorimeter p jet T and the sum of p T of tracks within the jet in data and MC simulation. The size of the centrality-dependent uncertainty in the JES reaches 0.5% in the most central collisions. Each component that contributes to the JES uncertainty is varied separately by ±1 standard deviation for each interval in p jet T , and the response matrix is recomputed accordingly. The data are unfolded with these matrices. The resulting uncertainty on the fragmentation functions increases with increasing z and particle p T at fixed p jet T and decreases with increasing p jet T . The uncertainty in the fragmentation functions due to the JER is evaluated by repeating the unfolding procedure with modified response matrices, where an additional contribution is added to the resolution of the reconstructed p jet T using a Gaussian smearing procedure. The smearing factor is evaluated using an in situ technique in 13 TeV pp data involving studies of dijet energy balance [44,45]. An additional uncertainty is included to account for differences between the heavy-ion-style jet reconstruction and that used in analyses of 13 TeV pp data. The size of the resulting uncertainty on the fragmentation functions due to the JER typically reaches 10% for the highest charged-particle z and p T bins and decreases with decreasing charged-particle z and p T at fixed p jet T . The positive and negative uncertainties from the JER are symmetrized.
The unfolding uncertainty is estimated by generating the response matrices from the MC distributions without reweighting in p jet T , D(z), and D(p T ). An additional uncertainty is assigned for the non-closure of the unfolded distributions in simulations, as described in Section 5. The magnitude of the uncertainty due to the unfolding and the non-closure is typically below 2% and 5%, respectively.
The systematic uncertainty associated with the estimation of the UE contribution on the fragmentation functions has two components. First, the parameter that excludes random cones from the estimate is varied. Random cones are assumed to be associated with a hard process and excluded if the centroid of the cone is ∆R < 0.8 from a reconstructed jet with p T > 90 GeV. The exclusion requirement is changed to ∆R < 1.2 to estimate the sensitivity of the UE contributions. The size of the resulting uncertainty on the fragmentation function is everywhere smaller than 3% and it decreases in higher charged-particle z or p T . The second component of the UE uncertainty arises from a difference when the UE from the cone method is compared with an alternative UE estimation. The UE is alternatively evaluated using an efficiency-corrected differential yield of charged particles d 4 n ch /dη ch dφ ch dp ch T d∆Ψ, where ∆Ψ is the difference in azimuth of the charged particle from the second-order event plane, evaluated in minimumbias Pb+Pb events. To each event considered, a weight is assigned such that the event sample obtained from the minimum-bias trigger has the same centrality distribution as the sample collected by the jet trigger. The resulting uncertainty on the fragmentation functions is smaller than 10% at low z or p T and it rapidly decreases in higher charged-particle z or p T bins. The uncertainties related to the track reconstruction and selection originate from several sources. Uncertainties related to the fake rate, the material description in simulation, and the track transverse momentum are obtained from studies in data and simulation described in Ref. [46]. The systematic uncertainty on the fake-track rate is 30% in both collision systems [46]. The contamination of fake tracks is less than 2%, and the resulting uncertainty on the fragmentation functions is at most 0.5%. The sensitivity of the tracking efficiency to the description of the inactive material in the MC samples is evaluated by varying the material description. This resulting uncertainty in the track reconstruction efficiency is between 0.5% and 2% over the track p T range used in the analysis. An additional uncertainty takes into account a possible residual misalignment of the tracking detectors in pp and Pb+Pb data-taking. The alignment in these data sets is checked in situ using Z → µ + µ − events, and a track-p T dependent uncertainty arises from the finite size of this sample. The resulting uncertainties on the fragmentation functions are typically smaller than 1%, except at large z, where they are as large as 4%. An additional uncertainty on the tracking efficiency due to the high local track density in the core of jets is 0.4% [38] for all p jet T ranges in this analysis. The uncertainty due to the track selection is evaluated by repeating the analysis with an additional requirement on the significance of the distance of closest approach of the track to the primary vertex. This uncertainty affects the track reconstruction efficiencies, track momentum resolution, and rate of fake tracks. The resulting uncertainty typically varies from 1% at low track p T to 5% at high track p T . Finally, the track-to-particle matching requirements are varied. This variation affects the track reconstruction efficiency, track momentum resolution, and rate of fake tracks. The resulting systematic uncertainty in the fragmentation functions is less than 0.5%.
Example systematic uncertainties on the D(z) and D(p T ) distributions for jets in the 126-158 GeV p jet T range measured in the two collision systems are presented in Figure 4. All track-related systematic uncertainties are added in quadrature and presented as a total tracking uncertainty. The systematic uncertainties from each source are assumed to be uncorrelated, so they are combined in quadrature to obtain the total systematic uncertainty.
The correlations between the various systematic components are considered in evaluating the ratios of Pb+Pb to pp fragmentation functions. The unfolding and the MC non-closure are each taken to be uncorrelated between the two collision systems. All other uncertainties are taken to be correlated. For the correlated uncertainties, the ratios are re-evaluated by applying the variation to both collision systems; the resulting variations of the ratios from their central values are used as the correlated systematic uncertainty. The uncorrelated uncertainties are added in quadrature. Each systematic uncertainty is assumed to be fully correlated with itself between different rapidity bins. The systematic uncertainty from each source, except the non-closure of the unfolded distributions and the residual misalignment of the tracking detectors, is bin-to-bin correlated. The total systematic uncertainties of the R D(z) and R D(p T ) distributions are shown in Figure 5 for one selected p jet T range.

Results
In this section, results are presented of the measurement of the D(z) and D(p T ) distributions for jet p T between 126 and 398 GeV and six centrality intervals in Pb+Pb collisions; the same distributions are presented in pp collisions for the same p jet T ranges. In order to study the effects of hot dense matter on the jet fragmentation process, ratios of Pb+Pb fragmentation functions to pp fragmentation functions are evaluated.
The D(z) and D(p T ) distributions in pp collisions are shown in Figure 6. The corresponding distributions in Pb+Pb collisions are shown in Figures 7 through 11.         In order to quantify the difference in the fragmentation functions between Pb+Pb and pp collisions, the ratios of D(z) and D(p T ) distributions measured in Pb+Pb collisions to those measured in pp collisions, R D(z) and R D(p T ) , are shown in Figures 12 and 13, respectively. In each figure, the shaded boxes indicate systematic uncertainties and the vertical bars show the statistical uncertainties.
The shapes of the R D(z) and R D(p T ) distributions are similar for all centralities: inside the jets; the yields of particles with low p T or z are enhanced; there is a reduction for particles with intermediate p T or z; and the yields of particles with high p T or z are enhanced. This is qualitatively consistent with previous measurements of jet fragmentation at √ s NN = 2.76 TeV [11-13]; a quantitative comparison is provided in Section 8. The magnitudes of the deviations of the ratios from unity decrease with decreasing collision centrality. In the most central collisions, the size of the enhancement is as large as 70% at low p T or z and 30% at high p T or z. The depletion of charged-particle yields at intermediate p T and z is as large as 20%.  Figures 16 and 17 show R D(p T ) distributions for the same jet rapidity, centrality, and p jet T ranges. In all rapidity ranges, the R D(z) and R D(p T ) distributions have the same qualitative shape and centrality dependence as the rapidity-inclusive results presented above.

Discussion
In this section, the results from the previous section are further discussed and compared to theoretical models.
In order to make a direct comparison with measurements at 2.76 TeV, Figure 18 overlays the R D(z) and R D(p T ) distributions measured in 2.76 TeV collisions [13] on those obtained in this analysis at 5.02 TeV. The two measurements at the two collision energies quantitatively agree over the entire z and chargedparticle p T range of the measurement; no significant collision energy dependence is observed (the lowest point in the D(p T ) ratios differs by less than two standard deviations when the statistical and systematic uncertainties are combined). In order to determine how the fragmentation functions depend on p jet T , the fragmentation functions from three p jet T intervals are compared in Figure 19. The D(p T ) and D(z) distributions are closely related to each other, differing, primarily, in the normalization by p jet T in the definition of z (see Eq. (1)). Therefore, a comparison of the modifications of the fragmentation functions as a function of p jet T can show whether the size of modifications scales with charged-particle z or with p T . The former would be expected for fragmentation effects, and the latter might indicate some scale in the QGP. The large p jet T range available in this measurement allows these two scenarios to be distinguished. Figure 19 shows that the excess of soft particles observed in central Pb+Pb collisions exhibits a much smaller p jet T dependence for the D(p T ) ratios than for the D(z) ratios; the transition from enhancement to suppression for soft fragments occurs at p T around 4 GeV for all p jet T values investigated in this analysis. The same comparison can be made for the hard particles. In this case, Figure 19 shows that the enhancement of hard fragments with z 0.3 is nearly independent of p jet T . The fragmentation functions have been calculated within a hybrid model of jet quenching, which uses perturbative techniques for the high-Q 2 processes in jet evolution and strong coupling for the low momentum scales associated with the QGP [47,48]. Within this model, there is a length scale, L res , which can be interpreted as the minimum distance required to resolve a parton as separate from the others in the showering process when it occurs in the QGP medium. The scale L res can be expressed in terms of the temperature of QGP, T, as L res = R res /πT where R res is a parameter of the model. The fragmentation functions measured here are compared with calculations from this model in Figure 20 for two values of R res . The calculations with R res = 3 are qualitatively consistent with the measurement at high z and p T . At low z and p T , the results of the calculations are below the data, in agreement with prior observations in comparisons to related observables [49]. Also shown in Figure 20 is a calculation from Ref. [18] which is a phenomenological model, the Effective Quenching (EQ) Model, incorporating energy-loss effects through two downward shifts in the p jet T spectrum: one for quark-initiated jets and a larger one for gluon-initiated jets. In this case, the jets fragment as in vacuum, but R D(z) differs from unity due to an increase in the fraction of quark jets in Pb+Pb collisions relative to pp collisions at a fixed p jet T . Since quark jets are more likely to produce high-z particles than gluon jets [50,51] this causes R D(z) > 1 at high z in the model predictions. The EQ model does not have a description of the soft processes from soft gluon radiation or the response of the hot QCD matter to the jet passing through it, so the comparison with data is only appropriate at z > 0.1.  collisions compared to pp collisions, the difference between the two distributions is evaluated for the p jet T and centrality intervals used in this analysis: where "cent" represents one of the six centrality intervals, and the values of p T,min and p T,max are boundaries of the low p T enhancement region, chosen to be 1.0 and 4.2 GeV, respectively. In addition, the p T -weighted difference between the same quantities is also computed: The P ch T | cent represents the total transverse momentum carried by particles in the low p T enhancement region. The dependence of N ch | cent and P ch T | cent on p jet T and centrality is presented in Figure 23. Overall, both quantities are found to increase as a function of p  Figure 25 shows the same ratios for the 0-10% centrality interval compared with calculations from the Hybrid model [48] and the Effective Quenching model [18]. Both calculations are consistent with the data for jets with |y jet | < 1.2 with larger deviations in rapidity interval 1.2 < |y jet | < 2.1.

Summary
This paper presents an analysis of 0.49 nb −1 of Pb+Pb and 25 pb −1 of pp collisions at √ s NN = 5.02 TeV using data collected with the ATLAS detector at the LHC in 2015. The analysis measures the fragmentation functions of jets into charged particles and the distributions of charged-particle transverse momenta within R = 0.4 anti-k t jets with |y jet | <2.1 and with p jet T from 126 to 398 GeV. The studies are performed as a function of the event centrality, jet rapidity and jet transverse momentum for charged particles with transverse momentum greater than 1 GeV .
Centrality-dependent modifications to these fragmentation functions in Pb+Pb collisions are observed when compared with those measured in pp collisions. The magnitude of these modifications increases with increasing collision centrality. The ratios of fragmentation functions evaluated in Pb+Pb collisions to those in pp collisions exhibit enhancements both for transverse momentum less than 4 GeV and for z 0.3. Between these two enhancements there is a suppression of the fragmentation functions in Pb+Pb collisions compared to pp collisions. The enhancement of yields of low and high transverse momentum fragments is as large as 70% and 30%, respectively, in central collisions. The depletion of fragment yields with intermediate p T and z is as large as 20%. The difference in charged-particle multiplicity and total transverse momentum in Pb+Pb compared to pp collisions for 1.0 < p T < 4.2 GeV range increases with increasing centrality and jet transverse momentum. No significant dependence of the high-z enhancement on the transverse momentum of the jet is observed. A small increase in the modification of yields of fragments with low and intermediate z is observed in forward jets compared to those at central rapidity. These measurements provide new information about the jet transverse momentum and rapidity dependence of the modifications to jet fragmentation in Pb+Pb collisions and, together with other jet measurements in heavy-ion collisions, will constrain models of jet quenching in the QGP created in heavy-ion collisions.      [ The ATLAS Collaboration