Measurement of the nuclear modification factor of b-jets in 5.02 TeV Pb+Pb collisions with the ATLAS detector

Abstract

This paper presents a measurement of b-jet production in Pb+Pb and pp collisions at $\sqrt{s_{_\text {NN}}}=5.02$ TeV with the ATLAS detector at the LHC. The measurement uses 260 $\text {pb}^{-1}$ of pp collisions collected in 2017 and 1.4 $\text {nb}^{-1}$ of Pb+Pb collisions collected in 2018. In both collision systems, jets are reconstructed via the anti-$k_{t}$ algorithm. The b-jets are identified from a sample of jets containing muons from the semileptonic decay of b-quarks using template fits of the muon momentum relative to the jet axis. In pp collisions, b-jets are reconstructed for radius parameters $R= 0.2$ and $R= 0.4$, and only $R= 0.2$ jets are used in Pb+Pb collisions. For comparison, inclusive $R= 0.2$ jets are also measured using 1.7 $\text {nb}^{-1}$ of Pb+Pb collisions collected in 2018 and the same pp collision data as the b-jet measurement. The nuclear modification factor, $R_\text {AA}$, is calculated for both b-jets and inclusive jets with $R= 0.2$ over the transverse momentum range of 80–290 GeV. The nuclear modification factor for b-jets decreases from peripheral to central collisions. The ratio of the b-jet $R_\text {AA}$ to inclusive jet $R_\text {AA}$ is also presented and suggests that the $R_\text {AA}$ for b-jets is larger than that for inclusive jets in central Pb+Pb collisions. The measurements are compared with theoretical calculations and suggest a role for mass and colour-charge effects in partonic energy loss in heavy-ion collisions.

1 Introduction

The primary physics aim of the heavy-ion programme at the Large Hadron Collider (LHC) is to produce and study the quark–gluon plasma (QGP), the high-temperature state of quantum-chromodynamic (QCD) matter in which quarks and gluons are no longer confined within protons and neutrons (for a recent review, see Ref. [1]). Measurements of jets arising from hard-scattered partons produced in the early stages of heavy-ion collisions provide information about the short-distance-scale interactions of high-energy partons with the QGP. The overall rate of jets in central Pb+Pb collisions at a given transverse momentum, ${{p}_{\textrm{T}}}$, is found to be significantly lower than expectations based on $pp$ collisions, up to a ${{p}_{\textrm{T}}}$ of approximately 1 TeV [2,3,4]. The showering partons which give rise to jets undergo medium-induced gluon radiation and elastic scattering off the QGP constituents; both processes broaden the angular distribution of the energy from the parton shower [5]. This leads to the measurement of lower-energy jets than in $pp$ collisions as some of the parton’s energy is moved outside of the jet cone.

The interactions between high-energy partons and the QGP are expected to depend on the parton’s QCD colour charge and mass. Thus, jets originating from $b$-quarks, $b$-jets, (for a recent review, see Ref. [6]) are of particular interest because the quark mass is large compared to light quarks. Additionally, the QCD colour charge is controlled; this contrasts with inclusive jets, which mostly originate from a mixture of light quarks and gluons. The $b$-jets produced in hadronic collisions consist of jets in which the $b$-quark is directly produced in the hard scattering (e.g. $gg \rightarrow b\bar{b}$) and jets in which a hard-scattered gluon splits into a $b\bar{b}$ pair ($g \rightarrow b\bar{b}$) as part of the showering process.

Inclusive jets and $b$-jets are expected to be sensitive to different effects in the QGP. Medium-induced gluon radiation is expected to be suppressed for heavy quarks by the dead-cone effect [7]. As the velocity of the $b$-quark increases the importance of the dead-cone effect decreases. Calculations for Pb+Pb collisions show that the dead-cone effect reduces the suppression of $B$-mesons up to transverse momenta of approximately 50 GeV [8]. This does not simply translate to a specific jet ${{p}_{\textrm{T}}}$ range because $b$-quarks in jets from gluon splitting tend to have lower momentum than those directly from hard scattering at the same jet ${{p}_{\textrm{T}}}$. Differences in the interactions between the QGP and the developing parton shower between $b$-jets and inclusive jets could also arise from differences in how quarks and gluons interact with the QGP [9]. Measurements of heavy-quark jets in heavy-ion collisions are also expected to be sensitive to the mixture of radiative and collisional energy loss in the QGP [10,11,12]. Due to these different effects, which are expected to lead to differences between $b$-jets and inclusive jets in Pb+Pb collisions, it is important to compare the $b{{{\hbox {-jet}}}}$ and inclusive-jet suppression in Pb+Pb collisions over as wide a kinematic range as possible.

Prior measurements have investigated the effects of energy loss for heavy quarks. Many measurements are based on electrons or muons from the semileptonic decay of charm and bottom hadrons [13,14,15,16,17]. Other measurements have partially [18,19,20,21] or fully [22,23,24,25,26,27] reconstructed the heavy-flavour hadrons. One limitation of these measurements is that they do not provide direct information about the total jet energy. This is important because the fragmentation functions of $b$-jets are very different from those of inclusive jets [28]; thus, it is not possible to directly compare the suppression of $b$-quarks with that of light quarks and gluons without full jet measurements. CMS has measured the suppression of $b$-jets in Pb+Pb collisions at 2.76 TeV and found it to be consistent with that of inclusive jets over the jet transverse momentum, $p_{\textrm{T}}^{\textrm{jet}}$, range of 70–250 GeV [29], although the uncertainties are large.

For the measurements reported in this article, jets are clustered with the ${\hbox {anti-}}k_{t}$ algorithm [30] using radius parameters $R= 0.2$ (in both Pb+Pb and $pp$ collisions) and $R = 0.4$ (only in $pp$ collisions). At generator level, a jet is considered as a $b{{{\hbox {-jet}}}}$ if a $b$-hadron with ${{p}_{\textrm{T}}}> 5$ GeV is found within an angular distance¹ of $\Delta R = 0.3$ from the jet axis. The $b$-jets are identified statistically from a sample of jets containing muons. The muons from $b$-hadron decays are distinguished from muons arising from charm- and light-hadron decays by the difference between their transverse momenta, $p_{{\textrm{T}}}^{\mu }$, relative to the jet+$\mu$ axis. The jet+$\mu$ axis is defined in the transverse plane by

$$\begin{aligned} \vec {u}_\text {T}^{\text {jet}+\mu } = \frac{\vec p_\text {T}^{\mu }+\vec p_\text {T}^{\textrm{jet}}}{\left| \vec p_\text {T}^{\mu }+\vec p_\text {T}^{\textrm{jet}}\right| } \end{aligned}$$

(1)

(where the vector notation indicates the vector quantity is meant instead of the magnitude of the vector), and the transverse momentum of the muon relative to the jet axis, $p_\textrm{T}^\textrm{rel}$, is defined as

$$\begin{aligned} p_\textrm{T}^\textrm{rel}\ = \left| \vec p_\textrm{T}^{\mu } \times \vec {u}_\text {T}^{\text {jet}+\mu } \right| . \end{aligned}$$

(2)

This method was used previously by ATLAS for $pp$ collisions at 7 TeV and the results were compared with those based on the reconstruction of secondary vertices [31]. Only $R= 0.2$ jets are used in Pb+Pb collisions because they have better angular resolution and have a smaller contribution from accidental jet–muon overlaps in Pb+Pb collisions; both of these features improve the performance of the $p_\textrm{T}^\textrm{rel}$ method.

This paper reports measurements of inclusive jet and $b{{{\hbox {-jet}}}}$ cross-sections in $pp$ collisions and the per-event yield for inclusive jets and $b$-jets in Pb+Pb collisions at 5.02 TeV. In both collision systems, the spectra are corrected to the generator level (just before the $b$-hadron decays for $b$-jets). For $R= 0.2$ jets the nuclear modification factor, $R_{\textrm{AA}}$, is calculated for $b$-jets as

where $\text {d}^2 N_\text {AA}^{b{{{\hbox {-jet}}}}}/\text {d}{{p}_{\textrm{T}}}\text {d}y$ is the yield of $b$-jets in Pb+Pb collisions for the centrality, ${{p}_{\textrm{T}}}$, and rapidity range of interest; $\text {d}^{2}\sigma _{pp}^{b{{{\hbox {-jet}}}}}/\text {d}{{p}_{\textrm{T}}}\text {d}y$ is the $b{{{\hbox {-jet}}}}$ cross-section in $pp$ collisions; $\langle T_{\text {AA}}\rangle$ is the nuclear thickness function [32]; and $N_\text {evt}$ is the number of minimum-bias (MB) Pb+Pb events for the centrality selection under consideration. Analogously, the nuclear modification factor for inclusive jets is defined as

where the yield and cross-section are for inclusive jets and the other quantities remain the same.

The measurements are performed in Pb+Pb collisions at $\sqrt{s_{_\text {NN}}}=5.02$ TeV collected during 2018, with an integrated luminosity of 1.4 $\text {nb}^{-1}$ for the $b{{{\hbox {-jet}}}}$ sample and 1.7 $\text {nb}^{-1}$ for the inclusive jet sample, and in $pp$ collisions at $\sqrt{s}=5.02$ TeV collected during 2017, with an integrated luminosity of 260 ${\textrm{pb}^{-1}}$. The measurements are presented for jets with $80<{{p}_{\textrm{T}}}<290$ GeV and $|y|<2.1$.

The paper is structured as follows. Section 2 describes the ATLAS detector, and Sect. 3 discusses the selection procedure applied to the data. The data analysis is presented in Sect. 4 and the systematic uncertainties are presented in Sect. 5. The results and a summary are presented in Sects. 6 and 7.

2 ATLAS detector

The ATLAS detector [33] at the LHC covers nearly the entire solid angle around the collision point. It consists of an inner tracking detector surrounded by a thin superconducting solenoid, electromagnetic and hadronic calorimeters, and a muon spectrometer incorporating three large superconducting toroidal magnets.

The inner-detector system (ID) is immersed in a 2 T axial magnetic field and provides charged-particle tracking in the range $|\eta | < 2.5$. The high-granularity silicon pixel detector covers the vertex region, and is composed of four layers including the insertable B-layer [34, 35]. It is followed by the silicon microstrip tracker, which usually provides four two-dimensional measurement points per track. These silicon detectors are complemented by the transition radiation tracker (TRT), which enables radially extended track reconstruction up to $|\eta | = 2.0$.

The calorimeter system covers the pseudorapidity range $|\eta | < 4.9$. Within the region $|\eta |< 3.2$, electromagnetic calorimetry is provided by barrel and endcap high-granularity lead/liquid-argon (LAr) electromagnetic calorimeters, with an additional thin LAr presampler covering $|\eta | < 1.8$ to correct for energy loss in material upstream of the calorimeters. The hadronic calorimeters have three sampling layers longitudinal in shower depth in $|\eta | < 1.7$ and four sampling layers in $1.5< |\eta | < 3.2$, with a slight overlap in $\eta$. The solid angle coverage is completed with forward copper/LAr and tungsten/LAr calorimeter modules (FCal) optimised for electromagnetic and hadronic measurements respectively.

The muon spectrometer (MS) comprises separate trigger and high-precision tracking chambers measuring the deflection of muons in a magnetic field generated by the superconducting air-core toroids. The field integral of the toroids ranges between 2.0 and 6.0 T   m across most of the detector. A set of precision chambers covers the region $|\eta | < 2.7$ with three layers of monitored drift tubes, complemented by cathode strip chambers in the forward region, where the background is highest. The muon trigger system covers the range $|\eta | < 2.4$ with resistive plate chambers in the barrel, and thin gap chambers in the endcap regions.

The zero-degree calorimeters (ZDCs) are located symmetrically at $z = \pm 140$ m and cover $|\eta | > 8.3$ during the Pb+Pb data-taking period. The ZDCs use tungsten plates as absorbers and quartz rods sandwiched between the tungsten plates as the active medium. In Pb+Pb collisions the ZDCs primarily measure spectator neutrons, which are 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 energy of a single neutron.

A two-level trigger system is used to select interesting events [36]. The first-level trigger is implemented in hardware and uses a subset of detector information, including ZDC coincidences in Pb+Pb collisions, to reduce the event rate to a design value of at most 100 kHz. This is followed by a software-based high-level trigger which reduces the event rate to several kHz. An extensive software suite [37] is used in the reconstruction and analysis of real and simulated data, in detector operations, and in the trigger and data acquisition systems of the experiment.

3 Data selection and Monte Carlo samples

All events are required to have at least one reconstructed vertex, to have been collected during stable beam conditions, and to satisfy detector and data-quality requirements [38].

These events were selected by two sets of triggers: single-jet triggers, and muon–jet triggers requiring a muon spatially matched to a jet at the trigger level [39]. In both Pb+Pb and $pp$ collisions, the muon–jet triggers require a muon with ${{p}_{\textrm{T}}}>$ 4 GeV matched to an $R = 0.4$ jet with various ${{p}_{\textrm{T}}}$ thresholds. The spatial matching of the jet to the muon is made within a cone of size $\Delta R =0.5$ around the jet axis; this matching is fully efficient. All events are in a kinematic range where the jet trigger is fully efficient.

The overlap area of two colliding nuclei in Pb+Pb collisions is characterized by the total transverse energy deposited in the FCal [40]. This analysis uses three centrality intervals which are defined according to successive percentiles of the $\sum E_\text {T}^\text {FCal}$ distribution obtained in MB collisions. The centrality regions used in this analysis, going from the most central (largest $\sum E_\text {T}^\text {FCal}$) collisions to the peripheral (lowest $\sum E_\text {T}^\text {FCal}$) collisions are 0–20%, 20–50%, and 50–80%. The values of $\langle T_{\textrm{AA}}\rangle$ and the number of binary nucleon–nucleon collisions, $\langle N_\text {coll}\rangle$, for each centrality interval are evaluated by a Monte Carlo (MC) Glauber model analysis [32, 41] of the $\sum E_\text {T}^\text {FCal}$ distribution, and are shown in Table 1.

Table 1 The $\langle T_{\textrm{AA}}\rangle$ and $\langle N_\text {coll}\rangle$ values and uncertainties in each centrality bin. These are the results from the Glauber modelling of the summed transverse energy in the forward calorimeters, $\sum E_\text {T}^\text {FCal}$

A small fraction of the triggered Pb+Pb events $(< 0.5\%)$ contain multiple collisions, known as ‘pile-up’. The expected anti-correlation between $\sum E_\text {T}^\text {FCal}$ and the number of neutrons detected in the ZDCs is used to reject these pile-up events. The pile-up contribution is not rejected in $pp$ collisions.

This analysis uses several MC samples to evaluate the performance of the detector and the analysis procedure, and to correct the measured distributions for detector effects. All MC samples were produced with the full ATLAS detector simulation based on the $\textsc {GEANT4}4$ toolkit [42, 43]. Dijet samples were generated with Pythia 8 [44], using the parameter values of the A14 tune [45] with the NNPDF23lo set of parton distribution functions (PDFs) [46]. These were generated both with no additional requirements and with the requirement of a muon with ${{p}_{\textrm{T}}}>$ 3 GeV at the generator level (such as from heavy-flavour hadron decays). The EVTGEN  package [47] was used to fully simulate $b$-hadron decays.

In $pp$ collisions, pile-up from additional interactions in the same bunch crossing was generated by Pythia 8, using the parameter values of the A3 tune [48] with the NNPDF23lo PDFs; the distribution of the number of extra collisions was matched to that of data. In Pb+Pb collisions the simulated events were overlaid with events from a dedicated sample of Pb+Pb data events. This sample was recorded with a combination of MB triggers and total energy triggers to increase the number of events from central collisions. This ‘data overlay’ sample was reweighted on an event-by-event basis to obtain the same centrality distribution as in the jet-triggered data sample.

For muon reconstruction studies, prompt $(pp \rightarrow J/\psi \rightarrow \mu \mu )$ and non-prompt $(pp \rightarrow bb\rightarrow J/\psi \rightarrow \mu \mu )$ samples of $J/\psi$ events were produced with Pythia 8 and corrected for electromagnetic radiation with Photos [49]. The A14 tune and CTEQ6L1 PDFs [50] were used. Generator-level $pp$ Herwig++ [51] events, using the UEEE5 tune [52] and the CTEQ6L1 PDFs [50], were used to study systematic uncertainties.

4 Analysis

4.1 Jet reconstruction

All jets are reconstructed using procedures which follow those used by ATLAS for previous jet measurements in Pb+Pb collisions [2, 53]. Jets are reconstructed using the ${\hbox {anti-}}k_{t}$ algorithm [30] implemented in the FastJet software package [54]. In both $pp$ and Pb+Pb collisions, jets with $R=0.2$ and $R = 0.4$ are formed by clustering calorimetric towers of spatial size $\Delta \eta \times \Delta \phi =0.1\times \pi /32$. The energies in the towers are obtained by summing the energies of calorimeter cells at the electromagnetic energy scale [55] within the tower boundaries. In Pb+Pb collisions, a background subtraction procedure is applied to estimate, within each event, the underlying event (UE) average transverse energy density, $\rho (\eta ,\phi )$, where the $\phi$ dependence is due to global azimuthal correlations in the particle production from hydrodynamic flow [40]. The modulation accounts for the contribution to the UE of the second-, third-, and fourth-order azimuthal anisotropy harmonics characterized by values of flow coefficients $v_n^\textrm{UE}$ [40]. Additionally, the UE is also corrected for $\eta$- and $\phi$-dependent non-uniformities of the detector response by correction factors derived in MB Pb+Pb data. In $pp$ collisions, the same background subtraction procedure is applied without the $\phi$-dependent modulation and without the correction for $\eta$- and $\phi$-dependent non-uniformities to remove the pile-up contribution to the jet.

An iterative procedure is used to remove the impact of jets on the estimated $\rho$ and $v_n^\textrm{UE}$ values. The first estimate of the average transverse energy density of the UE, $\rho (\eta )$, is evaluated in 0.1 intervals of $\eta$, excluding towers within $\Delta R =0.4$ of ‘seed’ jets. In the first subtraction step, the seeds are defined to be an union of $R = 0.2$ jets and $R=0.4$ track-jets. Track-jets are reconstructed by applying the ${\hbox {anti-}}k_{t}$ algorithm with $R = 0.4$ to charged particles with ${p}_{\hbox {T}}> 4$ GeV. The $R=0.2$ jets must pass a requirement on the minimum value of the tower $E_{\textrm{T}}$ and on a ratio of maximum tower $E_{\textrm{T}}$ to average tower $E_{\textrm{T}}$, while the track-jets are required to have ${p}_{\hbox {T}}> 7$ GeV. The background is then subtracted from each tower constituent and the jet kinematics are recalculated. After the first iteration, the $\rho$ and $v_n$ values are updated by excluding from the UE determination the regions within $\Delta {R} = 0.4$ of both the track-jets and the newly reconstructed $R=0.2$ jets with ${{p}_{\textrm{T}}}> 25$ GeV (8 GeV) in Pb+Pb ($pp$) collisions. The updated $\rho$ and $v_n^\textrm{UE}$ values are used to update the jet kinematic properties in the second iteration.

Jet $\eta$- and ${p}_{\hbox {T}}$-dependent correction factors derived in simulations are applied to the measured jet energy to correct for the calorimeter energy response [56]. 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 [57]. This calibration is followed by a cross-calibration which relates the jet energy scale of jets reconstructed by the procedure outlined in this section to the jet energy scale in 13 TeV $pp$ collisions [58].

Jets are defined at the truth level in the MC sample before detector simulation by applying the ${\hbox {anti-}}k_{t}$ algorithm with the appropriate R value to stable particles with a proper lifetime greater than 30 ps, but excluding muons and neutrinos, which do not leave significant energy deposits in the calorimeter. The $\Delta R$ between the truth jet and reconstructed jet is required to be $\Delta R< 0.15$ (0.30) for $R= 0.2$ ($R = 0.4$) jets. The corrections for muons and neutrinos from semileptonic decays are discussed below.

4.2 Muon reconstruction

Muon candidates in both Pb+Pb and $pp$ collisions are formed by combining charged-particle tracks reconstructed in the ID and the MS that pass the ‘tight’ selection requirements detailed in Ref. [59], except the requirement on the number of TRT hits.

Muons are selected with $p_{{\textrm{T}}}^{\mu }> 4$ GeV and $|\eta ^{\mu }|<2.4$ requirements. If the muon is required in the trigger, the data is corrected to account for the muon trigger inefficiency, and the reconstructed muon must be within $\Delta R =0.01$ of the trigger muon object. The muon selection is the same in both $pp$ collisions and Pb+Pb collisions. Muons and jets are judged to be associated if $\Delta R(\text {jet},\mu )$ is less than the jet radius, R. If more than one muon passes this selection, the muon with the largest momentum is used.

The muon trigger efficiency with respect to reconstructed muons is estimated from $pp$ data by using the tag-and-probe method [16, 60] in bins of $p_{{\textrm{T}}}^{\mu }$ and $\eta ^{\mu }$. A small centrality dependence of the muon trigger efficiency was observed in Pb+Pb data; this is corrected by an extra factor, which is the Pb+Pb to $pp$ data-driven efficiency ratio, applied to Pb+Pb data as a function of the centrality. In $pp$ collisions, the trigger efficiency plateaus at 78% for $p_{{\textrm{T}}}^{\mu }> 6$ GeV and $|\eta ^{\mu }|<1.05$, and at 90% for $p_{{\textrm{T}}}^{\mu }> 9$ GeV and $|\eta ^{\mu }|>1.05$. In Pb+Pb collisions, the efficiency is lower than in $pp$ collisions by 9% in the most central collisions; the efficiency in peripheral Pb+Pb collisions is the same as in $pp$ collisions.

The muon reconstruction efficiency is estimated from simulated prompt and non-prompt $J/\psi \rightarrow \mu \mu$ events using the tag-and-probe method [16, 59], in fine bins of $p_{{\textrm{T}}}^{\mu }$ and $\eta ^{\mu }$. Mis-modelling of the reconstruction performance in simulation, quantified by the ratio of measured efficiencies in $pp$ data and the simulation, is accounted for by applying a multiplicative correction which changes the efficiency by less than 5%. For $p_{{\textrm{T}}}^{\mu }> 10$ GeV in $pp$ collisions, the muon reconstruction efficiency plateaus at 90% for $|\eta ^{\mu }|<1.05$ and at 95% for $|\eta ^{\mu }|>1.05$. No difference between the muon reconstruction efficiencies in $pp$ and Pb+Pb collisions was observed and the same values were used in both collision systems.

4.3 b-jet yield reconstruction

4.3.1 Templates and fitting

At the generator level, the jet flavour is defined by matching jets to hadrons with ${{p}_{\textrm{T}}}> 5$ GeV. The jet is considered a $b{{{\hbox {-jet}}}}$ if a $b$-hadron is found within $\Delta R =0.3$ of the jet axis; otherwise, if a c-hadron is found within the same distance the jet is labelled as a c-jet. All other jets are considered to be light-jets.

Since the muons in the momentum range of interest generally do not stop in the calorimeter, their momentum is not included in the calorimetric jet ${{p}_{\textrm{T}}}$. In order to better characterize the jets, the jet+$\mu$ scale is defined as the ${{p}_{\textrm{T}}}$ of the sum of the muon and jet four-vectors. The $p_\textrm{T}^\textrm{rel}$ distributions in the data and MC samples are constructed for selections in this jet+$\mu$ object at the reconstruction level, $p_{\textrm{T, reco}}^{\text {jet}+\mu }$. In order to form a jet+$\mu$ pair, jets are required to have a minimum calorimetric $p_{\textrm{T, reco}}^\text {jet}$ of 40 GeV (58 GeV) in $pp$ (Pb+Pb) collisions.

In semileptonic $b$-hadron decays, the ${{p}_{\textrm{T}}}$ of the lepton relative to the jet+$\mu$-axis (see Eqs. (1) and (2)), $p_\textrm{T}^\textrm{rel}$, is used to distinguish between b-, c-, and light-jets[31]. Due to the large mass of $b$-hadrons relative to hadrons containing charm or light quarks, their decay products are more energetic. Consequently, muons originating from $b$-hadron decays have a harder $p_\textrm{T}^\textrm{rel}$ spectrum than muons in c-jets and light-jets.

The templates distributions of $p_\textrm{T}^\textrm{rel}$ for b- and c-jets are extracted from the muon-filtered MC samples. The $b$-hadron mixture is taken from Tevatron measurements [61], which are consistent with those from the LHCb Collaboration [62]. The c-hadron mixture is taken from default Pythia 8 simulation. The light-jet template is obtained from track-jet pairs in the data inclusive-jet sample as in Refs. [31, 63] (in the same centrality class as the jet+$\mu$ pair for Pb+Pb collisions). The tracks are selected with the standard selection, with no additional requirements on the distance of closest approach to the vertex. The ${{p}_{\textrm{T}}}$ spectra of the tracks are reweighted to reproduce the muon ${{p}_{\textrm{T}}}$ spectrum from light-jets from the inclusive dijet MC sample.

In $pp$ collisions, these three templates are inputs to a template fit of the data $p_\textrm{T}^\textrm{rel}$ spectrum to obtain the fractions of b-, c-, and light-jets. A binned maximum-likelihood fit is performed using the RooFit framework [64]. The model for the fit is defined as a normalized sum of three templates: light-jets ($F_{\textrm{l},i}$($p_\textrm{T}^\textrm{rel}$)), c-jets ($F_{\textrm{c},i}$($p_\textrm{T}^\textrm{rel}$)), and $b$-jets($F_{\textrm{b},i}$($p_\textrm{T}^\textrm{rel}$)). The $pp$ fit model, $M^{pp}$, is structured as

$$\begin{aligned}{} & {} M^{pp}_{i}\!\left( p_\textrm{T}^\textrm{rel}\right) = f_{\textrm{b},i}F_{\textrm{b},i}\!\left( p_\textrm{T}^\textrm{rel}\right) + \left( 1-f_{\textrm{b},i}\right) \\{} & {} \quad \times \left[ f_{\textrm{c},i}F_{\textrm{c},i}\!\left( p_\textrm{T}^\textrm{rel}\right) + \left( 1-f_{\textrm{c},i}\right) F_{\textrm{l},i}\!\left( p_\textrm{T}^\textrm{rel}\right) \right] \end{aligned}$$

where $f_{\textrm{b},i}$ is the fraction of $b$-jets in the total sample, $f_{\textrm{c},i}$ is the fraction of c-jets in the non-$b{{{\hbox {-jet}}}}$ sample, and the index i denotes the ${{p}_{\textrm{T}}}$ bin. The fit model has two free parameters, $f_{\textrm{b},i}$ and $f_{\textrm{c},i}$ for each $p_{\textrm{T, reco}}^{\text {jet}+\mu }$ bin.

In Pb+Pb collisions the model has an additional term which accounts for combinatoric muon–jet pairs. These are muon–jet pairs which pass the $\Delta R$ matching requirements, but the muon and jet do not come from the same hard scattering. These are the most common in central Pb+Pb collisions, where the multiplicity of jets and muons is the highest. In $pp$ collisions, this effect is negligibly small. For the combinatoric pairs, both the shape, $F_{\textrm{mix},i}$, and the amplitude, $f_{\textrm{mix},i}$, are determined by event mixing in the Pb+Pb data. This procedure uses a muon from one event and a jet from another event; all the analysis selections are applied, including the requirement on the $\Delta R$ separation between the muon and jet. Events which are mixed together are required to have $\sum E_\text {T}^\text {FCal}$ values that agree within 0.05 TeV and vertex z-positions that agree within 10 mm. The nominal data sample and mixed-event sample are found to agree well in the large $\Delta R$ region where there is no correlated signal. The Pb+Pb model, $M^\text {Pb+Pb}$, is structured as

$$\begin{aligned} M^\text {Pb+Pb}_{i}\!\left( p_\textrm{T}^\textrm{rel}\right)= & {} \left( 1-f_{\textrm{mix},i}\right) \times M^{pp}_{i}\!\left( p_\textrm{T}^\textrm{rel}\right) \\{} & {} + f_{\textrm{mix},i}\times F_{\textrm{mix},i}\!\left( p_\textrm{T}^\textrm{rel}\right) . \end{aligned}$$

The background fraction is largest in central collisions and is always less than 1.5%.

The data $p_\textrm{T}^\textrm{rel}$ distributions are resampled 500 times and the mean and width of the range of results are used to set the central value and uncertainties in the fit. The statistical uncertainty of the templates is taken to be part of the statistical uncertainty of the yield. Example fits for $pp$ and Pb+Pb collisions are shown in Figs. 1 and 2, for low and high $p_{\textrm{T, reco}}^{\text {jet}+\mu }$ respectively.

Fig. 1

The $p_\textrm{T}^\textrm{rel}$ distributions in $pp$ collisions (top left) and 50–80% (top right), 20–50% (bottom left), and 0–20% (bottom right) centrality Pb+Pb collisions. The data are shown for the range $80.7< p_{\textrm{T, reco}}^{\text {jet}+\mu }< 95.0$ GeV. The stacked histograms show the fit results. Middle and bottom panels of each plot show the pulls and data-to-fit ratios, respectively. The error bars in the ratio plots show the statistical uncertainties of the data and MC samples

Fig. 2

The $p_\textrm{T}^\textrm{rel}$ distributions in $pp$ collisions (top left) and 50–80% (top right), 20–50% (bottom left), and 0–20% (bottom right) centrality Pb+Pb collisions. The data are shown for the range $113.4< p_{\textrm{T, reco}}^{\text {jet}+\mu }< 142.3$ GeV. The stacked histograms show the fit results. Middle and bottom panels of each plot show the pulls and data-to-fit ratios, respectively. The error bars in the ratio plots show the statistical uncertainties of the data and MC samples

The raw $b{{{\hbox {-jet}}}}$ ${{p}_{\textrm{T}}}$ spectrum is constructed in each Pb+Pb centrality bin and in $pp$ collisions by taking each $f_{\textrm{b},i}$ value and multiplying it by the number of jets in i-th bin in the total $p_{\textrm{T, reco}}^{\text {jet}+\mu }$ spectrum.

4.3.2 Comparing muon momentum distributions in the data and simulations

The $b{{{\hbox {-jet}}}}$ fragmentation function in Pb+Pb collisions has not been measured. Significant modification to the $b$-hadron longitudinal momentum due to jet quenching could bias the $b{{{\hbox {-jet}}}}$ template. In order to ensure that the simulations reproduce the data sufficiently well, a check of the longitudinal momentum distribution of the muon with respect to the jet+$\mu$ axis is performed. The fitting results as a function of centrality and $p_{\textrm{T, reco}}^{\text {jet}+\mu }$ from the previous subsection are used to weight the jet+$\mu$ pairs from the MC samples (used for the charm and bottom templates) and those used in constructing the data-driven light-jet template to have the same flavour fractions as the data. Muons which have a transverse distance to the primary vertex, $d_0$, greater than 0.25 mm are selected; this requirement further enhances the heavy-flavour fractions relative to the light jets. For these jet+$\mu$ pairs, the distribution of

$$\begin{aligned} z\equiv p_{{\textrm{T}}}^{\mu }\cos \!\left( \theta \right) / p_{\textrm{T, reco}}^{\text {jet}+\mu }\, \end{aligned}$$

where $\theta$ is the angle between the muon and the jet+$\mu$ axis, is constructed. This combined distribution from the pairs in the charm-, bottom-, and light-jet templates is compared with the data distribution for the same quantity. Since the light-jet template is data-driven, it should agree between the data and the template, and thus the comparison between data and simulations is sensitive to the degree to which the MC modelling of the charm and bottom templates agrees with the data. These distributions are shown in Fig. 3 for $pp$ and 0–20% centrality Pb+Pb collisions. The data and the simulations (including the data-driven light-jet template) agree in both $pp$ and Pb+Pb collisions. Thus, within the current uncertainties, the longitudinal momentum fraction of the muon is unmodified in Pb+Pb collisions and the $p_\textrm{T}^\textrm{rel}$ distributions are largely sensitive to the $b$-hadron decay kinematics.

Fig. 3

The $z$ distributions for data and simulations (including the data-driven light-jet template) for $pp$ collisions (top row) and 0–20% centrality Pb+Pb collisions (bottom row) for two $p_{\textrm{T, reco}}^{\text {jet}+\mu }$ selections. In each figure the top panel shows the distributions themselves and the bottom panel shows the ratio of data to simulations. The error bars in the ratio plots show the statistical uncertainties on the data and MC samples

4.4 Corrections to the raw spectra

The raw jet spectra are constructed from all the measured jets for the inclusive jets and from the muon–jet pairs, scaled by the $f_{\textrm{b},i}$ values, for the $b$-jets after correction for the muon trigger efficiency. The raw spectra are unfolded to account for bin migrations due to the finite jet energy resolution, and any reconstruction inefficiency for the jets and muons (and muon and neutrino energies in the $b{{{\hbox {-jet}}}}$ sample). Both the $b{{{\hbox {-jet}}}}$ and inclusive jet ${{p}_{\textrm{T}}}$ spectra are unfolded using the one-dimensional (1D) Bayesian unfolding [65] from the RooUnfold software package [66]. The $b{{{\hbox {-jet}}}}$ response matrices are built from the muon-filtered MC samples using truth-level $b$-jets (including the muon and neutrino momenta) that are matched to reconstructed jets in simulations and have $|y_\text {truth}^{b{{{\hbox {-jet}}}}}|<2.1$. The inclusive jet response matrices are built from the dijet MC samples using all truth jets and not including any muon or neutrino momenta.

Fig. 4

The relative systematic uncertainties for several categories, as a function of b-jet ${{p}_{\textrm{T}}}$ for (top) the $pp$ cross-section for $R = 0.4$ (left) and $R= 0.2$ (right) jets, and (bottom) the Pb+Pb per-event-yield in peripheral (left) and central (right) collisions for $R= 0.2$ jets

Fig. 5

Relative size of systematic uncertainties for several categories, as a function of jet ${{p}_{\textrm{T}}}$ for $b{{{\hbox {-jet}}}}$ $R_{\textrm{AA}}$ (left column) and inclusive jet $R_{\textrm{AA}}$ (right column), in peripheral (top row), semi-central (middle row) and central (bottom row) Pb+Pb collisions

The response matrices are generated separately for $pp$ collisions and for each centrality interval in Pb+Pb collisions. To better represent the data, the response matrices are reweighted along the truth-$p_{\textrm{T}}^{\textrm{jet}}$ axis by the reconstruction-level data to simulation ratio. The number of iterations in the unfolding was chosen such that the result is stable when changing the number of iterations while minimizing the amplification of statistical uncertainties. Four iterations are used in all cases presented here. The reconstructed $p_{\textrm{T, reco}}^{\text {jet}+\mu }$ values are required to be larger than 72 GeV in Pb+Pb collisions and 64 GeV in $pp$ collisions for both $R= 0.2$ and $R = 0.4$ jets. The inclusive-jet minimum reconstructed ${{p}_{\textrm{T}}}$ is required to be larger than 72 GeV for both Pb+Pb and $pp$ collisions. Reconstructed jets below these thresholds that match to truth-${{p}_{\textrm{T}}}$values in the measurement region are corrected via the unfolding procedure. Fully unfolded results are presented starting from 80 GeV for both categories of jets.

As described in Sect. 3, the main MC sample used in the $b{{{\hbox {-jet}}}}$ analysis required the truth-muon ${{p}_{\textrm{T}}}$ to be larger than 3 GeV. The missing part of the cross-section from $b$-jets with muons below this threshold is addressed through an acceptance correction based on the Pythia 8 dijet MC samples which is applied after unfolding. The size of the correction is approximately 22% (25%) at 80 GeV and 19% (23%) at 250 GeV for $R= 0.2$ ($R = 0.4$) jets.

4.5 Observables

The $b{{{\hbox {-jet}}}}$ cross-section in $pp$ collisions is defined as

(3)

where $L_\text {int}$ is the integrated luminosity and $\mathcal {B}$ is the branching ratio which includes direct $(b\rightarrow \mu )$ and cascade $(b\rightarrow c\rightarrow \mu )$ semileptonic decays. The value of $\mathcal {B}$ is $(20.6\pm 0.6)$% [28]. The quantity is the $b{{{\hbox {-jet}}}}$ yield after the unfolding and acceptance correction described above, and $\Delta {{p}_{\textrm{T}}}$ and $\Delta y$ are the widths of the ${{p}_{\textrm{T}}}$and y bins. Following the same approach, the per-event yield of $b$-jets measured in Pb+Pb collisions is calculated as

(4)

where $N_\text {evt}$ is the number of MB events in the centrality class ‘cent’. The inclusive jet cross-section and per-event yields are defined in the same way as in Eq. (3) and Eq. (4) but without the $\mathcal {B}$ factor.

5 Systematic uncertainties

The systematic uncertainties of the jet cross-sections and per-event yields common to both the inclusive jet and the $b{{{\hbox {-jet}}}}$ measurements arise from the jet energy scale and resolution, the unfolding procedure, luminosity ($pp$ only), and $\langle T_{\textrm{AA}}\rangle$ determination (Pb+Pb only). There are additional contributions to the $b{{{\hbox {-jet}}}}$ measurements from the modelling of the $p_\textrm{T}^\textrm{rel}$ distributions, the fitting procedure, the branching ratio of $b$-hadrons to muons, and the muon performance. The uncertainties are discussed in detail below.

The systematic uncertainty of the jet energy scale (JES) has five parts. First, there is a centrality-independent baseline component that is determined from in situ studies of the calorimeter response for jets reconstructed with the procedure used in 13 TeV $pp$ collisions [55, 67], including an additional term for $b{{{\hbox {-jet}}}}$  energy scale [55]. The second is a centrality-independent component which accounts for the relative energy scale difference between the jet reconstruction procedure used in this paper and that in 13 TeV $pp$ collisions [58]. Potential inaccuracies in the MC description of the relative abundances of jets initiated by quarks and gluons and of the calorimetric response to quark and gluon jets are accounted for by a third, JES flavour, component; this component is estimated independently for the inclusive-jet and $b{{{\hbox {-jet}}}}$ energy scale by varying the quark and gluon fractions from their Pythia 8 values to those extracted from Herwig++. The fourth, centrality-dependent, component (in Pb+Pb collisions only) accounts for a different structure [68], and possibly a different detector response, for jets in Pb+Pb collisions that is not modelled by the simulations. It is evaluated by the method used for 2015 and 2011 data [58] that compares the calorimetric jet ${{p}_{\textrm{T}}}$ and the sum of the transverse momentum of charged particles within the jet in data and MC samples. The size of the centrality-dependent uncertainty in the JES reaches 1.2% in the most central collisions; it is smaller for more peripheral collisions. The systematic uncertainties from the JES discussed above are derived for $R=0.4$ jets and applied to both $R = 0.4$ and $R= 0.2$ jets. An additional component applies only to $R= 0.2$ jets and accounts for a potential uncertainty difference between $R = 0.4$ and $R = 0.2$ jets. The uncertainty is assessed by comparing the ratio of $R= 0.2$ jet ${{p}_{\textrm{T}}}$ to $R = 0.4$ jet ${{p}_{\textrm{T}}}$ in data and simulations.

The uncertainty due to the jet energy resolution (JER) is evaluated by repeating the unfolding procedure with modified response matrices, where an additional contribution is added to the resolution of the reconstructed ${p}_{\hbox {T}}$ using a Gaussian smearing procedure. The smearing factor is evaluated using an in situ technique in 13 TeV $pp$ data that involves studies of dijet ${p}_{\hbox {T}}$ balance [69, 70]. Additionally, an uncertainty is included to account for differences between the tower-based jet reconstruction and the jet reconstruction used in analyses of 13 TeV $pp$ data, as well as differences in calibration procedures. Similarly to the JES, an additional uncertainty in the JER accounting for differences between $R = 0.2$ and $R = 0.4$ jets is added. The resulting uncertainty from the JER is symmetrized.

Uncertainties related to muons are associated with the trigger and reconstruction efficiency measurement. The systematic uncertainties are estimated by varying the tag-and-probe method as described in Refs. [59, 60]. Additionally, the statistical uncertainty of the factor used to correct the centrality dependence of the trigger efficiency in Pb+Pb collisions and the difference between the data-driven reconstruction efficiencies in Pb+Pb and $pp$ collisions are taken as muon systematic uncertainties that apply only to Pb+Pb collisions.

The uncertainties in modelling the $p_\textrm{T}^\textrm{rel}$ distributions come from several sources, which are common to $pp$ and Pb+Pb collisions. For the fraction of $b$-hadrons which arise from gluon splitting, the analysis uses that from Pythia 8 simulation as the central value. In $pp$ collisions, the gluon-splitting fraction is reweighted to the value obtained from the Herwig++ sample [71]. In Pb+Pb collisions, there is no information about the modification of the gluon-splitting contribution to $b$-jets. Additionally, the fragmentation functions of $b$-jets in Pb+Pb collisions have not been measured. In order to cover both of these effects, the gluon-splitting fraction is conservatively varied between zero and 100%. For the c-jets, the gluon-splitting uncertainty is estimated by varying the Pythia 8 value of the gluon-splitting fraction by a factor of two, based on measurements in Ref. [72]. The fraction of muons which arise from $b$-hadron decays which include an intermediate $D$-meson is varied in accord with Ref. [28]. The fractions of the various $b$-hadron species are varied according to the world average values in Ref. [61]. The fraction of c-baryons was varied according to ALICE measurement [27] with negligible impact on the results. The modelling of the muon momentum in the $b$-hadron rest frame is crucial for the $p_\textrm{T}^\textrm{rel}$ method. The modelling in Pythia 8 is used for the central values. The Pythia 8 distributions were compared with the measurement from DELPHI [73] and the difference between Pythia 8 and the DELPHI measurement is used as a systematic uncertainty. The uncertainty in the light-jet template is evaluated by using muons with a distance of closest approach to the collision vertex smaller than 0.01 mm, to minimize the contribution from heavy-flavour jets, and remaking the templates. In $R= 0.2$ jets in $pp$ collisions, the dominant components of modelling uncertainties stem from the light-jet template and the fraction of $b$-hadrons; for $R = 0.4$ jets, the light-jet template uncertainty dominates. For Pb+Pb collisions, the dominant modelling uncertainties stem from the light-jet template and gluon-splitting components.

The uncertainty in the unfolding procedure is determined in all cases by constructing response matrices from the MC distributions without the reweighting factors that are used to match the MC distributions to those in data. The uncertainty due to using a particular MC model in unfolding $b$-jets is addressed by reweighting the 2D ($p_{\textrm{T}}^{\textrm{jet}}$, $p_{{\textrm{T}}}^{\mu }$) distribution in Pythia 8 to that observed in Herwig++. The inclusive jet analysis has an uncertainty associated with the non-closure of the unfolding procedure when the MC sample is divided and one portion is used in place of the data and the other portion used to generate a response matrix; in the $b{{{\hbox {-jet}}}}$ analysis the closure was found to be consistent with unity within the statistical uncertainties and no uncertainty is added.

Uncertainties from the finite size of the MC samples are combined with the data statistical uncertainties. A systematic uncertainty associated with the template fit procedure is addressed by allowing the templates to deform in Pb+Pb collisions by convolving the nominal template with a Gaussian function where the width parameter is free. Deformations of the templates account for mismodelling due to the large UE, such as via the jet position resolution, muon momentum resolution, or other effects. A similar uncertainty was applied in Ref. [16]. The small non-closure of the fitting procedure, when the $f_{\textrm{b},i}$ extraction is tested in simulations, is also included in the systematic uncertainties.

For each uncertainty component discussed above, the entire analysis procedure is repeated with the variation under consideration and the resulting changes are added in quadrature to form the total systematic uncertainty of the measurement. A summary of the systematic uncertainties for the $R = 0.4$ and $R= 0.2$ $b{{{\hbox {-jet}}}}$ cross-sections is shown in Fig. 4. For the $R = 0.4$ jets, the largest contribution is the modelling component, while for $R= 0.2$ jets, the jet and modelling components have similar magnitudes.

The integrated luminosity determined for 2017 $pp$ data was calibrated using data from dedicated beam-separation scans, also known as van der Meer scans [74]. Sources of systematic uncertainty similar to those examined in the 2012 $pp$ luminosity calibration [74] were studied in order to assess the systematic uncertainties for the 2017 data. The combination of these systematic uncertainties results in a relative uncertainty of 1.6%. The uncertainty of the mean nuclear thickness function arises from geometric modelling uncertainties (nucleon–nucleon inelastic cross-section, Woods–Saxon parameterization of the nucleon distribution) and the uncertainty in the fraction of selected inelastic Pb+Pb collisions. The values of these uncertainties are taken from Ref. [2]. The branching ratio for $b$-hadrons into muons is $(20.6 \pm 0.6)\%$ and is taken from Ref. [28].

The uncertainties which are common to $pp$ and Pb+Pb collisions are treated as correlated when determining the uncertainty in the $R_{\textrm{AA}}$ value, with the exception of gluon splitting, where the uncertainties account for possible production and fragmentation mechanism differences between $pp$ and Pb+Pb collisions, and the unfolding. Similarly, the uncertainties which are common to inclusive jets and $b$-jets cancel out when ratios of cross-sections or $R_{\textrm{AA}}$ values are taken; the remaining uncertainties are the JES flavour component, the $b{{{\hbox {-jet}}}}$-specific JES uncertainty, and the unfolding. A summary of the systematic uncertainties for inclusive jet and $b{{{\hbox {-jet}}}}$ $R_{\textrm{AA}}$ is shown in Fig. 5. The uncertainties in the $R_{\textrm{AA}}^{b{{{\hbox {-jet}}}}}/R_{\textrm{AA}}^\text {inclusive jet}$ ratio are shown in Fig. 6; the uncertainties in the gluon-splitting contribution and deformation of the templates in Pb+Pb collisions dominate in most cases.

Fig. 6

Relative size of systematic uncertainties of the ratio $R_{\textrm{AA}}^{b{{{\hbox {-jet}}}}}/R_{\textrm{AA}}^\text {inclusive jet}$ shown for the most relevant components, as a function of jet ${{p}_{\textrm{T}}}$ for 50–80% (top), 20–50% (middle), and 0–20% (bottom) centrality Pb+Pb collisions

Fig. 7

(Left) Differential cross-section for $b{{{\hbox {-jet}}}}$ production for $R= 0.2$ and $R = 0.4$ jets with $|y|<2.1$ as a function of ${{p}_{\textrm{T}}}$ in 5.02 TeV $pp$ data. (right) Ratio of the predictions to the measured b-jet cross-section in $pp$ collisions at 5.02 TeV for $R = 0.4$ (top) and $R= 0.2$ (bottom) jets. The $R = 0.4$ jets are compared with calculations from Ref. [75] and $R= 0.2$ jets are compared with calculations from Refs. [11, 12, 75, 76]. Both cross-sections are compared with Pythia 8 and Herwig++ calculations. The bands around unity represent the total uncertainty of the data. The text provides additional discussion

Fig. 8

(Left) Cross-section of $R= 0.2$ $b{{{\hbox {-jet}}}}$ and inclusive jet production in $pp$ collisions at 5.02 TeV, and (right) the $b{{{\hbox {-jet}}}}$ to inclusive jet cross-section ratio, together with Pythia 8 simulation and measurements from the CMS Collaboration at 7 TeV for $R = 0.5$ jets [78]. For the current measurement, the boxes represent the systematic uncertainties and the error bars represent the statistical uncertainties. For the CMS data the bars represent the total uncertainty

Fig. 9

Per-event yields scaled by $\langle T_{\textrm{AA}}\rangle$ in Pb+Pb collisions for three centrality classes for $R= 0.2$ $b$-jets (left) and inclusive jets (right). The boxes show the systematic uncertainties and the bars represent the statistical uncertainties. The different centrality classes are offset by the factors shown on the plot for clarity. The values of the $pp$ cross-sections are shown by the black lines; they are offset by the factors shown on the plot. The size of the $T_{\textrm{AA}}$ uncertainties are noted in the legend

6 Results

6.1 Cross-section in $pp$ collisions

Figure 7 shows the b-jet cross-section as a function of b-jet ${{p}_{\textrm{T}}}$ in $pp$ collisions at 5.02 TeV for $R= 0.2$ and $R = 0.4$ jets with $|y|<2.1$. Additionally, $b{{{\hbox {-jet}}}}$ cross-sections for both $R= 0.2$ and $R = 0.4$ jets are compared with theory and generator calculations. The data is in good agreement with the calculations by Li and Vitev [75] for both $R = 0.4$ and $R= 0.2$ jets. This calculation is based on semi-inclusive jet functions where the cross-section is expressed in terms of the parton distribution functions, the hard kernel and the jet functions. The jet functions have terms in $\ln (R)$, which are resummed. The $\textsc {SHERPA}\,\,2.2.4$ calculations [12] for $R= 0.2$ jets underestimate the measured $b{{{\hbox {-jet}}}}$ cross-sections by an amount which increases with ${{p}_{\textrm{T}}}$. A calculation [11, 76] which is based on Pythia 8 where the PDFs and $\alpha _{s}(\text {M}_{Z})$ is set with $\textrm{pSet} = 8$ [44] is approximately consistent with the upper edge of the uncertainty band of the $R= 0.2$ jet yields. The Pythia 8 calculation using the NNPDF23lo PDF [46] and the A14 tune [45] is 20–30% higher than data for both $R = 0.4$ and $R= 0.2$ jets. The Herwig++ calculations [77] using the NNPDF30nlo PDF underestimate the measured cross-sections at both jet radii. There is no uncertainty estimate for the generator calculations, and the sensitivity of the results to the choice of parameter values has not been investigated.

Figure 8 shows a comparison of the $b{{{\hbox {-jet}}}}$ and inclusive jet cross-sections for $R= 0.2$ jets and their ratio. The figure also shows the same ratio from the Pythia 8 MC sample; it agrees with the data. The cross-section ratio measured at 7 TeV for $R = 0.5$ jets with $|y|<0.5$ by the CMS Collaboration [78] is also shown; the jet radii in this measurement are larger than in the present analysis but the results are qualitatively similar. No significant ${{p}_{\textrm{T}}}$ dependence of the $b{{{\hbox {-jet}}}}$ to inclusive jet cross-section ratio is observed.

Fig. 10

$b{{{\hbox {-jet}}}}$ (filled points) $R_{\textrm{AA}}$ for three centrality classes compared with the inclusive jet $R_{\textrm{AA}}$ (open points) at 5.02 TeV. Both $R_{\textrm{AA}}$ measurements are compared with theory calculations [11, 12, 76]. The boxes represent the systematic uncertainties and the error bars represent the statistical uncertainties. For the LIDO calculation, the width of the band shows the variation of the $\mu _\text {min}$ parameter from $1.3\pi T$ (lower edge) to $1.8\pi T$ (upper edge). The boxes at unity represent the scale uncertainties from $\langle T_{\textrm{AA}}\rangle$ and the luminosity determination

6.2 Per-event yields and $R_{\textrm{AA}}$ in Pb+Pb collisions

The inclusive jet and $b{{{\hbox {-jet}}}}$ per-event yields in Pb+Pb collisions scaled by $\langle T_{\textrm{AA}}\rangle$ are shown in Fig. 9 for the three centrality classes used in this analysis overlaid with the values of the $pp$ cross-sections. Figure 10 shows a direct comparison of the inclusive jet and $b{{{\hbox {-jet}}}}$ $R_{\textrm{AA}}$ for each centrality class. The $R_{\textrm{AA}}$ values for both types of jets decrease going from peripheral to central collisions. Both are consistent with unity in peripheral collisions. In central and semi-central collisions the $R_{\textrm{AA}}$ central values for inclusive jets are lower than those for $b$-jets. A difference between the slopes of the inclusive jet and $b{{{\hbox {-jet}}}}$ differential cross-sections could cause the $R_{\textrm{AA}}$ values to differ between the two categories of jets; however, the ratio of the $b{{{\hbox {-jet}}}}$ to inclusive jet cross-section, shown in Fig. 8, does not vary with ${{p}_{\textrm{T}}}$.

Fig. 11

Ratio of $b{{{\hbox {-jet}}}}$ $R_{\textrm{AA}}$ to the inclusive jet $R_{\textrm{AA}}$ for each centrality class. Ratios are compared with theory calculations [11, 12, 76]. The boxes represent the systematic uncertainties and the error bars represent the statistical uncertainties. For the LIDO calculation, the width of the band shows the variation of the $\mu _\text {min}$ parameter from $1.3\pi T$ (upper edge) to $1.8\pi T$ (lower edge)

The $R_{\textrm{AA}}$ values for both inclusive jets and $b$-jets are compared with two theory calculations. The first calculation is the LIDO model [11, 76, 79], which includes both energy loss and diffusion of heavy quarks. The dead-cone effect for $b$-jets is included and the medium is implemented via (2+1)D viscous hydrodynamics with averaged initial conditions. The parameter which controls the coupling between the jet and the medium, $\mu _\text {min}$, is varied between $1.3\pi T$ and $1.8 \pi T$, where T is the temperature of the QGP in the model, in the calculation shown here. The choice of $\mu _\text {min}$ values is motivated by comparisons with other jet measurements. These parameters have been shown [79] to provide a reasonable description of measurements of the $R_{\textrm{AA}}$ values of B- and D-mesons [22, 23]. The calculation by Dai et al. in Ref. [12] is based on a Langevin transport model describing the evolution of $b$-quarks and their collisional energy loss and a higher-twist description of radiative energy loss for both heavy and light partons. This model also includes the dead-cone effect for $b$-jets and also uses a (2+1)D viscous hydrodynamic medium with averaged initial conditions. The parameter controlling the coupling of the jet to the medium, $q_0$, is set to be 1.2 $GeV^2/$fm. The LIDO model shows good agreement with the data for both inclusive jet and $b{{{\hbox {-jet}}}}$ $R_{\textrm{AA}}$, although with the inclusive jets on the low side and the $b{{{\hbox {-jet}}}}$ case on the high side, for all three centralities considering model and data uncertainties. The calculation in Ref. [12] is below the measured $R_{\textrm{AA}}$ in central and semi-central collisions for both $b$-jets and inclusive jets.

In order to assess any difference between the $b{{{\hbox {-jet}}}}$ and inclusive jet $R_{\textrm{AA}}$, the ratio $R_{\textrm{AA}}^{b{{{\hbox {-jet}}}}} / R_{\textrm{AA}}^\text {inclusive jet}$ is presented in Fig. 11 for each centrality class. This provides a more precise comparison of the $R_{\textrm{AA}}$ values than in Fig. 10. The results suggest that in central collisions the suppression of $b$-jets is less than that of inclusive jets, with an overall significance of approximately $1.7\sigma$. The calculation by Dai et al. [12] agrees well with the ratio in all centrality classes, while LIDO model calculations tend to overestimate the double ratio at low ${{p}_{\textrm{T}}}$, especially in central collisions, but agree well with the data at higher ${{p}_{\textrm{T}}}$. In the LIDO model, the difference between the inclusive jet and $b{{{\hbox {-jet}}}}$ $R_{\textrm{AA}}$ values is not expected to be entirely a result of the large mass of the $b$-quark, and qualitatively similar differences are seen between the two $R_{\textrm{AA}}$ values in that model with and without the inclusion of the dead-cone effect [79].

Differences in the internal structure of the $b$-jets, such as those that come from gluon-splitting processes, are also expected to be important in determining the $b{{{\hbox {-jet}}}}$ $R_{\textrm{AA}}$ value. Based on the Pythia 8 MC samples, the fractions of jets which are initiated by gluons in the inclusive jet and $b{{{\hbox {-jet}}}}$ samples have opposite trends with ${{p}_{\textrm{T}}}$. The fraction increases from 23% to 45% over the range of 80 to 250 GeV for the $b$-jets and decreases from 61% to 47% over the same range for the inclusive jets.

These results are compatible with previous results from CMS [29] (although both the collision energy and jet radius are different) but have better precision. Additional measurements of the fragmentation functions of $b$-jets in heavy-ion collisions would improve the uncertainties in future measurements using the method used in this article.

7 Summary

This paper reports cross-sections for $b$-jets and inclusive jets in Pb+Pb and $pp$ collisions, both at $\sqrt{s_{_\text {NN}}}=5.02$ TeV, and recorded by the ATLAS detector at the LHC. The measurement uses three datasets: 1.4 ${\textrm{nb}^{-1}}$ and 1.7 ${\textrm{nb}^{-1}}$ of Pb+Pb collisions collected in 2018 for $R= 0.2$ $b$-jets and inclusive jets respectively, and 260 ${\textrm{pb}^{-1}}$ of $pp$ collisions collected in 2017 for $R= 0.2$ and $R = 0.4$ $b$-jets and $R= 0.2$ inclusive jets. The $b{{{\hbox {-jet}}}}$ cross-section in $pp$ collisions is compared with a theoretical calculation and Monte Carlo generator predictions. The $b{{{\hbox {-jet}}}}$ and inclusive jet per-event yields for $R= 0.2$ jets and the corresponding nuclear modification factor, $R_{\textrm{AA}}$, are also reported for Pb+Pb collisions at the same per-nucleon collision energy. The $R_{\textrm{AA}}$ values are found to decrease with increasing collision centrality for both $b$-jets and inclusive jets. In order to more directly compare the suppression of $b$-jets and inclusive jets, the ratio of the $R_{\textrm{AA}}$ values is presented. The central values of this ratio suggest that the $R_{\textrm{AA}}$ for $b$-jets is larger than that for inclusive jets in central Pb+Pb collisions. The observed differences may arise primarily from the different mixture of quark and gluon jets in the inclusive jets and b-jets, and the b-quark mass effect may be subdominant in the kinematic range measured here. However, the current systematic uncertainties do not permit a more definitive statement. This highlights the need for more precise measurements of this quantity, mainly the necessity to measure the fragmentation functions of $b$-jets in heavy-ion collisions. The measurements are compared with theoretical calculations and suggest a role for mass and colour-charge effects in partonic energy loss in heavy-ion collisions.