Measurement of the nuclear modification factor of $b$-jets in 5.02 TeV Pb+Pb
collisions with the ATLAS detector CERN-EP-2022-043 This paper presents a
measurement of $b$-jet production in Pb+Pb and $pp$ collisions at
$\sqrt{s_{{}_{\textrm{NN}}}}=5.02$ TeV with the ATLAS detector at the LHC. The
measurement uses 260 $\textrm{pb}^{-1}$ of $pp$ collisions collected in 2017
and 1.4 $\textrm{nb}^{-1}$ 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
$\textrm{nb}^{-1}$ of Pb+Pb collisions collected in 2018 and the same $pp$
collision data as the $b$-jet measurement. The nuclear modification factor,
$R_{\textrm{AA}}$, 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 $R_{\textrm{AA}}$ to inclusive jet
$R_{\textrm{AA}}$ is also presented and suggests that the $R_{\textrm{AA}}$
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. EPJC

###### Contents

  1. 1 Introduction
  2. 2 ATLAS detector
  3. 3 Data selection and Monte Carlo samples
  4. 4 Analysis
    1. 4.1 Jet reconstruction
    2. 4.2 Muon reconstruction
    3. 4.3 $b\text{-jet}$ yield reconstruction
      1. 4.3.1 Templates and fitting
      2. 4.3.2 Comparing muon momentum distributions in the data and simulations
    4. 4.4 Corrections to the raw spectra
    5. 4.5 Observables
  5. 5 Systematic uncertainties
  6. 6 Results
    1. 6.1 Cross-section in $pp$ collisions
    2. 6.2 Per-event yields and $R_{\mathrm{AA}}$ in Pb+Pb collisions
  7. 7 Summary

##  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, $p_{\text{T}}$, is found to be significantly lower than
expectations based on $pp$ collisions, up to a $p_{\text{T}}$ 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
$b\text{-quarks}$, $b\text{-jets}$, (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 $b\text{-jets}$ produced in hadronic collisions consist of jets in
which the $b\text{-quark}$ is directly produced in the hard scattering (e.g.
$gg\to b\bar{b}$) and jets in which a hard-scattered gluon splits into a
$b\bar{b}$ pair ($g\to b\bar{b}$) as part of the showering process.

Inclusive jets and $b\text{-jets}$ 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 $b\text{-quark}$ 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
$p_{\text{T}}$ range because $b\text{-quarks}$ in jets from gluon splitting
tend to have lower momentum than those directly from hard scattering at the
same jet $p_{\text{T}}$. Differences in the interactions between the QGP and
the developing parton shower between $b\text{-jets}$ 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 $b\text{-jets}$ and inclusive jets in
Pb+Pb collisions, it is important to compare the $b\text{-jet}$ 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]
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 $b\text{-jets}$ are
very different from those of inclusive jets [25]; thus, it is not possible to
directly compare the suppression of $b\text{-quarks}$ with that of light
quarks and gluons without full jet measurements. CMS has measured the
suppression of $b\text{-jets}$ in Pb+Pb collisions at 2.76 TeV and found it to
be consistent with that of inclusive jets over the jet transverse momentum,
$p_{\mathrm{T}}^{\mathrm{jet}}$, range of 70–250 GeV [26], although the
uncertainties are large.

Jets are clustered with the anti-$k_{t}$ algorithm [27] 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 $b\text{-jet}$
if a $b$-hadron with $p_{\text{T}}>5$ GeV is found within an angular
distance111ATLAS uses a right-handed coordinate system with its origin at the
nominal interaction point (IP) in the centre of the detector, and the $z$-axis
along the beam pipe. The $x$-axis points from the IP to the centre of the LHC
ring, and the $y$-axis points upward. Cylindrical coordinates $(r,\phi)$ are
used in the transverse plane, $\phi$ being the azimuthal angle around the
$z$-axis. The pseudorapidity is defined in terms of the polar angle $\theta$
as $\eta=-\ln\tan(\theta/2)$. The rapidity is defined as
$y=0.5\ln[(E+p_{z})/(E-p_{z})]$ where $E$ and $p_{z}$ are the energy and
$z$-component of the momentum along the beam direction, respectively.
Transverse momentum and transverse energy are defined as
$p_{\text{T}}=p\sin\theta$ and $E_{\text{T}}=E\sin\theta$, respectively. The
angular distance between two objects with relative differences $\Delta\eta$ in
pseudorapidity and $\Delta\phi$ in azimuth is given by $\Delta
R=\sqrt{(\Delta\eta)^{2}+(\Delta\phi)^{2}}$. of $\Delta R=0.3$ from the jet
axis. The $b\text{-jets}$ 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, $p_{{\mathrm{T}}}^{\mu}$, relative to the jet+$\mu$
axis. The jet+$\mu$ axis is defined in the transverse plane by

$\vec{u}_{\textrm{T}}^{\textrm{jet}+\mu}=\frac{\vec{p}_{\textrm{T}}^{\mu}+\vec{p}_{\textrm{T}}^{\mathrm{jet}}}{\left|\vec{p}_{\textrm{T}}^{\mu}+\vec{p}_{\textrm{T}}^{\mathrm{jet}}\right|}$
(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, $p_{\mathrm{T}}^{\mathrm{rel}}$, is defined as

$\mbox{$p_{\mathrm{T}}^{\mathrm{rel}}$}\
=\left|\vec{p}_{\mathrm{T}}^{\mu}\times\vec{u}_{\textrm{T}}^{\textrm{jet}+\mu}\right|.$
(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 [28]. 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 $p_{\mathrm{T}}^{\mathrm{rel}}$ method.

This paper reports measurements of inclusive jet and $b\text{-jet}$ cross-
sections in $pp$ collisions and the per-event yield for inclusive jets and
$b\text{-jets}$ 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 $b\text{-jets}$). For $R=0.2$ jets the nuclear modification factor,
$R_{\mathrm{AA}}$, is calculated for $b\text{-jets}$ as

$\mbox{$R_{\mathrm{AA}}$}^{b\text{-jet}}\equiv\left.\left.\frac{1}{N_{\textrm{evt}}}\frac{\textrm{d}^{2}N_{\textrm{AA}}^{b\text{-jet}}}{\textrm{d}p_{\text{T}}\textrm{d}y}\right|_{\textrm{cent}}\middle/\langle
T_{\textrm{AA}}\rangle\frac{\textrm{d}^{2}\sigma_{pp}^{b\text{-jet}}}{\textrm{d}p_{\text{T}}\textrm{d}y}\right.$

where
$\textrm{d}^{2}N_{\textrm{AA}}^{b\text{-jet}}/\textrm{d}p_{\text{T}}\textrm{d}y$
is the yield of $b\text{-jets}$ in Pb+Pb collisions for the centrality,
$p_{\text{T}}$, and rapidity range of interest;
$\textrm{d}^{2}\sigma_{pp}^{b\text{-jet}}/\textrm{d}p_{\text{T}}\textrm{d}y$
is the $b\text{-jet}$ cross-section in $pp$ collisions; $\langle
T_{\textrm{AA}}\rangle$ is the nuclear thickness function [29]; and
$N_{\textrm{evt}}$ 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

$\mbox{$R_{\mathrm{AA}}$}^{\mathrm{inclusive\>jet}}\equiv\left.\left.\frac{1}{N_{\textrm{evt}}}\frac{\textrm{d}^{2}N_{\textrm{AA}}^{\mathrm{inclusive\>jet}}}{\textrm{d}p_{\text{T}}\textrm{d}y}\right|_{\textrm{cent}}\middle/\langle
T_{\textrm{AA}}\rangle\frac{\textrm{d}^{2}\sigma_{pp}^{\mathrm{inclusive\>jet}}}{\textrm{d}p_{\text{T}}\textrm{d}y}\right.$

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

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

##  2 ATLAS detector

The ATLAS detector [30] 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 $2\text{\,}\mathrm{T}$ 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 [31,
32]. 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 $6.0\text{\,}\mathrm{T}\text{\,}\mathrm{m}$
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 [33]. 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 [34] 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 [35].

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 [36]. In both Pb+Pb and $pp$ collisions, the muon–jet triggers require a
muon with $p_{\text{T}}>$ 4 GeV matched to an $R=0.4$ jet with various
$p_{\text{T}}$ 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 [37]. This analysis uses
three centrality intervals which are defined according to successive
percentiles of the $\sum E_{\textrm{T}}^{\textrm{FCal}}$ distribution obtained
in MB collisions. The centrality regions used in this analysis, going from the
most central (largest $\sum E_{\textrm{T}}^{\textrm{FCal}}$) collisons to the
peripheral (lowest $\sum E_{\textrm{T}}^{\textrm{FCal}}$) collisions are
0–20%, 20–50%, and 50–80%. The values of
$\langle\mbox{$T_{\mathrm{AA}}$}\rangle$ and the number of binary
nucleon–nucleon collisions, $\langle N_{\textrm{coll}}\rangle$, for each
centrality interval are evaluated by a Monte Carlo (MC) Glauber model analysis
[38, 29] of the $\sum E_{\textrm{T}}^{\textrm{FCal}}$ distribution, and are
shown in Table 1.

Table 1: The $\langle\mbox{$T_{\mathrm{AA}}$}\rangle$ and $\langle N_{\textrm{coll}}\rangle$ values and uncertainties in each centrality bin. These are the results from the Glauber modelling of the summed transverse energy in the forward calorimeters, $\sum E_{\textrm{T}}^{\textrm{FCal}}$. Centrality [%] |  $\langle\mbox{$T_{\mathrm{AA}}$}\rangle$ $[\textrm{mb}^{-1}]$ | $\langle N_{\textrm{coll}}\rangle$
---|---|---
0–20 | 18.84$\pm$0.18 | 1319$\pm$90
20–50 | 5.41$\pm$0.15 | 378$\pm$23
50–80 | 0.69$\pm$0.05 | 48$\pm$4

A small fraction of the triggered Pb+Pb events $(<0.5\%)$ contain multiple
collisions, known as ‘pile-up’. The expected anti-correlation between $\sum
E_{\textrm{T}}^{\textrm{FCal}}$ 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 [39, 40]. Dijet
samples were generated with Pythia 8 [41], using the parameter values of the
A14 tune [42] with the NNPDF23lo set of parton distribution functions (PDFs)
[43]. These were generated both with no additional requirements and with the
requirement of a muon with $p_{\text{T}}>$ 3 GeV at the generator level (such
as from heavy-flavour hadron decays). The EvtGen package [44] 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
[45] 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 [46]. The A14 tune
and CTEQ6L1 PDFs [47] were used. Generator-level $pp$ Herwig++ [48] events,
using the UEEE5 tune [49] and the CTEQ6L1 PDFs [47], are 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, 50]. Jets are
reconstructed using the anti-$k_{t}$ algorithm [27] implemented in the FastJet
software package [51]. 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 [52] 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 [37]. 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 $v_{n}^{\mathrm{UE}}$ [37]. 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 $v_{n}^{\mathrm{UE}}$ 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
anti-$k_{t}$ algorithm with $R=0.4$ to charged particles with $p_{\text{T}}>4$
GeV. The $R=0.2$ jets must pass a requirement on the minimum value of the
tower $E_{\mathrm{T}}$ and on a ratio of maximum tower $E_{\mathrm{T}}$ to
average tower $E_{\mathrm{T}}$, while the track-jets are required to have
$p_{\text{T}}>7$ 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 $p_{\text{T}}>25$ GeV (8 GeV) in
Pb+Pb ($pp$) collisions. The updated $\rho$ and $v_{n}^{\mathrm{UE}}$ values
are used to update the jet kinematic properties in the second iteration.

Jet $\eta$\- and $p_{\text{T}}$-dependent correction factors derived in
simulations are applied to the measured jet energy to correct for the
calorimeter energy response [53]. 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 [54]. 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 [55].

Jets are defined at the truth level in the MC sample before detector
simulation by applying the 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 $\mbox{$\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. [56], except the requirement
on the number of TRT hits.

Muons are selected with $\mbox{$p_{{\mathrm{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 $\mbox{$\Delta
R$}(\textrm{jet},\mu)$ 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 [57, 16] in bins of
$p_{{\mathrm{T}}}^{\mu}$ 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 $\mbox{$p_{{\mathrm{T}}}^{\mu}$}>6$ GeV
and $|\eta^{\mu}|<1.05$, and at 90% for $\mbox{$p_{{\mathrm{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 [56,
16], in fine bins of $p_{{\mathrm{T}}}^{\mu}$ 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 $\mbox{$p_{{\mathrm{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 $b\text{-jet}$ yield reconstruction

####  4.3.1 Templates and fitting

At the generator level, the jet flavour is defined by matching jets to hadrons
with $p_{\text{T}}>5$ GeV. The jet is considered a $b\text{-jet}$ 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
$p_{\text{T}}$. In order to better characterize the jets, the jet+$\mu$ scale
is defined as the $p_{\text{T}}$ of the sum of the muon and jet four-vectors.
The $p_{\mathrm{T}}^{\mathrm{rel}}$ distributions in the data and MC samples
are constructed for selections in this jet+$\mu$ object at the reconstruction
level, $p_{\mathrm{T,reco}}^{\textrm{jet}+\mu}$. In order to form a jet+$\mu$
pair, jets are required to have a minimum calorimetric
$p_{\mathrm{T,reco}}^{\textrm{jet}}$ of 40 GeV (58 GeV) in $pp$ (Pb+Pb)
collisions.

In semileptonic $b$-hadron decays, the $p_{\text{T}}$ of the lepton relative
to the jet+$\mu$-axis (see Eqs. (1) and (2)), $p_{\mathrm{T}}^{\mathrm{rel}}$,
is used to distinguish between $b$-, $c$-, and light-jets [28]. 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
$p_{\mathrm{T}}^{\mathrm{rel}}$ spectrum than muons in c-jets and light-jets.

The templates distributions of $p_{\mathrm{T}}^{\mathrm{rel}}$ for $b$\- and
c-jets are extracted from the muon-filtered MC samples. The $b$-hadron mixture
is taken from Tevatron measurements [58], which are consistent with those from
the LHCb Collaboration [59]. 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. [60, 28] (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 $p_{\text{T}}$ spectra of the tracks are
reweighted to reproduce the muon $p_{\text{T}}$ 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 $p_{\mathrm{T}}^{\mathrm{rel}}$ spectrum to obtain the fractions of $b$-,
$c$-, and light-jets. A binned maximum-likelihood fit is performed using the
RooFit framework [61]. The model for the fit is defined as a normalized sum of
three templates: light-jets
($F_{\mathrm{l},i}$($p_{\mathrm{T}}^{\mathrm{rel}}$)), c-jets
($F_{\mathrm{c},i}$($p_{\mathrm{T}}^{\mathrm{rel}}$)), and $b\text{-jets}$
($F_{\mathrm{b},i}$($p_{\mathrm{T}}^{\mathrm{rel}}$)). The $pp$ fit model,
$M^{pp}$, is structured as

$M^{pp}_{i}\\!\left(\mbox{$p_{\mathrm{T}}^{\mathrm{rel}}$}\right)=\mbox{$f_{\mathrm{b},i}$}\mbox{$F_{\mathrm{b},i}$}\\!\left(\mbox{$p_{\mathrm{T}}^{\mathrm{rel}}$}\right)+\left(1-\mbox{$f_{\mathrm{b},i}$}\right)\left[\mbox{$f_{\mathrm{c},i}$}\mbox{$F_{\mathrm{c},i}$}\\!\left(\mbox{$p_{\mathrm{T}}^{\mathrm{rel}}$}\right)+\left(1-\mbox{$f_{\mathrm{c},i}$}\right)\mbox{$F_{\mathrm{l},i}$}\\!\left(\mbox{$p_{\mathrm{T}}^{\mathrm{rel}}$}\right)\right]$

where $f_{\mathrm{b},i}$ is the fraction of $b\text{-jets}$ in the total
sample, $f_{\mathrm{c},i}$ is the fraction of c-jets in the non-$b\text{-jet}$
sample, and the index $i$ denotes the $p_{\text{T}}$ bin. The fit model has
two free parameters, $f_{\mathrm{b},i}$ and $f_{\mathrm{c},i}$ for each
$p_{\mathrm{T,reco}}^{\textrm{jet}+\mu}$ 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,
$F_{\mathrm{mix},i}$, and the amplitude, $f_{\mathrm{mix},i}$, 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 $\sum
E_{\textrm{T}}^{\textrm{FCal}}$ 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, $M^{\textrm{Pb+Pb}}$, is structured as

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

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

The data $p_{\mathrm{T}}^{\mathrm{rel}}$ 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 Figures 1 and 2, for
low and high $p_{\mathrm{T,reco}}^{\textrm{jet}+\mu}$ respectively.

Figure 1: The $p_{\mathrm{T}}^{\mathrm{rel}}$ 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
$80.7<\mbox{$p_{\mathrm{T,reco}}^{\textrm{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.

Figure 2: The $p_{\mathrm{T}}^{\mathrm{rel}}$ 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
$113.4<\mbox{$p_{\mathrm{T,reco}}^{\textrm{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 $b\text{-jet}$ $p_{\text{T}}$ spectrum is constructed in each Pb+Pb
centrality bin and in $pp$ collisions by taking each $f_{\mathrm{b},i}$ value
and multiplying it by the number of jets in i-th bin in the total
$p_{\mathrm{T,reco}}^{\textrm{jet}+\mu}$ spectrum.

####  4.3.2 Comparing muon momentum distributions in the data and simulations

The $b\text{-jet}$ 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 $b\text{-jet}$ 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
$p_{\mathrm{T,reco}}^{\textrm{jet}+\mu}$ 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

$\mbox{$z$}\equiv\mbox{$p_{{\mathrm{T}}}^{\mu}$}\cos\\!\left(\theta\right)/\mbox{$p_{\mathrm{T,reco}}^{\textrm{jet}+\mu}$}\
,$

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
Figure 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
$p_{\mathrm{T}}^{\mathrm{rel}}$ distributions are largely sensitive to the
$b$-hadron decay kinematics.

Figure 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 $p_{\mathrm{T,reco}}^{\textrm{jet}+\mu}$
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 $f_{\mathrm{b},i}$
values, for the $b\text{-jets}$. 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 muon and neutrino energies in the
$b\text{-jet}$ sample). Both the $b\text{-jet}$ and inclusive jet
$p_{\text{T}}$ spectra are unfolded using the one-dimensional (1D) Bayesian
unfolding [62] from the RooUnfold software package [63]. The $b\text{-jet}$
response matrices are built from the muon-filtered MC samples using truth-
level $b\text{-jets}$ (including the muon and neutrino momenta) that are
matched to reconstructed jets in simulations and have
$|y_{\textrm{truth}}^{b\text{-jet}}|<2.1$. The inclusive jet response matrices
are built from the dijet MC samples using all truth jets and not including any
muon or neutrino momenta.

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-$p_{\mathrm{T}}^{\mathrm{jet}}$ 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
$p_{\mathrm{T,reco}}^{\textrm{jet}+\mu}$ 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 $p_{\text{T}}$ is
required to be larger than 72 GeV for both Pb+Pb and $pp$ collisions.
Reconstructed jets below these thresholds that match to truth-$p_{\text{T}}$
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 Section 3, the main MC sample used in the $b\text{-jet}$
analysis required the truth-muon $p_{\text{T}}$ to be larger than 3 GeV. The
missing part of the cross-section from $b\text{-jets}$ 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 $b\text{-jet}$ cross-section in $pp$ collisions is defined as

$\frac{\textrm{d}^{2}\sigma_{b\text{-jet}}}{\textrm{d}p_{\text{T}}\textrm{d}y}=\frac{1}{L_{\textrm{int}}\mathcal{B}}\frac{N^{\textrm{Unfol.}}_{\textrm{$b\text{-jet}$}}}{\Delta
p_{\text{T}}\Delta y}\ ,$ (3)

where $L_{\textrm{int}}$ 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 $(20.6\pm 0.6)$% [25]. The quantity
$N^{\textrm{Unfol.}}_{\textrm{$b\text{-jet}$}}$ is the $b\text{-jet}$ yield
after the unfolding and acceptance correction described above, and $\Delta
p_{\text{T}}$ and $\Delta y$ are the widths of the $p_{\text{T}}$ and $y$
bins. Following the same approach, the per-event yield of $b\text{-jets}$
measured in Pb+Pb collisions is calculated as

$\left.\frac{1}{N_{\textrm{evt}}}\frac{\textrm{d}^{2}N_{\textrm{$b\text{-jet}$}}}{\textrm{d}p_{\text{T}}\textrm{d}y}\right|_{\textrm{cent}}=\left.\frac{1}{N_{\textrm{evt}}\mathcal{B}}\frac{N^{\textrm{Unfol.}}_{\textrm{$b\text{-jet}$}}}{\Delta
p_{\text{T}}\Delta y}\right|_{\textrm{cent}},$ (4)

where $N_{\textrm{evt}}$ 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 $b\text{-jet}$ measurements arise
from the jet energy scale and resolution, the unfolding procedure, luminosity
($pp$ only), and $\langle\mbox{$T_{\mathrm{AA}}$}\rangle$ determination (Pb+Pb
only). There are additional contributions to the $b\text{-jet}$ measurements
from the modelling of the $p_{\mathrm{T}}^{\mathrm{rel}}$ 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 [52, 64], including an additional
term for $b\text{-jet}$ energy scale [52]. 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 [55]. 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 $b\text{-jet}$ 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 [65], 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 [55] that compares the
calorimetric jet $p_{\text{T}}$ 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 $p_{\text{T}}$ to $R=0.4$ jet $p_{\text{T}}$ 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
$p_{\text{T}}$ 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 $p_{\text{T}}$ balance [66, 67]. 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.

Muon-related uncertainties 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. [56, 57]. 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 $p_{\mathrm{T}}^{\mathrm{rel}}$
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 [68]. In Pb+Pb collisions, there is no information
about the modification of the gluon-splitting contribution to $b\text{-jets}$.
Additionally, the fragmentation functions of $b\text{-jets}$ 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. [69]. The fraction of muons which arise from $b$-hadron
decays which include an intermediate $D$-meson is varied in accord with Ref.
[25]. The fractions of the various $b$-hadron species are varied according to
the world average values in Ref. [58]. The modelling of the muon momentum in
the $b$-hadron rest frame is crucial for the $p_{\mathrm{T}}^{\mathrm{rel}}$
method. The modelling in Pythia 8 is used for the central values. The Pythia 8
distributions were compared with the measurement from DELPHI [70] 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
$b\text{-jets}$ is addressed by reweighting the 2D
$p_{\mathrm{T}}^{\mathrm{jet}}$$-$$p_{{\mathrm{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 $b\text{-jet}$
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
$f_{\mathrm{b},i}$ 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
$R=0.2$ $b\text{-jet}$ cross-sections is shown in Figure 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
[71]. Sources of systematic uncertainty similar to those examined in the 2012
$pp$ luminosity calibration [71] 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. [25].

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

Figure 4: The relative systematic uncertainties for several categories, as a
function of $b$-jet $p_{\text{T}}$ 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.

Figure 5: Relative size of systematic uncertainties for several categories, as
a function of jet $p_{\text{T}}$ for $b\text{-jet}$ $R_{\mathrm{AA}}$ (left
column) and inclusive jet $R_{\mathrm{AA}}$ (right column), in peripheral (top
row), semi-central (middle row) and central (bottom row) Pb+Pb collisions.

Figure 6: Relative size of systematic uncertainties of the ratio
$\mbox{$R_{\mathrm{AA}}$}^{b\text{-jet}}/\mbox{$R_{\mathrm{AA}}$}^{\textrm{inclusive
jet}}$ shown for the most relevant components, as a function of jet
$p_{\text{T}}$ for 50–80% (top), 20–50% (middle), and 0–20% (bottom)
centrality Pb+Pb collisions.

##  6 Results

###  6.1 Cross-section in $pp$ collisions

Figure 7 shows the b-jet cross-section as a function of b-jet $p_{\text{T}}$
in $pp$ collisions at 5.02 TeV for $R=0.2$ and $R=0.4$ jets with $|y|<2.1$.
Additionally, $b\text{-jet}$ 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 [72] 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 $b\text{-jet}$ cross-sections by
an amount which increases with $p_{\text{T}}$. A calculation [73, 11] which is
based on Pythia 8 where the PDFs and $\alpha_{s}(\textrm{M}_{Z})$ is set with
$\mathrm{pSet}=8$ [41] 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 [43] and the A14 tune [42] is 20–30% higher than data for both
$R=0.4$ and $R=0.2$ jets. The Herwig++ calculations [74] 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 7: (left) Differential cross-section for $b\text{-jet}$ production for
$R=0.2$ and $R=0.4$ jets with $|y|<2.1$ as a function of $p_{\text{T}}$ 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. [72]
and $R=0.2$ jets are compared with calculations from Refs. [72, 73, 11, 12].
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.

Figure 8 shows a comparison of the $b\text{-jet}$ 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 [75] is also shown; the jet radii in this measurement are larger
than in the present analysis but the results are qualitatively similar. No
significant $p_{\text{T}}$ dependence of the $b\text{-jet}$ to inclusive jet
cross-section ratio is observed.

Figure 8: (left) Cross-section of $R=0.2$ $b\text{-jet}$ and inclusive jet
production in $pp$ collisions at 5.02 TeV, and (right) the $b\text{-jet}$ 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 [75]. 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.

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

The inclusive jet and $b\text{-jet}$ per-event yields in Pb+Pb collisions
scaled by $\langle\mbox{$T_{\mathrm{AA}}$}\rangle$ are shown in Figure 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 $b\text{-jet}$ $R_{\mathrm{AA}}$ for each centrality class. The
$R_{\mathrm{AA}}$ 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 $R_{\mathrm{AA}}$
central values for inclusive jets are lower than those for $b\text{-jets}$. A
difference between the slopes of the inclusive jet and $b\text{-jet}$
differential cross-sections could cause the $R_{\mathrm{AA}}$ values to differ
between the two categories of jets; however, the ratio of the $b\text{-jet}$
to inclusive jet cross-section, shown in Figure 8, does not vary with
$p_{\text{T}}$.

Figure 9:  Per-event yields scaled by $\langle\mbox{$T_{\mathrm{AA}}$}\rangle$
in Pb+Pb collisions for three centrality classes for $R=0.2$ $b\text{-jets}$
(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 $T_{\mathrm{AA}}$ uncertainties are
noted in the legend.

The $R_{\mathrm{AA}}$ values for both inclusive jets and $b\text{-jets}$ are
compared with two theory calculations. The first calculation is the LIDO model
[73, 11, 76], which includes both energy loss and diffusion of heavy quarks.
The dead-cone effect for $b\text{-jets}$ 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,
$\mu_{\textrm{min}}$, 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 $\mu_{\textrm{min}}$ values is motivated by comparisons with other
jet measurements. These parameters have been shown [76] to provide a
reasonable description of measurements of the $R_{\mathrm{AA}}$ values of
$B$\- and $D$-mesons [77, 22]. The calculation by Dai et al. in Ref. [12] is
based on a Langevin transport model describing the evolution of
$b\text{-quarks}$ 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 $b\text{-jets}$ and also uses a
(2+1)D viscous hydrodynamic medium with averaged initial conditions. The
medium parameter, $q_{0}$, is set to be 1.2 $\text{GeV}^{2}/$fm. The LIDO
model shows good agreement with the data for both inclusive jet and
$b\text{-jet}$ $R_{\mathrm{AA}}$, although with the inclusive jets on the low
side and the $b\text{-jet}$ case on the high side, for all three centralities
considering model and data uncertainties. The calculation in Ref. [12] is
below the measured $R_{\mathrm{AA}}$ in central and semi-central collisions
for both $b\text{-jets}$ and inclusive jets.

Figure 10: $b\text{-jet}$ (filled points) $R_{\mathrm{AA}}$ for three
centrality classes compared with the inclusive jet $R_{\mathrm{AA}}$ (open
points) at 5.02 TeV. Both $R_{\mathrm{AA}}$ measurements are compared with
theory calculations [73, 11, 12]. 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
$\mu_{\textrm{min}}$ parameter from $1.3\pi T$ (lower edge) to $1.8\pi T$
(upper edge). The boxes at unity represent the scale uncertainties from
$\langle\mbox{$T_{\mathrm{AA}}$}\rangle$ and the luminosity determination.

In order to assess any difference between the $b\text{-jet}$ and inclusive jet
$R_{\mathrm{AA}}$, the ratio
$\mbox{$R_{\mathrm{AA}}$}^{b\text{-jet}}/\mbox{$R_{\mathrm{AA}}$}^{\textrm{inclusive
jet}}$ is presented in Figure 11 for each centrality class. This provides a
more precise comparison of the $R_{\mathrm{AA}}$ values than in Figure 10. The
results suggest that in central collisions the suppression of $b\text{-jets}$
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 $p_{\text{T}}$, especially in central
collisions, but agree well with the data at higher $p_{\text{T}}$. In the LIDO
model, the difference between the inclusive jet and $b\text{-jet}$
$R_{\mathrm{AA}}$ values is not expected to be entirely a result of the large
mass of the $b\text{-quark}$, and qualitatively similar differences are seen
between the two $R_{\mathrm{AA}}$ values in that model with and without the
inclusion of the dead-cone effect [76].

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

These measurements highlight the importance of measuring the fragmentation
functions of $b\text{-jets}$ in heavy-ion collisions. These results are
compatible with previous results from CMS [26] (although both the collision
energy and jet radius are different) but have better precision.

Figure 11: Ratio of $b\text{-jet}$ $R_{\mathrm{AA}}$ to the inclusive jet
$R_{\mathrm{AA}}$ for each centrality class. Ratios are compared with theory
calculations [73, 11, 12]. 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
$\mu_{\textrm{min}}$ parameter from $1.3\pi T$ (upper edge) to $1.8\pi T$
(lower edge).

##  7 Summary

This paper reports cross-sections for $b\text{-jets}$ and inclusive jets in
Pb+Pb and $pp$ collisions, both at $\sqrt{s_{{}_{\textrm{NN}}}}=5.02$ TeV, and
recorded by the ATLAS detector at the LHC. The measurement uses three
datasets: 1.4 ${\mathrm{nb}^{-1}}$ and 1.7 ${\mathrm{nb}^{-1}}$ of Pb+Pb
collisions collected in 2018 for $R=0.2$ $b\text{-jets}$ and inclusive jets
respectively, and 260 ${\mathrm{pb}^{-1}}$ of $pp$ collisions collected in
2017 for $R=0.2$ and $R=0.4$ $b\text{-jets}$ and $R=0.2$ inclusive jets. The
$b\text{-jet}$ cross-section in $pp$ collisions is compared with a theoretical
calculation and Monte Carlo generator predictions. The $b\text{-jet}$ and
inclusive jet per-event yields for $R=0.2$ jets and the corresponding nuclear
modification factor, $R_{\mathrm{AA}}$, are also reported for Pb+Pb collisions
at the same per-nucleon collision energy. The $R_{\mathrm{AA}}$ values are
found to decrease with increasing collision centrality for both
$b\text{-jets}$ and inclusive jets. In order to more directly compare the
suppression of $b\text{-jets}$ and inclusive jets, the ratio of the
$R_{\mathrm{AA}}$ values is presented. The central values of this ratio
suggest that the $R_{\mathrm{AA}}$ for $b\text{-jets}$ is larger than that for
inclusive jets in central Pb+Pb collisions. Thus within this model, the
differences may arise from the inclusive jet mixture of quarks and gluons,
where gluons lose more energy, and the $b$-quark mass effect may be
subdominant at these $p_{\text{T}}$ values. 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 $b\text{-jets}$ 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.

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