#  $Z^{0}$ Boson Associated b-jet Production in High-Energy Nuclear Collisions

Sa Wang  Key Laboratory of Quark & Lepton Physics (MOE) and Institute of
Particle Physics, Central China Normal University, Wuhan 430079, China  Wei
Dai  School of Mathematics and Physics, China University of Geosciences, Wuhan
430074, China  Ben-Wei Zhang<EMAIL_ADDRESS>Key Laboratory of Quark
& Lepton Physics (MOE) and Institute of Particle Physics, Central China Normal
University, Wuhan 430079, China  Institute of Quantum Matter, South China
Normal University, Guangzhou 510006, China  Enke Wang  Institute of Quantum
Matter, South China Normal University, Guangzhou 510006, China  Key Laboratory
of Quark & Lepton Physics (MOE) and Institute of Particle Physics, Central
China Normal University, Wuhan 430079, China

###### Abstract

The production of vector boson tagged heavy quark jets provides potentially
new tools to study jet quenching, especially the mass hierarchy of parton
energy loss. In this work, we present the first theoretical study on
$Z^{0}\,+\,$b-jet in heavy-ion collisions. Firstly utilizing a Monte Carlo
transport model, our simulations give nice descriptions of the azimuthal angle
correlation $\Delta\phi_{jZ}$, transverse momentum imbalance $x_{jZ}$ for
$Z^{0}\,+\,$jet as well as the nuclear modification factor $R_{AA}$ of
inclusive b-jet in Pb+Pb collisions. Then we calculate the azimuthal angular
correlation $\Delta\phi_{bZ}$ of $Z^{0}\,+\,$b-jet and $\Delta\phi_{bb}$ of
$Z^{0}\,+\,2\,$b-jets in central Pb+Pb collisions at $\sqrt{s_{NN}}=$ 5.02
TeV. We find that the medium modification of the azimuthal angular correlation
for $Z^{0}\,+\,$b-jet has a weaker dependence on $\Delta\phi_{bZ}$, as
compared to that for $Z^{0}\,+\,$jet. With the high purity of quark jet in
$Z^{0}\,+\,$(b-)jet production, we calculate the momentum imbalance
distribution of $x_{bZ}$ of $Z^{0}\,+\,$b-jet in Pb+Pb collisions. We observe
a smaller shifting of the mean value of momentum imbalance for
$Z^{0}\,+\,$b-jet in Pb+Pb collisions $\Delta\left\langle
x_{bZ}\right\rangle$, as compared to that for $Z^{0}\,+\,$jet. In addition, we
investigate the nuclear modification factors of tagged jet cross sections
$I_{AA}$, and show a much stronger suppression of $I_{AA}$ in $Z^{0}\,+\,$jet
than that of $Z^{0}\,+\,$b-jet in central Pb+Pb collisions.

##  I Introduction

The formation of quark-gluon plasma (QGP), produced in the early stages of the
high-energy nucleus-nucleus collisions both at the Relativistic Heavy Ion
Collider (RHIC) and the Large Hadron Collider (LHC), offers a new possibility
to test Quantum Chromodynamics (QCD) under such extreme hot and dense de-
confined state of nuclear matter. The high-$p_{T}$ partons (quarks and gluons)
produced in the initial hard scattering will strongly interact with the QGP
and dissipate their energy to the medium, referred as the jet quenching effect
Wang:1991xy ; Gyulassy:2003mc ; Qin:2015srf ; Vitev:2008rz ; Vitev:2009rd .
Consequently, the ’quenched jet’ observables are used to quantify the
properties Burke:2013yra  of the hot and dense QCD matter by investigating
their medium modifications in heavy-ion collisions (HIC) relative to their p+p
baselines.

Recently the associated production of vector boson ( photon $\gamma$ or
electroweak boson such as $Z^{0}$ and $W^{+/-}$) and jets ($V+$jet) has been
extensively studied both in theory (Catani:2002ny, ; Boughezal:2015ded, ;
Ridder:2015dxa, ) and experiment (Aad:2013ysa, ; Aad:2013zba, ;
Aaboud:2016sdm, ; Chatrchyan:2013mwa, ; Chatrchyan:2013oda, ;
Khachatryan:2014zya, ) to test the fundamental properties of QCD and improve
the constraints on the parton distribution function (PDF) in proton. More
importantly, due to the fact that vector boson would not involve the strong
interaction with the medium and gauge the initial energy of the tagged jets,
V+jet are recognized ideal probe to the properties of the quark-gluon
plasma(QGP) (Neufeld:2010fj, ; Wang:1996yh, ; Dai:2012am, ; Luo:2018pto, ;
Zhang:2018urd, ; Chen:2018fqu, ; Kang:2017xnc, ; Casalderrey-Solana:2015vaa, ;
KunnawalkamElayavalli:2016ttl, ; Zhang:2018urd, ).

In particular, new measurements on associated production of $Z^{0}$ boson and
b-jet (denoted as $Z^{0}\,+\,$b-jet) in p+p collisions at the LHC have been
performed by ATLAS and CMS Aad:2011jn ; Chatrchyan:2012vr ; Chatrchyan:2013zja
; Chatrchyan:2013zna ; Aad:2014dvb ; Chatrchyan:2014dha ; Khachatryan:2016iob
, since the final state b-jet associated with $Z^{0}$ boson is the dominant
background of the associated production of Higgs and $Z^{0}$ bosons
($Z^{0}+H\rightarrow Z+b\bar{b}$) within the standard model(SM)
Chatrchyan:2012ww  and can test many physics scenarios beyond the SM which
predict new generation mechanism of b quarks and $Z^{0}$ bosons
Chatrchyan:2013zja . It is noted that in heavy-ion collisions, on one hand,
the $Z^{0}$ boson tagged b-jet, since the initial energy of b quark is well
gauged by the vector boson and then its energy loss can be directly obtained,
better suits to explore quenching of heavy flavor jet than the inclusive
b-jet. On the other hand, the dead cone effect of heavy flavor quarks has
attracted intense investigations Dokshitzer:2001zm ; Zhang:2003wk ;
Armesto:2003jh ; Sharma:2009hn , though by comparing the $R_{AA}$ of inclusive
jet with b-jet as well as the $p_{T}$ imbalance ($x_{J}$) of inclusive di-jet
with $b\bar{b}$ di-jet, experimental measurements have not found direct
evidence Chatrchyan:2013exa ; Khachatryan:2016jfl ; Sirunyan:2018jju , except
some indication that B mesons, via $B\rightarrow J/\Psi$ channel, might
undergo smaller suppression than D-mesons and light hadrons in A+A collisions
Khachatryan:2016ypw . The possible reasons for this puzzle can be manifold,
while the large contribution of gluon-initiated b-parton processes may
contaminate many attempts of solving the puzzle. Previous studies in Ref.
Kartvelishvili:1995fr ; Neufeld:2010fj  have shown that the dominant
contribution of $Z^{0}$ tagged jet is quark-initiated jet, and the study of
$Z^{0}\,+\,$b-jet in HIC, especially their different medium modifications from
that of $Z^{0}\,+\,$jet provide a very useful tool to directly address the
mass effect between light-quark jet and massive bottom jet. Nevertheless, so
far studies on the associated production of b-jet and $Z^{0}$ boson in
nucleus-nucleus collisions are still lacking.

With this in mind, in this work, we present a Monte Carlo transport simulation
including elastic (collisional) and inelastic (radiative) interaction of the
energetic parton in the hot/dense QCD medium, while taking the next-to-leading
order (NLO) plus parton shower (PS) generated initial hard parton spectrum as
input, to study the in-medium modification of the vector boson $Z^{0}$ tagged
b-jets. This framework has been employed to describe the heavy-flavor jet
production of high-energy nuclear collisions in our previous studies
Dai:2018mhw ; Wang:2019xey ; Wang:2020bqz ; Yan:2020zrz . We will first show
our numerical results of $Z^{0}\,+\,$jet and compare them to the available
experimental data to test the application of our model. Then we proceed to
calculate the angular correlations of $Z^{0}\,+\,$b-jet in A+A collisions ,
and demonstrate the modifications on these correlations are sensitive to the
initial b-jet $p_{T}$ distribution instead of the azimuthal angle. Unlike the
case in $Z^{0}\,+\,$jet, the requirement of b-tagging excludes the
contribution from multiple jets so that the azimuthal angular correlations of
$Z^{0}\,+\,$b-jet show distinct pattern modifications. With the high purity of
light-quark jet in $Z^{0}\,+\,$jet events, we expect to address the mass
dependence of the jet quenching effects between $Z^{0}\,+\,$jet and
$Z^{0}\,+\,$b-jet.

The rest of our paper is organized as follows: in Sec. II we present the
productions of $Z^{0}\,+\,$b-jet in p+p collisions calculated by Monte Carlo
event generator and their comparisons with the experimental data. In Sec. III
we discuss our treatments of the jet in-medium evolution in A+A collisions. In
Sec. IV we give the simulated results and discussions of azimuthal angular
correlation, transverse momentum imbalance, and nuclear modification factor of
$Z^{0}\,+\,$b-jet in HIC. Finally, Sec. V summarizes our study.

##  II The Associated Production of $Z^{0}$ Boson and b-jet in p+p Collisions

Before we move into the study on $Z^{0}\,+\,$b-jet production in heavy-ion
collisions, we should address its production in p+p collisions. Fig. 1 shows a
few processes Aad:2011jn  contributing to the associated production of a
$Z^{0}$ boson and b-jet . In the Fig. 1(a) and (b), an initial bottom quark
from the parton distribution function(PDF) derived from the gluon distribution
of one beam particle evolves the hard scattering and then turn into a b-jet
and a $Z^{0}$ boson in the finial-state. In the bottom two diagrams, the
$b\bar{b}$ pairs originate from the hard scattering and then turn into two
b-jets associated with a emitted $Z^{0}$ boson in the final-state.

Figure 1: Feynman diagrams contributing to the associated production of b-jet
with $Z^{0}$ boson.

In this work, we use the MC@NLO event generator SHERPA (Gleisberg:2008ta, ) to
obtain the initial $Z^{0}\,+\,$b-jet production in p+p collision. The tree-
level matrix elements are calculated by the internal module Amegic
(Krauss:2001iv, )and Comix (Gleisberg:2008fv, ), while one-loop virtual
correction calculated by the external program BlackHat. The parton shower
based on Catani-Seymour (Schumann:2007mg, ) subtraction method is matched with
NLO QCD matrix elements by MC@NLO method (Frixione:2002ik, ). The NLO PDF from
NNPDF3.0 (Ball:2014uwa, ) with 5-flavour scheme has been chosen in the
calculations. FASTJET (Cacciari:2011ma, ) with anti-$k_{\rm T}$ algorithm is
used in the event selection and the final state jet reconstruction.

To confront our calculated results by SHERPA with the unfolded experimental
data in p+p collisions, the same configurations as the selection of ATLAS
collaboration (Khachatryan:2016iob, ) have been set up in our simulations. The
$Z^{0}$ boson is reconstructed based on its decay channels $Z^{0}\rightarrow
e^{+}e^{-}$ and $Z^{0}\rightarrow\mu^{+}\mu^{-}$ . The transverse momentum of
the electron and muon candidate are required to be lager than 20 GeV. To
exclude the barrel-endcap transition region, the electrons are selected within
the pseudorapidity region $|\eta|<1.44$ or $1.57<|\eta|<2.4$ while muons
within $|\eta|<2.4$. And according to the requirement of experiment, the
events are considered only when the invariant mass of the electron or muon
pairs lie in the mass region $70<M_{ll}<111$ GeV. The jets associated with the
$Z^{0}$ boson are reconstructed by FASTJET using anti-$k_{T}$ algorithm with
cone size $\Delta R=0.5$. To reduce the contribution from the underlying
event, the reconstructed jets must be in the pseudorapidity region $|\eta^{\rm
jet}|<2.4$ and have $p_{T,\rm jet}>30$ GeV. The contribution from the
underlying event is less than $5\%$ estimated by ATLAS (Khachatryan:2016iob, )
because the production of softer jets is greatly suppressed by this
requirement.

Figure 2: Differential cross section of $Z^{0}\,+\,$b-jet simulated by
SHERPA(blue line) in p+p collision at $\sqrt{s}=8$ TeV as a function of
transverse momentum of the highest-$p_{T}$ b-jet (up panel) and transverse
momentum of $Z^{0}$ boson (bottom panel) compared to the ATLAS data
(Khachatryan:2016iob, ).

Figure 3: Differential cross section of $Z^{0}\,+\,$b-jet simulated by
SHERPA(blue line) in p+p collision at $\sqrt{s}=8$ TeV as a function of
azimuthal angular difference $\Delta\phi_{bZ}=|\phi_{b}-\phi_{Z}|$ of $Z^{0}$
boson and b-jet (up panel), and azimuthal angular difference
$\Delta\phi_{bb}=|\phi_{b1}-\phi_{b2}|$ of the two b-jets (bottom panel)
compared to the ATLAS data (Khachatryan:2016iob, ).

For events with at least one b-jet, we show the differential cross sections
simulated by SHERPA as a function of leading b-jet $p_{T}$ and $Z^{0}$ boson
$p_{T}$ in Fig. 2. In the up panel of Fig. 3, the azimuthal angle correlation
between the leading b-jet and $Z^{0}$ boson
($\Delta\phi_{bZ}=|\phi_{b}-\phi_{Z}|$) is also calculated to compare with
ATLAS data. In addition, for events with two b-jets (henceforth
$Z^{0}\,+\,2\,$b-jets), we also plot the differential cross sections as a
function of the azimuthal angle between the two b-jets
($\Delta\phi_{bb}=|\phi_{b1}-\phi_{b2}|$). We find that our calculations
simulated by SHERPA are in nice agreement with the experimental measurements.

##  III In-medium jet evolution

In high energy nuclear collisions, a droplet of an exotic state of nuclear
matter, the QGP is expected to be formed. The high $p_{T}$ partons produced in
the hard scattering propagating in the QGP would suffer both collisional and
radiative energy loss as a result of the in-medium interaction. A complete
treatment of the heavy-flavored jets propagating in the QGP needs a
simultaneous description of space-time evolution for both light and heavy
partons (Rapp:2018qla, ; Djordjevic:2013xoa, ; Kang:2016ofv, ; Sharma:2009hn,
). The Boltzmann and Langevin transport equations had been generally employed
to describe the parton evolution in the expanding QCD medium
(Svetitsky:1987gq, ; Nahrgang:2013saa, ; Scardina:2017ipo, ; Cao:2013ita, ;
Cao:2016gvr, ; He:2011zx, ; Dai:2018mhw, ; Wang:2019xey, ). Due to the lack of
unified theoretical approach, there are usually two methods to combine the
vacuum parton shower with the medium-induced radiation in heavy-ion
collisions. The first one is to introduce a medium modified splitting function
for the in-medium parton shower, like Q-PYTHIA Armesto:2009fj  and JEWEL
Zapp:2013vla . Here we employ the alternative treatment, Arnold-Moore-Yaffe
(AMY) scheme Arnold:2002ja , as implemented in LBT He:2015pra  and MARTINI
Schenke:2009gb . We take the p+p events produced by SHERPA with full vacuum
parton shower as the input, sample their initial spatial positions by a MC-
Glauber model Miller:2007ri , and then simulate the subsequent in-medium
evolution.

###  III.1 Collisional energy loss

The movement of a heavy quark with large mass ($M\gg T$) propagating in the
hot and dense nuclear matter and suffering a large number of random kicks from
the medium, can be modelled as a Brown motion (Svetitsky:1987gq, ). Hence a
discrete Langevin equation could be utilized to describe the propagating of
heavy quarks in the QCD medium (Moore:2004tg, ; Cao:2013ita, ; Dai:2018mhw, ;
Wang:2019xey, )

$\displaystyle\vec{x}(t+\Delta t)=\vec{x}(t)+\frac{\vec{p}(t)}{E}\Delta t$ (1)
$\displaystyle\vec{p}(t+\Delta t)=\vec{p}(t)-\Gamma(p)\vec{p}\Delta
t+\vec{\xi}(t)-\vec{p}_{g}\,,$ (2)

where $\Delta t$ is the time step in the Monte Carlo simulation, and
$\Gamma(p)$ is drag coefficient representing the dissipation effect and
control the strength of quasi-elastic scattering. $\vec{\xi}(t)$ is the
stochastic term which obeys a Gaussian-form possibility distribution:

$\displaystyle W[\vec{\xi}(t)]=N\exp[\frac{\vec{\xi}(t)^{2}}{2\kappa\Delta
t}]\,,$ (3)

and leads to:

$\displaystyle\left\langle\xi_{i}(t)\right\rangle$ $\displaystyle=0$ (4)
$\displaystyle\left\langle\xi_{i}(t)\xi_{j}(t^{\prime})\right\rangle$
$\displaystyle=\kappa\Delta t\delta_{ij}(t-t^{\prime})\,,$ (5)

The diffusion coefficient $\kappa$ is relative to the drag coefficient
$\Gamma$ by the fluctuation-dissipation relation Kubo :

$\displaystyle\kappa=2\Gamma ET=\frac{2T^{2}}{D_{s}}\,,$ (6)

where $D_{s}$ is the spacial diffusion coefficient which fixed at $2\pi
TD_{s}=4$ in this work based on the Lattice calculation Rapp:2018qla ;
Banerjee:2011ra . The last term $-\vec{p}_{g}$ is the recoil momentum due to
the medium induced gluon radiation which will be discussed in the following
section. At each time step, we boost partons to the local rest frame of the
expanding medium to update the four-momentum and then boost them back to the
laboratory frame to update the spatial position until the local temperature
bellow $T_{c}=165$ MeV. The space-time evolution profile of the bulk medium in
Pb+Pb collision is provided by the VISHNEW Shen:2014vra  code.

Meanwhile, the calculation to leading logarithmic accuracy at Hard-Thermal-
Loop approximation Neufeld:2010xi ; Huang:2013vaa  is employed in our
framework to describe the collisional energy loss of light quarks and gluon:

$\displaystyle\frac{dE}{dz}=-\frac{\alpha_{s}C_{s}\mu_{D}^{2}}{2}ln{\frac{\sqrt{ET}}{\mu_{D}}}$
(7)

where $z$ is the transport path of the partons along to the propagating
direction. Instead of considering the exact probability of the elastic
scattering, the mean effect of the elastic energy loss is implemented in our
simulation.

###  III.2 Medium induced gluon radiation

The inelastic scattering also plays an important role in the in-medium energy
loss of energetic partons Zakharov:2007pj ; Qin:2007rn . In our work, the
Higher-Twist radiated gluon spectra Guo:2000nz ; Zhang:2003yn ; Zhang:2004qm ;
Majumder:2009ge  has been implemented to simulate the medium-induced gluon
radiation when parton propagate in the dense and hot QCD matter:

$\displaystyle\frac{dN}{dxdk^{2}_{\perp}dt}=\frac{2\alpha_{s}C_{s}P(x)\hat{q}}{\pi
k^{4}_{\perp}}\sin^{2}(\frac{t-t_{i}}{2\tau_{f}})(\frac{k^{2}_{\perp}}{k^{2}_{\perp}+x^{2}m^{2}})^{4}$
(8)

where $x$ and $k_{\perp}$ are the energy fraction and the transverse momentum
of the radiative gluon. $\alpha_{s}$ is the strong coupling constant which
fixed at $\alpha_{s}=0.3$ in our calculation, $C_{s}$ is the quadratic Casimir
in color representation, and $P(x)$ is splitting function Wang:2009qb  for the
splitting processes $q\rightarrow q+g$ and $g\rightarrow g+g$
respectively($g\rightarrow q+\bar{q}$ process is negligible due to its low
possibility He:2011zx ),

$\displaystyle P_{q\rightarrow qg}(x)=$
$\displaystyle\frac{(1-x)(1+(1-x)^{2})}{x}$ (9) $\displaystyle P_{g\rightarrow
gg}(x)=$ $\displaystyle\frac{2(1-x+x^{2})^{3}}{x(1-x)}$ (10)

$\tau_{f}$ is the radiated gluon formation time defined as
$\tau_{f}=2Ex(1-x)/(k^{2}_{\perp}+x^{2}m^{2})$, and $t-t_{i}$ is the time
interval between two inelastic scattering. And $\hat{q}$ is the jet transport
coefficient Chen:2010te :

$\displaystyle\hat{q}(\tau,r)=\hat{q}_{0}\frac{\rho^{QGP}(\tau,r)}{\rho^{QGP}(\tau_{0},0)}\frac{p^{\mu}u_{\mu}}{p^{0}}$
(11)

where $\hat{q}_{0}$ denotes the value of $\hat{q}$ at the center of the bulk
medium at the initial time $\tau_{0}=0.6~{}\rm fm/c$, with $\hat{q}_{0}=1.2$
GeV2/fm at $T~{}(\tau_{0}=0.6~{}\rm fm/c)$ extracted from the identified
hadron suppression (Ma:2018swx, ), and $\rho^{QGP}(\tau,r)$ is the parton
number density where the parton probed. To take into account of the radial
flow effect Baier:2006pt , the four momentum of parton $p^{\mu}$ and the four
flow velocity of medium in the collision frame $u^{\mu}$ act as a modification
for $\hat{q}$ in a expanding nuclear medium. The last term represents the
dead-cone effect Armesto:2003jh ; Zhang:2004qm  which suppresses the gluon
radiation of heavy quarks at small angle owing to their large mass.

An imposed cut-off of radiated gluon energy fraction
$x_{min}=\frac{\mu_{D}}{E}$ has been taken to avoid the divergence near
$x\rightarrow 0$, where $\mu_{D}^{2}=4\pi\alpha_{s}(1+\frac{n_{f}}{6})T^{2}$
is the Debye screening mass induced by the GQP medium. Therefore, following
the method introduced in (Cao:2016gvr, ), one could estimate the mean number
of radiated gluon$\left\langle N(t,\Delta t)\right\rangle$ during a time step
$\Delta t$ by integrating the phase space of $x$, $k_{\perp}$ and $t$ in Eq.
(8) :

$\displaystyle\left\langle N(t,\Delta t)\right\rangle=\int_{t}^{t+\Delta
t}dt\int_{x_{\rm
min}}^{1}dx\int_{0}^{(xE)^{2}}dk^{2}_{\perp}\frac{dN}{dxdk^{2}_{\perp}dt}$

By assuming that the multiple gluon radiation is a Possion process, we obtain
the probability distribution of the radiation number $P(n,t,\Delta t)$ during
a time step, as well as the total inelastic scattering probability
$P_{rad}(t,\Delta t)$:

$\displaystyle P(n,t,\Delta t)=\frac{{\left\langle N(t,\Delta
t)\right\rangle}^{n}}{n!}e^{-\left\langle N(t,\Delta t)\right\rangle}$ (13)
$\displaystyle P_{rad}(t,\Delta t)=1-e^{-\left\langle N(t,\Delta
t)\right\rangle}$ (14)

Figure 4: Radiative energy loss of partons with initial energy
$E_{0}=50~{}$GeV in a static medium with temperature $T=400~{}$MeV: Comparison
of Monte Carlo simulation and semi-analytical calculation.

In our Monte Carlo simulation, during every time step, the $P_{rad}(t,\Delta
t)$ would be firstly evaluated to decide whether the radiation occurs. If
accepted ,and the Possion distribution function $P(n,t,\Delta t)$ would be
used for the sampling of the radiated gluon number. At last, the four momentum
of the radiated gluon could be sampled based on the spectrum
$dN/dxdk_{\perp}^{2}$ expressed in Eq. 8. In Fig. 4, for a consistent
comparison between our Monte Carlo simulation and the analytical calculation
based on Eq. 8, we estimate the radiative energy loss of gluon, light quark,
charm and bottom in a static medium ($T=400$ MeV). Here we fix the parton
energy (50 GeV) at each evolution time step and also restore the initial time
$t_{i}$ in Eq. (8) to be 0 same as the treatment in Ref. Cao:2017hhk , since
their variations during the Monte Carlo simulation are not automatically
included in the analytical calculation. We find MC results agree well with the
analytical calculations and show a clear mass hierarchy for different parton
species.

##  IV Numerical results and discussions

In this section, to estimate the medium modification of jet observables in
nucleus-nucleus collisions, we use the p+p events provided by SHERPA as input
of our simulation within the hydrodynamic background to study the in-medium
jet evolution. Before proceed to $Z^{0}$ tagged b-jet, we firstly calculate
the azimuthal angular correlation ($\Delta\phi_{jZ}=|\phi_{\rm
jet}-\phi_{Z}|$) and transverse momentum balance ($x_{jZ}=p_{T}^{\rm
jet}/p_{T}^{Z}$) of $Z^{0}\,+\,$jet, as well as the nuclear modification
factor $R_{AA}$ of inclusive b-jet, and compare our theoretical results with
the available experimental data. Then we calculate the $Z^{0}\,+\,$b-jet
observables, including azimuthal angle correlation between $Z^{0}$ boson and
b-jet ($\Delta\phi_{bZ}=|\phi_{\textit{\rm b-jet}}-\phi_{Z}|$), angle
separation between the Z-tagged two b-jets
($\Delta\phi_{bb}=|\phi_{b1}-\phi_{b2}|$), transverse momentum
($x_{jZ}=p_{T}^{\textit{\rm b-jet}}/p_{T}^{Z}$), and nuclear modification
factor $I_{AA}$, which defined as:

$\displaystyle I_{AA}=\frac{1}{\left\langle
N_{coll}^{AA}\right\rangle}\frac{\frac{dN^{AA}}{dp^{\rm jet}_{T}}|_{p_{T}^{\rm
min}<p_{T}^{Z}<p_{T}^{\rm max}}}{\frac{dN^{AA}}{dp^{\rm jet}_{T}}|_{p_{T}^{\rm
min}<p_{T}^{Z}<p_{T}^{\rm max}}}.$ (15)

Here $\left\langle N_{coll}^{AA}\right\rangle$ denotes the averaged number of
the binary nucleon-nucleon collision in A+A collisions calculated in Glauber
model Miller:2007ri .

Figure 5: Distributions of the azimuthal angle difference $\Delta\phi_{jZ}$
between the $Z^{0}$ boson and the jet both in p+p and $0-30\%$ Pb+Pb
collisions at 5.02 TeV compared with CMS data Sirunyan:2017jic . The
distributions are scaled by the $Z^{0}$ events number $N$ in p+p collisions.

In Fig. 5, we show our calculated $\Delta\phi_{jZ}$ distributions both in p+p
and $0-30\%$ Pb+Pb collisions at 5.02 TeV compared with the CMS experimental
data Sirunyan:2017jic  . The same configurations in the jet reconstruction are
used as that of the CMS. All final state jets are reconstructed by FASTJET
using anti-$k_{T}$ algorithm with $R=0.3$ and required $p_{T}^{\rm jet}>30$
GeV, the selected $Z^{0}$ boson are reconstructed by the electron or muon
pairs based on its decay channels ($Z^{0}\rightarrow e^{+}e^{-}$ and
$Z^{0}\rightarrow\mu^{+}\mu^{-}$), and required $p_{T}^{Z}>60$ GeV. Note that
these distributions are normalized by the number of $Z^{0}$ events, and the
jet transverse momentum are smeared by Gaussian form Sirunyan:2017jic  to take
into account the detector resolution effects. It is shown that the
distribution of azimuthal angle correlation in Pb+Pb collisions suffers a
suppression at small $\Delta\phi_{jZ}$ region relative to the p+p baseline,
which is consistent with CMS measurement. However, at large angle region
($\Delta\phi_{jZ}\sim\pi$, where $Z^{0}$ boson and jet are almost back-to-
back), this suppression is not very apparent. The underlying reason for this
behavior has been discussed in detailed in Chen:2018fqu ; Luo:2018pto , namely
small $\Delta\phi_{jZ}$ region is dominated by the multiple jets processes and
large $\Delta\phi_{jZ}$ region by soft/collinear radiation. Usually, the jet
energy of multiple-jet processes is relatively low and easier to be shifted
below the jet selection threshold ($p_{T}^{\rm jet}>30$ GeV) because of parton
energy loss Zhang:2018urd .

Figure 6: Distributions of the transverse momentum balance $x_{jZ}$ of
$Z^{0}\,+\,$jet both in p+p and $0-30\%$ Pb+Pb collisions at 5.02 TeV compared
with CMS data Sirunyan:2017jic . The distribution is normalized by the $Z^{0}$
events number and $Z^{0}\,+\,$jet pairs are required with
$\Delta\phi_{jZ}>7\pi/8$.

In Fig. 6, we compute $x_{jZ}$ distribution for $Z^{0}\,+\,$jet both in p+p
and $0-30\%$ Pb+Pb collisions and find that our calculations can give quiet
decent description of the CMS data. Note here that selected $Z^{0}\,+\,$jet
pairs are required to be almost back-to-back ($\Delta\phi_{jZ}>7\pi/8$).
Relative to the p+p baseline, in Pb+Pb collisions, we find that $x_{jZ}$
distribution is shifted toward smaller value, which shows an enhancement at
$0<x_{jZ}<0.7$ and suppression at $0.7<x_{jZ}<2$. It’s understood that
$x_{jZ}$ represents the transverse momentum imbalance of $Z^{0}$ and jet, for
each $Z^{0}\,+\,$jet pair, the values of $x_{jZ}$ decrease by the jet energy
loss, and thus are shifted to smaller $x_{jZ}$ observed in the final-state.

Besides, shown in Fig. 7, we investigate the nuclear modification factor
$R_{AA}$ of inclusive b-jet in Pb+Pb collisions at $\sqrt{s_{NN}}=2.76$ TeV
comparing with the CMS measurements Chatrchyan:2013exa  to test our model
calculations. And we find that, both in central and peripheral Pb+Pb
collisions, our theoretical calculations give fairly well description on the
nuclear modification of b-jet yield.

Figure 7: The nuclear modification factor $R_{AA}$ of b-jet. The p+p baseline
is provided by SHERPA and theoretical calculations are comparing with CMS data
Chatrchyan:2013exa  at centralities of $0-10\%$, $10-30\%$, $30-50\%$,
$50-100\%$.

The nice agreement between our model calculations and the data of
$Z^{0}\,+\,$jet and inclusive b-jet facilitates our studies on the medium
modification of $Z^{0}\,+\,$b-jet in nuclear-nuclear collisions. In Fig. 8, we
calculate the azimuthal angular correlation of $Z^{0}$ boson and b-jet in p+p
and $0-10\%$ Pb+Pb collisions at 5.02 TeV. The b-jets associated with the
$Z^{0}$ boson are reconstructed by FASTJET using anti-$k_{T}$ algorithm with
cone size $\Delta R=0.5$, $|\eta^{\rm jet}|<2.4$ and $p_{T}^{\rm jet}>30$ GeV
both in p+p and Pb+Pb collisions. Note that these distributions are normalized
by initial $Z^{0}\,+\,$b-jet event number (in p+p collision) to address the
medium modification. We observe an overall suppression in Pb+Pb collisions
relative to the p+p baseline. We show their ratio PbPb/pp in the middle panel
of Fig. 8, and find that the suppression for $Z^{0}\,+\,$b-jet has a much
weaker dependence on $\Delta\phi_{bZ}$, as compared to that for
$Z^{0}\,+\,$jet which shows stronger suppression at small $\Delta\phi_{jZ}$
region where multiple jets dominate. Obviously, all selected jet must first be
b quark tagged, this requirement significantly reduces the contribution from
multiple jets processes when we consider the azimuthal angular
($\Delta\phi_{bZ}$) distribution.

To address the key factor which leads the flat suppression on
$\Delta\phi_{bZ}$ distribution, we estimate the averaged b-jet transverse
momentum $\left\langle p_{T}\right\rangle$ as a function of $\Delta\phi_{bZ}$,
which can be calculated by :

$\displaystyle\left\langle
p_{T}\right\rangle(\Delta\phi)=\frac{\int\frac{d\sigma}{dp_{T}d\Delta\phi}p_{T}dp_{T}}{\int\frac{d\sigma}{dp_{T}d\Delta\phi}dp_{T}}\,.$
(16)

The decreasing of the selected event number in A+A collisions results from the
in-medium energy loss, which shifts lower $p_{T}$ jet below kinematic
selection cut. The initial $\left\langle p_{T}\right\rangle$ distribution
actually reflect the $\Delta\phi_{bZ}$ dependence of this shift. It turns out
that the distribution of $\left\langle p_{T}\right\rangle$ versus
$\Delta\phi_{bZ}$ is nearly a constant value at 55 GeV as shown in the bottom
panel of Fig. 8, which leads to the rather flat suppression on
$\Delta\phi_{bZ}$ distribution.

Figure 8: Distributions of the azimuthal angle correlation of $Z^{0}$ boson
and b-jet both in p+p and $0-10\%$ Pb+Pb collisions at 5.02 TeV. The
distributions are scaled by the $Z^{0}\,+\,$b-jet event number $N$ in p+p
collisions.  Figure 9: Distributions of the azimuthal angular separation
$\Delta\phi_{bb}$ of the two b-jets tagged by $Z^{0}$ boson both in p+p and
$0-10\%$ Pb+Pb collisions at 5.02 TeV. The distributions are scaled by the
$Z^{0}\,+\,2\,$b-jets event number $N$ in p+p collisions.

As we mentioned in Sec. II, the azimuthal angular separation $\Delta\phi_{bb}$
of the two b-jets tagged by $Z^{0}$ boson is also a useful observable to
distinguish the contribution from subprocesses where $Z^{0}$ boson is emitted
from one of the final state b quark or gluon splitting ($g\rightarrow
b\bar{b}$) Chatrchyan:2013zja , as shown in diagrams (c) and (d) of Fig. 1.
Note that these two categories of contributions correspond to the regions
where the two b-jets are almost back-to-back ($\Delta\phi_{bb}\sim\pi$) or
collinear ($\Delta\phi_{bb}\sim 0$). What interests us is how the
$\Delta\phi_{bb}$ distribution of these two categories of
$Z^{0}\,+\,2\,$b-jets would be modified in the QGP. As shown in the top panel
of Fig. 9, we plot these two $\Delta\phi_{bb}$ distribution both in p+p and
$0-10\%$ Pb+Pb collisions at 5.02 TeV, and also plot the ratio PbPb/pp in the
middle panel. We observe an upward trend of the ratio from 0.5 to 0.7 as
$\Delta\phi_{bb}$ increasing. To figure out the distribution dependence on
$\Delta\phi_{bb}$, we estimate the $\left\langle p_{T}\right\rangle$ of the
leading and sub-leading b-jet shown in the bottom panel of Fig. 9. We find
that, for both leading and subleading b-jets, $\left\langle
p_{T}\right\rangle$ is increasing with $\Delta\phi_{bb}$, which then results
in the increasing of the distribution ratio with $\Delta\phi_{bb}$.

Figure 10: Gluon initiated jet fraction as a function of transverse momentum
of inclusive jet (red dash), inclusive b-jet (blue dash), $Z^{0}\,+\,$jet (red
solid), $Z^{0}\,+\,$b-jet (blue solid).

The associated production of $Z^{0}\,+\,$jet may shed new light on the mass
dependence of the jet quenching effect in the nuclear matter, owing to its
high purity sample of light-quark-initiated jets. In Ref.
Kartvelishvili:1995fr , the contributions from light-quark-jet and gluon-jet
in the $Z^{0}\,+\,$jet production is $\approx 70\%$ and $\approx 30\%$. To
verify this point, in Fig. 10, we estimate the gluon-jet fraction in four
categories of jets in p+p collisions at 5.02 TeV: inclusive jet, inclusive
b-jet, $Z^{0}$ tagged jet and $Z^{0}$ tagged b-jet. We find that at
$p_{T}^{\rm jet}\sim 50$ GeV the gluon-jet fraction is $\approx 50\%$ in
inclusive jet and $\approx 30\%$ in $Z^{0}$ tagged jet. The Z-tagging
requirement considerably decreases the gluon jet contribution by $40\%$
especially at lower $p_{T}$. More importantly, for inclusive b-jet and Z
tagged b-jet, the contributions from gluon-jet are greatly suppressed due to
the requirement of b-quark tagging and show almost equal values.

Figure 11: Distributions of $x_{jZ}$ ($x_{bZ}$) of $Z^{0}\,+\,$jet and $Z^{0}\,+\,$b-jet both in p+p and $0-10\%$ Pb+Pb collisions at 5.02 TeV.  |  $Z^{0}$\+ jet |  $Z^{0}+$ b-jet
---|---|---
${\left\langle x_{J}\right\rangle}_{pp}$ |  0.987 |  0.941
${\left\langle x_{J}\right\rangle}_{PbPb}$ |  0.851 |  0.849
$\Delta x_{J}$ |  0.136 |  0.092
Table 1: The comparison of mean value of momentum imbalance $x_{J}$ of $Z^{0}$
\+ jet and $Z^{0}$ \+ b-jet both in p+p and $0-10\%$ Pb+Pb collisions at
$\sqrt{s_{NN}}=5.02$ TeV, as well as the shifting of mean value of momentum
imbalance $\Delta x_{J}={\left\langle x_{J}\right\rangle}_{pp}-{\left\langle
x_{J}\right\rangle}_{PbPb}$.

In Fig. 11, we plot the scaled $x_{jZ}$ ($x_{bZ}$) distribution of
$Z^{0}\,+\,$jet and $Z^{0}\,+\,$b-jet in both p+p and $0-10\%$ Pb+Pb
collisions at 5.02 TeV. Note all selected jets (b-jets) must be in the region
$\Delta\phi_{jZ}>7\pi/8$ ($\Delta\phi_{bZ}>7\pi/8$) to guarantee they are
mostly back-to-back with the $Z^{0}$ boson. We observe the distributions of
$x_{J}$ shift towards smaller value both in $Z^{0}\,+\,$jet and
$Z^{0}\,+\,$b-jet in Pb+Pb collisions relative their p+p baseline, due to the
energy loss of the tagged jets. To make a more intuitive comparison between
$Z^{0}\,+\,$jet and $Z^{0}\,+\,$b-jet, we show the ratio of $x_{J}$
distribution in Pb+Pb to that in p+p (see the bottom panel of Fig. 11), which
have positive values at $0.2<x_{J}<0.8$ and negative values at
$0.8<x_{J}<1.6$. And then we find the absolute value of the ratio (PbPb-pp) of
$Z^{0}\,+\,$jet is larger than that of $Z^{0}\,+\,$b-jet. Furthermore, we
estimate the shifting of mean value of momentum imbalance $\Delta{\left\langle
x_{jZ}\right\rangle}={\left\langle x_{jZ}\right\rangle}_{pp}-{\left\langle
x_{jZ}\right\rangle}_{PbPb}$ for $Z^{0}\,+\,$jet, and $\Delta{\left\langle
x_{bZ}\right\rangle}={\left\langle x_{bZ}\right\rangle}_{pp}-{\left\langle
x_{bZ}\right\rangle}_{PbPb}$ for $Z^{0}\,+\,$b-jet shown in Tab. I

$\left\langle
x_{J}\right\rangle=\frac{1}{\sigma}\int\frac{d\sigma}{dx_{J}}x_{J}dx_{J}\,,$
(17)

with $J$ denotes different processes. It reads that $\Delta x_{jZ}$ ($\sim
0.136$) for $Z^{0}\,+\,$jet is significantly larger than $\Delta x_{bZ}$
($\sim 0.092$) for $Z^{0}\,+\,$b-jet, which clearly indicates a stronger
modification on light-quark jet than b-jet.

The nuclear modification factors of tagged jet cross section $I_{AA}$ is
practically another good observable to address the mass hierarchy and flavor
dependence of jet quenching effects. The comparison of $I_{AA}$ between
$Z^{0}\,+\,$jet and $Z^{0}\,+\,$b-jet would provide more reliable evidence to
the mass effect of jet quenching. For this purpose, we present the
calculations of $I_{AA}$ of $Z^{0}\,+\,$jet and $Z^{0}\,+\,$b-jet in $0-10\%$
Pb+Pb collisions at 5.02 TeV as a function of jet $p_{T}$ within three
$p^{Z}_{T}$ windows shown in Fig. 12. The shapes of $I_{AA}$ are sensitive to
the kinematical $p^{Z}_{T}$ cut both for $Z^{0}$ \+ jet and $Z^{0}$ \+ b-jet
due to the fact that $p^{Z}_{T}$ region gauges the initial jet energy.
Especially, comparing $I_{AA}$ of tagged b-jet and light jet in $0-10\%$ Pb+Pb
collisions for each $p^{Z}_{T}$ cut, one observes a visible stronger
suppression for $Z^{0}$ \+ jet relative to that of $Z^{0}\,+\,$b-jet. This
indicates that the in-medium energy loss of $Z^{0}$ tagged b-jet is smaller
than that of $Z^{0}$ tagged light-quark jet.

##  V Summary

Vector boson-tagged heavy quark jets are potentially new promising tools to
study the jet quenching effect. In this work, we presented a Monte Carlo
transport simulation, which takes into account the elastic and inelastic jet
interactions within a hydrodynamic background, to study the in-medium
modification of the vector boson tagged b-jets. The NLO+PS event generator
SHERPA has been used to provide the p+p baseline of $Z^{0}\,+\,$b-jet
production, which shows a nice agreement with the ATLAS measurements. This
framework has been proven to give nice descriptions of medium modifications on
$\Delta\phi_{jZ}$ and $x_{jZ}$ of $Z^{0}\,+\,$jet, as well as the $R_{AA}$ of
inclusive b-jet, measured in Pb+Pb collisions at the LHC.

The angular correlation between the vector boson and heavy quark tagged jets
may be a new promising observable to study the in-medium jet interaction. We
presented the first calculations of the azimuthal angular correlation
$\Delta\phi_{bZ}$ of $Z^{0}\,+\,$b-jet both in p+p and $0-10\%$ Pb+Pb
collisions at $\sqrt{s_{NN}}=$ 5.02 TeV. We observed a flat suppression factor
versus $\Delta\phi_{bZ}$, unlike the case in $Z^{0}\,+\,$jet, since the
requirement of b-tagging excludes the contribution from multiple-jet
processes. In addition, for $Z^{0}\,+\,2\,$b-jets ,we also calculated another
interesting observable, the azimuthal angle $\Delta\phi_{bZ}$ distribution
between the two b-jets in p+p and Pb+Pb collisions, and observed stronger
suppression at small $\Delta\phi_{bb}$ where dominated by gluon splitting
processes relative to at large $\Delta\phi_{bb}$. We estimated the
$\left\langle p_{T}\right\rangle$ distribution and demonstrate the
modification patterns on these azimuthal angle are sensitive to the initial
b-jets $p_{T}$ distribution versus $\Delta\phi_{bZ}$ (or $\Delta\phi_{bb}$).
These investigates may help us to understand the experimental measurements on
jet angular correlations a the LHC in recent years.

Furthermore, with the high purity of quark jet in $Z^{0}\,+\,$(b-)jet events,
we addressed the mass dependence of $Z^{0}$ tagged (b-)jet productions. We
calculate $x_{jZ}$ of $Z^{0}\,+\,$jet and $x_{bZ}$ of $Z^{0}\,+\,$b-jet, and
show the ordering of the shift of the mean momentum imbalance for
$Z^{0}\,+\,$b-jet and $Z^{0}\,+\,$jet in heavy-ion collisions to be
$\Delta\left\langle x_{bZ}\right\rangle<\Delta\left\langle
x_{jZ}\right\rangle$. We also calculate another observable, the nuclear
modification factors of tagged jet cross section $I_{AA}$ for $Z^{0}\,+\,$jet
and $Z^{0}\,+\,$b-jet, and observe stronger suppression of $I_{AA}$ for
$Z^{0}\,+\,$jet relative to those for $Z^{0}\,+\,$b-jet. In general, these
comparisons can be validated at the LHC with the current experimental
technologies, and may provide a key to unlock the puzzle of mass hierarchy of
jet quenching.

Figure 12: The nuclear modification factor as a function of transverse
momentum of the tagged jet within three $p_{T}^{Z}$ ranges: $40-60$ GeV,
$60-80$ GeV, $80-120$ GeV in $0-10\%$ centrality Pb+Pb collisions at
$\sqrt{s_{NN}}=5.02$ TeV.

###### Acknowledgements.

This research is supported by the NSFC of China with Project Nos. 11935007,
11805167, and partly supported by China University of Geosciences (Wuhan) (No.
162301182691).

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