EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-EP-2016-092 LHCb-PAPER-2016-011 30 May 2016

Measurement of forward $W$

and $Z$ boson production

in association with jets in proton-proton collisions at $\sqrt{s}=8$ TeV

The LHCb collaboration†††Authors are listed at the end of this paper.

The production of $W$ and $Z$ bosons in association with jets is studied in
the forward region of proton-proton collisions collected at a centre-of-mass
energy of 8 TeV by the LHCb experiment, corresponding to an integrated
luminosity of 1.98 $\pm$ 0.02 fb-1. The $W$ boson is identified using its
decay to a muon and a neutrino, while the $Z$ boson is identified through its
decay to a muon pair. Total cross-sections are measured and combined into
charge ratios, asymmetries, and ratios of $W+$jet and $Z$+jet production
cross-sections. Differential measurements are also performed as a function of
both boson and jet kinematic variables. All results are in agreement with
Standard Model predictions.

Published in JHEP 05 (2016) 131

© CERN on behalf of the LHCb collaboration, licence CC-BY-4.0.

##  1 Introduction

Measurements of vector boson production in association with jets in the
forward region at the Large Hadron Collider (LHC) can be used to test the
Standard Model (SM) and provide constraints on the parton density functions
(PDFs). LHCb is the only detector at the LHC with precision tracking coverage
in the forward region, allowing sensitivity to PDFs at a different range of
Bjorken-$x$ compared to ATLAS and CMS [1]. LHCb measurements typically probe
PDFs at $x$ as low as $10^{-4}$ and at high $x$ [2].

This article reports total and differential cross-section measurements of $W$
and $Z$ production in association with jets, hereafter referred to as ${W}{j}$
and ${Z}{j}$, respectively.111Here, the notation $Z$ additionally includes
contributions from virtual photon production and its interference with $Z$
boson production, ${Z}/\gamma^{*}$. The measurements are performed using data
collected during 2012 at a centre-of-mass energy of $\sqrt{s}$
=8$\mathrm{\,Te\kern-1.00006ptV}$, corresponding to an integrated luminosity
of 1.98 $\pm$ 0.02 fb-1. The $W$ and $Z$ bosons are identified through the
${W}\rightarrow\mu\nu_{\mu}$ and ${Z}\rightarrow\mu\mu$ decay channels. This
work extends measurements of the ${Z}{j}$ production cross-section at
7$\mathrm{\,Te\kern-1.00006ptV}$ [3, 4] and ratios of the production cross-
sections at 7 and 8$\mathrm{\,Te\kern-1.00006ptV}$ [5]. It also complements
previous studies of inclusive electroweak boson production at LHCb, where the
electroweak bosons decay to muons [6, 7, 8].

This analysis makes use of the same fiducial acceptances for electroweak
bosons as previously employed in Ref. [7]. For $W$ boson decays, this
corresponds to requiring that the muon has a pseudorapidity, $\eta^{\mu}$, in
the range $2.0<\eta^{\mu}<4.5$ and transverse momentum,
$p_{\textrm{T}}^{\mu}$, greater than 20$\mathrm{\,Ge\kern-1.00006ptV}$.222This
article uses natural units, where the speed of light ($c$) and the reduced
Planck constant ($\hbar$) are set to unity, $c=\hbar=1$. For $Z$ boson decays,
both muons are required to fulfil these kinematic requirements, and in
addition, the dimuon invariant mass, $M_{\mu\mu}$, is required to be in the
range $60<M_{\mu\mu}<120\mathrm{\,Ge\kern-1.00006ptV}$. The fiducial criteria
for these measurements require at least one jet to have transverse momentum
$p_{\textrm{T}}^{\textrm{jet}}>20\mathrm{\,Ge\kern-1.00006ptV}$, and jet
pseudorapidity, $\eta^{\textrm{jet}}$, in the range
$2.2<\eta^{\textrm{jet}}<4.2$. The jet is also required to be separated by a
radius $\Delta R$ of 0.5 from the charged lepton(s) produced in the boson
decay, where $\Delta R$ is the sum in quadrature of the difference in
pseudorapidity and the difference in azimuthal angle between the jet and the
lepton. In addition, the ${W}{j}$ measurement requires that the transverse
component of the vector sum of the muon and jet momenta,
$p_{\textrm{T}}^{\mu+j}$, is greater than 20$\mathrm{\,Ge\kern-1.00006ptV}$.
Jets are reconstructed using the anti-$k_{\text{T}}$ algorithm [9], with the
$R$ parameter set to 0.5. Jet energies are defined at the hadron level, and do
not include the contribution of neutrinos in the jet.

All measurements are performed for the jet with the largest transverse
momentum in the event. The ${W}{j}$ measurement is made differentially as a
function of $p_{\textrm{T}}^{\textrm{jet}}$, $\eta^{\textrm{jet}}$, and the
pseudorapidity of the muon produced by the $W$ boson decay, $\eta^{\mu}$. For
the ${Z}{j}$ measurement, the differential cross-sections are determined as a
function of $p_{\textrm{T}}^{\textrm{jet}}$, $\eta^{\textrm{jet}}$, the boson
rapidity, $y^{{Z}}$, and the difference in azimuthal angle between the $Z$
boson and the jet, $|\Delta\phi|$. The jet transverse momentum distributions
and the $|\Delta\phi|$ distribution tend to be sensitive to higher-order
effects within perturbative quantum chromodynamics (QCD) [10], while
measurements of the (pseudo)rapidity distributions are sensitive to the PDFs
that parameterise the structure of the proton. The ratio of the ${W^{+}}{j}$
to the ${W^{-}}{j}$ cross-sections is measured, as is the ratio of the
${W}{j}$ cross-sections to the ${Z}{j}$ cross-section. Finally, the charge
asymmetry of ${W}{j}$ production is measured as a function of $\eta^{\mu}$.

##  2 Detector and simulation

The LHCb detector [11, 12] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. The detector includes a high-precision tracking
system consisting of a silicon-strip vertex detector surrounding the $pp$
interaction region, a large-area silicon-strip detector located upstream of a
dipole magnet with a bending power of about $4{\mathrm{\,Tm}}$, and three
stations of silicon-strip detectors and straw drift tubes placed downstream of
the magnet. The tracking system provides a measurement of momentum, $p$, of
charged particles with a relative uncertainty that varies from 0.5% at low
momentum to 1.0% at 200$\mathrm{\,Ge\kern-1.00006ptV}$. The minimum distance
of a track to a primary vertex (PV), the impact parameter, is measured with a
resolution of $(15+29/\mbox{$p_{\mathrm{T}}$}){\,\upmu\mathrm{m}}$, where
$p_{\mathrm{T}}$ is the component of the momentum transverse to the beam, in
$\mathrm{\,Ge\kern-1.00006ptV}$. Different types of charged hadrons are
distinguished using information from two ring-imaging Cherenkov detectors.
Photons, electrons and hadrons are identified by a calorimeter system
consisting of scintillating-pad (SPD) and preshower detectors, an
electromagnetic calorimeter and a hadronic calorimeter. Muons are identified
by a system composed of alternating layers of iron and multiwire proportional
chambers. The online event selection is performed by a trigger, which consists
of a hardware stage, based on information from the calorimeter and muon
systems, followed by a software stage, which applies a full event
reconstruction.

In this paper, candidate events are required to pass the hardware trigger,
which selects muons with a transverse momentum
$\mbox{$p_{\mathrm{T}}$}>1.76\mathrm{\,Ge\kern-1.00006ptV}$ and the subsequent
software trigger, where a muon with
$\mbox{$p_{\mathrm{T}}$}>10\mathrm{\,Ge\kern-1.00006ptV}$ is required to be
present. A global event cut (GEC) is also applied at the hardware stage, which
requires that the number of hits in the SPD sub-detector should be less than
600.

Simulated $pp$ collisions are generated using Pythia 8 [13, *Sjostrand:2006za]
with a specific LHCb configuration [15]. Decays of hadronic particles are
described by EvtGen [16], in which final-state radiation is generated using
Photos [17]. The interaction of the generated particles with the detector, and
its response, are implemented using the Geant4 toolkit [18,
*Agostinelli:2002hh] as described in Ref. [20].

Results are compared to theoretical calculations performed at
$\mathcal{O}(\alpha_{s}^{2})$ in perturbative QCD using the Powheg [10, 21]
and aMC@NLO [22] generators, interfaced with Pythia in order to simulate the
parton shower, where the NNPDF3.0 [23, 24] PDF set is used to describe the
dynamics of the colliding protons. Additional fixed-order predictions are
generated using Fewz [25] at $\mathcal{O}(\alpha_{s}^{2})$ with the NNPDF3.0,
CT14 [26] and MMHT14 [27] PDF sets.

##  3 Event selection

Events are selected containing one or two high-$p_{\mathrm{T}}$ muons produced
in association with a high-$p_{\mathrm{T}}$ jet. Jets are reconstructed at
LHCb using a particle flow algorithm [3] and clustered using the
anti-$k_{\text{T}}$ algorithm as implemented in Fastjet [28]. Additional
selection requirements are placed on the jet properties in order to reduce the
number of spurious jets selected. The jet energies are calibrated on an event-
by-event basis. These calibrations are determined from both data and
simulation, and are applied as a function of the jet $p_{\mathrm{T}}$,
azimuthal angle, pseudorapidity, charged particle fraction and the number of
reconstructed PVs in the event [3]. To reduce contamination from multiple $pp$
interactions, charged particles reconstructed within the vertex detector are
only clustered into a jet if they are associated to the same PV as the final
state muon(s).

The measured muons and jets are required to satisfy the fiducial requirements
outlined in Sec. 1. An exception is the requirement on the $p_{\mathrm{T}}$ of
the vector sum of the momentum of the muon and jet,
$p_{\textrm{T}}^{\mu+j}>20\mathrm{\,Ge\kern-1.00006ptV}$, in ${W}{j}$ events.
In the selection, the muon is replaced by the jet, $\mu\textrm{-jet}$, which
contains the signal muon after performing a jet reconstruction with relaxed
jet selection requirements. The modified fiducial requirement,
$p_{\textrm{T}}^{\mu\textrm{-jet}+j}>20\mathrm{\,Ge\kern-1.00006ptV}$,
improves the suppression of the background from di-jets, which tend to be
balanced in transverse momentum. An acceptance factor is introduced (see Sec.
5), which corrects the results to correspond to the fiducial regions defined
in Sec. 1.

As ${W}{j}$ events contain just one final-state muon and consequently suffer
from a higher background, additional requirements are placed on the sample.
The background to the ${W}{j}$ sample from ${Z}{j}$ events where both muons
are produced in the LHCb acceptance is suppressed by rejecting events
containing a second muon with $p_{\mathrm{T}}$ in excess of
20$\mathrm{\,Ge\kern-1.00006ptV}$. Backgrounds from semileptonic decays of
heavy-flavour hadrons are suppressed by requiring that the impact parameter of
the muon track with respect to the PV should be less than 0.04$\mathrm{\,mm}$.
Additionally, the sum of the energy associated with the track in the
electromagnetic and hadronic calorimeters is required to be less than 4% of
the muon momentum. In total, 8 162 ${Z}{j}$ and 133 746 (99 683) ${W^{+}}{j}$
(${W^{-}}{j}$) candidates are selected.

##  4 Purity determination

The selected data samples contain background contributions from three distinct
processes:

  * •

QCD multi-jet production, which can produce muons in the final state, either
due to the misidentification of hadrons, or through the semileptonic decay of
heavy-flavour hadrons where a high-$p_{\mathrm{T}}$ jet is also present in the
event.

  * •

Electroweak processes, such as $Z\rightarrow\tau\tau$, $W\rightarrow\tau\nu$
or, in the case of ${W}{j}$ production, $Z\rightarrow\mu\mu$, can produce
events that mimic the signal. Contributions are also expected from electroweak
diboson and top quark production.

  * •

A small background contribution from “fake jets” is present when the data
sample contains events where the reconstructed and identified jet is not
associated with genuine particles, but is instead due to detector effects,
such as the presence of fake or misreconstructed particles, or to particles
produced in a different $pp$ collision to that producing the $W$ or $Z$ boson.

###  4.1 ${W}{j}$ sample purity

The QCD background to the ${W}{j}$ sample is determined by performing an
extended maximum likelihood fit to the distribution of the muon transverse
momentum $p_{\textrm{T}}^{\mu}$, divided by the transverse momentum of the
$\mu\textrm{-jet}$, $p_{\textrm{T}}^{\mu\textrm{-jet}}$ (where the
$\mu\textrm{-jet}$ is defined in Sec. 3). This variable acts as a measure of
muon isolation, with a value close to unity when little activity is present in
the vicinity of the candidate muon and a value closer to zero as the
multiplicity in the surrounding region increases. Consequently, it provides
strong discrimination between muons produced in electroweak processes, which
tend to be produced in isolation, and those produced in QCD processes, which
are typically surrounded by additional particles. Two separate components are
accounted for in the fit:

  * •

The template shape describing all electroweak processes, including the signal,
is taken from simulation. The shape of the isolation variable is approximately
independent of $p_{\textrm{T}}^{\mu}$, and consequently provides a good
description of all electroweak processes. The simulated shape is corrected for
mismodelling by applying correction factors obtained from a comparison of
${Z}{j}$ events in data and simulation. The ${W}{j}$ signal contribution is
subsequently separated from the other electroweak backgrounds as described
below.

  * •

The QCD background template is obtained using a di-jet enriched data sample,
obtained by requiring
$p_{\textrm{T}}^{\mu\textrm{-jet}+j}<20\mathrm{\,Ge\kern-1.00006ptV}$. The
small contribution from signal events in the template is subtracted using
simulation where the normalisation is obtained from the bin corresponding to
$p_{\textrm{T}}^{\mu}/p_{\textrm{T}}^{\mu\textrm{-jet}}>0.95$ in the signal
region. The template shape is then corrected for differences in the
$p_{\textrm{T}}^{\mu\textrm{-jet}}$ distribution between the background and
signal regions.

The fits are performed in bins of $\eta^{\textrm{jet}}$,
$p_{\textrm{T}}^{\textrm{jet}}$, and $\eta^{\mu}$ separately for positively
and negatively charged ${W}{j}$ candidates. The background from $Z$ decays to
muons and $\tau$ leptons, where a single muon is present in the final state,
is determined from simulation where the sample is normalised to the number of
fully reconstructed ${Z}\rightarrow\mu\mu$ decays observed in data. The small
contribution from $WW$, $t\bar{t}$ and single top events is determined using
next-to-leading order (NLO) predictions obtained from MCFM [29]. Finally, the
background from ${W}\rightarrow\tau\nu$ decays is determined by first
obtaining the ratio of $W\rightarrow\tau\nu$ to $W\rightarrow\mu\nu$ events
expected from simulation and normalising to the remaining signal after all
other backgrounds have been determined. The background from fake jets is
evaluated using simulation.

The contribution from QCD processes is found to vary between 30–70% in
different bins of $\eta^{\textrm{jet}}$, $p_{\textrm{T}}^{\textrm{jet}}$ and
$\eta^{\mu}$ while the contribution from electroweak processes (including di-
boson and top production) amounts to 5–10% of the selected samples. The
contribution from fake jets represents approximately 0.8–0.9% of the samples.
The overall purity of the ${W^{+}}{j}$ (${W^{-}}{j}$) sample is determined to
be 46.7(36.5)% where the total contributions, obtained by summing over the
yields in the $\eta^{\textrm{jet}}$ bins, are shown in Fig. 1.

Figure 1: The contributions to the selected (left) ${W^{+}}{j}$ and (right)
${W^{-}}{j}$ samples are shown, where the QCD background is obtained by a fit
to the $p_{\textrm{T}}^{\mu}$/$p_{\textrm{T}}^{\mu\textrm{-jet}}$ spectrum and
the electroweak background is determined as described in the text. The
contributions shown are the sum of the individual contributions in bins of
$\eta^{\textrm{jet}}$, where the charge asymmetry typical of ${W}{j}$
production in $pp$ collisions is evident.

###  4.2 ${Z}{j}$ sample purity

The contribution from semileptonic decays of heavy-flavour particles to the
${Z}{j}$ sample is determined by selecting a background-enhanced sample using
two approaches, where either the muons are not isolated from other activity in
the event or where they do not form a good vertex. The efficiency with which
the requirements select background events is evaluated by comparing the number
of events selected by the two approaches as in Ref. [7]. The total
contribution is estimated to be approximately 0.7%. The misidentification of
hadrons as muons is evaluated as in Ref. [7], by considering the contribution
from events where both muons fulfil all the selection criteria, but with both
muons required to have the same sign charge; and gives a contribution of
approximately 0.4%. Decays of the $Z$ boson to $\tau$ pairs can contribute if
both $\tau$ leptons subsequently decay to muons. The contribution from this
source is determined from simulation to be approximately 0.1%. The number of
events containing di-boson or top production is again calculated using
simulation, normalised to NLO predictions from MCFM and is determined to be
negligible. The contribution from fake jets is determined from simulation to
amount to approximately 0.9% of the selected sample. The overall purity of the
${Z}{j}$ sample is determined to be 97.8%.

##  5 Cross-section measurement

The cross-section, $\sigma_{i}$, for $W$ and $Z$ boson production in
association with one or more jets in the $i^{th}$ phase space bin is given by

$\sigma_{i}=U_{i}\frac{\mathcal{A}_{i}\cdot\rho_{i}\cdot
N_{i}}{\varepsilon^{\rm muon}_{i}\cdot\varepsilon^{\rm
jet}_{i}\cdot\varepsilon^{\rm sel}_{i}\cdot\mathcal{L}}\,,$ (1)

where $U_{i}$ is an unfolding correction which accounts for resolution effects
causing migrations between different bins of phase space. The number of
candidates selected in bin $i$ is given by $N_{i}$ while $\rho_{i}$ represents
the signal purity. The acceptance factor, $\mathcal{A}_{i}$, accounts for
differences between the fiducial region of the measurement and the kinematic
requirements placed on the muons and jets. The efficiencies for reconstructing
the muons and the jet are given by $\varepsilon^{\rm muon}_{i}$ and
$\varepsilon^{\rm jet}_{i}$, respectively, while the efficiency of any
additional event selection is given by $\varepsilon^{\rm sel}_{i}$.

The instantaneous luminosity is measured continuously during the acquisition
of physics data by recording the rates of several selected standard processes.
The effective absolute cross-section of these processes is measured during
dedicated calibration periods, using both van der Meer scans [30, 31] and
beam-gas imaging methods specific to the LHCb detector [32]. Both methods give
consistent results and are combined to give the final luminosity calibration
with an uncertainty of 1.2% [33]. The integrated luminosity of the data sample
used, $\mathcal{L}$, is obtained from the accumulated counts of the calibrated
rates and amounts to 1.98 $\pm$ 0.02 fb-1.

The efficiency to reconstruct and select muons in the event is evaluated using
the same techniques employed in the inclusive $W$ and $Z$ boson measurements
at LHCb [6, 7, 8]. In particular, a data-driven tag-and-probe study is
performed on selected inclusive ${Z}\rightarrow\mu\mu$ events in data and the
efficiency of reconstructing, triggering and identifying the muons is
measured. These efficiencies are applied as a function of the pseudorapidity
of the muon(s) in the event. The efficiency to reconstruct and identify the
jet in the event $\varepsilon^{\rm jet}_{i}$, is evaluated from simulation.
This efficiency increases with $p_{\mathrm{T}}$, from about 90% for jets with
$p_{\mathrm{T}}$ of 20$\mathrm{\,Ge\kern-1.00006ptV}$ to saturate at about 95%
for higher $p_{\mathrm{T}}$ jets. It is dominated by the probability that the
jet passes the requirements designed to reject fake jets. In the case of the
${W}{j}$ sample, the efficiency of the additional requirements placed on the
event, including a veto on extra muons, is evaluated using a “pseudo-${W}{j}$”
sample, where ${Z}{j}$ events are selected but one muon is masked in order to
mimic the neutrino in ${W}{j}$ events. Corrections are applied based on a
comparison of the efficiency of the requirements in ${W}{j}$ and
“pseudo-${W}{j}$” events in simulation. The efficiency of the GEC requirement
at the hardware stage of the trigger is again evaluated in a similar fashion
to the inclusive analyses, where the efficiency is measured in a ${Z}{j}$
sample selected with a looser trigger requirement [6, 7, 8]. This efficiency
is evaluated separately in each kinematic bin considered in the analysis, but
shows little variation with the variables that describe the jet kinematics.

The unfolding correction, $U_{i}$, corrects for differences observed in the
number of events produced and measured in a given bin due to the finite
resolution of the detector, where the differences are primarily caused by
migrations in the $p_{\textrm{T}}^{\textrm{jet}}$ and $\eta^{\textrm{jet}}$
distributions. The correction is determined from simulation as the ratio of
events produced in a specific bin to those recorded by the detector in the
same bin. The correction varies between 0.9 and 1.0, where the largest
corrections are seen at low $p_{\textrm{T}}^{\textrm{jet}}$ and in the highest
and lowest $\eta^{\textrm{jet}}$ bins.

For the ${Z}{j}$ sample, the acceptance factor, $\mathcal{A}_{i}$, is
identically equal to unity as the selection mirrors the fiducial acceptance
exactly. In the case of the ${W}{j}$ selection, the requirement of
$p_{\textrm{T}}^{\mu\textrm{-jet}+j}>20\mathrm{\,Ge\kern-1.00006ptV}$ differs
from the fiducial requirement of
$p_{\textrm{T}}^{\mu+j}>20\mathrm{\,Ge\kern-1.00006ptV}$. Consequently, the
acceptance factor accounts for differences between these two variables arising
from extra activity that may be present in the neighbourhood of the signal
muon. This factor is evaluated using simulation, which is reweighted in bins
of jet $p_{\mathrm{T}}$ and pseudorapidity to match next-to-leading order
predictions obtained from aMC@NLO. The acceptance factor varies between 0.95
and 1.00 in different bins of phase space.

##  6 Systematic uncertainties

Several sources of systematic uncertainty have been evaluated. The uncertainty
on the estimated purity of the ${W}{j}$ sample is evaluated by repeating the
fit using alternative templates. The fit is performed for a number of
different scenarios:

  * •

the data-driven corrections are not applied to the simulated ${W}{j}$ shape,

  * •

the simulated ${W}{j}$ shape is replaced by the “pseudo-${W}{j}$” data sample,

  * •

the subtraction of signal events from the background template is performed by
obtaining the normalisation from simulation instead of the data-driven method
outlined in Sec. 4.1.

The uncertainty on the contributions from electroweak templates is taken to be
the statistical precision on the ${Z}{j}$ and ${W}{j}$ samples used to perform
the data-driven normalisation. For the ${Z}{j}$ sample, the uncertainty on the
misidentification background is given by the sum in quadrature of the
statistical precision and the accuracy of the method, obtained by comparing
the two approaches described in Sec. 4.2. This gives an uncertainty of
approximately 30% on the misidentification background. The uncertainty on the
contribution from semileptonic decays of heavy-flavour hadrons is about 20%,
consisting of the sum in quadrature of the statistical uncertainty on the
evaluated contribution, and the variation in the background level found by
changing the requirements used in selecting the background-enhanced region.
The uncertainty due to the presence of fake jets is taken to be the
statistical uncertainty of approximately 30% on the determination of the fake-
jet contribution. A similar level of agreement is observed between data and
simulation by comparing kinematic distributions in regions with enhanced fake-
jet populations.

The uncertainty in the muon reconstruction efficiency is determined by re-
evaluating the cross-section with the total efficiency varied by one standard
deviation around the central value. An additional 1% systematic uncertainty is
also applied to account for differences in efficiencies observed between
inclusive $Z$ events and ${Z}{j}$ events. The uncertainty on the jet
reconstruction efficiency is evaluated by comparing the differences in
efficiency between ${Z}{j}$ data and simulation where the quality requirements
are varied about their nominal values. This results in an uncertainty of 1.9%.
The uncertainty on the selection efficiency, 1%, includes the statistical
uncertainty due to the limited size of the “pseudo-${W}{j}$” data sample and
the uncertainty on the corrections evaluated from simulation for differences
between ${W}{j}$ and “pseudo-${W}{j}$” events. The uncertainty on the GEC
efficiency is taken to be the sum in quadrature of the accuracy of the method,
0.3% [7, 8], and the difference observed between ${W^{+}}{j}$, ${W^{-}}{j}$
and ${Z}{j}$ events in simulation, typically smaller than 0.2%. The
uncertainty on the efficiency with which jets are selected is evaluated by
varying the selection requirements and determining how the fraction of events
rejected agrees between data and simulation, using the methods described in
Ref. [3]. Agreement is typically seen at the level of about 1.7%. This is
taken as an uncertainty on the modelling of the efficiencies in simulation,
and is combined in quadrature with the statistical precision with which the
efficiencies are determined.

The uncertainty on the acceptance factor, $\mathcal{A}_{i}$, is determined by
comparing the values obtained with and without NLO reweighting performed, and
by comparing the acceptance calculated in “pseudo-${W}{j}$” events in data and
simulation. These individual differences, contributing 0.5% and 0.3%,
respectively, are added in quadrature with the statistical precision of the
determination.

Two contributions to the uncertainty on the unfolding correction, $U_{i}$, are
considered. The variation of the corrections is evaluated by comparing the
difference in the number of ${Z}{j}$ events between the bin-by-bin corrections
employed in the analysis and a Bayesian unfolding [34, 35] with two
iterations. The difference is typically 0.8–1.5%, depending on the
distribution considered. This is larger than the variation seen when changing
the number of iterations in the Bayesian approach, and it is also larger than
the effect of reweighting the bin-by-bin corrections to the jet transverse
momentum distributions produced by different event generators. An additional
uncertainty due to the resolution of the jet pseudorapidity in data is also
considered and obtained by comparing the difference between the jet
pseudorapidity calculated using just the charged component of the jet and
using both the charged and neutral components in ${Z}{j}$ data and simulation.
A good level of agreement is observed within the statistical precision of
0.5%. The two contributions are added in quadrature and taken as the
systematic uncertainty associated with the unfolding corrections.

Different sources for the jet energy scale uncertainty are considered. The
energy scale associated with tracks is known and simulated to an accuracy of
better than 1% [12]. The calorimeter energy scales are modelled to an accuracy
of better than 10%. This is confirmed by comparing the fraction of
$p_{\textrm{T}}^{\textrm{jet}}$ carried by neutral final-state particles
between data and simulation, and evaluating how much the calorimeter response
can be varied before disagreement is observed. The jet energy resolution at
LHCb is modelled in simulation to an accuracy of about 10% [3, 5]. The
analysis is repeated with the simulated $p_{\textrm{T}}^{\textrm{jet}}$
smeared by 10%; the change in the final result of approximately 0.3% is
assigned as the relevant uncertainty. Combining these effects yields an energy
scale uncertainty of about 3%, consistent with previous studies [3]
considering the $p_{\mathrm{T}}$ balance in $Z$ +1-jet events. In order to
determine the effect on the measurement, the analysis is repeated with the
energy scale varied to cover possible differences between data and simulation.
The variation in the measured cross-sections lies between 4 and 11%, depending
on the bin and sample considered. This is assigned as the energy scale
uncertainty.

A summary of the different contributions to the systematic and total
uncertainty for the measured quantities which will be outlined in Sec. 7 is
given in Table 1. In the case of ${Z}{j}$ measurements, the systematic
uncertainty is dominated by the knowledge of the jet energy scale, while for
${W}{j}$ measurements a similarly large uncertainty is present due to the
determination of the sample purity.

##  7 Results

The total cross-sections for ${W}{j}$ and ${Z}{j}$ production are obtained by
summing over the measured cross-sections in bins of $\eta^{\textrm{jet}}$. All
statistical uncertainties are taken to be uncorrelated, while uncertainties
arising from common sources and/or methods are taken to be fully correlated
between different bins. The cross-sections are calculated to be

$\displaystyle\sigma_{{W^{+}}{j}}$ $\displaystyle=$ $\displaystyle 56.9\pm
0.2\phantom{0}\pm 5.1\phantom{0}\pm 0.7\mathrm{\,pb}\,,$
$\displaystyle\sigma_{{W^{-}}{j}}$ $\displaystyle=$ $\displaystyle 33.1\pm
0.2\phantom{0}\pm 3.5\phantom{0}\pm 0.4\mathrm{\,pb}\,,$
$\displaystyle\sigma_{{Z}{j}}$ $\displaystyle=$ $\displaystyle 5.71\pm 0.06\pm
0.27\pm 0.07\mathrm{\,pb}\,,$

where the first uncertainties are statistical, the second are systematic, and
the third are due to the luminosity determination. The ratios of ${W}{j}$ and
${Z}{j}$ production are determined to be

$\displaystyle R_{{W}{Z}}$ $\displaystyle=$ $\displaystyle 15.8\pm
0.2\phantom{0}\pm 1.1\,,$ $\displaystyle R_{{W^{+}}{Z}}$ $\displaystyle=$
$\displaystyle 10.0\pm 0.1\phantom{0}\pm 0.6\,,$ $\displaystyle
R_{{W^{-}}{Z}}$ $\displaystyle=$ $\displaystyle 5.8\phantom{0}\pm
0.1\phantom{0}\pm 0.5\,,$ $\displaystyle R_{{W^{\pm}}}$ $\displaystyle=$
$\displaystyle 1.72\pm 0.01\pm 0.06\,,$

where $R_{{W}{Z}}$, $R_{{W^{+}}{Z}}$ and $R_{{W^{-}}{Z}}$ represent,
respectively, the ratio of the ${W}{j}$, ${W^{+}}{j}$ and ${W^{-}}{j}$ cross-
sections to the ${Z}{j}$ cross-section, and $R_{{W^{\pm}}}$ represents the
ratio of the ${W^{+}}{j}$ to ${W^{-}}{j}$ cross-sections. The asymmetry of
${W^{+}}{j}$ and ${W^{-}}{j}$ production, $A({W}{j})$, is given by

$\displaystyle
A({W}{j})\equiv(\sigma_{{W^{+}}{j}}-\sigma_{{W^{-}}{j}})/(\sigma_{{W^{+}}{j}}+\sigma_{{W^{-}}{j}})$
$\displaystyle=$ $\displaystyle 0.264\pm 0.003\pm 0.015\,.$

In the above results, the first uncertainties are statistical and the second
are systematic.

Table 1: Summary of the different contributions to the total uncertainty on $\sigma_{{W^{+}}{j}}$, $\sigma_{{W^{-}}{j}}$, $\sigma_{{Z}{j}}$ and their ratios given as a percentage of the measured observable. Source | $\sigma_{{W^{+}}{j}}$ | $\sigma_{{W^{-}}{j}}$ | $\sigma_{{Z}{j}}$ | $R_{{W}{Z}}$ | $R_{{W^{\pm}}}$
---|---|---|---|---|---
Statistical | 0.4 | 0.5 | 1.1 | 1.2 | 0.7
Muon reconstruction | 1.3 | 1.3 | 0.6 | 0.9 | 0.0
Jet reconstruction | 1.9 | 1.9 | 1.9 | 0.0 | 0.0
Selection | 1.0 | 1.0 | 0.0 | 1.0 | 0.0
GEC | 0.5 | 0.5 | 0.4 | 0.2 | 0.1
Purity | 5.5 | 7.0 | 0.4 | 6.0 | 2.5
Acceptance | 0.6 | 0.6 | 0.0 | 0.6 | 0.0
Unfolding | 0.8 | 0.8 | 0.8 | 0.0 | 0.2
Jet energy | 6.5 | 7.7 | 4.3 | 3.4 | 1.2
Total Systematic | 8.9 | 10.7 | 4.8 | 7.0 | 3.3
Luminosity | 1.2 | 1.2 | 1.2 | – | –

The results are compared to theoretical predictions calculated using the
aMC@NLO and Powheg generators in Fig. 2. The uncertainty on the theoretical
predictions due to higher-order effects is calculated by varying the
renormalisation and factorisation scales independently by a factor of two
around the nominal scale [36]. Additional uncertainties arise from the
description of the PDFs, and the value of the strong coupling, $\alpha_{s}$.
The total theoretical uncertainty is obtained by combining the PDF and
$\alpha_{s}$ uncertainties in quadrature, and adding the result to the scale
uncertainty linearly. The measurements are represented by bands where the
inner band represents the statistical uncertainty and the outer band the total
uncertainty. In the cross-section measurements, the scale uncertainty
dominates the theoretical uncertainty, while it largely cancels in the ratios
and asymmetry. The data and predictions are further compared differentially
for ${W}{j}$ production in Figs. 3 and 4, and for ${Z}{j}$ production in Figs.
5 and 6, with good agreement seen in all distributions.

Figure 2: Summary of the measurements performed in the fiducial region, as
defined in Sec. 1. The measurements are shown as bands, while the theoretical
predictions are presented as points. For the experimental measurements, the
inner band represents the statistical uncertainty, while the outer band
represents the total uncertainty. For the theory points, the inner error bar
represents the scale uncertainty, while the outer bar represents the total
uncertainty. The cross-sections and ratios are shown normalised to the
measurement, while the asymmetry is presented separately.

Figure 3: ${W}{j}$ bin-averaged differential cross-sections as a function of
$\eta^{\mu}$ (left) and $\eta^{\textrm{jet}}$ (right). The measurements are
shown as bands representing the statistical and total uncertainties, while the
theoretical predictions are shown as points (displaced horizontally for
presentation) representing the same bin-averaged cross-sections as the data.
The inner error bar represents the scale uncertainty, and the outer error bar
represents the total uncertainty. The ratio of the predicted to measured
cross-sections is shown below the distribution. The ${W^{+}}{j}$ and
${W^{-}}{j}$ cross-sections are seen to overlap in the final bin in
$\eta^{\mu}$. Figure 4: ${W}{j}$ bin-averaged differential cross-sections as a
function of $p_{\textrm{T}}^{\textrm{jet}}$. The experimental and theoretical
components are shown as in Fig. 3. The ratio of the predicted to measured
cross-sections is shown below the distribution.

Figure 5: The measured bin-averaged differential ${Z}{j}$ production cross-
section is shown as a function of (left) $\eta^{\textrm{jet}}$ and (right)
$p_{\textrm{T}}^{\textrm{jet}}$. The experimental measurements are shown as
bands, while the theoretical predictions are shown as points, horizontally
displaced for presentation. The ratio of the predicted to measured cross-
sections is shown below the distribution.

Figure 6: The measured bin-averaged differential ${Z}{j}$ production cross-
section is shown as a function of (left) $y^{{Z}}$ and (right) azimuthal
separation between the $Z$ boson and the jet. The experimental measurements
are shown as bands, while the theoretical predictions are shown as points,
horizontally displaced for presentation. The ratio of the predicted to
measured cross-sections is shown below the distribution.

Further to the total and differential production cross-sections, measurements
of the charge ratio and asymmetry of ${W}{j}$ production are also performed as
a function of lepton pseudorapidity and are compared to Powheg and aMC@NLO in
Fig. 7. Due to the cancellation of scale uncertainties, these distributions
are expected to show sensitivity to the PDFs and consequently are also
compared in Fig. 8 to fixed-order calculations performed with Fewz separately
for the NNPDF3.0, CT14 and MMHT14 PDF sets. The fixed-order predictions are
expected to give a good description of the ratios and asymmetries as the
effects of higher-order terms and hadronisation largely cancel between the
positively and negatively charged ${W}{j}$ predictions. In general, good
agreement is seen between the data and the predictions, although the data
presents a slightly larger ratio and asymmetry, particularly in the first bin
of $\eta^{\mu}$. However, when the spread of predictions obtained using
different PDF sets is considered, the deviations are not significant.

Figure 7: Ratio (left) and asymmetry (right) of ${W^{+}}{j}$ to ${W^{-}}{j}$
production as a function of the lepton pseudorapidity. The experimental
measurements are shown as bands, while the theoretical predictions are shown
as points, horizontally displaced for presentation. The ratio of the
predictions to the experimentally measured values is shown below the
distribution for the charge ratio, while their difference is shown for the
charge asymmetry.

Figure 8: Ratio (left) and asymmetry (right) of ${W^{+}}{j}$ to ${W^{-}}{j}$
production as a function of the lepton pseudorapidity compared to NLO
calculations performed with the Fewz generator and three different PDF sets.
The experimental measurements are shown as bands, while the theoretical
predictions are shown as points, horizontally displaced for presentation. The
same comparisons are shown below the distribution as described in Fig. 7.

##  8 Conclusions

Measurements of the forward $W$ and $Z$ boson cross-sections in association
with jets at $\sqrt{s}$ = 8$\mathrm{\,Te\kern-1.00006ptV}$ are presented. The
$W$ bosons are reconstructed in the decay ${W}\rightarrow\mu\nu_{\mu}$ and the
$Z$ bosons in the decay ${Z}\rightarrow\mu\mu$. Total cross-sections are
presented in the forward fiducial region in addition to measurements of the
charge ratio and asymmetry of ${W}{j}$ production and the ratio of ${W}{j}$ to
${Z}{j}$ production. Differential cross-sections are presented as a function
of $p_{\textrm{T}}^{\textrm{jet}}$, $\eta^{\textrm{jet}}$, $\eta^{\mu}$ in the
case of ${W}{j}$ production, and for ${Z}{j}$ production, where a full
reconstruction of the final state is possible, measurements are presented as a
function of $p_{\textrm{T}}^{\textrm{jet}}$, $\eta^{\textrm{jet}}$, $y^{{Z}}$,
and the azimuthal separation of the $Z$ boson and the jet, $|\Delta\phi|$. The
${W}{j}$ charge ratio and asymmetry are presented as a function of
$\eta^{\mu}$. All measurements are observed to be in agreement with
predictions obtained at $\mathcal{O}(\alpha_{s}^{2})$ interfaced with a parton
shower in order to achieve NLO plus leading-log accuracy. The measurements of
the charge ratio and asymmetry of ${W}{j}$ production are also compared to
predictions obtained at $\mathcal{O}(\alpha_{s}^{2})$ in fixed order
perturbative QCD and show good agreement.

## Acknowledgements

We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC
(China); CNRS/IN2P3 (France); BMBF, DFG and MPG (Germany); INFN (Italy); FOM
and NWO (The Netherlands); MNiSW and NCN (Poland); MEN/IFA (Romania); MinES
and FANO (Russia); MinECo (Spain); SNSF and SER (Switzerland); NASU (Ukraine);
STFC (United Kingdom); NSF (USA). We acknowledge the computing resources that
are provided by CERN, IN2P3 (France), KIT and DESY (Germany), INFN (Italy),
SURF (The Netherlands), PIC (Spain), GridPP (United Kingdom), RRCKI and Yandex
LLC (Russia), CSCS (Switzerland), IFIN-HH (Romania), CBPF (Brazil), PL-GRID
(Poland) and OSC (USA). We are indebted to the communities behind the multiple
open source software packages on which we depend. Individual groups or members
have received support from AvH Foundation (Germany), EPLANET, Marie
Skłodowska-Curie Actions and ERC (European Union), Conseil Général de Haute-
Savoie, Labex ENIGMASS and OCEVU, Région Auvergne (France), RFBR and Yandex
LLC (Russia), GVA, XuntaGal and GENCAT (Spain), Herchel Smith Fund, The Royal
Society, Royal Commission for the Exhibition of 1851 and the Leverhulme Trust
(United Kingdom).

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J.M. Otalora Goicochea2, A. Otto39, P. Owen54, A. Oyanguren68, P.R. Pais40, A.
Palano14,d, F. Palombo22,q, M. Palutan19, J. Panman39, A. Papanestis50, M.
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D. Pinci26, A. Pistone20, A. Piucci12, S. Playfer51, M. Plo Casasus38, T.
Poikela39, F. Polci8, A. Poluektov49,35, I. Polyakov32, E. Polycarpo2, G.J.
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J.D. Price53, J. Prisciandaro38, A. Pritchard53, C. Prouve47, V. Pugatch45, A.
Puig Navarro40, G. Punzi24,p, W. Qian56, R. Quagliani7,47, B. Rachwal27, J.H.
Rademacker47, M. Rama24, M. Ramos Pernas38, M.S. Rangel2, I. Raniuk44, G.
Raven43, F. Redi54, S. Reichert10, A.C. dos Reis1, C. Remon Alepuz68, V.
Renaudin7, S. Ricciardi50, S. Richards47, M. Rihl39, K. Rinnert53,39, V. Rives
Molina37, P. Robbe7, A.B. Rodrigues1, E. Rodrigues58, J.A. Rodriguez Lopez64,
P. Rodriguez Perez55, A. Rogozhnikov67, S. Roiser39, V. Romanovskiy36, A.
Romero Vidal38, J.W. Ronayne13, M. Rotondo23, T. Ruf39, P. Ruiz Valls68, J.J.
Saborido Silva38, E. Sadykhov32, N. Sagidova31, B. Saitta16,f, V. Salustino
Guimaraes2, C. Sanchez Mayordomo68, B. Sanmartin Sedes38, R. Santacesaria26,
C. Santamarina Rios38, M. Santimaria19, E. Santovetti25,j, A. Sarti19,k, C.
Satriano26,s, A. Satta25, D.M. Saunders47, D. Savrina32,33, S. Schael9, M.
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Shapoval17,44,g, Y. Shcheglov31, T. Shears53, L. Shekhtman35, V. Shevchenko66,
A. Shires10, B.G. Siddi17, R. Silva Coutinho41, L. Silva de Oliveira2, G.
Simi23,o, M. Sirendi48, N. Skidmore47, T. Skwarnicki60, E. Smith54, I.T.
Smith51, J. Smith48, M. Smith55, H. Snoek42, M.D. Sokoloff58, F.J.P. Soler52,
D. Souza47, B. Souza De Paula2, B. Spaan10, P. Spradlin52, S. Sridharan39, F.
Stagni39, M. Stahl12, S. Stahl39, P. Stefko40, S. Stefkova54, O. Steinkamp41,
O. Stenyakin36, S. Stevenson56, S. Stoica30, S. Stone60, B. Storaci41, S.
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J. Walsh24, J. Wang60, D.R. Ward48, H.M. Wark53, N.K. Watson46, D. Websdale54,
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Wotton48, K. Wraight52, S. Wright48, K. Wyllie39, Y. Xie63, Z. Xu40, Z. Yang3,
H. Yin63, J. Yu63, X. Yuan35, O. Yushchenko36, M. Zangoli15, K.A. Zarebski46,
M. Zavertyaev11,c, L. Zhang3, Y. Zhang7, Y. Zhang62, A. Zhelezov12, Y.
Zheng62, A. Zhokhov32, V. Zhukov9, S. Zucchelli15.

1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université Savoie Mont-Blanc, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9I. Physikalisches Institut, RWTH Aachen University, Aachen, Germany
10Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
11Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
12Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
13School of Physics, University College Dublin, Dublin, Ireland
14Sezione INFN di Bari, Bari, Italy
15Sezione INFN di Bologna, Bologna, Italy
16Sezione INFN di Cagliari, Cagliari, Italy
17Sezione INFN di Ferrara, Ferrara, Italy
18Sezione INFN di Firenze, Firenze, Italy
19Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
20Sezione INFN di Genova, Genova, Italy
21Sezione INFN di Milano Bicocca, Milano, Italy
22Sezione INFN di Milano, Milano, Italy
23Sezione INFN di Padova, Padova, Italy
24Sezione INFN di Pisa, Pisa, Italy
25Sezione INFN di Roma Tor Vergata, Roma, Italy
26Sezione INFN di Roma La Sapienza, Roma, Italy
27Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
28AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
29National Center for Nuclear Research (NCBJ), Warsaw, Poland
30Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
31Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
32Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
33Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
34Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
35Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
36Institute for High Energy Physics (IHEP), Protvino, Russia
37Universitat de Barcelona, Barcelona, Spain
38Universidad de Santiago de Compostela, Santiago de Compostela, Spain
39European Organization for Nuclear Research (CERN), Geneva, Switzerland
40Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
41Physik-Institut, Universität Zürich, Zürich, Switzerland
42Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
43Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
44NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
45Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
46University of Birmingham, Birmingham, United Kingdom
47H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
48Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
49Department of Physics, University of Warwick, Coventry, United Kingdom
50STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
51School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
52School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
53Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
54Imperial College London, London, United Kingdom
55School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
56Department of Physics, University of Oxford, Oxford, United Kingdom
57Massachusetts Institute of Technology, Cambridge, MA, United States
58University of Cincinnati, Cincinnati, OH, United States
59University of Maryland, College Park, MD, United States
60Syracuse University, Syracuse, NY, United States
61Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
62University of Chinese Academy of Sciences, Beijing, China, associated to 3
63Institute of Particle Physics, Central China Normal University, Wuhan,
Hubei, China, associated to 3
64Departamento de Fisica , Universidad Nacional de Colombia, Bogota, Colombia,
associated to 8
65Institut für Physik, Universität Rostock, Rostock, Germany, associated to 12
66National Research Centre Kurchatov Institute, Moscow, Russia, associated to
32
67Yandex School of Data Analysis, Moscow, Russia, associated to 32
68Instituto de Fisica Corpuscular (IFIC), Universitat de Valencia-CSIC,
Valencia, Spain, associated to 37
69Van Swinderen Institute, University of Groningen, Groningen, The
Netherlands, associated to 42

aUniversidade Federal do Triângulo Mineiro (UFTM), Uberaba-MG, Brazil
bLaboratoire Leprince-Ringuet, Palaiseau, France
cP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
dUniversità di Bari, Bari, Italy
eUniversità di Bologna, Bologna, Italy
fUniversità di Cagliari, Cagliari, Italy
gUniversità di Ferrara, Ferrara, Italy
hUniversità di Genova, Genova, Italy
iUniversità di Milano Bicocca, Milano, Italy
jUniversità di Roma Tor Vergata, Roma, Italy
kUniversità di Roma La Sapienza, Roma, Italy
lAGH - University of Science and Technology, Faculty of Computer Science,
Electronics and Telecommunications, Kraków, Poland
mLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
nHanoi University of Science, Hanoi, Viet Nam
oUniversità di Padova, Padova, Italy
pUniversità di Pisa, Pisa, Italy
qUniversità degli Studi di Milano, Milano, Italy
rUniversità di Urbino, Urbino, Italy
sUniversità della Basilicata, Potenza, Italy
tScuola Normale Superiore, Pisa, Italy
uUniversità di Modena e Reggio Emilia, Modena, Italy


# Temporal Quantum Control with Graphene

Alejandro Manjavacas<EMAIL_ADDRESS>IQFR - CSIC, Serrano 119, 28006
Madrid, Spain  Sukosin Thongrattanasiri  IQFR - CSIC, Serrano 119, 28006
Madrid, Spain  Darrick E. Chang  ICFO-Institut de Ciencies Fotoniques,
Mediterranean Technology Park, 08860 Castelldefels (Barcelona), Spain  F.
Javier García de Abajo<EMAIL_ADDRESS>IQFR - CSIC, Serrano 119, 28006
Madrid, Spain

###### Abstract

We introduce a novel strategy for controlling the temporal evolution of a
quantum system at the nanoscale. Our method relies on the use of graphene
plasmons, which can be electrically tuned in frequency by external gates.
Quantum emitters (e.g., quantum dots) placed in the vicinity of a graphene
nanostructure are subject to the strong interaction with the plasmons of this
material, thus undergoing time variations in their mutual interaction and
quantum evolution that are dictated by the externally applied gating voltages.
This scheme opens a new path towards the realization of quantum-optics devices
in the robust solid-state environment of graphene.

###### pacs:

78.67.Wj,73.20.Mf,42.50.-p

Controlling the temporal quantum evolution of a physical system by means of
external macroscopic stimuli will make versatile quantum-information devices
viable Cirac et al. (1997). Various methods for quantum control have been
proposed Cirac et al. (1997), and despite efforts and progress in fields such
as ion traps Amini et al. (2010); Ritter et al. (2012), scalable systems
remain an open challenge. Recently, solid-state quantum devices are attracting
growing interest because they provide a robust platform for implementing
scalable temporal control. Here we show that doped graphene nanostructures
combined with two-level atoms or quantum dots provide a robust platform for
achieving the desired goal of full temporal quantum control. The interaction
between the quantum dots is strongly mediated by plasmons in the graphene,
which can be electrostatically tuned through engineered gates Li et al.
(2008); Chen et al. (2011). The quantum evolution of the dots is then
manipulated by modulating over time the electric potential that we apply to
the gates. We provide realistic simulations demonstrating excellent control
over the decay of individual and interacting dots. Any desired decay profile
can be produced by resorting on the unprecedentedly fast electro-optical
modulation of graphene gra  and its strong interaction with neighboring
quantum emitters Yamamoto et al. (2011). This constitutes a radically new path
towards controlling quantum systems in nanoscale solid-state environments by
means of conventional electronics, with the capacity of bringing quantum
devices closer to reality.

Quantum mechanics rules the temporal evolution at small length and energy
scales, giving rise to non-intuitive properties such as quantum superposition
and entanglement. These phenomena provide an extra handle to process
information Duan and Monroe (2008a); Kimble (2008), to improve optical
metrology and break the imaging diffraction limit Dowling (2008), and to
substantially alter the statistics of light Mandel (1986); Ourjoumtsev et al.
(2006); Afek et al. (2010), with a vast range of potential applications,
including quantum computing and cryptography Nielsen and Chuang (2004) and
sensing Dowling (2008). However, controlling the quantum evolution at such
length and energy scales remains a challenge that can only be partially
addressed by using elaborate setups, for example in the context of cavity-QED
Cirac et al. (1997); van Enk et al. (1997); Ritter et al. (2012). Recently,
surface plasmons have been identified as a potential candidate to mediate the
interaction between externally controlled signals (e.g., laser beams) and
small quantum systems (e.g., quantum dots), allowing one to explore exciting
phenomena such as single photon transistors Chang et al. (2007), entanglement
Gonzalez-Tudela et al. (2011) and quantum blockade Manjavacas et al. (2012)
among other feats. In the following, we show that graphene can also be used to
control the evolution of a quantum system interacting with it through a
classical electric-potential signal. Specifically, we show that direct control
over two-level emitters is possible, thus eliminating the need to involve
additional excited or metastable states and external control fields upon which
other methods are relying Cirac et al. (1997); Duan and Monroe (2008b). Our
results can be used for a robust, solid-state implementation of photon storage
Gorshkov et al. (2007) and cascaded quantum systems in which one system drives
the evolution of another system Gardiner (1993); Carmichael (1993). These are
basic elements needed for distributed quantum networks and processors, and
quantum state transfer.

The emergence of graphene as a tunable plasmonic material, in which plasmons
can be literally switched on and off by applying external potentials gra ,
opens a natural way to control the quantum evolution of small systems through
plasmon-mediated interactions, which are in turn modulated by external fields.
Charge carriers in graphene, the so-called massless Dirac Fermions, follow a
linear dispersion relation that lead to several exciting properties. Among
them, low-energy plasmons exist in the atomically thin carbon film when it is
electrically charged, and their frequencies scale as $n^{1/4}$ with the doping
electron density Jablan et al. (2009) $n$. Therefore, the plasmon frequencies
can be controlled by changing $n$, which is in turn proportional to the
magnitude of the perpendicular electric field. This field can be supplied by
biasing the graphene with respect to a nearby gate, and thus, the voltage
applied through the gates directly modifies in a predictable way the plasmon
frequencies.

Figure 1: Temporal control over the quantum evolution of an optical emitter
via interaction with a doped graphene nanostructure. (a) We consider a two-
level optical emitter placed above a graphene nanodisk. The emitter is excited
by a laser pulse. Electrical doping of the graphene through a bias potential
$V$ allows the nanodisk to support plasmons. The plasmon frequency is
proportional to $\sqrt{|V|}$, and thus, it can be controlled over time by
modulating $V(t)$. The coupling of the emitter to the plasmon and therefore
the quantum evolution of the emitter state are both controlled by $V$. (b)
Different time scales are involved in the evolution of the emitter-graphene
system, represented here through the emitter natural decay rate $\Gamma_{0}$,
the enhanced decay rate $\Gamma_{11}$ produced by interaction with the
graphene plasmons, and the plasmon decay rate and frequency typical of doped
graphene. It is important to stress that the plasmon lifetime is short
compared to the electric modulation rates of the doping potential, in the GHz
range, which is in turn fast compared to the lifetime of the excited emitter.
(c) The increase in emission rate $\Gamma_{11}/\Gamma_{0}$ is shown as a
function of emission frequency for two different doping levels, quantified
through the graphene Fermi energy $E_{\rm F}$. The temporal profile of the
emission rate $R(t)$ can be controlled by suitably modulating $E_{\rm F}$ over
time as shown in (d) and (e) for $\hbar\omega_{0}=0.22\,$eV.

In this Letter, we show that the interaction between one or more quantum
emitters with graphene plasmons strongly modifies the evolution of the system,
which can be temporally controlled by switching on and off the plasmons
through electrostatic gating. We first illustrate this concept by analyzing
the interaction of a doped graphene nanodisk with a single quantum emitter
(Fig. 1). We choose a nanodisk of $100$ nm in diameter, with a level of doping
characterized by a Fermi energy Castro Neto et al. (2009) $E_{\rm F}=\hbar
v_{F}\sqrt{\pi|n|}$, where $v_{F}\approx 10^{6}$m/s is the Fermi velocity.
This determines the plasmon energy $\hbar\omega_{p}\propto\sqrt{E_{\rm F}}$
and the plasmon decay rate Koppens et al. (2011) $\Gamma_{p}\propto 1/E_{\rm
F}$. The emitter is placed $60$ nm above the center of the graphene nanodisk,
and is modeled as a two-level system with a characteristic transition energy
$\hbar\omega_{0}$ and a natural decay rate $\Gamma_{0}$. The different time
scales that characterize the evolution of the system are plotted in Fig. 1(b).
We choose a value of $\Gamma_{0}\sim 10^{3}-10^{4}$ s-1 typical of slowly
emitting atoms such as erbium. When the emitter is placed close to the
nanodisk, its decay rate is enhanced up to $\Gamma_{11}\sim
10^{4}\,\Gamma_{0}$ due to resonant interaction with the graphene plasmons.
These decay rates are well below the frequencies at which the doping level can
be modulated with currently available electronics ($\sim$ GHz), which is in
turn much smaller than the plasmon frequency $\omega_{p}$ and the decay rate
$\Gamma_{p}$. With this choice of parameters, we ensure that the emitter-
plasmon interaction remains in the weak coupling regime, instantaneously
following any modulation of the doping level. Under such conditions we can
trace out the plasmonic degrees of freedom, and therefore the dynamics of the
quantum emitter is completely described by the reduced density matrix $\rho$,
whose temporal evolution is given by Ficek and Tanas (2002)

$\displaystyle\frac{d\rho}{dt}=\frac{i}{\hbar}\left[\rho,\mathcal{H}\right]+\frac{\Gamma_{11}}{2}\left[2\sigma\rho\sigma^{{\dagger}}-\sigma^{{\dagger}}\sigma\rho-\rho\sigma^{{\dagger}}\sigma\right],$
(1)

where $\sigma$ is the annihilation operator of the emitter excited state, and
the Hamiltonian reduces to
$\mathcal{H}=\hbar\omega_{0}\sigma^{{\dagger}}\sigma$.

Fig. 1(c) shows the normalized decay rate $\Gamma_{11}/\Gamma_{0}$ as a
function of emission energy $\hbar\omega_{0}$ for two different doping levels
corresponding to Fermi energies of $0.3$ and $0.4$ eV, respectively. Then,
choosing an emitter of transition energy $\approx 0.22$ eV (highest peak of
black curve), we can switch the normalized decay rate from $\Gamma^{\rm
on}_{11}/\Gamma_{0}>10^{4}$ down to $\Gamma^{\rm off}_{11}/\Gamma_{0}\sim 10$,
just by shifting the graphene doping level from $0.3$ to $0.4$ eV. This
clearly shows the feasibility of controlling the temporal evolution of the
quantum emitter by modulating the doping level of the graphene nanodisk
between on- and off-resonance conditions. We explore this possibility in more
detail in Figs. 1(d-e) for three different doping modulation profiles:
rectangular (red curve), triangular (blue curve) and gaussian (green curve).
The emitter is initially prepared in the excited state, and we study the
plasmon generation rate
$R=\left\langle\Gamma_{11}\sigma^{{\dagger}}\sigma\right\rangle$, normalized
to $\Gamma_{11}^{\rm on}$. This magnitude measures the number of plasmons
generated per unit time, and it is calculated here neglecting decay channels
others than plasmon generation. This assumption is well justified by the large
values of $\Gamma^{\rm on/off}_{11}/\Gamma_{0}$. As shown in Fig. 1(e), each
different doping profile results in a totally different evolution of the
quantum emitter, which reflects the complete temporal control achievable with
the system under study. Actually, it is not difficult to obtain the analytical
relation existing between $R$ and the single-emitter decay rate. Assuming the
emitter in its excited state at time $t=t_{0}$, this relation reduces to

$\displaystyle\Gamma_{11}\left(t\right)=\frac{R\left(t\right)}{1-\int_{t_{0}}^{t}dt^{\prime}R\left(t^{\prime}\right)}.$

With the only constraint that $\Gamma_{11}\left(t\right)\leq\Gamma_{11}^{\rm
on}$, a desired profile $R\left(t\right)$ can be achieved with the temporal
evolution of $\Gamma_{11}\left(t\right)$ prescribed by this equation, which is
in turn obtained by directly modulating the doping voltage, and therefore
$E_{\rm F}$, over time using the Lorentzian dependence of $\Gamma_{11}$ on
$E_{\rm F}$ discussed in detail in the Supplemental Material (SM) sup .

Figure 2: Temporal control over the interaction between quantum dots mediated
by graphene. (a) Two emitters are excited and their decay and mutual
interaction is modulated electrically through the plasmons of a neighboring
graphene disk. (b) Interaction rate $\Gamma_{12}$ as a function of the
azimuthal angle between the positions of the dots (see inset) when plasmons of
$0.108$ eV energy are switched on ($E_{\rm F}=0.3\,$eV) and off ($E_{\rm
F}=0.4\,$eV), and the emitter dipoles are along radial directions. The
interaction rate is normalized to the on-resonance single-emitter decay rate
$\Gamma_{11}^{\rm on}$. The imposed temporal evolution of the doping (c) is
used to control the emission rate (d) with the two emitters initially prepared
in their excited states. When only one of the emitters is initially excited,
the degree of entanglement for the doping profile of (e) is quantified through
the Wootters concurrence (f).

When a second quantum emitter is placed close to the graphene, the interaction
between the two emitters can be also controlled over time. We investigate this
possibility by studying the system depicted in Fig. 2(a), where two identical
emitters are placed $100$ nm above a graphene nanodisk of $500$ nm in
diameter. The emitters are separated by a distance of $200$ nm and oriented
along orthogonal radial directions, so that they can be independently excited
by light plane waves linearly polarized along orthogonal directions. Their
temporal evolution is determined by the generalization of Eq. (1),

$\displaystyle\frac{d\rho}{dt}=\frac{i}{\hbar}\left[\rho,\mathcal{H}\right]+\sum_{i,j=1}^{2}\frac{\Gamma_{ij}}{2}\left[2\sigma_{i}\rho\sigma_{j}^{{\dagger}}-\sigma_{i}^{{\dagger}}\sigma_{j}\rho-\rho\sigma_{i}^{{\dagger}}\sigma_{j}\right],$
(2)

where
$\mathcal{H}=\hbar\omega_{0}\sum_{i=1}^{2}\sigma_{i}^{{\dagger}}\sigma_{i}$
and $\Gamma_{12}=\Gamma_{21}$ is the interaction rate. This magnitude is
plotted in Fig. 2(b) as a function of the azimuthal angle between the
emitters, normalized to the on-resonance single-emitter decay rate
$\Gamma_{11}^{\rm on}$. When the doping level of the graphene nanodisk matches
the on-resonance value, $\Gamma_{12}$ remains nearly equal to $\Gamma^{\rm
on}_{11}$ for all angles. In contrast, the normalized interaction rate
$\Gamma_{12}/\Gamma^{\rm on}_{11}$ drops below $0.01$ when the doping is tuned
to off-resonance conditions. Therefore, it is possible to switch on and off
the interaction between the emitters. Fig. 2(d) quantifies this possibility
through the temporal evolution of the plasmon generation rate
$R=\left\langle\sum_{i,j=1}^{2}\Gamma_{ij}\sigma_{i}^{{\dagger}}\sigma_{j}\right\rangle$
associated with the doping profile shown in Fig. 2(c). The two emitters are
assumed to be initially prepared in the excited state. Without interaction,
they decay rather slowly and independently, resulting in a negligible value of
$R$. This situation changes dramatically when the doping level is switched to
the on-resonance condition, so that the emitters interact strongly and decay
faster, producing a sudden jump in the plasmon generation rate.

The high degree of control displayed by this system can be exploited to
temporally modulate different properties of the quantum emitters, such as
their degree of entanglement. In Fig. 2(f) we plot the temporal evolution of
the Wootters concurrence Wootters (1998) associated with the doping profile of
Fig. 2(e) when only one of the emitters is initially excited. The concurrence
directly measures the degree of entanglement (1 for a maximally entangled
state). Our system is capable of reaching a value of $C$ close to 0.5, which
is understandable because a single excited emitter can be re-written as a
mixture of a symmetric (superradiant) state and anti-symmetric (subradiant)
state, but the subradiant state lives for very long time, such that there is
50% chance of creating a long-lived entangled state. Larger degree of
entanglement can be achieved by various means, such as utilizing quantum
emitters with more than two levels, heralded schemes based on photon
detection, or coherent dipole-dipole interactions between the emitters.

A similar scheme allows the on-demand modulation of the emission in systems
consisting of many emitters, in which superradiance R. H. Dicke (1954); Gross
and Haroche (1982) can be produced and controlled electrostatically sup .

In summary, here we demonstrate that electrical modulation of the plasmon
frequency in graphene provides an ingenious solution to achieve temporal
control over the evolution of quantum emitters placed in the vicinity of a
graphene nanostructure. This leads to a new paradigm in quantum information
processing technologies and serves as a platform on which to test quantum
phenomena controlled by means of externally applied, classical electrostatic
potentials.

This work has been supported by the Spanish MICINN (MAT2010-14885 and
Consolider NanoLight.es) and the European Commission
(FP7-ICT-2009-4-248909-LIMA and FP7-ICT-2009-4-248855-N4E). DEC acknowledges
support from Fundacio Privada Cellex Barcelona. A.M. acknowledges financial
support through FPU from the Spanish ME.

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