SMP-19-009

SMP-19-009

# Measurement of differential cross sections for the production of a boson in
association with jets in proton-proton collisions at $\sqrt{s}=13\TeV$

###### Abstract

A measurement is presented of the production of bosons that decay into two
electrons or muons in association with jets, in proton-proton collisions at a
centre-of-mass energy of 13. The data were recorded by the CMS Collaboration
at the LHC with an integrated luminosity of 35.9$\fbinv$. The differential
cross sections are measured as a function of the transverse momentum () of the
boson and the transverse momentum and rapidities of the five jets with largest
. The jet multiplicity distribution is measured for up to eight jets. The
hadronic activity in the events is estimated using the scalar sum of the of
all the jets. All measurements are unfolded to the stable particle-level and
compared with predictions from various Monte Carlo event generators, as well
as with expectations at leading and next-to-leading orders in perturbative
quantum chromodynamics.

##  0.1 Introduction

The production of bosons in proton-proton ($\Pp\Pp$) collisions is described
by the Drell–Yan process [1], where a quark and an antiquark from the
colliding protons annihilate into a boson. At the CERN LHC, this boson is
commonly produced with accompanying parton radiation via quantum
chromodynamics (QCD), which provides a superb opportunity to gain a better
theoretical understanding of both strong and electroweak physics in a jet
environment. Specifically, events containing boson decays into oppositely
charged lepton pairs (electrons and muons, but not taus) allow a sensitive
evaluation of the accuracy of perturbative QCD [2, 3, 4] at the highest
accessible energies for a broad range of kinematic configurations.

A precise understanding of the $\Pp\Pp\to\PZ(\to\ell^{+}\ell^{-})$ process is
also critical in other standard model (SM) measurements, where it is an
important background in studies of the properties of Higgs boson, and in
analyses focusing on physics beyond the SM such as searches for dark matter
and supersymmetric particles. The clean and readily identifiable signature and
relatively large production rate for this process provide an opportunity to
accurately constrain the parton distribution functions (PDFs) and probe the
strong coupling strength .

In addition to these motivations, $\PZ\to\ell^{+}\ell^{-}+\text{jets}$
production serves as an important experimental benchmark. It is a key
ingredient in calibrating the specific parts of the detector and the
properties of reconstructed objects, e.g. the jet energy scale. Comparisons of
$\PZ$+jets events with the expectations from Monte Carlo (MC) event generators
and with reliable higher-order calculations can improve confidence in their
predictions.

Measurements of differential cross sections for the production of bosons in
association with jets have previously been reported by the ATLAS, CMS, and
LHCb Collaborations in $\Pp\Pp$ collisions at centre-of-mass energies of 7 [5,
6, 7, 8, 9], 8 [10, 11, 12], and 13 [13, 14], and by the CDF and
Collaborations at the Fermilab Tevatron in proton-antiproton collisions at
1.96 [15, 16].

In this paper we present measurements of the differential cross sections for
the production of bosons in association with jets recorded by the CMS
Collaboration in 2016 with an integrated luminosity of 35.9$\fbinv$. This is
an update and an expansion of a previous CMS paper [13] that used 2015 data
with an integrated luminosity of 2.19. The events with both electron and muon
final states are combined and reconstructed as a pair of oppositely charged
leptons that are required to have an invariant mass between 71 and 111. This
mass range optimizes the signal acceptance, rejection of background, and the
relative fraction of boson and virtual photon contributions.

The new analysis provides measurements of events with up to eight jets
inclusively and five jets differentially, compared with the earlier
measurement [13] of events with up to six jets inclusively and three jets
differentially. Additionally, the ranges for all the observables are extended
to larger values of transverse momentum (), and the double-differential cross
sections with respect to the leading jet and the boson are measured.

The cross sections are measured as a function of jet multiplicity
($N_{\text{jets}}$) and the individual jet kinematic variables: rapidity ($y$)
and where the jets are ordered in decreasing . Jet kinematic variables are
presented for events with jet multiplicities up to five jets. The term
“inclusive” refers to distributions for events with at least $N_{\text{jets}}$
jets and the term “exclusive” for distributions where the events contain
exactly $N_{\text{jets}}$ jets. The cross sections are also measured as a
function of the scalar sum of the jets () for events having up to five jets.

##  0.2 The CMS detector

The central feature of the CMS apparatus is a superconducting solenoid of 6m
internal diameter, providing a magnetic field of 3.8T. Within the solenoid
volume are a silicon pixel and strip tracker, a lead tungstate crystal
electromagnetic calorimeter (ECAL), and a brass and scintillator hadron
calorimeter (HCAL), each composed of a barrel and two endcap sections. Forward
calorimeters extend the pseudorapidity ($\eta$) coverage provided by the
barrel and endcap detectors. Muons are detected in gas-ionization chambers
embedded in the steel flux-return yoke outside the solenoid.

The silicon tracker measures charged particles within the pseudorapidity range
$\abs{\eta}<2.5$. During the LHC running period when the data used in this
article were recorded, the silicon tracker consisted of 1440 silicon pixel and
15 148 silicon strip detector modules. For nonisolated particles of
$1<\pt<10\GeV$ and $\abs{\eta}<1.4$, the track resolutions are typically 1.5%
in and 25–90 (45–150)in the transverse (longitudinal) impact parameter [17].

The ECAL consists of 75 848 lead tungstate crystals, which provide the
coverages $\abs{\eta}<1.48$ in the barrel region and $1.48<\abs{\eta}<3.00$ in
two endcap regions. Preshower detectors consisting of two planes of silicon
sensors interleaved with a total of $3X_{0}$ of lead are located in front of
each endcap ECAL. In the barrel section of the ECAL, an energy resolution of
about 1% is achieved for unconverted or late-converting photons that have
energies in the range of tens of . The remaining barrel photons have a
resolution of about 1.3% up to $\abs{\eta}=1$, rising to about 2.5% at
$\abs{\eta}=1.4$. In the endcaps, the resolution of unconverted or late-
converting photons is about 2.5%, while the remaining endcap photons have a
resolution between 3 and 4% [18].

In the region $\abs{\eta}<1.74$, the HCAL cells have widths of 0.087 in
pseudorapidity and 0.087 radians in azimuth ($\phi$). In the $\eta$-$\phi$
plane, and for $\abs{\eta}<1.48$, the HCAL cells map on to $5\times 5$ arrays
of ECAL crystals to form calorimeter towers projecting radially outwards from
close to the nominal interaction point. For $\abs{\eta}>1.74$, the coverage of
the towers increases progressively to a maximum of 0.174 in $\Delta\eta$ and
$\Delta\phi$. Within each tower, the energy deposits in ECAL and HCAL cells
are summed to define the calorimeter tower energies, subsequently used to
provide the energies and directions of hadronic jets. When combining
information from the entire detector, the jet energy resolution amounts
typically to 15% at 10, 8% at 100, and 4% at 1, to be compared to about 40,
12, and 5% obtained when the ECAL and HCAL calorimeters alone are used.

Muons are measured in $\abs{\eta}<2.4$, with detection planes made using three
technologies, drift tubes, cathode strip chambers, and resistive plate
chambers, used in conjunction with the tracker [19].

Events of interest are selected using a two-tiered trigger system [20]. The
first level, composed of custom hardware processors, uses information from the
calorimeters and muon detectors to select events at a rate of around 100kHz
within a latency of less than 4. The second level, known as the high-level
trigger, consists of a farm of processors running a version of the full event
reconstruction software optimized for fast processing, and reduces the event
rate to around 1kHz before data storage. A more detailed description of the
CMS detector, together with a definition of the coordinate system used and the
relevant kinematic variables is reported in Ref. [21].

##  0.3 Event samples

Candidate events are selected online using single-lepton triggers that require
at least one isolated electron (muon) with ${\pt^{l}}>25(24)\GeV$ and
$\abs{\eta^{l}}<2.4$. The total trigger efficiency for events within the
acceptance of this analysis is greater than 90%. Simulated events for both
signal and background processes are produced using various MC event
generators, with the CMS detector response modelled using [22]. These events
are then reconstructed with the same algorithms used to reconstruct collision
data, and the simulated samples are normalized to the integrated luminosity of
the data sample using their respective cross sections. For the simulation of
the signal, we use a sample generated at next-to-leading order (NLO) with
versions 2.2.2 and 2.3.2 (denoted mg5_amc) [23] using the FxFx merging scheme
[24, 25]. Parton showering and hadronization are simulated with 8 (version
8.212) [26] using the CUETP8M1 tune [27]. The matrix element includes boson
production with up to two additional jets generated at NLO with mg5_amc,
effectively yielding leading order (LO) accuracy for $\PZ$+3 jets.

The production of $\PZ(\to\ell^{+}\ell^{-})+\text{jets}$ can be mimicked by
various background sources: decays of bosons resulting from top quark pair
production ($\PQt\PAQt$), diboson (WW, WZ, ZZ), triboson (ZZZ, WWZ, WZZ)
production, and bosons produced in association with jets, as well as
$\PZ+\text{jets}$ events in which the boson decays into
$(\PZ\to\PGt^{+}\PGt^{-})+\text{jets}$ where the leptons decay leptonically.
Background processes are split into two categories: resonant and nonresonant.
Resonant background arises from events with genuine bosons (WZ, ZZ, tribosons,
etc.) and is estimated using MC samples. The nonresonant background that comes
from events that do not have a boson in the final state (such as $\PQt\PAQt$)
is estimated using data events with both an electron and a muon. Events with
$\PZ\to\PGt^{+}\PGt^{-}$ are considered background and are estimated using the
mg5_amc signal sample at NLO.

Background samples corresponding to electroweak diboson and triboson
production [28] are generated at NLO with [29, 30, 31, 32] or mg5_amc, both
interfaced with 8 or mg5_amc.

The $\PW\PZ$ and $\PZ\PZ$ diboson and triboson samples are normalized to the
NLO cross section calculated with MCFM 6.6 [33]. The cross section used to
normalize $\PW\PW$ samples is obtained at next-to-NLO [34].

The simulated event samples include multiple $\Pp\Pp$ collisions within a
bunch crossing (pileup). Since the number of pileup interactions varies with
the beam conditions, the samples are produced using an approximate pileup
distribution. The actual distribution is measured in data and a weight is
applied to each simulated event to correct for the difference.

##  0.4 Event reconstruction, object selection and corrections

The global event reconstruction, also called particle-flow (PF) event
reconstruction [35] reconstructs and identifies each individual particle in an
event, using an optimized combination of all subdetector information. In this
process, the identification of the particle type (photon, electron, muon,
charged or neutral hadron) plays an important role in the determination of the
particle direction and energy.

The primary vertex (PV) is the vertex corresponding to the hardest scattering
in the event, evaluated using tracking information alone, as described in
Section 9.4.1 of Ref. [36]. The jets are clustered using the jet finding
algorithm [37, 38] with the tracks assigned to candidate vertices as inputs,
and the associated missing transverse momentum, which is the negative vector
sum of the of those jets.

The particle-level objects are defined with a lifetime of $c\tau>1\cm$
(excluding neutrinos) and identified using the same algorithms as used for the
data. Leptons are stable particles coming from boson decays, dressed by adding
the momenta of all photons within $R<0.1$ of their directions.

Electron candidates within the geometrical acceptance of $\abs{\eta}<2.4$,
excluding the barrel-to-endcap ($1.444<\abs{\eta}<1.566$) transition regions
of the ECAL, are reconstructed by combining the information from the ECAL and
from the silicon tracker. The energy of electrons is determined from a
combination of the electron momentum at the primary interaction vertex as
determined by the tracker, the energy of the corresponding ECAL cluster, and
the energy sum of all bremsstrahlung photons spatially compatible with the
electron track. This ”supercluster” [39] reconstruction efficiency for
superclusters with an energy above 5is close to 100% [40]. To reduce the
electron misidentification rate, electron candidates are subject to additional
identification criteria that are based on the distribution of the
electromagnetic shower in the ECAL, a matching of the trajectory of an
electron track with the cluster in the ECAL, and compatibility of the track
with the selected primary vertex.

Muon candidates within the geometrical acceptance of $\abs{\eta}<2.4$ are
reconstructed with a global fit using both the inner tracking system and the
muon spectrometer [19]. The momentum of the muons is obtained from the
curvature of the corresponding track; for muons with $20<\pt<100\GeV$ the
resolution is 1.3–2.0% in the barrel, and better than 6% in the endcaps. The
resolution in the barrel is better than 10% for muons with up to 1 [19].

Jets are formed from the particles reconstructed by the PF algorithm using the
software package [38], and the anti-jet clustering algorithm [37] with a
distance parameter $R$ of 0.4. The jet four-momentum is defined as the sum of
the four-momenta of its constituents. The technique of charged hadron
subtraction [41] is used to reduce the pileup contribution by removing charged
particles that originate from pileup vertices. The jet four-momentum is
corrected for the difference observed in the simulation between jets built
from PF candidates and generator-level particles. The jet mass and direction
are kept constant when the corrections are applied. An offset correction is
applied to jet energies to include the contribution from additional $\Pp\Pp$
interactions within the same or previous bunch crossings. Further jet energy
corrections are applied for differences between the observed and simulated
number of pileup interactions, as obtained from zero-bias events and in the
balance in dijet, $\PZ+\text{jet}$, and $\gamma+\text{jet}$ events [42]. Tight
identification quality criteria, based on the fraction of energy carried by
charged and neutral hadrons, are applied to jets [43] to maximize the
reconstruction efficiency while reducing the instrumental background. Jets are
required to have $\abs{\eta}<2.4$, to be separated from all selected lepton
candidates by at least a distance $\Delta
R=\sqrt{\smash[b]{(\Delta\eta)^{2}+(\Delta\phi)^{2}}}=0.4$, and have a larger
than 30for the single-differential cross sections and 20for the double-
differential ones.

To compare the measured distributions with the theoretical predictions,
various experimental corrections are applied after subtracting the total
expected background from the observed number of events in each bin. The event
acceptance and selection efficiency are estimated with simulation and are used
to correct the data. Efficiency corrections are determined from the data using
the “tag-and-probe” technique [44] to adjust for the efficiency differences
between data and simulation for lepton reconstruction, identification,
isolation, and trigger. A correction for the detector resolution effects is
implemented using an unfolding technique (as discussed in Section 0.8).

After offline reconstruction, two leptons are required with the first having
$\pt>30\GeV$ and the second having $\pt>20\GeV$. We require that the two
electrons (muons) with highest form a pair of oppositely charged leptons with
an invariant mass in the range ($91\pm 20\GeV$). Electrons and muons are
considered isolated based on the scalar sum of the nearby PF candidates with a
distance $\Delta R<0.4$. For both electrons and muons, medium identification
criteria are applied [45, 46]. Corrections are applied to the muon momenta to
adjust for a residual misalignment in the CMS detector between data and
simulation [47].

##  0.5 Observables

In this paper, the cross sections are presented as a function of several
kinematic and angular observables to characterize the production mechanisms of
$\PZ(\to\ell^{+}\ell^{-})+\text{jets}$ events. The cross sections are measured
as a function of both the exclusive and inclusive jet multiplicities up to a
total number of eight jets in the final state. In addition, they are measured
as a function of the kinematic variables , $y$ and for $N_{\textsc{jets}}$
from one to five. Comparisons of jet multiplicity distributions with
predictions from various MC generators show how accurately these generators
describe different jet configurations.

The measurement of the distribution of boson transverse momentum $\pt$($Z$)
for events with at least one jet is vital for understanding the balance of the
between the jets and boson, and may be used for comparing theoretical
predictions that produce multiple soft gluon emissions in different ways.

The rapidity of boson $y$($Z$) is related to the momentum fraction $x$ carried
by the partons in the two colliding protons. Therefore, the $y$($Z$)
distribution directly reflects the PDFs of the interacting partons. At the
LHC, the $y$($Z$) distribution is expected to be symmetric around zero,
therefore it is appropriate to measure the distribution of $Z$ bosons as a
function of $\abs{y}$.

The distributions of jet and jet are important because they are sensitive to
the effects of higher-order corrections, and provide an accurate estimation of
the background from $\PZ$+jets process for SUSY searches.

The cross sections are measured as a function of the difference in rapidity
$\Delta{y}(j_{\text{i}},{j}_{\text{k}})$ and of the difference in azimuthal
angle $\Delta{\phi}(j_{\text{i}},j_{\text{k}})$ between the
$i\textsuperscript{th}$ and $k\textsuperscript{th}$ jets from the ordered list
of jets in the event. For correlations between the boson and jets, cross
sections are measured as a function of the difference and the sum in rapidity,
$y(\PZ)\pm y(\text{j}_{\text{k}})$, and the difference in phi
$\Delta{\phi}(\PZ,j_{\text{k}})$. The angular separation $\Delta{\phi}$
between the $Z$ boson and a jet is sensitive to soft gluon radiation. The
advantage of studying the $\Delta\phi$ distribution is that it depends only on
the directions of the boson and a jet.

Lastly, double-differential cross sections are measured as functions of
leading jet and $y$, leading jet and boson $y$, and boson $\pt$ and $y$. The
measured cross sections are corrected for detector effects and compared with
theoretical predictions at LO and NLO accuracy matched with the parton
showering as implemented in MC generators.

##  0.6 Phenomenological models and theoretical calculations

We compare the measured $\PZ+\text{jets}$ differential cross sections with
three predictions: mg5_amc at NLO, mg5_amc at LO, and the Geneva MC program.
The two mg5_amc calculations (version 2.2.2) [25] are interfaced with 8
(version 8.212) [26]. For the LO mg5_amc, the generator calculates LO matrix
elements (MEs) for five processes: $\Pp\Pp\to\PZ+N_{\text{jets}}$ with
$N=0,\ldots,4$. The NNPDF 3.0 LO PDF [48] is used and $\alpS(m_{\PZ})$ is set
to 0.130. The NLO mg5_amc prediction includes NLO ME calculations for pp
$\to\PZ$+$N_{\text{jets}}$ with $N$ up to 2. The NNPDF 3.0 NLO PDF set is used
and $\alpS(m_{\PZ})$ is set to 0.118. Both predictions use 8 to model the
initial- and final-state radiation, parton showers and hadronization with the
CUETP8M1 [27] tune that is done with the NNPDF 2.3 [49] LO PDF and
$\alpS(m_{\PZ})=0.130$. The ME and parton shower matching is done using the
-MLM [23, 50] scheme with the matching scale set at 19for the LO mg5_amc and
the FxFx [24] scheme with the matching scale set to 30for the NLO mg5_amc.

The third calculation uses the Geneva 1.0-RC3 (GE) simulation program [51,
52], where a next-to-leading-order (NNLO) calculation for Drell–Yan production
is combined with higher-order resummation. Logarithms of the 0-jettiness
resolution variable, $\tau$, also known as beam thrust and defined in Ref.
[53], are resummed at next-to-next-to-leading logarithmic (NNLL) accuracy,
including part of the next-to-NNLL corrections. The accuracy refers to the
$\tau$ dependence of the cross section and is denoted as
$\text{NNLL}^{\prime}_{\tau}$. The PDF set NNPDF3.1 NNLO [54] is used for this
calculation and $\alpS(m_{\PZ})$ is set to 0.118. The resulting parton-level
events are further combined with parton showering and hadronization provided
by 8 using the same tune as for mg5_amc.

In this analysis, uncertainties in the ME calculation for the mg5_amc and
Geneva predictions are estimated using the procedure recommended by the
authors of the respective generators. For the mg5_amc prediction, the
factorization ($\mu_{\text{F}}$) and renormalization ($\mu_{\text{R}}$) scales
are varied by a factor of 0.5 and 2 to estimate the uncertainty coming from
missing higher-order terms in the fixed-order calculation. An envelope of the
six variations is used with the two extremes (one scale varied by a factor 0.5
and the other by a factor 2) excluded. For the Geneva sample, $\mu_{\text{F}}$
and $\mu_{\text{R}}$ are simultaneously varied by 0.5 and 2, leading to two
combinations, their uncertainties are symmetrized by using the maximum of the
up and down uncertainties for both cases. The uncertainty from the resummation
in Geneva is estimated using six profile scales [55, 56], as described in Ref.
[51], and added in quadrature to the scale uncertainty. The PDF uncertainty in
the mg5_amc sample is estimated using the set of 100 replicas of the NNPDF 3.0
NLO PDF and the uncertainty in the value used in the ME calculation is
estimated by varying it by $\pm$0.001. The PDF and uncertainties are added in
quadrature to the ME calculation uncertainties. For both mg5_amc and Geneva,
all these uncertainties are obtained using the reweighting method [51, 57]
implemented in these event generators.

##  0.7 Background estimation

Background events are split into two categories: resonant and nonresonant. The
resonant background, which consists mainly of multiboson events with at least
one boson produced in the final state, is estimated using simulation. The
background from nonresonant events containing two leptons primarily from boson
decays such as those appearing in is estimated from data events. The decay
$\PZ\to\PGt^{+}\PGt^{-}$ is considered as a background and is estimated from
the mg5_amc signal MC sample. The backgrounds from events where one or two
jets are misidentified as a lepton, such as W+jets or multijets, is
negligible.

The method used for estimating the nonresonant background uses a control
region in data containing events with one electron and one muon
$\Pe^{\pm}\PGm^{\mp}$ passing all other signal region criteria. This control
region is used to estimate the nonresonant background in the signal region by
applying a transfer factor to account for cross section and lepton efficiency
differences between channels. Assuming lepton flavor symmetry, the cross
section for the $\Pe^{\pm}\PGm^{\mp}$ and either $\Pep\Pem$ or $\PGmp\PGmm$
channel differs only by a factor of 2. The difference in the efficiency
between electrons and muons is estimated using the total yields of the two
channels. Resonant signal and background are estimated in the control region
with the same signal simulation and subtracted to avoid double counting.

The kinematic properties of the boson and the leading jet, and measurement of
jet multiplicity are shown in Figs. 1–3 together with the results of the
simulation. Background samples corresponding to diboson electroweak production
is denoted as “VV”, and nonresonant background samples are denoted as “NRB” in
the figure legends. The fraction of background events is small compared with
the signal and amounts to approximately 1% for $\geq$0 jets increasing to 10%
at 5 or more jets. For variables, the background increases from 1% below 100to
10% in the high-tails.

Figure 1: The boson candidate $\pt$ (upper) and $\abs{y}$ (lower) for events
with at least one jet. The muon (left) and electron (right) channels are shown
separately. The background is estimated from both simulation and data driven
methods (such as nonresonant background, NRB) as described in Section 0.7. The
error bars around the data points represent the statistical uncertainties. The
distribution ratio of simulation to data is shown in the bottom frames, with
error bars that represent the total statistical uncertainties from the data
and simulation samples.

Figure 2: Inclusive (upper) and exclusive (lower) jet multiplicity
distributions. The muon (left) and electron (right) channels are shown
separately. The background is estimated from both simulation and data driven
methods (such as nonresonant background, NRB) as described in Section 0.7. The
error bars around the data points represent the statistical uncertainties. The
distribution ratio of simulation to data is shown in the bottom frames, with
error bars that represent the total statistical uncertainties from the data
and simulation samples.

Figure 3: First jet $\pt$ (upper) and $\abs{y}$ (lower) distributions. The
muon (left) and electron (right) channels are shown separately. The background
is estimated from both simulation and data driven methods (such as nonresonant
background, NRB) as described in Section 0.7. The error bars around the data
points represent the statistical uncertainties. The distribution ratio of
simulation to data is shown in the bottom frames, with error bars that
represent the total statistical uncertainties from the data and simulation
samples.

##  0.8 Unfolding procedure

In this analysis, unfolding is performed to remove detector effects and
estimate the particle level distributions in data. The mg5_amc MC sample is
used to extract the detector transformation, called the response matrix, that
feeds into the unfolding algorithm. The unfolding procedure consists of
performing a least-squares fit with optional Tikhonov regularization [58], as
implemented in the TUnfold software package [59]. In this analysis the best
value for the regularization parameter is chosen using the L-curve method
[60]. Closure tests are performed by checking the unfolded distributions with
the original data.

The momenta of the leading leptons are summed to obtain the particle-level
boson momentum. The particle-level objects are required to pass the same
kinematic selections as at detector level.

##  0.9 Systematic uncertainties

The sources of experimental uncertainty are divided into the following
categories: Jet energy scale (JES) and jet energy resolution (JER); lepton
efficiencies (identification, isolation, and track reconstruction); lepton
energy scale (LES) and resolution (LER); trigger efficiency; luminosity;
pileup; background and unfolding uncertainties. The uncertainties listed above
are assumed to be independent such that each is computed individually and
added in quadrature to obtain a total uncertainty. To compute the systematic
uncertainty from each source, the analysis is repeated using the source values
increased and decreased by 1$\sigma$ from the central value. This results in
bin-by-bin uncertainty contributions from each source in the unfolded
distributions.

The JES uncertainty originates mainly from the uncertainty in the single-
particle response. It is the dominant source of systematic uncertainty. It
affects the reconstruction of the transverse energy of the selected jets. In
this analysis, jet energy corrections (JEC) are applied to include
inefficiencies, nonlinearities and finite resolution in energy and position of
the reconstructed jets. The effect of the JES uncertainty is studied by
scaling the reconstructed jet energy up and down by and $\eta$-dependent scale
factors. A similar procedure is followed for the JER. The uncertainties in the
JES and JER vary from 1–11% as a function of jet multiplicity.

Scale factors for lepton efficiencies are applied on an object-by-object basis
so that the simulation samples reflect the inefficiencies observed in data.
The lepton identification, isolation, track reconstruction and trigger
efficiencies in simulation are corrected with scaling factors derived with a
tag-and-probe method and applied as a function of lepton and $\eta$. To
estimate the uncertainties, the total yield is recomputed with the scaling
factors varied up and down by the fit uncertainties. The uncertainty
associated with lepton efficiency in the electron (muon) channel is 1% (0.5%).

The LES and LER uncertainties make a small contribution to the overall lepton
uncertainties of $\sim$1% for each channel.

A normalization uncertainty of 2.5% is assigned to account for the imperfect
knowledge of the integrated luminosity [61]. This uncertainty is propagated to
the measured differential cross sections.

The uncertainty coming from the pileup model is estimated by varying the
amount of pileup events by 4.6% up and down [62] when reconstructing the
response matrices from the simulation. The difference in the unfolded data is
the uncertainty.

The uncertainty in the unfolding procedure comes from: (1) the statistical
uncertainty in the response matrix coming from the finite size of the MC
sample used to compute it; and (2) the possible event generator dependence of
the response matrix itself. Because of the finite binning, a different
distribution could lead to a different response matrix. This uncertainty is
estimated by weighting the MC to agree with the data in each distribution to
be unfolded and building a new response matrix. The weights are extracted from
the data-to-MC ratio of a fine-binned histogram at the reconstruction level.
The fine binning allows us to account for the effect of the distribution of
events within each measurement bin. The difference between the nominal results
and the results unfolded using the alternative response matrix is assumed the
systematic uncertainty. Statistical fluctuations in the response matrix are
propagated analytically in the TUnfold package.

Lastly, the background samples are varied by their corresponding cross section
uncertainty before being subtracted from data prior to unfolding.

##  0.10 Results

The measurements from the electron and muon channels are consistent with each
other within the statistical and systematic uncertainties, and hence they are
combined. To combine the two channels, a hybrid method based on the weighted
mean and the best linear unbiased estimates method [63, 64] is used to
calculate the cross section values and uncertainties. The covariance matrix of
the combination is calculated assuming that all uncertainty sources are
correlated between channels except the statistical components and those
associated with the lepton reconstruction and identification.

Figure 4 shows the measured cross sections as a function of $N_{\text{jets}}$
for a total number of up to eight jets in the final state. The trend of the
jet multiplicity represents the expectation of the perturbative QCD prediction
for an exponentially falling spectrum with the number of jets. The agreement
is good for the exclusive distributions for all the theoretical estimations,
remaining within the uncertainties and going up to the maximum number of
final-state partons included in the ME, namely four in the MC generators used
here. The Geneva generator predicts a steeper spectrum than observed due to
the lack of hard jets at ME level beyond two.

The size of the 2016 data samples allows us to determine the differential
cross sections for jet multiplicities up to eight jets, and to measure the
cross sections as a function of several kinematic observables up to five jets.
The combined single-differential cross sections are shown in Figs. 5–22, while
double-differential cross sections are given in Figs. 23–25. All results are
compared with theoretical predictions from mg5_amc at LO and at NLO. Since the
Geneva predictions are effectively LO in QCD at two jets, only the results
with at least one or two jets are compared with Geneva.

The jet $y$ and up to five leading jets can be seen in Figs. 5–9. For both
quantities, the data distributions are well reproduced by the simulations. The
mg5_amc at LO and NLO, describe the data well in general. The Geneva
prediction shows good agreement for the measured and $y$ of the first jet,
although it underestimates the cross section at low in the second jet.

In addition, the inclusive jet differential cross sections as a function of
for events with at least one, two and three jets, respectively, are presented
in Fig. 10. The mg5_amc predictions at both LO and NLO are compatible with the
measurement. The cross section at higher values of is slightly overestimated,
but the discrepancy is compatible with the theoretical and experimental
uncertainties. The slopes of the distributions for the first two jet
multiplicities predicted by Geneva do not describe the data.

The measured cross section as a function of the dijet mass is also shown in
Fig. 11. The three predictions considered here agree with the measurement
within the experimental uncertainties, except for a dijet mass below $\sim$50,
where the predictions made with Geneva show a deficit with respect to the
measurement. The mg5_amc at NLO generator has better agreement with the
measurement in this region. The mg5_amc at LO generator predicts a
distribution that falls more steeply for a dijet mass above $\sim$100.

The rapidity distributions of the boson and jets are reasonably well modelled
by the predictions, but the correlations between the rapidities, which have
been shown by measuring multidimensional differential cross sections and
distributions of rapidity differences and sums (Figs. 13–17), are not well
described by the multileg ME calculation. We have shown that the NLO multileg
event generator reproduces the rapidity difference distributions well. The
rapidity sum is also successfully described. For this variable the discrepancy
with the ME calculation could be due to a different choice of the parton
distribution functions. The azimuthal angles between the boson and the jet
(Figs. 18–20) and between the jets (Figs. 21–22) are well defined by the
predictions including the ME one.

The results for the double-differential cross sections are presented in Figs.
23–25 and are compared with the predictions described in Section 0.6. The
double-differential cross sections are shown for at least one jet as a
function of the leading jet and $y$ (Fig. 23), leading jet and boson $y$ (Fig.
24), boson and $y$ (Fig. 25). In general, all the predictions are in agreement
with the data, and the NLO mg5_amc prediction provides a better description
than the LO mg5_amc and Geneva predictions for double-differential cross
sections in Fig. 25. In the low-$\pt$ region Geneva gives good description as
expected by the resummation.

Overall, the mg5_amc at NLO predictions describe the data within theoretical
uncertainties over a range of kinematic variables. In the regions of NLO
accuracy, such as the first and second jet and $y$, the agreement is within
10% up to the scale.

The differential cross section results with covariance matrices are presented
in HEPData [65].

Figure 4: The measured differential cross section as a function of inclusive
(left) and exclusive (right) jet multiplicities. The measurement statistical
(resp. systematic) uncertainties are represented with vertical error bars
(resp. hashed areas). The measurement is compared to the NLO mg5_amc, LO
mg5_amc, and Geneva predictions described in Section 0.6. The predictions
uncertainties, estimated as explained in this section, are represented by
coloured areas in the ratio plots (light colour for the statistical part and
darker colour for the total). Only the statistical uncertainty is displayed
for the LO prediction.

Figure 5: The measured differential cross section as a function of leading jet
$\abs{y}$ (left) and (right) for events with at least one jet. The measurement
statistical (resp. systematic) uncertainties are represented with vertical
error bars (resp. hashed areas). The measurement is compared to the NLO
mg5_amc, LO mg5_amc, and Geneva predictions described in Section 0.6. The
predictions uncertainties, estimated as explained in this section, are
represented by coloured areas in the ratio plots (light colour for the
statistical part and darker colour for the total). Only the statistical
uncertainty is displayed for the LO prediction.

Figure 6: The measured differential cross section as a function of second jet
$\abs{y}$ (left) and (right) for events with at least two jets. The
measurement statistical (resp. systematic) uncertainties are represented with
vertical error bars (resp. hashed areas). The measurement is compared to the
NLO mg5_amc, LO mg5_amc, and Geneva predictions described in Section 0.6. The
predictions uncertainties, estimated as explained in this section, are
represented by coloured areas in the ratio plots (light colour for the
statistical part and darker colour for the total). Only the statistical
uncertainty is displayed for the LO prediction.

Figure 7: The measured differential cross section as a function of third jet
$\abs{y}$ (left) and (right) for events with at least three jets. The
measurement statistical (resp. systematic) uncertainties are represented with
vertical error bars (resp. hashed areas). The measurement is compared to the
NLO and LO mg5_amc predictions described in Section 0.6. The predictions
uncertainties, estimated as explained in this section, are represented by
coloured areas in the ratio plots (light colour for the statistical part and
darker colour for the total). Only the statistical uncertainty is displayed
for the LO prediction.

Figure 8: The measured differential cross section as a function of fourth jet
$\abs{y}$ (left) and (right) for events with at least four jets. The
measurement statistical (resp. systematic) uncertainties are represented with
vertical error bars (resp. hashed areas). The measurement is compared to the
NLO and LO mg5_amc predictions described in Section 0.6. The predictions
uncertainties, estimated as explained in this section, are represented by
coloured areas in the ratio plots (light colour for the statistical part and
darker colour for the total). Only the statistical uncertainty is displayed
for the LO prediction.

Figure 9: The measured differential cross section as a function of fifth jet
$\abs{y}$ (left) and (right) for events with at least five jets. The
measurement statistical (resp. systematic) uncertainties are represented with
vertical error bars (resp. hashed areas). The measurement is compared to the
NLO and LO mg5_amc predictions described in Section 0.6. The predictions
uncertainties, estimated as explained in this section, are represented by
coloured areas in the ratio plots (light colour for the statistical part and
darker colour for the total). Only the statistical uncertainty is displayed
for the LO prediction.

Figure 10: The measured differential cross section as a function of for events
with at least one (left), two (right), and three (bottom) jets. The
measurement statistical (resp. systematic) uncertainties are represented with
vertical error bars (resp. hashed areas). The measurement is compared to the
NLO mg5_amc, LO mg5_amc, and Geneva (for $N_{\text{jets}}\geq 1$ and
$N_{\text{jets}}\geq 2$) predictions described in Section 0.6. The predictions
uncertainties, estimated as explained in this section, are represented by
coloured areas in the ratio plots (light colour for the statistical part and
darker colour for the total). Only the statistical uncertainty is displayed
for the LO prediction. Figure 11: The measured differential cross section as a
function of dijet mass for events with at least two jets. The measurement
statistical (resp. systematic) uncertainties are represented with vertical
error bars (resp. hashed areas). The measurement is compared to the NLO
mg5_amc, LO mg5_amc, and Geneva predictions described in Section 0.6. The
predictions uncertainties, estimated as explained in this section, are
represented by coloured areas in the ratio plots (light colour for the
statistical part and darker colour for the total). Only the statistical
uncertainty is displayed for the LO prediction. Figure 12: The measured
differential cross section as a function of boson $\abs{y}$ for events with at
least one jet. The measurement statistical (resp. systematic) uncertainties
are represented with vertical error bars (resp. hashed areas). The measurement
is compared to the NLO mg5_amc, LO mg5_amc, and Geneva predictions described
in Section 0.6. The predictions uncertainties, estimated as explained in this
section, are represented by coloured areas in the ratio plots (light colour
for the statistical part and darker colour for the total). Only the
statistical uncertainty is displayed for the LO prediction.

Figure 13: The measured differential cross section as a function of the
leading and subleading jet rapidity difference (left) and sum (right) for
events with at least two jets. The measurement statistical (resp. systematic)
uncertainties are represented with vertical error bars (resp. hashed areas).
The measurement is compared to the NLO mg5_amc, LO mg5_amc, and Geneva
predictions described in Section 0.6. The predictions uncertainties, estimated
as explained in this section, are represented by coloured areas in the ratio
plots (light colour for the statistical part and darker colour for the total).
Only the statistical uncertainty is displayed for the LO prediction.

Figure 14: The measured differential cross section as a function of the Z
boson and leading jet rapidity difference (left) and sum (right) for events
with at least one jet. The measurement statistical (resp. systematic)
uncertainties are represented with vertical error bars (resp. hashed areas).
The measurement is compared to the NLO mg5_amc, LO mg5_amc, and Geneva
predictions described in Section 0.6. The predictions uncertainties, estimated
as explained in this section, are represented by coloured areas in the ratio
plots (light colour for the statistical part and darker colour for the total).
Only the statistical uncertainty is displayed for the LO prediction.

Figure 15: The measured differential cross section as a function of the Z
boson and leading jet rapidity difference (left) and sum (right) for events
with at least two jets. The measurement statistical (resp. systematic)
uncertainties are represented with vertical error bars (resp. hashed areas).
The measurement is compared to the NLO mg5_amc, LO mg5_amc, and Geneva
predictions described in Section 0.6. The predictions uncertainties, estimated
as explained in this section, are represented by coloured areas in the ratio
plots (light colour for the statistical part and darker colour for the total).
Only the statistical uncertainty is displayed for the LO prediction.

Figure 16: The measured differential cross section as a function of the Z
boson and subleading jet rapidity difference (left) and sum (right) for events
with at least two jets. The measurement statistical (resp. systematic)
uncertainties are represented with vertical error bars (resp. hashed areas).
The measurement is compared to the NLO mg5_amc, LO mg5_amc, and Geneva
predictions described in Section 0.6. The predictions uncertainties, estimated
as explained in this section, are represented by coloured areas in the ratio
plots (light colour for the statistical part and darker colour for the total).
Only the statistical uncertainty is displayed for the LO prediction.

Figure 17: The measured differential cross section as a function of the Z
boson and dijet rapidity difference (left) and sum (right) with two jets
inclusive. The measurement statistical (resp. systematic) uncertainties are
represented with vertical error bars (resp. hashed areas). The measurement is
compared to the NLO mg5_amc, LO mg5_amc, and Geneva predictions described in
Section 0.6. The predictions uncertainties, estimated as explained in this
section, are represented by coloured areas in the ratio plots (light colour
for the statistical part and darker colour for the total). Only the
statistical uncertainty is displayed for the LO prediction.

Figure 18: The measured differential cross section as a function of the Z
boson and leading jet azimuthal difference for events with at least one
(left), two (right), and three (bottom) jets. The measurement statistical
(resp. systematic) uncertainties are represented with vertical error bars
(resp. hashed areas). The measurement is compared to the NLO mg5_amc, LO
mg5_amc, and Geneva (for $N_{\text{jets}}\geq 1$ and $N_{\text{jets}}\geq 2$)
predictions described in Section 0.6. The predictions uncertainties, estimated
as explained in this section, are represented by coloured areas in the ratio
plots (light colour for the statistical part and darker colour for the total).
Only the statistical uncertainty is displayed for the LO prediction.

Figure 19: The measured differential cross section as a function of the Z
boson and subleading jet azimuthal difference for events with at least two
(left) and three (right) jets. The measurement statistical (resp. systematic)
uncertainties are represented with vertical error bars (resp. hashed areas).
The measurement is compared to the NLO mg5_amc, LO mg5_amc, and Geneva (for
$N_{\text{jets}}\geq 2$) predictions described in Section 0.6. The predictions
uncertainties, estimated as explained in this section, are represented by
coloured areas in the ratio plots (light colour for the statistical part and
darker colour for the total). Only the statistical uncertainty is displayed
for the LO prediction. Figure 20: The measured differential cross section as a
function of the Z boson and third jet azimuthal difference for events with at
least three jets. The measurement statistical (resp. systematic) uncertainties
are represented with vertical error bars (resp. hashed areas). The measurement
is compared to the NLO and LO mg5_amc predictions described in Section 0.6.
The predictions uncertainties, estimated as explained in this section, are
represented by coloured areas in the ratio plots (light colour for the
statistical part and darker colour for the total). Only the statistical
uncertainty is displayed for the LO prediction.

Figure 21: The measured differential cross section as a function of the
leading and subleading jet azimuthal difference for events with at least two
(left) and three (right) jets. The measurement statistical (resp. systematic)
uncertainties are represented with vertical error bars (resp. hashed areas).
The measurement is compared to the NLO mg5_amc, LO mg5_amc, and Geneva (for
$N_{\text{jets}}\geq 2$) predictions described in Section 0.6. The predictions
uncertainties, estimated as explained in this section, are represented by
coloured areas in the ratio plots (light colour for the statistical part and
darker colour for the total). Only the statistical uncertainty is displayed
for the LO prediction.

Figure 22: The measured differential cross section as a function of the
leading and third jet azimuthal difference (left) and subleading and third jet
azimuthal difference (right) for events with at least three jets. The
measurement statistical (resp. systematic) uncertainties are represented with
vertical error bars (resp. hashed areas). The measurement is compared to the
NLO and LO mg5_amc predictions described in Section 0.6. The predictions
uncertainties, estimated as explained in this section, are represented by
coloured areas in the ratio plots (light colour for the statistical part and
darker colour for the total). Only the statistical uncertainty is displayed
for the LO prediction.

Figure 23: Double differential cross sections as a function of leading jet and
$\abs{y}$ for events with at least one jet (upper left). The measurement
statistical (resp. systematic) uncertainties are represented with vertical
error bars (resp. hashed areas). The measurement is compared to the NLO
mg5_amc, LO mg5_amc, and Geneva predictions described in Section 0.6. The
predictions uncertainties, estimated as explained in this section, are
represented by coloured areas in the ratio plots (light colour for the
statistical part and darker colour for the total). Only the statistical
uncertainty is displayed for the LO prediction.

Figure 24: Double differential cross sections as a function of leading jet and
Z boson $\abs{y}$ for events with at least one jet. The measurement
statistical (resp. systematic) uncertainties are represented with vertical
error bars (resp. hashed areas). The measurement is compared to the NLO
mg5_amc, LO mg5_amc, and Geneva predictions described in Section 0.6. The
predictions uncertainties, estimated as explained in this section, are
represented by coloured areas in the ratio plots (light colour for the
statistical part and darker colour for the total). Only the statistical
uncertainty is displayed for the LO prediction.

Figure 25: Double differential cross sections as a function of Z boson and
$\abs{y}$ for events with at least one jet. The measurement statistical (resp.
systematic) uncertainties are represented with vertical error bars (resp.
hashed areas). The measurement is compared to the NLO mg5_amc, LO mg5_amc, and
Geneva predictions described in Section 0.6. The predictions uncertainties,
estimated as explained in this section, are represented by coloured areas in
the ratio plots (light colour for the statistical part and darker colour for
the total). Only the statistical uncertainty is displayed for the LO
prediction.

##  0.11 Summary

The production of bosons, decaying into a pair of electrons or muons, in
association with jets has been studied in proton-proton collisions at a
centre-of-mass energy of 13at the LHC in 2016 by the CMS experiment using a
data set with an integrated luminosity of 35.9. Differential cross sections
have been measured for bosons decaying to electrons or muons with transverse
momentum $\pt>25\GeV$ and pseudorapidity $\abs{\eta}<2.4$ requiring at least
one jet with $\pt>30\GeV$ and $\abs{\eta}<2.4$.

Differential cross sections have been measured as a function of the exclusive
and inclusive jet multiplicities ($N_{\text{jets}}$), the of the boson, and
kinematic variables that include jet transverse momenta, the scalar sum for up
to five inclusive $N_{\text{jets}}$, rapidity, dijet invariant mass
($M_{\text{jj}}$) and their sum values.

The results, corrected for detector effects through unfolding, are compared
with three theoretical predictions: (1) the expectations are computed from
particle-level simulations using merged leading-order (LO) calculations with
the -MLM parton-showers and matrix-element matching scheme; (2) using next-to-
leading-order (NLO) calculations and the FxFx merging scheme; and (3) the
Geneva MC program, where a next-to-NLO (NNLO) calculation for Drell–Yan
production is combined with higher-order resummation.

High precision is achieved in measuring the cross sections using the latest
experimental methods and larger sets of data than were available in previous
CMS publications. The increased number of events is extended the kinematic
range to higher values of and mass. The measurements presented in this paper
provide a detailed description of the topological structure of
$\PZ\to\ell^{+}\ell^{-}+\text{jets}$ events that is complementary to the
existing measurements of rates and associated jet multiplicities.

The kinematics of $\PZ$+jets events is studied in detail. The measured
differential cross sections and $N_{\text{jets}}$ distributions are within the
experimental and theoretical uncertainties. Some deviations are observed for
$N_{\text{jets}}>3$. Such discrepancies offer the possibility of using these
data to further improve the modeling. The results also indicate that
multiparton NLO calculations can be used to estimate the
$\PZ\to\ell^{+}\ell^{-}+\text{jets}$ contributions to measurements and
searches at the LHC.

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[a]João Barata

# Energy loss effects in EECs at LO

Yacine Mehtar-Tani

###### Abstract

In recent years, there has been an effort towards establishing a more complete
picture for jet substructure in the presence of the quark gluon plasma. Such a
program requires not only a more detailed description of medium induced
effects, but also the design of novel substructure observables. Very recently,
it has been noticed that Energy Energy correlators (EECs) might provide one
type of such observables. Although the full extent of their sensitivity to the
medium has not been completely explored, they are capable to resolve the
transverse structure of the jet. In particular, they are sensitive to the
critical angle separating coherent and decoherent jet evolution in the medium.
In this talk, we show for the first time the effects of medium induced
radiative energy loss in EECs at leading order in the number of vacuum-like
emissions. The calculation takes into account all order soft gluon emissions,
in the large $N_{c}$ limit and neglecting subdominant interfering
contributions.

##  1 Introduction

In recent years, there has been a strong interest in applying jet substructure
techniques to study medium induced modifications to jets. One of the most
interesting ideas being currently pursued relates to measuring the so called
critical coherence angle $\theta_{c}$, which determines the radiation pattern
of jets in medium, see e.g. [1, 2]. As far as we are aware, two recent
proposals were made to extract information related to this parameter: one
resorts to measuring the angular distribution associated to groomed emissions
[3], while the other uses the information related to energy flow correlations
inside jets [4]. Although these observables have very different properties, it
is interesting to note that while for the former the coherence angle effects
enter mainly through the energy loss mechanism, in the latter it emerges due
to the medium modified splitting kernel. Thus, our goal is to explore the
sensitivity of the energy flow correlations to radiative energy loss.

##  2 Energy Energy Correlators and medium modified cross-section

The idea of understanding high energy scattering events using energy flow
correlations was proposed a long time ago [5]. Recently, these ideas have been
extended to study the structure of QCD jets in vacuum [6] and in heavy ion
collisions [4, 7]. At leading order in the strong coupling constant and
assuming an isotropic and homogenous medium, one can define the two point
energy flow correlator (EEC) as [7]

$\displaystyle~{}\frac{d\Sigma}{d\theta}$
$\displaystyle\equiv\int_{0}^{1}dz\,z(1-z)\frac{1}{\sigma_{qg}}\frac{d\sigma_{qg}}{d\theta
dz}\,,$ (1)

where we have focused on a jet originating from a single quark, $z$ denotes
the energy sharing fraction for the outgoing gluon, $\theta$ is the angle
between the outgoing states and we neglect the initial quark cross-section in
what follows. In the vacuum and at leading accuracy, this object is fully
determined by

$\displaystyle\frac{1}{\sigma_{qg}}\frac{d\sigma_{qg}^{\rm vac}}{d\theta
dz}=\frac{\alpha_{s}C_{F}}{\pi}P(z)\frac{1}{\theta}\,,$ (2)

where the relevant splitting function is given by $P(z)=(1+(1-z)^{2})/z$. In
the presence of a dense QCD medium, the partonic cross-section gets
corrections due to the interactions with the medium constituents. Although the
general form of such a cross-section is known, it is a highly complex object
for practical applications, see e.g. [8]. Therefore, for phenomenology,
different limits of this are typically used. Here we shall consider two
limiting cases:

  1. 1.

Hard splitting: It is natural to consider the scenario where the outgoing
partons are very energetic, and thus they propagate close to their classical
trajectories. In this limit, it has been shown that one can write [9, 10]

$\displaystyle\frac{d\sigma_{qg}^{\rm med}}{d\theta
dz}=\frac{d\sigma_{qg}^{\rm vac}}{d\theta dz}\left(1+F_{\rm
med}(\theta,z)\right)\,,$ (3)

where the medium modifications are encapsulated in the $F_{\rm med}$ factor,
which reads

$\displaystyle F_{\rm
med}=\frac{2}{t_{f}}\int_{0}^{L}dt\,\left\\{\frac{\left(1-e^{-\frac{1}{4}\chi\hat{q}\theta^{2}t^{2}(L-t)}\right)}{t_{f}(\frac{1}{4}\chi\hat{q}\theta^{2}t^{2})}\cos\left(\frac{t}{t_{f}}\right)e^{-\frac{1}{12}\hat{q}\theta^{2}\zeta
t^{3}}-\sin\left(\frac{L-t}{t_{f}}\right)e^{-\frac{1}{12}\hat{q}\zeta\theta^{2}(L-t)^{3}}\right\\}\,.$
(4)

Here we have neglected subleading terms related to color interference, $N_{c}$
is the number of colors, $\hat{q}$ is the jet quenching parameter, $L$ the
medium length and $t_{f}=\frac{2}{z(1-z)p_{t}\theta^{2}}$, with $p_{t}$ the
total jet energy. We note that since the outgoing states are very energetic,
this leads to the kinematical constraint
$\min(z,1-z)p_{t}>\frac{1}{2}\,\hat{q}L^{2}$ [11]. We have also introduced the
variables $\zeta=\left(1+z^{2}+\frac{2z}{N_{c}^{2}-1}\right)$ and
$\chi=(1-2z+3z^{2})$. In Fig. 1 (left), we plot the integrand of Eq. 1 using
Eq. 4, as a function of the gluon energy fraction for several values of
$\theta$. We observe that for angles close to
$\theta_{c}\equiv\frac{2}{\sqrt{\hat{q}L^{3}}}$, the distribution has a flat
dependence in $z$, but at large angles it is dominated by very asymmetrical
topologies, with the gluon carrying a small amount of energy.

Figure 1: Integrand of the in-medium EEC: hard branching (left) and soft
splitting (right). Here we used $p_{t}=100$ GeV, $\hat{q}=0.2\,{\rm GeV}^{3}$
and $L=6$ fm. As a result $\theta_{c}=0.047$.

  2. 2.

Soft gluon splitting: In opposition to the previous case, we also consider the
exact soft gluon limit, i.e $z\ll 1$. In this case, the medium modified cross-
section is given by

$\displaystyle\frac{d\sigma_{qg}^{\rm med}}{d\theta dz}=(2\pi)^{2}\frac{\omega
dI}{d\omega
d^{2}{\boldsymbol{k}}}\frac{zp_{t}^{2}\theta}{2\pi}\Theta(z(1-z)E<\omega_{c})\,,$
(5)

where $2\omega_{c}=\hat{q}L^{2}$, $\omega=zp_{t}$,
$\theta\omega=|{\boldsymbol{k}}|$ and the single gluon spectrums reads (see
e.g. [12])

$\displaystyle(2\pi)^{2}\omega\frac{dI}{d\omega d^{2}{\boldsymbol{k}}}$
$\displaystyle=\frac{4\alpha_{s}C_{F}}{zp_{t}}\text{Re}\int_{0}^{L}ds\,\frac{e^{\frac{(zp_{t}\theta)^{2}{\cal
T}_{s}}{i-Q_{s}^{2}{\cal T}_{s}}}}{Q_{s}^{2}{\cal
T}_{s}-i}-\frac{8\alpha_{s}C_{F}}{(zp_{t}\theta)^{2}}\,\text{Re}\left(1-e^{-i(zp_{t}\theta)^{2}{\cal
T}_{L}}\right)\,.$ (6)

Here $Q_{s}^{2}=\hat{q}(L-s)$ and ${\cal T}_{s}=\frac{\tan(\Omega
s)}{2\omega\Omega}$, $\Omega=(1-i)/2\sqrt{\hat{q}/(zp_{t})}$. In Fig. 1
(right) we again plot the EEC integrand for this limit of the in-medium cross-
section. We observe that for angles close to $\theta_{c}$ the behavior differs
significantly from the one seen in Fig. 1 (left). However, at large angles, we
observe that the system is again dominated by very asymmetrical
configurations.

##  3 Jet energy loss

The above results do not take into account the possibility of transporting the
jet energy down to the temperature scale ($T$) by the emission of multiple
gluons, which are thought to dominate the energy loss mechanism in matter, see
e.g. [13]. To include this effect here we use the quenching weight
approximation [14], in the limit of two color sources being produced in the
medium [15]. This allows us to model the jet energy loss via the relation
($t_{c}=2/(\hat{q}\theta^{2})^{\frac{1}{3}}$) [16]

$\displaystyle\frac{d\Sigma(p_{t})}{d\theta}$
$\displaystyle=\int_{0}^{1}dz\,z(1-z)\frac{1}{\sigma_{qg}}\frac{d\sigma_{qg}}{d\theta
dz}$ $\displaystyle\times
Q_{q}[\Theta(t_{f}>L)+\Theta(t_{f}<L)\Theta(\theta<\theta_{c})+Q_{g}\Theta(t_{f}<t_{c})\Theta(\theta>\theta_{c})]\,.$
(7)

Here $Q_{q(g)}$ denote the single quark (gluon) quenching factor, which can be
computed, under some assumptions, as the modified Laplace transform over the
single gluon spectrum

$\displaystyle Q=\exp-\left\\{\int
d\omega\int\frac{d^{2}{\boldsymbol{k}}}{(2\pi)^{2}}(2\pi)^{2}\frac{dI}{d\omega
d^{2}{\boldsymbol{k}}}\left(1-e^{-\eta\omega}\right)\right\\}\,,$ (8)

with $Q$ standing for either the quark or gluon quenching weights. Here, we
further model this object by assuming 1) all radiation between $T$ and the
scale where gluon emission becomes abundant
($\omega_{s}=\alpha_{s}^{2}C_{A}^{2}/\pi^{2}\omega_{c}$) is lost to the medium
2) gluon radiation above this limit is only lost if emitted at angles larger
than $2R$, with $R$ the jet radius. Then, one can write that111Here:
$\eta=\frac{n}{p_{t}}$,
$\alpha\equiv\frac{\chi}{\eta^{2}}=\left(\frac{2p_{t}R}{\sqrt{\hat{q}L}n}\right)^{2}$
and $n\sim 6$ is the spectral index of the jet spectra.

$\displaystyle Q(p_{t},R)$
$\displaystyle=\exp\Bigg{\\{}-\Bigg{(}\frac{2\alpha_{s}C_{i}}{\pi}\Bigg{[}\sqrt{\frac{2\omega_{c}}{T}}(1-e^{-\eta
T})-\sqrt{\frac{2\omega_{c}}{\omega_{s}}}(1-e^{-\eta\omega_{s}})$
$\displaystyle+\sqrt{2\eta\pi\omega_{c}}(\operatorname{erf}(\sqrt{\omega_{s}\eta})-\operatorname{erf}(\sqrt{T\eta}))\Bigg{]}\Bigg{)}$
$\displaystyle-\Bigg{(}\frac{\alpha_{s}C_{i}}{\pi}\sqrt{2\omega_{c}\eta}\int_{\eta\omega_{s}}^{\infty}dx\frac{1}{x^{\frac{3}{2}}}\left(1-e^{-x}\right)\left(e^{-x^{2}\alpha}-\alpha
x^{2}\Gamma_{0}(\alpha x^{2})\right)\Bigg{)}\Bigg{\\}}$ (9)

In Figs. 2 and 3, we show the numerical results after taking into account
radiative energy loss, for hard (Fig. 2) and soft splittings (Fig. 3). For the
hard branchings, we observe that the energy loss does not change the shape of
the distribution, unless a kinematical cut is imposed. In this case, energy
loss leads to a dip with respect to the vacuum at large angles. For soft
splittings, we observe a qualitatively similar behavior, despite the fact that
this limit is more sensitive to the energy loss. We note that in this limit
the validity of the description is stretched by integrating over the full $z$
domain.

Figure 2: EEC distribution including energy loss using $p_{t}=100$ ($250$) GeV
(right), $R=0.3$, $T=0.3$ GeV and the same parameters as in the previous
plots. Dashed lines denote the vacuum results, solid red and blue curves
correspond to including the medium modified cross-section. The vacuum lines
including energy loss, convolute the vacuum kernel with the energy loss model.
The black line corresponds to the full result while imposing the constraint
$\min(z,1-z)p_{t}>\omega_{c}$.

Figure 3: Same distributions as in Fig. 2, but for soft gluon branching.

##  4 Conclusion and Discussion

In this talk, we discussed the effects of energy loss in EECs measured inside
jets produced in heavy ion collisions. These exploratory results indicate that
energy loss effects might be important in the description of EECs, but a more
careful treatment of the in-medium cross-sections is needed.

## Acknowledgements

J. B. is grateful to Alba Soto-Ontoso, Paul Caucal, Andrey Sadofyev and Robert
Szafron for helpful discussions. We also are grateful to the authors of [4]
for clarifications on their work. Y. M.-T. and J. B.’s work has been supported
by the U.S. Department of Energy under Contract No. DE-SC0012704. Y. M.-T.
acknowledges support from the RHIC Physics Fellow Program of the RIKEN BNL
Research Center.

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