⟨⟩

2015 041 24 Feb

Measurement of dijet $k_{\rm T}$ in p–Pb collisions at
$\sqrt{s_{\mathrm{NN}}}=5.02$ TeV ALICE Collaboration††thanks: See Appendix A
for the list of collaboration members ALICE Collaboration

A measurement of dijet correlations in p–Pb collisions at
$\sqrt{s_{\mathrm{NN}}}=5.02$ TeV with the ALICE detector is presented. Jets
are reconstructed from charged particles measured in the central tracking
detectors and neutral energy deposited in the electromagnetic calorimeter. The
transverse momentum of the full jet (clustered from charged and neutral
constituents) and charged jet (clustered from charged particles only) is
corrected event-by-event for the contribution of the underlying event, while
corrections for underlying event fluctuations and finite detector resolution
are applied on an inclusive basis. A projection of the dijet transverse
momentum, $k_{\rm
Ty}=p_{\mathrm{T,jet}}^{\mathrm{ch+ne}}\;{\mathrm{sin}}(\Delta\varphi_{{\mathrm{dijet}}})$
with $\Delta\varphi_{{\mathrm{dijet}}}$ the azimuthal angle between a full and
charged jet and $p_{\mathrm{T,jet}}^{\mathrm{ch+ne}}$ the transverse momentum
of the full jet, is used to study nuclear matter effects in p–Pb collisions.
This observable is sensitive to the acoplanarity of dijet production and its
potential modification in p–Pb collisions with respect to pp collisions.
Measurements of the dijet $k_{\rm Ty}$ as a function of the transverse
momentum of the full and recoil charged jet, and the event multiplicity are
presented. No significant modification of $k_{\rm Ty}$ due to nuclear matter
effects in p–Pb collisions with respect to the event multiplicity or a PYTHIA8
reference is observed.

##  1 Introduction

Dijets produced in $2\rightarrow 2$ leading-order (LO) scattering processes
are balanced in transverse momentum and back-to-back in azimuth. In proton-
proton collisions a small acoplanarity appears due to intrisic transverse
momentum $k_{\rm T}$ from partonic Fermi motion [1] and initial state gluon
radiation [2, 3]. At large momentum transfer between the incoming partons, the
phase space for hard gluon radiation in the parton shower or from next-to-
leading-order (NLO) processes increases, resulting in acoplanarity of the
dijet system [4, 5, 6]. This also results in an imbalance of the jet
transverse momenta also referred to as a broadening of the dijet transverse
momentum. The relative contribution of hard QCD radiation to the dijet $k_{\rm
T}$ can be varied by applying kinematic and acceptance selections to the dijet
sample.

In p–Pb collisions the dijet kinematics are potentially modified due to
nuclear matter effects which are expected to induce a momentum imbalance and
acoplanarity of dijet pairs with respect to pp collisions, so-called
transverse momentum broadening [7]. For instance, multiple scatterings inside
the nucleus of the initial- and final-state partons in hard scatterings can
lead to such a transverse momentum broadening.

In heavy-ion collisions, jets produced in hard scattering processes are used
to probe the properties of the produced medium. Highly energetic partons
propagate through the medium, which modifies the parton shower resulting in a
modified fragmentation pattern of the final hadronization products [8, 9].
Heavy-ion jet measurements are compared to measurements in pp collisions to
determine the effect of hot nuclear matter on jet observables [10, 11, 12,
13]. In the context of such studies, measurements in p–Pb collisions serve as
a benchmark to study hard scattering processes in a nuclear target.

Measurements presented in [14] of the dijet transverse momentum imbalance and
dijet azimuthal angle distributions show results which are comparable to
results obtained with pp data and independent of the event activity. This
letter presents a measurement of dijet acoplanarity in p–Pb collisions at
$\sqrt{s_{\mathrm{NN}}}=5.02$ TeV, recorded with the ALICE detector at the
Large Hadron Collider (LHC). The jet azimuthal correlations are measured at
mid-rapidity for jet transverse momentum between 15 and 120 GeV/$c$. Jets
entering in the acceptance of electomagnetic calorimeter are reconstructed
from charged and neutral particles (full jet) while the recoil jet is
reconstructed from charged particles only (charged jet). Measurements are
presented as a function of the full and associated charged jet transverse
momentum in two event multiplicity classes which are correlated to the
centrality of the p–Pb collisions [15].

##  2 Experimental Setup and Data Sample

Collisions of proton and lead beams were provided by the LHC in the first
months of 2013. The beam energies were 4 TeV for the proton beam and 1.58 TeV
per nucleon for the lead beam, resulting in collisions at a center of mass
energy $\sqrt{s_{\mathrm{NN}}}=5.02$ TeV. The nucleon-nucleon center-of-mass
system moves in rapidity with respect to the ALICE reference frame by -0.465
in the direction of the proton beam [16]. In the following $\eta$ refers to
the pseudorapidity in the ALICE reference frame.

The V0 detectors, two arrays of scintillator tiles covering the full azimuth
within $2.8<\eta<5.1$ (V0A) and $-3.7<\eta<-1.7$ (V0C), were used for online
minimum bias event triggering, offline event selection and characterization of
events in different particle multiplicity classes. The minimum bias trigger
required a signal from a charged particle in both the V0A and V0C. The total
integrated luminosity of the minimum bias event sample is 37 ${\mathrm{\mu
b}}^{-1}$.

The electromagnetic calorimeter (EMCal) in ALICE [17] covers 100 degrees in
azimuth, $1.4<\varphi<\pi$, and $|\eta|<0.7$. For the analyzed data set an
online jet patch trigger of $32\times 32$ adjacent towers, corresponding to an
area of approximately $0.2$ $\mathrm{rad}$ was used. This jet patch trigger
fired if an integrated patch energy of at least $10$ GeV (low-energy trigger)
or $20$ GeV (high-energy trigger) was found. The low-energy triggered event
sample provided a significant overlap in jet energy between the minimum bias
and high-energy trigger event samples, allowing assessment of the trigger
biases. The event sample obtained with the low-energy trigger corresponds to a
total integrated luminosity of 21 ${\mathrm{\mu b}}^{-1}$. The event sample
with the high energy threshold has a total integrated luminosity of 1.6
${\mathrm{nb}}^{-1}$.

The position of the primary vertex was determined using reconstructed charged
particle tracks in the ALICE tracking systems, Inner Tracking System (ITS)
[18] and Time Projection Chamber (TPC) [19]. The algorithm to reconstruct the
primary vertex is fully efficient for events with at least one primary track
within $|\eta|<1.4$ [20]. To ensure a high tracking efficiency uniform in
$\eta$, events are accepted if the coordinate of the vertex along the beam
direction is within $\pm 10$ cm from the center of the detector.

The total event sample is divided into two multiplicity classes based on the
total charge deposited in the V0A detector [21]. For the data sample used in
this analysis, the V0A detector is located in the direction of the Pb remnants
and thus sensitive to the fragmentation of the nucleus limiting a correlation
in the definition of the multiplicity class with the dijet measurement at
midrapidity. Two multiplicity classes 0-40% and 40-100% are used in this
analysis. The higher multiplicity class (0-40%) corresponds to
$\langle{\mathrm{d}}N/{\mathrm{d}}\eta\rangle_{|\eta|<0.5}=37.2\pm 0.8$ and
the lower multiplicity class (40-100%) to
$\langle{\mathrm{d}}N/{\mathrm{d}}\eta\rangle_{|\eta|<0.5}=9.4\pm 0.2$.

##  3 Jet reconstruction and dijet $\mathbf{{\it k}_{Ty}}$

###  3.1 Jet Reconstuction

Jets are reconstructed with the anti-$k_{\rm T}$ jet algorithm of the FastJet
package [22, 23] combining charged tracks measured in the central tracking
detectors, ITS and TPC, and neutral fragments measured with the EMCal [17].
Tracks from the combined ITS and TPC track reconstruction algorithm are used.
Quality criteria for track selection follow the same strategy as in [24]. The
tracking efficiency is 70% for tracks with a transverse momentum
$p_{\mathrm{T,track}}=0.15$ GeV/$c$ and increases to 85% at
$p_{\mathrm{T,track}}=1$ GeV/$c$ and above. The $p_{\mathrm{T}}$ resolution of
tracks is 0.8% (3.8%) for $p_{\mathrm{T,track}}=1$ GeV/$c$ ($50$ GeV/$c$).
EMCal clusters are formed by a clustering algorithm that combines signals from
adjacent EMCal towers, with cluster size limited by the requirement that each
cluster contains only one local energy maximum. Energy deposited by charged
particles in the EMCal is subtracted from the measured energy in the EMCal
clusters which prevents counting the charged energy twice [25, 20]. ALICE also
reconstructs jets from charged particles only. These jets are referred to as
‘charged jets’, while jets reconstructed from charged and neutral fragments
are called ‘full jets’ in this letter.

In this analysis, anti-$k_{\rm T}$ jets are reconstructed using the boost-
invariant $p_{\mathrm{T}}$ recombination scheme and a jet resolution parameter
of $R=0.4$. A jet is only accepted if it is fully contained in the acceptance
in which the constituents are measured: for charged jets in the full azimuth
and $|\eta_{\mathrm{jet}}^{\mathrm{ch}}|<0.9-R$ while for full jets
$1.4+R<\varphi_{\mathrm{jet}}^{\mathrm{ch+ne}}<\pi-R$ and
$|\eta_{\mathrm{jet}}^{\mathrm{ch+ne}}|<0.7-R$. It was verified that reducing
the acceptance with 0.05 on all edges, in $\eta_{\mathrm{jet}}$ and
$\varphi_{\mathrm{jet}}$, has a negligible effect on the measurement. Tracks
with $p_{\mathrm{T,track}}>0.15$ GeV/$c$ and neutral constituents with $E_{\rm
T}>0.3$ GeV are considered. The minimum required area for jets with a
resolution parameter $R=0.4$ is equal to $0.3$ ($\approx 60\%$ of the area of
a rigid cone with $R=0.4$). This selection does not affect the jet finding
efficiency for jets (full and charged) with transverse momentum
$p_{\mathrm{T,jet}}>15$ GeV/$c$. In addition, jets containing a track with
$p_{\mathrm{T,track}}>100$ GeV/$c$, for which the track momentum resolution
exceeds 6.5%, are tagged and rejected. This last requirement has negligible
effect in the reported range of jet momenta. The measurement is corrected to
particle level as will be explained in Section 3.3.

The measured transverse momentum of the anti-$k_{\rm T}$ jet is corrected for
the contribution of the underlying event by subtracting the average background
momentum density, $\rho$ for full jets and $\rho_{{\mathrm{ch}}}$ for charged
jets, multiplied by the area of the considered jet. The contribution of the
underlying event to the charged jets is estimated using clusters reconstructed
with the $k_{\rm T}$ jet algorithm using only charged tracks. This is achieved
by calculating event-by-event the median charged background density,
$\rho_{{\mathrm{ch}}}$, from all $k_{\rm T}$ clusters in the event with in
addition a correction for the sparsely populated p–Pb events [26, 27]. The
average $\rho_{{\mathrm{ch}}}$ in minimum-bias events is equal to $1.9$
GeV/$c$ for the 0–40% multiplicity class and $0.7$ GeV/$c$ for the 40–100%
event multiplicity class with a negligible statistical uncertainty in both
cases. Finally, $\rho_{{\mathrm{ch}}}$ is multiplied by a scale factor to
account for the neutral energy to estimate $\rho$. The scale factor is
determined by measuring the ratio between the energy of all the EMCal clusters
and the charged tracks pointing into the EMCal acceptance in the minimum-bias
event sample. The extracted scale factor $1.28$ is independent of event
multiplicity. The influence of background fluctuations is quantified and
corrected for on an inclusive basis, see Section 3.3.

###  3.2 Dijet $\mathbf{{\it k}_{Ty}}$

Each measured full jet is correlated with the charged jet of highest
transverse momentum in the opposite hemisphere. Only pairs for which the full
jet has a larger transverse momentum than the associated charged jet are
considered. Furthermore only dijets pairs with
$|\Delta\varphi_{{\mathrm{dijet}}}-\pi|<\pi/3$, with
$\Delta\varphi_{{\mathrm{dijet}}}$ the angle between the jet axis of the full
and charged jet, are considered in the analysis. The selection in
$\Delta\varphi_{{\mathrm{dijet}}}$ rejects 5–8% of the dijet pairs depending
on the kinematic selection of the full and associated charged jet. The
azimuthal acoplanarity of dijets is studied by measuring the transverse
component of the $k_{\rm T}$ vector of the dijet system, $k_{\rm Ty}$, defined
as

$k_{\rm
Ty}=p_{\mathrm{T,jet}}^{\mathrm{ch+ne}}\;{\mathrm{sin}}(\Delta\varphi_{{\mathrm{dijet}}}),$
(1)

with $p_{\mathrm{T,jet}}^{\mathrm{ch+ne}}$ the transverse momentum of the full
jet. It should be noted that this definition differs from the one used in
previous publications, for example [28]. Since
${\mathrm{d}}N{\mathrm{/}}{\mathrm{d}}k_{\rm Ty}$ is a symmetric distribution
around zero, $|k_{\rm Ty}|$ is reported throughout the paper. For events from
the minimum-bias sample full jets with
$p_{\mathrm{T,jet}}^{\mathrm{ch+ne}}>20$ GeV/$c$ are considered, while in the
jet-triggered data samples (see Section 2) only jets with
$p_{\mathrm{T,jet}}^{\mathrm{ch+ne}}>40$ GeV/$c$ for the low energy trigger
and $p_{\mathrm{T,jet}}^{\mathrm{ch+ne}}>60$ GeV/$c$ for the higher energy
trigger are used. In these kinematic regimes the triggers are fully efficient
and no fragmentation bias is observed with respect to the minimum-bias jet
sample. The $|k_{\rm Ty}|$ distributions are reported at particle level
involving a correction for detector effects and $p_{\mathrm{T}}$-smearing due
to the underlying event, see Section 3.3.

The dijet sample is biased due to the requirement that the full jet has a
larger transverse momentum than the associated charged jet, while the full jet
momentum is used to estimate $k_{\rm Ty}$. In an unbiased measurement, the
full jet would correspond to the leading jet of the event in only 50% of the
cases. A PYTHIA [29, 30] study was performed in which the particle-level jet
was defined as the jet containing all final state particles (no kinematic
selection on constituents and full azimuthal acceptance). Applying the
selection of this analysis to detector-level reconstructed jets, results in a
correct tagging of the leading jet in 70% of the dijet events. This results in
a slightly harder $|k_{\rm Ty}|$ distribution with a 10% smaller yield at low
$|k_{\rm Ty}|$ and 20% higher yield at large $|k_{\rm Ty}|$. The results of
the p–Pb data analysis will be compared to particle-level PYTHIA with the same
dijet selection incorporating the mentioned bias.

The dijet acoplanarity is measured as a function of the transverse momentum of
the full jet while the kinematic interval of the associated charged jet is
also varied to explore $k_{\rm Ty}$ for more or less balanced dijets in
transverse momentum. In addition $k_{\rm Ty}$ distributions are also presented
for two event multiplicity classes.

###  3.3 Corrections and Systematic Uncertainties

The measured dijet $|k_{\rm Ty}|$ distributions are corrected to the particle
level, defined as the dijet $|k_{\rm Ty}|$ from jets clustered from all prompt
particles produced in the collision including all decay products, except those
from weak decays of light flavor hadrons and muons. Both full and charged jets
are accepted at particle level in the full azimuthal acceptance and in the
pseudorapidity range of $|\eta_{{\mathrm{jet}}}|<0.5$. The correction to
particle level is based on a data-driven method to correct for the influence
of the underlying event fluctuations and on simulated PYTHIA events (tune
Perugia-2011 [31]) transported through the ALICE detectors layout with GEANT3
[32]. The correction procedure takes into account the $p_{\mathrm{T}}$ and
angular resolution of the measured dijets.

Detector-level jets are defined as jets reconstructed from reconstructed
tracks and EMCal clusters after subtraction of the charged energy deposits.
The jet energy scale and resolution are affected by unmeasured particles
(predominantly ${\mathrm{K}}^{0}_{{\mathrm{L}}}$ and neutrons), fluctuations
of the energy deposit by charged tracks in the EMCal, the EMCal energy scale
and the charged particle tracking efficiency and $p_{\mathrm{T}}$ resolution.
A response matrix as a function of $p_{\mathrm{T,jet}}$ of the full and
associated charged jet, $\Delta\varphi_{{\mathrm{dijet}}}$ and $k_{\rm Ty}$ is
created after matching the detector-level to the particle-level jets as
described in [33].

The $p_{\mathrm{T}}$-smearing due to fluctuations of the underlying event is
estimated with the random-cones technique which is also applied in the
analysis of Pb–Pb data [24]. Cones with a radius equal to the resolution
parameter $R$ are placed in the measured p–Pb events at random positions in
the $\eta-\varphi$ plane ensuring the cone is fully contained in the detector
acceptance. The fluctuations of the background are characterized by the
difference between the summed $p_{\mathrm{T}}$ of all the tracks and clusters
in the random cone (RC) and the estimated background:
${\mathrm{\delta}}p_{\mathrm{T}}=\sum_{i}^{{\mathrm{RC}}}p_{\rm
T,i}-A\cdot\rho$, where $A$ is the area of the random cone ($A=\pi R^{2}$) and
the subscript $i$ indicates a cluster or track pointing inside the random
cone. A random cone can overlap with a jet but to avoid oversampling in small
systems like p–Pb, a partial exclusion of overlap with the leading jet in the
event is applied. This is achieved by excluding random cones overlapping with
a leading jet with a given probability, $p=1/N_{{\mathrm{coll}}}$ where
$N_{{\mathrm{coll}}}$ is the number of binary collisions.
$N_{{\mathrm{coll}}}$ is taken from estimates applying a Glauber fit to the
multiplicity measured in the V0A detector resulting in values between $14.7$
and $1.52$ depending on the event activity measured in the V0A detector. The
width of the background fluctuations for full (charged) jets varies between
$2.12$ ($1.59$) and $0.73$ ($0.56$) GeV/$c$ depending on the multiplicity of
the event.

The influence of background fluctuations is added to the response extracted
from detector simulation through a Monte Carlo model assuming that the
background fluctuation for the full and associated charged jet are
uncorrelated within 20% wide bins of V0A multiplicity classes. Within these
selected multiplicity classes the variation of the background fluctuations is
negligible. Since the Monte Carlo model does not generate full events and only
accounts for the ${\mathrm{\delta}}p_{\mathrm{T}}$ smearing on a jet-by-jet
basis, additional jet finding inefficiencies and worsening of angular
resolution due to the background fluctuation are not taken into account. These
effects are negligible since the contribution of the underlying event to the
jets in p–Pb collisions is small. No correction for the angular resolution of
the charged jet due to missing neutral fragments is applied. This effect
increases the width of $\Delta\varphi_{{\mathrm{dijet}}}$ by $\sim 0.03$ and
is present, and of the same magnitude, in the p–Pb data and the PYTHIA
reference.

The most probable correction to the jet energy, taking into account detector
effects and background fluctuations, for fully reconstructed jets is 28% at
$p_{\mathrm{T,jet}}^{\mathrm{ch+ne}}=20$ GeV/$c$ and decreases to 20% for jets
with $p_{\mathrm{T,jet}}^{\mathrm{ch+ne}}>40$ GeV/$c$. The uncertainty on the
jet energy scale is evaluated by changing the tracking efficiency in data and
full detector simulation [34], varying the double counting correction for the
hadronic energy deposit in the EMCal and by using different estimates of the
underlying-event fluctuations. The final uncertainty on the jet energy scale
is 4%. The jet energy resolution for full jets is 22% at
$p_{\mathrm{T,jet}}^{\mathrm{ch+ne}}=20$ GeV/$c$ and decreases gradually to
18% at $p_{\mathrm{T,jet}}^{\mathrm{ch+ne}}=120$ GeV/$c$. The influence of the
uncertainties on the jet energy scale and resolution on the dijet $|k_{\rm
Ty}|$ measurement are discussed in the following.

The measured $|k_{\rm Ty}|$ distributions are corrected to the particle level
by applying bin-by-bin correction factors, which are parametrized by a linear
fit to the ratio between the particle- and detector-level $|k_{\rm Ty}|$
distributions for a given dijet selection. The correction factors take into
account the effects of feed-in and feed-out of the selected kinematic and
angular intervals of the full and associated charged jets. These effects
slightly change the shape of the $|k_{\rm Ty}|$ distributions resulting in
correction factors which vary between $0.9$ for small $|k_{\rm Ty}|$ to $1.2$
at large $|k_{\rm Ty}|$. The correction is relatively small, because while
feed-in from lower $p_{\mathrm{T,jet}}^{\mathrm{ch+ne}}$ narrows the $|k_{\rm
Ty}|$ distribution, feed-in from higher $p_{\mathrm{T,jet}}^{\mathrm{ch+ne}}$
broadens the distribution resulting in a cancellation. Similarly the feed-out
to high and low $p_{\mathrm{T,jet}}^{\mathrm{ch+ne}}$ has a small effect on
the observable. By using a linear fit to the correction factors the
statistical fluctuations of the detector simulation are not propagated to the
measurement. The 95% confidence limit of the parametrization using the linear
fit is included in the systematic uncertainty of the measurement. Correction
factors are extracted as a function of V0A event multiplicity class and
kinematic intervals of the full and associated charged jet.

The dominant systematic uncertainty on the measurement originates from the
extraction of bin-by-bin correction factors. The uncertainty of the
parametrization of the correction factors results in 10–20% correlated
systematic uncertainty on the dijet $|k_{\rm Ty}|$ yields. An additional 2.5%
uncertainty arises from the uncertainty on the tracking efficiency which is
4%. Systematic uncertainties originating from the charged hadron energy
deposit in EMCal towers, background fluctuations and the average momentum
density $\rho$ were evaluated and found to be negligible.

##  4 Results

The dijet $|k_{\rm Ty}|$ distributions are presented as functions of the full-
jet transverse momentum, the transverse momentum of the associated charged jet
and event multiplicity classes. The results are compared to the predictions of
the PYTHIA8.176 event generator with tune 4C and $K=0.7$ [29, 30]. This tune
has been found to give a reasonable description of jet production at the LHC.
The final state particles are shifted in pseudorapidity with $\eta_{\rm
shift}=-0.465$ to mimic the rapidity shift of the laboratory frame due to the
energy difference of the proton and Pb beams.

###  4.1 Evolution with full-jet transverse momentum

Figure 1 shows the corrected $|k_{\rm Ty}|$ distributions for several
kinematic intervals for the full jet, from
$p_{\mathrm{T,jet}}^{\mathrm{ch+ne}}=20$ GeV/$c$ to
$p_{\mathrm{T,jet}}^{\mathrm{ch+ne}}=120$ GeV/$c$, in the 0–40% V0A
multiplicity class. The associated charged jet has a minimum transverse
momentum, $p_{\mathrm{T,assoc\;jet}}^{\mathrm{ch}}$, of 15 GeV/$c$ and is
always of lower transverse momentum than the full jet. The mean $|k_{\rm Ty}|$
increases with the transverse momentum of the full jet.

Figure 1: Dijet $|k_{\rm Ty}|$ distributions in p–Pb collisions in the 0-40%
V0A multiplicity event class for several kinematic intervals of the full jet
($p_{\mathrm{T,jet}}^{\mathrm{ch+ne}}$). The measurement is compared to
PYTHIA8 (tune 4C, $K=0.7$) with and without initial state radiation. The lower
panels show the ratio between the measurement and PYTHIA8 including initial
state radiation.

Increasing the transverse momentum of the full jet extends the kinematic reach
of $|k_{\rm Ty}|$ by opening phase-space for more gluon radiation. This
results in a harder $|k_{\rm Ty}|$ distribution which drops at large $|k_{\rm
Ty}|$ because the kinematic limit $|k_{\rm Ty}|_{\rm
max}=p_{{\mathrm{T,jet,max}}}^{{\mathrm{ch+ne}}}\;{\mathrm{sin}}(2\pi/3)$ is
reached. The p–Pb data points and the PYTHIA8 calculation show a similar
dependence on $p_{\mathrm{T,jet}}^{\mathrm{ch+ne}}$. The lower panels of Fig.
1 show the ratio between data and PYTHIA8, including initial state radiation
(ISR), which is observed to be consistent with unity for all transverse
momentum ranges studied. In the upper panels PYTHIA without the initial state
radiation option is shown in addition (dashed line). Without ISR the amount of
QCD radiation (which includes NLO corrections) is reduced, resulting in a
steeper $|k_{\rm Ty}|$ spectrum. The effect is most pronounced for the
$p_{\mathrm{T,jet}}^{\mathrm{ch+ne}}>40$ GeV/$c$ where the p–Pb measurement is
in agreement with full PYTHIA simulation but differs significantly from PYTHIA
without ISR. This observation suggests that the dijet $|k_{\rm Ty}|$ spectrum
for large $Q^{2}$ processes is highly sensitive to the increased available
phase-space of QCD radiation processes. Measurements presented in [14] of the
dijet transverse momentum imbalance for more energetic jets than the
measurement presented here also show results which are comparable to simulated
pp reference and independent of the forward transverse energy.

###  4.2 Evolution with event multiplicity and $\mathbf{{\it
p}_{T,assoc\;jet}^{ch}}$

In addition to the measurement of $|k_{\rm Ty}|$ in the highest multiplicity
p–Pb events, the $|k_{\rm Ty}|$ distribution is also measured in the lower
multiplicity V0A event class 40-100%. If strong nuclear effects are present
they are expected to be stronger in the high multiplicity events due to the
larger number of participants in the collision. A comparison is shown in the
left panel of Fig. 2. The systematic uncertainties between the two
measurements are fully correlated since they originate from the uncertainty on
the jet energy scale of the full jet. The consistency between the $|k_{\rm
Ty}|$ distributions in the high and low multiplicity event class was evaluated
by taking the ratio and performing a constant fit taking into account only the
statistical errors. The fit is within 1.2$\sigma$ consistent with unity. This
result shows that in the measured kinematical region, possible nuclear matter
effects and/or shadowing in the $|k_{\rm Ty}|$ distributions in p–Pb
collisions are not observed for dijets at midrapidity.

Figure 2: Distributions of $|k_{\rm Ty}|$ for two V0A event classes (left
panel) and three $p_{\mathrm{T,assoc\;jet}}^{\mathrm{ch}}$ ranges (right
panel).

The sensitivity to dijet acoplanarity is enhanced by selecting more
$p_{\mathrm{T}}$ imbalanced jet pairs. The $|k_{\rm Ty}|$ distribution for
full jets with $70<p_{\mathrm{T,jet}}^{\mathrm{ch+ne}}<120$ GeV/$c$ for
various $p_{\mathrm{T,assoc\;jet}}^{\mathrm{ch}}$ ranges is shown in the right
panel of Fig. 2. The $|k_{\rm Ty}|$ distribution tends to become steeper if
jets are more balanced indicating that the influence of QCD radiation
decreases. This behavior supports the previous observation (Sec. 4.1) that the
dijet $|k_{\rm Ty}|$ observable for highly energetic jets is over a wide range
of $|k_{\rm Ty}|$ mainly sensitive to QCD radiation processes rather than
elastic scatterings.

###  4.3 Evolution and characterization via $\mathbf{\langle|{\it
k}_{Ty}|\rangle}$

The measured $|k_{\rm Ty}|$ distributions are further characterized by
reporting the mean ($\langle|k_{\rm Ty}|\rangle$) of the distribution. To
avoid that the extracted moment is biased by statistical fluctuations for
large values of $|k_{\rm Ty}|$, the distributions are extrapolated using a
template generated with PYTHIA8 (tune 4C, $K=0.7$), which agrees well with the
p–Pb measurement (see Fig. 1). The PYTHIA $|k_{\rm Ty}|$ distribution is
normalized to minimize the $\chi^{2}$ between data and PYTHIA. The transition
from the data to the normalized template is fixed at 60% of the kinematic
limit $|k_{\rm Ty}|_{\rm max}$. The transition point is varied to estimate the
systematic uncertainty from this extrapolation procedure. In addition, the
normalization of the PYTHIA template is varied by one standard deviation of
the fit uncertainty. This results in an additional systematic uncertainty on
the extraction of $\langle|k_{\rm Ty}|\rangle$. For low
$p_{\mathrm{T,jet}}^{\mathrm{ch+ne}}$ the uncertainty on the extracted mean is
equal to 2.9% and increases to 8.1% for the highest
$p_{\mathrm{T,jet}}^{\mathrm{ch+ne}}$ values.

Figure 3: Mean of the $|k_{\rm Ty}|$ distributions as a function of the full
jet transverse momentum $p_{\mathrm{T,jet}}^{\mathrm{ch+ne}}$ (left) and the
associated charged jet transverse momentum
$p_{\mathrm{T,assoc\;jet}}^{\mathrm{ch}}$ (right) compared to PYTHIA8.

The left panel of Fig. 3 shows the mean of the measured $|k_{\rm Ty}|$
distributions as a function of the full jet transverse momentum and is
compared to the PYTHIA values. The measured moment in p–Pb collisions agrees
within the uncertainties of the measurement with the PYTHIA8 expectation. The
mean increases with $p_{\mathrm{T,jet}}^{\mathrm{ch+ne}}$ since the additional
$k_{\rm T}$ due to radiative QCD processes increases with
$p_{\mathrm{T,jet}}^{\mathrm{ch+ne}}$.

| 0-40% | 40-100% | PYTHIA8 pp
---|---|---|---
$\langle|k_{\rm Ty}|\rangle$ (GeV/$c$) | $14.7\pm 0.8\pm 0.3$ | $13.6\pm 1.1\pm 0.5$ | $15.1\pm 0.1$
Table 1: Mean of the $|k_{\rm Ty}|$ distributions for
$60<p_{\mathrm{T,jet}}^{\mathrm{ch+ne}}<80$ GeV/$c$ and
$15<p_{\mathrm{T,assoc\;jet}}^{\mathrm{ch}}<p_{\mathrm{T,jet}}^{\mathrm{ch+ne}}$
GeV/$c$ in a high (0-40%) and low (40-100%) V0A multiplicity event class. The
first quoted uncertainty is statistical while the second is systematic. The
last column corresponds to the values from the PYTHIA8 calculation at particle
level with the same kinematic selection. The uncertainty on the PYTHIA
calculation is statistical.

The right panel of Fig. 3 shows the evolution of $\langle|k_{\rm Ty}|\rangle$
as a function of $p_{\mathrm{T,assoc\;jet}}^{\mathrm{ch}}$ for
$70<p_{\mathrm{T,jet}}^{\mathrm{ch+ne}}<120$ GeV/$c$. The mean,
$\langle|k_{\rm Ty}|\rangle$, is compared to the earlier presented PYTHIA8
tune in Sec. 4.1 and is in agreement within the uncertainties of the
measurement. The mean for $60<p_{\mathrm{T,jet}}^{\mathrm{ch+ne}}<80$ GeV/$c$
is reported for two multiplicity event classes in Tab. 1. No significant
difference is observed as a function of the multiplicity measured with V0A.

##  5 Conclusion

The dijet acoplanarity in p–Pb collisions was studied by measuring dijet
transverse momentum $|k_{\rm Ty}|$. The evolution of $|k_{\rm Ty}|$ as
function of the transverse momentum of the full jet, associated charged jet
and event multiplicity was presented. The $|k_{\rm Ty}|$ spectra for different
full and associated charged jet transverse momentum ranges in the 0-40% V0A
event multiplicity class were found consistent with the PYTHIA prediction. The
observed increase with jet energy from the mean $|k_{\rm Ty}|$ of $6.6\pm
0.4{\mathrm{(stat.)}}\pm 0.2{\mathrm{(syst.)}}$ GeV/$c$ to $18.8\pm
1.3{\mathrm{(stat.)}}\pm 1.5{\mathrm{(syst.)}}$ GeV/$c$ as well as the
observed narrowing of $|k_{\rm Ty}|$ for more balanced jets suggests that the
dijet $|k_{\rm Ty}|$ spectrum for large $Q^{2}$ processes is mainly sensitive
to the increased available phase-space for QCD radiation processes.
Furthermore the dijet acoplanarity was found to be consistent (within
$1.2\sigma$) in the two event multiplicity classes analyzed in this study,
indicating that in the measured kinematical region no strong nuclear matter
effects in p–Pb collisions are observed. Since these results indicate that
nuclear $k_{\rm T}$ effects are small, the $p_{\mathrm{T}}$ imbalance of jet
correlations in Pb-Pb results [35, 36] are unlikely to originate from multiple
scatterings in the nuclear target.

## Acknowledgements

The ALICE Collaboration would like to thank all its engineers and technicians
for their invaluable contributions to the construction of the experiment and
the CERN accelerator teams for the outstanding performance of the LHC complex.
The ALICE Collaboration gratefully acknowledges the resources and support
provided by all Grid centres and the Worldwide LHC Computing Grid (WLCG)
collaboration. The ALICE Collaboration acknowledges the following funding
agencies for their support in building and running the ALICE detector: State
Committee of Science, World Federation of Scientists (WFS) and Swiss Fonds
Kidagan, Armenia, Conselho Nacional de Desenvolvimento Científico e
Tecnológico (CNPq), Financiadora de Estudos e Projetos (FINEP), Fundação de
Amparo à Pesquisa do Estado de São Paulo (FAPESP); National Natural Science
Foundation of China (NSFC), the Chinese Ministry of Education (CMOE) and the
Ministry of Science and Technology of China (MSTC); Ministry of Education and
Youth of the Czech Republic; Danish Natural Science Research Council, the
Carlsberg Foundation and the Danish National Research Foundation; The European
Research Council under the European Community’s Seventh Framework Programme;
Helsinki Institute of Physics and the Academy of Finland; French CNRS-IN2P3,
the ‘Region Pays de Loire’, ‘Region Alsace’, ‘Region Auvergne’ and CEA,
France; German Bundesministerium fur Bildung, Wissenschaft, Forschung und
Technologie (BMBF) and the Helmholtz Association; General Secretariat for
Research and Technology, Ministry of Development, Greece; Hungarian Orszagos
Tudomanyos Kutatasi Alappgrammok (OTKA) and National Office for Research and
Technology (NKTH); Department of Atomic Energy and Department of Science and
Technology of the Government of India; Istituto Nazionale di Fisica Nucleare
(INFN) and Centro Fermi - Museo Storico della Fisica e Centro Studi e Ricerche
”Enrico Fermi”, Italy; MEXT Grant-in-Aid for Specially Promoted Research,
Japan; Joint Institute for Nuclear Research, Dubna; National Research
Foundation of Korea (NRF); Consejo Nacional de Cienca y Tecnologia (CONACYT),
Direccion General de Asuntos del Personal Academico(DGAPA), México, :Amerique
Latine Formation academique – European Commission(ALFA-EC) and the EPLANET
Program (European Particle Physics Latin American Network) Stichting voor
Fundamenteel Onderzoek der Materie (FOM) and the Nederlandse Organisatie voor
Wetenschappelijk Onderzoek (NWO), Netherlands; Research Council of Norway
(NFR); National Science Centre, Poland; Ministry of National
Education/Institute for Atomic Physics and Consiliul Naţional al Cercetării
Ştiinţifice - Executive Agency for Higher Education Research Development and
Innovation Funding (CNCS-UEFISCDI) - Romania; Ministry of Education and
Science of Russian Federation, Russian Academy of Sciences, Russian Federal
Agency of Atomic Energy, Russian Federal Agency for Science and Innovations
and The Russian Foundation for Basic Research; Ministry of Education of
Slovakia; Department of Science and Technology, South Africa; Centro de
Investigaciones Energeticas, Medioambientales y Tecnologicas (CIEMAT),
E-Infrastructure shared between Europe and Latin America (EELA), Ministerio de
Economía y Competitividad (MINECO) of Spain, Xunta de Galicia (Consellería de
Educación), Centro de Aplicaciones Tecnológicas y Desarrollo Nuclear (CEADEN),
Cubaenergía, Cuba, and IAEA (International Atomic Energy Agency); Swedish
Research Council (VR) and Knut $\&$ Alice Wallenberg Foundation (KAW); Ukraine
Ministry of Education and Science; United Kingdom Science and Technology
Facilities Council (STFC); The United States Department of Energy, the United
States National Science Foundation, the State of Texas, and the State of Ohio;
Ministry of Science, Education and Sports of Croatia and Unity through
Knowledge Fund, Croatia. Council of Scientific and Industrial Research (CSIR),
New Delhi, India

## References

  * [1] R. Feynman, R. Field, and G. Fox, “Correlations Among Particles and Jets Produced with Large Transverse Momenta” Nucl.Phys. B128 (1977) 1–65.
  * [2] L. Apanasevich et al., “Evidence for parton ${k}_{T}$ effects in high- ${p}_{T}$ particle production” Phys. Rev. Lett. 81 (Sep, 1998) 2642–2645.
  * [3] L. Apanasevich et al., “${k}_{T}$ effects in direct-photon production” Phys. Rev. D 59 (Feb, 1999) 074007.
  * [4] D0 Collaboration, V. Abazov et al., “Measurement of dijet azimuthal decorrelations at central rapidities in $p\bar{p}$ collisions at $\sqrt{s}=1.96$ TeV” Phys.Rev.Lett. 94 (2005) 221801, arXiv:hep-ex/0409040 [hep-ex].
  * [5] CMS Collaboration, V. Khachatryan et al., “Dijet Azimuthal Decorrelations in $pp$ Collisions at $\sqrt{s}=7$ TeV” Phys.Rev.Lett. 106 (2011) 122003, arXiv:1101.5029 [hep-ex].
  * [6] ATLAS Collaboration, G. Aad et al., “Measurement of Dijet Azimuthal Decorrelations in $pp$ Collisions at $\sqrt{s}=7$ TeV” Phys.Rev.Lett. 106 (2011) 172002, arXiv:1102.2696 [hep-ex].
  * [7] J. Albacete, N. Armesto, R. Baier, G. Barnafoldi, J. Barrette, et al., “Predictions for p+Pb Collisions at $\sqrt{s_{NN}}$ = 5 TeV” Int.J.Mod.Phys. E22 (2013) 1330007, arXiv:1301.3395 [hep-ph].
  * [8] M. Gyulassy and M. Plumer, “Jet quenching in dense matter” Phys.Lett. B243 (1990) 432–438.
  * [9] R. Baier, Y. L. Dokshitzer, S. Peigne, and D. Schiff, “Induced gluon radiation in a QCD medium” Phys.Lett. B345 (1995) 277–286, arXiv:hep-ph/9411409 [hep-ph].
  * [10] ATLAS Collaboration, G. Aad et al., “Measurements of the Nuclear Modification Factor for Jets in Pb+Pb Collisions at $\sqrt{s_{\mathrm{NN}}}=2.76$ TeV with the ATLAS Detector” arXiv:1411.2357 [hep-ex].
  * [11] ATLAS Collaboration, G. Aad et al., “Measurement of inclusive jet charged-particle fragmentation functions in Pb+Pb collisions at $\sqrt{s_{NN}}=2.76$ TeV with the ATLAS detector” Phys.Lett. B739 (2014) 320–342, arXiv:1406.2979 [hep-ex].
  * [12] CMS Collaboration, S. Chatrchyan et al., “Measurement of jet fragmentation in PbPb and pp collisions at $\sqrt{s_{NN}}=2.76$ TeV” Phys.Rev. C90 no. 2, (2014) 024908, arXiv:1406.0932 [nucl-ex].
  * [13] CMS Collaboration, S. Chatrchyan et al., “Modification of jet shapes in PbPb collisions at $\sqrt{s_{NN}}=2.76$ TeV” Phys.Lett. B730 (2014) 243–263, arXiv:1310.0878 [nucl-ex].
  * [14] CMS Collaboration, S. Chatrchyan et al., “Studies of dijet transverse momentum balance and pseudorapidity distributions in pPb collisions at $\sqrt{s_{\mathrm{NN}}}=5.02$ TeV” Eur.Phys.J. C74 no. 7, (2014) 2951, arXiv:1401.4433 [nucl-ex].
  * [15] ALICE Collaboration, J. Adam et al., “Centrality dependence of particle production in p-Pb collisions at $\sqrt{s_{\rm NN}}$= 5.02 TeV” arXiv:1412.6828 [nucl-ex].
  * [16] ALICE Collaboration, B. Abelev et al., “Pseudorapidity density of charged particles in $p$-Pb at $\sqrt{s_{NN}}=5.02$ TeV” Phys.Rev.Lett. 110 (2013) 032301, arXiv:1210.3615 [nucl-ex].
  * [17] ALICE EMCal Collaboration, U. Abeysekara et al., “ALICE EMCal Physics Performance Report” arXiv:1008.0413 [physics.ins-det].
  * [18] ALICE Collaboration, K. Aamodt et al., “Alignment of the ALICE Inner Tracking System with cosmic-ray tracks” JINST 5 (2010) P03003, arXiv:1001.0502 [physics.ins-det].
  * [19] J. Alme, Y. Andres, H. Appelshauser, S. Bablok, N. Bialas, et al., “The ALICE TPC, a large 3-dimensional tracking device with fast readout for ultra-high multiplicity events” Nucl.Instrum.Meth. A622 (2010) 316–367, arXiv:1001.1950 [physics.ins-det].
  * [20] ALICE Collaboration, B. Abelev et al., “Performance of the ALICE Experiment at the CERN LHC” Int.J.Mod.Phys. A29 (2014) 1430044, arXiv:1402.4476 [nucl-ex].
  * [21] ALICE Collaboration, B. Abelev et al., “Multiplicity dependence of jet-like two-particle correlations in p-Pb collisions at $\sqrt{s_{NN}}$ = 5.02 TeV” Phys.Lett. B741 (2014) 38–50, arXiv:1406.5463 [nucl-ex].
  * [22] M. Cacciari, G. P. Salam, and G. Soyez, “FastJet User Manual” Eur.Phys.J. C72 (2012) 1896, arXiv:1111.6097 [hep-ph].
  * [23] M. Cacciari and G. P. Salam, “Dispelling the $n^{3}$ myth for the $k_{\mathrm{t}}$ jet-finder” Phys.Lett.B 641 (2006) 57–61, hep-ph/0512210.
  * [24] ALICE Collaboration, B. Abelev et al., “Measurement of Event Background Fluctuations for Charged Particle Jet Reconstruction in Pb-Pb collisions at $\sqrt{s_{NN}}=2.76$ TeV” JHEP 1203 (2012) 053, arXiv:1201.2423 [hep-ex].
  * [25] ALICE Collaboration, B. Abelev et al., “Measurement of the inclusive differential jet cross section in $pp$ collisions at $\sqrt{s}=2.76$ TeV” Phys.Lett. B722 (2013) 262–272, arXiv:1301.3475 [nucl-ex].
  * [26] CMS Collaboration, S. Chatrchyan et al., “Measurement of the underlying event activity in $pp$ collisions at $\sqrt{s}=0.9$ and 7 TeV with the novel jet-area/median approach” JHEP 1208 (2012) 130, arXiv:1207.2392 [hep-ex].
  * [27] ALICE Collaboration, J. Adam et al., “Measurement of charged jet production cross sections and nuclear modification in p-Pb collisions at $\sqrt{s_{\mathrm{NN}}}=5.02$ TeV” arXiv:1503.00681 [nucl-ex].
  * [28] A. Angelis et al., “A measurement of the transverse momenta of partons, and of jet fragmentation as a function of $\sqrt{s}$ in p-p collisions” Physics Letters B 97 no. 1, (1980) 163 – 168.
  * [29] T. Sjostrand, S. Mrenna, and P. Skands, “PYTHIA 6.4 physics and manual” JHEP 05 (2006) 026, hep-ph/0603175.
  * [30] T. Sjostrand, S. Mrenna, and P. Z. Skands, “A Brief Introduction to PYTHIA 8.1” Comput.Phys.Commun. 178 (2008) 852–867, arXiv:0710.3820 [hep-ph].
  * [31] P. Z. Skands, “Tuning Monte Carlo Generators: The Perugia Tunes” Phys.Rev. D82 (2010) 074018, arXiv:1005.3457 [hep-ph].
  * [32] R. Brun, F. Carminati, and S. Giani, “GEANT Detector Description and Simulation Tool”.
  * [33] ALICE Collaboration, B. Abelev et al., “Measurement of charged jet suppression in Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV” JHEP 1403 (2014) 013, arXiv:1311.0633 [nucl-ex].
  * [34] ALICE Collaboration, B. B. Abelev et al., “Charged jet cross sections and properties in proton-proton collisions at $\sqrt{s}$ = 7 TeV” arXiv:1411.4969 [nucl-ex].
  * [35] ATLAS Collaboration, G. Aad et al., “Observation of a Centrality-Dependent Dijet Asymmetry in Lead-Lead Collisions at $\sqrt{s_{NN}}=2.77$ TeV with the ATLAS Detector at the LHC” Phys.Rev.Lett. 105 (2010) 252303, arXiv:1011.6182 [hep-ex].
  * [36] CMS Collaboration, S. Chatrchyan et al., “Observation and studies of jet quenching in PbPb collisions at nucleon-nucleon center-of-mass energy = 2.76 TeV” Phys.Rev. C84 (2011) 024906, arXiv:1102.1957 [nucl-ex].

##  Appendix A The ALICE Collaboration

J. Adamorg39&D. Adamováorg82&M.M. Aggarwalorg86&G. Aglieri Rinellaorg36&M.
Agnelloorg110&N. Agrawalorg47&Z. Ahammedorg130&S.U. Ahnorg67&I.
Aimoorg93,org110&S. Aiolaorg135&M. Ajazorg16&A. Akindinovorg57&S.N.
Alamorg130&D. Aleksandrovorg99&B. Alessandroorg110&D. Alexandreorg101&R.
Alfaro Molinaorg63&A. Aliciorg104,org12&A. Alkinorg3&J. Almeorg37&T.
Altorg42&S. Altinpinarorg18&I. Altsybeevorg129&C. Alves Garcia Pradoorg118&C.
Andreiorg77&A. Andronicorg96&V. Anguelovorg92&J. Anielskiorg53&T.
Antičićorg97&F. Antinoriorg107&P. Antonioliorg104&L. Aphecetcheorg112&H.
Appelshäuserorg52&S. Arcelliorg28&N. Armestoorg17&R. Arnaldiorg110&T.
Aronssonorg135&I.C. Arseneorg22&M. Arslandokorg52&A. Augustinusorg36&R.
Averbeckorg96&M.D. Azmiorg19&M. Bachorg42&A. Badalàorg106&Y.W. Baekorg43&S.
Bagnascoorg110&R. Bailhacheorg52&R. Balaorg89&A. Baldisseriorg15&F. Baltasar
Dos Santos Pedrosaorg36&R.C. Baralorg60&A.M. Barbanoorg110&R. Barberaorg29&F.
Barileorg33&G.G. Barnaföldiorg134&L.S. Barnbyorg101&V. Barretorg69&P.
Bartaliniorg7&J. Bartkeorg115&E. Bartschorg52&M. Basileorg28&N. Bastidorg69&S.
Basuorg130&B. Bathenorg53&G. Batigneorg112&A. Batista Camejoorg69&B.
Batyunyaorg65&P.C. Batzingorg22&I.G. Beardenorg79&H. Beckorg52&C.
Beddaorg110&N.K. Beheraorg48,org47&I. Belikovorg54&F. Belliniorg28&H. Bello
Martinezorg2&R. Bellwiedorg120&R. Belmontorg133&E. Belmont-Morenoorg63&V.
Belyaevorg75&G. Bencediorg134&S. Beoleorg27&I. Berceanuorg77&A.
Bercuciorg77&Y. Berdnikovorg84&D. Berenyiorg134&R.A. Bertensorg56&D.
Berzanoorg36,org27&L. Betevorg36&A. Bhasinorg89&I.R. Bhatorg89&A.K.
Bhatiorg86&B. Bhattacharjeeorg44&J. Bhomorg126&L. Bianchiorg27,org120&N.
Bianchiorg71&C. Bianchinorg133,org56&J. Bielčíkorg39&J. Bielčíkováorg82&A.
Bilandzicorg79&S. Biswasorg78&S. Bjelogrlicorg56&F. Blancoorg10&D.
Blauorg99&C. Blumeorg52&F. Bockorg73,org92&A. Bogdanovorg75&H. Bøggildorg79&L.
Boldizsárorg134&M. Bombaraorg40&J. Bookorg52&H. Borelorg15&A. Borissovorg95&M.
Borriorg81&F. Bossúorg64&M. Botjeorg80&E. Bottaorg27&S. Böttgerorg51&P. Braun-
Munzingerorg96&M. Bregantorg118&T. Breitnerorg51&T.A. Brokerorg52&T.A.
Browningorg94&M. Brozorg39&E.J. Bruckenorg45&E. Brunaorg110&G.E. Brunoorg33&D.
Budnikovorg98&H. Bueschingorg52&S. Bufalinoorg110,org36&P. Buncicorg36&O.
Buschorg92,org126&Z. Butheleziorg64&J.T. Buxtonorg20&D. Caffarriorg36,org30&X.
Caiorg7&H. Cainesorg135&L. Calero Diazorg71&A. Calivaorg56&E. Calvo
Villarorg102&P. Cameriniorg26&F. Carenaorg36&W. Carenaorg36&J. Castillo
Castellanosorg15&A.J. Castroorg123&E.A.R. Casulaorg25&C. Cavicchioliorg36&C.
Ceballos Sanchezorg9&J. Cepilaorg39&P. Cerelloorg110&B. Changorg121&S.
Chapelandorg36&M. Chartierorg122&J.L. Charvetorg15&S. Chattopadhyayorg130&S.
Chattopadhyayorg100&V. Chelnokovorg3&M. Cherneyorg85&C. Cheshkovorg128&B.
Cheynisorg128&V. Chibante Barrosoorg36&D.D. Chinellatoorg119&P.
Chochulaorg36&K. Choiorg95&M. Chojnackiorg79&S. Choudhuryorg130&P.
Christakoglouorg80&C.H. Christensenorg79&P. Christiansenorg34&T.
Chujoorg126&S.U. Chungorg95&Z. Chunhuiorg56&C. Cicaloorg105&L.
Cifarelliorg12,org28&F. Cindoloorg104&J. Cleymansorg88&F. Colamariaorg33&D.
Colellaorg33&A. Colluorg25&M. Colocciorg28&G. Conesa Balbastreorg70&Z. Conesa
del Valleorg50&M.E. Connorsorg135&J.G. Contrerasorg39,org11&T.M.
Cormierorg83&Y. Corrales Moralesorg27&I. Cortés Maldonadoorg2&P.
Corteseorg32&M.R. Cosentinoorg118&F. Costaorg36&P. Crochetorg69&R. Cruz
Albinoorg11&E. Cuautleorg62&L. Cunqueiroorg36&T. Dahmsorg91&A.
Daineseorg107&A. Danuorg61&D. Dasorg100&I. Dasorg100,org50&S. Dasorg4&A.
Dashorg119&S. Dashorg47&S. Deorg118&A. De Caroorg31,org12&G. de
Cataldoorg103&J. de Cuvelandorg42&A. De Falcoorg25&D. De
Gruttolaorg12,org31&N. De Marcoorg110&S. De Pasqualeorg31&A.
Deistingorg96,org92&A. Delofforg76&E. Dénesorg134&G. D’Erasmoorg33&D. Di
Bariorg33&A. Di Mauroorg36&P. Di Nezzaorg71&M.A. Diaz Corcheroorg10&T.
Dietelorg88&P. Dillensegerorg52&R. Diviàorg36&Ø. Djuvslandorg18&A.
Dobrinorg56,org80&T. Dobrowolskiorg760&D. Domenicis Gimenezorg118&B.
Dönigusorg52&O. Dordicorg22&A.K. Dubeyorg130&A. Dublaorg56&L. Ducrouxorg128&P.
Dupieuxorg69&R.J. Ehlersorg135&D. Eliaorg103&H. Engelorg51&B.
Erazmusorg112,org36&F. Erhardtorg127&D. Eschweilerorg42&B. Espagnonorg50&M.
Estienneorg112&S. Esumiorg126&J. Eumorg95&D. Evansorg101&S. Evdokimovorg111&G.
Eyyubovaorg39&L. Fabbiettiorg91&D. Fabrisorg107&J. Faivreorg70&A.
Fantoniorg71&M. Faselorg73&L. Feldkamporg53&D. Feleaorg61&A.
Felicielloorg110&G. Feofilovorg129&J. Ferenceiorg82&A. Fernández
Téllezorg2&E.G. Ferreiroorg17&A. Ferrettiorg27&A. Festantiorg30&J.
Figielorg115&M.A.S. Figueredoorg122&S. Filchaginorg98&D. Finogeevorg55&F.M.
Fiondaorg103&E.M. Fioreorg33&M.G. Fleckorg92&M. Florisorg36&S.
Foertschorg64&P. Fokaorg96&S. Fokinorg99&E. Fragiacomoorg109&A.
Francesconorg36,org30&U. Frankenfeldorg96&U. Fuchsorg36&C. Furgetorg70&A.
Fursorg55&M. Fusco Girardorg31&J.J. Gaardhøjeorg79&M. Gagliardiorg27&A.M.
Gagoorg102&M. Gallioorg27&D.R. Gangadharanorg73&P. Ganotiorg87&C. Gaoorg7&C.
Garabatosorg96&E. Garcia-Solisorg13&C. Gargiuloorg36&P. Gasikorg91&M.
Germainorg112&A. Gheataorg36&M. Gheataorg61,org36&P. Ghoshorg130&S.K.
Ghoshorg4&P. Gianottiorg71&P. Giubellinoorg36&P. Giubilatoorg30&E. Gladysz-
Dziadusorg115&P. Glässelorg92&A. Gomez Ramirezorg51&P. González-Zamoraorg10&S.
Gorbunovorg42&L. Görlichorg115&S. Gotovacorg114&V. Grabskiorg63&L.K.
Graczykowskiorg132&A. Grelliorg56&A. Grigorasorg36&C. Grigorasorg36&V.
Grigorievorg75&A. Grigoryanorg1&S. Grigoryanorg65&B. Grinyovorg3&N.
Grionorg109&J.F. Grosse-Oetringhausorg36&J.-Y. Grossiordorg128&R.
Grossoorg36&F. Guberorg55&R. Guernaneorg70&B. Guerzoniorg28&K.
Gulbrandsenorg79&H. Gulkanyanorg1&T. Gunjiorg125&A. Guptaorg89&R.
Guptaorg89&R. Haakeorg53&Ø. Haalandorg18&C. Hadjidakisorg50&M. Haiducorg61&H.
Hamagakiorg125&G. Hamarorg134&L.D. Hanrattyorg101&A. Hansenorg79&J.W.
Harrisorg135&H. Hartmannorg42&A. Hartonorg13&D. Hatzifotiadouorg104&S.
Hayashiorg125&S.T. Heckelorg52&M. Heideorg53&H. Helstruporg37&A.
Herghelegiuorg77&G. Herrera Corralorg11&B.A. Hessorg35&K.F. Hetlandorg37&T.E.
Hildenorg45&H. Hillemannsorg36&B. Hippolyteorg54&P. Hristovorg36&M.
Huangorg18&T.J. Humanicorg20&N. Hussainorg44&T. Hussainorg19&D.
Hutterorg42&D.S. Hwangorg21&R. Ilkaevorg98&I. Ilkivorg76&M. Inabaorg126&C.
Ionitaorg36&M. Ippolitovorg75,org99&M. Irfanorg19&M. Ivanovorg96&V.
Ivanovorg84&V. Izucheevorg111&P.M. Jacobsorg73&C. Jahnkeorg118&H.J.
Jangorg67&M.A. Janikorg132&P.H.S.Y. Jayarathnaorg120&C. Jenaorg30&S.
Jenaorg120&R.T. Jimenez Bustamanteorg62&P.G. Jonesorg101&H. Jungorg43&A.
Juskoorg101&P. Kalinakorg58&A. Kalweitorg36&J. Kaminorg52&J.H. Kangorg136&V.
Kaplinorg75&S. Karorg130&A. Karasu Uysalorg68&O. Karavichevorg55&T.
Karavichevaorg55&E. Karpechevorg55&U. Kebschullorg51&R. Keidelorg137&D.L.D.
Keijdenerorg56&M. Keilorg36&K.H. Khanorg16&M.M. Khanorg19&P. Khanorg100&S.A.
Khanorg130&A. Khanzadeevorg84&Y. Kharlovorg111&B. Kilengorg37&B.
Kimorg136&D.W. Kimorg43,org67&D.J. Kimorg121&H. Kimorg136&J.S. Kimorg43&M.
Kimorg43&M. Kimorg136&S. Kimorg21&T. Kimorg136&S. Kirschorg42&I. Kiselorg42&S.
Kiselevorg57&A. Kisielorg132&G. Kissorg134&J.L. Klayorg6&C. Kleinorg52&J.
Kleinorg92&C. Klein-Bösingorg53&A. Klugeorg36&M.L. Knichelorg92&A.G.
Knospeorg116&T. Kobayashiorg126&C. Kobdajorg113&M. Kofaragoorg36&M.K.
Köhlerorg96&T. Kolleggerorg42,org96&A. Kolojvariorg129&V. Kondratievorg129&N.
Kondratyevaorg75&E. Kondratyukorg111&A. Konevskikhorg55&C.
Kouzinopoulosorg36&O. Kovalenkoorg76&V. Kovalenkoorg129&M.
Kowalskiorg36,org115&S. Koxorg70&G. Koyithatta Meethaleveeduorg47&J.
Kralorg121&I. Králikorg58&A. Kravčákováorg40&M. Krelinaorg39&M. Kretzorg42&M.
Krivdaorg101,org58&F. Krizekorg82&E. Kryshenorg36&M. Krzewickiorg96,org42&A.M.
Kuberaorg20&V. Kučeraorg82&T. Kugathasanorg36&C. Kuhnorg54&P.G. Kuijerorg80&I.
Kulakovorg42&J. Kumarorg47&L. Kumarorg78,org86&P. Kurashviliorg76&A.
Kurepinorg55&A.B. Kurepinorg55&A. Kuryakinorg98&S. Kushpilorg82&M.J.
Kweonorg49&Y. Kwonorg136&S.L. La Pointeorg110&P. La Roccaorg29&C. Lagana
Fernandesorg118&I. Lakomovorg36,org50&R. Langoyorg41&C. Laraorg51&A.
Lardeuxorg15&A. Lattucaorg27&E. Laudiorg36&R. Leaorg26&L. Leardiniorg92&G.R.
Leeorg101&S. Leeorg136&I. Legrandorg36&R.C. Lemmonorg81&V. Lentiorg103&E.
Leograndeorg56&I. León Monzónorg117&M. Leoncinoorg27&P. Lévaiorg134&S.
Liorg7,org69&X. Liorg14&J. Lienorg41&R. Lietavaorg101&S. Lindalorg22&V.
Lindenstruthorg42&C. Lippmannorg96&M.A. Lisaorg20&H.M. Ljunggrenorg34&D.F.
Lodatoorg56&P.I. Loenneorg18&V.R. Logginsorg133&V. Loginovorg75&C.
Loizidesorg73&X. Lopezorg69&E. López Torresorg9&A. Loweorg134&P.
Luettigorg52&M. Lunardonorg30&G. Luparelloorg26,org56&P.H.F.N.D. Luzorg118&A.
Maevskayaorg55&M. Magerorg36&S. Mahajanorg89&S.M. Mahmoodorg22&A.
Maireorg54&R.D. Majkaorg135&M. Malaevorg84&I. Maldonado Cervantesorg62&L.
Malininaorg65&D. Mal’Kevichorg57&P. Malzacherorg96&A. Mamonovorg98&L.
Manceauorg110&V. Mankoorg99&F. Mansoorg69&V. Manzariorg103,org36&M.
Marchisoneorg27&J. Marešorg59&G.V. Margagliottiorg26&A. Margottiorg104&J.
Marguttiorg56&A. Marínorg96&C. Markertorg116&M. Marquardorg52&N.A.
Martinorg96&J. Martin Blancoorg112&P. Martinengoorg36&M.I. Martínezorg2&G.
Martínez Garcíaorg112&M. Martinez Pedreiraorg36&Y. Martynovorg3&A.
Masorg118&S. Masciocchiorg96&M. Maseraorg27&A. Masoniorg105&L.
Massacrierorg112&A. Mastroserioorg33&H. Masuiorg126&A. Matyjaorg115&C.
Mayerorg115&J. Mazerorg123&M.A. Mazzoniorg108&D. Mcdonaldorg120&F.
Meddiorg24&A. Menchaca-Rochaorg63&E. Meninnoorg31&J. Mercado Pérezorg92&M.
Meresorg38&Y. Miakeorg126&M.M. Mieskolainenorg45&K. Mikhaylovorg57,org65&L.
Milanoorg36&J. Milosevicorg22,org131&L.M. Minerviniorg103,org23&A.
Mischkeorg56&A.N. Mishraorg48&D. Miśkowiecorg96&J. Mitraorg130&C.M.
Mituorg61&N. Mohammadiorg56&B. Mohantyorg130,org78&L. Molnarorg54&L. Montaño
Zetinaorg11&E. Montesorg10&M. Morandoorg30&D.A. Moreira De Godoyorg112&S.
Morettoorg30&A. Morrealeorg112&A. Morschorg36&V. Mucciforaorg71&E.
Mudnicorg114&D. Mühlheimorg53&S. Muhuriorg130&M. Mukherjeeorg130&H.
Müllerorg36&J.D. Mulliganorg135&M.G. Munhozorg118&S. Murrayorg64&L.
Musaorg36&J. Musinskyorg58&B.K. Nandiorg47&R. Naniaorg104&E. Nappiorg103&M.U.
Naruorg16&C. Nattrassorg123&K. Nayakorg78&T.K. Nayakorg130&S.
Nazarenkoorg98&A. Nedosekinorg57&L. Nellenorg62&F. Ngorg120&M.
Nicassioorg96&M. Niculescuorg61,org36&J. Niedzielaorg36&B.S. Nielsenorg79&S.
Nikolaevorg99&S. Nikulinorg99&V. Nikulinorg84&F. Noferiniorg104,org12&P.
Nomokonovorg65&G. Noorenorg56&J. Normanorg122&A. Nyaninorg99&J.
Nystrandorg18&H. Oeschlerorg92&S. Ohorg135&S.K. Ohorg66&A. Ohlsonorg36&A.
Okatanorg68&T. Okuboorg46&L. Olahorg134&J. Oleniaczorg132&A.C. Oliveira Da
Silvaorg118&M.H. Oliverorg135&J. Onderwaaterorg96&C. Oppedisanoorg110&A. Ortiz
Velasquezorg62&A. Oskarssonorg34&J. Otwinowskiorg96,org115&K. Oyamaorg92&M.
Ozdemirorg52&Y. Pachmayerorg92&P. Paganoorg31&G. Paićorg62&C.
Pajaresorg17&S.K. Palorg130&J. Panorg133&A.K. Pandeyorg47&D. Pantorg47&V.
Papikyanorg1&G.S. Pappalardoorg106&P. Pareekorg48&W.J. Parkorg96&S.
Parmarorg86&A. Passfeldorg53&V. Paticchioorg103&B. Paulorg100&T.
Peitzmannorg56&H. Pereira Da Costaorg15&E. Pereira De Oliveira Filhoorg118&D.
Peresunkoorg75,org99&C.E. Pérez Laraorg80&V. Peskovorg52&Y. Pestovorg5&V.
Petráčekorg39&V. Petrovorg111&M. Petroviciorg77&C. Pettaorg29&S.
Pianoorg109&M. Piknaorg38&P. Pillotorg112&O. Pinazzaorg104,org36&L.
Pinskyorg120&D.B. Piyarathnaorg120&M. Płoskońorg73&M. Planinicorg127&J.
Plutaorg132&S. Pochybovaorg134&P.L.M. Podesta-Lermaorg117&M.G.
Poghosyanorg85&B. Polichtchoukorg111&N. Poljakorg127&W. Poonsawatorg113&A.
Poporg77&S. Porteboeuf-Houssaisorg69&J. Porterorg73&J. Pospisilorg82&S.K.
Prasadorg4&R. Preghenellaorg36,org104&F. Prinoorg110&C.A. Pruneauorg133&I.
Pshenichnovorg55&M. Puccioorg110&G. Pudduorg25&P. Pujahariorg133&V.
Puninorg98&J. Putschkeorg133&H. Qvigstadorg22&A. Rachevskiorg109&S.
Rahaorg4&S. Rajputorg89&J. Rakorg121&A. Rakotozafindrabeorg15&L.
Ramelloorg32&R. Raniwalaorg90&S. Raniwalaorg90&S.S. Räsänenorg45&B.T.
Rascanuorg52&D. Ratheeorg86&K.F. Readorg123&J.S. Realorg70&K.
Redlichorg76&R.J. Reedorg133&A. Rehmanorg18&P. Reicheltorg52&M.
Reicherorg56&F. Reidtorg92,org36&X. Renorg7&R. Renfordtorg52&A.R.
Reolonorg71&A. Reshetinorg55&F. Rettigorg42&J.-P. Revolorg12&K.
Reygersorg92&V. Riabovorg84&R.A. Ricciorg72&T. Richertorg34&M. Richterorg22&P.
Riedlerorg36&W. Rieglerorg36&F. Riggiorg29&C. Risteaorg61&A. Rivettiorg110&E.
Roccoorg56&M. Rodríguez Cahuantziorg11,org2&A. Rodriguez Mansoorg80&K.
Røedorg22&E. Rogochayaorg65&D. Rohrorg42&D. Röhrichorg18&R. Romitaorg122&F.
Ronchettiorg71&L. Ronfletteorg112&P. Rosnetorg69&A. Rossiorg36&F.
Roukoutakisorg87&A. Royorg48&C. Royorg54&P. Royorg100&A.J. Rubio
Monteroorg10&R. Ruiorg26&R. Russoorg27&E. Ryabinkinorg99&Y. Ryabovorg84&A.
Rybickiorg115&S. Sadovskyorg111&K. Šafaříkorg36&B. Sahlmullerorg52&P.
Sahooorg48&R. Sahooorg48&S. Sahooorg60&P.K. Sahuorg60&J. Sainiorg130&S.
Sakaiorg71&M.A. Salehorg133&C.A. Salgadoorg17&J. Salzwedelorg20&S.
Sambyalorg89&V. Samsonovorg84&X. Sanchez Castroorg54&L. Šándororg58&A.
Sandovalorg63&M. Sanoorg126&G. Santagatiorg29&D. Sarkarorg130&E.
Scapparoneorg104&F. Scarlassaraorg30&R.P. Scharenbergorg94&C. Schiauaorg77&R.
Schickerorg92&C. Schmidtorg96&H.R. Schmidtorg35&S. Schuchmannorg52&J.
Schukraftorg36&M. Schulcorg39&T. Schusterorg135&Y. Schutzorg112,org36&K.
Schwarzorg96&K. Schwedaorg96&G. Scioliorg28&E. Scomparinorg110&R.
Scottorg123&K.S. Seederorg118&J.E. Segerorg85&Y. Sekiguchiorg125&I.
Selyuzhenkovorg96&K. Senosiorg64&J. Seoorg66,org95&E. Serradillaorg10,org63&A.
Sevcencoorg61&A. Shabanovorg55&A. Shabetaiorg112&O. Shaduraorg3&R.
Shahoyanorg36&A. Shangaraevorg111&A. Sharmaorg89&N. Sharmaorg60,org123&K.
Shigakiorg46&K. Shtejerorg9,org27&Y. Sibiriakorg99&S. Siddhantaorg105&K.M.
Sielewiczorg36&T. Siemiarczukorg76&D. Silvermyrorg83,org34&C.
Silvestreorg70&G. Simatovicorg127&G. Simonettiorg36&R. Singarajuorg130&R.
Singhorg78&S. Singhaorg78,org130&V. Singhalorg130&B.C. Sinhaorg130&T.
Sinhaorg100&B. Sitarorg38&M. Sittaorg32&T.B. Skaaliorg22&M. Slupeckiorg121&N.
Smirnovorg135&R.J.M. Snellingsorg56&T.W. Snellmanorg121&C. Søgaardorg34&R.
Soltzorg74&J. Songorg95&M. Songorg136&Z. Songorg7&F. Soramelorg30&S.
Sorensenorg123&M. Spacekorg39&E. Spiritiorg71&I. Sputowskaorg115&M.
Spyropoulou-Stassinakiorg87&B.K. Srivastavaorg94&J. Stachelorg92&I.
Stanorg61&G. Stefanekorg76&M. Steinpreisorg20&E. Stenlundorg34&G.
Steynorg64&J.H. Stillerorg92&D. Stoccoorg112&P. Strmenorg38&A.A.P.
Suaideorg118&T. Sugitateorg46&C. Suireorg50&M. Suleymanovorg16&R.
Sultanovorg57&M. Šumberaorg82&T.J.M. Symonsorg73&A. Szaboorg38&A. Szanto de
Toledoorg118&I. Szarkaorg38&A. Szczepankiewiczorg36&M. Szymanskiorg132&J.
Takahashiorg119&N. Tanakaorg126&M.A. Tangaroorg33&J.D. Tapia
Takakiidp5843168,org50&A. Tarantola Peloniorg52&M. Tariqorg19&M.G.
Tarzilaorg77&A. Tauroorg36&G. Tejeda Muñozorg2&A. Telescaorg36&K.
Terasakiorg125&C. Terrevoliorg30,org25&B. Teyssierorg128&J.
Thäderorg96,org73&D. Thomasorg116&R. Tieulentorg128&A.R. Timminsorg120&A.
Toiaorg52&S. Trogoloorg110&V. Trubnikovorg3&W.H. Trzaskaorg121&T.
Tsujiorg125&A. Tumkinorg98&R. Turrisiorg107&T.S. Tveterorg22&K.
Ullalandorg18&A. Urasorg128&G.L. Usaiorg25&A. Utrobicicorg127&M.
Vajzerorg82&M. Valaorg58&L. Valencia Palomoorg69&S. Valleroorg27&J. Van Der
Maarelorg56&J.W. Van Hoorneorg36&M. van Leeuwenorg56&T. Vanatorg82&P. Vande
Vyvreorg36&D. Vargaorg134&A. Vargasorg2&M. Vargyasorg121&R. Varmaorg47&M.
Vasileiouorg87&A. Vasilievorg99&A. Vauthierorg70&V. Vecherninorg129&A.M.
Veenorg56&M. Veldhoenorg56&A. Velureorg18&M. Venaruzzoorg72&E.
Vercellinorg27&S. Vergara Limónorg2&R. Vernetorg8&M. Verweijorg133&L.
Vickovicorg114&G. Viestiorg300&J. Viinikainenorg121&Z. Vilakaziorg124&O.
Villalobos Baillieorg101&A. Vinogradovorg99&L. Vinogradovorg129&Y.
Vinogradovorg98&T. Virgiliorg31&V. Vislaviciusorg34&Y.P. Viyogiorg130&A.
Vodopyanovorg65&M.A. Völklorg92&K. Voloshinorg57&S.A. Voloshinorg133&G.
Volpeorg36,org134&B. von Hallerorg36&I. Vorobyevorg91&D. Vranicorg96,org36&J.
Vrlákováorg40&B. Vulpescuorg69&A. Vyushinorg98&B. Wagnerorg18&J.
Wagnerorg96&H. Wangorg56&M. Wangorg7,org112&Y. Wangorg92&D. Watanabeorg126&M.
Weberorg36&S.G. Weberorg96&J.P. Wesselsorg53&U. Westerhofforg53&J.
Wiechulaorg35&J. Wikneorg22&M. Wildeorg53&G. Wilkorg76&J.
Wilkinsonorg92&M.C.S. Williamsorg104&B. Windelbandorg92&M. Winnorg92&C.G.
Yaldoorg133&Y. Yamaguchiorg125&H. Yangorg56&P. Yangorg7&S. Yanoorg46&Z.
Yinorg7&H. Yokoyamaorg126&I.-K. Yooorg95&V. Yurchenkoorg3&I. Yushmanovorg99&A.
Zaborowskaorg132&V. Zaccoloorg79&A. Zamanorg16&C. Zampolliorg104&H.J.C.
Zanoliorg118&S. Zaporozhetsorg65&A. Zarochentsevorg129&P. Závadaorg59&N.
Zaviyalovorg98&H. Zbroszczykorg132&I.S. Zguraorg61&M. Zhalovorg84&H.
Zhangorg7&X. Zhangorg73&Y. Zhangorg7&C. Zhaoorg22&N. Zhigarevaorg57&D.
Zhouorg7&Y. Zhouorg56&Z. Zhouorg18&H. Zhuorg7&J. Zhuorg7,org112&X. Zhuorg7&A.
Zichichiorg12,org28&A. Zimmermannorg92&M.B. Zimmermannorg53,org36&G.
Zinovjevorg3&M. Zyzakorg42

## Affiliation notes

0Deceased

idp5843168Also at: University of Kansas, Lawrence, Kansas, United States

## Collaboration Institutes

org1A.I. Alikhanyan National Science Laboratory (Yerevan Physics Institute)
Foundation, Yerevan, Armenia

org2Benemérita Universidad Autónoma de Puebla, Puebla, Mexico

org3Bogolyubov Institute for Theoretical Physics, Kiev, Ukraine

org4Bose Institute, Department of Physics and Centre for Astroparticle Physics
and Space Science (CAPSS), Kolkata, India

org5Budker Institute for Nuclear Physics, Novosibirsk, Russia

org6California Polytechnic State University, San Luis Obispo, California,
United States

org7Central China Normal University, Wuhan, China

org8Centre de Calcul de l’IN2P3, Villeurbanne, France

org9Centro de Aplicaciones Tecnológicas y Desarrollo Nuclear (CEADEN), Havana,
Cuba

org10Centro de Investigaciones Energéticas Medioambientales y Tecnológicas
(CIEMAT), Madrid, Spain

org11Centro de Investigación y de Estudios Avanzados (CINVESTAV), Mexico City
and Mérida, Mexico

org12Centro Fermi - Museo Storico della Fisica e Centro Studi e Ricerche
“Enrico Fermi”, Rome, Italy

org13Chicago State University, Chicago, Illinois, USA

org14China Institute of Atomic Energy, Beijing, China

org15Commissariat à l’Energie Atomique, IRFU, Saclay, France

org16COMSATS Institute of Information Technology (CIIT), Islamabad, Pakistan

org17Departamento de Física de Partículas and IGFAE, Universidad de Santiago
de Compostela, Santiago de Compostela, Spain

org18Department of Physics and Technology, University of Bergen, Bergen,
Norway

org19Department of Physics, Aligarh Muslim University, Aligarh, India

org20Department of Physics, Ohio State University, Columbus, Ohio, United
States

org21Department of Physics, Sejong University, Seoul, South Korea

org22Department of Physics, University of Oslo, Oslo, Norway

org23Dipartimento di Elettrotecnica ed Elettronica del Politecnico, Bari,
Italy

org24Dipartimento di Fisica dell’Università ’La Sapienza’ and Sezione INFN
Rome, Italy

org25Dipartimento di Fisica dell’Università and Sezione INFN, Cagliari, Italy

org26Dipartimento di Fisica dell’Università and Sezione INFN, Trieste, Italy

org27Dipartimento di Fisica dell’Università and Sezione INFN, Turin, Italy

org28Dipartimento di Fisica e Astronomia dell’Università and Sezione INFN,
Bologna, Italy

org29Dipartimento di Fisica e Astronomia dell’Università and Sezione INFN,
Catania, Italy

org30Dipartimento di Fisica e Astronomia dell’Università and Sezione INFN,
Padova, Italy

org31Dipartimento di Fisica ‘E.R. Caianiello’ dell’Università and Gruppo
Collegato INFN, Salerno, Italy

org32Dipartimento di Scienze e Innovazione Tecnologica dell’Università del
Piemonte Orientale and Gruppo Collegato INFN, Alessandria, Italy

org33Dipartimento Interateneo di Fisica ‘M. Merlin’ and Sezione INFN, Bari,
Italy

org34Division of Experimental High Energy Physics, University of Lund, Lund,
Sweden

org35Eberhard Karls Universität Tübingen, Tübingen, Germany

org36European Organization for Nuclear Research (CERN), Geneva, Switzerland

org37Faculty of Engineering, Bergen University College, Bergen, Norway

org38Faculty of Mathematics, Physics and Informatics, Comenius University,
Bratislava, Slovakia

org39Faculty of Nuclear Sciences and Physical Engineering, Czech Technical
University in Prague, Prague, Czech Republic

org40Faculty of Science, P.J. Šafárik University, Košice, Slovakia

org41Faculty of Technology, Buskerud and Vestfold University College,
Vestfold, Norway

org42Frankfurt Institute for Advanced Studies, Johann Wolfgang Goethe-
Universität Frankfurt, Frankfurt, Germany

org43Gangneung-Wonju National University, Gangneung, South Korea

org44Gauhati University, Department of Physics, Guwahati, India

org45Helsinki Institute of Physics (HIP), Helsinki, Finland

org46Hiroshima University, Hiroshima, Japan

org47Indian Institute of Technology Bombay (IIT), Mumbai, India

org48Indian Institute of Technology Indore, Indore (IITI), India

org49Inha University, Incheon, South Korea

org50Institut de Physique Nucléaire d’Orsay (IPNO), Université Paris-Sud,
CNRS-IN2P3, Orsay, France

org51Institut für Informatik, Johann Wolfgang Goethe-Universität Frankfurt,
Frankfurt, Germany

org52Institut für Kernphysik, Johann Wolfgang Goethe-Universität Frankfurt,
Frankfurt, Germany

org53Institut für Kernphysik, Westfälische Wilhelms-Universität Münster,
Münster, Germany

org54Institut Pluridisciplinaire Hubert Curien (IPHC), Université de
Strasbourg, CNRS-IN2P3, Strasbourg, France

org55Institute for Nuclear Research, Academy of Sciences, Moscow, Russia

org56Institute for Subatomic Physics of Utrecht University, Utrecht,
Netherlands

org57Institute for Theoretical and Experimental Physics, Moscow, Russia

org58Institute of Experimental Physics, Slovak Academy of Sciences, Košice,
Slovakia

org59Institute of Physics, Academy of Sciences of the Czech Republic, Prague,
Czech Republic

org60Institute of Physics, Bhubaneswar, India

org61Institute of Space Science (ISS), Bucharest, Romania

org62Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México,
Mexico City, Mexico

org63Instituto de Física, Universidad Nacional Autónoma de México, Mexico
City, Mexico

org64iThemba LABS, National Research Foundation, Somerset West, South Africa

org65Joint Institute for Nuclear Research (JINR), Dubna, Russia

org66Konkuk University, Seoul, South Korea

org67Korea Institute of Science and Technology Information, Daejeon, South
Korea

org68KTO Karatay University, Konya, Turkey

org69Laboratoire de Physique Corpusculaire (LPC), Clermont Université,
Université Blaise Pascal, CNRS–IN2P3, Clermont-Ferrand, France

org70Laboratoire de Physique Subatomique et de Cosmologie, Université
Grenoble-Alpes, CNRS-IN2P3, Grenoble, France

org71Laboratori Nazionali di Frascati, INFN, Frascati, Italy

org72Laboratori Nazionali di Legnaro, INFN, Legnaro, Italy

org73Lawrence Berkeley National Laboratory, Berkeley, California, United
States

org74Lawrence Livermore National Laboratory, Livermore, California, United
States

org75Moscow Engineering Physics Institute, Moscow, Russia

org76National Centre for Nuclear Studies, Warsaw, Poland

org77National Institute for Physics and Nuclear Engineering, Bucharest,
Romania

org78National Institute of Science Education and Research, Bhubaneswar, India

org79Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark

org80Nikhef, National Institute for Subatomic Physics, Amsterdam, Netherlands

org81Nuclear Physics Group, STFC Daresbury Laboratory, Daresbury, United
Kingdom

org82Nuclear Physics Institute, Academy of Sciences of the Czech Republic, Řež
u Prahy, Czech Republic

org83Oak Ridge National Laboratory, Oak Ridge, Tennessee, United States

org84Petersburg Nuclear Physics Institute, Gatchina, Russia

org85Physics Department, Creighton University, Omaha, Nebraska, United States

org86Physics Department, Panjab University, Chandigarh, India

org87Physics Department, University of Athens, Athens, Greece

org88Physics Department, University of Cape Town, Cape Town, South Africa

org89Physics Department, University of Jammu, Jammu, India

org90Physics Department, University of Rajasthan, Jaipur, India

org91Physik Department, Technische Universität München, Munich, Germany

org92Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg,
Heidelberg, Germany

org93Politecnico di Torino, Turin, Italy

org94Purdue University, West Lafayette, Indiana, United States

org95Pusan National University, Pusan, South Korea

org96Research Division and ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum
für Schwerionenforschung, Darmstadt, Germany

org97Rudjer Bošković Institute, Zagreb, Croatia

org98Russian Federal Nuclear Center (VNIIEF), Sarov, Russia

org99Russian Research Centre Kurchatov Institute, Moscow, Russia

org100Saha Institute of Nuclear Physics, Kolkata, India

org101School of Physics and Astronomy, University of Birmingham, Birmingham,
United Kingdom

org102Sección Física, Departamento de Ciencias, Pontificia Universidad
Católica del Perú, Lima, Peru

org103Sezione INFN, Bari, Italy

org104Sezione INFN, Bologna, Italy

org105Sezione INFN, Cagliari, Italy

org106Sezione INFN, Catania, Italy

org107Sezione INFN, Padova, Italy

org108Sezione INFN, Rome, Italy

org109Sezione INFN, Trieste, Italy

org110Sezione INFN, Turin, Italy

org111SSC IHEP of NRC Kurchatov institute, Protvino, Russia

org112SUBATECH, Ecole des Mines de Nantes, Université de Nantes, CNRS-IN2P3,
Nantes, France

org113Suranaree University of Technology, Nakhon Ratchasima, Thailand

org114Technical University of Split FESB, Split, Croatia

org115The Henryk Niewodniczanski Institute of Nuclear Physics, Polish Academy
of Sciences, Cracow, Poland

org116The University of Texas at Austin, Physics Department, Austin, Texas,
USA

org117Universidad Autónoma de Sinaloa, Culiacán, Mexico

org118Universidade de São Paulo (USP), São Paulo, Brazil

org119Universidade Estadual de Campinas (UNICAMP), Campinas, Brazil

org120University of Houston, Houston, Texas, United States

org121University of Jyväskylä, Jyväskylä, Finland

org122University of Liverpool, Liverpool, United Kingdom

org123University of Tennessee, Knoxville, Tennessee, United States

org124University of the Witwatersrand, Johannesburg, South Africa

org125University of Tokyo, Tokyo, Japan

org126University of Tsukuba, Tsukuba, Japan

org127University of Zagreb, Zagreb, Croatia

org128Université de Lyon, Université Lyon 1, CNRS/IN2P3, IPN-Lyon,
Villeurbanne, France

org129V. Fock Institute for Physics, St. Petersburg State University, St.
Petersburg, Russia

org130Variable Energy Cyclotron Centre, Kolkata, India

org131Vinča Institute of Nuclear Sciences, Belgrade, Serbia

org132Warsaw University of Technology, Warsaw, Poland

org133Wayne State University, Detroit, Michigan, United States

org134Wigner Research Centre for Physics, Hungarian Academy of Sciences,
Budapest, Hungary

org135Yale University, New Haven, Connecticut, United States

org136Yonsei University, Seoul, South Korea

org137Zentrum für Technologietransfer und Telekommunikation (ZTT),
Fachhochschule Worms, Worms, Germany


# Cartesian products of two $CR$ sets

Sayan Goswami
<EMAIL_ADDRESS>Institute of Mathematical Sciences, A CI of
Homi Bhabha National Institute, CIT Campus, Taramani, Chennai 600113, India.

###### Abstract

In this article, we prove the product of two CR sets is again a CR set. This
gives a negative answer to [5, Question 4.2.]. We will use combinatorial
arguments to prove our result.

Keywords: CR-sets, product space, $IP_{r}^{*}$ sets
Subject class: 05D10, 22A15, 54D35

## 1 Introduction

For any nonempty set $X$, let $\mathcal{P}_{f}(X)$ be the set of all nonempty
finite subsets of $X.$ Arithmetic Ramsey theory deals with the monochromatic
patterns found in any given finite coloring of the integers or of the natural
numbers ${\mathbb{N}}$. Here “coloring” means disjoint partition and a set is
called “monochromatic” if included in one part of the partition. A cornerstone
result in this field of research is Van der Waerden’s Theorem [8], which
states that for any finite coloring of the natural numbers one always finds
arbitrarily long monochromatic arithmetic progressions. One can show that if a
set $A$ contains arithmetic progressions of arbitrary length (A.P. rich in
short), then if $A=B\cup C$ is a 2-coloring of $A$, one of $B$ and $C$ is A.P.
rich. This property is called partition regularity.

The notions of piecewise syndetic set have been extensively studied in Ramsey
theory and its connection with the Algebra in the Stone-Čech Compactification
of ${\mathbb{N}}.$ For every $A\subseteq{\mathbb{N}},$ and $n\in{\mathbb{N}}$,
let $-n+A=\\{m:m+n\in A\\}.$ A set $A$ is said to be piecewise syndetic if
there exists a finite set $F$ such that for every finite subset $G$ of
${\mathbb{N}}$, there exists $y\in{\mathbb{N}}$ such that
$G+y\subset\cup_{t\in F}(-t+A)$. This can be shown that piecewise syndetic
sets are partition regular. For general semigroup one can define this notion
naturally. For details, readers can see [7]. In [2], authors introduced a more
general notion of large sets: combinatorially rich sets for commutative
semigroups. Later in [5], authors introduced the notions of combinatorially
rich sets for general semigroups and proved that these sets are partition
regular. It can be easily proved that these sets are A.P. rich (in commutative
semigroups).

######  Definition 1.1.

[5, Definition 2.2] Let $(S,\cdot)$ be an arbitrary semigroup. A set
$A\subseteq S$ is said to be a combinatorially rich set (a CR-set) if and only
if for each $k\in\mathbb{N}$, there exists $r\in\mathbb{N}$ such that whenever
$F\in{\mathcal{P}}_{f}(\,{}^{\hbox{${\mathbb{N}}$}}\\!S\,)$ with $|F|\leq k$,
there exist $m\in{\mathbb{N}}$, $a\in S^{m+1}$, and $t(1)<t(2)<\ldots<t(m)\leq
r$ in ${\mathbb{N}}$ such that for each $f\in F$,

$a(1)\cdot f\big{(}t(1)\big{)}\cdot a(2)\cdot f\big{(}t(2)\big{)}\cdot
a(3)\cdots a(m)\cdot f\big{(}t(m)\big{)}\cdot a(m+1)\in A.$

Call $r=r(A,k).$

Using the algebra of the Stone-Čech compactification of discrete semigroups,
in [6], authors proved several large sets are preserved under taking cartesian
products. Later combinatorial proofs of these results were recovered in [3].
Inspired by these studies, in [5, Question 4.2.], authors asked if there
exists two infinite semigroups $S$ and $T$, and two CR-sets $A,B$ in $S$ and
$T$ respectively such that $A\times B$ is not a CR-set in $S\times T$. Here we
show that such a possibility can’t occur. We prove the following theorem:

######  Theorem 1.2.

If $S$ and $T$ be two semigroups and $A\subseteq S$, $B\subseteq T$ are two
CR-sets, then $A\times B$ is CR set in $S\times T$.

## 2 Proof of our result

A set $A$ in ${\mathcal{P}}_{f}({\mathbb{N}})$ is said to be an $IP$ set if
there is a sequence $\langle H_{n}\rangle_{n}$ in
${\mathcal{P}}_{f}({\mathbb{N}})$ such that $\max H_{n}<\min H_{n+1}$ and
$A=FU(\langle H_{n}\rangle_{n})=\left\\{\bigcup_{n\in
K}H_{n}:K\in{\mathcal{P}}_{f}({\mathbb{N}})\right\\}.$ A consequence of
Hindman’s theorem [4] says that for every finite coloring of
${\mathcal{P}}_{f}({\mathbb{N}})$, there exists a monochromatic $IP$ set. For
every $r\in{\mathbb{N}}$, set $B$ is called an $IP_{r}$ set if there exists
sequence $\langle H_{n}\rangle_{n=1}^{r}$ in ${\mathcal{P}}_{f}({\mathbb{N}})$
such that $\max H_{n}<\min H_{n+1}$ and $B=FU(\langle
H_{n}\rangle_{n=1}^{r})=\left\\{\bigcup_{n\in
K}H_{n}:K(\neq\emptyset)\subseteq\\{1,2,\ldots,r\\}\right\\}.$ A set is called
$IP^{*}$ (resp. $IP_{r}^{*}$) if that set intersects with every $IP$ sets
(resp. $IP_{r}$ sets). Let us recall the following lemma essential for our
purpose.

######  Lemma 2.1.

Let $r,s\in\mathbb{N}$. Let $A$ and $B$ be $IP_{r}^{*}$ set (resp.
$IP_{s}^{*}$ set) in ${\mathcal{P}}_{f}({\mathbb{N}})$. Then there exists
$l=l(r,s)\in{\mathbb{N}}$ such that $A\cap B$ is an $IP_{l}^{*}$ set.

###### Proof.

[1, Propostion 2.5]. ∎

We also need the following lemma.

######  Lemma 2.2.

Let $(S,\cdot)$ be an arbitrary semigroup, and $k\in\mathbb{N}$. Let $A$ be a
CR-set in $S$, and let $F\in{\mathcal{P}}_{f}({}^{\hbox{${\mathbb{N}}$}}\\!S)$
with $|F|\leq k$. Let
$\Theta=\\{L\in{\mathcal{P}}_{f}({\mathbb{N}})\\!\\!:L=\\{t(1),t(2),\ldots,t(m)\\}_{<}\text{
and }(\exists a\in S^{m+1})(\forall f\in F)(a(1)\cdot f\big{(}t(1))\cdot
a(2)\cdots a(m)\cdot f\big{(}t(m)\big{)}\cdot a(m+1)\in A)\\}.$ Let
$r=r(A,k)$, and $\langle H_{n}\rangle_{n=1}^{r}$ be a sequence in
${\mathcal{P}}_{f}({\mathbb{N}})$ such that $\max H_{n}<\min H_{n+1}$ for each
$n\in\\{1,2,\ldots,r-1\\}$. Then there exists $K\subseteq\\{1,2,\ldots,r\\}$
such that $\bigcup_{n\in K}H_{n}\in\Theta$. In other words, for each
$k\in{\mathbb{N}},$ $\Theta$ is an $IP_{r}^{*}$ set, where $r=r(k)$.

###### Proof.

For each $n\in\\{1,2,\ldots,r\\}$, let $\alpha_{n}=|H_{n}|$ and write

$H_{n}=\\{b(n,1),b(n,2),\ldots,b(n,\alpha_{n})\\}_{<}\,.$

Pick $d\in S$ and for $f\in F$, define
$g_{f}\in{}^{\hbox{${\mathbb{N}}$}}\\!S$ by,

$g_{f}(n)=f\big{(}b(n,1)\big{)}\cdot d\cdot f\big{(}b(n,2)\big{)}\cdot
d\cdots\cdot d\cdot f\big{(}b(n,\alpha_{n})\big{)}\,\text{ if
}n\in\\{1,2,\ldots,r\\},$

and $g_{f}(n)=d$ otherwise. Now $\\{g_{f}\\!\\!:f\in
F\\}\in{\mathcal{P}}_{f}({}^{\hbox{${\mathbb{N}}$}}\\!S)$ with
$|\\{g_{f}\\!\\!:f\in F\\}|\leq k.$ So pick $m\in{\mathbb{N}}$, $a\in
S^{m+1}$, and $t(1)<t(2)<\ldots<t(m)\leq r$ in ${\mathbb{N}}$ such that
$a(1)\cdot g_{f}\big{(}t(1)\big{)}\cdot a(2)\cdots a(m)\cdot
g_{f}\big{(}t(m)\big{)}\cdot a(m+1)\in A$. Let
$K=\big{\\{}t(1),t(2),\ldots,t(m)\big{\\}}$. Then $K$ is as required. ∎

Now we are in the position to prove our main theorem.

###### Proof of Theorem 1.2:.

Let $A\subseteq S$ and $B\subseteq T$ be two CR-set. Let $k\in{\mathbb{N}}$
and $F\in{\mathcal{P}}_{f}(^{\mathbb{N}}(S\times T))$ with $|F|\leq k.$ Let
$G=\\{\pi_{1}\circ f:f\in F\\}$ and $H=\\{\pi_{2}\circ f:f\in F\\}$. Then
$G\in{\mathcal{P}}_{f}(^{\mathbb{N}}S)$ and
$H\in{\mathcal{P}}_{f}(^{\mathbb{N}}T)$ with $|G|=|H|\leq k.$ Now from Lemma
2.2, there exists $r,s\in{\mathbb{N}}$ such that
$\Theta_{1}=\\{L\in{\mathcal{P}}_{f}({\mathbb{N}})\\!\\!:L=\\{t(1),t(2),\ldots,t(m)\\}_{<}\text{
and }(\exists a\in S^{m+1})(\forall f\in G)(a(1)\cdot f\big{(}t(1))\cdot
a(2)\cdots a(m)\cdot f\big{(}t(m)\big{)}\cdot a(m+1)\in A)\\}$ and
$\Theta_{2}=\\{L\in{\mathcal{P}}_{f}({\mathbb{N}})\\!\\!:L=\\{t(1),t(2),\ldots,t(p)\\}_{<}\text{
and }(\exists b\in T^{p+1})(\forall f\in H)(b(1)\cdot f\big{(}t(1))\cdot
b(2)\cdots b(p)\cdot f\big{(}t(p)\big{)}\cdot b(p+1)\in B)\\}$ are
$IP_{r}^{*}$ and $IP_{s}^{*}$ sets respectively. So
$\Theta=\Theta_{1}\cap\Theta_{2}$ is an $IP_{l}^{*}$ set, where $l=l(r,s)$ is
coming from Lemma 2.1.

Hence there exists $\\{t(1),t(2),\ldots,t(k)\\}_{<}$ in $\\{1,2,\ldots,l\\}$,
$a\in S^{k+1}$, and $b\in T^{k+1}$ such that for all $f\in F$,

$(a(1),b(1))\cdot f\big{(}t(1))\cdot(a(2),b(2))\cdots(a(k),b(k))\cdot
f\big{(}t(k)\big{)}\cdot(a(k+1),b(k+1))\in A\times B.$

This completes the proof.

∎

## References

  * [1] V. Bergelson, and D. Robertson: Polynomial recurrence with large intersection over countable fields, Israel Journal of Mathematics, 214 (2016), no. 1, 109-120.
  * [2] V. Bergelson, and D. Glasscock: On the interplay between additive and multiplicative largeness and its combinatorial applications, J. Combin. Theory Ser. A, 172 (2020).
  * [3] S. Goswami: Cartesian product of some combinatorially rich sets, INTEGERS, (20)2020 #A64.
  * [4] N. Hindman: Finite sums from sequences within cells of partitions of ${\mathbb{N}}$, J. Comb. Theory (Series A) 17 (1974), 1-11.
  * [5] N. Hindman, H. Hosseini, D. Strauss, and M. Tootkaboni: Combinatorially rich sets in arbitrary semigroups, Semigroup Forum, 107 (2023), 127-143.
  * [6] N. Hindman, and D. Strauss: Cartesian products of sets satisfying the Central Sets Theorem, Topology Proc. 35 (2010), 203-223.
  * [7] N. Hindman, and D. Strauss: Algebra in the Stone-Čech Compactification: Theory and Applications, 2nd edition, Walter de Gruyter & Co., Berlin, 2012.
  * [8] B.L. van der Waerden: Beweis einer baudetschen vermutung. Nieuw. Arch. Wisk., 15:212–216,1927.