EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-EP-2017-203 LHCb-PAPER-2017-025 September 5, 2017

Bose-Einstein correlations of same-sign charged pions in the forward region in
$pp$ collisions at $\sqrt{s}$ = 7 TeV

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

Bose-Einstein correlations of same-sign charged pions, produced in proton-
proton collisions at a 7 TeV centre-of-mass energy, are studied using a data
sample collected by the LHCb experiment. The signature for Bose-Einstein
correlations is observed in the form of an enhancement of pairs of like-sign
charged pions with small four-momentum difference squared. The charged-
particle multiplicity dependence of the Bose-Einstein correlation parameters
describing the correlation strength and the size of the emitting source is
investigated, determining both the correlation radius and the chaoticity
parameter. The measured correlation radius is found to increase as a function
of increasing charged-particle multiplicity, while the chaoticity parameter is
seen to decrease.

Published in JHEP 12 (2017) 025

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

##  1 Introduction

Multiparticle production within the process of hadronisation can be
investigated by measuring Bose-Einstein correlations (BEC) between
indistinguishable bosons [1, 2]. The technique to study the BEC effect in
particle physics is the analogue of the Hanbury-Brown-Twiss (HBT) intensity
interferometry [3, 4, 5]. The production of identical bosons that are close in
phase space is enhanced by the presence of BEC. The measurements of the
quantum interference effect between indistinguishable particles emitted by a
finite-size source are useful to understand the space-time properties of the
hadron emission volume.

Since the first observation of BEC in identically charged pions produced in
$p\bar{p}$ collisions [6], the effect has been studied for multiboson systems
produced in leptonic, hadronic and nuclear collisions [7, 8, 9, 10, 11, 12,
13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31,
32]. At the LHC, the BEC effect has been studied by the ALICE, ATLAS and CMS
collaborations in proton-proton [26, 27, 28, 29, 30], proton-lead [31] and
lead-lead [31, 32] collisions.

Dependences of the BEC effect upon various observables have been studied,
including charged-particle multiplicity, average transverse momentum of the
particle pair and boson mass. The latter has been reported by the LEP
experiments [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], and can
be interpreted within some theoretical models [33, 34, 35, 36].

In this paper, the first study of the BEC effect in $pp$ collisions in the
forward region is presented. The BEC parameters characterising the correlation
radius and the chaoticity of the correlation source are measured.

##  2 BEC measurement

Quantum interference effects are probed by studying the Lorentz invariant
quantity $Q$ [2, 37] of two indistinguishable particles of rest mass $m$ and
four-momenta $q_{1}$ and $q_{2}$

$Q=\sqrt{-(q_{1}-q_{2})^{2}}=\sqrt{M^{2}-4m^{2}},$ (1)

which gives a measure of the phase-space separation of the two-particle system
of invariant mass $M$.

###  2.1 Two-particle correlation function

The BEC effect is expected to manifest itself as an enhancement in the two-
particle correlation function in the low-$Q$ region below
$\sim$0.5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$, expressed as [38]

$C_{2}(Q)=\frac{\rho_{2}(Q)}{\rho_{2}^{0}(Q)},$ (2)

where $\rho_{2}(Q)$ is the two-particle density function for like-sign pairs
of indistinguishable particles, as defined in Ref. [38], and $\rho_{2}^{0}(Q)$
is the corresponding density function without the BEC effect, which is
constructed as described in Sec. 2.2. The densities $\rho_{2}(Q)$ and
$\rho_{2}^{0}(Q)$ are normalised to unity, such that they can be interpreted
as probability density functions. The correlation function $C_{2}(Q)$ is
commonly parameterised as a Fourier transform of the source density
distribution, $C_{2}(Q)=N(1+\lambda e^{-|RQ|^{\alpha_{\mathrm{L}}}})$ [39],
where the parameter $R$, the correlation radius, can be interpreted as the
radius of the spherically symmetric source of the emission volume, $N$
accounts for the overall normalisation and $\lambda$ is the chaoticity
parameter, which accounts for the partial incoherence of the source [40]. The
chaoticity parameter can vary from zero, in the case of a completely coherent
source, to unity for an entirely chaotic source. The Levy index of stability
[39], $\alpha_{\mathrm{L}}$, accounts for the assumed density distribution.
The radial distribution of the static source corresponding to the case of
$\alpha_{\mathrm{L}}=1$ is used in the present analysis

$C_{2}(Q)=N(1+\lambda e^{-RQ})\times(1+\delta\cdot Q),$ (3)

where the $\delta$ parameter accounts for long-range correlations, e.g.
related to the transverse momentum conservation. This extended
parameterisation follows better the $Q$ distribution in data, including in the
low-$Q$ region below $\sim$0.5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ [41].

The correlation function is, to first order, independent of the single-
particle acceptance and efficiency. By construction of the correlation
function, the effects due to the detector occupancy, acceptance and material
budget are accounted for by dividing the $Q$ distribution for like-sign pion
pairs by a reference distribution.

###  2.2 Reference sample

The reference sample used to construct the $\rho_{2}^{0}(Q)$ density function,
present in the denominator of Eq. (2), should reflect the distribution without
the BEC effect while maintaining all other correlations. A number of reference
samples can be constructed but none fully satisfies the above conditions. The
reference sample may be constructed using experimental data, or with simulated
events incorporating the detector interactions.

A data-driven “event-mixed” reference sample [42] is used in the present
analysis. This approach is based on the choice of two identical bosons, each
originating from different events, which naturally do not contain the BEC
effect. However, this method of constructing boson pairs may not contain other
correlations present in the same-sign boson data sample, such as correlations
due to Coulomb interactions or long-range effects.

Alternative methods have been considered for constructing the reference
sample. For example, the reference sample could consist of opposite-sign
charged bosons originating from the same $pp$ interaction. As in the event-
mixed reference sample, the main advantage of the opposite-sign approach is
that the reference distribution is derived directly from data. However, the
opposite-sign charge pairs may also originate from resonances which result in
local enhancements in the $Q$ spectrum. Furthermore, correlations arising from
the attraction of opposite charges are present in such a sample. Another
method is to employ the simulated $Q$ distribution without the BEC effect. In
this case, the crucial requirement is a good level of agreement between data
and simulated samples in the distributions of crucial variables, e.g. the
particle momenta. The absence of the Coulomb and spin effects in generators
based on the Lund Model [43] may impinge on the correctness of this method.

###  2.3 Double ratio

To account for imperfections in the reference distribution derived from the
data a “double ratio” $r_{\rm d}$ is commonly used in BEC studies

$r_{\rm
d}(Q)\equiv\frac{C_{2}(Q)^{\mathrm{data}}}{C_{2}(Q)^{\mathrm{simulation}}},$
(4)

where $C_{2}(Q)^{\mathrm{data}}$ denotes the correlation function in the data
constructed using the event-mixed reference sample, while
$C_{2}(Q)^{\mathrm{simulation}}$ indicates the correlation function in the
simulation without the BEC effect, using an event-mixed sample built with
simulated events in the same way as for data. The correlation function in the
simulation without the BEC effect includes the simulated long-range
correlations that are also present in data. Therefore, if the long-range
correlations are correctly modelled, a constant $r_{\rm d}(Q)$ distribution is
expected in the high-$Q$ region up to
$\sim$2.0${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. In the present analysis
the BEC effect is measured by fitting the $r_{\rm d}(Q)$ distribution with the
event-mixed reference sample, using the parameterisation given in Eq. (3).

###  2.4 Coulomb correction

Final-state interactions involving both electromagnetic (Coulomb) and strong
forces are present in the low-$Q$ region below
$\sim$0.5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$, and may potentially
affect the distributions of the analysed observables. In the low-$Q$ region,
the Coulomb repulsion between two identically charged hadrons alters the
correlation function $C_{2}(Q)$ by decreasing the BEC effect. This effect is
corrected for with the Gamov penetration factor [44, 45], $G_{2}(Q)$, by
applying a weight per particle pair $1/G_{2}(Q)$, where
$G_{2}(Q)=\frac{2\pi\zeta}{e^{2\pi\zeta}-1}$, $\zeta=\pm\frac{\alpha m}{Q}$,
and $m$ and $\alpha$ denote the particle rest mass and the fine-structure
constant, respectively. The sign of $\zeta$ is positive for same-charge and
negative for opposite-charge pairs of hadrons.

The Coulomb interactions are not present in the simulated samples used in the
analysis. This effect therefore has to be corrected for in the data.

##  3 Detector and dataset

The LHCb detector [46] is a single-arm forward spectrometer designed for the
study of particles containing $b$ or $c$ quarks. The detector includes a high-
precision tracking system consisting of a silicon-strip vertex detector (VELO)
[47] surrounding the $pp$ interaction region and covering the pseudorapidity
range $2<\eta<5$, a large-area silicon-strip detector located upstream of a
dipole magnet with a bending power of about $4{\rm\,Tm}$, and three stations
of silicon-strip detectors and straw drift tubes [48] placed downstream of the
magnet. The tracking system provides a measurement of momentum, $p$, with a
relative uncertainty that varies from 0.5% at low momentum to 1% at
$200$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The minimum distance of a track
to a primary vertex (PV), the impact parameter (IP), is measured with a
resolution of $(15+29/\mbox{$p_{\mathrm{T}}$}){\,\upmu\mathrm{m}}$, where
$p_{\mathrm{T}}$ is the component of $p$ transverse to the beam,
in${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. Different types of charged hadrons
are distinguished using information from two ring-imaging Cherenkov detectors
[49]. Photon, electron and hadron candidates are identified by a calorimeter
system consisting of scintillating-pad and preshower detectors, an
electromagnetic calorimeter and a hadronic calorimeter. Muons are identified
by a system composed of alternating layers of iron and multiwire proportional
chambers [50]. The trigger [51] consists of a hardware stage, based on
information from the calorimeter and muon systems, followed by a software
stage, which applies a full event reconstruction.

In the present analysis, a dataset of no-bias and minimum-bias triggered
events collected in 2011 at a centre-of-mass energy of $\sqrt{s}$ = 7 TeV is
used. The no-bias trigger selects events randomly, while the minimum-bias
trigger requires at least one reconstructed VELO track. The data were
collected with an average number of visible interactions per bunch
crossing111A visible interaction corresponds to the PV reconstructed with at
least five VELO tracks. (pile-up) of 1.4 [52]. In order to eliminate biases
related to the trigger requirements, a sample of “independent $pp$
interactions” is constructed as described in Sec. 4.

In the simulation, $pp$ collisions are generated using Pythia 8 [53] with a
specific LHCb configuration [54] and without including the BEC effect. Decays
of hadronic particles are described by EvtGen [55], in which final-state
radiation is generated using Photos [56]. The interaction of the generated
particles with the detector and its response are implemented using the Geant4
toolkit [57, *Agostinelli:2002hh], as described in Ref. [59]. To study
systematic effects, an additional sample is simulated using Pythia 6.4 [60]
with the Perugia0 [61] tune.

##  4 Selection and model fitting

The analysis uses a sample of events that may contain multiple $pp$
collisions. In the absence of trigger requirements each $pp$ interaction in
the event can be analysed separately. Therefore, if the event is selected by
the no-bias trigger, all PVs are accepted. In the case of events with multiple
$pp$ collisions selected by the minimum-bias trigger, the related biases are
suppressed by randomly removing one of the PVs containing the track(s) on
which the trigger is fired.

The correlation function is constructed using pairs of same-sign pions. The
particle identification (PID) is based on the output of a neural network
employing subdetector information that quantifies the probability for a
particle to be of a certain kind [62]. Such probabilities are calibrated to
account for differences between data and the simulation that is used to train
the neural network. The corrected values are derived from the data
distributions using dedicated PID calibration samples [49]. A high purity of
the pion sample has to be ensured, but without suppressing low-momentum pions
which mostly contribute to the signal region at low $Q$. The optimal limit on
the pion identification probability is applied at the point where the signal
enhancement in the low-$Q$ region below
$\sim$0.5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ for data begins to
saturate. The pion purity with this selection remains high ($\sim$98%).
Additional vetoes on the kaon and proton identification probabilities are also
imposed.

The following single particle requirements are applied. The selection requires
that all pion candidates must have reconstructed track segments in the VELO,
with $2<\eta<5$, and tracking stations downstream of the magnet. Each track
must have a good-quality track fit, $p_{\mathrm{T}}$
$>0.1$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and no associated signal in the
muon stations. Both pion candidates must be assigned to the same PV. Particles
are assigned to the PV for which the $\chi^{2}$ value of the impact parameter,
$\chi^{2}_{\text{IP}}$, is the smallest, where $\chi^{2}_{\text{IP}}$ is
defined as the difference in the vertex-fit $\chi^{2}$ of a given PV
reconstructed with and without the track under consideration. A loose
requirement on the track IP, IP $<$ 0.4 mm, is applied to retain most of the
particles originating from a given PV. In order to reduce the contamination
from fake and clone tracks,222Fake tracks are wrongly reconstructed tracks
which combine the hits deposited by multiple particles in the tracking
detectors. Clone tracks are two or more tracks reconstructed by mistake from
the hits deposited in detectors by a single particle. in the case where the
tracks have all the same hits deposited in the VELO subdetector, only the
track with the best $\chi^{2}$ is retained. In addition, fake tracks are
removed using the requirements on the track $\chi^{2}$ and the output of a
dedicated neural network [62].

In the region $Q<0.05$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$, the
separation in momentum between two particles is degraded and is not well
simulated. The discrepancy between data and simulated track pairs tends to
increase as $Q$ approaches zero. Investigations using simulation indicate that
there is a significant fraction of pion pairs containing fake and clone tracks
in the region $Q<0.05$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ for all
activity classes. The double ratio is approximately constant and close to
unity in the high-$Q$ region up to $Q\sim
2.0$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ (see Fig. 2), which indicates
that the long-range correlations are modelled accurately in this region.
Consequently, the fits to the $r_{\rm d}$ distributions are restricted to the
range $0.05<Q<2.0$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$.

The BEC effect is expected to be largest in the low-$Q$ region below
$\sim$0.5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$, where it may be affected
by same-sign clone tracks. Such clone pion pairs should manifest themselves as
an enhancement in the distribution of the differences of the tangents of the
track momenta of the two particles, where the tangents are measured in the
$xz$ and $yz$ planes before the magnet, with the $z$ axis defined along the
beam direction. The tangents are used to estimate the number of clone tracks
remaining after the final selection, and the clone tracks can be suppressed
with a requirement on the difference between the tangents of the two particles
in a pair. Pion pairs are removed from the analysis if both $|\Delta t_{x}|$
and $|\Delta t_{y}|$ are less than 0.3 mrad, where $\Delta t_{x}$ and $\Delta
t_{y}$ are the differences of the tangents of the track momenta of the two
particles in the $xz$ and $yz$ planes. After applying these requirements, the
effect of the clone particles is found to be negligible in the region
$Q>0.05$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$.

The BEC parameters are studied as a function of the charged-particle
multiplicity. However, the measured charged-particle multiplicities cannot be
directly used to compare results among different experiments, mainly because
the detector acceptances may not overlap and the reconstruction efficiencies
may differ. This is why activity classes are introduced, reflecting the total
multiplicity in the full solid angle. Three activity classes are defined in
the range $2<\eta<5$ according to the multiplicity of reconstructed VELO
tracks assigned to a PV, which is a good probe of the total multiplicity.
These activity classes are illustrated in Fig. 1. The low activity class
corresponds to a fraction of 48% of PVs with lowest multiplicities (from 5 to
10 tracks). The medium activity class contains the 37% of PVs with higher
multiplicities (from 11 to 20 tracks). Finally, the high activity class
contains 15% of the highest multiplicity PVs ($\geq$ 21 tracks). Using this
classification, the comparison among different experiments is largely
independent from specific features of the detectors.

Although the activity classes have advantages in comparing results among
various experiments characterised by different rapidity ranges, an unfolding
procedure is performed to relate the reconstructed charged-particle
multiplicities to those predicted by Pythia 8 with a specific LHCb
configuration [54]. The multiplicity distributions are corrected using a
Bayesian unfolding technique [63]. An unfolding matrix reflecting the
probability of reconstructing a certain number of charged particles from a
single PV in the range $2<\eta<5$ with generated charged-particle multiplicity
$N_{ch}$ is populated using simulation and applied to the data. It is found
that the corrected multiplicities agree well with the unfolded multiplicities
previously determined by LHCb in Ref. [64]. The activity classes correspond to
the following generated charged-particle multiplicitiy intervals:
$N_{ch}\in[8,18]$ (low activity), $N_{ch}\in[19,35]$ (medium activity) and
$N_{ch}\in[36,96]$ (high activity).

Figure 1: Multiplicity of reconstructed VELO tracks assigned to a PV for the
2011 no-bias sample. Different colours indicate three activity classes defined
as fractions of the full distribution. The minimum value of the track
multiplicity to accept reconstructed PV is five.

The distributions of the double ratio of correlation functions in data and
simulation for like-sign pion pairs, determined using the event-mixed
reference sample, are fitted in the range
$0.05<Q<2.0$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ for the three different
activity classes using the parameterisation of Eq. (3). The results of the
binned maximum likelihood fit to the double ratio are summarised in Sec. 6.

##  5 Systematic uncertainties

The properties of the correlation function and the construction of the double
ratio make the fitted BEC parameters insensitive to the choice of the
selection requirements to a large extent. However, due to imperfections in the
reference sample and possible differences between data and simulation related
to the generation model, as well as subtle reconstruction effects (like the
reconstruction of close tracks sharing the same VELO hits or the track
reconstruction in the high-occupancy detector regions), some second-order
distortions in the double ratio may appear. The systematic uncertainties on
the fit parameters, $R$ and $\lambda$, of the exponential model are determined
by performing the analysis with modifications designed to estimate the
systematic effects on individual contributions to the $r_{\rm d}(Q)$
distribution.

The leading source of systematic uncertainty is due to differences in the
event generators used to determine the correlation function for the
simulation. To study this effect, a sample of minimum-bias events produced
using the Pythia 6.4 generator with Perugia0 tuning is used to construct the
double ratio. The corresponding contribution to the systematic uncertainty is
taken as the difference between the central values of the results obtained
using the Pythia 8 and Pythia 6.4 datasets.

Another important source of systematic uncertainty is related to the PV
multiplicity in the event. The constructed double ratio may be distorted in
events containing multiple PVs, due to imperfections in the construction of
the reference sample. To estimate the associated systematic uncertainty, the
sample is divided into three subsamples containing events with one, two, and
three or more PVs. For each subsample, the fit is performed and the maximum
difference for each measured parameter is taken as a systematic uncertainty.
The systematic uncertainty resulting from the PV reconstruction efficiency is
also considered. To account for the effect of pile-up in the data and
inefficiencies in the PV reconstruction, a systematic uncertainty is estimated
as the difference between the nominal fit results and the results obtained
from a fit to the data in which the PV reconstruction has been repeated after
removing randomly a subset of the tracks from the event.

After applying the track quality requirements, the fraction of remaining fake
tracks is determined from simulation to be at the level of 1%. To determine
the systematic uncertainty due to the presence of such fake tracks, the double
ratio is refitted with looser track quality requirements. A similar
uncertainty is obtained from a second method in which sets of randomly
selected uncorrelated tracks are added. The observed change in BEC parameters
is negligible with respect to the statistical uncertainty.

The fraction of like-sign pion pairs containing a clone track after the
selection is determined to be below 1%. The systematic uncertainty due to the
presence of clone tracks is estimated by fitting the double ratio $r_{\rm
d}(Q)$, after applying a tight requirement on the Kullback-Leibler distance
[65] such that the clone contribution is fully removed in simulation. The
effect is found to be negligible for all activity classes.

The systematic uncertainty due to the calibration of the particle
identification in the simulation is estimated by comparing several variants of
the calibration procedure with the acceptance evaluated in different binning
schemes for the particle momentum, pseudorapidity and track multiplicity. The
largest difference after refitting the double ratios is taken as a systematic
uncertainty.

As the requirement on the pion identification probability alters the
contamination of pions due to misidentification, it can influence the values
of the $R$ and $\lambda$ parameters. The contribution of this effect to the
systematic uncertainty is estimated by refitting $r_{\rm d}(Q)$ with the
requirement on the pion identification probability changed to increase the
fraction of misidentified pions by 50%.

The systematic uncertainty derived from the fit range in the low-$Q$
(high-$Q$) region is determined by changing the lower (upper) limit of the $Q$
value by $\pm$0.01${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$
($\pm$0.2${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$). The fits to the double
ratio with two different lower (upper) limits of $Q$ are performed for the
three activity classes and the largest difference is taken as a systematic
uncertainty.

The systematic uncertainty due to Coulomb corrections is estimated by varying
the corrections by $\pm$20%. The variation in the fit parameters is found to
be less than 0.1%, and is therefore neglected. It is also found that imposing
different requirements on the particle IPs has no significant influence on the
measured correlation radius or chaoticity parameter. The fractions of kaon-
kaon and proton-proton like-sign pairs misidentified as a pion pair in the
pion sample in the BEC signal region of
$Q<1.0$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ are found to be negligible.
Pairings of different particle types have a negligible effect.

Other effects like the fit binning, the resolution of the $Q$ variable,
different magnet polarities, beam-gas interactions and residual acceptance
effects related to possible differences between data and simulation in the
low-$Q$ region below $\sim$0.2${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$, are
also studied and found to be negligible.

The contributions to the systematic uncertainty are listed in Table 1.
Correlations of the systematic uncertainties between different activity
classes are negligible.

Table 1: Fractional systematic uncertainties on the $R$ and $\lambda$ parameters for the three activity classes, as described in the text. The total uncertainty is the sum in quadrature of the individual contributions. Source | Low activity | Medium activity | High activity
---|---|---|---
|  $\Delta R$ [%] |  $\Delta\lambda$ [%] |  $\Delta R$ [%] |  $\Delta\lambda$ [%] |  $\Delta R$ [%] |  $\Delta\lambda$ [%]
Generator tunings | 6.6 | 4.3 |  8.9 | 3.5 | 6.5 | 1.5
PV multiplicity | 5.9 | 5.8 |  6.1 | 4.5 | 3.9 | 4.3
PV reconstruction | 1.8 | 0.1 |  1.4 | 1.2 | 0.1 |  $<$0.1
Fake tracks | 0.4 | 1.1 |  1.7 | 3.9 | 1.1 | 0.8
PID calibration | 1.3 | 0.3 |  0.8 | 0.6 | 2.7 | 0.9
Requirement on pion PID | 2.9 | 1.8 |  1.6 | 0.1 | 1.3 | 0.1
Fit range at low-$Q$ | 1.2 | 1.0 |  1.2 | 1.5 | 1.8 | 2.7
Fit range at high-$Q$ | 1.8 | 0.1 |  2.1 | 0.8 | 2.4 | 1.4
Total | 9.8 | 7.6 | 11.4 | 7.3 | 8.8 | 5.6

##  6 Results

The results of fits to the double ratios for the correlation radius,
chaoticity parameter and $\delta$ parameter for the three different activity
classes are summarised in Table 2, including statistical and systematic
uncertainties, and are presented in Fig. 2.

The dependences of the correlation radius and the chaoticity parameter on the
activity class are shown in Figs. 3 and 4, respectively. As the activity class
increases, the $R$ parameter also increases, while the $\lambda$ parameter
decreases. This confirms previous observations at LEP [19] and in the other
LHC experiments [26, 28, 29, 30]. There are no theoretical predictions for the
BEC effect in $pp$ interactions, however the observed trends are qualitatively
predicted within some theoretical models [41, 66, 67, 68].

Due to the different pseudorapidity coverage of LHCb with respect to other LHC
experiments, the comparison of the measured BEC parameters for a given
multiplicity out of a $pp$ interaction is not straightforward. In the case of
unfolded multiplicities in different pseudorapidity ranges quoted by
experiments, the correspondence can be found using relations obtained from
simulated events. The results for $pp$ collisions at 7 TeV published by the
ATLAS experiment [30] are quoted for unfolded multiplicities in the
pseudorapidity range $|\eta|<2.5$ and $p_{\mathrm{T}}$
$>0.1$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. Pythia 8 is used to determine
the relation for the multiplicity bins defined in the LHCb ($2<\eta<5$) and
ATLAS ($|\eta|<2.5$ and $p_{\mathrm{T}}$
$>0.1$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$) acceptances. The data indicate
that the LHCb results for both $R$ and $\lambda$ are slightly below the ATLAS
ones at 7 TeV. In order to perform a more detailed comparison it would be
necessary to measure the BEC parameters using a full three-dimensional
analysis [69].

It should be noted that the fit quality using the parameterisation, Eq. (3),
is poor (see Fig. 2). The $\chi^{2}$ values are equal to 591, 623 and 621 for
386 degrees of freedom for low, medium and high activity classes,
respectively. The difference between the fitted function and the data points,
visible in the whole $Q$ range, is particularly large in the low-$Q$ BEC
signal region below 0.2${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. This
indicates that the approximate parameterisation of Eq. (3) does not reproduce
the measured distribution properly. Such an effect is observed also by other
experiments [29, 30]. This may introduce an additional systematic uncertainty
in the theoretical interpretation of the fit results.

Table 2: Results of fits to the double ratio $r_{\rm d}(Q)$ for the three different activity classes and corresponding $N_{ch}$ bins, using the parameterisation of Eq. (3). Statistical and systematic uncertainties are given separately. Activity | $N_{ch}$ |  $R$ [fm] | $\lambda$ |  $\delta$ [GeV-1]
---|---|---|---|---
Low | [8,18] | 1.01 $\pm$ 0.01 $\pm$ 0.10 | 0.72 $\pm$ 0.01 $\pm$ 0.05 | 0.089 $\pm$ 0.002 $\pm$ 0.044
Medium | [19,35] | 1.48 $\pm$ 0.02 $\pm$ 0.17 | 0.63 $\pm$ 0.01 $\pm$ 0.05 | 0.049 $\pm$ 0.001 $\pm$ 0.009
High | [36,96] | 1.80 $\pm$ 0.03 $\pm$ 0.16 | 0.57 $\pm$ 0.01 $\pm$ 0.03 | 0.026 $\pm$ 0.001 $\pm$ 0.010

##  7 Summary and conclusions

Using a data sample collected by the LHCb experiment in proton-proton
collisions at a centre-of-mass energy of 7 TeV, the Bose-Einstein correlations
between two indistinguishable pions are studied in the forward acceptance
region of $2<\eta<5$ for single pions with transverse momentum
$p_{\mathrm{T}}$ $>0.1$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. An enhancement
of pairs of same-sign charged pions with small relative momentum related to
the BEC effect is observed. An event-mixed reference sample is used to
determine the signal and the double ratio distributions are fitted using an
exponential parameterisation. The results confirm that the effective size of
the emission region increases as a function of increasing charged-particle
multiplicity, while the chaoticity parameter decreases, as previously observed
at LEP and at the other LHC experiments. The $R$ and $\lambda$ parameters
measured in the forward region in three different charged-particle
multiplicity bins are slightly lower with respect to those measured by ATLAS
for corresponding $pp$ interaction multiplicities.

Figure 2: Results of the fit to the double ratio for like-sign pion pairs with
event-mixed reference samples and the Coulomb effect subtracted for the three
activity classes: (a) low, (b) medium and (c) high activity. The blue solid
line denotes the fit result using the parameterisation of Eq. (3). Only
statistical uncertainties are shown. Figure 3: Correlation radius $R$ as a
function of activity. Error bars indicate the sum in quadrature of the
statistical and systematic uncertainties. The points are placed at the centres
of the activity bins. Figure 4: Chaoticity parameter $\lambda$ as a function
of activity. Error bars indicate the sum in quadrature of the statistical and
systematic uncertainties. The points are placed at the centres of the activity
bins.

## Acknowledgements

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

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LHCb collaboration

R. Aaij40, B. Adeva39, M. Adinolfi48, Z. Ajaltouni5, S. Akar59, J. Albrecht10,
F. Alessio40, M. Alexander53, A. Alfonso Albero38, S. Ali43, G. Alkhazov31, P.
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Appleby56, F. Archilli43, P. d’Argent12, J. Arnau Romeu6, A. Artamonov37, M.
Artuso61, E. Aslanides6, G. Auriemma26, M. Baalouch5, I. Babuschkin56, S.
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Barsuk7, W. Barter56, F. Baryshnikov32, V. Batozskaya29, V. Battista41, A.
Bay41, L. Beaucourt4, J. Beddow53, F. Bedeschi24, I. Bediaga1, A. Beiter61,
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A. Oblakowska-Mucha28, V. Obraztsov37, S. Ogilvy19, R. Oldeman16,f, C.J.G.
Onderwater71, A. Ossowska27, J.M. Otalora Goicochea2, P. Owen42, A.
Oyanguren70, P.R. Pais41, A. Palano14,d, M. Palutan19,40, A. Papanestis51, M.
Pappagallo14,d, L.L. Pappalardo17,g, W. Parker60, C. Parkes56, G. Passaleva18,
A. Pastore14,d, M. Patel55, C. Patrignani15,e, A. Pearce40, A. Pellegrino43,
G. Penso26, M. Pepe Altarelli40, S. Perazzini40, P. Perret5, L. Pescatore41,
K. Petridis48, A. Petrolini20,h, A. Petrov68, M. Petruzzo22,q, E. Picatoste
Olloqui38, B. Pietrzyk4, M. Pikies27, D. Pinci26, F. Pisani40, A. Pistone20,h,
A. Piucci12, V. Placinta30, S. Playfer52, M. Plo Casasus39, F. Polci8, M. Poli
Lener19, A. Poluektov50,36, I. Polyakov61, E. Polycarpo2, G.J. Pomery48, S.
Ponce40, A. Popov37, D. Popov11,40, S. Poslavskii37, C. Potterat2, E. Price48,
J. Prisciandaro39, C. Prouve48, V. Pugatch46, A. Puig Navarro42, H. Pullen57,
G. Punzi24,p, W. Qian50, R. Quagliani7,48, B. Quintana5, B. Rachwal28, J.H.
Rademacker48, M. Rama24, M. Ramos Pernas39, M.S. Rangel2, I. Raniuk45,†, F.
Ratnikov35, G. Raven44, M. Ravonel Salzgeber40, M. Reboud4, F. Redi55, S.
Reichert10, A.C. dos Reis1, C. Remon Alepuz70, V. Renaudin7, S. Ricciardi51,
S. Richards48, M. Rihl40, K. Rinnert54, V. Rives Molina38, P. Robbe7, A.
Robert8, A.B. Rodrigues1, E. Rodrigues59, J.A. Rodriguez Lopez66, P. Rodriguez
Perez56,†, A. Rogozhnikov35, S. Roiser40, A. Rollings57, V. Romanovskiy37, A.
Romero Vidal39, J.W. Ronayne13, M. Rotondo19, M.S. Rudolph61, T. Ruf40, P.
Ruiz Valls70, J. Ruiz Vidal70, J.J. Saborido Silva39, E. Sadykhov32, N.
Sagidova31, B. Saitta16,f, V. Salustino Guimaraes1, C. Sanchez Mayordomo70, B.
Sanmartin Sedes39, R. Santacesaria26, C. Santamarina Rios39, M. Santimaria19,
E. Santovetti25,j, G. Sarpis56, A. Sarti26, C. Satriano26,s, A. Satta25, D.M.
Saunders48, D. Savrina32,33, S. Schael9, M. Schellenberg10, M. Schiller53, H.
Schindler40, M. Schlupp10, M. Schmelling11, T. Schmelzer10, B. Schmidt40, O.
Schneider41, A. Schopper40, H.F. Schreiner59, K. Schubert10, M. Schubiger41,
M.-H. Schune7, R. Schwemmer40, B. Sciascia19, A. Sciubba26,k, A. Semennikov32,
E.S. Sepulveda8, A. Sergi47, N. Serra42, J. Serrano6, L. Sestini23, P.
Seyfert40, M. Shapkin37, I. Shapoval45, Y. Shcheglov31, T. Shears54, L.
Shekhtman36,w, V. Shevchenko68, B.G. Siddi17,40, R. Silva Coutinho42, L. Silva
de Oliveira2, G. Simi23,o, S. Simone14,d, M. Sirendi49, N. Skidmore48, T.
Skwarnicki61, E. Smith55, I.T. Smith52, J. Smith49, M. Smith55, l. Soares
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S. Stefkova55, O. Steinkamp42, S. Stemmle12, O. Stenyakin37, M. Stepanova31,
H. Stevens10, S. Stone61, B. Storaci42, S. Stracka24,p, M.E. Stramaglia41, M.
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S. T’Jampens4, A. Tayduganov6, T. Tekampe10, G. Tellarini17,g, F. Teubert40,
E. Thomas40, J. van Tilburg43, M.J. Tilley55, V. Tisserand4, M. Tobin41, S.
Tolk49, L. Tomassetti17,g, D. Tonelli24, F. Toriello61, R. Tourinho Jadallah
Aoude1, E. Tournefier4, M. Traill53, M.T. Tran41, M. Tresch42, A. Trisovic40,
A. Tsaregorodtsev6, P. Tsopelas43, A. Tully49, N. Tuning43,40, A. Ukleja29, A.
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M. Winn7, J. Wishahi10, W. Wislicki29, M. Witek27, G. Wormser7, S.A. Wotton49,
K. Wraight53, K. Wyllie40, Y. Xie65, Z. Xu4, Z. Yang3, Z. Yang60, Y. Yao61, H.
Yin65, J. Yu65, X. Yuan61, O. Yushchenko37, K.A. Zarebski47, M.
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1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université Savoie Mont-Blanc, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6Aix Marseille Univ, CNRS/IN2P3, CPPM, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9I. Physikalisches Institut, RWTH Aachen University, Aachen, Germany
10Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
11Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
12Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
13School of Physics, University College Dublin, Dublin, Ireland
14Sezione INFN di Bari, Bari, Italy
15Sezione INFN di Bologna, Bologna, Italy
16Sezione INFN di Cagliari, Cagliari, Italy
17Universita e INFN, Ferrara, Ferrara, Italy
18Sezione INFN di Firenze, Firenze, Italy
19Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
20Sezione INFN di Genova, Genova, Italy
21Universita & INFN, Milano-Bicocca, Milano, Italy
22Sezione di Milano, Milano, Italy
23Sezione INFN di Padova, Padova, Italy
24Sezione INFN di Pisa, Pisa, Italy
25Sezione INFN di Roma Tor Vergata, Roma, Italy
26Sezione INFN di Roma La Sapienza, Roma, Italy
27Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
28AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
29National Center for Nuclear Research (NCBJ), Warsaw, Poland
30Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
31Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
32Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
33Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
34Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
35Yandex School of Data Analysis, Moscow, Russia
36Budker Institute of Nuclear Physics (SB RAS), Novosibirsk, Russia
37Institute for High Energy Physics (IHEP), Protvino, Russia
38ICCUB, Universitat de Barcelona, Barcelona, Spain
39Universidad de Santiago de Compostela, Santiago de Compostela, Spain
40European Organization for Nuclear Research (CERN), Geneva, Switzerland
41Institute of Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL),
Lausanne, Switzerland
42Physik-Institut, Universität Zürich, Zürich, Switzerland
43Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
44Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
45NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
46Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
47University of Birmingham, Birmingham, United Kingdom
48H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
49Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
50Department of Physics, University of Warwick, Coventry, United Kingdom
51STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
52School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
53School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
54Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
55Imperial College London, London, United Kingdom
56School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
57Department of Physics, University of Oxford, Oxford, United Kingdom
58Massachusetts Institute of Technology, Cambridge, MA, United States
59University of Cincinnati, Cincinnati, OH, United States
60University of Maryland, College Park, MD, United States
61Syracuse University, Syracuse, NY, United States
62Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
63University of Chinese Academy of Sciences, Beijing, China, associated to 3
64School of Physics and Technology, Wuhan University, Wuhan, China, associated
to 3
65Institute of Particle Physics, Central China Normal University, Wuhan,
Hubei, China, associated to 3
66Departamento de Fisica , Universidad Nacional de Colombia, Bogota, Colombia,
associated to 8
67Institut für Physik, Universität Rostock, Rostock, Germany, associated to 12
68National Research Centre Kurchatov Institute, Moscow, Russia, associated to
32
69National Research Tomsk Polytechnic University, Tomsk, Russia, associated to
32
70Instituto de Fisica Corpuscular, Centro Mixto Universidad de Valencia -
CSIC, Valencia, Spain, associated to 38
71Van Swinderen Institute, University of Groningen, Groningen, The
Netherlands, associated to 43

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