Measurement of Bose-Einstein Correlations in pp Collisions at sqrt(s)=0.9 and 7 TeV

Bose-Einstein correlations between identical particles are measured in samples of proton-proton collisions at 0.9 and 7 TeV centre-of-mass energies, recorded by the CMS experiment at the LHC. The signal is observed in the form of an enhancement of number of pairs of same-sign charged particles with small relative momentum. The dependence of this enhancement on kinematic and topological features of the event is studied.


Introduction
In particle collisions, the space-time structure of the hadron emission region can be studied using measurements of Bose-Einstein correlations (BEC) between pairs of identical bosons. Since the first observation of BEC in proton-antiproton interactions fifty years ago [1], a large number of measurements have been performed by experiments using different initial states [2,3]. At the CERN Large Hadron Collider (LHC), BEC were observed for the first time by CMS using data at centre-of-mass energies √ s = 0.9 and 2.36 TeV, collected in 2009 [4]; measurements by ALICE at 0.9 TeV were reported in [5]. The present paper reports measurements using data taken in 2010 at 0.9 TeV, with a sample increase by a factor 15, and at 7 TeV, for the first time. The analysis method is similar to that in [4], where more details can be found. In this article the results at the two energies are compared and additional studies are performed.
Constructive interference affects the joint probability for the emission of a pair of identical bosons with four-momenta p 1 and p 2 . Experimentally, the proximity in phase space between final-state particles is quantified by the Lorentz-invariant quantity Q = −(p 1 − p 2 ) 2 = M 2 − 4m 2 π , where M is the invariant mass of the two particles, assumed to be pions with mass m π . The BEC effect is observed as an enhancement at low Q of the ratio of the Q distributions for pairs of identical particles in the same event, to that for pairs of particles in a reference sample that by construction is expected to include no BEC effect: The ratio is fitted with the parameterization R(Q) = C [1 + λΩ(Qr)] · (1 + δQ).
In a static model of particle sources, Ω(Qr) is the Fourier transform of the space-time region emitting bosons with overlapping wave functions, characterized by an effective size r. The parameter λ measures the strength of BEC for incoherent boson emission from independent sources, δ accounts for long-distance correlations, and C is a normalization factor. The correlation function is often parameterized as an exponential Ω(Qr) = e −Qr or with a Gaussian form Ω(Qr) = e −(Qr) 2 . Other forms have also been used ([6] and references therein), and several of these are mentioned below. In addition a formulation aimed at describing the time evolution of the source [7] is considered and compared with the data.

Data and Track Selection
A detailed description of the CMS detector can be found in [8]. The central feature of the CMS apparatus is a superconducting solenoid of 6 m internal diameter, providing an axial magnetic field of 3.8 T. The inner tracking system is the most relevant detector for the present analysis. It is composed of a pixel detector with three barrel layers at radii between 4.4 and 10.2 cm and a silicon strip tracker with 10 barrel detection layers extending outwards to a radius of 1.1 m. Each system is completed by two endcaps, extending the acceptance up to a pseudorapidity |η| = 2.5. The transverse-momentum (p T ) resolution, for 1 GeV charged particles, is between 0.7% at η = 0 and 2% at |η| = 2.5.
Minimum-bias events were selected by requiring activity in both beam scintillator counters [9]. Charged particles are required to have |η| < 2.4 and p T > 200 MeV, ensuring that particles emitted from the interaction region cross all three barrel layers of the pixel detector and thus have good two-track separation. To achieve a high purity of the primary track selection, the

Definition of Signal and Reference Samples
trajectories are required to be reconstructed in fits with more than five degrees of freedom (N dof ) and χ 2 /N dof < 5.0. The transverse impact parameter with respect to the collision point is required to be less than 0.15 cm. The innermost measured point of the track must be within 20 cm of the beam axis, in order to reduce contamination from electrons and positrons from photon conversions in the detector material and secondary particles from decay of long-lived hadrons.
For this analysis a total of 4.2 million events were selected at √ s = 0.9 TeV, with 51.5 million tracks passing the selection criteria. At 7 TeV, 2.7 million events with 51.7 million tracks were selected from data taken during low-intensity runs. Neither of the two energy samples is affected by event pileup. Several minimum-bias Monte Carlo (MC) samples were generated, followed by detailed detector simulation based on the GEANT4 package [

Definition of Signal and Reference Samples
All pairs of same-sign charged particles with Q between 0.02 and 2 GeV are used for the measurement. The lower limit is chosen to avoid cases of tracks that are duplicated or not well separated, while the upper limit extends far enough beyond the signal region (confined to Q < 0.4 GeV) to allow verification of a good match between signal and reference samples. The Q resolution in the signal region is better than 10 MeV. Coulomb interactions between charged particles modify their relative momentum distribution. This effect, which differs for pairs with same charge (repulsion) and opposite charge (attraction), is corrected using Gamow factors [16]. As discussed in [4], the reference sample in the denominator of Eq. (1) can be defined in several ways: opposite-charge pairs; opposite-hemisphere pairs, where particles are paired after inverting the three-momentum of one of them, this procedure being applied to pairs with same and opposite charges; rotated particles, where pairs are constructed by inverting the x and y components of the three-momentum of one of the two particles; pairs from mixed events. In the case of pairs from mixed events, particles from different events are combined with the following methods: i) events are mixed at random; ii) events with similar charged-particle multiplicities in the same η regions are selected; iii) events with an invariant mass of all charged particles similar to that of the signal are used to form the pairs. In this paper, we use the sample obtained by pairing same-sign charged particles from different events that have similar charged-particle multiplicities in the same η regions. This method avoids the possible effect of remaining correlations [17] between particles within the same event. The r.m.s. spread of the results obtained from the different samples is taken as a conservative systematic uncertainty. In [4] an additional, "combined" reference sample was obtained by summing the Q distributions of the seven corresponding reference samples. It has been checked that the results obtained with the "combined" reference sample are compatible within errors with those presented here.
In order to reduce possible biases in the construction of the reference sample, a double ratio R is defined, where the subscripts "MC" and "MC, ref" refer to the corresponding distributions from the simulated events, generated without BEC effects. Figure 1 shows the distributions of the double ratio R for Q > 0.02 GeV and both centre-ofmass energies, computed using the tune Z2 of the PYTHIA6.422 simulation, which best describes the measured track distributions (in particular the charged-particle multiplicity). The shapes fitted with the exponential parameterization Ω(Qr) = e −Qr in Eq. (2) are superimposed and the results of the fits are given in Table 1. The values of the two parameters r and λ are basically related to different features of the distributions: the width of the peak at small Q to r and the height to λ; they are, however, strongly correlated, with correlation coefficients of about 86%. The fit quality is poor, as can be seen from the values of χ 2 /N dof . Gaussian parameterizations, which are used by some experiments, provide values of χ 2 /N dof larger than 9, which confirms the observation in [4] that an exponential parameterization is preferred. Compared to an exponential shape, alternative functions, as defined in [18,19], and the Lévy parameterization, Ω(Qr) = e −(Qr) α [20], yield fits of only slightly better quality. However, the latter better describes the data at low Q, with values of α < 1 as in [4]. As a cross-check of the stability of the measurement of the width of the peak at small Q and of the fact that it does not depend on the fit quality, the average values (first moment) of the Ω(Qr) distributions over the same interval in Q are found to be consistent for the different functions. More discussion on the shape of the R distribution and on the fit quality can be found at the end of this section. As discussed  3), for data at √ s = 0.9 (left) and 7 TeV (right). The reference sample is obtained from same-sign charged particles from mixed events with similar multiplicities, and the MC simulation is PYTHIA6.4 tune Z2. The lines are the results of fits using the exponential parametrization for Ω(Qr), with the values of the parameters given in the text. The error bars on the data points are statistical only.  Table 2, and the total systematic uncertainties are obtained from their quadratic sum. It was checked that reducing the fit range to 0.04 < Q < 2 GeV, thus excluding the first two points at low Q in Fig. 1, gives consistent results within errors. The BEC parameters are thus measured to be As will be shown below, the increase of r is related to the different average charged-particle multiplicities at the two energies, while the value of the λ parameter is stable. The BEC signal is studied as a function of the charged-particle multiplicity in the event, N ch , as in [4], and of the pair average transverse momentum k T , defined as half of the absolute vector sum of the two transverse momenta, k T = |p T,1 + p T,2 |/2. A dependence on k T has been observed at the Tevatron [22] and at RHIC [23], where it is associated with the system collective expansion. Figure 2 shows the double ratio R as a function of Q for different values of N ch and k T . The k T dependence of the r and λ parameters for three intervals of multiplicity, obtained with the exponential parameterization, is shown in Fig. 3 and given in Table 3 for the 0.9 and 7 TeV data. The effective radius r is observed to increase with multiplicity, for all reference samples and for all MC models and tunes, in agreement with previous results. At low multiplicity, r is approximately independent of k T , and it decreases with k T as N ch increases. The λ parameter decreases with increasing multiplicity and k T . The systematic uncertainties are estimated to be the same as for the overall measurements (4.2% and 6.3% for λ, 7.9% and 10.1% for r, at 0.9 and 7 TeV, respectively). It should be noted that these uncertainties are  Figure 2: Distributions of the double ratio R as a function of Q, for three intervals in k T and three intervals in charged-particle multiplicity in the event, N ch , for √ s = 7 TeV. The lines are the results of fits using the exponential parametrization for Ω(Qr), with the values of the parameters given in Table 3. The error bars are statistical only.
As was noted above and can be deduced from the χ 2 /N dof in Table 1, none of the quoted functions is able to provide a good description of the R distributions. This is due to an anticorrelation effect between same-sign charged particles for Q values just above the signal region (dip with R < 1), as shown in Fig. 5. This anticorrelation is observed in the double ratio at both energies with any choice of reference sample and MC simulation. It shows little sensitivity to  Figure 3: Values of the parameters r (top) and λ (bottom), as a function of k T in three intervals of charged-particle multiplicity in the event, N ch , for √ s = 0.9 (left) and 7 TeV (right). The points are presented at the position corresponding to the mean value of k T in the considered interval of N ch . The error bars are statistical only (in some cases they are smaller than the marker size). The systematic uncertainties are discussed in the text. Table 3: Results of fits using the exponential parametrization for Ω(Qr) to the double ratios R, for three intervals in k T and three intervals in charged-particle multiplicity in the event, N ch , for √ s = 0.9 and 7 TeV. The errors are statistical only. The systematic uncertainties, which are point-to-point correlated, are estimated from the relative uncertainties affecting the overall measurements (see text). k T , while it decreases with increasing charged-particle multiplicity in the event, as shown in  Figure 5: Detail of the distribution of the double ratio R for √ s = 0.9 (left) and 7 TeV (right). The dotted lines correspond to fits with Eq. (4), and the solid lines to exponential fits. Note the enlarged scale on the y axis. The error bars are statistical only. Fig. 6 for the 7 TeV data. This detailed observation is made possible by the large data samples studied here, and constitutes the first evidence of this effect at the LHC. Such a structure was observed in e + e − collisions at LEP [25]. The presence of a region of anticorrelation between same-sign charged particles has been explained in [7]. In this reference, a parameterization for R(Q) has been proposed, aimed at describing the time evolution of the source: Here r 0 is related to the proper time of the onset of particle production, and r enters in both the exponential and the oscillation factors. As visible in Fig. 5, fits with Eq. (4) are of good quality, much better than with an exponential function, and the χ 2 /N dof is 1.1 both for the 0.9 and 7 TeV data samples. The depth of the dip in the anticorrelation region is measured as the difference ∆ between the baseline curve defined as C · (1 + δQ) and the value of R defined by Eq. (4) at its minimum. Results are shown in Fig. 7. The depths are found to decrease with N ch consistently for the two centre-of-mass energies. The systematic errors have been computed from the r.m.s. spread of the results obtained with the various reference samples and MC simulations. It has been checked that these results are robust: when the fitting range is extended to Q = 5 GeV the results are consistent within errors and the trend is similar.

Conclusions
Bose-Einstein correlations have been measured using data collected with the CMS experiment in proton-proton collisions at the LHC, with centre-of-mass energies of 0.9 and 7 TeV. The signal is observed as an enhancement of pairs of same-sign charged particles with small relative momentum. The parameters are obtained from fits using the exponential parametrization for Ω(Qr) to the distribution of Q. In agreement with previous results, an increase of the effective emission radius r with charged-particle multiplicity in the event is observed, which accounts for the increase of r from 0.9 to 7 TeV. The parameter r is nearly independent of the average transverse momentum of the pair of particles, and decreases with k T in events with large charged-particle multiplicity. Anticorrelations between same-sign charged particles are observed for Q values above the signal region as previously reported with LEP data. The anticorrelation effects decrease with increasing charged-particle multiplicity in the event.