Measurement of Two-Particle Correlations of Hadrons in $e^{+}e^{-}$ Collisions at Belle

The measurement of two-particle angular correlation functions in high-multiplicity $e^+e^-$ collisions at $\sqrt{s}=10.52$ GeV is reported. In this study, the $89.5~{\text{fb}}^{-1}$ of hadronic $e^+e^-$ annihilation data collected by the Belle detector at KEKB are used. Two-particle angular correlation functions are measured in the full relative azimuthal angle ($\Delta \phi$) and three units of pseudorapidity ($\Delta \eta$), defined by either the electron beam axis or the event-shape thrust axis, and are studied as a function of charged-particle multiplicity. The measurement in the thrust axis analysis, with mostly outgoing quark pairs determining the reference axis, is sensitive to the region of additional soft gluon emissions. No significant anisotropic collective behavior is observed with either coordinate analyses. Near-side jet correlations appear to be absent in the thrust axis analysis. The measurements are compared to predictions from various event generators and are expected to provide new constraints to the phenomenological models in the low-energy regime.

The measurement of two-particle angular correlation functions in high-multiplicity e + e − collisions at √ s = 10.52 GeV is reported.In this study, the 89.5 fb −1 of hadronic e + e − annihilation data collected by the Belle detector at KEKB are used.Two-particle angular correlation functions are measured in the full relative azimuthal angle (∆φ) and three units of pseudorapidity (∆η), defined by either the electron beam axis or the event-shape thrust axis, and are studied as a function of chargedparticle multiplicity.The measurement in the thrust axis analysis, with mostly outgoing quark pairs determining the reference axis, is sensitive to the region of additional soft gluon emissions.No significant anisotropic collective behavior is observed with either coordinate analyses.Near-side jet correlations appear to be absent in the thrust axis analysis.The measurements are compared to predictions from various event generators and are expected to provide new constraints to the phenomenological models in the low-energy regime.
PACS numbers: 12.38.-t,12.38.Mh,25.75.-qTwo-particle angular correlations have been extensively studied in search of quark-gluon plasma (QGP) formation and its properties in nucleus-nucleus collisions [1][2][3][4] over the last several decades.In such collisions, a ridge-like structure of the correlation function, residing in a particular phase space where particle pairs have large differences in pseudorapidity but small differences in azimuthal angle, is observed.This signal in relativistic heavy ion collisions is interpreted as the macroscopic consequence of the hydrodynamical expansion of the perfect-fluid-like QGP state with the presence of initial density fluctuations [5][6][7].The ridge-like signal was also observed in high charged-particle multiplicity events in proton-proton, proton-nucleus, deuteron-nucleus, and helium-nucleus collisions [8][9][10][11][12][13][14][15][16].Recently, data on ultraperipheral PbPb photonuclear collisions [17] also resulted in significant second-and third-order flow coefficients, which is an approach to quantify the two-particle azimuthal anisotropy with Fourier harmonics.Essentially, the ridge-like signal is reported in all collision systems involving at least one hadron.However, the physical origin of azimuthal anisotropies in these smaller collision systems is not yet fully understood [18,19].In hadronhadron collisions, the complexity introduced by the initial state cannot be easily factored out.A large number of theoretical models based on different underlying mechanisms such as partonic initial-state correlations [20], finalstate interactions [21,22], and hydrodynamic medium expansion [23] have been proposed to explain the observed ridge-like signal in these small systems.
To break down the question, high charged-particle multiplicity events produced in the even smaller electron-ion and electron-positron collision systems are proposed to provide accessibility to understanding the cause of this special collective behavior [24].As an example, a color dipole configuration (two color strings aligned in parallel with a gap in between) in the e + e − collision system can exhibit anisotropy in the initial parton geometry and generate ridge-like correlations.Recently, experimental studies have been extended to such smaller collision systems, e.g., electron-proton [25] and electronpositron (e + e − ) [26] collisions.No significant ridge-like signal was observed in these measurements.These results have stimulated discussions on the ways to search for and understand possible collectivity signatures in e + e − [27][28][29] and electron-ion collisions [30,31].However, the data samples used for the search in the e + e − ALEPH archived data [26] is small, which motivates the examination of a high-statistics data to study the highest multiplicity tail at Belle.
Taking advantage of the clean environment in e + e − collisions and high-statistics data collected with the Belle detector at KEKB [32], the analysis is performed for the first time at a center-of-mass energy of √ s = 10.52 GeV, which is 60 MeV lower than the Υ(4S) resonance.Overall, a data sample of 89.5 fb −1 is utilized in this analysis, which is the full dataset of collisions at √ s = 10.52 GeV.
This analysis closely follows the previous analysis procedure with ALEPH archived data [26].Although the average event multiplicity is lower than the ALEPH data, the two-particle correlation analysis is performed on the largest off-resonance Belle dataset, whose results can solidify previous findings.The Belle hadronic-event dataset is about four times larger than that with ALEPH archived data, enabling the collectivity search to move forward from a scan amongst the 0.5% highest multiplicity events of the total distribution to that of 0.02% percentile.Moreover, the measurement of the two-particle correlation function in the low-energy regime can provide additional inputs to the phenomenological fragmentation models.
The Belle detector is a large-solid-angle magnetic spectrometer that consists of a silicon vertex detector, a 50layer central drift chamber, an array of aerogel threshold Cherenkov counters, a barrel-like arrangement of timeof-flight scintillation counters, and an electromagnetic calorimeter (ECL) comprising CsI(Tl) crystals located inside a superconducting solenoid coil that provides a 1.5 T magnetic field.An iron flux-return located outside of the coil is instrumented to detect K 0 L mesons and muons.The detector is described in detail elsewhere [33].
The hadronic-event selection [34], including requirements on event multiplicity and energy sum in the ECL, is adopted to suppress contamination from two-photon, radiative Bhabha, and other QED events.Particles used in the calculation of the correlation functions are pri-mary charged tracks, defined as prompt tracks or decay products of intermediate particles with proper lifetime τ < 1 cm/c.The corresponding selection on experimental data is tracks produced from the interaction point (including from short-lifetime particle decays) and tracks from long-lifetime particle decays with V r < 1 cm, where V r is the distance in the transverse plane of the decay vertex from the interaction point.The latter can intersect with other track on a common space-point and has dihadron invariant mass consistent with the mass of a K 0 S (0.480-0.516GeV/c 2 ) or Λ 0 / Λ0 (1.111-1.121GeV/c 2 ) candidate; otherwise, the track is deemed as a prompt track and accepted in the primary particle selection.Charged tracks are required to be within the detector acceptance, i.e., with polar angles ranging from 17 • -150 • (+z is defined opposite to the e + beam), and have transverse momenta in the center-of-mass frame greater than 0.2 GeV/c.Impact parameter requirements are adopted to select charged tracks within ±2 cm with respect to the interaction point in the transverse plane, and ±5 cm along the z direction.For a pair of neighboring lowmomentum tracks with the absolute value of cosine opening angle greater than 0.95 and transverse momenta below 0.4 GeV/c, the one with less momentum is deemed as a duplicated track and is hence removed.Tracks from photon conversion candidates are vetoed, with the latter selected with the invariant mass less than 0.25 GeV/c 2 and the decay-vertex radius greater than 1.5 cm.
To eliminate the effects of the nonuniform detection efficiency and misreconstruction bias, efficiency correction factors are derived by the Belle Monte Carlo (MC) sample.The Belle MC is simulated based on evtgen [35] and pythia6 [36], where hadronic q q (q = u, d, s, c) fragmentation as well as low-multiplicity e + e − → τ + τ − and two-photon processes are taken into account.The detector response is simulated with GEANT3 [37].The MC sample is further reweighted to match event multiplicity and thrust distributions of the data in order to correct for the imperfection in MC simulation.In order to study the multiplicity dependence of the correlation function, the events are binned into multiplicity classes using the reconstructed track multiplicity, denoted N rec trk , by counting tracks after all selections.For low-multiplicity events with N rec trk < 12, only a sample size of 11.5 fb −1 is used.The multiplicity classes used in this study, their corresponding fraction of data, and the mapping of average reconstructed multiplicities N rec trk to average multiplicities after efficiency correction N corr trk are listed in Table I.
The Belle experiment is operated with the KEKB asymmetric energy collider, colliding the 8 GeV electron beam and the 3.5 GeV positron beam.We boost events to their center-of-mass frame to perform the angular correlation analysis.Following a procedure similar to what has already been established in Ref. [10], the two-particle correlation function observable is defined as The correlation function is expressed in terms of the particle pair's angular difference ∆η = ±(η i − η j ) and ∆φ = ±(φ i − φ j ), where i, j label the track pair's indices.The calculation is based on an assumption of correlations being symmetric about the origin (∆η, ∆φ) = (0, 0); hence, four entries are counted for a given pair.The percharged-particle associated track-pair yield is denoted as N pair , and is reweighted by efficiency correction factors of both particles.The signal correlation S(∆η, ∆φ) and the background correlation B(∆η, ∆φ) can be explicitly written out as where N same (N mix ) counts the number of track-pairs formed by matching the i-th charged particle of a given event with the j-th particle in the same event ("mixed event" [8]).A mixed event in this work is a combination of tracks from three random events and is normalized by a factor of 1/3.Three random events are chosen such that their N rec trk 's are the same as that of the event they match to.The B(0, 0) factor is incorporated in the calculation of the correlation function, serving as the normalization of the artificially constructed B(∆η, ∆φ).This factor is obtained by extrapolating the function value to the origin of B(∆η, ∆φ).An additional correction on the correlation function is applied to deal with the effects introduced by using finite-bin histogramming to approximate the density function.The bin-size effect is modeled by a second-order polynomial and the magnitude of the correlation function is calibrated.To unfold back to the truth level, final correlations are corrected with the bin-by-bin method [38], accounting for residual reconstruction effects after efficiency corrections.
The two-particle correlation function is explored in two coordinate systems: beam and thrust axis coordinates in the e + e − center-of-mass frame.The former is the same as that presented in most of the two-particle correlation studies, while in the latter, initiated by Ref. [26], the event thrust axis [39] is used as the reference axis, with missing momentum of the event included.The construction of mixed events in the thrust axis analysis is identical to that in the beam axis analysis, requiring the multiplicity matching only.In the thrust axis coordinate analysis, the kinematics (p T , η, φ) of a mixed event are calculated with respect to the thrust axis of its matched physical event.To adjust the kinematics distribution of the mixed event to physical kinematic (p T , η, φ) spectra, a reweighting correction is adopted.
In the e + e − annihilation process, when the interacting system is located in between or along the color string connecting the q q pair, measuring with a coordinate system defined by the event thrust axis provides a more direct picture.From the viewpoint of relativistic fluid dynamics [7], conventional beam-axis measurements are sensitive to features within the plane transverse to the collision axis, probing any anisotropic behavior of the QCD medium, which are widely studied as the phenomena of elliptic or triangular flow [6,40,41].The insensitive region of the two-particle correlation function in the beam axis analysis is at the beam pipe, where a particle pair with a large pseudorapidity difference is excluded from the finite ∆η region of interest (e.g., |∆η| ≤ 3.0 in this analysis).In addition, the on-axis track-pair correlation is too deformed to form an obvious correlation structure, since the φ coordinate is ill-represented near both poles of the spherical coordinate.Correspondingly, the insensitivity of the thrust-axis correlation function is at its reference thrust axis which quark-initiated dijets are close to; however, they are sensitive in the mid-rapidity region, where additional soft gluons emit apart from the leading quark-antiquark dijet-like structure.The sensitivity to the finer structure allows one to check in details if there are special correlations among the color activity in the small system.
In Fig. 1, correlation functions with multiplicity N rec trk ≥ 12 are shown for both beam and thrust axis coordinates.In the beam axis coordinate view, the peak near the origin (∆η, ∆φ) = (0, 0) has contributions from pairs originating in the same jet, while the structure at ∆φ ≈ π is from back-to-back correlations.These features reflect the two-particle correlation of dijet-like q q events, which mainly contribute in e + e − collisions.In contrast, for the thrust axis coordinates, the dominant structure is the hill-like bump near (∆η, ∆φ) ≈ (0, π), while a sizeable near-side correlation is lacking.The decrease of the nearside-peak correlation is because that leading two jets are brought to insensitive regions around poles of the reference thrust axis.As a result of balance for the event thrust calculation, track pairs amongst on-axis jets tend to yield larger ∆φ angular differences.Compared to collisions at high center-of-mass energies, jets are composed of fewer constituents and have broader shapes at the low energy regime.This makes it hard to form a significant near-side-peak correlation.We calculated the magnitude of the near-side-peak correlation with respect to different collision energies with sherpa 2.2.5 [42] simulation of e + e − → hadrons at the leading order, and found results suggesting a significant correspondence.Evidence for the ridge signal can be best examined in the azimuthal differential yield Y (∆φ) by averaging the correlation function over the long-range region with 1.5 ≤ |∆η| < 3.0.A "zero yield at minimum"(ZYAM) method [43] is further implemented to separate any enhanced near-side correlation around ∆φ = 0 distinct from a constant correlation.The constant contribution along ∆φ, denoted as C ZYAM , is estimated by the minimum of the fit with a third-order Fourier series to the data points.A fit with a third-order polynomial plus a cosine term, and with a fourth-order polynomial are also checked to estimate C ZYAM in parallel.
The systematic uncertainties due to the selection and correction operations are calculated with respect to longrange Y (∆φ).Hadronic-event selection is examined by tightening the energy sum requirement in the ECL from E sum > 0.18 √ s to 0.23 √ s.The primary-particle selection systematic is estimated by making variations of the proper lifetime requirement τ < 1 cm/c and the vertex V r < 1 cm.In general, results from both variations differ by less than 1% (or by 1-6% for the high-multiplicity bin).A 0.35% uncertainty is quoted for the track reconstruction efficiency, which is evaluated by comparing partially and fully reconstructed D * → π slow D 0 (→ π + π − K 0 S (→ π + π − )) decays [44].In the beam axis analysis, the systematic uncertainties are mainly from the primaryparticle selection and the tracking efficiency, which are in general on the order of 0.3%-0.4%,while the primaryparticle selection increases to a 6.3% uncertainty for highmultiplicity events with N rec trk ≥ 14.For high-multiplicity event classes in the thrust axis analysis, dominant sources of systematic uncertainties are due to the event selection (< 2%) and the primary-particle selection (< 4%), where the estimation of uncertainties suffers from the need for large statistics to derive a precise reweighting factor for the efficiency correction and for the mixed events.On the other hand, for low-multiplicity classes, the dominant source of uncertainty is due to tracking.Other uncertainties originate from MC reweighting, the B(0, 0) factor, mixed events reweighting, scaling corrections due to bin effects and residual bin effects, all of which are checked to be small in different multiplicity bins (of order O(10 −4 )), with the largest one contributing up to 0.3% uncertainty.
Figure 2 shows the measurement of long-range Y (∆φ) after performing the ZYAM method, along with a comparison of predictions from Belle MC, herwig 7.2.2 [45], and sherpa 2.2.5 [42] event generators.The region with small azimuthal angle difference (∆φ ≈ 0) is where possible ridge signals would be visible as a nonzero value.In the beam axis coordinates, all generators are consistent with data in the near-side ridge region, but herwig and sherpa undershoot the data in the away-side region.In the thrust axis analysis, the Belle simulation, with specific tunes adapted to Belle data, gives again a better description of these correlation data.A larger discrepancy from the data is seen in the herwig simulation.An excess of correlations is showing up in the near-side ridge-prone region and there is also an overshoot in the away-side region.
The significance of any ridge signal can be quantified by integrating over Y (∆φ) from ∆φ = 0 to where the ZYAM fit minimum occurs.Ridge yields smaller than an order of 10 −10 are measured.Since there is no obvious ridge-like structure in either the beam or thrust axis analysis, a bootstrap procedure [46] is implemented and the confidence limit of the integrated ridge yield is reported.In the bootstrap procedure, each azimuthal differential yield distribution is varied according to their statistical and systematic uncertainties, and the yield distribution is sampled 2 × 10 6 times.For the ZYAM subtraction, three fit templates (a third-order Fourier series, a third-order polynomial plus a cosine term, and a fourth-order polynomial) are attempted, of which the most conservative confidence limit is quoted.In Fig. 3, the 95% confidence level upper limits as a function of N corr trk are reported.For the ridge yield upper limit less than 10 −7 , we report the confidence levels of the ridge signal exclusion, instead.
In summary, the first measurement of two-particle correlations of hadrons in beam and thrust axis coordinate systems, performed using e + e − collision data at √ s = 10.52 GeV from Belle is reported.A strong exclusion of ridge yield in the beam axis coordinate analysis is set.In the thrust axis coordinate analysis, there is no significant near-side-peak correlation, different from what was observed in hadronic collisions or high-energy e + e − collisions.A 95-97% confidence level upper limit or a 5σ exclusion is set versus N corr trk for the absence of any ridge yield in our measurement.Though there is no hint Thrust axis limits are shifted to the right by 0.5 for presentation purposes.The label "> 5σ" indicates the 5σ confidence level upper limit.
of collectivity signals in the low-energy e + e − collision system, the measurement can be provided as a reference for tunes of fragmentation models in the soft QCD scale.Belle MC samples based on evtgen(with a pythia6 interface for e + e − → q q generation), along with herwig and sherpa event generators are examined.Similar to the conclusion from the previous analysis with ALEPH archived data [26], the results in this study are found to be better described by pythia6 than sherpa or herwig.

FIG. 2 :
FIG. 2: Comparison of ZYAM-subtracted Y (∆φ) in the range 1.5 ≤ |∆η| < 3.0 for the multiplicity 12 ≤ N rec trk < 14 (top) and N rec trk ≥ 14 (bottom), where subtracted constants of data C Data ZYAM are quoted.In each panel, results for beam (left) and thrust (right) axis analyses are displayed.The colored bands show simulation predictions from Belle MC (green), herwig (blue), and sherpa (violet).The error bars on the data represent the statistical uncertainties, and the gray boxes are systematic uncertainties.For visual purposes, the minimal statistical uncertainty of the MC colored bands is set to be 3 × 10 −4 , and beam-axis ZYAM-subtracted yields are scaled by a factor of 0.025.

FIG. 3 :
FIG.3: Upper limits of the ridge yield as a function of N corr trk in beam axis (red) and thrust axis (black) coordinate frames.Thrust axis limits are shifted to the right by 0.5 for presentation purposes.The label "> 5σ" indicates the 5σ confidence level upper limit.

TABLE I :
Average multiplicities and corrected multiplicities of different N rec trk intervals.