Multiplicity distributions in the forward rapidity region in proton-proton collisions at the Large Hadron Collider
Measured multiplicity distributions of primary charged particles produced in the forward rapidity region of the () collisions at the centre-of-mass energy, = 7 TeV at the Large Hadron Collider (LHC) have been analyzed in terms of the Negative Binomial Distribution (NBD) function. Like the multiplicity distributions in the mid-rapidity region for the collisions at = 7 TeV, the distributions for the minimum bias events in the forward region also are better described with the superposition of two-NBDs, as proposed by a two-component model of particle production from two processes, the ”” and the ””. However, the multiplicity distribution for the ”hard-QCD” events in a large pseudorapidity window does not oblige the two-component model.
The major experiments at the Large Hadron Collider ref01 (), depending on specific physics requirements, have detector setups of different geometrical acceptance for detecting several kinds of particles in different kinematic ranges. Beside the specific physics goals, all these detector setups facilitate study of the physics of the collisions, in general, by implicitly recording information on particle productions in terms of a few basic observables. Sometimes, comparisons of data recorded in different acceptance of detectors of these experiments could provide insights to the particle production mechanisms in different phase-space of collisions, due to different kinematic origin. In this respect, out of the four major experiments at the LHC, the Large Hadron Collider Beauty (LHCb) experiment has a unique standing. While A Large Ion Collider Experiment (ALICE), the Compact Muon Solenoid (CMS) experiment and A Toroidal LHC Apparatus (ATLAS) experiment primarily address the mid-rapidity physics by measuring majority of the produced charged particles in the mid-rapidity region, the LHCb setup allows measurement of charged particles in the forward rapidity region, facilitating the study of forward physics.
The multiplicity distribution of primary charged particles produced in collisions is one of the most basic observables, characterizing the final states of multi-particle production process in a high energy physics experiment. All the LHC experiments have measured ref02 (); ref03 (); ref04 (); ref05 (); ref06 () multiplicity distributions in proton-proton () - collisions at the available LHC energies in different kinematic ranges and for different classes of events. In the context of the present work, these LHC experiments, in spite of the differences in detector acceptance, have a common observation - the multiplicity distributions of produced particles at the new LHC energies have been found to be underestimated by several of the standard event generators / models (like PYTHIA, PHOJET etc.) in use. This observation has made the study of multiplicity distribution at LHC energies all the more interesting.
In this article, we analyze the primary charged particle multiplicity distributions in the forward rapidity region in proton-proton () - collisions at = 7 TeV, as measured by the LHCb experiment at LHC, in terms of parameters of the Negative Binomial Distribution(NBD) function. The two-parameter NBD function, as given below in Eq. - (1) played a significant role in describing multiplicity distributions of produced charged particles in the mid-rapidity region in (and ) collisions for a wide range of the centre-of-mass energy, including = 7 TeV.
where is the average multiplicity and the parameter is related to dispersion , () by
The NBD function could describe the charged particle multiplicity distributions in () collisions at = 540 GeV at the Super Proton Synchrotron (SPS) ref07 () at CERN in the full pseudorapidity () space as well as in limited pseudorapidity intervals (for high momentum low mass particles, the rapidity can be approximated to the pseudorapidity, , where is the polar angle of the particle with respect to the counterclockwise beam direction). At = 900 GeV SPS energy, however, the single NBD function could describe the data only for small pseudorapidity intervals at the mid-rapidity region. With the appearance of sub-structures in multiplicity distributions at higher energies and in larger pseudorapidity intervals, the weighted superposition or convolution of more than one function including one NBD function, as proposed by several models, ref08 (); ref09 (); ref10 (); ref11 () representing more than one source or process of particle productions could explain the data better. Such sub-structure in SPS data at 900 GeV and in Tevatron data at 1.8 TeV ref12 () could be well explained by weighted superposition of two NBD functions ref09 (), as given by Eq. - 3. The multiplicity distributions of primary charged hadrons in Non-Single Diffractive (NSD) events in - collisions at = 7 TeV in the mid-rapidity region also could be well explained ref13 () by the two-NBD function.
where is the fraction of ”soft” events and is a function of only. The other parameters, functions of both, the and the , have usual meanings as described for Eq. - (1) with suffixes in parameters indicating respective components.
At this point, discussing other models or approaches of multi-particle production involving NBDs would be relevant. The framework of the weighted superposition mechanism of different classes of events has been extended from the two-component to a three component model ref14 () for explaining possible new physics at LHC at = 14 TeV. The third component would attribute to the eventual new class of high multiplicity events which would be manifested by the appearance of a new elbow structure’ in the tail of multiplicity distribution of the collisions at the highest planned centre-of-mass energy at the LHC. So far, the multiplicity distributions for the collisions up to = 7 TeV are available and these distributions have no such new structure, in the tail of the distributions, which calls for application of the model. A very recent theoretical approach ref15 (), following Glasma flux tube model, has shown that the multiplicity distribution of multi-particle production could be described by convolution of a number of NBD functions as a natural consequence of several impact parameters of the collisions. The model reproduces the multiplicity distributions data of collisions in small pseudorapidity window ( 0.5) at the LHC energies. The scope of the present work is, however, restricted to the analysis of the LHCb data in terms of a single NBD and a superposition of two NBDs, as prescribed by the two-component model of Ref. ref09 ().
According to the two-component model of Ref.- ref09 (), the multiplicity distribution of hadronic collisions can be explained by weighted superposition of two NBDs, representing two classes of events, “semihard - events with minijets or jets” and “soft - events without minijets or jets”. It is note worthy that the ”semihard” events involving hard parton-parton scatterings (due to high momentum transfer) resulting in QCD jets of high transverse momentum above a certain threshold is also referred to as ”hard-QCD” events.
The LHCb experiment has measured ref06 () multiplicity distributions of primary charged particles produced in the collisions at = 7 TeV in the pseudo rapidity coverages, and for two classes of events: the minimum bias and the hard-QCD. The hard-QCD events were chosen out of the minimum bias events by selecting events with at least one particle with transverse momentum greater than 1 GeV/c. The multiplicity distributions for both the event-classes were measured for small pseudo rapidity windows of width scanning over the - range of the detector coverage as well as for the wide - window, (). We analyze these distributions in the forward-rapidity in terms of the NBD that has been successful in describing the mid-rapidity data. We discuss the results, comparing with observations in similar analysis of data at the same at the mid-rapidity region. For the mid-rapidity region, we consider the distributions, measured by the CMS experiment ref04 (), as there exists ref13 () similar phenomenological study of the CMS data in terms of the NBD-formalism.
Iii Analysis and Discussions
We fit the multiplicity distributions of primary charged particles, as measured ref09 () by the LHCb, in the five pseudorapidity windows of width in the - range, for the minimum bias events with the NBD function as given in Eq. -1. The Table - 1 contains the values of the parameters obtained by the best fits, along with the the corresponding values of .
As can be seen from the values, the single NBD function is far from a satisfactory description of the multiplicity distribution for the minimum bias events. The fitted values of the NBD parameters, however, show consistent dependence on the position of the - bin. The average multiplicity () decreases and the -parameter increases indicating broader distributions in the psedurapidity bins in the more forward regions.
The multiplicity distributions for the minimum bias events could be better described by the weighted superposition of two NBDs as can be seen from the plots in Fig. 1, where the multiplicity distributions along with the single NBD and the two-NBD fits have been plotted. The deviation for the single NBD fits is more for the distributions in the more forward region. Though from the plots in the Fig. 1 and the , as listed in the Table - 2, it is clear that two-NBD describes the minimum bias data better, the values of the fit parameters with large errors in these small rapidity intervals, as tabulated, are not suitable to reveal systematic behavior of the parameters. At this point, we recollect that the multiplicity distributions of the charged hadrons produced in Non-Single Diffractive (NSD) events of collisions at = 7 TeV ref04 () in overlapping pseudorapidity bins of different widths, = = 0.5 to 2.4, also fit better ref13 () to the two-NBD than a single NBD function. It is worth mentioning, however, that the Clan-structure description of Ref.- ref09 () failed to match the mid-rapidity LHC data ref12 (); ref13 ().
In the context of the hard-QCD events, it may be noted that the LHCb experiment selected an event with at least one particle with transverse momentum greater than 1 GeV/c in the range 2.5 4.5, as a ”hard” event. Similar approach was adopted ref16 () by the CDF experiment at Tevatron, Fermilab where two isolated sub-samples, soft and hard, were analyzed separately to reveal that the properties of the soft sample were invariant as a function of the centre-of-mass energy. The CDF experiment isolated events considering the events with no particle of transverse energy, 1.1 GeV as ”soft” event. Though, none of the other experiments of LHC has measured multiplicity distribution for the so-called hard-QCD events, the invariance of multiplicity distribution of soft events as a function of has been revealed ref13 () in the analysis of the data of the CMS experiment ref04 () in terms of two-NBD. Considering that the two-component model of particle productions is valid in the forward region and that the criterion for isolating the hard-QCD events is proper, one may expect the multiplicity distributions for the hard-QCD events to be well described by a single NBD function only.
We fit a single NBD function to the multiplicity distributions of the produced primary charged particles for the hard-QCD events of LHCb experiments ref05 () in small non-overlapping pseudo rapidity bins. The relevant plots are depicted in the Fig. 2. The plots in the Fig. 2 show that the single NBD function fits reasonably well to the multiplicity distributions in small - windows. The values of for the respective plots are given in Table - 3. For two of - windows, however, the values of are not satisfactory. The values of the parameters and , as tabulated in Table - 3 show systematic trend, the decreases and the increases with shift of the -window more towards forward rapidity. On the whole, the single NBD appears to describe the multiplicity distributions for the hard-QCD events in small -windows in the forward region.
We continue to fit the NBD function to the multiplicity distributions for the wider pseudo rapidity range, in the - range for both the event classes. As can be seen from the values obtained from the best-fit methods and tabulated in the Tables - 4, both the distributions do not fit a single NBD function. This led us to the consideration of the weighted superposition of two-NBD in describing both the event-classes. In case of hard-QCD events, of course, the terminology of the two-component model in respect of the Eq. - 3 becomes irrelevant and it is just the functional form of the equation that we are interested in. In the Tables - 5, we denote the two components of the multiplicity distribution of the hard-QCD events with suffixes 1 and 2.
The Fig. 3 depicts the primary charged particle multiplicity distributions for the minimum-bias and the hard-QCD events along with corresponding best fits with a single NBD function and with the superposition of two-NBDs. The values of the fit parameters, obtained by the best fits in terms of , as well as the values are tabulated in the Tables - 5, for the two-NBD. The Fig. 3 and the values tabulated in Tables - 4 and 5 clearly indicate that the multiplicity distributions for both the minimum bias and the the so-called hard-QCD events, indeed are better described by the two-NBD than a single NBD function for the large pseudorapidity bin, in the - range .
Iv Summary and Remarks
We have analyzed the multiplicity distributions of primary charged particles in collisions at = 7 TeV, as measured by the LHCb experiment at the LHC. The LHCb has measured the multiplicity distributions in several small ( 0.5) pseudorapidity windows mostly in the forward - region, , as well as in a large - window ( 2.5) for two classes of events, the minimum bias and the hard-QCD. The distributions have been analyzed in terms of the NBD function.
For the minimum bias events, we observe that the distributions in both the small and the large pseudorapidity windows could be better described by the weighted superposition of two-NBDs than a single NBD function - a feature similar to what has been exhibited by the multiplicity distributions of primary charged hadrons produced in the mid-rapidity region in the collisions at = 7 TeV.
The reasonable good fits of the single NBD to the multiplicity distributions for the ”hard” events in small - windows also are more-or-less in agreement with the two component model of the so-called ”hard” and ”soft” particle productions. But, the need of a similar function formed by the weighted superposition of two NBDs in describing multiplicity distribution of the ”hard” events in the large - window contradicts the concept of the discussed two-component model.
On the basis of the finding that the multiplicity distribution of ”hard” events in large - window deviates appreciably from a single NBD and requires weighted superposition of two NBDs, one may conclude that the discussed two-component model ref09 () does not conform fully with the multiplicity distribution in the forward-rapidity region of collisions at = 7 TeV. The finding could be attributed either to biased selection criterion of the ’hard events’ or to the possibility of different particle production mechanism in different phase space.
It is worth discussing at this point that there exists no specific orthogonal variable, as yet, to separate the ”soft” and the ”hard” events in collisions. Isolating the ”hard” (”soft”) events on the basis of having at least one (no) particle with the transverse momentum or transverse energy greater than a certain given value is a data driven approach and may inherit some biases, which need corrections. The selection criterion of the hard interaction events at LHCb resulted the geometrical acceptance no longer independent of momentum and the distributions were accordingly corrected by the collaboration ref06 (). In this study, we have analyzed the corrected distributions.
To conclude on the possibility of different particle production mechanism at different phase space, direct comparison of similar analysis in the mid-rapidity and in the forward-rapidity is essential. Our study with the minimum bias events at the forward rapidity and comparison of the related results with similar study with the mid-rapidity data ref13 () do not indicate to the possibility of different particle production mechanism, in the framework of the two-component model. The results of our analysis with the ’hard-QCD’ events, on the other hand, could not be compared with the mid-rapidity data as there exists no measured multiplicity distribution for ”hard” events in the mid-rapidity region at LHC. In the present scenario, similar analysis of isolated ”hard” events of collisions in the mid-rapidity region would be useful to obtain a comprehensive picture on the role of the discussed two-component model vis-a-vis the multiplicity distributions in different phase space in collisions at the LHC energies.
Also, other theoretical and phenomenological approaches ref15 (); ref17 (); ref18 (), successful in describing the data at the mid-rapidity, may be compared with the LHCb data at the forward rapidity region. In reference ref17 (), the CMS data ref04 () at the mid-rapidity have been successfully described in the framework of Independent Pair Parton Interaction (IPPI) ref19 () and in terms of Quark Gluon String Model (QGSM) ref20 (); ref21 () that fits better to the data than the IPPI model. The mid-rapidity data have been analyzed ref22 () also in the light of another multiple scattering model of particle production, the Dual Parton Model (DPM).
Note: During the review process of this article, we came across an article ref23 () that reports analysis of minimum bias multiplicity distributions measured by all the experiments at the LHC by weighted superposition of three NBD functions.
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