Dual Parton Model for the Charged Multiplicity in p-p Collisions at 13, 13.6 TeV and for future LHC energy of 27 TeV

The number of charged particles produced in a high energy particle interaction gives a measure of an important quantity known as multiplicity. One can validate the predictions from a theoretical or phenomenological model by studying the multiplicity distribution (MD) of the charged particles obtained from the experimental observations. Interest in MD was stimulated by a paper of Koba, Nielson and Olesen [1] in 1972 in which they established a scaling behaviour of the MD. They showed that the multiplicity follows a universal scaling behaviour at high energies and termed it as the KNO scaling. The scaled distribution, with the parameters $z=n/\langle{n}\rangle$ and $\Psi(z)=\langle{n}\rangle P_{n}$ is independent of energy, where, $P_{n}$ is the probability of producing $n$ charged hadrons with a mean value of $\langle{n}\rangle$. The very first KNO violation was observed at the CERN collider in $\overline{p}p$ collisions at $\sqrt{s}$ = 540 GeV [2]. Soon after, the violation was also reported by different experiments involving particle interactions at other energies with different species of particles  [3, 4, 5, 6]. In the present work, analysis of the multiplicity distributions in $pp$ interactions at the Large Hadron Collider (LHC) energies is described in the framework of the Dual Parton Model (DPM) [7, 8, 9, 10]. The use of DPM to describe the soft interactions started with a single Pomeron exchange. Later, multiple Pomeron exchange diagrams were included and referred to as multiple scattering DPM. The DPM makes use of the dual topological unitarization (DTU) scheme [11, 12, 13, 14, 15] to make unitarity cuts on the Pomeron exchange diagrams. In this model a weight is associated with the cross section of each diagram appearing in the DTU expansion. The use of DPM to define the hadron multiplicity is described briefly in the Section II.

The DPM has been observed to describe the experimental charged particle multiplicities not only at ISR energies but could also account for scaling violation observed at the UA5 experiment [8, 9]. The success of the DPM is limited not only to the hadron-hadron collisions, but it has been extended to the hadron-nucleus [16, 17, 18, 19] and nucleus-nucleus [20, 21, 17] collisions as well. The charged particle multiplicities data from the non-single diffractive (NSD) $pp$ and $AA$ collisions at the LHC energy of 2.76 TeV have also been found to follow the predictions for non-diffractive (ND) charged multiplicities calculated from the DPM [22, 23]. The contribution from the double diffractive (DD) processes is small [24, 25] which makes the NSD experimental data and ND calculations from the model comparable.

The ability of the DPM to describe the hadronic spectra from high energy collisions of particles of various species motivates us to study the $pp$ collisions in the framework of the DPM. The LHC has started producing data for $pp$ collisions at $\sqrt{s}$ = 13.6 TeV, which is the highest energy that has been ever achieved. In this paper, the experimental charged particle multiplicities from the CMS collaboration at $\sqrt{s}$ = 0.9, 2.36, 7 TeV [26] are studied using the DPM. A comparison of the charged particle multiplicities from the DPM to the experimental values is also presented in the KNO form and are found to be in agreement within the experimental limits. The average charged multiplicities for these collisions are calculated using DPM and are compared to the experimental results. For each of the collision energies, simulated data is produced using PYTHIA 8 and compared to the predictions from the DPM and to the results from the CMS experiment.

The main highlights of this paper are the predictions of the charged particle multiplicities, KNO distributions and average multiplicities for the present LHC RUN3 energy of 13.6 TeV and the future LHC energy of 27 TeV in addition to the LHC RUN2 energy of 13 TeV using the DPM. These calculations are presented for the inelastic $pp$ scatterings in two central pseudo-rapidity intervals of 0.5 and 2.4. Simulated data are produced for the NSD $pp$ collisions at the energies of 13, 13.6 and 27 TeV in the two psuedo-rapidity intervals. A detailed comparison of the KNO distributions and average multiplicities is presented in this paper using DPM calculations and predictions from PYTHIA 8.

In the DPM, an inelastic scattering is considered as an exchange of a Pomeron between colliding hadrons, which results in two strings when a unitarity cut is made on the Pomeron. In a multiple scattering DPM, multiple Pomerons are exchanged which give rise to twice the number of strings when DTU expansion scheme is used. The weights, $\sigma_{k}$ with which different multiple scattering amplitudes combine in the multiple scattering DPM can be calculated using the eikonal model. The weights $\sigma_{k}$ are proportional to the probability of observing $k$ inelastic collisions at given collision energy $\sqrt{s}$ and are written below [23, 27, 28].

$$\sigma_{\textit{k}}(\xi)=\frac{\sigma_{P}}{\textit{k}Z}\left[1-e^{-Z}\sum_{i=0}^{\textit{k}-1}\frac{Z^{i}}{i!}\right],k\geq 1$$

(1)

where $\xi=ln(\frac{\textit{s}}{\textit{s}_{0}}$), with $s_{0}$ = 1 GeV², $\sigma_{P}=8\pi\gamma_{P}e^{\Delta\xi}$, $Z=\frac{2C_{E}\gamma_{P}e^{\Delta\xi}}{R^{2}+\alpha_{P}^{{}^{\prime}}\xi}$. Here $\sigma_{P}$ is the Born term given by Pomeron exchange with intercept $\alpha_{P}(0)=1+\Delta$. The values of other parameters are given below and are taken from [23]
$R^{2}=3.3$ GeV^-2, $\gamma_{P}=0.85$ GeV^-2, $C_{E}=1.8$, $\Delta=0.19$, $\alpha_{P}^{\prime}=0.25$ GeV^-2.

One of the most important parameters in Equation (1) is $\Delta$ appearing in the Pomeron intercept, which is taken to be 0.19 and is motivated from the studies done on the LHC $pp$ data in [23] and $\gamma^{*}p$ data in [29]. The rise in the single particle inclusive cross section per unit pseudo-rapidity with energy is governed by $\Delta$. Parameters $R$ and $\alpha_{P}^{\prime}$ control the $t$-dependence of the elastic peak [28, 30]. The parameter $\gamma_{P}$ is determined from the absolute normalization of the total ND inelastic cross section. A higher value of $C_{E}=1.8$ is needed to include the high-mass diffraction states.

From the AGK cancellation [31, 32], one writes

$$\sigma_{P}(\xi)=\sum_{\textit{k}\geq{1}}\textit{k}\sigma_{\textit{k}}(\xi).$$

(2)

The non-diffractive inelastic scattering cross section, ${\sigma_{\textit{ND}}^{\textit{pp}}}$ can be obtained from the weights $\sigma_{\textit{k}}$ as

$${\sigma_{\textit{ND}}^{\textit{pp}}}=\sum_{\textit{k}\geq 1}\sigma_{\textit{k}}(\xi).$$

(3)

Ends of each chain are linked to the valence or sea quarks of the colliding hadrons. The energy flow within the chains is governed by the parton distribution functions of the quarks in the colliding hadrons. For computing the hadronic spectra in the inelastic collision, the fragmentation functions are required. However, for the central rapidity region, the average hadronic multiplicities can be computed without using the fragmentation functions but using the rapidity position of each chain.

If we know the number of 2$k$ chains in the $k$ inelastic scatterings with $\sigma_{k}$ combining cross-section, the underlying mechanism of the particle production can be predicted. The average number of inelastic collisions is denoted by $<k>$ and is calculated as

$$<k>=\frac{{\sum_{k\geq{1}}k\sigma_{k}(\xi)}}{{\sum_{k\geq{1}}\sigma_{k}(\xi)}}=\frac{\sigma_{P}(\xi)}{\sigma_{ND}^{pp}}.$$

(4)

Small clusters of hadrons are produced as a result of fragmentation of chains in the inelastic scatterings. Any cluster would contain on an average $K$ number of charged particles. A value of $K=$ 1.4 is taken as it is found to describe well the hadronic spectra in $pp$ collisions [33, 34]. The probability of discovering $n_{c}$ clusters of hadrons in $k$ inelastic collisions is assumed to be given by the Poisson distribution [22]

$$\ P^{k}_{n_{c}}=e^{-k<n_{c}>_{0}}\frac{(k<n_{c}>_{0})^{n_{c}}}{n_{c}!}.$$

(5)

Here, $<n_{c}>_{0}$ is the mean cluster multiplicity in a single inelastic scattering and it is related to the corresponding mean charged particle multiplicity $<n>_{0}$, as $<n_{c}>_{0}=<n>_{0}/K$.

The value of $<n>_{0}$ in a given central pseudo-rapidity region can be obtained by integrating the charged multiplicity per unit pseudo-rapidity in an individual scattering, $\frac{dN_{0}^{\textit{pp}}}{d\eta}$, as shown below

$$\ <n>_{0}=\int_{\eta-\eta_{0}}^{\eta+\eta_{0}}\frac{dN_{0}^{pp}}{d\eta}d\eta\sim 2\eta_{0}\frac{dN_{0}^{pp}}{d\eta}\hskip 2.27621pt(\eta^{*}=0)=3\eta_{0}$$

(6)

The quantity $\frac{dN_{0}^{\textit{pp}}}{d\eta}\hskip 2.27621pt(\eta^{*}=0)=1.5$, is independent of the collision energy at mid-rapidity and at high energy as described in [23]. Using $<k>$, the value of differential charged multiplicity in a given psuedo-rapidity interval is calculated as

$$\frac{dN^{pp}}{d\eta}=<k>\frac{dN_{0}^{pp}}{d\eta}$$

(7)

The total cluster multiplicity is then given by the equation

$$\ P_{n_{c}}=\frac{\sum_{\textit{k}\geq{1}}\sigma_{\textit{k}}P^{k}_{n_{c}}}{\sigma_{\textit{ND}}^{\textit{pp}}}.$$

(8)

The values of $\frac{dN^{pp}}{d\eta}$, $\sigma^{pp}_{\textit{ND}}$ and $\sigma^{pp}_{\textit{tot}}$ from the DPM at various collision energies reported in [23] are reproduced as shown in the Table 1. Their values at the collision energies of 13, 13.6 (ongoing RUN3) and future LHC collision energy of 27 TeV are also calculated and shown in the given table.

The KNO form of the multiplicity distributions are written in terms of $\Psi(z)$

$$\Psi(z)=\langle{n}\rangle P_{n}=\langle{n_{c}}\rangle P_{n_{c}}$$

(9)

where,

$$z=n/\langle{n}\rangle=n_{c}/\langle{n_{c}}\rangle.$$

(10)

Here, $n$ is the number of emitted charged particles and $\langle{n}\rangle$ is it’s mean value. The latter can be calculated from the average cluster multiplicity using the equation $\langle{n}\rangle=<k><n>_{0}=3\eta_{0}<k>$. The relation between $n$ and $n_{c}$ is given by using $n_{c}=n/K$, also $\langle{n_{c}}\rangle=\langle{n}\rangle/K$.

The charged hadron multiplicity in NSD processes in $pp$ collisions has been measured by the CMS collaboration [26] at various center of mass energies, $\sqrt{s}=0.9,2.36$ and 7 TeV in restricted pseudo-rapidity intervals of $|\eta|<0.5$ and $|\eta|<2.4$. These experimental results are available on the HEPData [35] and have been used for the present study. In addition, for detailed comparison, we used the Monte Carlo (MC) event generator PYTHIA 8.306 for the generation of $10^{7}$ $pp$ NSD events at each $\sqrt{s}$ and in each pseudo-rapidity interval under study.

With the aim of providing a better description of the observables of the data, event generators have adjustable parameters to control the behaviour of the event modeling. These observables may be sensitive to partons from initial-state radiation (ISR) and final-state radiation (FSR), underlying events (UE) consisting of beam remnants, particles produced in multiple-parton interaction (MPI) etc. The processes of hadronization and MPI are particularly afflicted, as they involve non-perturbative QCD physics. A good modeling of hadronization is required. A set of parameters, which need to be adjusted to fit some aspects of the data, is referred to as a tune. The experimental data from an experiment is often fitted to the predictions from an event generator by tuning and optimizing these parameters. Description of PYTHIA as an event generator can be found in [36]. For the present analysis, PYTHIA 8 tunes, Monash [37] and 4C [38] are used for studying the charged hadron multiplicity distributions. The MDs have also been simulated for $\sqrt{s}$ = 13, 13.6 and 27 TeV using PYTHIA 8.3 for comparison with the theory and for predicting the multiplicity at the future LHC energy of 27 TeV.

A detailed analysis and comparison of the charged hadron multiplicities in $pp$ collisions at various center of mass energies at the LHC is presented. The analysis uses the data classified as non-single diffractive events obtained by the CMS experiment at $\sqrt{s}$ = 0.9, 2.36 and 7 TeV in the two pseudo-rapidity intervals, $|\eta|<0.5$ and $|\eta|<2.4$. These data are compared with the theoretical predictions of the Dual Parton Model and MC simulations of charged hadron production by using two different tunes of event generator PYTHIA 8. Out of the two tunes used, Monash is the default tune in PYTHIA and 4C is the tune used by the CMS experiment. Using these tunes and the theoretical calculations from the model, multiplicities are also obtained at $\sqrt{s}$ = 13, 13.6 and 27 TeV. The LHC RUN3 has just started taking data at 13.6 TeV. This analysis presents predictions for the charged hadron multiplicities at these energies and also for the future LHC energy of 27 TeV.

It is observed that the MDs in the $|\eta|<0.5$ interval agree with the experimental distributions at $\sqrt{s}$ = 0.9, 2.36 and 7 TeV. For the $|\eta|<2.4$ interval, PYTHIA 8 predictions underestimate the experimental MDs in the mid multiplicity region for both Monash and 4C tunes, as seen in the ratio plot. For the higher multiplicity region PYTHIA 8 predictions overestimate the experimental MDs. In addition, a shoulder structure can be observed in the lowest multiplicity region.

The mean multiplicities obtained from the data, theory and MC are in good agreement for $\sqrt{s}$ = 0.9 and 2.36 TeV. The theoretical mean multiplicity soon starts to deviate from MC and the data at higher energy. The $\langle{n}\rangle$ of the data and MC are in agreement within the error limit of experimental data at $\sqrt{s}=7$ TeV. However, theory systematically underestimates the experimental values from the CMS data, and PYTHIA overestimates. The deviation of the theoretical mean gets more pronounced at centre of mass energy beyond 7 TeV.

KNO distributions obtained from the data [26], theory and MC are presented for $\sqrt{s}$ = 0.9, 2.36 and 7 TeV in Figures 7 and 8. For $|\eta|<$ 0.5 theoretical distributions are found to deviate from the data at $z\sim 6$, while for MC distributions the deviation is seen much earlier at $z\sim 4$. Similarly, the KNO distributions at these energies in $|\eta|<$ 2.4 interval show deviation of theoretical distributions from the data at $z\sim 3$ and for MC distributions the deviation from the data starts at $z\sim 4$. Exceptionally, the theoretical distribution from the model at $\sqrt{s}$ = 7 TeV follows the data very closely while the MC distributions show deviation.

The data from the CMS are not available for analysis at $\sqrt{s}$ = 13 TeV, for the NSD events in the same transverse momentum range. The LHC RUN3 has started very recently and the data at the collision energy of 13.6 TeV is being collected by the experiments at the LHC.

We present the predictions for the mean multiplicities and the multiplicity distributions as estimated from the theoretical model and PYTHIA 8 for the two tunes for these energies. Mean multiplicity and the KNO distributions for the future LHC energy of 27 TeV are also predicted.

It is also observed that the KNO distributions obtained from the two tunes agree very closely, though the low $z$ region below the peak shows disagreement between the theory and the data. The peak of the KNO distribution in each case shifts towards smaller $z$ value as the collision energy increases from 0.9 TeV to 27 TeV.

The observations from the present study are indicative of the trends in the future data from the LHC. Comparison with the data when it becomes available makes an interesting study and may lead to a new direction in our current understanding.