The energy dependence of the multiplicity moments at the LHC

In this work, from the experimental data we evaluate the first C-moments of the multiplicity distributions recently measured in proton-proton collisions at the LHC and compare them with the predictions of two models: the Kharzeev-Levin model and the Bialas-Praszalowicz model. We divide the data into three sets according to their phase space coverage: I: $p_T>100 $ MeV and $|\eta|<0.5$; II: $p_T>100 $ MeV and $|\eta|<2.4$ and II: $p_T>500 $ MeV and $|\eta|<2.4$. The mean multiplicity grows with the energy according to a power law and the power is different for each set. The $C_n$ moments grow continuously with the energy, slowly in set I and faster in the other sets. Except for KL in set II, both models reproduce the main features of the data. The negative binomial parameter $k$ decreases continuously with the energy and there is no sign of change in this behavior.


I. INTRODUCTION
Particle production in hadronic collisions at very high energies is a very interesting phenomenon. On the experimental side particle multiplicities are easy to measure. On the theoretical side it is still a very challenging subject [1]. Most of the particles are produced with low and medium transverse momentum, where perturbative QCD cannot be applied and one has to use phenomenological models and/or Monte Carlo event generators. In [2] it was shown that these generators are able to reproduce the main features of the multiplicity distribuions but they are not yet able to describe the data with precision, especially in the large multiplicity region.
Over the past twenty years the QCD based theory of particle production has experienced a significant progress, especially because of the development of the Color Glass Condensate (CGC) formalism [3]. One of the most interesting predictions of the CGC is that at very high energies the multiplicity distributions will become narrower [4]. This may be observed looking at the behavior of the multiplicity moments C n , which should decrease with the energy. Ten years ago this prediction was confronted with the LHC data in [5], where a careful study of the moments was carried out. The conclusion was that the moments were continuously growing with the energy with no sign of change in this trend. Since then, new data appeared, taken at much higher energies, and it is time to check if there is anything special happening to the moments. Here it is important to mention that in the most recent experimental papers the multiplicity distributions P (n) were presented but the moments C n were not. The first goal of this work is thus to compute the moments from the multiplicity tables, which we took from the hepdata.net databasis (the corresponding links are given right after the related articles in the reference list). Then we will compare them with the predictions of two simple models. The first one, which we call Kharzeev-Levin (KL) was proposed in [6] and used in [7] to compute the moments of charged particle multiplicity distributions. This model is based on the Balitsky-Kovchegov (BK) equation with fixed dipole sizes. In the second model, called here Bialas -Praszalowicz (BP) Model [5,8], multiparticle production is described by a probability distribution, which is a superposition of a distribution of the number of sources and a Poisson distribution describing particle emission from each source. In the KL model there is always only one source, whereas in the BP model the number of sources grows with energy and can be large. In this aspect, these models are complementary.
The predictions of KL and BP models will be compared with the most recent data from the LHC on non single diffractive pp collisions, which can be grouped into three sets: • Set I: p T > 100 MeV, |η| < 0.5, and energies √ s = 900, 2360 and 7000 GeV, from CMS [2].
These data sets may contain particles produced through different production mechanisms. The particles measured in set I are produced mainly from gluons; those measured in set II come also from the fragmentation region (larger rapidities) and hence are produced also from the valence quarks. Due to the larger transverse momentum cut-off, set III contains more particles which are produced perturbatively. These differences might lead to a different behavior of some observables. In previous analyses [2], it has been observed that multiplicity distributions in set I satisfy the Koba-Nielsen-Olsen (KNO) scaling, whereas those in set II do not.

A. The Kharzeev -Levin Model
In Ref. [6,7], the authors developed a model for multiplicity distributions based on the BK equation, which we will call KL model. They propose the following evolution equation for the parton multiplicity distribution P n : which has the simple solution: where Y = ln(1/x) and ∆ is the BFKL Pomeron intercept, ∆ = 4 ln 2ᾱ s withᾱ s = α s N c /π. From the above expression we obtain the mean multiplicity: The variable x is defined here as in [15,16]: where q 0 is a constant. Inserting (4) into (3) we obtain: The energy scale q 0 can be a mass or the average transverse momentum and hence it might be different for different data sets, but it should not depend on the collision energy √ s. In what follows we will fix q 0 and ∆ from the fit of the available experimental data.

B. The Bialas -Praszalowicz Model
In the BP model the multiplicity distribution is given by: where t is a fraction of the average multiplicity, and F (t) the distribution of sources that contribute a fraction t to the multiplicity probability P BP (n). The function F is normalized: The above equation implies that n =n. In the BP model F is given by (8) and P BP turns out to be the negative binomial distribution (NBD), which is known to describe relatively well the data at lower energies [17]. Distribution (8) depends on one parameter k, which depends on the collision energy. The analysis of lower energy data shows that k decreases with increasing energy. When k = 1 the probability distribution P NBD becomes a geometrical distribution P (n) = n n /(1 + n ) n+1 When k is large (1/k → 0), the distribution P NBD tends to a Poisson distribution.

A. The mean multiplicity
We start fitting the data on the mean multiplicity as a function of the energy with the help of the parametrization (5), as shown in Fig. 1. The obtained values of q 0 and ∆ are presented in Table I. As it can be seen in the figure, the data are well reproduced by the power law (5). However, there is a striking difference in the size of the parameters. For set II, the power ∆ is much smaller and the corresponding energy dependence is much weaker than in the other sets. This is not an artifact of the fitting procedure and is rather a feature of the data. To illustrate this, in Fig. 1d we have have multiplied the data and the fitting curves by appropriate constants so that the curves start at the same point. The resulting plot shows clearly the different energy behavior of the three sets. Since in sets II and III the rapidity coverage is the same, the difference in the energy behavior of the mean multiplicity must be related to the difference in the lower p T cut. As noted in the introduction, the most natural explanation (if not the only one) for this energy behavior is the increase of the contribution of perturbative events, which are known to have a strong energy dependence. It is nevertheless remarkable how a relatively modest increase in p T (which is here still far from the typical few GeV region) can produce a visible effect.
The data of sets I and II have the same lower p T cut and very different rapidity coverage. In set I, we observe particles produced in the central (rapidity) region, which is dominated by gluons. In set II there is a larger contribution coming from the fragmentation region, where the valence quarks play an important role in particle production. The separation of central and fragmentation regions and its relevance for particle production was first discussed in [18] and later implemented as a model in [19]. In this region, partons from the projectile and from the target collide in a very asymmetric kinematical configuration. If a parton from the projetile carries a large momentum fraction of the proton, the one from the target carries a very small fraction of the target proton momentum. Therefore, data in this region of the phase space are more sensitive to the low-x QCD dynamics and to saturation effects. As it is well known, saturation tames the growth of observables (parton distribution functions, color dipole scattering amplitudes, hadron cross sections, etc...) with the energy. According to these ideas, it is tempting to interpret the weaker energy dependence of the data of set II (as compared to set I) as a manifestation of low x saturation effects.  The moments are defined as:  [2]. b) Upper-right: Set II. Data from [10,11] (circles) and from [9] (triangles). c) Lower-left: Set III. Data from [10,12,13] (circles) and [14] (squares). d) Lower-right: mean multiplicities adjusted with (5) for all sets. The points in each set were multiplied by a constant.
They have been measured at the LHC and explicitly given in Refs. [2,20]. For higher energies we can calculate them from the hepdata.net databasis, as we did in the case of the mean multiplicities. In the KL model we can obtain the moments from the definition (9) and the multiplicity distribution (2). The first of them are given by: Since the mean multiplicities are given by (5), there are no free parameters in the calculation with the KL model. In the BP model the moments are obtained from (6) and the first moments C n are given by: In the above expressions, we need to know k to compute the moment C 2 and then all the other moments. Instead of choosing values for k, we follow [5] and parametrize C 2 as Next, we fit C 2 to the data, fixing a and b, and finally we calculate k using (11). Actually, in order to find the parameters a and b we fit C 4 rather than C 2 . The obtained values are listed in Table II (13), for the three sets.
parameters a and b, which are energy independent, we can calculate all the first C moments, compare them with data and make predictions. This is shown in Fig. 2, where we compare the KL and BP moments with the three data sets. Looking first at the data (which are put together here for the first time) we observe that in all figures the moments grow with the energy. The moments from set I grow much slower and are even compatible with a constant value. This motivated the observation made in Ref. [2], where the authors claimed that data relative to small rapidities exhibit KNO scaling [1,21], while data of larger rapidity intervals do not. Comparing the moments obtained with sets II and III we see that the C n 's grow with energy in the same (strong) way. Both models give a reasonable description of data. The KL model has less freedom than the BP one, since in the latter, apart from < n >, we can also adjust C 2 to the data. This flexibility yields better fits. In [5] the mean multiplicity was calculated from the central rapidity density, dN/dη(η = 0), while here we have used (5). We believe that this procedure is more accurate. The results support the BP picture, in which multiple sources (the number of sources follows the distribution F ) produce a number of particles which follows a Poisson distribution and both distributions depend on the energy. On the other hand the success of the KL model would imply that particle production comes mostly from gluon cascading resulting ultimately from the BK equation. In view of its simplicity, the KL model does a reasonable job with sets I and III and fails badly with set II. In this set we have the bigger multiplicities and, as already pointed out in [7], with the expressions (10) we rapidly reach the asymptotic values of the "large < n >" limit. The resulting moments are large and very flat functions of √ s, in contradiction with the data. It is not clear which part of the model is wrong or incomplete. Having in mind the observations made in the case of the mean multiplicities, we would guess that there is a missing component in this version of the model, namely the contribution from the fragmentation region. Increasing the lower p T cut and selecting higher p T particles, we reduce the contribution of the fragmentation region yield (which is mostly forward and with low transverse momentum) and improve the agreement between the KL model and data, as is indeed seen in the comparison with set III data. Finally, it is worth mentioning that the discrepancy between the KL model and the set II data is visible not only in the C n moments but also in the entire distribution P (n), as shown in [22].
Finally, having fitted < n > and C 2 we return to (11) and plot 1/k as a function of √ s. The result, shown in Fig.  3 indicates that 1/k is an increasing function and there is no sign of a different behavior. These findings extend the conclusions found ten years ago in [5] to the present energies, which are two times higher.
reproduces reasonably well the moments of the data of sets I and III (where the model is indeed expected to be valid as we have mainly gluons in the perturbative regime). The failure in describing the data of set II is arguably due to the absence of a component from the fragmentation region. In the BP model (12) and assuming a logarithmic growth (13) of C 2 moment, we have been able to reproduce the multiplicity moments over the wide range of energies for different rapidity intervals. The input growth of C 2 with energy can be translated through (11) into a decrease of the parameter k. This behavior is consistent with lower energies and does not exhibit the change predicted in [4].