References

Effective-energy universality approach

describing total multiplicity centrality dependence

in heavy-ion collisions

Experimental Physics Department, CERN, 1211 Geneva 23, Switzerland

Department of Physics, The University of Texas at Arlington, Arlington, TX 76019, USA

Instituto de Ciencias Nucleares, Universidad Nacional Autonoma de Mexico, Mexico City, 04510, Mexico

Discipline of Physics, School of Basic Science, Indian Institute of Technology, Indore 452020, India

Department of Physics, New York University, New York, NY 10003, USA

Physics Department, Manhattan College, Riverdale, NY 10471, USA

The recently proposed participant dissipating effective-energy approach is applied to describe the dependence on centrality of the multiplicity of charged particles measured in heavy-ion collisions at the collision energies up to the highest LHC energy of 5 TeV. The effective-energy approach relates multihadron production in different types of collisions, by combining, under the proper collision energy scaling, the constituent quark picture with Landau relativistic hydrodynamics. The measurements are shown to be well described in terms of the centrality-dependent effective energy of participants and an explanation of the differences in the measurements at RHIC and LHC are given by means of the recently introduced hypothesis of the energy-balanced limiting fragmentation scaling. A similarity between the centrality data and the data from most central collisions is proposed pointing to the central character of participant interactions independent of centrality. The findings complement our recent investigations of the similar midrapidity pseudorapidity density measurements extending the description to the full pseudorapidity range in view of the considered similarity of multihadron production in nucleon interactions and heavy-ion collisions.

Recently, we have shown [2] that the centrality dependence of the midrapidity pseudorapidity density in heavy-ion collisions in the range of the center-of-mass (c.m.) energy, , spanning GeVs to TeVs, is well described in the framework of the approach of the dissipating energy of constituent quark participants, or, for brevity, the effective-energy approach, proposed by two of us [3, 4], exploiting a concept of centrality-dependent effective energy of nucleon participants [5]. We have also shown [6] that within this approach one describes as well the total (mean) multiplicity data and the pseudorapidity distribution in heavy-ion collisions up to 2.76 TeV collision energy. Within this description, an explanation of the so-called RHIC “puzzle”, namely of the difference between the nucleon-participant-normalized total multiplicity independence of centrality and the midrapidity pseudorapidity density increase with centrality, is suggested. This difference is found to be taken into account by introducing the energy-balanced limiting fragmentation scaling allowing to well describe the multiplicity data and the pseudorapidity spectra independent of centrality. No such a difference is measured at the LHC at TeV energies, where both the normalized midrapidity density and multiplicity are observed to decrease with the centrality, as indeed is expected within the effective energy approach. In this paper, we show that the recent measurements by ALICE [7] of the centrality dependence of the multiplicity of charged particles in PbPb collisions at  TeV, the highest collision energy ever reached, are well described in the picture of the participant dissipating effective-energy approach. The study extends our findings [2] of the similar data of midrapidity pseuodorapidity densities to the full pseudorapidity range, then the total multiplicity, a variable providing crucial information of the multihadron production dynamics [8, 9, 10].

Let us give a brief description of the effective-energy approach. Within this approach, which interrelates different types of collisions [3], the particle production process is quantified in terms of the amount of effective energy deposited by interacting constituent quark participants into the small Lorentz-contracted volume formed at the early stage of a collision. The whole process of the particle production is then considered as the expansion of an initial state and the subsequent break-up into particles. This resembles the Landau relativistic hydrodynamic model of multiparticle production [11]. In the effective-energy approach, the Landau hydrodynamics is employed in the framework of constituent (or dressed) quarks, in accordance with the additive quark model [12, 13]. Then, in collisions, a single constituent quark from each nucleon is assumed to contribute in a collision. The remaining quarks are treated as spectators and do not participate in the particle production, while resulting in formation of leading particles carrying away a significant part of the collision energy. On the contrary, in the most central (head-on) heavy-ion collisions, all three constituent quarks from each of the participating nucleons are treated to contribute so that the whole energy of the nucleons becomes available for the particle production. Thus, the bulk measurements in head-on heavy-ion collisions at are expected to be similar to those from collisions at the properly scaled c.m. energy , i.e. taken at .

Combining the above-discussed ingredients one obtains the relationship between the charged particle rapidity density per participant pair, , in heavy-ion collisions and in interactions:

 ρ(η)ρpp(η)=2NchNpartNppch√LppLNNexp[η22(1Lpp−1LNN)], √spp=3√sNN. (1)

Here, and are the (total) mean multiplicities in nucleus-nucleus and nucleon-nucleon collisions, respectively, and is the number of nucleon participants. The relation of the pseudorapidity density and the mean multiplicity is applied in its Gaussian form as obtained in Landau hydrodynamics. The factor is defined as . According to the approach considered, is the proton mass, , in nucleus-nucleus collisions and represents the constituent quark mass in collisions set to .

Solving Eq. (1) for the multiplicity at a given midrapidity density pseudorapidity at , and for the rapidity density and the multiplicity at , one finds:

 2NchNpart=Nppchρ(0)ρpp(0)√1−2ln3ln(4.5√sNN/mp) , √sNN=√spp/3. (2)

In the further development [5], one considers this dependence in terms of centrality. The centrality is regarded as the degree of the overlap of the volumes of the two colliding nuclei, characterized by the impact parameter, and is closely related to the number of nucleon participants. Hence, the largest number of participants contribute to the most central heavy-ion collisions. The centrality is related to the amount of the energy released in the collisions, i.e. to the effective energy, , of the participants. In the framework of the approach considered, this effective energy can be defined as a fraction of the c.m. energy available in a collision according to the centrality, :

 εNN=√sNN(1−α). (3)

 2NchNpart=Nppchρ(0)ρpp(0)√1−2ln3ln(4.5 εNN/mp), εNN=√spp/3. (4)

for central collisions of nuclei at the effective c.m. energy . Here and are taken at .

Figure 1 shows the effective-energy calculations by Eq. (4) compared to the charged particle multiplicity, , as a function of number of participants as measured in heavy-ion collisions at of TeV energies by the ALICE experiment at the LHC [14, 7] and, at GeV energies, by the PHOBOS experiment at RHIC [15]. In the calculations, the midrapidity density and the multiplicity are taken from the existing data [10, 16], and the values are taken from the heavy-ion collision data [15, 18, 17] where available. Where no data exist, the corresponding experimental c.m. energy fits are used. The linear-log [10] and power-law [19] -fits for at  53 GeV and at  53 GeV, respectively, along with the power law c.m. energy fits for [6] and [17] are used.

One can see from Fig. 1 that within the dissipating participants effective-energy approach, where the collisions are derived by the centrality-defined effective c.m. energy , the calculations well reproduce the centrality dependence obtained in the TeV-energy region from LHC, slightly underestimating a couple of the most peripheral measurements. However, for the RHIC data, the deviation between the measurements and the calculations is seen already for middle values. The difference in the behaviour of the data obtained at RHIC and at LHC becomes more clearer as soon as one multiplies the RHIC 200 GeV data by a factor 2.87 in order to match the ALICE 2.76teV data from highly central collisions, In the meantime, one can observe that there is almost no difference between the 2.76 TeV and 5.02 TeV LHC data, where the lower-energy measurements are multiplied by 1.25 to match the higher-energy ones.

The differences observed have been discussed in [6] and an explanation has been given by introducing the energy-balanced limiting fragmentation scaling for the pseudorapidity spectra in non-central collisions. By means of this scaling, the pseudorapidity distributions in heavy-ion collisions at RHIC energies are shown to be reproduced resulting into the centrality independence of the multiplicity, see Fig. 1. As noticed above, the energy-balanced limiting fragmentation provides also an explanation of the RHIC “puzzle” which is shown to occur due to the fact that, in contrast to the midrapidity densities, the total multiplicity gets additional contribution due to the difference between the collision energy and the effective energy shared by the nucleon participants.

Let us briefly discuss the findings in [6] leading to the energy-balanced limiting fragmentation hypothesis.

As it is outlined above, in the picture of the effective energy approach, the global observables are defined by the energy of the participating constituent quarks pumped into the overlapped zone of the colliding nuclei. Hence, the bulk production is driven by the initial energy deposited at zero time at rapidity , similar to the Landau hydrodynamics. Then, as is expected and indeed found in [2, 4, 5, 3], pseudorapidity density (and pseudorapidity transverse energy density) at midrapidity is well reproduced for all centralities.

Meantime, the calculations for the multiplicity to be obtained after the integration in Eq. (1) over the full- interval, where, within the considered approach, the values are taken from the most central collisions at the c.m. energy equal to , i.e. similar to the Landau hydrodynamics, the collisions of nuclei are treated head-on-like. As soon as the initial energy of participating quarks determines the particle production, the calculated pseudorapidity distributions and the mean multiplicities for highly-central heavy-ion collisions are expected to reproduce the data and this been shown for all available c.m. energies [6]. However, for non-central collisions at relatively low c.m. energies, one may expect the calculations for the central -region to be narrower with the wide fragmentation region. This was indeed observed at RHIC energies.

To take into account the fragmentation region, one recalls the hypothesis of the limiting fragmentation scaling [20], which supposes that at high enough energies, the (pseudo)rapidity density spectra for given interacting particle types become independent of the c.m. energy in the fragmentation region when shifted by the beam rapidity : . Then, within the effective-energy approach, one would expect similar behaviour to hold for the calculated pseudorapidity distribution when it is shifted by . Indeed, the calculated pseudorapidity distribution was found to match the measured one, independent of centrality, as the corresponding shift was applied [6]. Due to this hypothesis, called energy-balanced limiting fragmentation scaling, the calculated pseudorapidity density is found to be corrected outside the central- region.

Then, to describe the measured (unshifted) -distribution one needs to shift the calculated distribution, Eq. (1), by the energy difference, ), in the fragmentation region, or by , see Eq. (3), to balance the energy difference. This was found [6] to put the calculated distribution in agreement with the measurements for non-central collisions with no deficit in the fragmentation region. It is clear that in head-on or very central collisions, tends to zero. Where no pseudorapidity density distributions are available in measurements at , so no integration is possible using Eq. (1), the energy-balanced limiting fragmentation scaling is applied to reproduce the calculated : the measured distribution from a non-central heavy-ion collision is shifted by , i.e. . is calculated by adding the difference between the integral from the obtained shifted distribution and the measured multiplicity to the calculation of Eq. (4) in a non-central heavy-ion collision.

Using this ansatz, the values of , calculated for each centrality for the RHIC measurements, found to reproduce well the pre-LHC data. as shown in Fig. 1.

From the above, the RHIC “puzzle” gets its explanation in the sense that the midrapidity pseudorapidity densities are defined by the centrally colliding nucleon participants while the multiplicity gets an additional fraction of the energy as the difference between the collision c.m. energy shared by all nucleons and the effective energy of the participants.

Within this picture, the all-centrality multiplicity measurements at RHIC are shown to be complementary to the head-on measurements, i.e. to follow the same behaviour as the multiplicities head-on data while taken at [6], confirming the above consideration.

Moreover, as seen from Fig. 1, in contrast to the RHIC measurements, almost no additional contribution is needed for the calculations to describe the LHC mean multiplicity data at TeV energies. Given this observation, one can conclude that in TeV-energy heavy-ion collisions, the multihadron production obeys a head-on collision regime, for all the centralities measured. This observation is supported by the one made in [6] where the dependence of the centrality multiplicity data are shown to well follow the dependence of the head-on multiplicity data without taking into account the energy balance. This may be treatead pointing to apparently different regime of hadroproduction occurring in heavy-ion collisions as moves from GeV to TeV energies. This conclusion finds its support in our results [2] of describing the midrapidity density -dependence as soon as the LHC data are included.

As mentioned above, the centrality data on multiplicity (similar to the pseudorapidity and transverse energy densities at midrapidity [5]) are found to be complementary to the head-on measurements in sense of the similarity of the vs dependences, well within the effective-energy expectations. In this sense, one would expect the fit to the head-on multiplicity data to describe the dependence as soon as the centrality dependence is considered to be driven by the effective energy . Indeed, Fig. 1 demonstrates this feature, where the fit by ALICE to head-on measurements in heavy-ion collisions [7] is shown to well follow the LHC centrality data expressed in terms of , see Eq. (3), considering the correspondence of centrality and in the experimental measurements. One may also notice how well the head-on fit follows the calculations within the effective energy approach. This confirms the central-collision character of the interactions of the participating nucleons, as assumed above. Small deviations for low-energy data are due to slight inconsistency of the ALICE fit with the data at below a few tens GeV.

Let us note in conclusion that the picture of the effective-energy approach is shown as well to explain [3, 4] the similarity of the measurements in other collisions, such the scaling between the charged particle mean multiplicity in and collisions [21] and the universality of both the multiplicity and the midrapidity density measured in the most central nuclear collisions and in annihilation [22]; see [9, 10] for discussion. In the latter case, colliding leptons are considered to be structureless and deposit their total energy into the Lorentz-contracted volume, This is shown to be supported by the observation that the multiplicity and -distributions in interactions are well reproduced by data as soon as the inelasticity is set to 0.35, i.e. effectively 1/3 of the hadronic interaction energy[10]. For recent discussion on the universality of hadroproduction up to LHC energies, see [16].

Summarizing, the effective-energy dissipation approach based on the picture combining the constituent quark model together with Landau relativistic hydrodynamics in sense of the universality of the multihadron production in hadronic and nuclear collisions is shown to well describe the total (mean) multiplicity data in GeV to TeV c.m. energy heavy-ion collisions. In particular, the centrality dependence of the multiplicity of charged particles measured up to 5.02 TeV are shown to be well reproduced. In addition, the centrality dependence of the data is shown to be in good agreement with the fit to the head-on c.m. energy dependence confirming the central character of the collisions independent of the centrality measured. The observation of differences between the RHIC and the LHC measurements suggests the change of the particle production regime as the c.m. energy of heavy-ion collisions moves from GeV to TeV energy region. This study complements and supports the results from our recent investigations [2] of the midrapidity pseudorapidity density centrality dependence measurements in the same energy range while now made for the full-rapidity interval. Foreseen measurements at LHC and future colliders at higher energies and with different types of colliding objects are of high interest in clarifying the features of the effective-energy approach and in view of better understanding of the mechanism of the hadroproduction process.

Raghunath Sahoo acknowledges the financial support from Project No. SR/MF/PS-01/2014-IITI(G) of Department of Science & Technology, Government of India. The work of Alexander Sakharov is partially supported by the US National Science Foundation under Grants No. PHY-1505463 and No. PHY-1402964.

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