Charm production in high multiplicity pp events

Recent experimental studies of the multiplicity dependence of heavy quark (HQ) production in proton-proton collisions at 7 TeV showed a strong non-linear increase of the HQ multiplicity as a function of the charged particle multiplicity. We try to understand this behavior using the EPOS3 approach. Two issues play an important role: multiple scattering, in particular its impact on multiplicity fluctuations, and the collective hydrodynamic expansion. The results are very robust with respect to many details of the modeling, which means that these data contain valuable information about very basic features of the reaction mechanism in proton-proton collisions.

Recently, several experimental groups investigated the dependence of heavy quark production on the event activity, both for open and hidden charm or bottom, in high energy protonproton collisions. We will focus here on D meson production, where the term "D meson multiplicity" refers in the following to the average multiplicity of D + , D 0 and D * + . The ALICE collaboration found a quite unexpected result [1]: When plotting the D meson multiplicity versus the charged particle multiplicity, both divided by the corresponding minimum bias mean values, one obtains a dependence which is very significantly more than linear (where "linear" means N D / N D = N ch / N ch ). The effect seems to be bigger for larger transverse momentum (p t ). Both D meson and charged particle multiplicity refer to central rapidities.
It is clear that the experimental observations are very interesting, and provide valuable insight into the very nature of the reaction mechanism in pp scattering, in particular in case of high event activity. So we try in this paper to provide an analysis of the phenomenon in the EPOS3 framework. Two key aspects are: Multiple scattering and collectivity.
EPOS3 [2] is a universal model in the sense that for pp, pA, and AA collisions, the same procedure applies, based on several stages: Initial conditions. A Gribov-Regge multiple scattering approach is employed ("Parton-Based Gribov-Regge Theory" PBGRT [3]), where the elementary object (by definition called Pomeron) is a DGLAP parton ladder, using in addition a CGC motivated saturation scale [4] for each Pomeron, of the form Q s ∝ N partŝ λ , where N part is the number of nucleons connected the Pomeron in question, andŝ its energy. The parton ladders are treated as classical relativistic (kinky) strings.
Core-corona approach. At some early proper time τ 0 , one separates fluid (core) and escaping hadrons, including jet hadrons (corona), based on the momenta and the density of string segments (First described in [5], a more recent discussion in [2]). The corresponding energymomentum tensor of the core part is transformed into an equilibrium one, needed to start the hydrodynamical evolution. This is based on the hypothesis that equilibration happens rapidly and affects essentially the space components of the energy-momentum tensor.
Viscous hydrodynamic expansion. Starting from the initial proper time τ 0 , the core part of the system evolves according to the equations of relativistic viscous hydrodynamics [2,6], where we use presently η/s = 0.08. A cross-over equation-of-state is used, compatible with lattice QCD [7,8]. The "core-matter" hadronizes on some hyper-surface defined by a constant temperature T H , where a so-called Cooper-Frye procedure is employed, using equilibrium hadron distributions, see [8]. After hadronization, there occur still hadronhadron rescatterings, realized via UrQMD [9].
The above procedure is employed for each event (event-by-event procedure). Heavy quarks (Q) are produced during the initial stage, in the PBGRT formalism, in the same way as light quarks. We have several parton ladders, each one composed of two spacelike parton cascades (SLC) and a Born process. The time-like partons emitted in the SLC or the Born process are in general starting points of time like cascades (TLC). In all these processes, whenever quark-antiquark production is possible, heavy quarks may be produced. We take of course into account the modified kinematics in case of non-zero quark masses (we use m c = 1.3, m b = 4.2). D meson production in the EPOS3 framework has been studied extensively, comparing to data and other calculation, in ref. [10].
We try to understand the dependence of the D meson multiplicity on the charged particle multiplicity, first for EPOS basic (without hydro). We study the case, where both multiplicities refer to central rapidities (|y| ≤ 0.5 for the D mesons, and |η| ≤ 1 for the charged particles). We use the variables N ch for the charged particle multiplicity, and N Di for the D meson multiplicities for different p t ranges (N D1 for 1 < p t [GeV/c] < 2, and N D8 for 8 < p t [GeV/c] < 12).
In EPOS3, we have in each individual event a certain number of parton ladders (cut Pomerons). Each ladder contributes (roughly, on the average) the same to both charged particle and charm production, so both corresponding multiplicities are proportional to the number N Pom of cut Pomerons: N Di ∝ N ch ∝ N Pom , which leads to a "natural" linear relation between the charged particle multiplicity N ch and the D meson multiplicities N Di (to first approximation).
We define normalized multiplicities, n = N/ N , both for charged particles (n ch ) and D meson multiplicities (n Di ). In the following, we consider fixed values n * ch of normalized charged multiplicities.
We will study the average normalized D meson multiplicity for the largest p t range, for some given n * ch , which may be expressed in terms of the Pomeron number distribution prob(N Pom , n ch * ) at fixed n * ch and the number n D8 (N Pom , n ch * ) of D mesons for fixed N Pom and n ch * , as The two curves representing prob(N Pom , n ch * ) and n D8 (N Pom , n ch * ) are shown in fig. 1 (left). We see in the figure that n D8 (N Pom , n ch * ) increases strongly towards small N Pom with an increasing slope. Let us compare the expression of eq. (1) with the corresponding sum (as a reference) where we use n D8 (N Pom , n ch * ) = n ch * , which would lead to n D8 (n * ch ) = n * ch . For large N Pom , the contribution to the sum in eq. (1) will be less than the reference case, but this is more than compensated at small N Pom . Therefore, we have n D8 (n * ch ) > n ch * , which is confirmed by the precise calculation shown in fig. 1 (right) as red point. Also shown is the complete curve n D8 (n ch ) as obtained from EPOS basic. Indeed, we get a more than linear increase.  Figure 2. (Color online) D meson multiplicities versus the charged particle multiplicity, both divided by the corresponding minimum bias mean values. The different symbols and the notations N D1 , N D2 , N D4 , N D8 refer to different p t ranges: 1-2, 2-4, [4][5][6][7][8][8][9][10][11][12], N ch refers to the charged particle multiplicity. We compare our calculations (lines) to ALICE data (points). Left plot : EPOS basic. Right plot : full EPOS.
The results of our calculation agree qualitatively with the trend in the data, namely a more than linear increase, in particular for high transverse momentum D mesons. But the effect is actually too small, as seen in fig. 2 (left), where we plot the D meson multiplicities versus the charged particle multiplicity, both for our calculation and data from ALICE [1].
But anyhow, EPOS basic (w/o hydro) reproduces neither spectra nor correlations, we have to consider the full approach, i.e. EPOS with hydrodynamical evolution (with or without hadronic cascade makes no difference). In fig. 2 (right), we plot again the D meson multiplicities versus the charged particle, EPOS3 compared to data, but here we refer to the calculations based on the full EPOS model (with hydro). We see a significant non-linear increase, much more pronounced as in the case of EPOS basic (without hydro), mainly due to the fact that the multiplicities from full EPOS are considerably below the results from EPOS basic. A much more detailed discussion will be provided in a separate publication.
To summarize: We analyzed the dependence of D meson multiplicities (in different p t ranges) on the charged particle multiplicity in proton-proton collisions at 7 TeV, using the EPOS3 approach. We find a non-linear increase. Two issues play an important role: Multiplicity fluctuations due to multiple scattering (realized via multiple Pomerons), and the collective hydrodynamic expansion. Multiplicity fluctuations are important since in particular high p t D meson production at given (large) charged particle multiplicity is very much favored for small Pomeron numbers, which is responsible for the strong increase of the D meson production with multiplicity. In addition, the effect is amplified when turning on the hydrodynamical expansion, due to a reduction of the charged particle multiplicity with respect to the model without hydro.