Can the triple-parton scattering be observed in open charm meson production at the LHC?

We investigate whether the triple-parton scattering effects can be observed in open charm production in proton-proton collisions at the LHC. We use so-called factorized Ansatz for calculations of hard multiple-parton interactions. The numerical results for each parton interaction are obtained within the $k_{T}$-factorization approach. Predictions for one, two and three $c\bar c$ pairs production are given for $\sqrt{s}= 7$ TeV and $\sqrt{s}= 13$ TeV. Quite large cross sections, of the order of milibarns, for the triple-parton scattering mechanism are obtained. We suggest a measurement of three $D^{0}$ mesons or three $\bar{D^{0}}$ antimesons by the LHCb collaboration. Confronting our results with recent LHCb experimental data for single and double $D^{0}$ (or $\bar{D^{0}}$) meson production we present our predictions for triple meson final state: $D^{0}D^{0}D^{0}$ or $\bar{D^{0}}\bar{D^{0}}\bar{D^{0}}$. We present cross sections for the LHCb fiducial volume as well as distributions for $D^{0}$ meson transverse momentum and three-$D^{0}$ meson invariant mass. The predicted visible cross sections, including the detector acceptance, hadronization effects and $c \to D^{0}$ branching fraction, is of the order of a few nanobarns. The counting rates including $D^{0} \to K^{-}\pi^{+}$ branching fractions are given for known or expected integrated luminosities.


I. INTRODUCTION
The multi-parton scattering effects got new impulse with the start of the LHC operation [1,2]. There are several ongoing studies of different processes. So far theoretical studies concentrated on double-parton scattering. Some time ago we have shown that charm production should be one of the best reaction to study double-parton scattering effects [3] (see also Ref. [4]). This was confirmed by the LHCb experimental data [5] and their subsequent interpretation [6][7][8].
Very recently also triple parton scattering was discussed in the context of multiple production of cc pairs [9]. Inspiringly large cross sections were presented there.
We decided to verify this result within our approach which was succesfully used previously for single and double D meson production [8,10]. Experimentally one measures rather D meson (or nonphotonic leptons). We wish to answer the question whether the triple-meson scattering could be seen in three D 0 or threeD 0 production. In order to answer the question one has to obtain cross section for meson production, taking into account c → D hadronization and acceptance of the existing detectors. Reliable predictions for triple D meson production should check consistency of model predictions for single and double D meson production with already existing experimental data. The triple-parton scattering (TPS) mechanism for pp → ccccccX reaction is schematically illustrated in Fig. 1. The corresponding inclusive TPS cross section in a general form [11][12][13] can be written as follows:

II. A SKETCH OF THE MODEL CALCULATIONS
) are the partonic cross sections for gg → cc mechanism and 1 3! is the combinatorial factor relevant for the case of the three identical final states. The above TPS hadronic cross section is expressed in terms of the so-called triple- The triple parton distribution functions (triple PDFs) shall account for all possible correlations between the partons, not only kinematical and spatial one, but also including spin and color/flavor correlations. The MPI theory in this general form is well established (see e.g. Refs. [14,15]) but not yet fully applicable for phenomenological studies.
The objects like triple PDFs (and even double PDFs in the case of DPS) are under intense theoretical studies but their adoption to real process calculations is still limited.
Therefore, in practice one usually follows the so-called factorized Ansatz, where the correlations between partons are neglected and longitudinal and transverse degrees of freedom are separated. According to these approximations the triple-gluon PDFs from Eq. (2.1) take the following form: is the product of single gluon PDFs and F( b i ) describe the gluon distributions in transverse plane. The transverse factors F( b i ) of the two triple-gluon PDFs in Eq.(2.2) are connected to the proton-proton overlap function via the following relation: and are usually assumed to be universal for all types of partons.
Taking all together, the formula for inclusive TPS cross section (Eq. (2.1)) can be simplified to the pocket form: where the triple-parton scattering normalization factor σ eff,TPS is related to the overlap function from Eq. (2.3) via the following expression: The normalization factor σ eff,TPS contains all unknowns about the TPS dynamics. Its pure geometrical interpretation comes from the practical approximations of the factorized Ansatz mentioned above. In principle, taking into account various parton correlations as well as multi-parton PDF sum rules [16] or including perturbative-parton-splitting contributions [17][18][19] may lead to a breaking of the pocket-formula. However, most of the violation sources are expected to vanish for processes driven by small-x partons (see e.g. Refs. [20,21]), that is exactly the case of charm production at high energies. As was shown by us, e.g. in Ref. [7], the factorized framework seems to be sufficient to explain the LHCb double charm data. Therefore, we think that it can be safely used, at least as a starting point, to draw practical conclusions also in the case of triple charm production.
In principle, the DPS normalization factor σ eff,DPS was extracted experimentally from several Tevatron and LHC measurements (see e.g. Refs. [1,2] and references therein) and its world average value is σ eff,DPS ≃ 15 ± 5 mb 1 . Such experimental inputs are not available for σ eff,TPS in studies of triple-parton scattering. However, as was shown in Ref. [9] for proton-proton collisions, the latter quantity can be expressed in terms of their more known DPS counterpart: The relation is valid for different (typical) parton transverse profiles of proton. In the numerical calculations below we take σ eff,DPS = 21 mb which is rather a conservative 1 A detailed study of the σ eff,DPS can be found in Ref. [22].
choice but it corresponds to the average value extracted by the LHCb experiment only from the double charm data [5]. This input gives us the value of σ eff,TPS ≃ 17 mb.
In this paper, each of the single-parton scattering cross sections σ SPS pp→cc in Eq. (2.4) is calculated in the k T -factorization approach [23] where higher-order QCD corrections are effectively included. In this framework exact kinematics is kept from the very beginning and additional hard dynamics coming form transverse momenta of incident partons is taken into account. It was shown in Ref. [10] that within this approach one can get a good description of the LHC inclusive charm data, similar to the case of next-to-leading order (NLO) collinear calculations. Likewise, the successful theoretical analyses of double charm production from Refs. [6][7][8] were also based on the k T -factorization.
According to this approach the differential SPS cross section for inclusive single cc pair production can be written as: where the extra, compared to collinear factorization, integrals over transverse momenta k it of initial state particles appear. Here, M g * g * →cc is the well-known gauge-invariant off-shell matrix element for g * g * → cc partonic subprocess and F g (x i , k 2 it , µ 2 ) are the socalled unintegrated (transverse momentum dependent) gluon PDFs (uPDFs).
The pocket-formula for TPS (Eq. 2.4) can then be written in the differential form: where for simplicity dξ ij stand for dy i dy j d 2 p i,t d 2 p j,t .
In the present paper we use the Kimber-Martin-Ryskin (KMR) uPDFs [24], generated where dξ a stand for dy a 1 dy a 2 dy a 3 d 2 p a 1,t d 2 p a 2,t d 2 p a 3,t taking a = c quark or D meson and p c i,t = p D i,t z i with meson momentum fractions z i ∈ (0, 1). The usual approximation here is that the quark rapidities y c 1 , y c 2 , y c 3 are unchanged in the fragmentation process which is known to be especially legitimate in the case of heavy flavors and meson transverse momenta larger than its mass (see e.g. Ref. [26]). In the numerical calculations here we use the commonly used in the literature scale-independent Peterson FF [27] with the parameter ε c = 0.05, which is the averaged value extracted from different e + e − experiments. In the last step the obtained cross sections for triple-meson production in the way sketched above are normalized with the corresponding fragmentation fraction BR(c → D 0 ) = 0.565 [28].

III. NUMERICAL RESULTS
We start with our predictions for multiple cc production. In order to compare our results to the results of Ref. [9] in Table I we show cross sections for the full phase space for two different collision energies. We get considerably larger cross section for triple cc production compared to the numbers read off from Fig. 1 of Ref. [9]. It is not obvious how to understand the difference, as rather different approaches were used in both cases. There are obvious uncertainties related to the choice of factorization/renormalization scales in both approaches. In our approach the region of very small transverse momenta of c orc quarks is the least certain, which is related to uncertain region of very small gluon transverse momenta. Small uncertainties for single-parton scattering are considerably magnified for triple-parton scattering. Large differences of cross section in the full phase space do not necessarily involve such differences for fiducial volume. We think that agreement of theoretical inclusive cross sections for D meson production in a fiducial volume is a necessary (and sufficient) condition for the best estimating of DPS and TPS effects.
In Fig. 2 we present our results for inclusive single D 0 meson production for the LHCb experiment at √ s = 7 and 13 TeV. In both cases we get good description of the measured  In Table II we show our predicted cross sections for two and three mesons within the fiducial volume of the LHCb detector. The DD pairs were already measured by the LHCb collaboration. The predicted value at √ s = 7 TeV is consistent with the measured one (see Table 12 in Ref. [5]). Whether the three mesons can be measured at the LHCb will be discussed in the following.
Now we wish to discuss some differential distributions for double and triple D 0 meson production. In Fig. 3 we show transverse momentum distribution of one of the two or one of the three D 0 mesons (all measured by the LHCb detector). In the used multiple The integrated cross sections for double and triple D 0 meson production (in nb) within the LHCb acceptance: 2 < y D 0 < 4 and 3 < p D 0 T < 12 GeV, calculated in the k T -factorization approach. The numbers include also the charge conjugate states.  the LHCb experimental data [5]. As for single scattering case (see Fig.2) we get a good agreement also here, so we hope our predictions for triple D 0 production (lowest curves) are (should be) reliable.  Ref. [5].  Finally in Table III we show the number of counts for different realistic values of the integrated luminosity for the LHCb experiment. The predicted numbers of events for double-and triple-D 0 production correspond to the cross sections from Table II. Here we have included in addition the relevant decay branching fraction BR(D 0 → K − π + ) = 0.0393 [31].

IV. CONCLUSIONS
In this letter we have made first estimation of the cross sections for triple D 0 production within the LHCb fiducial volume in order to verify triple parton scattering effects for triple cc production.
We have obtained rather large cross sections for triple cc production, larger than predicted very recently in Ref. [9] but still consistent within uncertainties of the two approaches. We have checked, however, our approach against inclusive single D 0 and double D 0 D 0 production as measured by the LHCb collaboration. In both cases we have obtained a fairly good agreement which gives us confidence for triple D 0 production.
We have presented both integrated cross section as well as differential distributions for double and triple D 0 production for √ s = 7 and 13 TeV. We have presented also predicted number of counts for different realistic values of integrated luminosity for the LHCb experiment. In the case of triple D 0 production we have predicted about 100 counts at √ s = 7 TeV and a few thousands of counts at √ s = 13 TeV for realistic integrated luminosities. We hope the LHCb collaboration will be able to verify our predictions soon.