Charm production asymmetry at the LHC

In this work we use the meson cloud model to study the energy dependence of the asymmetry in D meson production in p − p collisions. We find a good agreement with the recent data from LHCb on D−/D+ asymmetry. Although small, this asymmetry may shed light on the role played by the charm meson cloud of the proton.


Introduction
It is well known from experiments [1,2,3,4,5,6] that the production of D + /D − in hadronic collisions is not exactly symmetric. For large x F (Feynman momentum) it is clearly observed that there are more D − 's than D + 's produced in pp, Σ − p and π − p collisions. The recent data from COMPASS collaboration [7] have confirmed that charm production asymmetry is also observed in γp collisions, and the very recent data from LHCb collaboration [8] on pp collisions at 7 TeV show a very small (but different of zero) asymmetry. The origin of the observed asymmetries is still an open question and cannot be explained with usual perturbative QCD (pQCD) or with the string fragmentation model contained in PYTHIA. This fact motivated the construction of alternative models [9,10,11] that were able to obtain a reasonable description of the low energy data and make predictions for higher energies. Now we can, for the first time, compare the predictions of these models with the high energy data from LHCb and study the energy dependency of the production asymmetry.

Charm production asymmetry
The production asymmetry is quantified as: where N i represents the number of mesons i produced as a function of, e.g., the Feynman x F variable. In pQCD the most relevant elementary processes of charm production are q +q → c +c and g + g → c +c. At high energies the distribution of gluons is much larger than the one of any other parton and the second process dominates. After being produced the c andc quarks fragment independently and the resulting D + and D − (also D 0 andD 0 ) mesons will have the same rapidity y, p T and x F distributions. Consequently, in calculations using pQCD there is no production asymmetry (A = 0). This is indeed true for the bulk of charm production. Experimental results on hadronic collisions show that differences between the D + and D − x F distributions appear at large x F , with D − being harder. Given the valence quark content of the proton p(uud) and of the D − (dc), a natural explanation of the observed effect is that thec is "dragged" by the projectile valence d quark, forming the, somewhat harder, dc bound state. This process, that has been called "recombination" (or coalescence), is a non-perturbative process. Recombination models have been first proposed long time ago [12,13,14] and then used more recently [15,16] to study the accumulated experimental data and to make predictions for the RHIC collisions.
A possible way to implement the idea of recombination is to use models in which the proton is a fluctuating object. In this approach, before colliding with the target the proton projectile fluctuates into a " charm baryon -charm meson " virtual pair. The charmed meson can be liberated during the interaction, being present in the final state of the collision. This type of fluctuation happens all the time. Sometimes the proton fluctuates into a virtual pair "pionneutron" and then the virtual pair recombines back to its original form, the proton. It was shown [17] that this type of fluctuation is quite relevant to the understanding of hadron structure. This mechanism, in which the "meson cloud" plays a major role (see [18] for light mesons) is ususally called the Meson Cloud Model (MCM). A simple and accurate description of charm asymmetry production at lower energies ( √ s ≃ 10 − 40 GeV) within the framework of the MCM can be found in [19].
In Fig. 1 we show an illustration of D + /D − production in the meson cloud approach. Before the collision, the proton projectile fluctuates into a virtual meson-baryon (M B) pair. Fig. 1 and 1 b show the situation in which the meson (the baryon) scatters with the target producing a system X of particles. The system X contains all sorts of particles, including the mesons D + and D − , however there is no asymmetry since the D + and the D − are identically distributed. In what follows we shall call it "indirect production" (I) of D ± . The production asymmetry happens when the proton fluctuates into a pair Σ ++ c D − and the D − is liberated by the interaction with the target, as shown in Fig. 1c. We shall call it "direct production" (D) of D − . This last mechanism is, of course, very much suppressed but it is the responsible for the asymmetry because only D − is produced in this process. This happens because the valence quark content of the proton (uud) only allows its fluctuation into a baryon-meson pair where the meson is always D − (dc) and never D + (dc). The meson and the baryon in the cloud have fractional momentum y M and y B , with distributions called f M/M B (y M ) and f B/M B (y B ) respectively (we shall use for them the short notation f M and f B ). Of course, by momentum conservation, y M + y B = 1 and these distributions are related by [17,20]: The "splitting function" f M (y) represents the probability density to find a meson with momentum fraction y of the total cloud state M B. With f M and f B we can compute the differential cross section of D production. In the reaction pp → D − X the differential cross section of D − production is given by: where Φ 0 and Φ I refer respectively to "bare" and indirect contributions to D meson production and x F is the fractional longitudinal momentum of the outgoing meson. By "bare" contribution we mean the process in which the hadron projectile does not fluctuate into any cloud state. The bare contribution is responsible for most of the production and does not generate any asymmetry. The term Φ D represents the direct process depicted in Fig. 1c and is given by [21,20]: where f D ≡ f D − /Σ ++ c D − and σ Σp is the total p Σ ++ c cross section. The term Φ I in Eq. (3) can generate some asymmetry, however we can neglect its contribution when compared with the asymmetry generated by the term Φ D . Thus the D + /D − production asymmetry is given by [22]: where Φ D T is the total D + + D − x F distribution. The denominator of this expression can be replaced by a parametrization of the experimental data [1,2,3,4,5,6], being given by [22]: where n D = 5 for D mesons [22]. Integrating the above expression we obtain the total cross section for charged charm meson production σ D ± = 1/3 σ D 0 . Assuming isospin symmetry the charged and neutral (σ D 0 ) production cross sections are equal. Neglecting the contribution of other (heavier) charm states we can relate the D meson production cross section to the total c −c production cross section, σ cc , in the following way: From the above relation we can extract the parameter σ D 0 from the experimentally measured σ cc : Inserting (4) and (6) into (5) the asymmetry becomes: The behavior of the asymmetry, Eq. (9), is controlled by the splitting function f D (x F ). For the splitting function we will use the Sullivan approach [17,20]. The fractional momentum distribution of a pseudoscalar meson M in the Fock state |M B ′ (of a baryon |B ) is given by [17,20]: where t and m M are the four momentum square and the mass of the meson in the cloud state and with m B and m B ′ being the B and B ′ masses respectively. Following a phenomenological approach, we use for the baryon-meson-baryon form factor F M BB ′ the exponential form: where Λ M BB ′ is the form factor cut-off parameter. Considering the particular case where B = p, B ′ = Σ ++ c and M = D − , we insert (10) into (9) to obtain the final expression for the asymmetry in our approach: where Noticing that y = x F in the above equations, we can see that in the limit x F → 1, t max → −∞, and the integral in (12) goes to zero. In fact, it vanishes faster than the denominator and therefore A → 0. This behavior does not depend on the cut-off parameter but it depends on the choice of the form factor. For a monopole form factor we may obtain asymmetries which grow even at very large x F . Before presenting the results we emphasize that they depend only on two parameters: Λ and N . Whereas Λ affects the width and position of the maximum of the momentum distribution of the leading meson in the cloud (and consequently of the asymmetry), N is a multiplicative factor which determines the strength of the asymmetry.

Results and discussion
Although the recent data [8] are given in terms of the pseudo-rapidity η, in order to study the energy dependence it is more convenient to use the Feynman x F variable, which for η > 1 is given by x F ≃ 2 m T e η / √ s. If the transverse momentum of the final D meson is zero or very In order to study the energy dependence of the asymmetry we shall consider the two energies where we have experimental data: √ s 1 = 33 GeV and √ s 2 = 7000 GeV (see Table 1). Let us consider the asymmetry ratio R A = A( √ s 2 )/A( √ s 1 ). Using the definitions of the splitting function (10) in (4) and then in (5), many energy independent factors cancel out and we find: where in the last step we have used (8) and assumed that σ Σ = σ Σ ++ c p = const . σ p p . Moreover, we have neglected the energy dependence of n D . The above ratios can be estimated with the recently obtained experimental data listed in Table 1. For √ s 1 = 33 GeV and √ s 2 = 7000 GeV we obtain R A = 1/75. Therefore there is a strong decrease in the asymmetry when we increase the energy. This happens because the responsible for the asymmetry is meson emission, a nonperturbative process, which has a slowly growing cross section. In contrast, the symmetric processes are driven by the perturbative partonic interactions, which have strongly growing cross sections. With the data presented in Table 1 we can make the prediction for the order of magnitude of the D + /D − asymmetry in the forthcoming 14 TeV pp collisions. Using the last line of Table 1, setting √ s 2 = 14 TeV and susbtituting the numbers in (14) we find R A = 1/100 showing the decreasing trend of the asymmetry. Due to the lack of experimental data a direct comparison of D + /D − asymmetries in the same reaction, e.g. proton-proton, at different energies is difficult. However, we can relate and compare similar reactions. In [22] we related the two sets of data on asymmetries in protonproton collisions: the low energy one taken by the SELEX collaboration [6] and the high energy one obtained by the LHCb collaboration [8]. In Ref. [22], using the definition (13) we made an estimate for N D ( N D ≃ 32).  Comparison of the MCM asymmetry with experimental data [8] for D − /D + . Extension of Fig. 2 to the large x F region.
Since we do not have data on D meson asymmetry at √ s = 33 GeV we fixed the values of the cut off parameter by fitting the data of LHCb at √ s = 7 TeV. In Fig. 2 we show the curves corresponding to the different cut off parameters. The values of Λ used to draw the curves are quite reasonable. It is interesting to amplify the region of higher x F in Fig. 2, where we do not have experimental data. The amplifed curves are shown in Fig. 3. Having fixed all the parameters at the LHC energy, we can go back to the energy √ s = 33 GeV and compute the asymmetry. This is shown in Fig. 4. Comparing Figs. 3 and 4 we draw the most important conclusion of this work: the asymmetry definitely decreases at increasing energies, reaching at most 2 % at x F ≃ 0.4.  Finally, in Fig. 5 we show our prediction for the D − /D + asymmetry to be measured at √ s = 14 TeV.