Ds+ -- Ds- Asymmetry in Photoproduction

Considering of the possible difference in strange and antistrange quark distributions inside nucleon, we investigate the Ds+ -- Ds- asymmetry in photoproduction in the framework of heavy-quark recombination mechanism. We adopt two distribution models of strange sea, those are the light-cone meson-baryon fluctuation model and the effective chiral quark model. Our results show that the asymmetry induced by the strange quark distributions is distinct, which is measurable in experiments. And, there are evident differences between the predictions of our calculation and previous estimation. Therefore, the experimental measurements on the Ds+ -- Ds- asymmetry may impose a unique restriction on the strange-antistrange distribution asymmetry models.

High energy charmed hadron production plays an important role in studying strong interactions. Due to the large charm quark mass, the charm quark involving processes can often be factorized, i.e., into a hard process, which describes the interaction scale much larger than Λ QCD , the typical QCD renormalization scale; and a soft part, which demonstrates the hadronization effects. In recent years, a large production asymmetries between the charmed and anti-charmed mesons have been measured in fixed-target hadroand photo-production experiments [1,2,3,4,5,6,7]. Among these, photoproduction is thought to give more clean signature than the hadroproduction ones, because there is only one hadron in the initial state. However, it is intriguing to notice that the experimental observation on the asymmetries of the charmed hadron production are greatly in excess of the predictions of perturbative QCD(pQCD) [8,9,10,11].
For large transverse momentum processes, the QCD factorization theorem [12] enables us to calculate the cross sections of heavy hadron production by the pQCD. To be specific, for the D s photoproduction, the differential cross section may be expressed as the convolution of the parton distribution function, the partonic cross section, and the fragmentation function, like where f i/N is the distribution function of the parton i in a nucleon N; dσ(γ +i → c+c+X) is the pQCD calculable cross section of partonic subprocess; and D c→Ds is the nonperturbative fragmentation function. In this picture, c andc are produced symmetrically at leading order in α s . The asymmetry appears only in the next-to-leading order (NLO), or higher, corrections. However, the charm-anticharm asymmetries predicted by NLO correction are an order of magnitude smaller than the asymmetries observed in photoproduction experiments [13]. That is, the charmed and anticharmed photoproduction asymmetry may not be fully explained by the charm-anticharm quark production asymmetry. Moreover, the theoretical predictions through above mechanism (1) cannot account for the differences among the asymmetries of D mesons with different light quark flavors.
There have already been some early attempts to explain the observed asymmetries [14,15]. In these approaches, the asymmetry is supposed to appear due to the nonperturbative hadronization effects. Hence, they are all sensitive to unknown distribution functions of partons in the remnant of the target nucleon or photon after the hard scattering. In comparison, the heavy-quark recombination mechanism proposed by Braaten et al. [16] can give a more reasonable explanation to the D meson and Λ c production asymmetries. In the heavy-quark recombination mechanism, a light parton(q) that participates in the hardscattering process recombines with a heavy quark(c) or an antiquark(c) and subsequently hadronize into the final-state heavy-light meson. The recombination happens only when the light-quark in the final state has momentum of O(Λ QCD ) in the heavy-quark, or antiquark, rest frame. Namely, the light-quark and the heavy-quark recombine in a small phase space. By virtue of the heavy quark symmetry, SU(3) flavor symmetry and the large N c limit of QCD, the heavy-quark recombination mechanism gives a simple and predicative explanation for the asymmtries with two nonperturbative parameters. Since the light quark q in the recombination model can be either u, d or s quark, it can account for the difference of asymmetries among different light flavors.
In Ref. [17], Braaten et al. calculated the charm/anticharm production ratio and asymmetry of D mesons using the heavy-quark recombination mechanism and confronted their results to the experimental data. In their consideration, the asymmetry of D s meson comes from the process in which thec(c) and light valence-quark of nucleon recombine into a D − (D + ) meson, while the recoiling c(c) quark fragments to D + s (D − s ) meson. That is, the D + s − D − s asymmetry stems from the excess of u and d overū andd in the nucleon. Because the s ands content of the nucleon are identical in their assumption, the asymmetry of D s meson has the opposite sign as that of D + and D − meson and is relatively small. The experimental data on D s exist [5,7], but with very large errors, and hence do not tell whether there is an asymmetry or not. Different from their consideration, recent years, many studies show that there is striking strange/antistrange sea asymmetry in the momentum distribution inside the nucleon [18,19]. Stimulated by this idea, in this work, we calculate the D s meson production asymmetry induced by the strangeantistrange quark distribution asymmetry within nucleon by employing the heavy-quark recombination mechanism.
According to the heavy-quark recombination mechanism, the cross section of D s meson photoproduction may schematically formulated as: where (cs) n represents that the s in the final state has small relative momentum in thec rest frame, and n is the color and angular momentum quantum numbers of (cs) intermediate state. dσ[γ +s → (cs) n +c] is the perturbative QCD calculable partonic subprocess. The factor ρ[(cs) n → D s ] is the probability of the (cs) n state to evolve into a final state, here, the D s .
In our consideration the D s meson may be produced via two different schemes, i.e.
while in Ref. [17] only the second one, the recombination-fragmentation mechanism, was taken into account. In process (a), the (cs) n recombines into the D s meson directly; in process (b), (cq) n recombines into the D − s , D − orD 0 meson, and the recoiling c quark fragments to the D + s meson. The calculation of the partonic cross section dσ[γ + q → (cq) n + c] in pQCD is straightforward, and our calculation confirms the Ref. [17] results. That is, Here, κ = e q /e c is the ratio of the electric charge fractions of light and charm quarks. The Lorentz invariants are defined as S = (p q +p γ ) 2 , T = (p γ −p c ) 2 −m 2 c and U = (p γ −pc) 2 −m 2 c . p q , p γ and p c are the momenta of the light quark, photon, and c quark, respectively. It is noted that because the relatively small momentum of the light quark in (cs) n system, the higher angular momentum excited states are suppressed by power of Λ QCD /p T or Λ QCD /m c . For a leading order estimation, we take only the contributions from 1 S 0 and 3 S 1 states.
To calculate the total cross section (dσ[γ + N → D s + X]) of the production of D s meson, we need to know the distribution of strange sea quark. According to the lightcone meson-baryon fluctuation model [18], the strange quark-antiquark asymmetry of the nucleon sea is generated by the intrinsic strangeness fluctuations in the proton wavefunction. In this model, the asymmetry stems from the intermediate K + Λ configuration of the incident nucleon, which has the lowest off-shell light-cone energy and invariant mass [20]. The momentum distributions of the intrinsic strange and antistrange quarks in the K + Λ system can be parameterized as where f Λ/K + Λ (y), f K + /K + Λ (y) are the probabilities of finding Λ, K + in the (K + Λ) state; q s/Λ ( x y ), qs /K + ( x y ) are probabilities of finding strange and antistrange quark in Λ or K + states. To estimate these quantities, two simple functions of the invariant mass for the two-body wavefunction are given [21], where the α represents the typical internal momentum scale. In our analysis in this work, we simply adopt the Gaussian type wavefunction for use.
In recently, another model, which based on the effective chiral quark theory, for the distribution of strange sea quark is proposed by Ma et al. [19]. In this model, the strange quark distribution is determined by both the constituent quark distribution and the quark splitting function. For instance, ands Here, P jαs/i is the splitting function representing the probability of finding a constituent quark j together with a spectator Goldstone(GS) boson (α = π, K, η) in a parent constituent quark. u 0 and d 0 denote the constituent quark distributions; V i/α is the quark i distribution within the GS boson α.
Using the resultant production cross section, it is straightforward to calculate the asymmetry of the D + s − D − s photoproduction, which are defined as In our calculation, the two nonperturbative input parameters, ρ sm and ρ sf , are extracted from experiments by fiting to the E687 and E691 data [5,7]. and Here, the subscripts sm and sf stand for spin-matched and spin-flipped situations, respectively. Considering the possible uncertainties, the above values are in agreement with those extracted from the hadroproduction [22].
As for the fragmentation function of charm quark to the D s meson, we exploit the well-known Peterson fragmentation function, Here, D(z; ǫ) [23] is the Peterson function and f c→Ds [24] is the fragmentation probability. In practice, the parameter ǫ is set to be 0.06, as in Ref. [17], which falls in the region of experimental fitting allowance. All through this paper, the charm quark mass are set to be 1.5 GeV, and the factorization and renormalization scales are set to be at the meson transverse mass m T , the p 2 ⊥ + m 2 Ds . The parton distribution distribution function(PDF) of CTEQ6L [25] is used. For simplicity, we use the average photon beam energy of the E687 experiment, E γ = 200GeV , in the calculation. To be more accurate, the D * s feed down effect is included in our analysis, but those from other higher excited states are thought to be small and simply neglected.
In figures 1 and 2, based on the light-cone meson-baryon fluctuation model, the dependence of the inclusive cross sections of D + s and D − s production on the Feynman variable x F and the rapidity y respectively are shown. The corresponding effective chiral quark model results are presented in Figures 3 and 4. For comparison, the cross section of the leading order photon-gluon fusion process, which will give a symmetric D ± s production through fragmentation, is also presented. We find that the results appear to be insensitive to the variations of the factorization scale and the Peterson parameter ǫ. From Figs. 1 -4, we see that cross sections peak in the region of small x F and large rapidity y. Although the contributions from the recombination mechanism are not the dominant ones, they may give out the asymmetry. Both for the light-cone meson-baryon fluctuation model and the effective chiral quark model, in the region of the experimental cut x F > 0, the  corresponding cross sections for D + s are bigger than those for the D − s . Thus, the D s photoproduction asymmetry appears with the effect of strange-antistrange quark asymmetric distributions.
Our predicted D s production asymmetries, in comparison with Ref. [17], are shown in in Figure 5. We find that, by adopting the light-cone meson-baryon fluctuation model, the production asymmetry of D s meson is about 1.2 times larger than that in Ref. [17]; and by using the effective chiral quark model, it is about 80% bigger only. If we extend to the x F negative region, the asymmetries coming from the intrinsic strange sea momentum distribution are very big and diverge very much from the prediction of Ref. [17]. In this region, the asymmetry from the light-cone meson-baryon fluctuation model is even flipped relative to the positive x F region, while the effective chiral quark model and the fragmentation-recombination scheme give the results with the same sign as in the x F > 0 region. Since the obtained data are in the x F > 0 region, here we will not show the results in the x F < 0 region. From the figure 5, it is obvious that the three different asymmetry producing schemes(models) give results diverging with each other with the x F increase. This scaling difference leaves the experiments with an opportunity to decide the physical reality.
In summary, in this paper, the heavy-quark recombination mechanism was employed, which gets a first success in explaining the D meson and Λ c asymmetry production. We have studied the D + s − D − s asymmetry in the photoproduction. Our point is that this asymmetry can be induced not only by the recombination-fragmentation mechanism, the Eq.4, but also by the strange and antistrange distribution asymmetry inside the nucleon, the Eq.3. And, we find the latter effect is even bigger then the former one depending on the employed model and phase space region. Two QCD relevant strange quark distribution models were used, that is the light-cone meson-baryon fluctuation model and the effective chiral quark model. After including the process (3), the predicted asymmetry increases anyhow. However, what we discussed should be the dominant contributions to the D s photoproduction asymmetry. Although there remains some uncertainties, such as from the breakdown of SU(3) symmetry, the use of large N c limit and the NLO corrections to the leading order photon-gluon fusion, etc., our results are adequate to make qualitative predictions, and more detailed discussion on the relevant uncertainties can be found in Ref. [16,17]. It is noticed that the nowadays D s asymmetry experimental data are very limited and preliminary. With enough data collection in the future, it is expected that the experiment can inversely impose a strong restriction on the strange and antistrange quark distributions by measuring the D s production asymmetry.