Prediction of charm-production fractions in neutrino interactions

The way a charm-quark fragments into a charmed hadron is a challenging problem both for the theoretical and the experimental particle physics. Moreover, in neutrino induced charm-production, peculiar processes occur such as quasi-elastic and diffractive charm-production which make the results from other experiments not directly comparable. We present here a method to extract the charmed fractions in neutrino induced events by using results from $e^+e^-$, $\pi N$, $\gamma N$ experiments while taking into account the peculiarities of charm-production in neutrino interactions. As results, we predict the fragmentation functions as a function of the neutrino energy and the semi-muonic branching ratio, $B_\mu$, and compare them with the available data.


Introduction
The problem of how a charm-quark fragments into a charmed hadron is challenging both from the theoretical and the experimental point of view. Indeed, perturbative QCD is not applicable at energies comparable with the charm-quark mass. Therefore, parameterizations have to be used and the parameters are determined experimentally. In the experiments the direct identification of the charmed hadron in the final state is only possible through the visual observation of the hadron decay and the measurement of the kinematical variables, practically feasible only with nuclear emulsions.
In the following, we focus on the so-called charmproduction fractions (f h 's); i.e., the probability that E-mail address: pasquale.migliozzi@cern.ch (P. Migliozzi). a charm-quark fragments into a charmed hadron h (= D 0 , D + , D s , Λ c ).
In this Letter we review all existing data on charmproduction fractions as measured by e + e − , πN and γ N experiments and predict f h in neutrino induced charm-production. Indeed, data on charmed fractions in interactions induced by neutrinos are rather scarce. Only one experiment, E531 [1], measured f h with a statistics of 122 events with an identified charmed hadron in the final state.

Measurement of the D + /D 0 ratio
In this section we give an overview of the available data on D + /D 0 measurements in e + e − , πN, pN and γ N experiments. Furthermore, we also study the correlated variable F V , which is defined as V /(V + P ), where V and P signify vector and pseudoscalar charmed mesons, respectively. When needed, we recomputed F V by using the latest charmed-hadron branching-ratios (BR). Finally, we extract for the first time F V from neutrino induced charm-production data.
In the following we assume that D 0 and D + production cross-sections are equal at the parton level, as well as the direct production of D 0 and D + .
From here on, we indicate with D the sum of both prompt and decayed (D → D) D meson production.
Under the previous assumptions, the measurement of D + /D 0 and D /D ratios allows us to extract F V . By using the formulae given in Ref. [3], F V can be extracted from the relations where The basic principle to reconstruct e + e − events with charmed hadrons in the final state is common to all the experiments. Being a cc pair produced in the annihilation, one charmed hadron is used to tag the events, while the other one to study the decay properties. The decay modes used to tag the event are 1 where the corresponding branching ratios are also given in brackets [4].
Results on the total cross-sections for inclusive production of the charmed particles D 0 , D + , D 0 and D + at various √ s are shown in Table 1. A complete review of the probabilities (f (c → C)) that a c-quark fragments into a D , D 0 , D + and other charmed hadrons as measured in Z 0 decays is given in Ref. [9] and reported in Table 2.
From Tables 1 and 2 we can extract both R 1 and F V , the latter being estimated by using Eqs. (1) and (2). The results are given in Table 3 and show that within the experimental errors, both R 1 and F V are independent of the energy.
From these data we can conclude that, within the experimental errors, R 1 is both process-and energyindependent.

νN experiments
Recently two measurements which allowed us to extract for the first time F V from neutrino experiments became available. In Ref. [12] the CHORUS Collaboration presented a measurement of the production rate of D 0 based on a sample of about 26 000 ν µ chargedcurrent events interactions located and analyzed so far in the target emulsions. After reconstruction of the event topology in the vertex region, 283 D 0 decays were observed with an estimated background of 9.2 events from K 0 and Λ decays. The CHORUS Collaboration measured the D 0 production cross-section [13]. The total cross-section has been extracted by accounting for the D 0 decays into all neutrals. The value we used is BR(D 0 → all neutral) = (25 ± 5)% [14].
Therefore, the D 0 production cross-section normalized to ν µ charged-current (CC) interactions is at 27 GeV average ν µ energy. Notice that this measurement includes both D 0 prompt and D 0 from the decay of D mesons. Table 5 Summary of the available data on F V and predictions from different models. The error for the theoretical predictions is not shown being relevant only for the third digit The D + production in ν µ charged-current interactions has been measured, with a similar ν µ beam, by the BEBC [15] and NOMAD [16] experiments to be (1.22 ± 0.25) × 10 −2 and (0.79 ± 0.20) × 10 −2 , respectively. The weighted average D + production rate normalized to ν µ charged-current interactions is From the measured ratios (3) and (4), and by knowing B , we can compute R 2 = 0.25 ± 0.06. From the latter value and from Eq. (2) we can extract It is worth to notice that R 1 and F V extracted from neutrino experiments can be compared straightway to e + e − , πN and γ N results, being D + and D 0 either produced promptly or from the decay of prompt D + and D 0 . Namely, processes peculiar of ν interactions do not affect R 1 and F V . 2

Summary and discussion of all available data on R 1 and F V
From results reported in the previous sections, we can argue that within the experimental errors R 1 is constant over a wide range of energies ( √ s ∼ 4-90 GeV) and independent of the process. The constant behavior of R 1 down to √ s ∼ 4 GeV can be derived with simple arguments: 2 In neutrino interactions D ( )+ may also be produced diffractively but, due to the V cd suppression, its rate is expected to be about (1.6 ± 0.3) × 10 −4 with respect to CC interactions and therefore negligible. For these reasons we assume that R 1 measured in e + e − (see Table 3) can be used in neutrino induced charm-production, too.
As an important by product of our study we have also extracted F V from different processes and at several energies (see Table 5). The simplest model to predict F V is based on the spin-counting. Namely, vector mesons are spin-one states ( 3 S 1 ), while pseudoscalar mesons are spin-zero states ( 1 S 0 ), therefore F V = 0.75. The discussion of more refined models (UCLA, JETSET, HERWIG and others) is beyond the purposes of this Letter. For details we refer to [17].
From Table 5 we can see that the measured F V is independent of the processes and of the energy. This means that the probability for a c-quark to fragment into a D or a D meson is universal and does not depend neither on the process nor on the energy. Notice that the UCLA model is the best in describing available e + e − data.

Measurement of the D s to D 0 and Λ c to D 0 ratios
The ratio D s /D 0 has been measured in e + e − , πN and γ N experiments. A summary of the available data is given in Table 6. Table 6 Summary of the available R s and R c measurements extracted from e + e − , νN and γ N experiments. The πN data have been taken from Refs. [10,11] and references therein, while γ N results from Ref. [ [5] 0.176 ± 0.076 -10.55 [7] 0.148 ± 0.052 0.148 ± 0.052 10.55 [8] 0 The decay mode used to tag the event is D + s → φπ + → (K − K + )π + whose BR, as reported by the Particle Data Group, are [4]: From Table 6 we can see that the D s /D 0 ratio is, within the errors, independent of the energy and of the process.
Data on the ratio Λ c /D 0 are very poor. Indeed, it has been measured only in e + e − experiments (see Table 6).
The decay mode used to tag the event is Λ c → pK − π + whose BR, as reported by the Particle Data Group, is [4]: BR Λ c → pK − π + = 0.050 ± 0.013.
From Table 6 we can see that, although with a smaller statistical accuracy, both R s and R c are, within the errors, independent of the energy.

Summary and discussion of all available data on R s and R c
Although from Table 6 it seems that R s is constant over a wide range of energies ( √ s ∼ 4-90 GeV) and independent of the process, some comments are in order.
Given the quark composition of the D + s (cs) meson, it has to be created always together with at least one K meson. Therefore, being m K ≈ 500 MeV, we expect that the threshold effect for D s production is more pronounced than for D mesons, when at least one π has to be produced (m π ≈ 100 MeV). To account for the different threshold effect at low energies, in the following we do not use the R s values measured at the Z 0 peak. Furthermore, under the assumption that √ s in collider experiments can be replaced with W in fixed target experiments, W being the final state hadronic mass, and noting that neutrino-induced charm events at present experiments are characterized by values of W in the range 4-10 GeV, we can argue a R s value of R s = 0.171 ± 0.029 which corresponds to the weighted average of the first three results in Table 6.
The available measurements on R c are very poor. Nevertheless, as it will be discussed in Section 3, we do not use the R c value to predict the charmed fractions in events induced by neutrinos.

The method
From the previous sections we can argue that once a charm-quark has been produced in deep-inelastic interactions (i.e., the energy of the process is higher than the threshold), it has a probability to produce a charmed hadron C h which is, within the experimental errors, independent of the process and of the energy. Therefore, as far as the deep-inelastic scattering (DIS) is concerned, we can write the charm production rate as In the case of neutrino induced charm-production, Eq. (5) is not correct. Indeed, in this case we also have to account for diffractive and quasi-elastic charmproduction. Therefore, the inclusive charm-production Table 7 Prediction of charm-production fractions in neutrino induced events as a function of the neutrino energy If one accepts that R 1 , R s and R c from e + e − (with √ s = 4.1-90 GeV) and other experiments can be used to described the fragmentation of charm-quarks produced in DIS neutrino interactions with average final state hadronic mass W ∼ 10 GeV, then From Eq. (7) charm-production fractions in neutrino interactions can be written as .
Notice that, given the poor knowledge on R c and on the quasi-elastic charm-production cross-section, we derive f Λ c by using the normalization constrain In order to estimate the charmed fractions and their energy dependence, we use the following inputs: • the inclusive charm-production rate derived in Ref. [18]; • the energy dependence of the D 0 production rate reported in Ref. [12] properly scaled to account for the effect discussed in Section 2.1.3; • the energy dependence of the diffractive D s production given in [19], properly scaled in order to reproduce the average diffractive charmproduction cross-section as measured by BEBC and NuTeV [20] (

Results on f h and B µ and comparison with the data
By using the method described in the previous section, we derived the charm-production fractions, as a function of the neutrino energy, as reported in Table 7 and shown in Fig. 1. Our results are in good agreement with the charm-production fractions extracted from the E531 data, see Fig. 1. It is worth noticing that f Λ + c shows a dependence on the energy, higher values at low neutrino energies, consistent with the expectations. Indeed, quasi-elastic charmproduction, which yields only Λ c , is expected to largely contribute to σ c (ν) for E ν < 25 GeV [2]. In Table 7 the expected charm-production fractions in the CHORUS experiment are also given.

PDG [4]
V cd = 0.219-0.225 V cd = 0.222 ± 0.003 (From unitarity at 90%) a very important quantity, being the input variable needed to extract from the dimuon data the element of the CKM matrix V cd .
Recently, a direct measurement of B µ has been performed by the CHORUS Collaboration by using a statistics of about 1000 charm events reconstructed in the nuclear emulsions. Out of these, (88 ± 10 ± 8) dimuon events have been reconstructed, which correspond to [13] B µ = (9.3 ± 1.3)%.
This measurement has to be compared with our prediction obtained by convoluting the charm-production fractions with the CHORUS neutrino flux B µ = (8.8 ± 1.0)%.
In Table 8 we derived, by using the V cd value obtained by imposing the unitarity constraint to the CKM matrix and the measurements of B µ |V cd | 2 from various experiments, B µ . Given the fact that the different experiments exploit different neutrino energy spectra, we can probe the sensitivity of B µ to the neutrino energy. As expected, the higher the neutrino beam energy the larger the value of B µ .

Conclusions
We have presented a method to extract the charmed fractions in neutrino induced events. The method relies on the fact that, apart from processes peculiar to neutrinos such as quasi-elastic and diffractive charmproduction, the charm-production and fragmentation mechanism is believed to be process-independent. We have verified this natural assumption going through a complete review of the available data from different experiments. Moreover, the D + over D 0 ratio is constant over the large energy range spanned by the collider and fixed target experiments reviewed in this Letter. By using recent data from neutrino experiments, we have assessed the consistency of this ratio with the one predicted by other experiments. On the other side, the energy-independent behavior of the ratio itself is clear from the review of the experiments.
Threshold effects for the D s production are seen and accounted for. By introducing the diffractive charm-production and using the unity constraint, we have predicted the charm fragmentation as a function of the neutrino energy in the range useful to present neutrino experiments. In particular, a prediction for the CHORUS experiment has been made. The determination of the fragmentation function also allows the prediction of the semi-muonic branching ratio of charmed hadrons. The prediction given is in good agreement with a recent measurement made by the CHORUS experiment.