SLAC–PUB–95–7068

LBL–37666

Charmed Hadron Asymmetries in the Intrinsic Charm Coalescence Model
^{*}^{*}*
This work was supported in part by the Director, Office of Energy
Research, Division of Nuclear Physics of the Office of High Energy
and Nuclear Physics of the U. S. Department of Energy under Contract
Numbers DE-AC03-76SF0098 and DE-AC03-76SF00515.

[8ex]

R. Vogt

[2ex] Nuclear Science Division

Lawrence Berkeley Laboratory

Berkeley, California 94720

and

Physics Department

University of California at Davis

Davis, California 95616

[2ex]

and

[2ex]

S. J. Brodsky

[2ex]

Stanford Linear Accelerator Center

Stanford University

Stanford, California 94309

[12ex]

(Submitted to Nuclear Physics B.)

[3ex]

ABSTRACT

Fermilab experiment E791, measuring charmed hadron production in interactions at 500 GeV with high statistics, has observed a strong asymmetry between the hadroproduction cross sections for leading mesons which contain projectile valence quarks and the nonleading charmed mesons without projectile valence quarks. Such correlations of the charge of the meson with the quantum numbers of the beam hadron explicitly contradict the factorization theorem in perturbative QCD which predicts that heavy quarks hadronize through a jet fragmentation function that is independent of the initial state. The E791 experiment also measures and production asymmetries as well as asymmetries in pair production. We examine these asymmetries and the fractional longitudinal momentum, , distributions for single and pairs of charmed hadrons within a two-component model combining leading-twist and fusion subprocesses with charm production from intrinsic heavy quark Fock states. A key feature of this analysis is intrinsic charm coalescence, the process by which a charmed quark in the projectile’s Fock state wavefunction forms charmed hadrons by combining with valence quarks of similar rapidities.

1. Introduction

The E791 experiment, studying 500 GeV interactions with carbon and platinum targets, employs an open geometry spectrometer with a very open trigger and a fast data acquisition system to record the world’s largest sample of hadroproduced charm [1]. This large data set allows detailed investigations of charmed hadron production including , , , , and pairs.

One of the most striking features of charm hadroproduction is the leading particle effect: the strong correlation between the quantum number of the incident hadron and the quantum numbers of the final-state charmed hadron. For example, more than are produced at large in [1, 2, 3, 4, 5]. There is also evidence of leading particle correlations in [6, 7, 8] and [9] production in collisions and production in hyperon-nucleon interactions [10, 11]. Such correlations are remarkable because they explicitly contradict the factorization theorem in perturbative QCD which predicts that heavy quarks hadronize through a jet fragmentation function that is independent of the initial state.

Leading charm production can be quantified by studies of the production asymmetries between leading and nonleading charm production. In interactions, both and , which share valence quarks with the , are “leading” while and , which do not, are “nonleading” at . The harder leading distributions suggest that hadronization at large involves the recombination of the charmed or anticharmed quarks with the projectile spectator valence quarks. The asymmetry is defined as

(1) |

The measured asymmetry increases from nearly zero at small to around [4, 5], indicating that the leading charm asymmetry is primarily localized at large . Thus the asymmetry reflects the physics of only a small fraction of the total cross section. The neutral ’s were not used in the analysis since these states can also be produced indirectly by the decay of nonleading mesons.

In a recent paper [12], we discussed a QCD mechanism which produces a strong asymmetry between leading and nonleading charm at large A key feature of this analysis is coalescence, the process by which a produced charmed quark forms charmed hadrons by combining with quarks of similar rapidities. In leading-twist QCD, heavy quarks are produced by the fusion subprocesses and . The factorization theorem [13] predicts that fragmentation is independent of the quantum numbers of both the projectile and target. Thus one expects to leading order. However, it is possible that the forward-moving heavy quarks will coalescence with the spectator valence quarks of the projectile to produce leading hadrons in the final state. In a gauge theory one expects the strongest attraction to occur when the spectator and produced quarks have equal velocities [14]. Thus the coalescence probability should be largest at small relative rapidity and relatively low transverse momentum where the invariant mass of the system is minimal, and its amplitude for binding is maximal.

The coalescence of charmed quarks with projectile valence quarks may also occur in the initial state. For example, the or wavefunctions can fluctuate into or Fock states. These states are produced in QCD from amplitudes involving two or more gluons attached to the charmed quarks. The most important fluctuations occur at minimum invariant mass where all the partons have approximately the same velocity. These fluctuations can have very long lifetimes in the target rest frame, , where is the projectile momentum. Since the charm and valence quarks have the same rapidity in these states, the heavy quarks carry a large fraction of the projectile momentum. Furthermore the comoving heavy and light quarks can readily coalesce to produce leading charm correlations at a large combined longitudinal momentum. Such a mechanism can dominate the hadroproduction rate at large . This is the underlying assumption of the intrinsic charm model [15].

The intrinsic charm fluctuations in the wavefunction are initially far off the light-cone energy shell shell. However, they become on shell and materialize into charmed hadron when a light spectator quark in the projectile Fock state interacts in the target [16]. Since such interactions are strong, the charm production will occur primarily on the front face of the nucleus in the case of a nuclear target. Thus an important characteristic of the intrinsic charm model is its strong nuclear dependence; the cross section for charm production via the materialization of heavy Fock states should have a nuclear dependence at high energies similar to that of inelastic hadron-nucleus cross sections.

In this work, we concentrate on the charmed hadrons and meson pairs channels studied by E791 in order to further examine the relationship between fragmentation and coalescence mechanisms. The calculations are made within a two-component model: leading-twist fusion and intrinsic charm [12, 17, 18]. We find that the coalescence of the intrinsic charmed quarks with the valence quarks of the projectile is the dominant mechanism for producing fast and mesons. On the other hand, when the charmed quarks coalesce with sea quarks, there is no leading charmed hadron. We discuss the longitudinal momentum distributions and the related asymmetries for and production, as well as pairs. (We have only applied our model to pairs in the forward hemisphere in order to provide a clear definition of the asymmetry.)

As expected, the asymmetries predicted by the intrinsic charm coalescence model are a strong function of . We find that production in the proton fragmentation region ( in collisions) is dominated by the coalescence of the intrinsic charm quark with the valence quarks of the proton. Coalescence is particularly important in pair production. The production of and, at , by coalescence must occur within still higher particle number Fock states.

Leading particle correlations are also an integral part of the Monte Carlo program PYTHIA [19] based on the Lund string fragmentation model. In this model it is assumed that the heavy quarks are produced in the initial state with relatively small longitudinal momentum fractions by the leading twist fusion processes. In order to produce a strong leading particle effect at large , the string has to accelerate the heavy quark as it fragments and forms the final-state heavy hadron. Such a mechanism goes well beyond the usual assumptions made in hadronization models and arguments based on heavy quark symmetry, since it demands that large changes of the heavy quark momentum take place in the final state.

In this paper we shall compare the predictions of the intrinsic
charm coalscence model with those of PYTHIA [19]. The
comparison of the data with these models should distinguish the
importance of higher heavy quark Fock state fluctuations in the
initial state and the coalescence process from the strong string
hadronization effects postulated in the PYTHIA model.

2. Leading-Twist Production

In this section we briefly review the conventional leading-twist model for the production of single charmed hadrons and pairs in interactions. We will also show the corresponding distributions of charmed hadrons predicted by the PYTHIA model [19].

Our calculations are at lowest order in . A constant factor is included in the fusion cross section since the next-to-leading order distribution is larger than the leading order distribution by an approximately constant factor [20]. Neither leading order production nor the next-to-leading order corrections can produce flavor correlations [21].

The single charmed hadron distribution, , has the factorized form [18]

(2) |

where and are the initial partons, 1 and 2 are the charmed quarks with GeV, and 3 and 4 are the charmed hadrons. The convolution of the subprocess cross sections for annihilation and gluon fusion with the parton densities is included in ,

(3) |

where and are the interacting hadrons. For consistency with the leading-order calculation, we use current leading order parton distribution functions, GRV LO, for both the nucleon [22] and the pion [23].

The fragmentation functions, , describe the hadronization of the charmed quark where is the fraction of the charmed quark momentum carried by the charmed hadron, produced roughly collinear to the charmed quark. Assuming factorization, the fragmentation is independent of the initial state (leptons or hadrons) and thus cannot produce flavor correlations between the projectile valence quarks and the charmed hadrons. This uncorrelated fragmentation will be modeled by two extremes: a delta function, , and the Peterson function [24], as extracted from data. The Peterson function predicts a softer distribution than observed in hadroproduction, even at moderate [18], since the fragmentation decelerates the charmed quark, decreasing its average momentum fraction, , approximately 30% by the production of mesons. The delta-function model assumes that the charmed quark coalesces with a low- spectator sea quark or a low momentum secondary quark such that the charmed quark retains its momentum [18]. This model is more consistent with low charmed hadroproduction data [25, 26, 27] than Peterson fragmentation.

The parameters of the Peterson function we use here are taken from studies of production [28]. The distributions are very similar to the distributions but the distribution appears to be somewhat softer [29]. Although there is some uncertainty in the exact form of the Peterson function for charmed baryons and mesons, it always produces deceleration.

In Fig. 1 we show the single inclusive distributions calculated for (a) delta function and (b) Peterson function fragmentation in interactions at 500 GeV. The results are normalized to the total single charmed quark cross section. The parton distributions of the pion are harder than those of the proton at large , producing broader forward distributions. As expected, the delta-function fragmentation results in harder distributions than those predicted by Peterson fragmentation for . However, as shown in [4], the conventional fusion model, even with delta-function fragmentation, cannot account for the shape of leading distributions.

The charmed hadron distributions from PYTHIA, obtained from a run with events at 500 GeV using all default settings and the GRV LO parton distributions, are shown in Fig. 1(c) for and and 1(d) for and hadrons. The distributions are normalized to the number of charmed hadrons per event. PYTHIA is based on the Lund string fragmentation model [19] in which charmed quarks are at string endpoints. The strings pull the charmed quarks toward the opposite string endpoints, usually beam remnants. When the two string endpoints are moving in the same general direction, the charmed hadron can be produced with larger longitudinal momentum than the charmed quark, accelerating it. In the extreme case where the string invariant mass is too small to allow multiple particle production, this picture reduces to final-state coalescence and the string endpoints determine the energy, mass, and flavor content of the produced hadron [30]. Thus a or can inherit all of the remaining projectile momentum while and production is forbidden. The coalescence of a charmed quark with a valence diquark results in the secondary peak at in the distribution shown in Fig. 1(c). The baryon is a leading charmed hadron in the proton fragmentation region since it can have two valence quarks in common with the proton. Also in the proton fragmentation region, the is somewhat harder than the . This is evidently a secondary effect of coalescence with a diquark: the can pull the charmed quark to larger in the wake of the , producing more at negative than . Such coalescence correlations are not predicted for the or the in the proton fragmentation region. In the pion fragmentation region, the predicted distributions are harder than the distributions.

The charmed pair distribution is

(4) | |||||

where and . Figure 2 shows the forward distribution of pairs from interactions at 500 GeV with (a) delta function and (b) Peterson function fragmentation for both charmed quarks. The distributions are normalized to the total pair production cross section. Note that the pair distributions are harder than the single distributions in Fig. 1. The perturbative QCD calculation cannot distinguish between leading and nonleading hadrons in the pair distributions.

In Fig. 2(c), the pair distributions from PYTHIA have been classified as doubly leading , , nonleading-leading , , and doubly nonleading , . The same classification for pairs is shown in Fig. 2(d). The distributions are normalized to the number of charmed pairs per event. We have not considered or pairs. Note that the leading particle assignments are only valid for . The assignments are more meaningful for the ’s since they may be assumed to be directly produced. The neutral mesons in Fig. 2(c) arise in part from charged decays. Thus the pair distributions have a shoulder at from decays to which is absent in the distributions. Note also that the pairs are most numerous since both charged and neutral pairs contribute to the distribution. Almost three times as many neutral ’s are produced than charged ’s. Only some of this difference can arise from decays since charged and neutral ’s are produced in nearly equal abundance.

Other final-state coalescence models have been proposed, including a
valence spectator recombination model [31] and the valon
model [32]. Two important unknowns in these models are the
correlation between the charmed quark and the valence spectator in
the recombination function and the -particle parton distributions
of the spectator and participant valence quarks. In this work we
will not compare to either of these models but simply note that they
can also produce charmed hadrons and hadron pairs by final-state
coalescence.

3. Intrinsic Heavy Quark Production

The wavefunction of a hadron in QCD can be represented as a superposition of Fock state fluctuations, e.g. , , , …components where for a and for a proton. When the projectile scatters in the target, the coherence of the Fock components is broken and the fluctuations can hadronize either by uncorrelated fragmentation or coalescence with spectator quarks in the wavefunction [15, 16]. The intrinsic heavy quark Fock components are generated by virtual interactions such as where the gluons couple to two or more projectile valence quarks. The probability to produce fluctuations scales as relative to leading-twist production [33] and is thus higher twist. Intrinsic Fock components are dominated by configurations with equal rapidity constituents so that, unlike sea quarks generated from a single parton, the intrinsic heavy quarks carry a large fraction of the parent momentum [15].

The frame-independent probability distribution of an –particle Fock state is

(5) |

where , assumed to be slowly varying, normalizes the probability, . The delta function conserves longitudinal momentum. The dominant Fock configurations are closest to the light-cone energy shell shell and therefore have minimal invariant mass, , where is the effective transverse mass of the particle and is the light-cone momentum fraction. Assuming is proportional to the square of the constituent quark mass, we adopt the effective values GeV, GeV, and GeV [17, 18].

The intrinsic charm production cross section can be related to and the inelastic cross section by

(6) |

The factor of arises because a soft interaction is needed to break the coherence of the Fock state. The NA3 collaboration [34] separated the nuclear dependence of production in interactions into a “hard” contribution with a nearly linear dependence at low and a high “diffractive” contribution scaling as , characteristic of soft interactions. One can fix the soft interaction scale parameter, GeV, by the assumption that the diffractive fraction of the total production cross section [34] is the same for charmonium and charmed hadrons. Therefore, we obtain b at 200 GeV and b [12] with % from an analysis of the EMC charm structure function data [35]. A recent reanalysis of this data with next-to-leading order calculations of leading twist and intrinsic charm electroproduction confirms the presence of an % intrinsic charm component in the proton for large [36]. Note that a larger would not necessarily lead to a larger . Since we have fixed from the NA3 data, increasing would decrease accordingly.

We now calculate the probability distributions, for charmed hadrons and pairs resulting from both uncorrelated fragmentation and coalescence of the quarks in the intrinsic charmed Fock states. These light-cone distributions are frame independent. In a hadronic interaction, these states are dissociated and materialize with the corresponding differential cross section

(7) |

In the case of collisions, the fluctuations of the
state produces charmed
hadrons at in the center of mass while the fluctuations of
the state produces charmed hadrons at
.

3.1 Single charmed hadrons

There are two ways of producing charmed hadrons from intrinsic states. The first is by uncorrelated fragmentation, discussed in Section 2. Additionally, if the projectile has the corresponding valence quarks, the charmed quark can also hadronize by coalescence with the valence spectators. The coalescence mechanism thus introduces flavor correlations between the projectile and the final-state hadrons, producing e.g. ’s with a large fraction of the momentum. In the pion fragmentation region, , and have contributions from both coalescence and fragmentation while and can only be produced from the minimal Fock state by fragmentation. In the proton fragmentation region, , the and may be produced by both coalescence and fragmentation.

If we assume that the quark fragments into a meson, the distribution is

(8) |

where , 5 for pion and proton projectiles in the configuration. This mechanism produces mesons carrying 25-30% of the projectile momentum with the delta function and 17-20% with the Peterson function. The distributions are shown in Fig. 3(a) and 3(b) for proton and pion projectiles normalized to the total probability of the Fock state configuration assuming that is the same for protons and pions. Less momentum is given to the charmed quarks in the proton than in the pion because the total momentum is distributed among more partons. These distributions are assumed for all intrinsic charm production by uncorrelated fragmentation.

The coalescence distributions, on the other hand, are specific for the individual charmed hadrons. The coalescence contribution to leading production is

(9) |

With the additional momentum of the light valence quark, the takes 40-50% of the momentum. In the proton fragmentation region, the quark can coalesce with valence and quarks to produce leading ’s,

(10) |

carrying 60% of the proton momentum. The distribution, shown in Fig. 3(c), is also normalized to .

Coalescence may also occur within higher fluctuations of the intrinsic charm Fock state. For example, at , and can be produced by coalescence from and configurations. We previously studied production from states [37]. Assuming that all the measured pairs [38, 39] arise from these configurations, we can relate the cross section,

(11) |

to the double intrinsic charm production probability, , where is the fraction of intrinsic pairs that become ’s. The upper bound on the model, pb [38], requires [37, 40]. This value of can be used to estimate the probability of light quark pairs in an intrinsic charm state. We expect that the probability of additional light quark pairs in the Fock states to be larger than ,

(12) |

leading to and .

Then with a pion projectile at , the coalescence distribution from a six-particle Fock state is

(13) |

also shown in Fig. 3(c) and normalized to . Since half of the quarks are needed to produce the , it carries 50% of the pion momentum. The mesons arising from coalescence in the state,

(14) |

carry -40% of the hadron momentum, as shown in Fig. 3(d) and normalized to . The and
inherit less total momentum than the leading since the Fock
state momentum is distributed over more partons. Thus as more
partons are included in the Fock state, the coalescence
distributions soften and approach the fragmentation distributions,
Eq. (8), eventually producing charmed hadrons with less momentum
than uncorrelated fragmentation from the minimal
state if a sufficient number of pairs are included.
There is then no longer any advantage to introducing more light
quark pairs into the configuration–the relative probability will
decrease while the potential gain in momentum is not significant.
We thus do not consider production by
coalescence at since a minimal nine-parton Fock state is
required.

3.2 pair production

In any state, pairs may be produced by double fragmentation, a combination of fragmentation and coalescence, or, from a pion projectile only, double coalescence. We discuss only pair production at so that is the minimal Fock state. In the proton fragmentation region, with no valence antiquark, the leading mesons would be and . Therefore no doubly leading pairs can be produced from the five parton state: two intrinsic pairs are needed, i.e. states, automatically softening the effect. However, doubly leading meson-baryon pairs such as and may be produced by coalescence in the state. These combinations might be interesting to study in interactions.

The pairs resulting from double fragmentation,

carry the lowest fraction of the projectile momentum. The distributions are shown in Fig. 4(a). When both charmed quarks fragment by the Peterson function, % of the momentum is given to the pair while the average is 62% with delta function fragmentation. If, e.g. a is produced by coalescence while the is produced by fragmentation,

These distributions, with rather large momentum fractions, 71% for Peterson fragmentation and 87% for the delta function, are shown in Fig. 4(b). When the projectile has a valence antiquark, as in the pion, pair production by double coalescence is possible,

(17) |

All of the momentum of the four-particle Fock state is transferred
to the pair, i.e. . We also consider double coalescence from a pion in a six
particle Fock state. In this case, 74% of the pion momentum is
given to the pair, similar to the result for Peterson fragmentation
with coalescence, Eq. (16), as could be expected from our
discussion of production in this model.

4. Predictions of the Two-Component Model

We now turn to specific predictions of the distributions and asymmetries in our model. The distribution is the sum of the leading-twist fusion and intrinsic charm components,

(18) |

where is related to in Eq. (7). Note that when we discuss uncorrelated fragmentation, the same function, either the delta or Peterson function, is used for both leading twist fusion and intrinsic charm. The intrinsic charm model produces charmed hadrons by a mixture of uncorrelated fragmentation and coalescence [12, 18]. Coalescence in the intrinsic charm model is taken to enhance the leading charm probability over nonleading charm. Since we have not made any assumptions about how the charmed quarks are distributed into the final-state charmed hadron channels, an enhancement by coalescence is not excluded as long as the total probability of all charmed hadron production by intrinsic charm does not exceed . Because little experimental guidance is available to help us separate the charm production channels, we have not directly addressed the issue here. Thus the above distributions are normalized to the total cross section for the pair distributions and to the single charm cross section for the single charmed hadrons. This is naturally an overestimate of the cross sections in the charm channels, but more complete measurements are needed before the relative strengths of the , , , , etc. contributions to the charm cross section can be understood.

In the case of nuclear targets, the model assumes a linear
dependence for leading-twist fusion and an dependence for
the intrinsic charm component [34]. This dependence is
included in the calculations of the production asymmetries while the
distributions are given for interactions. The
intrinsic charm contribution to the longitudinal momentum
distributions is softened if the dependence is included.

4.1 Single charmed hadrons

We now consider the single charmed hadron distributions produced in interactions at 500 GeV over the entire range. Since the production mechanisms are somewhat different for positive and negative , particularly for the , we will discuss the pion and proton fragmentation regions separately.

We begin with production in the proton fragmentation region, negative . As we have already indicated, the can be produced by coalescence from the configuration with an average of 60% of the center-of-mass proton momentum. The can only be produced by fragmentation from a five-particle Fock state and, if a nine-particle Fock state is considered, the coalescence distribution will not be significantly harder than the fragmentation distribution shown in Fig. 3(a) since four additional light quarks are included in the minimal proton Fock state.

Therefore coalescence is only important for the , leading naturally to an asymmetry between and . We will assume that the same number of and are produced by fragmentation and that any excess of production is solely due to coalescence. Then, at ,

(19) | |||||

(20) |

The fragmentation distribution is taken from Eq. (8), the coalescence distribution from Eq. (10). The parameter is related to the integrated ratio of to production. We have assumed three values of : 1, 10, and, as an extreme case, 100. The results for the two uncorrelated fragmentation functions are shown in Fig. 5(a) and 5(b). Intrinsic charm fragmentation produces a slight broadening of the distribution for delta function fragmentation over leading-twist fusion (increasing to a shoulder for the Peterson function). The distribution, strongly dependent on , is considerably broadened.

The value is compatible with early low statistics measurements of charmed baryon production [41] where equal numbers of and were found in the range . The data is often parameterized as , where

(21) |

For , we predict for the delta
function and 7.3 for the Peterson function. The difference is due
to the steeper slope of the fusion model with the Peterson function.
We find rather large values of since the average
is dominated by the leading-twist fusion component at low
. If we restrict the integration to , then
decreases to 1.4 independent of the fragmentation
mechanism^{†}^{†}†The parameterization is only good if
the distribution is monotonic. However, our two-component
distribution does not fit this parameterization over all
. At low , the leading-twist component dominates. If
only the high part is included, the value of is a more accurate reflection of the shape of the intrinsic
charm component.. The data on production measured in
collisions at the ISR with GeV are consistent
with this prediction. For , was found [6] while for , [7]. Hard charmed baryon distributions have
also been observed at large in interactions at the
Serpukhov spectrometer with an average neutron energy of 70 GeV,
for [8]. Charmed
hyperons produced by a 640 GeV neutron beam
[42] do not exhibit a strong leading behavior, . This is similar to the delta function prediction for
when . On the other hand, charmed hyperons
produced with a beam [10, 11] are leading
with for [10]. Thus in
the proton fragmentation region is compatible with the shape
of the previously measured distributions. When we
compare the cross section in the proton fragmentation
region with that of leading-twist fusion, the coalescence mechanism
increases the cross section by a factor of 1.4-1.7 over the fusion
cross section and by 30% over the cross
section.

The extreme value, , was chosen to fit the forward production cross section measured at the ISR, b [6], assuming that the charmed quark and cross sections are equal, already an obvious overestimate. This choice produces a secondary peak in the distributions at , the average momentum from coalescence. Such a large value of implies that the intrinsic charm cross section is considerably larger than the leading-twist cross section.

The shape of the distribution with is similar to that due to diquark coalescence in PYTHIA [19], shown in Fig. 1(c), except that the PYTHIA distribution peaks at due to the acceleration induced by the string mechanism. While a measurement of the cross section over the full phase space in the proton fragmentation region is lacking, especially for interactions at , no previous measurement shows an increase in the distributions as implied by these results. However, the reported production cross sections are relatively large [6, 7, 8, 42], between 40 b and 1 mb for GeV. In particular, the low energy cross sections are much larger than those reported for the total cross section at the same energy. This is not yet understood.

A few remarks are in order here. Some of these experiments [7, 8] extract the total cross section by extrapolating flat forward distributions back to and also assume associated production, requiring a model of production. On the other hand, the reported total cross sections are usually extracted from measurements at low to moderate and would therefore hide any important coalescence contribution to charmed baryon production at large . High statistics measurements of charmed mesons and baryons over the full forward phase space () in interactions would help resolve both the importance of coalescence and the magnitude of the total production cross section.

We also chose as an intermediate value. In this case, a secondary peak is also predicted but the cross section at is only a factor of two to three larger than the fusion cross section rather than the factor of 21 needed to fit the ISR data at . The magnitude of the second maximum is also less than the fusion cross section in the central region.

An important test of the production mechanism for charm hadroproduction is the and asymmetry, defined as

(22) |

If is assumed to arise
only from initial state coalescence, we can estimate the parameter
from the E791 500 GeV data. Our calculated asymmetries
for the three values^{‡}^{‡}‡We have only shown the delta
function results. Those with the Peterson function are quite
similar. The slope increases slightly but the point where does not shift. are compared
with the results from PYTHIA in Fig. 5(c). It is, as expected,
closest to our model with although the asymmetry predicted
by PYTHIA does not increase as abruptly as in our model.

Preliminary data from E791 [43] indicate a significant asymmetry for as small as , albeit with large uncertainty. The intrinsic charm model in its simplest form can only produce such asymmetries if , against intuition. Alternatively, a softer distribution from coalescence would make a larger asymmetry at lower , thus allowing a smaller . Such a softening could be due to either an important contribution to production from a configuration or a different assumption about the wavefunction [44]. In a more realistic model, both initial and final state coalescence will play some role in production. Final-state coalescence, added to the leading-twist fusion prediction, would also require a smaller .

At there is no asymmetry in interactions since both the baryon and antibaryon can be produced by fragmentation from a state and by coalescence from a state (,). Then

(23) |

The coalescence contribution, obtained from Eq. (13), produces a small shoulder in the distributions at . We extract for , in good agreement with the NA32 measurement, [27]. The same mechanism can account for both and production in the fragmentation region since no asymmetry is observed [43], which is also in accord with the NA32 result, [27]. To look for subtle coalescence effects as well as to understand the large difference between and production at it is important to compare the shape of the momentum distributions in addition to the asymmetries.

Assuming that equal numbers of and are produced, the distributions are

(24) |

The coalescence contribution is given by Eq. (14) and the distributions are shown in Fig. 6. The shoulder at predicted in charmed baryon production is absent for production. Coalescence does not produce a significant enhancement of ’s since . The average momentum gain over uncorrelated fragmentation of the state is small. The forward distributions in Fig. 6 are only slightly harder than those from leading-twist production. This is also true for where the , distributions are not significantly different from the distributions even though coalescence is included in the production of the and not in . When we compare our distribution with the parameterization , we extract at , in agreement with the NA32 measurement, [27].

We find for all in
our model since the production mechanisms are everywhere identical
for the particle and antiparticle. In contrast to the intrinsic
charm model, the excess predicted by PYTHIA in the proton
region, shown in Fig. 1(d), leads to a backward asymmetry similar
to , shown in Fig. 6(c).

4.2 production

We simplify our discussion of pair production by several respects. We have assumed that equal numbers of mesons and, separately, primary mesons are produced by fragmentation and that any production excess is the result of coalescence. Thus the and distributions are equivalent within the model and the analysis applies for both types of pairs unless stated otherwise. Therefore we shall also implicitly assume that all secondary ’s produced by decays can be separated from the primary ’s. We do not consider or pairs.

We use the same pair classification as in our discussion of the pair distributions from PYTHIA, shown in Fig. 2(c) and 2(d). Then the probability distributions for intrinsic charm pair production are:

(25) | |||||

(26) | |||||

(27) |

In the above, we have assumed that there is a 20% production enhancement due to coalescence, e.g.

(28) |

This assumption is different from our calculation of in [12] where we assumed that the and