ЯДЕРНАЯ ФИЗИКА, 2004, том 67, № 5, с. 1033-1041
= ЭЛЕМЕНТАРНЫЕ ЧАСТИЦЫ И ПОЛЯ
OFF-PEAK INCLUSIVE J/ф AND ASSOCIATED J/ф + с + с AND J/ф + nc PRODUCTION IN e+e- ANNIHILATION AT BELLE
© 2004 S. P. Baranov*
Lebedev Institute of Physics, Russian Academy of Sciences, Moscow Received February 21, 2003
We consider the inclusive and associated production of J/ф mesons at BELLE conditions. In the framework of QCD perturbation theory and nonrelativistic bound-state formalism, the different production mechanisms are analyzed in detail. The calculations are compared with recent experimental data, and significant disagreement is found in a few cases. For these channels, the predictions of a nonperturbative model are also explored. We find that the J/ф polarization has strong dependence on the production dynamics, so that it may serve as a sensitive indicator in the future experimental studies.
1. INTRODUCTION
Investigation of the J/ty production mechanisms in hadron—hadron and electron—hadron collisions makes up an intensively discussed subject in high-energy physics over the last decade. A complementary topic is the production of J/ty particles (as well as other quarkonia states) in e+e- annihilation. In comparison with other reactions, the latter case shows at least two interesting features. First, the theoretical calculations are not subject to the uncertainties coming from the parton distributions in the initial beams. Hence, the attention can be focused on the J/ty formation mechanism on its own. In particular, this provides much better conditions for the inspection of basic theoretical inputs, such as the applicability of the perturbation approach and the nonrelativistic heavy-quark bound-state formalism. Another remarkable feature of the e+e- annihilation processes is that the production of two heavy-quark pairs (say, the associated production of J/ty and D mesons) is not suppressed in comparison with the production of a single quark pair (say, the inclusive J/ty production).
In the present paper, we give theoretical predictions on the inclusive and associated production of J/ty mesons and confront them with the data [1, 2] collected recently by the BELLE Collaboration at KEK. The collaboration reports on the measurement of a number of cross sections: the inclusive production of J/ty mesons, the associated production of J/ty and D mesons, and the associated production of J/ty and mesons. The measurements have been carried out at the invariant beam energy ^/s = 10.6 GeV.
E-mail: baranov@sci.lebedev.ru
The outline of the paper is the following. In Section 2 we explain the theoretical framework which is based on the standard perturbation theory and nonrelativistic bound-state formalism. Here we also discuss the main sources of theoretical uncertainties. The numerical results are displayed in Section 3. The conclusions are summarized in Section 4. The technical details and the explicit expressions for the matrix elements used in calculations are collected in the Appendix.
2. THEORETICAL BACKGROUNDS
At the quark level, the processes of interest may be interpreted as
e+ + e- ^ 7* ^ J/ty + g + g, (1)
e+ + e- ^ y* J/ty + c + c, (2)
e+ + e- ^ y* ^ J/ty + nc, (3)
The presence of the final-state gluons in reaction (1) is motivated by the necessity to get rid of the energy excess (as yfs > m^), where at least two gluons are needed to meet the color and charge parity conservation. The production of unbound charmed quarks in reaction (2) is assumed to be followed by fragmentation, that results in the formation of charmed hadrons. The fragmentation probabilities may be regarded as model parameters, or can be taken from independent experimental measurements. On the contrary, in the case of reaction (3) the formation of the hadronic final state is completed already at the quark level, and no fragmentation is needed. The corresponding Feynman diagrams are displayed in Fig. 1. The full gauge-invariant set comprises six diagrams of the type (a) for reaction (1), four diagrams of the type (b)
BARANOV b
Fig. 1. Feynman diagrams representing the production of Jparticles via the color-singlet (a—c) and color-octet (a—f) mechanisms. Diagram (g) represents the nonperturbative model with point-like ^cc coupling (see the text).
for reaction (3), and four diagrams of the type (c) for reaction (2).
The evaluation of the relevant matrix elements (see Appendix) has been performed in accord with the standard Feynman rules. To guarantee the proper spin and angular orbital momentum structure of the quarkonium states under consideration, the color-singlet spin projection operators [3] have been introduced in the amplitudes of the processes (1)—(3). The gauge invariance of the matrix elements has been explicitly tested by substituting the virtual photon momentum for its polarization vector.
The most important theoretical uncertainties refer to the value of the meson wave function ^(0) and to the choice of the renormalization scale /2 in the running coupling constant as(/2). One can also consider the production of P-wave states (followed by their radiative decay %c ^ ^7) and take into account the hypothetical color-octet contributions. Now we will discuss these uncertainties in more detail.
The value of the J/^-meson wave function is thought to be known [4] from the experimentally measured leptonic decay with :
(0)l2 =r^
m
16na2 e2
1 -
16as ~3tT
Here, the effect of including or neglecting strong radiative corrections (the second term in the above formula) approaches a factor of 2. This brings the largest theoretical uncertainty. In calculations we accept the choice with radiative corrections, (0)|2 = = 0.07 GeV3, and ascribe the same value to the wave function of meson.
Although the possible definitions of the renormalization scale / may be formally very different (say, yfs/2 or m^, or the two-body invariant mass of J/ip meson with a co-produced particle, etc.), they lead to rather close numerical values of /. Typically, in the kinematic conditions under study, they range from 3 to 5 GeV, and so, the variations in the strong coupling constant do not exceed 15% (making an effect for the cross section of about 30%). In calculations we set as = const = 0.25.
As far as the contributions from P waves are concerned, they only can be important for the process (2). In general, the production of P waves is suppressed in comparison with that of S waves by the inequality (0)|2/m2 < l^s(0)|2, although this suppression is partly compensated by the large number of spin degrees of freedom in the %c family. Besides the effect of the wave function, the contributions from P-wave states to the processes (1) and (3) are subject to an additional suppression owing to the radiation of extra
2
gluons (which is required by the color and charge parity conservation). When estimating the contributions from xc states we take |^(0)|2 = 0.006 GeV5, as is predicted by the potential model [5].
Last, we discuss the hypothetical color-octet production channels which are represented by Feynman diagrams shown in Figs. id, 1e, 1f. Using the standard spectroscopic notation for the ccc states we write
e+ + e" — 7* — 3J + g, (4)
e+ + e" — 7* — 3Sf + q + q, (5)
e+ + e" — 7* — 3P8 + q + q, (6)
The color-octet production scheme [6] implies that the ccc quark pair is perturbatively created in a hard subprocess as an octet color state and subsequently evolves into a physical quarkonium state via emitting soft (nonperturbative) gluons, which may be interpreted as a series of classical color-dipole transitions: 3Pj — J/^ + g, 3Sj — J/^ + g + g. The nonper-turbative transition probabilities are regarded as free parameters, which are assumed to obey a definite hierarchy in powers of v, the relative velocity of the quarks in the bound system under study. This freedom is commonly used to estimate the color-octet parameters by adjusting them to experimental data. Only the diagrams of Fig. 1d, which contribute to the subprocess (4) may be expected to be of any importance in the present kinematic conditions. In comparison with all other production channels (both color-singlet and color-octet), these diagrams are of formally leading order in as, that partly compensates the suppression by powers of v. In our numerical estimates, we use the nonperturbative matrix elements taken from [7] and set the light-quark mass equal to 300 MeV.
As an alternative to the nonrelativistic treatment of bound states, we also consider a nonperturbative model proposed in [8]. This model does not rely on the concept of the quarkonium wave function; instead, a pointlike charmed quark—meson interaction is introduced in the form C^c = g^ccC7^ce^, with e^ being the J/^-meson polarization vector. The interaction strength is regulated by the coupling constant ai which has been set by the authors of [8] as high as ai = glcc/(4n) = 1/4. The relevant Feynman diagram is displayed in Fig. 1g.
3. RESULTS AND DISCUSSION
We start the discussion by recalling the experimental results reported by BELLE: aexp(e+e" —
— J/^X) = 1.47 ± 0.1 ± 0.11 pb [1], CTexp(e+e" —
— J/^D*+X) = 0.53+0;J9 ± 0.14 pb [2],
Fig. 2. The energy dependence of the cross sections corresponding to different production mechanisms: solid curve — color-singlet J+ g + g channel (1); dash-dotted curve — color-singlet J/^ + nc channel (3); thick dashed curve — color-singlet J/^ + c + c channel (2); thick dotted curve — color-octet 3P8 + g channel (4); upper and lower thin dotted curves — color-octet 3SJ + q + + c and 3SJ + c + c channels (5); thin dashed curve — predictions of the nonperturbative model of [8] for the J/^> + c + c channel, rescaled by the factor of 10-2.
CTexp (e+e" — J/^D°X) = 0.87+0;38 ± 0.20 pb [2], CTexp (e+e" — J/^nc)Br(nc — >4 charged) =
= 0.033+0;006 ± 0.009 pb [2].
According to the Lund model [9], cc fragmentation produces charmed mesons at the rate 0.26 per event for D*+, and 0.59 per event for D0, where both
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