P H Y S I C A L R E V I E W D V O L U M E 2 3 , N U M B E R 9 1 M A Y 1 9 8 1

Off-resonance production of heavy vector quarkonium states in e +e - annihilation

Wai-Yee Keung Brookhaven National Laboratory, Upton, New York 11973

(Received 5 January 198 1)

The inclus~ve production of the heavy quarkonium states V ( = lli, 'T .) in e + e - annihilation IS studied in the framework of quantum-chromodynamic perturbative theory. The rate and spectrum allow the study of the jet structure produced via two intermediate gluons. The similar process of the Z 0 decay into VX is found to be extremely rare.

The study of heavy vector quarkonium states ($, T, . . .) provides tests' of the quantum chromo- dynamics (QCD) and i t s related models. Most information2 about $ and 2' has been obtained from their resonances in et e-annihilation and their de- cay channels. In this paper, we consider z) and T produced off resonance and study their produc- tion mechanisms. Such a process in the lowest order of QCD can be visualized in the parton pic- ture a s e'e--$gg (Tgg') with two gluon jets re- coiling against the produced $ (2"). (See Fig. 1.) Hereafter, we use Vas a generic symbol for all heavy vector quarkonia including the presumed states made of top-quark-antiquark pair ( t T ) . The The Vvertex i s determined by the wave function at the origin @ (0) in the nonrelativistic limit. Our perturbative calculation therefore provides a test of QCD. It has been argued3 that other dia- grams such a s Figs. 2(a) and 2(b) may contribute significantly. In Fig. 2(a), the heavy-quark pair 88 produced along with one hard gluon brerns- strahlung eventually turns into the Vmeson in the final state. There a r e uncertainties about the effects of the soft gluons released from the color- octet Qg in order to form an observable color- singlet V. We expect that such color rearrange- ment has been correctly described in the process of Fig. 1. In Fig. 2(b), the nonperturbative nature of the VQQ vertex introduces uncertainties in the evaluation of this process and i ts overall strength.

FIG. 1. Lowest-order QCD diagrams for s-em- Vgg The five c r o s s e d d iagrams a r e not shown.

Our perturbative calculation i s therefore the minimal expected contribution to the V inclusive production in e'e- annihilation. Future high-sta- t is t ics e'e- experiments could help to settle this important issue regarding possible nonperturba- tive contributions.

We define the scaling variables for the process ete-- y * - - + d l ) +dB):

( b FIG. 2. Diagrams for inclusive V production in e+e-

annihilation (a) through one hard gluon bremsstrahlung and (b) through nonperturbative VQQ vertex. Crossed d iagrams a r e now shown.

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Here Ei i s the energy measured in the efe- c.m. r ( y * - Vgg) of the virtual photon y* a s follows: f rame and M i s the mass of the Vmeson. The differential c ros s section after integrating over do/dx,dx, = dl?(?'* - Vgg)/dxvdxI ( 2 ) the angles between the plane of Vgg and the direc- tion of e+e- beam i s related to the "decay width" . with

The matrix element above i s related to the cor- responding matrix element4 of the crossed-chan- nel reaction V- y *gg with appropriate variable replacements. The allowed region of the phase space i s

Fo r $ and T, the wave functions at the origin @ ( O ) were measured through the leptonic decay widths Fee,

I The process eie- - VX can in principle allow a study of the final s tate from two gluon jets over a full range of kinematics Mx : (c - M). The invariant mass squared of the hadronic recoil sys- tem i s M; = s ( 1 - xy +A). Figure 4 shows the spectrum of M, in eie-- VX.

It i s natural to consider the Z0 resonance in e'e- annihilation a s a source for Vgg production. The differential decay ra te i s given by Eq. ( 3 ) with the following substitutions:

a - a G,M,'T ,

The values of ( a ( 0 ) /2 for 4 and T a r e therefore 6-60 . 0.04 and 0.4 GeV3, respectively.' We also set the Here we use the standard electroweak model5 and strong coupling constant a, = 0.3 in our calcula- set sin2 @, = 0.23. There is no contribution from tions. For the tf production, we assume the axial-vector coupling a s a result of charge M, = 4 0 GeV and approximate the wave-function value I @ ( O ) /2 from the calculation based on a Coulomb-type binding potential

Our results a r e shown in Fig. 3. The c ros s sec- tions for the inclusive V production a r e small but not unmeasurably so.

- t C ' " " " I 1 I i l l l l l l

FIG. 3 . Ratio of total cte--VX cross section to the FIG. 4. Hadronic mass spectrum u4du/d( i~,/&) pre- pure &ED e'e-- yfy- c ross section versus v% for the dicted for the process efe--VX. The cases shown have cases V being $, T, and 5 . & = 2 ~ , 3 M , and lOM.

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conjugation. The decay width r ( Z O - Vgg) is very smal l because of the smal lness of the vector coupling in the standard electroweak model. Resul ts a r e

1.1 x 1 0 3 , v=+ r(zO - vgg)/r(zO - pp) = 4.7 x v= T (8)

2.2 x 1 0 3 , V = g .

In conclusion, we study the p r o c e s s e'e- - VX with two gluon jets a s the recoiling sys tem X. Rates and spec t ra a r e predicted. The decay mode Z0 - VX in the Z0 resonance is extremely r a r e .

This work was supported by the U. S. Depart- ment of Energy under Contract No. DE-ACO2- 76CH00016.

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