Volume 240, number 1,2 PHYSICS LETTERS B 19 April 1990

Q U A R K O N I U M P R O D U C T I O N AT HERA

K.J. ABRAHAM NIKHEF-H, Postbus 41882, NL-1009 DB Amsterdam, The Netherlands

Received 22 January 1990

We have calculated the production cross-sections for "f and J/~g (both the 3S~ states) via photon-gluon fusion at HERA ener- gies. We find sizeable cross-sections even after phase space cuts needed to guarantee inelasticity are imposed.

1. Introduction

It has been long recognised that the pho ton-g luon fusion mechanism at the HERA coll ider ( x / s = 314 GeV) can be used to study the gluon structure func- t ion of the photon at low values of Bjorken x [ 1 ]. In this paper we consider the product ion of two S wave vector quarkonia ( r and J / ~ ) at HERA via this mechanism. The basic process is an electron and a gluon interacting via a virtual quark loop producing a vector quarkonium and a bremsstrahlung gluon along with the scattered electron in the final state.

As has been emphasized by Baier and Riickl [ 2 ] in their analysis of the EMC data on inelastic J / ~ pro- duction, the cross section for this process is of O ( a 2) and is therefore sensitive to the scale at which as is defined. It is also poin ted out in ref. [2] that phase space cuts are needed to just i fy the use of the par ton model.

Unlike previous work on the subject at HERA energies [3] we do not exclusively use the Weizs~icker-Williams approx imat ion for the virtual photon distr ibut ion. Since there is no ment ion in ref. [ 3 ] of the scale at which the strong coupling constant is defined or of any kinematical cuts we are unable to precisely compare results but they seem to agree.

2. Calculations

The Feynman ampl i tude for the process is calcu- lated from the generic d iagram shown in fig. 1. The

e,b g~

_ J

Fig, I.

remaining diagrams differ only by permuta t ions of the gluon lines. The mass of the incoming and scat- tered electron has been explicitly included both in the matr ix element and the kinematics routines. All al- gebraic manipula t ions and Dirac spinor traces were per formed using the symbolic algebra program F O R M of J.A.M. Vermaseren. The results, however, are too long to be presented here.

The bound state formal ism which we used to gen- erate the Feynman rule for the quarkonium is due to Guber ina et al. [4 ] and has been used and discussed extensively since [5,2]. An impor tan t simplifying feature of this formal ism is the assumption of weak binding, which sets the mass of the quarkonium to the sum of the masses of its quark constituents, re- ducing the number of independent masses. It is worth noting that in this formal ism the wave function of the bound state is a colour singlet, which forces the intro- duct ion of the bremsstrahlung gluon for the purposes of colour conservation.

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Volume 240, number 1,2 PHYSICS LETTERS B 19 April 1990

As a check on the calculation we have compared our amplitude for the process

virtual photon + gluon--, quarkonium + gluon

with the one obtained in ref. [ 6 ] and find agreement after a difference in normalisation has been taken into account. As a further check we extracted from our calculation the square of the matrix element for photo-production of J /~ from on-shell photons, which agrees with calculations of Berger and Jones [7].

The cross-section was calculated by folding the output generated by FORM with the gluon structure function of GliJck, Hoffman and Reya [ 8 ] and inte- grating numerically over phase space and Bjorken x. The numerical integration was performed by Monte Carlo simulation using the routine VEGAS [ 9 ].

For the sake of comparison we also calculated the same cross-sections using the Weizs~icker-Williams approximation. This was done by folding the matrix element for photo-production of J /~ given in ref. [7 ] with the following photon distribution:

~ ( 1 + ( l_ -x )2 ) log (~mS e2),

where as usual c~ is the fine structure constant, x is the energy fraction of the photon, and me the electron mass. The results of these calculations are presented in the following section.

3. Results

We used the following values for masses and decay widths: mass of the J/~¢=3.096 GeV, mass of the "f= 9.46 GeV, width of the decay J / /W-, ~t + ~t - = 4.72 keV, width of the decay Y--, ~t+~t-= 1.34 keV. The strong coupling constant was defined at the mass of the resonance produced using the value A = 400 MeV and setting the number of massless quark flavours to three as required by the evolution equation for the gluon structure function in ref. [ 8]. With these val- ues of the parameters, the cross-sections are (with no phase space cuts imposed) 2.64 nb for J /~ and 10.2 pb for Y if the Weizs~icker-Williams approximation is not used, and 2.05 nb for J/t~ and 9.24 pb for Y in the Weizs~icker-Williams approximation.

We see that the Weizs~icker-Williams approxima-

tion is fairly accurate for "f but less reliable for J/~g production. As have been pointed out by Donnachie and Gaemers [10], the validity of this approxima- tion lessens as the contribution of off-shell photons to the total cross-section increases. This contribution is intuitively larger for J/~g than for Y due to phase space considerations and the low x behaviour of the gluon structure function, quite independent of the production dynamics. Seen in this light, the relative discrepancy between the lepto-production and effec- tive photon approximation cross-sections is to be expected.

There is of course a contribution to J/~g produc- tion from cascade decays of b-13 pairs which have a sizeable production cross-section [ 11 ]. However, due to the small branching ratio B~J/t~ this is not a very serious background. Furthermore, experimental res- olution of isolated events is expected to be sufficient to distinguish the two sources of production ~l.

Assuming an annual integrated luminosity for HERA of 200 pb- 1 and branching ratios into muon pairs of 6.9% for J/~¢ and 2.6% for ~, it is clear that the cross-sections above represent appreciable event rates. However, we has assumed implicitly that per- turbative QCD is valid. This is probably justified even in the absence of cuts for Y production given its large mass, but probably not so for J /~ production. Hence it is necessary to impose cuts to guarantee inelastic- ity. As has been elegantly shown in ref. [ 7 ], it suffices to require that the invariant mass of the bremsstrah- lung gluon and the spectator partons in the proton be sufficiently large so that recombination into a proton in the final state is impossible, which corresponds to the fact that the proton has been broken up, which in turn justifies the use of the parton model. If we re- quire that this invariant mass be 10 GeV 2 or greater, we find that the cross-section for J/t~ production falls to 2.44 pb. Changes in the invariant mass cut of a few percent typically change the total cross-section by about the same amount. The effect of QCD radiative corrections on both the cross sections discussed is possibly sizeable. Initial state radiation from the in- coming electron may also substantially affect the cross section. We now conclude with a summary of our main results.

~ We thank J. Engelen for a clarification of this point.

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Volume 240, number 1,2 PHYSICS LETTERS B 19 April 1990

4. Conclusions

The cross-sections for the production of~f and J/~¢ (both the 3S~ resonances) through photon-gluon fu- sion at HERA energies are found to be 10.2 pb and 2.64 nb respectively, if no cuts are imposed. Once inelasticity cuts are imposed the J/W cross-section falls to 2.44 nb while Y production is assumed to be ine- lastic even in the absence of cuts. Given the experi- mental resolution expected, production of J /~ from cascade decays of B mesons is not a serious back- ground. QCD radiative corrections could be substantial.

Acknowledgement

It is pleasure to thank the members of the NIKHEF ZEUS group, particularly, J. Engelen and S.J. de Jong, for numerous clarifications on experimental details. We have also benefited from discussions with K.J.F.

Gaemers and J. Smith on theoretical issues. The FORTRAN kinematics routines used for three body phase space were written by J.A.M. Vermaseren and S.J. de Jong.

References

[ 1 ] S.M. Tkaczyk, W.J. Stirling and D.H. Saxon, in: Proc. HERA Workshop (1987) Vol. 1, pp. 265-281.

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B 174 (1980) 317. [ 5 ] K.J. Abraham, Z. Phys. C 44 (1989) 467. [6] J. Cleymans, G.J. Gounaris, J.G. K6rner and M. Kuroda,

Nucl. Phys. B 204 (1982) 6. [ 7 ] E. Berger and D. Jones, Phys. Rev. D 23 ( 1981 ) 1521. [8] M. Glfick, E. Hoffman and E. Reya, Z. Phys. C 13 (1982)

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Vermaseren, Z. Phys. C 41 (1988) 173.

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