vv- ’D 6 :> / ’}5O I i `§ J EERN |.IBR’HRIES» E5ENE\·’Fi llllllliillllliIIIHIIIIHIIIII|H||H|}|HI|!H|!|| tw. 0 PEBBISBEB TTP93—30 Oct. 1993 Testing J/ip Production and Decay Properties in e+e‘ Annihilation V.M. Driesen, .l.H. Kiihn and E. Mirkes Institut fiir Theoretische Teilchenphysik Universitat Karlsruhe Kaiserstr. l2, Postfach 6980 76128 Karlsruhe, Germany Abstract Within the framework of pertubative QCD we calculate inclusive J/¢ production using the “color singlet model” (CSM). Predictions for the total cross section including the leading corrections from ini tial state radiation and the 1/z' contribution are given. We analyze the production and decay properties by considering suitable lepton hadron correlations. It is shown that the production and the decay process of an J/1/1 are described by four structure functions, respec tively. We give analytical expressions for these structure functions and suggest energy dependent coefficients of angular distributions as a test forthe model. Finally we give detailed predictions for an energy of 10.5 GeV that could be tested at the CLEO—lI experi ment. The effect of arbitrarily polarized electron / positron beams is included. 1Supported by BMFT Contract O55KA94P 2Address after Sept. 1, 1993: Physics Dept., University of Wisconsin, Madison WI 53706, USA OCR Output OCR Output1 Introduction The production of heavy quarks has become an important tool to test the validity of pertubative QCD predictions. Tagging of heavy quarks allows to separate quark and gluon jets and to examine a variety of characteristic angulaq distributions. In the case of hadronic collisions heavy flavour productions serves as pn important tool to extract parton —— in particular gluon —— distributions. This holdb true fort the production of charm and bottom mesons and similarly of the nontelativistic bohind states J/1/J or 'I`. The latter have been observed abundantly in hadronic collisions. In electron positron annihilation, however, inclusive production of J/1/1 (not to speak of T) has been limited to a few data points at a cm energy of about 4.5 GeV [1] and preliminary results at the 'I`(4S) resonance and slightly- below These results demonstrate that the predictions for continuum production of J/1,0 are well compatible with experiment. In particular no resonant contribution from the T(4S) decay is required, consistent with theoretical expectations. The large data sample collected at CLEO II in the meantime will improve this situation considerably [3] in the near future. In the energy region \/5 = 4-5 GeV J/dr is produced together with a few mesons 1r1r,n,1y' in an I : 0 and C = + configuration. The rates of these exclusive reactions can be predicted [4, 5] individually on the basis of our knowledge of 1b' ——> J/tb + X transitions. For higher- energies, relevant e.g. for the CLEO experiment, two QCD models have been proposed to predict the inclusive J/rb cross section: i) The color evaporation model (CEM) where a cF: pair with invariant mass in the charmonium region is produced in conjuction with a gluon and hence in a color octet configuration Spin and color, appropriate for one of the charmonium states are arranged by soft gluon emission with probability one. ii) The model adequate for a more rigorous treatment is the color singlet model (CSM): The ci is produced in the 351 (J P C = l") configuration in a color singlet state and hence in conjunction with two gluons [7, 8, 9]. e+e` —> 7* —> J/‘~I* gg —} lll; gg (1) This reaction would allow particularly clean studies of purely gluon induced final states: Events with energetic J/gb and correspondingly low invariant mass MM of the two gluon system might allow to search for glueballs. Events with J/qbs of low momentum and hence large Mgg will involve two gluon jets which could be compared to qi} events at the equivalent cm energy. This could provide important clues on potential difference between quark and gluon fragmentation. In a first step the validity of the model has to be explored. Predictions for the total cross section are rather sensitive to the input assumptions, that is the wave function at the origin, the choice of 04, and the estimate of (uncalculated) higher OCR Output order corrections, and we estimate these uncertainties to be about 50%. Predictions for the energy distribution, the angular distribution and the J/gb polarization are far less sensitive to these uncertainties and will, therefore, be described in some detail. We willshow that ananalysis of the angular distribution of the intitial (e+e‘) and final (1112) state lepton pairs with respect to the (J/qbgg) system allows to test the underlying dynamics of the production and decay models of (polarized) J/1,b’s in a much more detailed way than by rate measurements alone. Technically the physics of these lepton-hadron correlations is described by contraction of the lepton tensor LW, with the hadron tensor H *“’ . The contraction may be written in a rather suggestive manner depending on whether the lepton pair is in the initial or final state. For our process one would write Lf? H#~=·‘~" Lggr (2) where Lie- acts as a polarizer of the 7* (and therefore for the gg)-system)), whereas Lg? acts as an analyzer of the polarization of the We will present compact analytical results for the relevant components of the hadron tensor and derive formulae which can easily be compared to fourthcoming experimental results. This paper expands on results that can be found in the literature already. The total cross section as well as the J/1,0 energy distribution has been derived in [7, 8, 9], angular distributions can be found in Here we shall present a more comprehensive treatment which includes a description of the angular distribution of the production plane and a complete description of the angular distribution of the leptons in the decay. We take into account the contribution from 1b' and incorporate the effect of initial state radiation. Finally we give angular. distributions for polarized electron and positron beams. Our paper is organized as follows: The formalism for the calculation of bound states will be introduced in section 2 and applied to the reaction under study. An gular distributions for J/1/¤ production are presented, including the predictions with polarized beams. Section 3 is concerned with the information that can be derived s from the analysis of J/it polarization through its decay to lepton pairs. In section 4 ~¢ we shall present numerical results and the comparison with the scarce data. Section f 5 contains our conclusions. ` 2 Structure Functions in J/ip Production The differential production cross section for the production process (1) is given by 1 l l 2 3 §;M§d¤_|M| d1>s<> (:2) where s = Q2 is the center of mass energy. The amplitude which describes the OCR Output gi 1 pl P 7 · Q gv K1 e v p2 Q, K2 Figure 1: Notation: "particle·, momentum” effective coupling of the virtual photon to the J/ip gg-system can be calculated within the bound state formalism of [10] to be Am = $—¥——<¤<¤m [wr — M,><—¢>] 2 <4> \/4WM,[, where @(0) denotes the radial wave function of the bound state, which can be calculated either from potential models or related to the leptonic decay rate ·1> 0 2 Z 5 P, M,/, and E denote -the J/1b momentum, mass and polarization. The amplitude O° can be obtained from the amplitude for the production of a free quark pair at threshold and two gluons: 0 _ #1 [¢§U? + 2% + M·»)¢(—H — 2% + M¢)¢I PK, PK, ¢(-F - 2% · 2% + M¢)¢I(—1? — 2% + M¢)¢Z (PK, + PK, + 2K,K,) PK, ¢§(I? + 2% + M¢)¢I(1? + 2% + 2% + M¢)¢ PK, (PK, —I- PK, + 2K,K,) + [1 H 2] (6) Here K1, K2, el, Eg are the momenta and polarization vectors of the two gluons and 6 is the polarization vector of the virtual photon. Coupling constants and color matrices contribute a factor (41r oz, e2/3)2(2/3) to the rate.·The differential cross section can be expressed in terms of the lepton tensor Liifi = 4(pitpzt + P1.»P2,, — gt-»1>11>z) (7) OCR Output and the hadron tensor V · Hp Z —9as·P P (1%) + -·Tr . 1/¤ [O°"*°(1§' — M,,,)»y‘$ Tr[<'?.."p(1? — M·1)‘r‘;) as follows dcos6dqSda d 2 U -()() °"+“1 cx 2I`7T MO1 +- 64166-¢A-*£.L==H»*~.d ** (1 " °°1”’2 ———- 9 2 211211 () The scaled energies of the two gluons and the J/1tv are denoted by :1:1,2:2 and ·y, respectively: :c1:2E1/\/E m2:2E2/\/E y=2E,,,/\/E 1-:Mj,/s (10) The three Euler angles 9, gb and oz fix the relative orientation of the gg) and the laboratory systems and will be specified below. All cross sections are given in terms of 0,,+,,- = 41raz/(3s). The general structure of the hadronic tensor H“" can be readily exhibited by writing down the general covariant expansion3 yu u qqA IJ A u A M A vyiv H Q?.1 H] gu ”’ +H2K1K1 ”+‘H3K2 K2 + 1141%;*1%;+ 1%;1%;+ )115 1%;*1%;( » 1%;*1%;* (11) where we have introduced the four momenta K? : Kf ·— wg". From the hermiti· city of the hadron tensor H"" : H"’* * we conclude that H1 ·— H4 are real and H5 is imaginary. Therefore, in Born approximation, H5=0. An equivalent representation of the hadron tensor is obtained in the helicity basis hmm· = ¤L(m)H""¢»(m') (12) (m,m' : +,0, T) where €»(i) = (0;i1, -11,0)/J? (13) @(0) = (0;0,0, 1) are the polarization vectors for the 7* defined with respect to the coordinate frame specified below.