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. `The matrix hmmi represents the polarization density matrix of the vitual photon. Its elements can be analyzed by considering angular distributions of 3The most general covariant expansion is obtained by adding four additional terms: H6 ¢(1w,q,Ki),Hv ¤(u, v,<1,Kz),Ha(K{‘€(v,K1,Kz,q) + (11 <—> v)) and H¤(K$‘¤(¤,K1,K¤,q) + (p. 4-+ However, parity conservation requires H6 — Hg:0. OCR Output thc initial state lepton pair with respect to the (J/¢gg) system. The general angular distributions in terms of two Euler angles 0 and ¢ is given by: da U 2 2 ..______Z ,__ . _ . da:1d::2d cos 0d¢ 161r [(1 + COS 0)UU+L + (1 3cOS 0)aL 14 ( ) -4- (2 sin20 cos(2¢))0·T + (2x/§sin(20) cos ¢)0·I Here 0 and ¢ are the polar and azimuthal angles of the incoming electron in the 7* Q rest frame and will be specified below. The third Euler angle has beenintegrated.} out. The angular coefficients UX are linearly related to the density matrix elements! in (12) (or alternatively to the structure functions H; in (11)):
aX:C'm`—'%a,J+,,HX X€{U+L,L,T,I} LU40 where
HU+L hm + hir + hn : HH + H" + H°3 (15) hg;) 3 H33
h+_ + h_+ : i(H" - HH (17)
(mo + hor - hn, — h.,_) I /uf§(H13 + Hm) (18) and A gi2 lh _ 128qf \q$(0) <)31r u M,) _ 27vr M3, (19) The structure functions JX depend on :1:1, :1:2 or alternatively on :1:1,y. At this point the relative orientation of the electron direction in the hadron frame has to be specified (Fig. 2). The z axis is defined by the direction of the J/1/2 momentum. The orientation of the :1: axis is given by the most energetic gluon. Thus the momenta of the outgoing hadrons span the :1: — z plane. Analytical expressions for the structure functions UX can be found in appendix A. Note that the cross section after integration over the angles is given by da dibldilbg UU+L ( ) in‘agreement4 with eqs. (8) and (9) in'[8] and (3) in Integrating the angular distribution in eq. (14) over the azimuthal angle d2 we obtain the angular distribution of the J/zp o -———*da : - 1 Z 0 1 s 3 2 0 dzldzzd cos 9 8 + COS )°”+L + ( COS m· 21 l ( ) §(aU+L + al,) (1 + cx cosz 6 (22) 4a.fter corrections ot two misprints: 6(1 — pz — y) has to be replaced by 6(1 + pz — y) and (1 — pz) by —2( 1 - M). OCR Output Figure 2: Definition of` the angles 9 and cb
where 1raf/UUI / — 30* cr UI (23) The corresponding coefficient for the distribution of longitudinally polarized J/1,b is denoted by 0zL and will be calculated below. Integrating over 9 gives
da 1 ;-<—- = — 1 2 dmldmzdqs 2w+Ll + B ¢ 4 <2 >
Sl where (25) GUM! »\:_’ The structure funictions are also suited to fully predict the angular distribution l for polarized beams. The cross section, which is still differential in 0:, is obtained through the replacements
(1+ cosz 0) =>— s;,s;,)(1+ cosz 6) + sing 0(sin(2cx) s+ + cos(2
with
S- Z S;1 5;: “` SL: $;r 5+ Z 3;; $$1 + 3;: 5;:
No new structure functions arise in this case. The spin vectors 5} and E2 are defined in a coordinate system where the z' axis is aligned with the electron direction, cx denotes the angle between the plane spanned by z' and P), and the sc', z' plane (Fig. 3). For the special case of natural transverse polarization E is conveniently aligned in the :v' direction and hence sz: : s+ = O and s- = -1.
F1) = Z
pi I Z OCR Output
Figure 3: Definition of the angle oz
llIIIII-l-l 3 Structure Functions in J/it Decay
In the previous section we have analyzed the production mechanism of an unpolarized J/gb by considering angular correlations between the initial state lepton pair and the (J/1/zgg)-system. However the angular distribution in eq. (14) remains valid also for the production of a polarized J/ip:
do"` o —-—-——:—i 20* 1-320* (C1d2BgdCOS0d¢ 161r V it >¤¤+L + < M 26 <> + (2 sin20 cos(2qS))a;’>· + (2x/2sin(20) cos 45):7*}
Here the superscript A 6 {U + L, L, T, I} refers to the polarization of the The structure functions aj} (X 6 {U + L, L, T, [}) can be calculated by suitable projec tions 7)},,, instead of 7?5,;» =L 79gT: —g,;,;· + P,;P,;i in eq. The polarization of the J/1/w determines the angular distribution of the leptons in the subseqent decay of the J/it for each of the production cross sections o·U+L, UL, UT and 0*;. In analogy to eqs. (12)-(18) the decay angular distribution can be disentangled by introducing suitable combinations of density matrix elements of the The general form of the decay lepton distribution for each production cross section JX in (14) is then given
day; o »——-———— : —- 1 *0 "+L 1-3 ze L .,d..,m.idX 16., V + M + < M 27 <> + (2 sinzd cos(2X))a§ + (2x/2sin(219) cos X)a§,»
Here 0*gare+L the production cross sections for an unpolarized J/1/1 as given in eq. (14), i.e. 0g+5L JX. Correspondingly 0,% are the cross sections for a longitudinally polarized J/1/1, where the projection (—g,;,;: -§— P,;P6· in eq. (8) is replaced by Pg,. E age; with
1 Eq, M;. 1 r E'“=—·PO0,E :—-P 28 the longitudinal polarization vector of the J/gb with respect to its direction of flight in the 7* rest frame. Similary projections with
Pg,. E EQEQ + E{EgC (29) and 1 ag,. E 7 (eye; + age; - age;} — age; (30) lead to the remaining polarization cross sections in (27). In this paper we restrict ourselves to the calculation of aff and 0%+which1; are suiiicient to describe the distribution in the polar angle 19. OCR Output The angles 19 and X in (27) are the polar and azimuthal angle of the decay leptons in the J/ap rest frame. The z axis is defined by the direction of the J/1b in the 7* rest frame. Integrating the angular distribution in eq. (27) over the azimuthal angle X we get d0’X (1 + cosz 19)ag++L (1-— 3 cosz 19)0·§'{( (31) da:1 dxzd cos 19 :+L(a§+ ag) I1 + AX cosz 19 (32) where 1 — 3a§é/,%+1; AX Z and AU+L E A (33)
The angular distribution of longitudinally polarized J/1/xs, which is defined in analogy to eqs. (22,23) is given by L _ 1- 3¤t/ 4 Numerical Results We used a running coupling constant a,(Q2) with A : 500 MeV and nf = 3. For the scale Q2 either the mass of the J/1,b (M), : 3.097 GeV) or the invariant mass of the two gluon system (Mag) is adopted. Mgg falls below 1 GeV in a small fraction of the phase space and in this region oz, is frozen at the value 1, if the second option zx,(M§g) is adopted 1 if (/Q2 < 1GeV For the fine structure constant we adopt the running coupling [11] (3) = -·;··— 1 " '”’·n(S) (36) 5 1r,,(s) = 0.00165 + 0.00299 log(1 + s/1GeV2)§ (eu+ E log(s/mf) — (37) For the coupling dictated by the wave function at the origins we use the potential VJ from [12] and obtain ¢>(0)|”/M3, : 2.832 - 10“ (38) 5In [8] the experimental value for Fe, and the lowest order formula (5) was used to extract ]¢(0)]°. Actually I‘,, recieves large negative QCD corrections of about -35%, which are of course included in the measurement. This lowered the previous prediction by a factor of about 0.65 compared to the present result. OCR Output 4.1 The total cross section For constant a,(M$) an analytical formula for the total cross section can be derived ((13) in [8l)¤ 0'TOT = I d231d2l>2O'U.+.L Z 0’,J~;-M J/(W O _;r, O _5 l O F l l O lO lO E lj 5 l O l 5 20 5 l O l 5 20 CMS-Energy [Ge\/l Cl\/!S—Energy [Ge\/l Figure 4: The total cross section for J/gb and gb'. Solid line: o4,(Mgg), dashed line: O(¤(M¢) results for the two choices of oe, are shown in Fig. 4. For a comparison with the data the production of the 1// (JPG : 1",M¢» : 3.686GeV) with the subsequent decay 1b' —> J/gb +X has to be added. This amounts to a relative contribution proportional l_`(gb’—> e+e") ——-————B I J Z 0.23. I`(J/wp ——> e+e‘) rw) _) The production cross section increases rapidly for smaller \/E, in particular for the choice with cx,(M3g). Initial state radiation leads, therefore, to an enhancement 10 OCR Output WO ·""""”\T_j·—--—---.`,____ N`-L.L_`_`—`. 2 4 6 810121416182022 CMS—En@rgy [GeV] 2O 15 "" TO c 52 -5 P l' —— 1 O 2 4 6 810121416182022 Ct/1S—Energy [Ge\/] Figure 5: Influence of initial state radiation on the total cross section (Fig. a) and relative size of the corrections (Fig. b) for o¤,(M3g) (solid and dotted lines) and &,(M;) (dashed and dotted lines). up to 12% in the region of interest, if we convolute the lower order cross section with the radiator function from [13]. The influence of initial state radiation as a function of energy is shown in Fig. 5. In Fig. 6 we compare theoretical predictions including initial state radiation and the gb' contribution with the data from the PLUTO and CLEO ll collaborations. Satis factory agreement is observed. We stress that uncalculated higher order corrections induce an uncertainty of the order 50%. For example, had we taken l`,, : 5.36keV and used the Born formula (19) to derive the wave function instead of eq. (38), the 11 OCR Output J/ww 10 J/w J/www J/w WO WO 2 4 6 8 1O 12 14 16 18 2O 22 Ctx/\S—~Energy [GeV] Figure 6: Comparison of the theoretical prediction with the data from [1, 2] prediction would have decreased by a factor 0.65. To reduce or remove the aforementioned uncertainties from the choice of a, and from the overall normalization, we now concentrate on energy and angular distributions which will be normalized to the total cross section. In section 4.3, we present numerical results for ratios of the structure functions UX, where the QCD corrections are expected to cancel out to a large extend. See [14] for an related example. Therefore, these quantities are particularly suited to test the validity of the model. 4.2 Energy distribution One of the first distributions that will be measured is the energy distribution of the It can be calculated analytically [8]: ddj A 2 C.2 12 OCR Output X/1,7*75 (1 +21·¤) 1·(1—4q·) 2 —2 { T i<1—T><2T—y>2+<1—1~> and is plotted in Fig. 7 for two CMS energies: 4.5 and 10.5 GeV. In contrast to the 4.5 GeV 2O 18 Mw 16 14 12 10 1.38 1.4 1.42 1.44 1.46 1.48 10.5 GeV 5 ;T·Y—·;· 4.5 3.5 2.5 1.5 0.5 0.6 0.7 0.8 0.9 1 1.1 Figure 7: Energy distribution of the J/ip for o1,(M;g) and a,(M1/2). CEM [8] the distribution in the CSM is nearly fiat. For convenience of the reader we also give the distribution in the scaled momentum z = 2|P,/,1 / \/E (Fig. 8). 4.3 Angular distributions The results for the ratios of the structure functions after integration over the Dalitz variables as function of \/E are shown in figure 9. For X/E = 10.5GeV the cross sections are given in table 1, their ratios in table 2 and the coefficients c1,o¢'“,A and AL calculated from these numbers in table 3. The influence of the scale in cx, is 13 OCR Output 7 4.5 Ce\/ > Bt — 11.. { 51------— M, 3 4 0 0.1 0.2 0.3 0.4 0.5 0.6 7 10.5 GeV >\ 6 cut 5111 2 3 4 0 0.1 0.2 0.5 0.4 0.5 0.6 0.7 0.5 0.9 1 0 1 2 5 4 5 F’,[Ge\/l Figure 8: Momentum distribution of the J/gb for a,(M;g) and 14 OCR Output O.5 _ 1 Msg r"` 0.4 ‘i` GL/UU1-Ij—`T—_—"";— Og, ____ _ .,.i—.—r--.---,_. O.2 Oo/ Og; ._- ...... -. .»..» .·~—.—. -?~- . OFT/O-u+n. O A > ¥—J_J 2 4 6 B NO `\2 W4 16 T8 20 CMS——Energy [GeV] Figure 9: Ratios of structure functions for a,(M;g) and QAM;) 0.8 0.Blj ‘._~ U6 "`·—`O-LL/Uu+LL L L O1. /Ou+L 0.6 /a--__`-`."" "" 0.4 0.4F ,»' UL/Gun. Gr/vw O2 0.2 2 4 6 81012‘|4161B2O 2 4 6 B TOT21416182O CMS—Energy [GeV] CMS—Energy [GeV] 1 L UL /OL OB 0_8 ULL/UL L O‘6 UU-+L /Uu+¤. 0.6% jr. O-u+L/0.u+L O.4 0.4}- ,· 0.2 0.2 2 4 5 B \0\214\51B2O 2 4 5 81012 14151820 CMS—Energy [GeV] CMS—Energy [GeV] Figure 10: Ratios of structure functions for polarized J/zh for a,(M§g) and cx,(M3,). implies that the polarizations of the virtual photon and of the J/1/J are strongly 15 OCR Output 0.2 4.5 CSV 0-2 10.5 GQV -0.2 -0.2 -0.4 -0.4 -0.5 -0.5 -0.8 -0.8 L I -1 - 1 .2 - 1 .2 1.375 1.4 1.4251.451.475 0.5 0.8 1 10.5 GeV 4.5 GeV 0.2 ¢ "`;" rg 0.2 if 0 2{ 0 0.2 -0.2 -0.4 -0.4 -0.5 -0.5 -0.8 -0.8 — 1 .2 — 1 .2 1.375 1.4 1.4251.451.475 0.5 0.8 1 Figure 11: a(y) and A(y) at 4.5 and 10.5 GeV correlated. At the T(4S) resonance (10.5 GeV) a J/gb with high momentum cannot originate from B decay. To distinguish between secondary J/gb from B decays and directly produced .]/1/:, a momentum cut at 2 GeV has been employed which corresponds to a cut at y :0.702 (to be compared with the phase space limits 0.59 $ y f 1.087). The structure functions calculated with this cut are also listed in table 1, their ratios in table 2. These ratios are typically larger than those without the cut. Since large ratios imply pronounced angular distributions and low momentum J/gb are distributed nearly isotropic, this result seems quite plausible. The data sample expected from the next round of data taking will presumably not yet allow to measure a(y) and A(y) from the two dimensional distribution in y and in the production or decay angle. We therefore suggest to split the events into a low energy (0.702 § y § 0.9) and high energy (0.9 § y $ 1.087) sample, corresponding to a “cut" at |P,),\=3.5 GeV (see also fig. 8). From the two dimensional distribution Fig. 12 it becomes evident, that the hadronic plane is preferentially aligned with the beam. This trend is visible for unpolarized J/1/1 already (a) and becomes even more pronounced for longitudinally polarized J/zb In the experimental analysis the 20% "contamination” from the 1/w' —> J/gb + X has to be considered, as we]1 as the effect of a cut in the J/gb energy. 16 OCR Output x 10 0.35 0.3 0.25 cos® 05 % ,5 05 6 O.5 Cb/7T x 1O O.55 0.3 0.25 0.2 O. 1 5 oos® 1 0.5 , 1.5 1 ¢ n / . . 05 5 O.5 Figure 12: Differential cross section do/d¢d cos 9 at 10GeV for unpolarized (Fig. a) and polarized (Fig. b) J/rb. 5 Summary Inclusive J/gb production in electron positron collisions has been evaluated in the nonrelativistic bound state model. Including the leading corrections from initial state radiation, the running QQED and the gb' contribution, satisfactory agreement has been obtained. The prediction of the total cross section is uncertain to about :1:50% as a consequence of relativistic and higher order QCD corrections which is in particular reflected in different assumptions on the choice of oz,. Energy and an 17 OCR Output OCR OutputOCR OutputOCR OutputOCR OutputOCR OutputOCR OutputOCR OutputReferences {1] Pluto Collaboration, J. Burmester et al.: Phys. Lett. B 68 (1977) 283. [2] R. A. Poling: Proceedings of the Joint International Lepton Photon Symposium and Europhysics Conference on HEP: LPHEP’91, p.546. [3] K. Honscheid: private communication. [4] J. H. Kiihn: Z. Phys. C 15 (1982) 143. [5] J. H. Kuhn, K. H. Streng: Z. Phys. C 17 (1983) 175. [6] H. Fritzsch, J. H. Kiihn: Phys. Lett. B 90 (1980) 164. [7] J. H. Kiihn, H. Schneider; Phys. Rev. D 24 (1981) 2996. [8} J. H. Kiihn, H. Schneider: Z. Phys. C 11 (1981) 263. [9] W. Y. Keung: Phys. Rev. D 23 (1981) 2072. [10] J. H. Kiihn, J. Kaplan, E. G. O. Saiiani: Nucl. Phys. B 157 (1979) 125. [11] H. Burkhardt, in “Radiative Corrections, Results and Perspectives”. N. Dombey and F. Boudjema eds. Plenum Press, New York (1990). [12] K. Igi and S. Ono, Phys. Rev. D 33 (1986) 3349. [13] G. Bonneau and F. Martin, Nucl. Phys. B27(1971)381. [14] E. Mirkes, Nucl. Phys. B387(1992)3. 23 OCR Output