CP violation in top pair production at an e+e− collider

NUCLEAR PHYSICS B

Nuclear Physics B408 (1993) 286—298 North-Holland

CF violation in top pair production at an e + e collider Darwin Chang~,Wai—Yee Keungc~~~ and Ivan Phillips a a

Department of Physics and Astronomy, Northwestern University, Evanston, IL 60208, USA b Institute of Physics, Academia Sinica, Taipei, Taiwan, R. 0. C. C Physics Department, University of Illinois at Chicago, Chicago IL 60680, USA d Theory Division, CERN, CH-1211, Geneva 23, Switzerland Received 25 January 1993 (Revised 11 May 1993) Accepted for publication 21 July 1993 We investigate possible CF violating effects in e~e annihilation into top quark pairs. One of the interesting observable effects is the difference in production rates between the two CP conjugate polarized states ILIL and tRtR. The result is an asymmetry in the energy spectra of the lepton and the anti-lepton from the heavy quark decays. Another CF-odd observable is the up—down asymmetry of the leptons with respect to the reaction plane. These two asymmetries measure complementarily the absorptive and dispersive form factors of the electric dipole moment. Finally, as an illustration, we calculate the size of the CF violating form factors in a model where the CF nonconservation originates from the Yukawa couplings of a neutral Higgs boson.

1. Introduction Since the top quark is widely believed to be within the reach of the present collider machines, it is not unreasonable for theorists to imagine what we can learn from the top quark. The best place to study the top quark in detail is in an e+ e collider. One of the facts one would like to learn from the discovery of the top quark is the origin of the still mysterious CF violation. In this paper we investigate a way CP violation can manifest itself in the top pair production of an e+ e collider. The top quark,due to its short lifetime, is believed to decay before it hadronizes [1]. Therefore the information about its polarization may be preserved in its decay products. Ifthat is the case, then one can investigate the source of CF nonconservation by measuring the CF violating observable involving a polarized top pair in the final state. This idea of detecting the rate asymmetry between different polarizedstates was recently proposed by Schmidt and Peskin [2,31. For t1 production through the virtual photonic or Z intermediate states, to the lowest order in the final state quark mass, the polarizations ofthe quarks are either tLtR or tRtL. (Note that we have adopted the notation that tL is the antiparticle of tR and should be left handed.) These two modes are CF self-conjugate. However, 0550-321 3/93/$ 06.00

© 1 993—Elsevier Science Publishers B.V. All rights reserved

D. Chang et a!. / CF violation at an e + e

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since the top quark is heavy, there will also be large percentage of tL~Land t~t~ modes which are CF conjugates of each other. Therefore one can consider a CF asymmetry in the event rate, N(tLi~L) N(tRi~R). Schmidt and Peskin have shown how to detect the asymmetry N (tLti,) N (tR t~)through the energy spectra of prompt leptons [2,3]. One assumes that the t quark decays semileptonically through the usual V A weak interaction. Knowing that the hadronization time is much longer than the decay time [11, one can analyze polarization dependence of its decay at the quark level. The top quark first decays into a b quark and a W+ boson, which subsequently becomes £ + ii. For heavy top quark, the W+ boson produced in top decay is predominantly longitudinal. Due to the V A interaction, the b quark is preferentially produced with left-handed helicity. So the longitudinal W~boson is preferentially produced along the direction of the top quark polarization. Therefore the anti-lepton £ + produced in the W~decay is also preferentially in that direction. In the rest frame of the t, the angular distribution [4] of the produced £ + has the form 1 + cos ti’, with ~i’as the angle between ~ + and the helicity axis of the t. Above the U threshold, the top quark is produced with nonzero momentum. As a result of the Lorentz boost, the anti-lepton £ + produced in the decay of the right handed top quark tR has a higher energythan that produced in the decay of the left handed top quark tL. Similarly, the lepton £ produced in the decay of ~L has a higher energy than that produced in the decay of 1R. Consequently, in the decay of the pair tLfL the lepton from i~Lhas a higher energy than the anti-lepton from tL; while in the decay of tRtR the anti-lepton has a higher energy. Therefore one can observe N (tL1L) N (tR~R) by measuring the energy asymmetry in the leptons. It turns out that this energy asymmetry is sensitive only to the absorptive parts of CF violating form factors. There is another equally interesting CF-odd effect in the azimuthal angular distribution, namely, the rate difference between the events with £± above the reaction plane and the events with £ ±below the reaction plane. Such an observable, like the previous one, is a direct measurement of CF violation and thus has no background from the CF conserving interactions. Unlike the previous case, this up—down asymmetry will probe the CF violating dispersive form factors. Though there have been many studies of CF violating observables in the literature [5,6], we feel that the above two observables are simple and intuitive in nature and are easily implemented in future experiments. Finally, we will use a generic neutral Higgs model with CF violation in the Yukawa couplings to illustrate how these observables can arise. Such mechanisms of CF violation are contained in many extensions of Standard Model including the simple two doublet model [7]. Throughout this paper, we focus our attention on CF non-conservation in the production mechanism only. There will be additional contributions if the usual V A decay amplitude is also modified by CF violating interactions [81. —

—

—

—

—

—

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2. CP violating form factors and amplitudes We start by writing down the general form factors of the t quark. The vertex amplitude ieT~for the virtual y’~or Z~turning into t(p) and f(p’) can be parametrized in the following expression: j. Ppt—Pjj c4y,~+ C~)’~5+ CdIY5 2m~ + We use the tree-level values for c~and Ca. They are 17

=

c~= ~, Z

C,,,

=

3

=

y,Z.

(1)

c~=0,

tl

~i I ii ~Xw,/VXw!,i—XW),

2

c~= —~/~xw(1—xw). Here XW ~ 0.23 is the electroweak mixing angle in the Standard Model. The Cd terms are the electric dipole form factors. The spinor structure can be rewritten into anotherform using iy5 (p p’ ),~= a,~(p + p’ )“y~.Other irrelevant terms, like the magnetic moments, are not listed in eq. (1). It can also be easily shown that, in the limit me = 0, Cd is the only relevant form factor for the CF violating quantities we are interested in. 2M(he, h~,h~,h The helicity amplitudes e 1) for the process ee~ U at the scattering angle 0 have been given in the literature [6]. For the initial configuration of eLeR, we have —

—~

M(—+—+)

=

[C~+rLc~—flrLc~](l+cos0),

M

=

[C~ + rLC~+

=

[2t(c~ + rLC~)—~(C~ +rLC~)P/tIsin0,

=

[2t(C~ + rLC~)+ ~

(—

+ +

M(— +

—)

——)

M(— + ++)

I1rLC~](1

—

cos 0),

+ rLC~)fl/t]sin0.

(3)

Here we have used the convention* that CF invariance, when c~~Z are turned off, is signified by the relation M(c,ö;A~) = M(—d,—o;—)~,—)L).

2 (4) =

The dimensionless variablesand are its defined by, to t =themt/~/~, z = mz/\/i, 1 4t2. The Z-propagator coupling left-handed electron pgives —erL!s with —

rL=

(~—xw)/[(l—z2)~/~(l—Xw)}.

(5)

The cross section is da(eLëp. *

—~

th,t)~)Id(cos0)= ~ir~2fl~M(—+,ht,ht)I2/s.

(6)

Note that we use a phase convention different from ref. [6]. One can see sign flips between our

eq. (3) and eq. (6.6) in ref. [6].

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A color factor 3 has been included explicitly in the above formula. Similarly, we obtain formulas for the initial configuration eRëL with rL replaced by rR, (7) TR = —Xw/[(l z2)~xw(l —xw)] , —

and cos0 by —cos0 in eq. (3). In case of an unpolarized ee~ machine, we must sum up initial configurations eLë1~and eReL, and include the spin average factor

~.

It is also interesting to note that, if the absorptive part ofa scattering amplitude can be ignored (which is certainly true at the tree level), then the unitarity ofthe S matrix implies that the scattering matrix is hermitian. This hermiticity allows one to write down the constraint due to the CFT invariance as M(a,O~)L,)~) = M*(_o,_c,;_2,_2).

(8)

We shall refer to this special case of CPT invariance as CFT invariance. The CFD invariance of course can be violated by the absorptive part ofthe scattering amplitude. 3. Leptonic energy asymmetry it is straightforward to see that the absorptive part Im C~or Im C~is required to produce the difference between configurations tilL and IRtR. This is expected because we need the absorptive part via the final state interactions to overcome the CPD constraint. The asymmetry integrated over the angular distribution is —

[N(t~~) —N(tRfR)] N(ti~all)

—

—

~h=L,R2P(CV+ rhC,~)(ImC~+rhImC~) >h=L,R(3 — fl2)(C~+ rhC~)2+ 2IJ2r~C~2(9)

Note that there is no CF violating contribution due to c~coupling in the numerator when the electron mass me is ignored. We can make use of this asymmetry parameter ô to illustrate the difference in the energy distributions of £ + or £ — from the t or t decays. The energy E 0 (~ +) distribution of a static t-quark £ + vb isItvery [4]in the Tw approximation whendecay mb is tnegligible. can simple be represented as narrow width f(x ) fxo(l —xo)/D ifm~/m? ~ X 0 ~ (10) —*

—

1~0

otherwise. for on-shellWin the decay. Here we denote the scaling x0 = 2E0 (1 +the )/mt 4 +variable ~(mw/mt)6. When t and the normalization factor D = ~(mw/mt) quark is not static, but moves at a speed /3 with helicity L or R, the distribution expression becomes a convolution, flx 0±(x—x0) fR,L(x,P) = f(xo) 2 2 dx0. (11) Jx/(1+fl) 2x0fi °

—

—

290

D. Chang et a!. /

CF violation at an e~e collider

0.4

(b)

•

~

x=4E(1~)/’~s

Fig. 1. The energy distributions of prompt leptons, for the case that mt = 120 GeV, = 300 GeV. Case (a) for N’[dN/dx(t~) + dN/dx(t)], and case (b) for [dN/dx(~~)

—

dN/dx(t)]/[dN/dx(t~)

+ dN/dx(t1] per unit of Imc~(solid) or Imc~ (dashed).

Here x = 2E (1 + ) /E~.The kernel above is related to the (1 ±cos~v)distribution mentioned in the introduction. Similar distributions for the 1 decay are related by CF conjugation at the tree-level. Using the polarization asymmetry formula in eq. (9), we can derive an expression for the difference in the energydistributions oft and~: ii dN dN 1 Nldx(e+) = ö[fL(x,/3) —fR(X,P)]. (12) Here distributions are compared at the same energy for the lepton and the antilepton, x(t) = x(t~) = x = 4E(t~)/~/iThe count N includes events with prompt leptons or anti-leptons from the top pair production. It is useful to compare eq. (12) with that of the overall energy distribution, —

1~ dN N ldx(t+)

dN

dx(t)]

~

—

~)J

~h~L,R4PrhC~(C~ + rhC~)[fR(x,fl) — fL(x,P)] 3 — fl2)(c~+ rhC~)2+ 2fl2r~C~2 ~h=L,R( +fL(x,fl)

+ fR(x,/i).

(13)

Here we only keep the dominant tree-level contribution. The first term of eq. (13) is due to the two helicity modes that are CF self-conjugate. Fig. I shows the overall prompt lepton energy distribution of eq. (13), and the ratio of the expressions in eq. (12) and eq. (13) per unit Im C~or Im C~.

D. Chang et a!. / CF violation at an e~e collider

291

Note that our sample includes events with one or at least one prompt £±. However, if we are willing to use a smaller sample of prompt dilepton events (from the simultaneous semileptonic decays ofboth t and T), a recent paper [9] has shown how one can use the kinematics of such events to analyze the helicities oft and I. Therefore such knowledge can be used to enhance the detectability of the asymmetry by cutting away the CF self-conjugate modes in the denominator of ö in eq. (9) [9]. However, the reduction in the size ofevent samples may be too high a price to pay. 4. CF-odd up—down asymmetry It is known that explicit CF violation requires the CF nonconserving vertex as well as additional complex amplitudes. In the above case, this complex structure comes from the absorptive part due to the final state interactions. However, the complex structure can also come from other sources. One of these is the azimuthal phase exp (iL’ q5) in the decay process. This will produce a CF-odd up—down asymmetry even with only the dispersive part of CF violating vertex, Re Cd. The angular distribution of £ + from the t decay is specified by the spin density matrix PA,~of the top quark, =

N(0)~ ~

tkI(he, hë,2,hj)Af*(he, ~

h1)

.

(14)

~

Here .iV is the normalization such that Trp dN(e~) = [(1 + cos w)p+÷+ (I xd9dcosW/(4ir).

—

1. p is hermitian by definition, t~’) sin cos ~‘)p~ + 2Re (p÷..e (15) =

The polar angles w and the azimuthal angle ~ of £ + are defined in the t rest frame F~,which is constructed by boosting the U center of mass frame F 0 along the motion of the top quark. In F0, the z-axis is along the top momentum Pt, the y axis is along Pe X Pt, and the x-axis is given by the right-handed rule. The production plane is the x—z plane. Similarly, the angular distribution of £ from the t decay is specified by the spin density matrix ,ö~,of the anti-top quark, —

=

dN(~)

=

.Af(0)~ ~

Af(h~,h~,ht))M~*(he, h~,h~,.Z’) .

(16)

[(1 +cos~)~~+ + ~ xdç~dcos~/(4ir).

(17)

The polar angles ~ and the azimuthal angle çi~of £ are defined in the t rest frame F1, which is similarly constructed by boosting the t~center of mass frame —

292

D. Chang et a!. / CP violation at an e~e collider

F0 along the motion of the anti-top quark. It is important to keep in mind that the three coordinate systems F0, F~and F1, have parallel directions of coordinate axes. In the phase convention such that eqs. (4) and (8) are satisfied when CF is conserved, the following identities, first noticed by Gounaris et al. [10]: 0)—.~,_A’ , (18) = ~ô( between the density matrix elements can be derived. Under CF conjugation, as we exchange £ and £ +, by definitionone should make the angular substitutions IV ~ti, ~ IV + q. In that case, one observes that the distribution in eq. (17) is transformed into eq. (15) provided eq. (18) is satisfied. On the other hand, the CFD invariance, when the absorptive amplitudes are ignored, implies —

—

—~

=

~(0)~-~,_~’,

(19)

This is very similar to the CF and CPT transformations in the case of ee~ WW~as analyzed before [10,11]. When the effect of the CF violating form factors are included in the analysis, one can form the following CF or CFT odd combinations:

—~

R(±)(0)~,= Re p(O)~,,~ ±Re,O(0Y...t,.~i and I(±)(O)A,A’ = Im p(0)~,~ ±Im ~ Among them, R (—) and I ( +) are CP D odd; R (—) and I (—) are CF odd and all the others CF and CPt even. The observation of R(—) requires final state interactions due to CFD. Therefore it does not need to involve the complex phase of the azimuthal dependence in eqs. (15) and (17). It can be decoded by analyzing the polar angular dependence of w or ~i?in these equations. In the collider center of mass frame these dependence can be translated into the energy dependence of the corresponding lepton in the final state, which has already been studied in the previous section. Here we will focus on I (—) which does not require final state interactions. Since p is hermitian, the only nonzero component of I (—) is I ÷. It can be related to Cd as (—

.Af(0)Im[p(0)~—~(O)_÷]= 2sinO ~

)

(C~+rhC~±flrhC~cos0)

h=R,L

xRe

(C,,~+

rhc~)P/t,

(20)

where the contributions h = R, L pick up the signs +, respectively. One can in principle make detailed angular analysis of the difference between eq. (15) and its CF conjugate in eq. (17) similar to what was done for the case of e—e+ ~ WW+ by Gounaris et al. [10]. However, in an effort to find simpler observables which may be more intuitive and may be easier to detect, we shall —

D.

Chang et al. / CF violation at an e+ e co!lider

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consider the following partially integrated observable. Let dN (tv, up) count events with £ + above the x—z plane, i.e. p~,(1k) > 0. Then, with other obvious notations, we define the following up—down asymmetry: A~.~.d(0) —

—

[dN(,up)+dN(t,up)]—[dN(~,down)+dN(t,down)] [dN(~+,up) + dN(t,up)] + [dN(~+,down) + dN(t,down)Y (21)

It is evaluated for each scattering angle 0. The branching fraction ofthe t semileptonic decay cancels in the ratio. Integrating on w~~ or ~, çö over up or down hemispheres, we obtain A°~’ (0) in a very simple form from eqs. (15) and (17), A~L(0)=Imö(0)...~÷—Imp(0)~...

(22)

As we sum up contributions from £±in each square bracket of eq. (21), the asymmetry is insensitive to the sign of charge, it is obvious that a non-vanishing value of A~’(0) is a genuine signal of CF violation. Although the angular distributions of the leptons derived from eq. (3) will have corrections from the strong interaction, the corrections Cannot fake the CF asymmetry as the effects due to the strong interaction cancel away in the differences. To enhance statistics, it is useful to measure the integrated up—down asymmetry, =

!fAd.(o)dld(coso)

(23)

From eqs. (20)—(23), we obtain =

3IV~h=L,R(C~ + rhC~)(ReC~ + rhC~)2 + rhReC~)P/t + 2fl2r~C~2 8 ~h=L,R(~P2)(’~

(24)

Note that information on Ct in eq. (20) is lost when integrating the whole range of cos 0. However, one can easily find other convolution in eq. (23) to recover the CF information due to Ct. In fig. 2, we plot the integrated up—down asymmetry A~”~ per unit Re C~or Re C~for various parameters. It seems that .4” per unit Re Cd increases with ~ however, Re Cd is usually energy dependent, characterized by the underlying scale ofnew physics. Above that scale, Re Cd diminishes very fast. Therefore, the optimal choice of ~,/? is about at the scale of new physics. Note that even if Re Cd is relatively constant and the angular asymmetry is larger at higher energy, the event rate will become smaller because ofthe nature ofthe s-channel production. To measure the up—down asymmetry, we need a good determination of the reaction plane. This is possible for those events in which one of the top quarks decays hadronically intojets. One may be concerned about the imperfect angular resolution ofhadronicjets. We argue that even ifthe angles ofjets are ambiguous

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2.5

2.0

—

—

mt=12OGeV up—down asymmetry 1

1.5

—

1.0

—

0.5

—

—

,~-‘~Re Cd

a. a 0.

~

0.0

—

Re c,~5—

~--:.:r

250

300

350 ‘/s (GeY)

400

450

500

Fig. 2. The weighted up—down asymmetry of prompt leptons, for the case that m, versus ,,,/~,per unit of Re c~(solid) or Re c~(dashed).

=

120 GeV,

at the level of 10 degrees or so, as long as some sort of orientation of the reaction plane can be defined, the asymmetry will not be smeared away by more than one order of magnitude. For example, one can simply discard events that the definition of up or down is made ambiguous by this smearing. The details of such a smearing effect will strongly depend on future detectors. 5. Higgs model Among various mechanisms for the CF violation, the one that may manifest itself most easily is the neutral Higgs mediated CF violation. Since the neutral Higgs couplings are typically proportional to the quark mass, the large mass of the top quark naturally gives large couplings to the neutral Higgs bosons. CF non-conservation occurs in the complex Yukawa coupling, £cpx

=

—(mt/v)i(AFL + A*&)tH + (m~/v)BHZ”Z~.

(25)

Here v = (V’~GF) h/2 ~ 246 GeY. The complex coefficient A is a combination of model-dependent mixing angles. Simultaneous presence of both the real part AR = Re A and the imaginary part A 1 = Im A guarantees CF asymmetry. For example, in the low energy regime, it can give rise to the electric dipole moment of elementary particles [12,13]. Here we will derive the CF violating form factors at high energy. They are induced at the one-loop level as shown in fig. 3. First, we calculate

D. Chang et a!. / CF violation at an e~e co!!ider

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>~,H

(a)

(b)

Fig. 3. Feynman diagrams for the process e~e —+ tI. The tree amplitude interferes with those one-loop amplitudes with (a) the final state interactions due to the exchange of a Higgs boson, or (b) the intermediate state of the ZH bosons.

the absorptive parts according to the Cutkosky rules. The leading contribution to Im C~comes from the rescattering of the top quark pair through the Higgs-boson exchange. Both CF violation and the final state effect are produced by the same one-loop graphs. ImC~= C~(_—~)~~ [1_ ~log

(i +

~-~-)j .

(26)

The dimensionless variables are defined by t = ~ z = mz/~/i~, /32 = 2 as before, as well ash = mH/~/~. For Im C~,there is a similar contribution. I —4t In addition, there could be a contribution due to the ZH intermediate state, fig. 1b, provided the kinematics is allowed, Im C~= £~ImC~ aA 2 2 [/Jz + (2t2 + 2t2h2 2t2z2 h2)L]. 1BC~t Ci,, 2(1 —xw)xwfl (27) —

—

Here /J~= 1 + h4 + z4

—

2z2

— 2h2

—

—

2h2z2, and the logarithmic factor

L— 11 O~lzhfiIJz —~

12~2

PP.

28 (

Our expression (27) agrees with that in ref. [14]. Note that, at the threshold (p = 0), Im C~vanishes but Im C~has a value 2+h2 mCd 8(1_xw)xwk~1_z2_h2 3(lz2h2)2 ~ z_ C~AIBC~PZ (1—z _________

In case mz + mH < ‘,/~< 2mt, we still have the absorptive part Im C~,which is simply given by replacing the logarithm factor L in eq. (27) by its continuation, L—

~[~l2h2 2 ~

PZ~/T

(29)

D. Chang et a!. / CF violation at an e~e collider

296

0.008

0.008

—

—

/

m,=12OGeV

/ 0.004

J

—

~‘

. ‘~ . —.

—

•1

0.002

—

0.000

—

—

Re

—

I 200

Cd’

Cd’

.

. — . —

.

.

—

Re c

—

4~

_.1

—

1=

—

I 300

400

500

‘Is (GeV)

Fig. 4. c~and c~versus ~ for the case that mH = 100 GeY and mt are chosen to be A1 = AR = B = 1.

=

120 GeV. The parameters

The dispersive parts are obtained by the dispersion relation ReC~(s)=

1 lv

p

/ JsO

‘~

Im

Jj’~F\

d’

S’—S

‘th’

.

(30)

The symbol P denotes the principal value of the singular integral. Note that the absorptive part is evaluated at s’ above the threshold 5~,which is 4m? or (mz + mH 2 for tt or ZH intermediate 2states It is understood and respectively. z 2, are defined with respect tothat s’. the dimensionless variables, Ii, liz, ~ 2, h Fig. 4 shows typical sizes of C~for various cases. It is of order l02_l03 in general. We must keep in mind that we only show the Higgs boson contribution in the perturbative regime. In scenarios that the Higgs bosons interact strongly, the CF violating effect in the colliders could be much larger.

6. Conclusion We have shown that the energy asymmetry or the up—down asymmetry of the prompt lepton events are sensitive to the CF violation in the top pair production in e+e annihilation. These two asymmetries are complementary to each other for measuring the absorptive and the dispersive parts of the CF violating form factors ~ The cross section for ee+ at ~ = 300 GeV, for mt = 120 GeV, is 1400 Ib, among which about a quarter is due to the channels tLk and tRfR (see fig. 5). At an optimal luminosity of l0~~ cm2s~,one year run —~ U

D. Chang et a!. / CF vio!ation at an e~e co!lider

297

1500

oe~- top pair 1250

—

1000

—

—

mt =

120 GeV

—

150 750

—

500

-

—

—

/

~

I

200

—.

—

-

/

400

200

‘-.

800

Total: LR+RL+LL+RR

800

1000

‘Is (0eV)

Fig. 5. The total cross sections are shown in the upper three curves for cases m, = 120 GeV (solid), 150 GeV (dashed), and 200 GeV (dash—dotted). The partial cross sections of the helicity configurations tLtL and tRtR are shown in the lower three curves correspondingly.

of the futuristic NLC [15] will produce 20000 prompt semi-leptonic events bb~~X. Note that we do not require both t and t to decay semileptonically. Therefore, only one power ofthe branching fraction 2/9 is involved. Such sample is good enough to analyze the CF violating form factor Cd at a level of ten percent. Although predictions from Higgs models is still smaller than this level of sensitivity, the measurement could provide directly the important, modelindependent information about the form factors of the top quark interactions. For a heavier top quark, ~ has to be raised above the corresponding threshold, but not so far away as to incur the 1/s suppression in event rates. D.C. wishes to thank the Theory Group at the Institute ofPhysics at Academia Sinica in Taipei, Taiwan and Department of Physics and Astronomy at University of Hawaii at Manoa for hospitality while this work was in progress. This work is supported by grants from Department of Energy and from National Science Council of Republic of China. References [1] I. Bigi and H. Krasemann, Z. Phys. C7 (1981) 127; J. Kuhn, Acta Phys. Austr. Suppi. XXIV (1982) 203; I. Bigi, Y. Dokshitzer, V. Khoze, J. Kuhn, and P. Zerwas, Phys. Lett. B18l (1986) 157 [2] C.R. Schmidt and M. E. Peskin, Phys. Rev. Lett. 69 (1992) 410; C.R. Schmidt, Phys. Lett. B293 (1992) 111

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