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Testing an anomalous top quark via precision electroweak measurements
Physics Letters B 269 ( 1991 ) 412-418 North-Holland
P H YSIC S k ET T ER$ B
Testing an anomalous top quark via precision electroweak measurements Maurizio Frigeni Dipartimento di Fisica, Universith di Pisa, 1-56100 Pisa, Italy and Riccardo Rattazzi t CERN, CH-1211 Geneva 23, Switzerland Received 24 June 1991
The effect on electroweak corrections of anomalous interactions of the top quark with the Z° is studied. Such anomalous terms are expected to arise in the context of dynamical symmetry breaking. It is shown that, depending on mr, the whole set of radiativecorrection parameters can be relevant. This set consists of three observables coming from the vacuum polarization graphs plus the correction to the Z°t3bvertex. The recent data coming from collider experiments are used to place bounds on the anomalous terms. The same observablesare evaluated in a simple renormalizable model in which the top quark mixes with an isosingletuptype quark.
1. It is well k n o w n that precision measurements at LEP can open a window on physics above the Fermi scale. The most relevant effects due to virtual heavy particles come typically through corrections to vector boson propagators, the so-called "oblique" corrections [ 1 ]. An exception to this rule is represented by the top quark, since large G y m 2t terms appear in the Zl~b vertex corrections [ 2 ]. However, this is because b and t fit into the same weak doublet. Any possible anomalous behavior of the top quark should show up both in oblique and vertex corrections. The purpose of this letter is to study the impact on precision measurements of anomalous non-universal interactions of the top quark with the Z °. We shall deal with two models where such extra couplings arise in completely different ways: in one case they are induced by dynamical gauge symmetry breaking [3], while in the other they simply appear via the mixing of the top quark with a heavy isosinglet up-type quark [4]. Inspired by an analogy to low-energy chiral symA. Della Riccia fellow. 412
metry breaking, it has been argued in ref. [ 3 ] that, if the electroweak symmetry breaking is dynamical, residual interactions between fermion a n d vector bosons should in general arise. One could also argue that such anomalous couplings depend somehow on the mass of the involved fermions. According to this picture one expects very small deviations for the light fermions (consistent with experimental observations), Conversely, large deviations are expected for the heavy top quark. These could give measurable effects in precision tests at LEP. In ref. [ 5 ] the contrib u t i o n to the p parameter and to the Z13b vertex were studied. The aim of this paper is to study the whole set of radiative-correction parameters, including e2 and e3 defined in ref. [ 6 ]. A comparison is made with the latest LEP data, thus updating the analysis in ref. [5]. Of course this model is only phenomenological and renormalizability is lost. In order to make a comparison with a renormalizable model, we consider the simplest possible case in which anomalous couplings of the top are present, consisting of the addition of a heavy isosinglet up-type quark. This choice is also of
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speculative interest. In fact, many grand-unified models require the addition of isosinglet fermions. Furthermore, the addition of such an isosinglet characterizes many models which try to explain the heaviness of the top quark [ 7 ]. This paper is organized as follows. In section 2 we recall the definition of the three parameters El,2,3 which determine the pattern of oblique radiative corrections in Z ° physics [6]. In section 3 we calculate these corrections in the dynamical-breaking model, and make a comparison with the recent experimental data coming from the four LEP experiments. In section 4 we c o m m e n t on the radiative effects in the case of the isosinglet. Finally, section 5 is devoted to our conclusions.
where i, j stand for W, Z, y. For heavy particles we can approximate the functions F 0 ( q 2 ) with Fij(0) = F v. Considering that A r r = A z v = 0, one is left with six constants Aww, Fww, Azz, Fzz, Fzv, Frr. Three linear independent combinations of these amount to a simple redefinition of c~, GF and Mz [8]. The combinations
2. The effect of the oblique corrections can be seen as a small modification of the tree level relations among the fundamental parameters of the electroweak theory. In general this effect can be described [6] with the introduction of three observables. One of these, Ap, determines the relative strength of charged and neutral current four-fermion interactions. The other two, Arw and Ak', describe the relative difference among the three possible definitions of the sine of the Weinberg angle: Sw, so and gw. More precisely, ZXrwis defined by
zXp= e, ,
Azz
Aww
e ~ - M2
M2w
E2 = Fww e3-
F3v
-,433 - A w w
M2w
,
(2.4b)
- F33 ,
- F 3 3 = cwFo3
Sw
(2.4a)
(2.4c)
Sw
affect the physical observables as (2.5a)
Ak'_ -c2<
+~3
c~-S~w
7 {I
sb
(2.5b)
'
--1
e2 qt- 2e3 .
(2.5c)
gA = T3( 1 + l A p ) ,
(2.2a)
In the standard model the leading contribution is due to a heavy top quark, and resides in e ~ - 3 G F m 2 / 8,,/2 g2. No large contribution arises in ~2 and E3. This picture is not essentially altered by supersymmetry [8 ]. On the other hand one expects large contributions to both el and E3 in technicolor models [9], while E2 is still less relevant. A last c o m m e n t on eq. (2.2) is in order. In general Ap and Ak' receive non-universal corrections due to light known particles and arising from box, vertex diagrams and imaginary parts of the propagators. In the standard model, these are small and reasonably well-known quantities, with the exception of the Zbb vertex corrections. In this case the left-handed Zbb vertex receives a correction proportional to m 2, so that (2.2) is modified by
g v = 1 + 4Qr~2w = 1 + 4 @ ( 1 + A k ' ) s 2 . gA
(2.2b)
gbA= T~[1 + l (Ap+ Apbertex) ] ,
s~c2 = ( 1 _
M2~M 2 ~o~ (Mz) M 2 , ] M2z - ,~f2 G F M z ( 1 - & r w )
sgco - l-Arw '
(2.1)
where c ~ ( M z ) = 128.8 is the running fine structure constant evaluated at s = M 2 and Gv= 1.166× 10 -5 ( G e V ) - 2 is the Fermi constant as measured in g decay. The other two observables are given through the on-shell couplings gv and gA of the Z ° to fermions,
The contribution of heavy particles to these observables via vacuum polarization graphs is quite simple. We can write the vacuum polarization tensor
gbv g~A = 1 + 4@~2w =l+4Qf(l+Ak,
1 b 2• ~APvertex)So
(2.6b)
In the standard model one has Ap b....... = - (xf2 GF/
as
H.~j" = -ig~'"[Av +q2F#(q2) ] ,
(2.6a)
(2.3)
47r2) m 2 . 413
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3. In ref. [3] it has been shown that, if the symmetry breakdown in the electroweak theory is dynamical, residual interactions between fermions and Goldstone bosons should in general arise. These extra terms can be shown to give non-universal interactions between the SU (2) × U ( 1 ) gauge fields and the fermions. These interactions are the analogue of the extra couplings of the pions to nucleons which arise in Q C D when S U ( 2 ) × S U ( 2 ) is dynamically broken. In Q C D the extra couplings are O(1 ) (for instance the ~lqN form factor at zero m o m e n t u m is characterized by an extra piece k =gA-- 1 --~0.25 ). This can be seen as a consequence of the fact that in Q C D there is only one mass scale AQCD. For the electroweak theory the situation is rather different, since there is a full hierarchy of mass scales in the fermion spectrum ranging from me ~ 0.5 MeV to mt > 80 GeV [ 10 ]. One could argue [ 3,5 ] that, if fermions participate in the symmetry breakdown in a non-trivial fashion, the structure of the mass spectrum may in some way be reflected in the extra couplings to gauge bosons. Indeed by assuming that the anomalous terms are of order mrx/~v for a fermion f, consistency is met with the experimental data on light fermions [ 3 ]. For the top quark, however, since it has not yet been detected, we cannot put direct bounds on its extra couplings. On the other hand, given the present lower bound on m~ and supposing k t o p ~ m t ~ , one could expect large deviations for the top. It is then natural, in order to study the effects of these anomalous couplings, to neglect any term ~ apart from those affecting the Z°tt vertex. We can then write the effective Z °tt lagrangian as g fz~, = ~ C w i Y ' [ P r ( l + k t ' ) + P R k R -- ~Sw 8 2 ] tZ ~ , (3.1) where PL,R are chirality projectors and kL,R parametrize the extra interaction. The lagrangian with a modified vertex (3.1) is non-renormalizable. As a matter of fact, radiative corrections coming from (3.1) bring a physically meaningful cutoff depen-
~ For example a simple form ~ ~ G v , for the anomalous couplings involving two fermions i, j and a vector boson, is consistent with our choice. 414
31 October 1991
dence. However, since eq. (3. t ) is deduced from a gauge-invariant non-linear chiral lagrangian by going to the unitary gauge [3], a gauge-invariant cutoff must be chosen. It is then useful to make the calculation in dimensional regularization and interpret 1/~ as in A [ 11 ]. A represents the typical scale of the interactions which break the gauge group. The expressions of e1.2.3 can be readily found as functions ofkL and kR: 3GFmZt ~:1 -- 8N~
X
7"[2
l + 2 [ 2 ( k R - - k L ) - - ( k R - - k L ) 211n
,
(3.2a) e2=-
GFM2w A2 x/AS"~ [ ½ k e + ~ ( k ~ + k 2 ) ] In rn~'
e3 = -
GFM2w - ~ [~ke-~kR+l(k2+k~)l
(3.2b)
A2 lnm~. (3.2c)
In deriving these expressions, only terms proportional to 1/e were retained. Finite anomalous contributions must be discarded in this cutoff dependent model. The expression for El was already given in ref. [5], and corresponds to the standard contribution with the addition of terms proportional to the axial anomalous coupling kA=kL--kR. Notice that for kA=2 eq. (3.2a) goes back to the standard model resuit. In fact kA= 2 corresponds to a simple change of sign of the standard axial current interaction. A relevant phenomenological feature of eq. (3.2a) is that the anomalous contribution can take negative values. This means that larger values of m t than in the standard case can be compatible with the experimental bound on Ap. The standard contribution due to the top quark has not been added to e2 and e3 in eqs. (3.2b), (3.2c) since it is not relevant compared to the present experimental precision. Notice that, ~2 and ~3 being derivative quantities, they receive contributions from both the vector kv and axial kA combinations o f couplings. Moreover, they depend only logarithmically on mr, so that the results we get from their comparison with the experimental data will be less affected
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by the ( u n k n o w n ) value o f the top mass. We recall the expression for Apbe~tex o b t a i n e d from this m o d e l [ 5 ], inclusive o f the s t a n d a r d model leading contribution:
b
,/2 aFm~
Z~kpverte x
×
- -
- -
4~ 2
l+(lkR-2kc)ln~-{
.
(Y2d)
Notice that the O ( m ~ ) contributions to the vertex and to e~ are no longer p r o p o r t i o n a l to each other (in the s t a n d a r d case ~o)~,~x = - 4z~o). As m, grows, and at fixed values o f the k's, e~ and ( o r ) Ap~e~t~xbecome the most relevant effects. However, for mt not very large ( m , < 2 0 0 G e V ) also the effect o f e2,3 is relevant. W h e n making a c o m p a r i s o n with the experimental d a t a all the observables must then be taken into account. In what follows we compare our formulae to the present data coming from precision m e a s u r e m e n t s in the four LEP experiments. The d a t a on the leptonic partial widths o f the Z ° and on the leptonic F - B a s y m m e t r i e s yield for the effective Z ° couplings [ 12 ] ~2 ga = ½+ ¼Ap= 0.4998 -+ 0.0017, gv --0.074--+0.012,
(3.3)
where a m e a n has been taken over the four LEP exp e r i m e n t s [14]. The error in (3.3) is mainly statistical. Using the value M z = 91.177 GeV, one has from eq. (2.1) s~ = 0 . 2 3 1 3 , and then from eq. (2.2) A k ' = 0 . 0 0 1 -+0.013. In addition, the value and UA2 [ l 4 ] yields
(3.4)
of Mw/Mz measured
Arw= - 0 . 0 1 5 - + 0 . 0 1 8 .
by CDF
(3.5)
A direct a p p l i c a t i o n o f eq. (2.5) gives the following b o u n d s for ~1,2,3: El = -- 0.0008 -+ 0 . 0 0 6 8 ,
31 October 1991
e2 = - - 0 - 0 0 8 + 0 . 0 1 1 , C3 = 0.000 +__0 . 0 0 9 ,
( 3.6 c o n t ' d )
where the errors on Ap, Ak', Arw have been propagated quadratically. A c o m p a r i s o n between eq. (3.2) and eq. (3.6) b o u n d s kc and kR ~3 An additional b o u n d is p r o v i d e d by the experimental value o f the ratio between hadronic and leptonic width [ 13 ], Fh = 2 0 . 9 4 _ + 0 . 1 2 ,
(3.7)
r~
whose expression in terms of our observables is given by Fh
R=F-
6 5 - - y-~0 --
1+ (1_4s2)2
1+
× (1 + 0 . 2 6 ~ 1 - 0.34e3 + 0.25 Ap~e~ex),
(3.8)
where the numerical coefficients were c o m p u t e d using so2 = 0.23, and C~sis the strong coupling constant. We show in figs. 1 and 2 which part o f the ( k b kR) plane is allowed by the above data, for different values o f m, and the cutoffA. In doing that we d o u b l e d the errors given above for the experimental quantities. Thus we m a y roughly consider our bounds to be at the two-sigma level (part o f the error is systematic). In order to take into account the uncertainties related to Q C D corrections, we also considered o~s=0.12_+ 0.01 [16]. The b o u n d s coming from ~o and R were already shown in ref. [5]. There, however, older LEP data were used and e2, t3 were not taken into account. This is surely correct for large enough values o f the top mass. However, for m o d e r a t e l y large values o f mt ( ~ 200 G e V ) , we expect the contributions from ez and % to be o f the same order. As a m a t t e r o f fact we see in fig. 1 ( m r = 150 G e V ) that the use o f all o f the observables implies stronger b o u n d s than the simple use o f R and ~ . F o r m , = 120 GeV, see fig. 2, the b o u n d from R becomes irrelevant. This is due to the
(3.6)
~2 The same data can be found in ref. [13]. In ref. [13], however, due to a different procedure for combining statistical and systematic errors, the quoted value for gA is gA=0.4998+ 0.0012.
,3 It should be noticed that in the interplay between the experimental data ( 3.3 )- ( 3.5 ) and eq. ( 3.6 ) some corrections due to known effects (such as those due to box and vertex diagrams) should be taken into account. However, a direct evaluation of these terms by using the formulae in ref. [ 15 ] shows that they are not larger than the errors quoted in eq. (3.6), 415
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ITltop = 4
PHYSICS LETTERS B
A
1 5 0 GeV,
....
i
I
. . . .
= 1
/ / ." / . )i....
TeV
31 October 1991
ITltop =
.I. . . . .
1E0 GeV,
A -- 1 TeV /
.....
•
KR
/
•
t ••/• /iii/i///
KR /"
•
J
E •
/
/
J
/
/•/
•
/
/y
0
/
•
2
/ / /
.;<
•
..
/ /
/.~.
-2
-2 /
(~. , , •
/ //:
. /i.' . ..Z •
/
/ /
../- 4 .-Z -4
.... -2
I .... 0
]
,
2
,
,
-4
r
4
-4
-2
0 KL
KL
Fig. 1. Regions in the ( k L , k R ) plane allowedby present-day LEP data, when compared to the computed values of i~1 (tWOdotted stripes centred respectivelyaround kA= 0 and kA= 2 ), R (region delimited by the dashed lines), e2 (internal part of the lower standing circle) and ea (internal part of the upper standing circle). The values m~= 150 GeV and A= 1 TeV have been assumed for the top quark mass and the cutoff scale respectively. the fact that, for negative values kL ~ kR < 0, the contribution from Ap)ertcx in eq. (3.8) is compensated by that of e3- At m t = 250 GeV the c o m b i n e d b o u n d s from e~ and R are more restrictive. We do not report the graph in this case. By choosing A = 3 TeV, the allowed region reduces to a strip with - - 0 . 4 < k L + kR< 1.2, - - 0 . 0 6 < k R - - k L < --0.02. Anyway, in a phenomenological model like the one at hand, one should take into account all the available information. It is well conceivable, for instance, that there exist small additional couplings not related to the top quark and changing the above picture, One example is an anomalous Zl~b vertex, which could give an extra c o n t r i b u t i o n to R, naively of the same order as that coming from the top in eq. (3.2d), if one assumes k b ~ r n b x ~ f . On the other hand, according to the same pattern, very small additional effects are expected for the light fermions e and g. This observation increases the importance of e3, since it is b o u n d e d through purely leptonic quantities [Ak' in eq. ( 3 . 4 ) ] . 416
Fig. 2. The same as fig. 1, for mr-- 120 GeV and A = 1 TeV.
The use of e2 and E3 is relevant also in view of the foreseen precision in the d e t e r m i n a t i o n of Ak' and Arw. (It should be noticed that, conversely, the precision on Ap is not expected to increase noticeably). By c o m b i n i n g the above quantities the el contribution can be cancelled,
( c 2 - s 2 ) A k ' - s 2 Arw = ( c 2 - s 2 ) (e3 - e 2 ) =e_
(3.9)
The e observable is proportional to the vector comb i n a t i o n kL+ kR, since it is determined by F3v in eq. (2.4) (in our model there are no anomalous contrib u t i o n s to Fww). The measure of the left-right asymmetry with an error a(ALR) = 0 . 0 0 3 [6] allows a det e r m i n a t i o n of Ak' with a ( A k ' ) = 0 . 0 0 1 6 . The measure of Mw at LEP II [17] with a ( M w ) = 100 MeV corresponds to a(Arw) =0.005. Thus the resulting error on E_ will allow us to b o u n d kL+kR at the two-sigma level as E < k L + k R < E + 1.2, where E depends on the experimental value of E_ (A = 3 TeV and m , = 150 GeV have been assumed). This b o u n d is quantitatively similar to the one that can already be obtained for m r = 2 5 0 GeV by using the data on R; it is, however, interesting since it is rather i n d e p e n d e n t on m,.
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4. In this section we examine briefly the effects on precision measurements due to the presence of an isosinglet up quark. Indicating by g~ and g~ the two components of this fermion and supposing that it mixes essentially with the top [4], the Yukawa interactions are given by
~ = h q 'LH t 'R + m ~g 'Lg 'R + m 2£ 'Lt 'R ,
(4.1)
where H is the Higgs field, m~ and m2 are bare mass terms and h is a Yukawa coupling. The fields q~_ and t[ are the top-bottom left gauge doublet and the top right singlet. The mass eigenstates tL and gL arising from the mass matrix after SSB are related to t[ and g~ via a mixing angle ~L ~4. Due to this mixing, the coupling of the top tL to the Z ° is non-standard. In addition there is a flavor-violating vertex Z°tg [ 4 ]. The net contribution to Ap is then [4] 3GF
/~P= 8~/2 ~ / C 7~" OS
[
4
+ 2 sin2~ecos2~L mZrn2g In mg2~
m2 - m 2
~2t j ,
a~M~w
~2 = 4x/~ ~r2
(4.2)
(4.3)
where Ap is given in eq. (4.2). Thus the experimental measurement of Fh/F~ cannot be used to get an ad-
~4 Another mixing angle ~R appears for the right-handed components. The value of this parameter is fixed by ~l_and by the mass eigenvalues m., m~.
m~
2
In ~zz - ½+sin2~L In mg m2
- COS2~Lsin2~c H ( m ~ / m ~ ) ) ,
GvM~v (
e3= 1 2 ~ 2
m~ +~+sin2~Llnm~
lnM-~z
m2
- 3 cos2~L sin'-~LH(m2/m2g)),
(4.4)
where the function H ( x ) is given by
H(x)-3 •
which, in contrast to eq. (3.2), is always positive. In addition eq. (4.2) always strengthens the bound on mt implied by the standard expression. In fact, it can be easily verified that the expression in the large parentheses in (4.2) is larger than min[m~, rag], 2 for any value of ~L. This property is shared by the more general formula for Ap given in ref. [4], which also takes into account a possible mixing with the lighter up quarks. It is interesting to see whether Apb~,~ evaluated in this model turns out proportional to Ap. The calculation follows the same lines as in the standard case. The leading contributions, proportional to the squared fermion masses, arise from the triangle graphs in which a pseudo-Goldstone boson is propagated. The result mimics the standard one b -4 ~P vertex -- - g A p
ditional bound on the parameters of this model. The contribution to e: and e3 is
5
~L in2 +sin4~, rag2
31 October 1991
, ( x+, (x_l)2 4X+x_l (l+x2-4x)
lnx
)
(4.5)
In analogy with the standard model case, the terms in eq. (4.4) are very small. 5. We have studied the electroweak radiative effects of anomalous top quark couplings to the Z ° boson. Such anomalous terms are expected to arise in the context of dynamical gauge symmetry breaking. In turns out that, for moderately large values of the top quark mass m~<150 GeV, the whole set of observables, consisting of el, e2, e3 and Apbertex, must be considered. In this range of mt the bounds on the anomalous couplings kL and kR coming from the experimental data are essentially determined by e~,2,3. These bounds are roughly given by - 2 < k v < 2 , - 0.15 < kA < 0.05. The foreseen measurements with polarized beams and at LEP II will permit to reduce the allowed range in kv of about a factor 3 (Akv ~ 1 ). For larger values of the top mass (m, > 200 GeV), the bounds on kL and kR given by the experimental values of Ap and R =F,/F~ are more significant than those coming from e2 and e3. It must, however, be noticed that the case of a very heavy top quark (m, > 250 GeV ), even if allowed in principle by eqs. (3.2a) and (3.2b), is not very plausible since, in order to be consistent with the experimental data, a considerable degree of fine tuning must be exercised on the anomalous kL and kR. We have also studied the whole set of radiative ob417
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s e r v a b l e s in a s i m p l e r e n o r m a l i z a b l e m o d e l in w h i c h the top q u a r k is m i x e d w i t h an isosinglet quark. It t u r n s out that the p a t t e r n is a n a l o g o u s to the stand a r d o n e in the i m p r o v e d B o r n a p p r o x i m a t i o n . T h e c o n t r i b u t i o n to Ap a n d the leading c o n t r i b u t i o n to the Z°13b v e r t e x c o r r e c t i o n are p r o p o r t i o n a l to the s a m e expression. T h e p a r a m e t e r s e2 a n d c3 r e m a i n irrelevant. We gratefully a c k n o w l e d g e s t i m u l a t i n g discussions w i t h R. B a r b i e r i a n d F. Z w i r n e r .
References [ 1 ] D.C. Kennedy and B.W. Lynn, Nucl. Phys. B 322 (1989) 1. [2] A.A. Akhundov et al., Nucl. Phys. B 276 (1988) 1; F. Diakonos and W. Wetzel, preprint HD-THEP-88-21 (1988); W. Beenakker and W. HoUik, Z. Phys. C 40 ( 1988 ) 569; B.W. Lynn and R. Stuart, Phys. Lett. B 252 (1990) 676. [3 ] R.D. Peccei and X. Zhang, Nucl. Phys. B 337 (1990) 269. [4] R. Rattazzi, Nucl. Phys. B 335 (1990) 301. [5]R.D. Peccei, S. Peris and X. Zhang, Nucl. Phys. B 349 (1991) 305. [6] G. Altarelli and R. Barbieri, Phys. Lett. B 253 (1991 ) 16l.
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[ 7 ] R. Barbieri and L. Hall, Nucl. Phys. B 319 ( 1989 ) 1; B.S. Balakrishna, A.L. Kagan and R.N. Mohapatra, Phys. Lett. B 205 (1988) 345. [ 8] R. Barbieri, M. Frigeni, F. Giuliani and H.E. Haber, Nucl. Phys. B 341 (1990) 309. [9]M. Peskin and T. Tacheuchi, Phys. Rev. Lett. 65 (1990) 964; M. Golden and L. Randall, preprint Fermilab-Pub 90-83-T (1990). [10] CDF Collab., F. Abe et al., preprint Fermilab-Pub-89/212E (1989). [11] T. Appelquist and C. Bernard, Phys. Rev. D 22 (1980) 201; A. Longhitano, Nucl. Phys. B 188 ( 1981 ) 118. [121 G. Altarelli, talk Annecy Meeting on LEP physics (February 1991). [13] H. Burkhardt and J. Steinberger, preprint CERN-PPE/9150, Annu. Rev. Nucl. Part. Sci. 41, to appear. [14] D. Froidevaux, in: Proc. Neutrino-90 Conf. (Geneva, 1990), to be published. [15] M. Consoli and W. Hollik, in: Z physics at LEP, eds. G. Altarelli et al., CERN report CERN 89-08 (1989). [161 Z. Kunszt and P. Nason, in: Z physics at LEP, eds. G. Altarelli et al., CERN report CERN 89-08 ( 1989); N. Magnoli, P. Nason and R. Rattazzi, Phys. Lett. B 252 (1990) 271; S. Bethke, preprint CERN-PPE/91-36, invited talk Workshop on Jet physics at LEP and HERA (Durham, UK, December 1990). [17] A. Bohm and W. Hoogland, eds., Proc. LEP200 Workshop (Aachen, 1986).
P H YSIC S k ET T ER$ B
Testing an anomalous top quark via precision electroweak measurements Maurizio Frigeni Dipartimento di Fisica, Universith di Pisa, 1-56100 Pisa, Italy and Riccardo Rattazzi t CERN, CH-1211 Geneva 23, Switzerland Received 24 June 1991
The effect on electroweak corrections of anomalous interactions of the top quark with the Z° is studied. Such anomalous terms are expected to arise in the context of dynamical symmetry breaking. It is shown that, depending on mr, the whole set of radiativecorrection parameters can be relevant. This set consists of three observables coming from the vacuum polarization graphs plus the correction to the Z°t3bvertex. The recent data coming from collider experiments are used to place bounds on the anomalous terms. The same observablesare evaluated in a simple renormalizable model in which the top quark mixes with an isosingletuptype quark.
1. It is well k n o w n that precision measurements at LEP can open a window on physics above the Fermi scale. The most relevant effects due to virtual heavy particles come typically through corrections to vector boson propagators, the so-called "oblique" corrections [ 1 ]. An exception to this rule is represented by the top quark, since large G y m 2t terms appear in the Zl~b vertex corrections [ 2 ]. However, this is because b and t fit into the same weak doublet. Any possible anomalous behavior of the top quark should show up both in oblique and vertex corrections. The purpose of this letter is to study the impact on precision measurements of anomalous non-universal interactions of the top quark with the Z °. We shall deal with two models where such extra couplings arise in completely different ways: in one case they are induced by dynamical gauge symmetry breaking [3], while in the other they simply appear via the mixing of the top quark with a heavy isosinglet up-type quark [4]. Inspired by an analogy to low-energy chiral symA. Della Riccia fellow. 412
metry breaking, it has been argued in ref. [ 3 ] that, if the electroweak symmetry breaking is dynamical, residual interactions between fermion a n d vector bosons should in general arise. One could also argue that such anomalous couplings depend somehow on the mass of the involved fermions. According to this picture one expects very small deviations for the light fermions (consistent with experimental observations), Conversely, large deviations are expected for the heavy top quark. These could give measurable effects in precision tests at LEP. In ref. [ 5 ] the contrib u t i o n to the p parameter and to the Z13b vertex were studied. The aim of this paper is to study the whole set of radiative-correction parameters, including e2 and e3 defined in ref. [ 6 ]. A comparison is made with the latest LEP data, thus updating the analysis in ref. [5]. Of course this model is only phenomenological and renormalizability is lost. In order to make a comparison with a renormalizable model, we consider the simplest possible case in which anomalous couplings of the top are present, consisting of the addition of a heavy isosinglet up-type quark. This choice is also of
0370-2693/91/$ 03.50 © 1991 Elsevier SciencePublishers B.V. All rights reserved.
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speculative interest. In fact, many grand-unified models require the addition of isosinglet fermions. Furthermore, the addition of such an isosinglet characterizes many models which try to explain the heaviness of the top quark [ 7 ]. This paper is organized as follows. In section 2 we recall the definition of the three parameters El,2,3 which determine the pattern of oblique radiative corrections in Z ° physics [6]. In section 3 we calculate these corrections in the dynamical-breaking model, and make a comparison with the recent experimental data coming from the four LEP experiments. In section 4 we c o m m e n t on the radiative effects in the case of the isosinglet. Finally, section 5 is devoted to our conclusions.
where i, j stand for W, Z, y. For heavy particles we can approximate the functions F 0 ( q 2 ) with Fij(0) = F v. Considering that A r r = A z v = 0, one is left with six constants Aww, Fww, Azz, Fzz, Fzv, Frr. Three linear independent combinations of these amount to a simple redefinition of c~, GF and Mz [8]. The combinations
2. The effect of the oblique corrections can be seen as a small modification of the tree level relations among the fundamental parameters of the electroweak theory. In general this effect can be described [6] with the introduction of three observables. One of these, Ap, determines the relative strength of charged and neutral current four-fermion interactions. The other two, Arw and Ak', describe the relative difference among the three possible definitions of the sine of the Weinberg angle: Sw, so and gw. More precisely, ZXrwis defined by
zXp= e, ,
Azz
Aww
e ~ - M2
M2w
E2 = Fww e3-
F3v
-,433 - A w w
M2w
,
(2.4b)
- F33 ,
- F 3 3 = cwFo3
Sw
(2.4a)
(2.4c)
Sw
affect the physical observables as (2.5a)
Ak'_ -c2<
+~3
c~-S~w
7 {I
sb
(2.5b)
'
--1
e2 qt- 2e3 .
(2.5c)
gA = T3( 1 + l A p ) ,
(2.2a)
In the standard model the leading contribution is due to a heavy top quark, and resides in e ~ - 3 G F m 2 / 8,,/2 g2. No large contribution arises in ~2 and E3. This picture is not essentially altered by supersymmetry [8 ]. On the other hand one expects large contributions to both el and E3 in technicolor models [9], while E2 is still less relevant. A last c o m m e n t on eq. (2.2) is in order. In general Ap and Ak' receive non-universal corrections due to light known particles and arising from box, vertex diagrams and imaginary parts of the propagators. In the standard model, these are small and reasonably well-known quantities, with the exception of the Zbb vertex corrections. In this case the left-handed Zbb vertex receives a correction proportional to m 2, so that (2.2) is modified by
g v = 1 + 4Qr~2w = 1 + 4 @ ( 1 + A k ' ) s 2 . gA
(2.2b)
gbA= T~[1 + l (Ap+ Apbertex) ] ,
s~c2 = ( 1 _
M2~M 2 ~o~ (Mz) M 2 , ] M2z - ,~f2 G F M z ( 1 - & r w )
sgco - l-Arw '
(2.1)
where c ~ ( M z ) = 128.8 is the running fine structure constant evaluated at s = M 2 and Gv= 1.166× 10 -5 ( G e V ) - 2 is the Fermi constant as measured in g decay. The other two observables are given through the on-shell couplings gv and gA of the Z ° to fermions,
The contribution of heavy particles to these observables via vacuum polarization graphs is quite simple. We can write the vacuum polarization tensor
gbv g~A = 1 + 4@~2w =l+4Qf(l+Ak,
1 b 2• ~APvertex)So
(2.6b)
In the standard model one has Ap b....... = - (xf2 GF/
as
H.~j" = -ig~'"[Av +q2F#(q2) ] ,
(2.6a)
(2.3)
47r2) m 2 . 413
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3. In ref. [3] it has been shown that, if the symmetry breakdown in the electroweak theory is dynamical, residual interactions between fermions and Goldstone bosons should in general arise. These extra terms can be shown to give non-universal interactions between the SU (2) × U ( 1 ) gauge fields and the fermions. These interactions are the analogue of the extra couplings of the pions to nucleons which arise in Q C D when S U ( 2 ) × S U ( 2 ) is dynamically broken. In Q C D the extra couplings are O(1 ) (for instance the ~lqN form factor at zero m o m e n t u m is characterized by an extra piece k =gA-- 1 --~0.25 ). This can be seen as a consequence of the fact that in Q C D there is only one mass scale AQCD. For the electroweak theory the situation is rather different, since there is a full hierarchy of mass scales in the fermion spectrum ranging from me ~ 0.5 MeV to mt > 80 GeV [ 10 ]. One could argue [ 3,5 ] that, if fermions participate in the symmetry breakdown in a non-trivial fashion, the structure of the mass spectrum may in some way be reflected in the extra couplings to gauge bosons. Indeed by assuming that the anomalous terms are of order mrx/~v for a fermion f, consistency is met with the experimental data on light fermions [ 3 ]. For the top quark, however, since it has not yet been detected, we cannot put direct bounds on its extra couplings. On the other hand, given the present lower bound on m~ and supposing k t o p ~ m t ~ , one could expect large deviations for the top. It is then natural, in order to study the effects of these anomalous couplings, to neglect any term ~ apart from those affecting the Z°tt vertex. We can then write the effective Z °tt lagrangian as g fz~, = ~ C w i Y ' [ P r ( l + k t ' ) + P R k R -- ~Sw 8 2 ] tZ ~ , (3.1) where PL,R are chirality projectors and kL,R parametrize the extra interaction. The lagrangian with a modified vertex (3.1) is non-renormalizable. As a matter of fact, radiative corrections coming from (3.1) bring a physically meaningful cutoff depen-
~ For example a simple form ~ ~ G v , for the anomalous couplings involving two fermions i, j and a vector boson, is consistent with our choice. 414
31 October 1991
dence. However, since eq. (3. t ) is deduced from a gauge-invariant non-linear chiral lagrangian by going to the unitary gauge [3], a gauge-invariant cutoff must be chosen. It is then useful to make the calculation in dimensional regularization and interpret 1/~ as in A [ 11 ]. A represents the typical scale of the interactions which break the gauge group. The expressions of e1.2.3 can be readily found as functions ofkL and kR: 3GFmZt ~:1 -- 8N~
X
7"[2
l + 2 [ 2 ( k R - - k L ) - - ( k R - - k L ) 211n
,
(3.2a) e2=-
GFM2w A2 x/AS"~ [ ½ k e + ~ ( k ~ + k 2 ) ] In rn~'
e3 = -
GFM2w - ~ [~ke-~kR+l(k2+k~)l
(3.2b)
A2 lnm~. (3.2c)
In deriving these expressions, only terms proportional to 1/e were retained. Finite anomalous contributions must be discarded in this cutoff dependent model. The expression for El was already given in ref. [5], and corresponds to the standard contribution with the addition of terms proportional to the axial anomalous coupling kA=kL--kR. Notice that for kA=2 eq. (3.2a) goes back to the standard model resuit. In fact kA= 2 corresponds to a simple change of sign of the standard axial current interaction. A relevant phenomenological feature of eq. (3.2a) is that the anomalous contribution can take negative values. This means that larger values of m t than in the standard case can be compatible with the experimental bound on Ap. The standard contribution due to the top quark has not been added to e2 and e3 in eqs. (3.2b), (3.2c) since it is not relevant compared to the present experimental precision. Notice that, ~2 and ~3 being derivative quantities, they receive contributions from both the vector kv and axial kA combinations o f couplings. Moreover, they depend only logarithmically on mr, so that the results we get from their comparison with the experimental data will be less affected
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by the ( u n k n o w n ) value o f the top mass. We recall the expression for Apbe~tex o b t a i n e d from this m o d e l [ 5 ], inclusive o f the s t a n d a r d model leading contribution:
b
,/2 aFm~
Z~kpverte x
×
- -
- -
4~ 2
l+(lkR-2kc)ln~-{
.
(Y2d)
Notice that the O ( m ~ ) contributions to the vertex and to e~ are no longer p r o p o r t i o n a l to each other (in the s t a n d a r d case ~o)~,~x = - 4z~o). As m, grows, and at fixed values o f the k's, e~ and ( o r ) Ap~e~t~xbecome the most relevant effects. However, for mt not very large ( m , < 2 0 0 G e V ) also the effect o f e2,3 is relevant. W h e n making a c o m p a r i s o n with the experimental d a t a all the observables must then be taken into account. In what follows we compare our formulae to the present data coming from precision m e a s u r e m e n t s in the four LEP experiments. The d a t a on the leptonic partial widths o f the Z ° and on the leptonic F - B a s y m m e t r i e s yield for the effective Z ° couplings [ 12 ] ~2 ga = ½+ ¼Ap= 0.4998 -+ 0.0017, gv --0.074--+0.012,
(3.3)
where a m e a n has been taken over the four LEP exp e r i m e n t s [14]. The error in (3.3) is mainly statistical. Using the value M z = 91.177 GeV, one has from eq. (2.1) s~ = 0 . 2 3 1 3 , and then from eq. (2.2) A k ' = 0 . 0 0 1 -+0.013. In addition, the value and UA2 [ l 4 ] yields
(3.4)
of Mw/Mz measured
Arw= - 0 . 0 1 5 - + 0 . 0 1 8 .
by CDF
(3.5)
A direct a p p l i c a t i o n o f eq. (2.5) gives the following b o u n d s for ~1,2,3: El = -- 0.0008 -+ 0 . 0 0 6 8 ,
31 October 1991
e2 = - - 0 - 0 0 8 + 0 . 0 1 1 , C3 = 0.000 +__0 . 0 0 9 ,
( 3.6 c o n t ' d )
where the errors on Ap, Ak', Arw have been propagated quadratically. A c o m p a r i s o n between eq. (3.2) and eq. (3.6) b o u n d s kc and kR ~3 An additional b o u n d is p r o v i d e d by the experimental value o f the ratio between hadronic and leptonic width [ 13 ], Fh = 2 0 . 9 4 _ + 0 . 1 2 ,
(3.7)
r~
whose expression in terms of our observables is given by Fh
R=F-
6 5 - - y-~0 --
1+ (1_4s2)2
1+
× (1 + 0 . 2 6 ~ 1 - 0.34e3 + 0.25 Ap~e~ex),
(3.8)
where the numerical coefficients were c o m p u t e d using so2 = 0.23, and C~sis the strong coupling constant. We show in figs. 1 and 2 which part o f the ( k b kR) plane is allowed by the above data, for different values o f m, and the cutoffA. In doing that we d o u b l e d the errors given above for the experimental quantities. Thus we m a y roughly consider our bounds to be at the two-sigma level (part o f the error is systematic). In order to take into account the uncertainties related to Q C D corrections, we also considered o~s=0.12_+ 0.01 [16]. The b o u n d s coming from ~o and R were already shown in ref. [5]. There, however, older LEP data were used and e2, t3 were not taken into account. This is surely correct for large enough values o f the top mass. However, for m o d e r a t e l y large values o f mt ( ~ 200 G e V ) , we expect the contributions from ez and % to be o f the same order. As a m a t t e r o f fact we see in fig. 1 ( m r = 150 G e V ) that the use o f all o f the observables implies stronger b o u n d s than the simple use o f R and ~ . F o r m , = 120 GeV, see fig. 2, the b o u n d from R becomes irrelevant. This is due to the
(3.6)
~2 The same data can be found in ref. [13]. In ref. [13], however, due to a different procedure for combining statistical and systematic errors, the quoted value for gA is gA=0.4998+ 0.0012.
,3 It should be noticed that in the interplay between the experimental data ( 3.3 )- ( 3.5 ) and eq. ( 3.6 ) some corrections due to known effects (such as those due to box and vertex diagrams) should be taken into account. However, a direct evaluation of these terms by using the formulae in ref. [ 15 ] shows that they are not larger than the errors quoted in eq. (3.6), 415
Volume 269, number 3,4
ITltop = 4
PHYSICS LETTERS B
A
1 5 0 GeV,
....
i
I
. . . .
= 1
/ / ." / . )i....
TeV
31 October 1991
ITltop =
.I. . . . .
1E0 GeV,
A -- 1 TeV /
.....
•
KR
/
•
t ••/• /iii/i///
KR /"
•
J
E •
/
/
J
/
/•/
•
/
/y
0
/
•
2
/ / /
.;<
•
..
/ /
/.~.
-2
-2 /
(~. , , •
/ //:
. /i.' . ..Z •
/
/ /
../- 4 .-Z -4
.... -2
I .... 0
]
,
2
,
,
-4
r
4
-4
-2
0 KL
KL
Fig. 1. Regions in the ( k L , k R ) plane allowedby present-day LEP data, when compared to the computed values of i~1 (tWOdotted stripes centred respectivelyaround kA= 0 and kA= 2 ), R (region delimited by the dashed lines), e2 (internal part of the lower standing circle) and ea (internal part of the upper standing circle). The values m~= 150 GeV and A= 1 TeV have been assumed for the top quark mass and the cutoff scale respectively. the fact that, for negative values kL ~ kR < 0, the contribution from Ap)ertcx in eq. (3.8) is compensated by that of e3- At m t = 250 GeV the c o m b i n e d b o u n d s from e~ and R are more restrictive. We do not report the graph in this case. By choosing A = 3 TeV, the allowed region reduces to a strip with - - 0 . 4 < k L + kR< 1.2, - - 0 . 0 6 < k R - - k L < --0.02. Anyway, in a phenomenological model like the one at hand, one should take into account all the available information. It is well conceivable, for instance, that there exist small additional couplings not related to the top quark and changing the above picture, One example is an anomalous Zl~b vertex, which could give an extra c o n t r i b u t i o n to R, naively of the same order as that coming from the top in eq. (3.2d), if one assumes k b ~ r n b x ~ f . On the other hand, according to the same pattern, very small additional effects are expected for the light fermions e and g. This observation increases the importance of e3, since it is b o u n d e d through purely leptonic quantities [Ak' in eq. ( 3 . 4 ) ] . 416
Fig. 2. The same as fig. 1, for mr-- 120 GeV and A = 1 TeV.
The use of e2 and E3 is relevant also in view of the foreseen precision in the d e t e r m i n a t i o n of Ak' and Arw. (It should be noticed that, conversely, the precision on Ap is not expected to increase noticeably). By c o m b i n i n g the above quantities the el contribution can be cancelled,
( c 2 - s 2 ) A k ' - s 2 Arw = ( c 2 - s 2 ) (e3 - e 2 ) =e_
(3.9)
The e observable is proportional to the vector comb i n a t i o n kL+ kR, since it is determined by F3v in eq. (2.4) (in our model there are no anomalous contrib u t i o n s to Fww). The measure of the left-right asymmetry with an error a(ALR) = 0 . 0 0 3 [6] allows a det e r m i n a t i o n of Ak' with a ( A k ' ) = 0 . 0 0 1 6 . The measure of Mw at LEP II [17] with a ( M w ) = 100 MeV corresponds to a(Arw) =0.005. Thus the resulting error on E_ will allow us to b o u n d kL+kR at the two-sigma level as E < k L + k R < E + 1.2, where E depends on the experimental value of E_ (A = 3 TeV and m , = 150 GeV have been assumed). This b o u n d is quantitatively similar to the one that can already be obtained for m r = 2 5 0 GeV by using the data on R; it is, however, interesting since it is rather i n d e p e n d e n t on m,.
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4. In this section we examine briefly the effects on precision measurements due to the presence of an isosinglet up quark. Indicating by g~ and g~ the two components of this fermion and supposing that it mixes essentially with the top [4], the Yukawa interactions are given by
~ = h q 'LH t 'R + m ~g 'Lg 'R + m 2£ 'Lt 'R ,
(4.1)
where H is the Higgs field, m~ and m2 are bare mass terms and h is a Yukawa coupling. The fields q~_ and t[ are the top-bottom left gauge doublet and the top right singlet. The mass eigenstates tL and gL arising from the mass matrix after SSB are related to t[ and g~ via a mixing angle ~L ~4. Due to this mixing, the coupling of the top tL to the Z ° is non-standard. In addition there is a flavor-violating vertex Z°tg [ 4 ]. The net contribution to Ap is then [4] 3GF
/~P= 8~/2 ~ / C 7~" OS
[
4
+ 2 sin2~ecos2~L mZrn2g In mg2~
m2 - m 2
~2t j ,
a~M~w
~2 = 4x/~ ~r2
(4.2)
(4.3)
where Ap is given in eq. (4.2). Thus the experimental measurement of Fh/F~ cannot be used to get an ad-
~4 Another mixing angle ~R appears for the right-handed components. The value of this parameter is fixed by ~l_and by the mass eigenvalues m., m~.
m~
2
In ~zz - ½+sin2~L In mg m2
- COS2~Lsin2~c H ( m ~ / m ~ ) ) ,
GvM~v (
e3= 1 2 ~ 2
m~ +~+sin2~Llnm~
lnM-~z
m2
- 3 cos2~L sin'-~LH(m2/m2g)),
(4.4)
where the function H ( x ) is given by
H(x)-3 •
which, in contrast to eq. (3.2), is always positive. In addition eq. (4.2) always strengthens the bound on mt implied by the standard expression. In fact, it can be easily verified that the expression in the large parentheses in (4.2) is larger than min[m~, rag], 2 for any value of ~L. This property is shared by the more general formula for Ap given in ref. [4], which also takes into account a possible mixing with the lighter up quarks. It is interesting to see whether Apb~,~ evaluated in this model turns out proportional to Ap. The calculation follows the same lines as in the standard case. The leading contributions, proportional to the squared fermion masses, arise from the triangle graphs in which a pseudo-Goldstone boson is propagated. The result mimics the standard one b -4 ~P vertex -- - g A p
ditional bound on the parameters of this model. The contribution to e: and e3 is
5
~L in2 +sin4~, rag2
31 October 1991
, ( x+, (x_l)2 4X+x_l (l+x2-4x)
lnx
)
(4.5)
In analogy with the standard model case, the terms in eq. (4.4) are very small. 5. We have studied the electroweak radiative effects of anomalous top quark couplings to the Z ° boson. Such anomalous terms are expected to arise in the context of dynamical gauge symmetry breaking. In turns out that, for moderately large values of the top quark mass m~<150 GeV, the whole set of observables, consisting of el, e2, e3 and Apbertex, must be considered. In this range of mt the bounds on the anomalous couplings kL and kR coming from the experimental data are essentially determined by e~,2,3. These bounds are roughly given by - 2 < k v < 2 , - 0.15 < kA < 0.05. The foreseen measurements with polarized beams and at LEP II will permit to reduce the allowed range in kv of about a factor 3 (Akv ~ 1 ). For larger values of the top mass (m, > 200 GeV), the bounds on kL and kR given by the experimental values of Ap and R =F,/F~ are more significant than those coming from e2 and e3. It must, however, be noticed that the case of a very heavy top quark (m, > 250 GeV ), even if allowed in principle by eqs. (3.2a) and (3.2b), is not very plausible since, in order to be consistent with the experimental data, a considerable degree of fine tuning must be exercised on the anomalous kL and kR. We have also studied the whole set of radiative ob417
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s e r v a b l e s in a s i m p l e r e n o r m a l i z a b l e m o d e l in w h i c h the top q u a r k is m i x e d w i t h an isosinglet quark. It t u r n s out that the p a t t e r n is a n a l o g o u s to the stand a r d o n e in the i m p r o v e d B o r n a p p r o x i m a t i o n . T h e c o n t r i b u t i o n to Ap a n d the leading c o n t r i b u t i o n to the Z°13b v e r t e x c o r r e c t i o n are p r o p o r t i o n a l to the s a m e expression. T h e p a r a m e t e r s e2 a n d c3 r e m a i n irrelevant. We gratefully a c k n o w l e d g e s t i m u l a t i n g discussions w i t h R. B a r b i e r i a n d F. Z w i r n e r .
References [ 1 ] D.C. Kennedy and B.W. Lynn, Nucl. Phys. B 322 (1989) 1. [2] A.A. Akhundov et al., Nucl. Phys. B 276 (1988) 1; F. Diakonos and W. Wetzel, preprint HD-THEP-88-21 (1988); W. Beenakker and W. HoUik, Z. Phys. C 40 ( 1988 ) 569; B.W. Lynn and R. Stuart, Phys. Lett. B 252 (1990) 676. [3 ] R.D. Peccei and X. Zhang, Nucl. Phys. B 337 (1990) 269. [4] R. Rattazzi, Nucl. Phys. B 335 (1990) 301. [5]R.D. Peccei, S. Peris and X. Zhang, Nucl. Phys. B 349 (1991) 305. [6] G. Altarelli and R. Barbieri, Phys. Lett. B 253 (1991 ) 16l.
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[ 7 ] R. Barbieri and L. Hall, Nucl. Phys. B 319 ( 1989 ) 1; B.S. Balakrishna, A.L. Kagan and R.N. Mohapatra, Phys. Lett. B 205 (1988) 345. [ 8] R. Barbieri, M. Frigeni, F. Giuliani and H.E. Haber, Nucl. Phys. B 341 (1990) 309. [9]M. Peskin and T. Tacheuchi, Phys. Rev. Lett. 65 (1990) 964; M. Golden and L. Randall, preprint Fermilab-Pub 90-83-T (1990). [10] CDF Collab., F. Abe et al., preprint Fermilab-Pub-89/212E (1989). [11] T. Appelquist and C. Bernard, Phys. Rev. D 22 (1980) 201; A. Longhitano, Nucl. Phys. B 188 ( 1981 ) 118. [121 G. Altarelli, talk Annecy Meeting on LEP physics (February 1991). [13] H. Burkhardt and J. Steinberger, preprint CERN-PPE/9150, Annu. Rev. Nucl. Part. Sci. 41, to appear. [14] D. Froidevaux, in: Proc. Neutrino-90 Conf. (Geneva, 1990), to be published. [15] M. Consoli and W. Hollik, in: Z physics at LEP, eds. G. Altarelli et al., CERN report CERN 89-08 (1989). [161 Z. Kunszt and P. Nason, in: Z physics at LEP, eds. G. Altarelli et al., CERN report CERN 89-08 ( 1989); N. Magnoli, P. Nason and R. Rattazzi, Phys. Lett. B 252 (1990) 271; S. Bethke, preprint CERN-PPE/91-36, invited talk Workshop on Jet physics at LEP and HERA (Durham, UK, December 1990). [17] A. Bohm and W. Hoogland, eds., Proc. LEP200 Workshop (Aachen, 1986).