The W-width in the two Higgs doublet model

NUCLEAR PHYSICS B ELSEVIER

Nuclear Physics B 449 (1995) 69-79

The W-width in the two Higgs doublet model Dong-Shin Shin Max-Planck-lnstitut fu'r Physik- Werner-Heisenberg-lnstitut, F6hringer Ring 6, 80805 Munich, Germany Received 21 November 1994; revised 1 May 1995; accepted 8 May 1995

Abstract The electroweak l-loop corrections to the decay width of the W boson into fermions F ( W + ---+ f , f 2 ) are discussed in the extended Higgs sector of the general two Higgs doublet model without any constraint. The relative corrections to the decay W + ~ e+pe which come exclusively from

W boson propagator are unobservably small in the whole parameter region with the maximal correction of - 6 × 10-3%. Due to the increasing of the Yukawa couplings with fermion masses, the relative corrections to the decays W+ ~ 7+~,~ and W + --+ cg become larger, but even after having raised the Yukawa couplings with help of tan/3, the maximal corrections lie under the percent region and thus give also no observable effect. We discuss also the Higgs sector of the minimal supersymmetric standard model for the decays W+ -~ ~-+~,~and W + ~ cg. Similarly to the cases in the general two Higgs doublet model, the corrections in this model give no sizable effect and hence are not significant.

1. Introduction The experiments at LEP and SLC have thus far confirmed all predictions of the Standard Model ( S M ) for fermionic final states, but did not show any indication of a Higgs boson with a mass below 60 GeV [ 1 ]. Thus the Higgs sector of the SM still remains as an unsatisfactory region where the experimental confirmation is lacking. On the other hand, the experimentally confirmed relation P - - M ~ c o s 20w - 1

(l.1)

is satisfied not only in the SM with a single Higgs doublet, but also in other versions of the SM with any number of Higgs doublets. Therefore we can, in principle, extend the Higgs sector of the SM to more Higgs doublets without violating the relation. At the same time, from the view of many open questions like the origin of C P violation or the hierarchy problem, the SM needs to be extended. If we try to solve 0550-3213/95/$09.50 ~ 1995 Elsevier Science B.V. All rights reserved SSDI 05 50-3213 ( 9 5 ) 00221-9

70

D.-S. Shin~NuclearPhysics B 449 (1995) 69-79

these problems, we are led to adding additional Higgs doublets to the Higgs sector of the SM. The strongest motivation for two Higgs doublet model (2HDM) may come from the minimal supersymmetric standard model (MSSM). In this model, in order to give masses to the two types of quarks, two Higgs doublets are needed; one Higgs doublet for the quarks with isospin 13 = +½ and the other for those with 13 = - 3 l [2]. The extension of the SM to the 2HDM results in the production of a new particle spectrum in the Higgs sector [2]. After diagonalizing the mass matrix, it has five physical mass eigenstates: a pair of charged Higgs bosons H ±, two neutral CP-even states H °, h ° ( 'scalars' ) and one neutral CP-odd state A ° ( 'pseudoscalar' ) with masses M~I~, MHo, Mho,MAO. In the procedure of generating the mass eigenstates, two Higgs mixing angles appear additionally as free parameters: tan/3 (the ratio of vacuum expectation values) and c~. In contrast to the SM which has only one free parameter, the neutral Higgs mass MH, the 2HDM has thus 6 free parameters. Besides a direct search for these particles, another way to get information about them is through the calculation of radiative corrections; they appear in the loop calculations as virtual states, and so influence physical predictions. The comparison of the calculations with precision measurements gives indirect information on the masses of these particles. In contrast to the measurements of the parameters of the Z boson where precise experimental data exist, the observables for the W boson are not known so well - but with the construction of LEP 200 this situation will improve. The precision for the mass will be about 0 . 1 % and for the width in the percent region. For an exact comparison between the measurements and the theoretical predictions, the calculation of the radiative corrections is necessary. These corrections to higher order of the perturbation theory for the decay of the W boson into fermions were already calculated both in the QCD and in the electroweak SM: the QCD corrections are about 4 % [3,4], while the electroweak corrections are under 1 % and practically not dependent on the Higgs boson [ 5]. Since the W decay width is not sensitive to the Higgs sector of the SM, it is interesting to investigate the effect of an extension of the Higgs sector from the SM to the 2HDM on the decay width through the radiative corrections. The aim of this work is to study the influence of the additional free parameters appearing through the extension on the W decay width. In the next section we give the necessary formulae for the decay width and the l-loop contributions to it. Thereafter, the results are presented quantitatively and discussed in Section 3.

2. The l-loop decay width We consider the decay of a W + boson into two fermions W+(q)

' f l ( P l ) + f2(P2).

The lowest order decay width is given by

fl and f2: (2.1)

D.-S. Shin~NuclearPhysics B 449 (1995) 69-79 F(oa) = O%m MW

6 2 sin20w

x [1

71

Nc l Vftf2 12 K M2w

ml2 + m~ 2M 2

(m12_-- m2) 2] 2M4w ]

(2.2)

where

t¢ = v/ ( M2w - m~ - m~ ) 2 - 4m2m~ .

(2.3)

Vfd2 denotes the element of the Kobayashi-Maskawa matrix for hadronic decays and is equal to one for leptonic decays. Nc is the color factor. If we use the tree level relation between the Fermi constant G u and the fine structure constant Ceern [6] 77"~em Gu = v/-~M 2

1 sin20w,

(2.4)

we can express F0('~) in (2.2) with Gu: Gu

3

1"(0c") - 6-~TrMwNc IV flY2 X[1

mZ+m~ 2M 2

12 K

M2

( m 2 - mZ)2J 5~,/~ •

(2.5)

Compared with (2.2), this form has the advantage of being independent of the mixing angle sin 20w. For the loop calculation we use the on-shell renormalization scheme discussed in [7,8]. Since the calculation at the l-loop level in the SM has already been done and the 2HDM concerns only an extension of the SM in the Higgs sector, we calculate explicitly only the corrections which appear additionally through the extension. If we try to calculate only these "non-standard" (NS) loop contributions, a small complication appears, since in the general case of the mixing in the neutral Higgs sector neither H ° nor h ° can be identified with the "standard" Higgs in a natural way. In order to obtain the pure non-standard contributions we must therefore subtract the contributions from the SM Higgs part. Without loss of generality we take M/4 = MH0 as the reference mass for the SM Higgs boson. At the l-loop level, the vertex of the decay matrix element has the following structure I.

F ~ = - i ~ 2 { y~L [ I + F L + ~ Z L - ~ H I1^ ( m l ) - ~22 ~ w (1^' Mw) +T~RFR + "P1--~-~ (HLLMw + HRR) } I We use the Feynmanrules and conventionsof [2].

2]] (2.6)

D.-S. Shin/Nuclear Physics B 449 (1995) 69-79

72

(a)

•

u~

u

, ~ /

W*

H°'h° "~r'~ 7 u

+

H...r ~d

wl ,J v Av A~,,,~,. ; n , n,.o.o ,,'~ ~w"

d~k~... d

u

H°,h°,A°"% d w-

u

W*

C".+ . . ~ _ u,

.if

H°h°A°

H+ " %

w•

u W+ d

u

J G+ . ~ ¢ a H°'h°"~

d

H°,ho.T/u

d

(b)

IsI

H ° ' h ° ' A ~ " % I-

H°'h°"%

W+

~

VI

L. '

I-

Fig. 1. Additional Higgs contributions to the vertex: (a) the decay into two quarks (b) the decay into two leptons. The diagrams are valid also for c and s quarks with the substitutions u ~ c, d ~ s and l states for e,/z and r.

with the gauge coupling g = e/sin Ow and the right- and left-handed projectors R, L = ½( 1 4-ys). The form factors FL, FR, HL and HR originate from the vertex corrections (Fig. 1). 6ZL is the field renormalization constant for the external fermions. 2 In our renormalization scheme, it is the only counter term of the vertex, since the combination 6 Z w - 6 Z w vanishes in the extra Higgs contributions. &ZL is fixed by requiring the residue of the renormalized propagator of the fermion with 13 = -½ to be 1: '~fz(

6ZL = - - 2 . c

v f 2 / ( . 2,,

2-

f2 t

m 2) - - m ~ [ X f = ' ( m ~ ) + "R , ' 2 ) + 2Xs

2 (m2)],

(2.7)

where the self-energy has been decomposed according to Xf(p)

= / ~ L S L f ( p 2)

+/kRXf(p 2) + m f X f s ( p 2 ) .

(2.8)

With the minimum number of field renormalization constants of our renormalization scheme where the 13 = +1 component is renormalized by the same renormalization constant 6ZL, the residue of this component is, in general, not equal to 1, yielding a finite wave function renormalization --3//1 1 (ml) 2 with ^

[/l (m 2) = ~$ZL+/71 (m12), Hl(m2)=.sf'(m~)

+ m i2l s f~ L ( m2, ) + ~vf'l(m~) + 2Ef;(m2)].

The NS Higgs contributions to the fermion self-energies are shown in Fig. 2. 2 For the notation used in this section see [7,8].

(2.9) (2.10)

D.-S. Shin~Nuclear Physics B 449 (1995) 69-79

73

(~) H°,h°,A °

H-

H°,h°,A°

H÷ l

d

d

d

d

u

d

u

u

u

u

d

u

(b)

H°,h°,A °

I-

I-

H-

I-

I-

H+

v~

I-

~l

I-

vf

Fig. 2. Additional Higgs contributions to the fermion self-energies: (a) self-energiesof quarks (b) self-energies of leptons. The diagrams are valid also for c and s quarks with the substitutions u ~ c, d --+ s and l states for e,/* and 7". Analogously, the external W boson gets a finite wave function renormalization 1 ^/ 2 ^/ 2 -~Zw(Mw) where ~w(Mw) is expressed by the derivative of the unrenormalized W self-energy Xw and the counter term given in [7,9]'

^/ 2 V/ ( i,42 ,~ Xw(Mw) = "w~'"wJ +

~ Z 2W ,

~z~ = --re(o)- 2 c~ s'z(°) c~ [~M~ sw M----~ + s ~ \ M ~

~a4~'~ a4~/

(2.11)

In our case of considering only the additional Higgs contributions £ r z ( 0 ) = 0, and the explicit expressions for (2.11) can be found in [ 10]. Using for the lowest order of the decay width F~ ") in (2.2), we have the following decay width in first order: FNS

=

F0(a) { 1 + 2ReFL + 28ZL

_

f I l (m~)

_

Xw(Mw)^,

2

+ G' 2ReFRGo + G-~2 2 R e H L o o + O3 2ReHRGo } - Fo('~) • (1 + a~s)

(2.12)

with

Go=2M2_m2

m~

(m~-m22) 2

a l = 6ml N2

m! [ _ l M G2 = -~w - 2w+m~+m ~

(m=, - m~= l

-2~

J

G3 = m2G2. 1721

Since Gl, G2 and G3 are suppressed strongly due to the small fermion masses, the terms FR, Hc and HR are negligible. Therefore, for the vertex corrections only the form factor FL is important.

74

D.-S. Shin~Nuclear Physics B 449 (1995) 69-79

If we use, instead of F0('~), the quantity F~ c") in (2.5) as the lowest order of the decay width, we have the decay width in so-called G~-scheme: FNS = F(o6~)(1 - ArNs + ~ s ) --/'0~")

( 1 + 6~[).

(2.13)

hrNs is the NS part of Ar which is the I-loop correction to the/z lifetime of (2.4) : G~

=

7"/'O~em x/~M 2

1 --(1 sin2 0w

+Ar)

(2.14)

with Ar = ArSM + ArNS.

(2.15)

Since the NS Higgs contributions to Ar in the vertex and box diagrams are completely negligible, we can restrict ArNs to the vector boson self-energies: hrNs =

(2.16)

2w(O) - SM 2 _ ~zW.

M2W

The explicit expressions can be found again in [ 10] and for the detailed discussion of ArSM we refer to the literature [ 11 ].

3. Numerical results and discussion

Among the 6 free parameters in the Higgs sector, we fixed the mass of our reference Higgs boson MHO to 100 GeV, since we are interested in the evaluation of the difference with respect to a given SM Higgs mass M H = MHO. In the concrete calculations we set ce =/3. With this choice of ~e, the diagrams involving the SM Higgs boson completely drop out of the entire non-standard part and hence we obtain the pure NS contributions which are not dependent on the SM. For the numerical evaluation of the virtual correction (2.12) we chose, according to the on-shell renormalization scheme, the quantities O~em,G~,, M z and m f a s fixed inputs [ 12]. On the other hand, since the input parameter M w is not yet known well experimentally, we determined it with help of the very well known Fermi constant G~ via the relation M~v\l-

M2 j = x/,-~Gu

1--ArsM--ArNs"

(3.1)

3.1. C o r r e c t i o n s to the d e c a y W + ~ e+l~e

The corrections to the decay W + --+ e+ve consist exclusively of those from the W propagator, since, due to the small mass of the electron, the vertex corrections are completely negligible. The relative corrections through the W propagator 6 w are given by (2.11): 8., = - X w^'( M ~ ) = - X w,( M w )2

- ~z w.

(3.2)

D.-S. Shin~Nuclear Physics B 449 (1995) 69-79

75

Table I Relative c o r r e c t i o n s to the d e c a y W + ---+ e + v e in the two H i g g s doublet model with tan fl = 0.7. The statements o f the con'ections are in [ % ] a n d the masses are given in units o f G e V

M ho

M Af~

MHI

100

300

100

300

100

o,

G u

MW

t~NS

•NS

300

80.472

- 3 . 0 5 x 10 - 2

--4.83 × 10 - 4

600

81.817"

- - 8 . 5 0 x 10°

--1.92 x 10 - 4

600

300

80.051

2.18 × 10°

--3.73 × 10 - 4

100

600

600

80.473

--3.47 x 10 - 2

- - 1 . 3 2 x 10 - 4

600

300

300

80.460

3.80 x 10 - 2

- 2 . 2 6 x 10 - 4

600

300

600

80.471

- 2 . 2 6 x 10 - 2

- 1 . 0 6 × 10 - 4

600

600

300

81.276"

- 4 . 8 1 x 10 °

- 1 . 5 4 x 10 - 4

600

600

600

80.467

- 1 . 8 0 x 10 - 4

- 7 . 8 9 x 10 - 5

These corrections get large contributions from the term ~Zw which is sensitive to mass splitting between the Higgs bosons. This form is valid in the a-scheme (2.12) where we use as input parameter, among others, the fine structure constant aem. However, for the calculation of the W decay width it is more suitable to use G~ as input parameter. In this G~-scheme (2.13), if we neglect the vertex corrections, the relative corrections are given exclusively by the W propagator: G/L ~---- --ArNs -- X^/w ( M~v) = - - Xw(O) - 6 M 2 / 2 ~NS M2w - X w ( M w)

(3.3)

which comes from (2.16) and (3.2). In this way, the term 6 Z w responsible for the large corrections drops out and the corrections are therefore small and stable. We list the evaluated relative corrections both in the a-scheme and in the G : s c h e m e together with the evaluated masses of the W boson in Table 1. We chose tan/3 = 0.73 and varied 3 Higgs masses. In the table, 6~s stands for the relative corrections in the a-scheme and 6NS for those in the Gu-scheme. On the other hand, the W boson mass is also an observable for which experimental measurements exist. The measured mass of the particle is at present given by 80.22 zk 0.26 GeV [12]. In the cases of the deviation of the evaluated W mass from the measured by more than two standard deviations we marked these with "*" . The corresponding parameter sets are therefore already excluded by Mw. The table shows that the corrections in the G~-scheme are very small in the whole parameter region and thus give no observable effect.

3 The results are h o w e v e r not d e p e n d e n t on the actual value o f tan/3.

D.-S. Shin~Nuclear Physics B 449 (1995) 69-79

76

Table 2 Relative corrections to the decay W+ ~ 7-+pr in the two Higgs doublet model. The statements of the corrections are in [%] and the masses are given in units of GeV

tan/3 0.7

70

M ho

M AO

Mr4 ~

a

Mw

G~

6NS

~NS

100

300

300

80.472

--3.05 X 10 -2

--4.65 X 10 - 4

100

300

600

81.817"

--8.50 × 10o

--1.84 × 10 -4

100

600

600

80.473

--3.47 X 10 -2

--1.22 X 10 - 4

600

300

300

80.460

3.80 × 10 - 2

--2.05 × 10 - 4

600

300

600

80.471

- 2 . 2 6 x 10 - 2

- 8 . 5 2 x 10 -5

600

600

600

80.467

- 1 . 5 7 × 10 -4

- 5 . 5 3 × 10 - 5

100

300

300

80.472

100

300

600

81.817"

1.50 x 10 -1 - 8 . 4 2 × 10°

1.80 × 10 -1 7.90 x 10 -2

100

600

600

80.473

6.15 x 10 -2

9.61 x 10 -2

600

300

300

80.460

2.51 × 10 - l

2.12 x 10 - l

600

300

600

80.471

1.89 × 10 -1

2.12 × 10 - l

600

600

600

80.467

2.36 × 10 -1

2.36 x 10 - l

Table 3 Relative corrections to the decay W+ --~ c~ in the two Higgs doublet model. The statements of the corrections are in [%] and the masses are given in units of GeV tan/3

M ho

M AO

Mn ~

0.7

100

300

100

300

70

3.2.

Gk~

Mw

&~s

6NS

300

80.472

--3.05 X 10 -2

--4.97 × 10 - 4

600

81.817"

--8.50 X 10°

--2.40 X 10 - 4

100

600

600

80.473

--3.47 X 10 - 2

--1.70 X 10 -4

600

300

300

80.460

3.80 × 10 - 2

- 2 . 3 3 x 10 - 4

600

300

600

80.471

- 2 . 2 6 x 10 - 2

- 1 . 1 9 × 10 - 4

600

600

600

80.467

- 1 . 8 7 × 10 - 4

- 8 . 5 8 × 10 -5

100

300

300

80.472

- 3 . 0 9 x 10 - 2

- 8 . 9 3 x 10 - 4

100

300

600

81.817"

-8.50 × to°

- 1 . 4 9 x 10 -3

100

600

600

80.473

- 3 . 6 0 × 10 -2

- 1 . 4 0 x 10 -3

600

300

300

80.460

3.75 x 10 - 2

- 6 . 8 7 × 10 - 4

600

300

600

80.471

- 2 . 3 0 x 10 -2

- 5 . 7 4 x 10 -4

600

600

600

80.467

- 7 . 3 4 × 10 -4

- 6 . 3 3 x 10 - 4

C o r r e c t i o n s to t h e d e c a y s W + - ~ 7"+uT a n d W + ~

With increasing fermion masses r e c t i o n s to t h e d e c a y s W + ~

c~

the Yukawa couplings also increase. Thus the cor-

r + u T a n d W + --~ c~ g e t , a p a r t f r o m t h e c o n t r i b u t i o n s f r o m

the W propagator, also contributions from the vertex. On the other hand, in the 2HDM w e h a v e t h e p o s s i b i l i t y to i n c r e a s e t h e Y u k a w a c o u p l i n g s w i t h h e l p o f t a n ft. S i n c e t h e

D.-S. Shin/Nuclear Physics B 449 (1995) 69-79

77

Table 4 Relative COlTections to the decay W + ~ r+vr in the minimal supersymmetric standard model. The statements of the corrections are in [%] and the masses are given in units of G e V tan/3 2

30

MAO

Mw

a 8NS

G~

8NS

20

80.538

--3.878 X 10 -1

- - I . 5 1 2 X 10 . 2

300

80.433

--2.061 X 10 - I

--4.641 X 10 . 3

600

80.384

--2.120 X 10 - I

--4.479 X 10 . 3

1000

80.344

--2.129 X 10 - l

--4.466 × 10 . 3

20

80.539

- 3 . 5 6 3 x 10 - j

1.322 x 10 . 2

85

80.488

- 5 . 0 9 6 x 10 - 2

4.232 x 10 . 2

600

80.378

- 1 . 5 6 6 x 10 -1

1.603 x 10 - 2

1000

80.338

- 1 . 7 2 0 x 10 - I

3.077 x 10 - 3

Table 5 Relative CO~Tections to the decay W + ---* cS in the minimal supersymmetric standard model. The statements of the corrections are in [% t and the masses are given in units of G e V tan ,8 2

30

M Ao

Mw

a

6NS

G,,,

6NS

20

80.538

--3.878 X 10 -1

--1.520 X 10 - 2

300

80.433

--2.062 X 10 - I

--4.770 X 10 - 3

600

80.384

- 2 . 1 2 1 x 10 - l

- 4 . 5 4 4 x 10 - 3

1000

80.344

- 2 . 1 2 9 x 10 -1

- 4 . 4 7 6 x l0 - 3

20

80.539

- 3 . 8 3 6 x 10 -1

- 1 . 4 0 8 x 10 . 2

300

80.427

- 1 . 6 2 1 x 10 - I

- 1 . 7 2 7 x 10 - 3

600

80.378

- I . 7 4 2 x 10 -1

- I . 5 4 5 × 10 - 3

1000

80.338

- I . 7 6 6 x 10 -1

- 1 . 5 5 0 x 10 - 3

Yukawa couplings on up-type ferrnions contain typically the term m---u- x ~ m W

u2 '

the situa-

tion vl >> v2 (tan/3 ~ 0) leads to the raising of the couplings on the up-type fermions, while the Yukawa couplings on down-type fermions contain the term m_& x v2 which mw Vl ' leads to the raising of the couplings for v2 >> vt (tan/3 >> 0). We took 0.7 < tan/3 < 70 as the allowed region for tan/3 and for the actual calculations chose the two limiting values tan/3 = 0.7, 70. The restriction tan/3 > 0.7 comes fi'om the K - K, B - / ~ mixing and the decay of the B meson if the charged Higgs bosons are heavy [ 13]. On the other hand, we can increase the Yukawa couplings on the downtype fermions to any size by increasing tan/3, but with tan/3 = 70 we already arrive at the maximal limit of the couplings for the applicability of perturbation theory. For example, for this value of tan/3 we get from the quadratic coupling of the b quark on the A ° boson

(

_9_)2 2 sin Ow tan/3~Vlw/

e2

200--.

4¢r

(3.4)

D.-S. Shin~Nuclear Physics B 449 (1995) 69-79

78

We list the evaluated relative corrections to the decays W + ~ ~-+uz and W + ~ cg as well as the evaluated masses of the W boson in Tables 2 and 3. The tables show that for tan fl = 70 we get substantially larger corrections, but their increases with help of tan/3 are not sufficient to raise the effects in percent region. The entire values remain in the region of a few per mil and are therefore not significant. The experimental uncertainty of the quark masses c (1.0 ~ 1.6 GeV) and s (100 200 MeV) does not play any role for the corrections. Even for tan/3 = 70 the change of the quantities by variation of the masses within the unknown regions is completely negligible. We calculated also the NS Higgs contributions to the decay width in MSSM where only two of the 6 free parameters in 2HDM are independent [2]. In the actual calculation we chose Mao and tan/3 as free parameters and evaluated the corrections for 20 GeV <

MAo _< 1 TeV and tan/9 = 2, 30. We list the results in Tables 4 and 5. As we see in the tables, the maximal corrections in this model are for the decay W + -~ T+v~ given by 4 . 2 3 . 10-2% for MAo -- 85 GeV, tan/3 = 30 and for the decay W + --~ c$ by - 1 . 5 2 . 1 0 - 2 % for MAo = 20 GeV, tariff = 2. Radiative corrections to the MSSM Higgs sector were not considered here. They are however also irrelevant in view of the small effects. From our calculations of the radiative corrections to the W decay width in the NS part of the Higgs sector we conclude that the corrections give no observable effects in the whole parameter region. The additional corrections lie below a percent, similarly to the minimal standard model. In the 2HDM, as in the SM, the decay width of the W boson is thus practically not dependent on the Higgs sector.

Acknowledgement I thank W. Hollik for suggesting this work and many useful discussions as well as P. Weisz for reading this paper.

References [ 1] W. Hollik, presented at XVI Int. Symp. on Lepton-PhotonInteractions,Comell, August 1993. [12] J.E Gunion, H.E. Haber, G. Kane and S. Dawson,The Higgs Hunter's Guide (Addison-Wesley,Reading, MA, 1990). [3] K.G. Chetyrkin, A.L. Kataev and EV. Tkachov, Phys. Lett. B 85 (1979) 277; M. Dine and J. Sapirstein, Phys. Rev. Lett. 43 (1979) 668; W. Celmaster and R. Gonsalves,Phys. Rev. Lett. 44 (1980) 560; S.G. Gorinshny,A.L Kataev and S.A. Latin, Phys. Lett. B 259 (1991) 144; L.R. Surguladzeand M.A. Samuel, Phys. Rev. Lett. 66 (1991) 560. [4] J. Jersak, E. Laerman and EM. Zerwas, Phys. Rev. D 25 (1980) 1218; T.H. Chang, K.J.E Gaemers and W.L. van Neerven,Nucl. Phys. B 202 (1982) 407; A. Djouadi, Z. Phys. C 39 (1988) 561. [5] A. Denner and T. Sack, Z. Phys. C - Particles and Fields 46 (1990) 653. [61 W. Hollik, Precision Tests of the Electroweak Theory, Lectures given at the CERN-JINR School of Physics 1989, CERN-TH.5667/90.

D.-S. Shin~Nuclear Physics B 449 (1995) 69-79

79

171 M. Boehm, W. Hollik and H. Spiesberger, Fortschr. Phys. 34 (1986) 687. 181 w. Hollik, Z. Phys. C - Particles and Fields 32 (1986) 291. [9] W. Hollik, Fortschr. Phys. 38 (1990) 165. [ 101 W. Hollik, Z. Phys. C - Particles and Fields 37 (1988) 569. I 11 I G. Burgers and E Jegerlehner, in Z physics at LEP 1, ed. G. Altarelli, R. Kleiss and C. Verzegnassi, Vol. 1, p. 55 xx. 112] Review of Particle Properties: Phys. Rev. D - Particles and Fields, Vol. 50 (August 1994). 1131 J.E Gunion and B. Grzadkowski, Phys. Lett. B 243 (1990) 301; A.J. Buras, P. Krawczyk, M.E. Lautenbacher and C. Salazar, Nucl. Phys. B 337 (1990) 284; V. Barger, J.L. Hewett and R.J.N. Phillips, Phys. Rev. D 41 (1990) 3421; D. Cocolicchio and J.-R. Cudell, Phys. Lett. B 245 (1990) 591.