QCD corrections, virtual heavy quark effects and electroweak precision measurements

Volume 214, number 4

PHYSICSLETTERSB

1 December 1988

QCD CORRECTIONS, VIRTUAL HEAVY QUARK EFFECTS AND ELECTROWEAK P R E C I S I O N M E A S U R E M E N T S

B.A. KNIEHL, J.H. K U H N and R.G. STUART Max-Planck-Institut J~r Physik und Astrophysik - Werner-Heisenberg-Institut fiir Physik, P.O. Box 40 12 12, D-8000 Munich, Fed. Rep. Germany

Received 4 July 1988

QCD correctionsto virtual heavyquark effects on electroweak parameters are calculated, which may affect planned precision measurements at SLC and LEP. The influence of toponium and Tb resonances is incorporated as well as the proper threshold behaviour of the imaginarypart of the vacuum polarization function. The shift of the W-bosonmass from these corrections and their influence on the polarization asymmetryare calculated and comparedto the envisagedexperimental precision.

1. Introduction Precision tests of the standard model of electroweak interactions will be one of the central themes at the forthcoming generation of e÷e - colliders SLC and LEP. The determination of the gauge boson masses Mw and Mz together with the measurement of various asymmetries will probe the theory at the one-loop level. These experiments will therefore be sensitive to the top quark mass. In particular they could pin down its value to about + 20 GeV in the high mass range of 100-200 GeV, a region not easily accessible by direct production experiments in the near future. Conversely, once production and decay of top mesons has been observed and their mass been determined, precision measurements could be sensitive to tiny effects from physics beyond the standard model and will help to fix, at least, a range for the mass of the Higgs boson. It has often been stressed that a potential limitation for the interpretation of precision measurements follows from inherent uncertainties in the hadronic vacuum polarization. The light quarks' contribution is related to experimental results through dispersion relations, and an uncertainty of (1-2) × 10 - 3 in Ar has been quoted [ 1 ] from this source. However, as we will see below also the hadronic correction to the vacuum polarization from the tb doublet cannot be neglected. This aspect has received renewed attention [ 2-4 ] with the lower limit on the top mass increasing steadily over the recent years [ 5 ], and the size of the aforementioned corrections growing correspondingly with the leading term proportional to m t2/ M w .2 Most calculations of the impact of heavy quarks neglect the influence of QCD corrections. However, once their effects become truely large (e.g. 0.07 in Ar for m t ~ 200 GeV) QCD corrections can no longer be neglected in this case also. Their influence is therefore studied in this paper. To evaluate QCD effects on heavy quarks' vacuum polarization, it is convenient to calculate first the imaginary part and, in a second step, the real part through dispersion relations - either analytically or numerically. The main task thus consists in a proper modelling of Im/-/u, the imaginary parts of the various polarization functions. Calculations where the perturbative O (o~s) result for Im/-/0 is used from threshold up to high q2 and which employ the choice of a constant value of ors(/~2) can - at least in principle - be performed analytically. However, several problems arise: ( 1 ) The choice of the scale/~2 in ors is uncertain. This is particularly severe since three potentially drastically different mass scales arise and, without further investigation, #2 could be expected of order mb2, M 2 or m 2. (2) The contributions from bb, tb and tt resonances (Y, Tb and 0) are absent. 0370-2693/88/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division )

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(3) The relation between the quark mass mt which appears in perturbative expressions and physical masses like the mass of the T-meson mT or of the toponium ground state is ambiguous, and this ambiguity is reflected in the different treatment of this problem in refs. [ 2-4 ]. In this paper a different approach is advocated. Im Hij, which (at least in principle could be determined experimentally, is constructed in such a way that it simulates this observable as close as possible. The real part is then derived numerically through dispersion relations. Specifically, narrow quarkonium resonances below threshold are incorporated with parameters calculated from the Richardson potential [ 6 ]. In the region very close ( + 0.5 GeV) to threshold Im H o.can be parametrized either by densely spaced resonances or alternatively by the continuum contribution as derived in perturbation theory, provided the scale of the running coupling constant is chosen as the quarks' relative momentum such that as increases towards the threshold to simulate the onset of confinement and, furthermore, m t is chosen as the constituent quark mass approximately 400 MeV less than mr [7 ]. (as does not exceed 0.20 with this choice even for X/~=2mx. ) This duality and the requirement of continuity justify the use of perturbation theory in a region where the QCD corrections are huge compared to the Born term. A few GeV above threshold QCD corrections are already small enough to justify a perturbative treatment without reference to duality. After the numerical evaluation of the real part o f H Uthe impact of QCD corrections on various observables is studied, in particular the contribution to Ar (and thus to the prediction of Mw as a function of a, Mz, G,, mH and m t) and to the polarization asymmetry ALR. Furthermore, QCD corrected constraints for the top quark mass from a precise determination of the Z-boson mass combined with existing neutral current data are evaluated. These results are compared with related work of refs. [2-4,8 ]. 2. The imaginary part of the vacuum polarization The four vacuum polarization functions Hww, Hzz, HAA and HzA which describe the W, Z and photon selfenergies and Z-A mixing are important for the subsequent discussion. The contributions from a tb doublet can be written in terms of vector and axial current polarization function H v and H A as follows: Hww(q 2, ml, m 2 ) = ~g2[IIV(q2, ml, m2)+HA(q 2, ml, m2)] ,

[v~IIV(q 2, m3 +IIA(q 2, m31,

//zz (q 2, m~, m2)= ~ i=l 2

HAA(q 2, m,, m2)=g2s2 ~, Q2HV(q2, mi) , i=l

g 2s //ZA (q 2, ml, m2) = ~c

QiviHV(q 2, m i ) ,

( 1)

i=l

where v~ = 211 - 4s2Qi. Here s and c denote the sin and cos of the weak mixing angle. The second mass argument has been dropped when the internal quarks are the same. To fLXthe notation, we list H v and H A for free quarks

IIV'A(q 2, ml, rn2)= ~

2q 2

( m 2l - m 22) 2q 2

~

( 1

[q2--~(ml -T- m2)~l (A--In mira2)

2-('~

] T'n~m2

+m 2.q2.x/~

mE_m21nml]l q2 m2/d '

where 2 - 2 ( q 2, ml2, m E) = [q2_ (ml +mE) E] [q2_ (ml -- m2)2 ]. 622

(2)

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2.1. Continuum region To fix the imaginary parts of the various Ha, it is sufficient to specify the imaginary parts o f H v and H A. The QCD corrections to order as have been derived in refs. [ 2,9 ]. We shall base our formulae on Im H v and Im H A from ref. [2 ], eq. 3.3 ~1. The choice of the scale in the running coupling constant ors in a priori not fixed from this calculation. To match simultaneously the behaviour for s >> m 2 in the high energy region (where one expects /t 2 of order s) and the onset of confinement with the corresponding increase of as in the threshold region [ 7 ], the scale in as is chosen proportional to the momentum (4p 2 =)./s): 12It

1

as(s, m 2, m 2 ) = 33--2NF(S) ln[2(s, m~, m2) /SA~cD] "

(3)

The constituent quark mass mt--- roT-- 400 MeV is closely related to the mass of the top meson. With this ansatz I m / / o is approximately dual to the contribution from the densely spaced resonances close to (but above and below) threshold. The numerical results are thus insensitive to the precise location of the transition between continuum and resonance region.

2.2. Resonances For quarkonium resonances Im/-/v can be approximated by a series of&functions

Im//V(s, m 2, m 2) =

4 2 • E zcMnf2vS(s-Mn)

(4)

n

fv and the precise locations of the S-wave resonances have been calculated as a function of mt in the nonrelativi stic quark model using the Richardson potential. The masses of resonances are then given by M0 = 2mt + Eainding for toponium and by MTb = mt + mb + EBinding for Tb, and mr = mt + 400 MeV [ 7 ] in accordance with the treatment of the continuum. There are five narrow Tb resonances below TB threshold and 12 (m t = 50 GeV) to about 25 ( m t = 250 GeV) 0 resonances below TT threshold. Including QCD corrections the vector coupling constant fv is given by f 2v

Arc I R ( 0 ) I 2 [ -

It

M 3

1-

16as(

1

3m~-m__________~2 in m_2~)l

(5)

8 ml +mE

where R (0) denotes the wavefunction at the origin. The QCD correction for the case of unequal masses (which has not been calculated elsewhere) is obtained from the threshold behaviour of the perturbative O (ors) correction to the imaginary part, Im/-/V(s, mr, m2)= Im/-/~'(s, ml, mE) where

1(~ 1 +

Vr ~-'~2

Ors

(1

-- ~4 (1

mEln~22)+O(v2)] 83 m mll - +mE

'

(6)

--~2)~/2(s'm2,m22) S

is the relative velocity. Up to the colour factor the first term is identical with Sommerfeld's Coulomb rescattering correction [ 10 ]. In the treatment of bound state annihilation this term is absorbed in the wavefunction. The remainder is attributed to the exchange of transverse gluons and interpreted as genuine perturbative vertex correction. Compared to (tt) resonances the relative weight of (tb) bound states is small. The relevant mass for ~ We have checked the agreement between ref. [ 2 ] and ref. [ 9 ].

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the evaluation of Rt6(0) is the reduced mass mr = mtmb/( mt + mb). For phenomenologically acceptable potentials [R (0) 12/m 2 is empirically approximately constant. The contribution from Tb resonances to Hww is proportional to IR (0) I2 / (mt + mb )2 and thus suppressed by a factor ~ (4mb/mt )2 compared to the corresponding contributions of (tt) and (b6) resonances to//zz and//AA. Nevertheless they are included in this treatment. The axial vector current couples to P-wave qCt resonances only. This leads to an additional suppression oc < v2/ c2> for nonrelativistic bound states, corresponding to the strong suppression of the continuum part of//Aoc f13 close to threshold. 1+ + resonances are therefore not included.

3. Dispersion relations As stated in the introduction, the real part of H o is obtained from the imaginary part through a numerical evaluation of dispersion relations. The proper subtractions are derived in ref. [ 2 ] and lead to the following expressions: A2

Re HV,A(q2, m~,m2,A) = -~I p

~

A2

dslmHV'A(S, ml,m2) s--q 2

~

1 iL= 1 ; _ ~ Im lIV(s, m~) ,

(rot + m 2 ) 2

(7)

4m 2

where an ultraviolet cutoffA has been introduced. With as decreasing oc 1/In s (or 4p 2) the divergent part reads N¢ _ HV'k(q 2, ml, m 2 , A ) = (33_2Nv)it 2 3(ml T m2)2 l n _A~CD

+ q2--9(ml:T-mz)Z+3(ml~m2) mlln m----~l - T - m 2 1 n - m2 / d

AQcnJ

(8)

whereas for constant tZs /-/V,A(q2, ml, m2,A) A2 ( ~-~) A2 ] Arc as [ 3 ( m t ~ m 2 ) 2 1 n 2 + q2--9(mlT-m2)2--3(m21--m~)ln In , - 121t2 rt ml m2 ml m2

(9)

These divergences cancel in the expressions for the observables discussed below.

4. Measurable quantities It has become customary to express the magnitude of radiative corrections in the relation between Mw, Mz, G~ and a through Ar:. Mw2 = M z2' ~I ( I + N / I

4trot

-x/~m2G~,(1 --Ar)) •

(10)

The influence of a heavy quark doublet on Ar can be expressed in terms of//ij,

c2[1-1ww(M 2)

Ar= ~-~Ke~, ~

624

Hzz(M2)) 1 ] M2z +//~,A(0)+ M---~w[Hww(0)-Re//ww(M2w) "

(11)

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The leading contribution from heavy quarks increases proportional to m 2 and is related to the p-parameter Ap(0) =

~zz(0) M2

/lww(0)

M2

(12)

The validity o f this approximation will be discussed below. Fig. 1a separately displays the continuum part o f the Q C D correction to Artb as a function o f mr. These hadronic corrections to Ar are thus larger than 2 X 10- 3 for top masses beyond 70 GeV and thus more important than the uncertainty from the light quark v a c u u m polarization. For comparison also the result for a constant as (M2z) = 0.13 is shown, and for a choice o f the scale /£, where the quark m o m e n t u m in eq. (3) is replaced by the energy, such that #2 = s. The difference between the three curves is significant for top masses beyond 100 GeV. It has been checked that the difference between the curves with/z 2 chosen as 4p 2 and s respectively cannot be traced to a different treatment very close to threshold but receives significant contributions from regions in s extending far above threshold. The different behaviour o f the three curves for Ar (and similarly Ap) can be qualitatively understood. The contribution to Ar is dominated by the region q2 close to 4m 2 in the dispersive integrals such that the ratio between the curves with constant ors ( M E ) and running ors(q2) is given approximately by as ( M E) /ors (4m 2 ). The choice ors (4p 2) enhances the top contribution in the threshold region. As far as the integrals are concerned this is equivalent to an effective lowering o f the top threshold and thus reduces the magnitude o f Artb. The requirement for continuity close to threshold and o f an R-value that increases with increasing s favours the choice l / 2 = 4 f f 2 A firm statement, however, could only come from a full calculation to O ( ot2). For m t < 100 GeV the differences are hardly noticeable; in the high mass region the difference between as (s) and a~ ( 4/I 2 ) amounts to I X 1 0 -3. Also shown in fig. la is the leading t e r m -c2/$2~ for constant ~2 and for running ors. For top quark masses below 100 GeV the approximation by the leading term fails badly ,3. The sign o f the correction agrees with ref. [ 3 ] but is opposite to the sign obtained in ref. [4 ]. The spikes at mt ~ 46 GeV and ~ 77 GeV originate from the sharp rise o f Im//~j at the Ti" and TB threshold which leads to a logarithmic singularity in the real part, as ,2 In this case agreement is found with ref. [ 3 ] where 1-- ~ ( 2 ~ 2+ 6 ) cts/n is obtained as correction factor for Ap. ~3 For the Born term the same observation has been made (see e.g. ref. [4] ). ,"2 -~-',/~p(O)

.....

:

........

:

------

: -S--T~0(fl}

Ar

(constant

~,)

(constant

~,)

.....

C2

........ : - 2 ~ r ........ : - ~ A p { 0 l :

Ar

:

(running

~.(s}l

(running (running

~.(sll ~,(4~2}} ~,(4~2]}

[running

:

C2 -s-'-~Ap [ O } Ar

,008

• 006

• 004

/,;;" /

...,~ r,~" ,~

'i'!'l'.l!t!l'!'l'l!l!r'!'l'l'J'i'llllllllll 50 100 150

..

a

°"

b

/// ,,-';.J jI o,,f ,// $1,, "°/ o,

ooo

o.°

200

I In t 250

(G-eg}

-.oo15

! !l'l'l'l'!'l'l!l'l'l'~'l'l'~!!'l'l!lll."! 50 l O0 150

IIIt 2O0

(GeV}

250

Fig. 1. QCD corrections for the tb doublet contribution to Ar and to the leading term ocAp( 0 ). (a) Continuum contribution for different choices of ors (b) Resonance contribution. 625

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Table 1 Continuum contribution of the tb doublet to Ar and ALRfor different choices of oq and full prediction for Ar and ALR. rnt [GeV]

8Arx 103

30 60 90 120 150 180 210 240

~ALRX 10 3

ArX 102

ALR

O (otot,(s))

0 (aot,(4p2))

O(ota~(s))

O(o~0t,(4p2) )

O(ota~(s))

0 (oto~(4p2))

0 (or (4p 2)

0.23 1.62 1.65 1.93 2.50 3.24 4.13 5.15

0.26 1.54 1.55 1.68 2.07 2.62 3.29 4.07

-0.57 -0.99 - 1.24 - 1.51 -2.15 -2.78 - 3.53 - 4.37

-0.61 - 1.01 - 1.17 - 1.45 - 1.83 -2.31 - 2.88 - 3.53

7.02 7.25 6.13 5.12 3.99 2.65 0.99 - 1.07

7.02 7.24 6.12 5.09 3.94 2.56 0.86 - 1.26

-0.160 -0.162 -0.168 -0.173 -0.183 -0.193 - 0.204 - 0.216

observed also in ref. [ 2 ]. For a selected set of top masses the c o n t i n u u m c o n t r i b u t i o n is also listed in table 1. The contributions from t o p o n i u m a n d Tb resonances to Ar as derived from eq. (7) a n d eq. (4) seem to diverge. However, this singularity is easily removed: The leading term in Ar for M0 ~ Mz reads 2 BAr= ( ~ _ ~ _ ~ )fv

2l 2• M7-M0

(13)

One may invert eq. (10) such that M z is given as a function of or, G~, Mw, a n d Ar. For the resulting shift in M z one then finds 6M 2

s2

M 2 _ 7~ f A r ,

(14)

which corresponds exactly to the mass shift from 0 - Z mixing obtained e.g. in ref. [ 11 ] when the singular denominator is replaced by [ ( M E - M z 2) 2+ F z2M z2] 1 . A corresponding argument applies to T b - W mixing. The contribution from narrow resonances alone is shown in fig. l b a n d the size of the resonant structure at mt ~ 46 GeV agrees with the shift of Mz derived in ref. [ 11 ], which a m o u n t s to 8 MeV at most. The full Ar which includes light quarks, leptons a n d bosons based on the formulae from refs. [ 12,13 ] has been evaluated for M z = 9 1 . 9 GeV, m H = 100 GeV a n d Mw chosen such that eq. (10) is fulfilled throughout. The result is displayed in fig. 2 and, for a selected set of top mass values, also listed in table I. For light quarks (u, d, c, s ) the q u a n t i t y / / k g (0) is calculated everywhere through the relation -

-

r&A (0) = / l k A (~) -- [HkA(~) - - H k A ( 0 ) ] •

Ar

• 06

.....

:

0(~]

.......

:

O[e, ax,( a 1 1

i

:

O[c~x.(4~ 2) )

• 04

.02

o.

- . 02

~l!!!t!l!L!l!!!l!t!l!l!!!l!~,l,l!!!~!l!l!l!~ 111t (GeV1 50

626

100

I50

200

250

Fig. 2. Complete prediction for Ar without and including (continuum and resonance) QCD corrections for two choices ofoq.

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02 is chosen such that/-/~,A (02 ) can be reliably calculated using perturbation theory, including QCD corrections through the factor 1 + c%/rt and the quark mass is set to zero. The same factor is used to correct the light quark contributions to the remaining/-/o. The quantity in square brackets is obtained from the experimental data for a(e+e - ~hadrons). ALR, the asymmetry in the e + e - annihilation cross section with right- or left-handed polarized beams, together with the measurement of Mw will be one of the most sensitive quantities for physics beyond the standard model or, alternatively, it will pin down the mass of the top quark. The influence of the tb doublet on ALR is given by

C2S4 I ~ZZ Re ( 2----~S 2 I I z A ( M z ) - I I z z ( M z ) ) + / - / k A ( 0 ) + /-/~2w( 1

~ALR=--64 -(1- -+V~) --'-~

c -s

2

2

0).

1•

(15)

The corresponding hadronic corrections are shown in fig. 3.For comparison the leading contribution from the p-parameter is also shown,

C2S4 8ALR--~ s+~o 64 (1 +V2) e 2 Ap(0) "

(16)

The same conventions are employed as in fig. 1 and, mutatis mutandis, the same discussion applies. The difference in the prediction of 8ALR between different choices of the scale #2 in c~, becomes relevant only for m t above ~ 150 GeV. For mt = 250 GeV the difference between H2= 4p 2 and s is 1 × 10-3 and thus below the anticipated experimental uncertainty of + 3 × 10- a. The correction itself, however, is for large mt well comparable to this uncertainty. The approximation by the leading term oc m 2 [ 3 ] differs from the true result by more than a factor of two for m t below 150 GeV. The complete prediction for ALR including the O (or) results from ref. [ 13 ] are displayed in fig. 3b and in table 1. The size of the hadronic corrections is again comparable to the anticipated experimental errors. Finally, the impact of QCD corrections is investigated on the upper limit on mt following the method of ref. [8]. As a starting point one uses the result for sin20= 1 - M 2 / M 2 = 0.233 + 0.010 (90% CL, experimental and

~4C2g4

- -

_

• ^

(constant (constant : 64c2s4 'tAp.t : l(+ - - ~ ' ~ x ~ A p ( 0 ) ( r u n n l n G ( runnin G : 2 • aA,.~ 64c~s" A ( 0 : [l+-~TQw~v P ) ( runnin G : 6A,., [ runnln G : l( + - ~ p t

u )

c<.) oz.) c<.(sl ) c~.( s ) ) c<.(4~ 2] ) c<,[4~'] )

.....

"]~po I

O.

: D(c<) : O(oox.) mz-91SGeV mK-IOOGeV

on Z °

resonance

-.IG -.001

a

-.17

-. 002

-.18

-.003

-.19

-.

004

-+20 -.21

-+005

-.

006

b

lllllllt llllllll[llllllllllllllllllt lllll N mL 50

100

150

200

250

(GeVI

i'i'.l 50

'l'.!!l¢l'l'l'l't'l'.]'.l!!! 100

tS0

r~t

lll'l.l~r 200

( ~'~V

]

250

Fig. 3. (a) QCD corrections for the tb doublet contribution to ALR and to the leading term ocz~o(0) (continuum contribution) for different choices of 0%. (b) Complete prediction for ALR without and including (continuum and resonance) QCD corrections for two choices ofc~s.

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mt (GeVI

.............. : O(e~]

m.-

250

_ ....

:

O[o<]

ran-

IOOGeV

_ .......

:

D(o<)

mn-i

O00GeV

~-.

_

.

.

IOGeV

o 1;, ........ ~1........ , ......... , . . . . . . . . . . . . . . mz IGeV] 89

89.5

90

90.5

91

91.5

92

92.5

93

93.5

1 December 1988

Fig. 4. Prediction from neutrino scattering for the allowed range of mr as a function ofmz without (tort= 10 GeV, 100 GeV, 1000 GeV) and including ( mn = 100 GeV) QCD corrections.

theoretlcal error a d d e d quadratically) d e r i v e d from n e u t r i n o - n u c l e o n scattering which includes the radiative corrections with m r = 30 G e V a n d m H = 100 GeV. The p r e d i c t e d range o f M z as a function o f mr is shown in fig. 4. F o r c o m p a r i s o n also the predictions without Q C D corrections are displayed for mH = 10, 100 a n d 1000 GeV. The influence o f Q C D effects on the present b o u n d s on mt is negligible. This will not change even after a precise d e t e r m i n a t i o n o f Mz, since the error is in this case d o m i n a t e d by the experimental error from neutrino scattering which will not d i m i n i s h in the foreseeable future.

5. Summary H a d r o n i c corrections to the v a c u u m polarization from the tb doublet with large mass splitting lead to contributions to Ar which are significantly larger than the uncertainty from light quarks. F o r mt above 150 G e V imp o r t a n t differences arise between the evaluation with constant a n d with running a s a n d even for the choices p : = 4/72 or s. A rigorous distinction between the latter two possibilities could only arise from a calculation to O ( a 2). The t r e a t m e n t a d v o c a t e d in this p a p e r includes contributions from q u a r k o n i u m 0 a n d Tb resonances below threshold a n d incorporates a s m o o t h threshold behaviour. The quark mass mt used in this context is directly related to the mass o f the top meson (mt = m T - 0.4 G e V ) . The change in Ar o f about ( 2 - 5 ) X 1 0 - 3 for mt between 60 and 250 G e V is reflected in a change in the p r e d i c t i o n for M w o f about 2 5 - 6 0 MeV, close to the expected experimental accuracy o f 100 MeV. Keeping only the quadratic t e r m in mt is i n a d e q u a t e for mt below 150 GeV. Also the polarization a s y m m e t r y is affected by these corrections a n d the change o f ~ 1.5 × 10-3 for mt = 200 G e V is again not far below the a n t i c i p a t e d experimental error o f 3 X 10-3. Limits on mt based on a d e t e r m i n a t i o n o f the Z-mass together with neutrino scattering experiments are practically unaffected.

Acknowledgement The authors thank A. D j o u a d i for helpful c o m m e n t s on the work o f ref. [ 9 ].

References [ 1] F. Jegedehner, Z. Phys. C 32 (1986) 195, and references therein; B.W. Lynn, G. Penso and C. Verzegnassi, Phys. Rev. D 35 ( 1987 ) 42. [ 2 ] T.H. Chang, K.J.F. Gaemers and W.L. van Neerven, Nucl. Phys. B 202 (1982) 407. 628

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[3] A. Djoudadi, preprint PM/87-53; A. Djouadi and C. Verzegnassi, Phys. Lett. B 195 (1987) 265. [ 4 ] F. Jegerlehner, preprint Bi-TP 87/16. [ 5 ] F. Takasaki, Intern. Symp. on Lepton and photon interactions at high energies; UA1 Collab., C. Albajar et al., Z. Phys. C 37 ( 1988 ) 505. [6] J.L. Richardson, Phys. Lett. B 82 (1979) 272. [7] S. Giisken, J.H. Kiihn and P.M. Zerwas, Phys. Lett. B 155 (1985) 185. [8] P. Langacker, W.J. Marciano and A. Sirlin, Phys. Rev. D 36 (1987) 2191. [9] L.J. Reinders, H. Rubinstein and S. Yazaki, Phys. Rep. 127 (1985) 1. [ 10] A. Sommerfeld, Atombau und Spektrallinien, Bd. 2 (Vieweg, Braunschweig, 1939 ). [ 11 ] J.H. Kfihn and P.M. Zerwas, Phys. Rep. 167 (1988) 321. [ 12] A. Sirlin, Phys. Rev. D 22 (1980) 971. [ 13 ] B.W. Lynn and R.G. Stuart, Nucl. Phys. B 253 ( 1985 ) 216.

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