Radiation of light fermions in heavy fermion production

NUCLEAR PHYSICS B ELSEVIER

Nuclear Physics B 452 (1995) 173-187

Radiation of light fermions in heavy fermion production * A.H. Hoang, J.H. Ktihn, T. T e u b n e r 1 Institut fiir Theoretische Teilchenphysik, UniversittitKarlsruhe, D-76128 Karlsruhe, Germany Received 15 May 1995; accepted 15 June 1995

Abstract The rate for the production of a pair of massive fermions in e+e - annihilation plus real or virtual radiation of a pair of massless fermions is calculated analytically. The contributions for real and virtual radiation are displayed separately. The asymptotic behaviour close to threshold and for high energies is given in a compact form. These approximations provide arguments for the appropriate choice of the scale in the O ( a ) result, such that no large logarithms remain in the final answer.

1. I n t r o d u c t i o n QED reactions leading to four-fermion final states f : f l f 2 f 2 with fermion masses ml and m2 have been considered in the literature a long time ago [ 1 ] and the final result was at that time expressed in the form o f a two-dimensional integral. The cross sections were calculated for leptons in the context o f QED. However, it is fairly evident that a large part o f the results can easily be transferred to the corresponding QCD reactions. The analogy becomes even closer when considering the mass assignments o f interest for actual physical reactions: either equal masses rnl = m2 or alternatively ml >> m2. This mass hierarchy applies e q u a l l y well to leptons and to quarks and hence to QCD calculations and will be exploited in this work. Apart o f the "exclusive" channel, where all four fermions are detected, also the inclusive rate for f l f l + anything is o f practical interest. To O ( a ) this calculation has WWork supported by BMFr 056 KA 93E 1E-mail: [email protected] 0550-3213/95/$09.50 @ 1995 Elsevier Science B.V. All rights reserved SSDI 0550-3213 (95) 00308-8

A.H. Hoang et al./Nuclear Physics B 452 (1995) 173-187

174

Fig. 1. Characteristic Feynman diagrams describing the production of a pair of massive and a (real or virtual) pair of massless fermions.

been presented in the classic book by Schwinger [2]. Higher-order results are available in the context of QCD in the limit of vanishing fermion mass [3,4] or including mass corrections through an expansion in m2/s. Terms of order m2/s where calculated up to ces3 [5], the m4/s2 terms up to oz2 [6]. This expansion is, however, inadequate close to threshold. In this kinematical region the full calculation up to order a~2 would be required. This is particularly desirable in view of the fact that the ambiguity in the scale /z 2 in ors leads to a large uncertainty in the leading-order correction. Close to threshold both the mass and the three-momentum of the produced fermions seem to be reasonable choices f o r / z 2, giving rise, however, to drastically different predictions. The analytical calculation of the cross section, including the full mass and energy dependence and all (real and virtual) gauge boson contributions seems like a difficult task. However, the subclass of diagrams involving the real and virtual radiation of light fermions is more accessible. In this paper the result for both real and virtual radiation will be presented for arbitrary ml2 and s, in the limit ml2 >> m22. Combining the two contributions, one arrives at a result which still exhibits mass singularities of the form ln(m2/s). They can be removed by adopting the MS scheme. Normalizing the coupling constant at scale br2 = s eliminates all large logarithms, at least away from the threshold region. This provides the first step in the calculation of corrections to order a 2s. The relatively compact analytical result can then be studied in the limit close to threshold as well as in the high energy region.

2. R e a l r a d i a t i o n

For definiteness the cross section for the production of the flflf2f2 final state in e+e - annihilation through a virtual photon will be considered, normalized relatively to the point cross section. The result is evidently also applicable to Z decays through the vector current. The relevant Feynman amplitudes can be derived from the four-fermion cuts of the diagrams with two closed fermion loops, with representative examples depicted in Fig. 1. The rate can be expressed by a two-dimensional integral [ 1 ]:

Rfd,:~A _ o-f,?,:,.:~- : oL.,(~)2 .o R , O'fl fl ,pt

(I)

A.H. Hoang et al.INuclear Physics B 452 (1995) 173-187

OR 4

3

('-2mfIv~)zdy(1-?)2 ~ d - -( Z21m-+Z2z~sZ ] antis

175

1 4m2sz

4m~/s

X { 2_.~ s"-I- ~-(1-y+z) s-

¼(1-y+z)2-1(l+z)y 1- y + z

1--y+z _~_4__~sy AV2(1,Y' Z)

x In

1 - y + z +~4_..~_sy A1/z(1,Y, z)

- V~1- 4m2syA1/2(I'Y'Z) x

2--~~' 4--~ {l+2ml~'~ ' + ~ +k s) z

1

g+

( 1 - ; ~ z) "--2": (1--- ~

] }

Aii,y,z) (2)

where

A(1,y,z) = l + y2 + z 2_2(y+ z + yz).

(3)

For arbitrary ml, m2 and s already the first integration leads to elliptic functions and fairly lengthy expressions. However, in the limit m~ >> m22 the radiation can be split into two parts: "soft" radiation with energy of the f2f2 system smaller than a cutoff A, with m2 << A << ml, and the remainder, denoted "hard" radiation. The two parts can be integrated separately, and their sum is given by pR= f~2)in 2 m.__~+ f~l) s

m2

ln,-2 + fCRO)"

(4)

s

As expected, the cutoff A cancels in the sum. The functions multiplying the second and first power of the logarithm are closely related to the well-known Schwinger result [2] for real radiation of a light vector particle with mass A << ml: °'Y~]lz' = 6 °-fly~,pt

f(R2)

ln~ +

- 2j n

(5)

The evaluation of the function without logarithmic enhancement constitutes the main effort of this section. The three functions f(0,1,2) are given by f~2) _

3 -- w2 24

[(l+w2) lnp+2w]'

(6)

A.H. Hoanget aL/NuclearPhysicsB 452 (1995)173-187

176

(3 -- W2) (I + W2) [Li2(p) + Li2(p 2) 6

fR1) =

+~

41nw+51np-61n(

)

w (3-w2) [ l n ( ~ _ ~ ) _ 2 1 n w ] 3-

lnp-2((2)

q 39--70w2q-23w4 144

lnp

+ w (--177+71w 2) 72 '

(7)

fR(o) (3-w2) ( l + w 2) I 4 L i 3 ( l _ p ) + 3 L i 3 ( p Z ) + 4 L i 3 ( l ~ p

=

6

+5Li3(1 _p2) _ ~ g"(3)

4w2 ) (Li2 ( p ) + Li2(p2)) + 2 In( 1----~

+(31n(~-w--~)-41np-41n(w2))~'(2)+ + i 2 lnp

)

+,6

1 - ~ l n 3 ( ~ -~) 1 - w2 ))

+lnwlnp(81nw+lllnp-121n(~-~-~))] +

(1_w2) (21-13w2) Li2(p) 24

4 ~157-58w2+17W4 (Li2(p) +Li2(p2)) 72 + 2 w (3_-w2) (_1n2¢ 4w2 - + 3 ( ( 2 ) ) + -33 + 218w 2 - 73w 4 ((2) " ---S-~ 1 ) 72 +-27-3w+72w

+

2 - 9 8 w 3 - 3 3 w 4+31w 5 in2p 72 w

- 1 5 - 34w 2 + 11 w4 15 +42w 2 - 13w4 18 lnw+ 24

ln(~--~)) lnp

+ 2451 - 766 W2 -- 205 W4 lnp + w (177 -- 71 w2) ln( 14W2w2) 864 36 + W (-2733 + I067W2) 432

(8)

A.H. Hoang et al./Nuclear Physics B 452 (1995) 173-187

177

where /

w =- ~/1 - 4m~/s , 1-w P --= 1 + w "

(9)

Li2, Li3 denote the di- and trilogarithms, ((2) and ((3) the Zeta function of the respective arguments [7]. The details of this calculation will be given elsewhere [8]. This result is directly applicable for example to the annihilation of e+e - into r+r - with the additional "final state" radiation of e+e - (or/~+/~-) which adds incoherently to the process with r radiated from the light fermion final state [9]. The behaviour close to threshold (w --+ 0) and in the high energy region (m}/s + 0) is of relevance for direct applications and for the discussion of our results presented below. One finds

~RW~°w3[ lln2m2s

94 ln~-~ ( 6 ln(2w) - 13)

__ _ _ 1 6 208 898 ] + 3 l n 2 ( 2 w ) - 9 l n ( 2 w ) - 4 ( ( 2 ) + 27 ] + O(w5), QR ~ 0

(10)

- - g1i n 2 -m~ ( l n x + l ) + I n m2 ''°2 ( 6 lnex - 13 l n x + 4 ((2)-- 53) s

s

i~

~g

1 ln3x+13 (133 2 ) 5 18 ~-~ ln2x - \ 1 - - ~ + ~ ( ( 2 ) lnx+g((3)+ + x

I

~

((2)

833 216

l m2 m2 ( 2 ;0) -- 3 ln2 --s + In '"2s In x -- - 3l

ln2x+~ln

x - - 54( ( 2 )

--

22_~1

-{- O ( X 2 )

(11)

where rn~ (12) s Close to threshold radiation of light fermions is strongly suppressed, similar to the radiation of photons as calculated in O(a). The leading logarithms in the high energy limit coincide with those given in [1]. The prediction for Rflflf2f~ (with ml/m2 = my/me = 3477.5 chosen for illustrative purpose) based on Eq. (1), is shown in Fig. 2 (solid line). Also shown are the high energy approximation, Eq. (11), (dashed line) including the linear term in x and the threshold approximation, Eq. (10), (dashed-dotted line). It is evident that the high energy approximation can be used for values of x = m~/s between 0 and about 0.125 corresponding to values of w from 1 down to 0.7 and hence surprisingly close to the threshold. For w below this value the threshold approximation provides an adequate description. X~

A.H. Hoang et al./Nuclear PhysicsB 452 (1995) 173-187

178

~

0.0008

'

'

'

I

. . . .

I

. . . .

&O006 ' ~

,\

&0004

I

. . . .

I

'

'

U-

full res--------ul;-

I ....

,,arinx

'

'

&O002 0 ~

,

,

,

[

,

~

i

i

0.05

0

l

t

i

,

0.1

,

Rflflf2Y2 based on the exact result m2/s with ml = mr and m2 = me.

Fig. 2. Production rate text as functions of

. . . .

I

0.15

I

m~

,-,-7"~

0.2

m

0.25

S

(solid line) and approximations described in the

3. Virtual corrections

Virtual corrections in the present context arise from the two particle cut of the "double bubble" diagrams (Fig. 1). The O(tr 2) corrections 8Au to the lowest-order vertex can be classified into contributions to the Dirac (F1) and the Pauli form factor (F2) 6A~ = yg

(~)2

Fl + ~

i

o'~

q,, ( ~ ) 2

F2

(13)

where o-~ = ~ [y~, y~ ] and q denotes the photon momentum flowing into the vertex. Using the dispersive methods applied already in [ 10,11] the calculation of F1,2 can be easily reduced to a one-dimensional integration. It results from the convolution of the massive vector boson exchange vertex correction to f l f l production with the absorptive part of the vacuum polarization of fermions with mass m2. Denoting the vector boson mass by A the convolution reads F1,2 = ~

4m2 ~

1+

~2~ 1,2~ )

(14)

4m~

with

1[

l w2 2(

Re FI(A2) = - ~ ww 1 + w2-t- - w2

ml2

32w2) 1w2) 2)] 1+

8w2

3 ] - 1 + 2 w 2 --3 + 7 w 2 + 2 w 4 A2 8W 4 m---~+ ~ ] + 2 w2 +

I

+~

, [ -- 3 + 7 w2 + 2

W 4 -t-

~

¢(p,~212)

A2 ~(~212)

(l_w2)2/3+2w ) "2W

In p

m--~lIn m-'~

oo

r

V

I

/~

h)

I

h)

h.) °

h.)

I " , ) I ~--'

~

--.,I

i

-t-

U,)

÷

Jr

i

i

i

-t-

÷

i

i

i

~s

÷

÷

I

I

Jr

,

~1 ~

,

~

II ~'~

÷

~

~~

I

I

I

I

"~

÷

÷

~-

i

r

J~

I

1

I

i

I

i

~mL

"~-t-

I

h.) I ~--,

I

f

O0 "-,I

h~

A.H. Hoang et aL /Nuclear Physics B 452 (1995) 173-187

180

The QED normalization/?1 (0) = 0 is evidently adopted. The evaluation of Eq. (14) is tedious for arbitrary ml and m2 and will be presented in detail in [8]. In the limiting case ml >> m2, however, the result is drastically simplified: ReFI= f~2) In2 ~m2 + f~l) In -m22 - + f~0) ,

(19)

-o2 + fz(o) ReF2 = f2(1) In m2

(20)

s

s

s

where f~2) = ~

1 [ l+w2 ] 2w lnp + 1 ,

--E

(21)

/

f~l)= l + w 2 L i z ( 1 - p ) + l n p ln(l+p)-~Inp 6w 8 - 6w + 11 w2 11 1 + 36 w lnp + g ln( 1 + p) + 1---8'

-35(2)

1 (22)

f~o)_ 1 3"w + w 2 [ -T3(1,0, w) +T3(1, -wl,w) -T3(1,w, 1)w + 2 lnwT2(1,w) 7r ( 1T~ 2

-l+p) ) - Li3(1- p ) - Li3( ~

' w l ) - G~- + Li3(

W4 (1 -- W 2)

) - 2 Li3(p) + ~1 5(3) ÷ ~1 (Li2(p) ln( 4 1 2p_ ) + 1 --Li2(p 2) lnw + 2 ln2 lnp In( ( l ~ p ) 2 ~ lnp ln2w

14 ln2pln(1-p2)+5ln3p+(31n(4p)-21n(l+p))5(2)] 8+11W2 [ 1 18W Li2(p) + ~lnp 2 + l n p l n w + 25(2)

]

1 1 - w2 131 - 132w+ 134w2 + ]-2 ln2(----4 --) + 216w lnp 11 1 67 +-~- ln(1 + p ) + 55(2) + ~--~, fO) _

(23)

1 -- W 2 12w lnp, 6w

-Li2(1-p)+~ln

(24) 2p-lnpln(l+p)-]~

lnp+35(2) (25)

and

A.H. Hoang et al./Nuclear Physics B 452 (1995) 173-187

181

1

fdx fdx fdx

T2(~,~)

arctan((x) x 2 + r/2

o

1

T~(n,()

ln(x 2 + s~2) X 2 _1_ 7/2

0

1

W3(n,e,X)

ln(x 2 +

b~2)

arctan(x x )

(26)

X2 -~ 712

0 1

G=fax arct~(x) x

_ 0.915965594177219...

(Catalan's constant).

0

Again the f u n c t i o n s f~l,2) a n d f2(1) are closely related to the very well-known logarithmically divergent and constant pieces of the corresponding one loop corrections for small A [2] : Re F1 a-+o 6

/~2)

In ~ + g

-- ~

,

Re/~ a~9 - 3 f2(1) .

(27)

The similarity of these relations with Eq. (5) is evident. Interesting special cases are again the behaviour close to threshold and for high energies. Let us discuss the former: ReFlW--+0 1 [ ~6w

m22 31n--+61nW-8s

]

((2)

+ ~ ln--+In2+s +~

1[

m22 -31n--+61nw-ll s

~ww((2)+ g

ln--+ln2+s

]

((2) w+O(w2),

(28)

~

(29)

-~((2)

w + O ( w 2).

The Coulombic behaviour ~ 1/w is evident from this result. For F1 the Coulomb singularity is modified by the logarithmic factor ln(m2/s) which is responsible for the "running" of the coupling constant in the O(o0 result. It is instructive to combine the vertex correction of O(cr) and O(ce2) in the region close to threshold. As an illustrative example we will examine the Dirac form factor for this case. The infrared divergent part of/~ vanishes for w --+ 0 and therefore Re/~I +

Re Fl

3((2) (~)[1+ 2 w

(~)1

w--m,

( g

rn22 -- In sw 2

8)] 3

A.H. Hoanget al./NuclearPhysicsB 452 (1995)173-187

182

The fine structure constant ae, defined at vanishing momentum transfer, is related to the MS coupling constant at subtraction point/x 2 by oe = oe~-ff(/z2) (1 + °~M-ff(/z2~).1 In m22"~ + C9(oe3). 7r 3 /~2 J

(31)

At this point it becomes obvious that the natural scale for cefi-gin the threshold region is given by the nonrelativistic momentum/x 2 = s w2 as far as the 1/w terms are concerned. [This holds true as long as w > or. Below this value the approximations used in this work are no longer applicable.] For the correction resulting from transverse photon exchange, which are not enhanced by l/w, the scale /z2 = s/4 is appropriate. This suggests the following form of the Dirac form factor in the threshold region Re/~1 + 3((2) 2 w

Re F1

w~0

(crM--g~w2)) [ 1 _ ( ~ ) ~ ] 8

32(aM-g(~s/4)) [ 1 - ( ~ ) 1 ]

+O(w)

(32,

where the scale in the O ( a 2) term is not yet determined. It is clear that the virtual corrections will dominate the rate close to the threshold. For high energies, on the other hand, one finds m2

ReF1 ~-~0 L In2 mA (lnx 12 s k

+ 1~/

1 m~( -- ln2 x +

+~-~ In --s

1

13 ~) - - lnx -- 8 ~'(2) +

13

+~-~ln3x-~-~lnax+.

1

-3 ((3) +x

~

(1 ~ ' ( 2 ) + 2133~ 16J

lnx

67 ( ( 2 ) + 5~

6 in 2m~s + g ln--s

lnx+

49 4 115 ] 1 ln2x + lnx((2)+ 12 3-6 g ~ ]

+O(x 2)

'

(33)

m2

ReF2~°x

- ~ln--s lnx+gln2x

-~-~lnx+g~'(2)

+ O ( x 2).

(34)

A.H. Hoanget al./Nuclear PhysicsB 452 (1995) 173-187

183

For the special case mx = m2 = m the integrals (14) lead to particularly simple results

1 [ ( - 1 + 2 w 2 ) ( 9 - 6 w 2 + 5 w 4) 31-23w2+30w 4 Re F1 - 12 w2 24 w3 ln3 p + 12 w2 ln2 p q-( 267-238w2+236w418w -I- ( 1 - 2 w 2 ) (9-6w2+5w4)2w 3 ( ( 2 ) ) lnp

+

9 - 2 1 w 2 - 10w 4 147 + 236w 2 ] w2 ((2) + 9 J '

1 - w2 ReF

2 ~

_

(1

_

4 w2

-- W2) 2

ln3p

11 - 9w 2 ln2p

8 w3

(93-68w 2

18w

+

(35)

~

3 (1 - w2) 2

~w~

((2)

)

17

lnp-

-5-

+

--3 + 5 w z ] w2 ((2)

J

(36)

with the high energy expansion ReF1

383 11 lnx--~-((2)+-108 5 853] ((2)+108j +O(x2)'

@~01 ln3x+19 1 (~_5 3--6 ~-~ln2xq-~---((2) +x

Re F2 ~ 0

[5

49 ln2x+]-~lnx-

)

25 l n x + 2 ( ( 2 ) -- ~ ] x [ - ~1 ln2x - ~-~

+O(x2).

(37)

(38)

The logarithmically enhanced and the constant parts are in agreement with [ 12]. Finally, the O(ce2) contribution of the virtual corrections to the rate for the case ml >> m2 is given by

oV = w ( 3 - w2) (Re Fl + Re F2) + w3 Re F2 .

(39)

4. The total rate

Combining real and virtual radiation one thus arrives at

~i¢ + o v = _ 51W l n - ~m-2+ f ~ ° ) + w ( 3 - w 2) (f~o) + f~°))+w3f~2°) .

(40)

The quadratic logarithm in mZ/s from the real and the virtual radiation cancel. A linear logarithm, however, remains. Its origin can be easily understood through the running of the coupling constant a. The prefactor W is identical to the correction function of O ( a ) derived by Schwinger [2]. Therefore the expression for the rate for the inclusive f l f l

A.H.Hoanget al./NuclearPhysicsB 452(1995)173-187

184

final state including O(a) photonic corrections plus photonic O(a 2) corrections due to one light fermion with mass m2 reads 1

1 l n " ' 2ms2 + f(R0) + w ( 3 - w 2) ( f~0)+f(0) ) +w3j2¢(o)1 + (~)2[ -~W where

W=--3[f{R1)~-W(3--W2)(f~l)-~f(21))-~w3f,(1)] 2] = (3--w2)2(l+w2)

[2Li2(p) +Li2(p2) +lnp (21n(1-p) +ln(I +P)) ]

w ( 3 - w (2 ln(1- p) + ln(1 + p)) (-l+w) +

(33-39w-17w2+7w 16

3) lnp

3w ( 5 - 3w 2) (41) + 8 [Note that the massive quark is not accounted for, consistent with the fact that virtual heavy fermion loops are not considered in Eq. (4). Adding virtual corrections Eq. (35, 36) and the real radiation, e.g. based on a numerical evaluation of Eq. (1) one would thus include "double bubble" diagrams with two massive fermions. This will be done in [ 8].] Relating again the fine structure constant oe to the MS coupling ce~--gat the scale/22, the mass singularities disappear and one finds =1

R=zW(3-w2) +

(a~-.__~2))

--

W

+ (ff~--ff__~_~2)) 2 [ - 3 1 Wln-+f(°)+/.t,w(23-w2)s

( f ~ 0 ) + f ~ 0 )) + w 3 ¢(o1 1 2 . (42)

The behaviour close to threshold for the choice/22 = s is easily read off from Eq. (42) : R w-~o 3

--W 2 + (o~s))[9((2)__6w]

+(~s))2[(31nw-5)((2)+(41n2+6)w

]

+O(w2).

(43)

The discussion following Eq. (30) applies equally well to this formula, since real radiation vanishes close to threshold.

A.H. Hoang et al./Nuclear Physics B 452 (1995) 173-187

185

In the high energy region one finds 4 R - s- - ; o

1 -

6 x2 -

8 X3

+ ( ~ )

{3+9x+x2

[ - 18 lnx + -~-

~

lnx + --~

X

[

27

+x 2 -31n 2x+-~-Inx-4((3)-18((2)[ 752 304 1282]} +x 3 -81n2x+--~-lnx---~-((2)+---~] +O(x4).

(44)

In Fig. 3 the comparison between the exact (Q(a2) correction for/z 2 = s (solid line), threshold approximations (dashed dotted lines) and high energy expansions (dashed lines) is performed. Eq. (44) provides an important consistency check on our result. It is straightforward to relate pole and MS definition for the remaining fermion mass ml taking again into account in O (ce2) only the contribution from one light virtual fermion:

3 m~(~2) 2] 9

(45)

Replacing the pole mass by the running mass at the scale/x 2 = s the logarithmic factor of the ~2 term disappears as expected from general considerations. The structure of the logarithms of the ~2 and --4 m terms coincides with the expectations from [5,6]. In fact, after replacing the abelian factors by the proper SU(3)-coefficients one obtains ( ~ )

q- nf

R = 1÷

(_~)z

s

[2 11] ~ ~'(3) -~- n f

J 2

+ -7-

L --6--22

x g In

s

(-~)2[ q- n f

3

+nf

((3) - 4 ( ( 2 ) + - -

4 ~2(s)~10 3 ln2 s +

2480 ] 27 J ~2(s) 176 83151} In -s 27 ((2) + (46)

186

A.H. Hoang et al. /Nuclear Physics B 452 (1995) 173-187

0 -2 -4 -6 -8 -10 -12 -14 0

0.05

0.1

0.15

0.2

0.25

mA s

Hg, 3. (9(a2s(s)) correction to the inclusive production rate R/l f! based on the exact result (solid line) and approximations described in the text.

where now the number of light fermions, n f, is displayed explicitly. The relation to the n f dependent terms of Eq. (27) in [6] is evident.

5. Summary The rate for the production of a pair of massive fermions in e+e - annihilation plus real and virtual radiation of a pair of light fermions has been calculated analytically. This result, together with [9] can be considered as a first step towards the evaluation of the production cross section for heavy fermions in O ( a 2 ) . The expansion of the result for energies close to threshold and for high energies and subsequent comparisons with earlier asymptotic formulas provide important cross checks. The transition to the MS scheme leads to interesting insights into the proper scale of the coupling constant.

Acknowledgement We would like to thank K. Chetyrkin and M. Je2abek for helpful discussions.

References [1] V.N. Baier, V.S. Fadin and V.A. Khoze, Soviet Physics JETP Lett. 23 (1966) 104. [2] J.S. Schwinger, Particles, sources and fields, (Addison-Wesley, 1970/73) and refs. therein. [3] K.G. Chetyrkin, A,L. Kataev and EV. Tkachov, Phys. Left. B 85 (1979) 277; M. Dine and J. Sapirstein, Phys. Rev. Lett. 43 (1979) 668; W. Celmaster and R.J. Gonsalves, Phys. Rev. Lett. 44 (1980) 560. [4] S.G. Gorishny, A.L. Kataev and S.A. Larin, Phys. Lett. B 259 (1991) 144; L.R, Surguladze and M.A. Samuel, Phys. Rev. Lett. 66 (1991) 560, 2416 (E). [5] K.G. Chetyrkin and J.H. Kiihn, Phys. Lett. B 248 (1990) 359. [6] K.G. Chetyrkin and J.H. Kiihn, Nucl, Phys. B 432 (1994) 337.

A.H. Hoang et al./Nuclear Physics B 452 (1995) 173-187 [ 7] [8] [9] [10] [11]

187

L. Lewin, Polylogarithms and associated functions (North-Holland, Amsterdam, 1981 ). A.H. Hoang, J.H. Kiihn and T. Teubner, in preparation. A.H. Hoang, M. Je~abek, J.H. Kiihn and T. Teubner, Phys. Lett. B 338 (1994) 330. B.A. Kniehl, M. Krawczyk, J.H. Kiihn and R.G. Stuart, Phys. Lett. B 209 (1988) 337. A.H. Hoang, M. Je~abek, J.H. Kiihn and T. Teubner, Phys. Lett. B 325 (1994) 495 [Erratum: B 327 (1994) 439]. [12] G.J.H. Burgers, Phys. Lett. B 164 (1985) 167.