Higher order corrections to QCD jets: Gluon-gluon processes

Volume 211, number 4

PHYSICS LETTERS B

8 September 1988

HIGHER O R D E R C O R R E C T I O N S TO QCD J E T S : G L U O N - G L U O N P R O C E S S E S F. AVERSA, P. C H I A P P E T T A 1, M. G R E C O and J.Ph. G U I L L E T 1.2

INFN, LaboratoriNazionali di Frascati, 1-00044Frascati, Italy Received 2 June 1988

We evaluate the full o (a~) cross sections for the processes g + g~ g + X and g + g~ q + X, relevant for jet production at large PT at very high energies. Some phenomenological applications of our results at SplbSand Tevatron energies are also presented.

In a recent letter we have presented [ 1 ] the first results o f a general calculation o f Q C D one-loop corrections to the production o f jets in hadronic collisions. Indeed, motivated by recent calculations [ 2 ] of matrix elements for o ( a 3) p a r t o n - p a r t o n scattering processes, we have given in ref. [1] the results for parton subprocesses involving only quarks o f different flavour. In view o f the well-known dominance o f gluon-gluon interactions at present hadron colliders we have considered as a next step the o ( a s ) radiative corrections to gluon-gluon scattering. In the present letter we present the evaluation o f the full Q C D 0 ( a 3) cross section for the real and virtual processes I

g+g~g+g+(g),

II

g + g - , q + C t + (g).

(1)

More precisely we evaluate the cross sections to order a 3 for the reactions g + g--.g + X and g + g--*q + X. Some phenomenological applications o f our results at the SpaS and Tevatron energies are also presented. The method followed is described in refs. [ 1,3,4]. Starting from the expressions o f matrix elements in n dimensions we perform the phase space integration o f the real processes, then we cancel the 1/e 2 divergences (e = n - 4 ) by adding the virtual contributions. The left-over 1/e terms, corresponding to mass singularities, are then absorbed into the structure and fragmentation functions beyond the leading order. The algebraic manipulations are done in parallel with two independent programmes, one using R E D U C E and the other one MACSYMA. The inclusive cross section for the one-hadron inclusive production at large transverse m o m e n t u m H. (K.) + H 2 (K2) ~ H 3 (K3) + X

(2)

is given [ 1 ] by 1

da

1 1 -- V+ VW

×

1 -- ( 1 -- V)/x3

dx3

1

dv VW/x3

f VW/x3

dw Dr,' H3 IAr2 l l T H I w (x3, 1,, r J~ pi

(X 1~

M 2 )FpjH2 (x2, M 2 )

v

\ dv jp,pj_p(s, v)d(1 -- w) + ~

(3)

Centre de Physique Throrique, CNRS, Luminy, F-13288 Marseille, France. 2 Institut de Physique Nuclraire, Universit6 de Lausanne, CH-1015 Lausanne, Switzerland. 0 3 7 0 - 2 6 9 3 / 8 8 / $ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )

465

Volume 2 11,number

PHYSICS

4

LETTERS

B

8 September

1988

where the hadronic variables Vand Ware related to the Mandelstam variables as V= 1 + T/S, W= - U/ (S+ T), and similarly for the partonic variables u, w and s, t, u. Furthermore, xi = VW/x,vw, x2 = ( 1 - V) /x3 ( 1 -v) and s=SxIx2. FE (x, M2) are the structure functions of a parton pi inside the hadron H at the factorization mass scale M*, and D,“l(x3, M: ), similarly, are the fragmentation functions of a parton pI into the hadron H. For the processes ( 1) under consideration, we restrict p, = pj= g and p/= g, q respectively. Furthermore the factorization mass scales M* and M: have been, in principle, kept distinct. The Born cross sections dcrk_p, (s, u) for the two subprocesses ( 1) are [ 5 ]

(4)

where averaging over spin and colour has been performed. Finally KP,P,_P,(s, v, w) are the partonic and (.u,(p2) is the running coupling constant evaluated at the renormalization scale p2:

as(P2)=

271 POln(f12/n2)

(

1_ Pi ln]ln(~2/~2)l B~ln(~21~2)

>’

K-factors

(5)

with Po=+N-fNF,

p, =+fN2--NNF-iCFNF,

and N=3 (NF) being the number of colours (flavours) with C,= (N2- 1)/2N. So far eq. ( 3) describes the one-hadron inclusive production, with the appropriate definition of the fragmentation function. In the case of inclusive jet production at large transverse momentum one can also use eq. (3) with the following simple phenomenological prescription. As stated above, in perturbation theory the distribution functions F z, (x) and D:,(x) are transformed, after cancellation of virtual and real singularities, into scalebreaking distributions Fp,(x, M2 ) and Dp,( x, M,2 ) beyond the leading order, which absorb the collinear left-over singularities. Correspondingly the partonic K-factors KP,P,_P,(s, v, w) have a residual dependence on the factorization scales M* and Mf , in addition to the renormalization scale p2, which has not been explicitly shown in to jet production from the our short-hand notation for Kprp,_P,(s, v, w). It is clear then that the contribution parton processes pi+p,+p,+ X can be simply calculated by replacing Dp,(x3, M:) with 6( 1 -x,), the sensitivity to the phenomenological jet algorithm being related to the variation with Mf of the partonic K-factors. Alternatively one can use, as in ref. [ 41, a description of jets h la Sterman and Weinberg [ 61, namely the jet being defined as an arbitrary set of hadrons contained within a cone of given opening angle 6. It is clear that the above fragmentation scale is of order Mf- E,,, sin 6. We have studied the production of jets also in this framework. For the time being we only give here explicit results for gluon-gluon subprocesses within the first algorithm. A more complete phenomenological analysis of jet production, including all parton subprocesses, will be given elsewhere. The inclusive production of a gluon in a g-g collision takes contributions from the processes g+g+g+

(g+g),

g+g+g+

(q+G),

(6)

where one integrates over the unobserved final partons. Furthermore, following refs. [ 1,3,4 1, after the cancellation of the 1/e2 divergences from real and virtual gluon emission, the collinear singularities are absorbed into Fg (x, Al*) and D,(x, Mf ) by adding the bremsstrahlung contributions from the initial and final parton legs as follows: 466

Volume 211, number 4

E3 ( ~ddak3)

- izsl gg~ (g~H3) + X

X ~1 (de°~

f __dX3 1dr_V~ dWwf°(xl)f°g(X2)D°g(X3)

l_.}__H (1-v){da°'~ t -do -

[" 1-v t,l- - s ,w

1- w v

8 September 1988

(s, v , ~ ( l - w , + oq [ 1 /do "°' ~-~n0 ( 1 - w) K~og(s,v,w)+-H~(w)I--I (ws, v) v \ dv ]~-gs

tv \ d S / ~

+

PHYSICS LETTERS B

\ dv Iqg_qg

+ - 2N~ -

1--v+vw sL'(1-

)

l

2Nv H ( )1-v
, ,<,o.o, _

,,s,

+vw) t. dv)._,,t.

( da°~ (S vW ) \ dv,l~_**\ ' l--~vw

+ 1-v+vw - -

,,w

1-v+vw

(da°'~ { 1-v t dv )q,~q,ltl-vw

)]} '

(7)

where [7] ,1

(M2)Pp,pj(X)+fp,pj(x), Hp,p~(x)=ln(M~'~ep,pj(x)+dppj(x), \u~)

Hp,pj(x)=ln -~-T

(8)

and Pp,pj(X),fp,pj(x) and dp,~,j(x) are, respectively, the Altarelli-Parisi kernels and the finite o(as) corrections to the structure and fragmentation functions. So far only fqq (x) and dqq(x) have been calculated explicitly [9]. As is well known, they contain next-toleading terms which become particularly large near the boundary of the phase space. Furthermore this kind of correction can be simply taken into account by the appropriate use of the correct kinematical limits in the various processes and in particular by incorporating an explicit dependence of the running coupling constant on the kinematic variables in the Altarelli-Parisi evolution equations [ 10 ]. For gluon initiated processes, we can similarly incorporate in thef's and d's the relevant kinematical factors, as is done for example in ref. [ 11 ], by " m u l t i p l y i n g " Ppips(X) by In[ (1 -x)/x] for fp,pj(X) and by In[ ( 1 - x ) x " ] for dp,pj(x). Furthermore imposing energy-momentum sum rules we obtain the following expressions forfo,pj(X) and dp,p~(X): { tin(I-x)]

xlnx

f,g(X)=2N x k -1--~--xxJ + - 1 - - - - ~ d~(x) =2N

( 5 N F _ 1;¢2

\24N - -½

+

{ Fin(I-x)] 2xlnx x L i----x _]+ + 1 ----~

)

8(1-x)

},

+ (7Nv +]n2-~)~(1-x)} \16N

fq,(x)= ½[x2 + (1-x)2l ln (~fl-) , dqg(x) = ½[x2+ ( 1 - x ) 2] ln[x2(1 - x ) ],

f,q(x)=cv[l+(1-X)Zln(~_~_ ) 4 ] X

dgq(x)

=Cv (1 + (lx-X)" In[ (1 -x)x 2] - 2 ) ,

(9)

"~ For a review see ref. [ 8 ].

467

fqq(x)=Cv

{

8 September 1988

PHYSICS LETTERS B

Volume 211, number 4

Fln(1-x)] 3 1 ( l + x : ' ) k i - - x _1+ 2 ( l - x ) +

{

dqq(x)=CF ( l + x 2) L 1 - x

ll_x+X2In x + 3 + 2 x 1

__3

I+X 2

+ +2 1-i-~-x l n x

( 9 + 1~Z2)5( 1 - - X ) } ,

(1-x)---~ + { ( 1 - x ) + ( ] z c 2 - 9 ) d ( 1 - x )

}

' (9cont'd)

where, for convenience, we have also reported the complete expressions [9 ] for the quark-quark case ~2. We think that the above definition of thefa's and da's are more physical ones, instead of the naive choicef,j = d a = 0 (i, j = g), because it reduces the large correction terms of kinematical origin, common to all processes involving gluons [ 11,12 ], and should be absorbed into the structure and fragmentation functions. From eqs. ( 7 ) - (9) we finally obtain an explicit expression for the (gg--, g) K-factor. Due to the very long and cumbersome expression o f Kgg~g(S, v, W), we only report in table 1 the coefficients of the distributions in w, with the same notation as in ref. [ 1 ],

K~Ms, v, w) _ 1 do° (s,v) v \ dv/~gg

(S)

+c2+e21n ~

{[

(s)

c,+G In ~-5 +el In ~ ( S )]

+~21n ~

1

.

(s)]

+G In ~

Fin(I-w,]

}

5(1-w)

(l-w)+ *c3[ iLw J+ + G ~ d s , v,w),

(lO)

where K'gg~g(s, v, w) is finite for w--, 1. Notice the explicit dependence on M], M 2 and ~t2. Comparing the gluon-gluon results of table 1 with the analogous ones of ref. [ 1 ], for the quark-quark case, one simply finds c3 (gg)/c3 ( q q ) = N / C F , in agreement with the expectations from the Sudakov form factor. Indeed, as shown in ref. [ 1 ], c3 is the coefficient of the leading (In k-~/k-~) term of the relative transverse momentum distributions. Notice that the result c3 (gg) = 4N in eq. (10) depends critically upon the definition of ~2 We disagree with ref. [ 11 ] on the coefficient of the 5( 1 - x ) term i n f w

Table 1

g~ = ] N v - ~ N 3~ = ½N[6 log v - 6 log( 1 - v ) + 11 ] - ]Nv

dt={N(121ogv + ll )-~Nv cl = N [ 9 0 ( 6 v 6 - 2 1 v S + 4 5 v 4 - 5 8 v 3 + 5 4 v 2 - 3 0 v + 10) log2v - 9 0 log( 1 - v ) (4v 6 - 1 6 v 5 + 37v 4 - 5 0 v 3 + 4 7 v 2 - 2 6 v + 8 ) log v - 3 0 ( r e - v + 1 ) ( 11 v 4 - 15v3+4v2+22v - 11 ) log v + 9 0 log2( 1 - v ) ( 2 v 6 - 7v5+ 14v 4 - 16v3+ 14v 2 - 7v+2) + 120~2v 6 + 914v6- 3 6 ~ 2 v ~ - 2 7 4 2 v 5 + 765~2v~ + 5349v4- 9 3 ~ 2 v 3 - 6 l 28v3 + 3~ ~ g ( ~ - v ) v ( v 2 - v + l ) ( 7v2 + 8v+ 7 ) + 855~c2v~ + 5349v 2 -450~z2v-2742v+ 1 2 0 x z + 9 1 4 ] / 1 8 0 ( v 2 - v + l )3 + [ - 9 m y ( v - 1 )~)2(V2"{- V-- 1 ) log2v +6NF(v 2 - v+ 1 ) ( 4 v 4 - 3 v 3 - v 2 + 8 v - 4 ) log v + 18NF log( 1 --V) (v-- 1 )V(2Vz--2V+ 1 ) log v --Nv( 83v6- 249vS + 187t2V4+ 444Va-- 36ZC2V3--473V3 + 27~2V2 + 444V2-- 9~2V--249V+ 83 )--6NF log(1 --v) v( v2--v+ l ) × (5v2--2v+ 5) +9NF log2( 1 --v)(v-- 1 )2V(U2--3v+ 1 ) ] / 7 2 ( v 2 - v + 1 )3 32 = 4n ~2 = 2 n c2 = n [ 1 2 ( 2 v 2 - 3 v + 3 ) log v - 1 1 ( v 2 - v+ 1 ) + 12 log( 1 - v ) v ] / 6 ( v 2 - v + 1 ) + ½NF c3 = 4n

468

Volume 211, number 4

PHYSICS LETTERSB

8 September1988

fgg and dgg made in eq. (9). The naive choice f u = d ~ , = 0 would in fact lead to c3(gg)= 10N, similarly to c3(qq) = 10CF, that could have been found in ref. [ 1 ] forfqq=dqq= 0. We present now, in analogy to eq. (7), our result for the case g + g ~ q + X : de E 3 ( d~

-

3 )gg~ (q~H3)+ X 1 [ dx3

dv

dw

o

o

~ Fg (x,)Fg (x 2)D °(x3)

da° ~ X { 1 (dO'O's\ dv .]gg.qq (s'v)6(1-w)+~nO(X-w)[ggg'q(s'v'W)+v 1 n ~ ( w ) (\--~vlgg~q(WS'V) 1

H { 1 - v "~(da°'~

{

+

1--V

UW) "Jr 1 "qg(W)(d°'°~

\ dv/q=_qg (ws, V)

s, 1

1- v

\ dv

,]qg~qg

( s,1

.o

j

_ vw" ~ +

1 I~qg(1--V+VW)(--'~ (S, VW + 1-v+vw \ d v J g , ~ g = \ l-v+vw

~

1

da ° \ dv Jgg_q~\

vw 1

)]} "

(11)

The same result holds for the reaction g + g-. (t + X. With the help of eqs. (8) and (9) we finally get an expression for Kgg~q(s, v, w), similar to eq. (10). The corresponding results for the coefficients cu) are shown in table 2. Following ref. [ 1 ], the relative transverse momentum distributions of two jets produced in a gluon-gluon collision, can be simply obtained from eq. (10) for gg->g+X and the analogous one for g g ~ q + X , using the general formula given in ref. [ 1 ], with the appropriate substitution of the Born cross sections and the coefficients c, in the Sudakov form factor and in the regular terms. We present now some phenomenological numerical consequences of our results. We define R as the ratio of the cross sections (3) to order oq2 + a s3 to the cross section including o ( a 2) only, for g g ~ g + X and g g ~ q + X . Thus R= 1 would imply the exact validity of the leading-order formula. To be inconsistent, the numerator of R has been calculated with as (/22) to two loops, while the Born cross section in the denominator has been evaluated at one-loop order only. As discussed above, we have taken D(x3, M~ )=d(1 -x3) in eq. (3), for the jet algorithm. Table 2 cl =NNv(2v2-2v+ 1 ) / 3 ( N v 2 - N v + C F ) - N F / 3 N ( N v 2 _ N v + C v ) + 1 I/6(Nv2-Nv+Cv)

g, = [6 log v - 6 log( 1 - v ) + 11 ]/6(Nv2-Nv+ CF) +NNv(2v2-2v+ 1 )/3(Nv2-Nv+ CF) -Nv/3N(Nv2-Nv+CF) 31 = [4(v2--v+ 1) log v + 3 ( v 2 - v + I ) ] / 4 ( N v 2 - N v + C F ) + ( - 4 l o g v - 3 ) / 8 N a ( N v 2 - N v + C v ) cl = [3(v 3-16v2+ 17v-8) log2v + 12 log( 1 - v ) (2v2-2v+ 1 ) log v + 3 ( v - 1 )(v2-9v+6) log v - 3 log( 1 - v ) v(vZ+7v-2) - 3 l o g 2 ( l - v ) ( v - 1)(vZ-2v+2)-gn2v2-39v2+gnZv+39v_4n2_ 18]/12(2v2_2v+I)(Nv2_Nv+Cv) + [ (vZ+ 1 ) logZv + ( v - 3) ( v - 1) log v +log2( 1 - v ) (v2-2v+2) +2v 2 +log( 1 - v ) v(v+2) - 2 v + 1 ]/4N2(2v2-2v+ 1 ) (Nv2-Nv+Cr)+5NF/12N(NvZ_Nv+CF) C2= 2 / ( Nv2- Nv+ Cv ) - 2N 2( 2vZ- 2v+ 1) / Nv2- Nv+ CF ~2 = (v 2- V+ 1)/(Nv2--Nv+ CF) -- 1/ (2NZ(NvZ-Nv+ Cv) ) c2=N2[(4vZ-6v+3)logv+log(1--v)

(2v-l)]/(Nv2-Nv+Cv)+[_31ogv_log(l_v)]/(Nv2

Nv+Cv)

c3 =2NZ(2v2-2v+ 1)/(Nv2-Nv+CF) - 2 / ( N v 2 _ N v + Cv)

469

Volume 211, number

4

PHYSICS

gg

6

....

->

I ....

a

g

LETTERS

B

8 September

gg - > g X

X

I ....

1988

I ....

I ....

6

'''1

....

I ....

I ....

o

,,,I

....

I ....

I ....

I ....

/

5

// //

4

// //

3

/ /

7

1

J ....

I ....

i ....

0.2

I ....

0.4

[ ....

0.6

0

0.8

0.2

0.4

0.6

I .... 0,8

7)

gg ....

->

i ....

Fig. 2. S a m e as in fig. l a for the full lines. T h e dashed line corresponds to the analogous case for x / @ = 1.8 T e V .

qX

i ....

I ....

I ....

b

gg 3.0

....

I ....

->

g

I ....

X I ....

I ....

// 2.5

t~

/ "1/ 2.0

/

L5

1.0 i

0

,

,

i

I

0.2

i

i

i

i

[

0.4

i

i

i

i

I

0.6

i

i

i

L

i

i

i

i

03 0.5

F i g . 1. T h e ratio R o f o (ct~ + ct 3 ) cross section to the cross section o f o(ot~2) only as a function o f r/--- 2pT/X/~, for g g ~ g X ( a ) a n d g g ~ q X ( b ). Full ( d a s h e d ) curves correspond to f u and d ~ given by eq. ( 9 ) ( r e s p e c t i v e l y f u = d u = 0 ) . T h e upper ( l o w e r ) curves correspond t o / 1 2 = M 2 = M r 2 =4p~(It2=M2=M2~T/4). T h e energy is x / S = 0 . 6 3 T e V .

0o

.... 0

I .... 0.2

I .... 0.4

L .... 0.6

I .... 0.8

r/

Fig. 3. D e p e n d e n c e o f R u p o n the factorization m a s s Mr. Full, dashed, dot-dashed lines correspond to Mf=!~r/4, pr/2 and PT respectively, at x / ~ = 0 . 6 3 T e V .

In figs. 1a and 1b we show for PI) collisions at v / S = 0.63 TeV and with the set of structure functions of Duke and Owens [ 13], the ratio R as a function of tl=--2pT/v/S, PT being the transverse momentum of the jet. The results have been obtained for 0jet= 90 °, but do not change significantly at 0jet= 60 °. The full curves (respectively dashed curves) correspond to f ~ and d~ given by eq. (9) (respectively fgg = dgg= 0). The upper (lower) curves correspond to #E=ME=M2 =4p2(#E=Ma=M2 = p ~ / 4 ) . As is clear from fig. 1, the increasing behaviour of R is much weakened for a "physical" definition of gluon structure functions, when the main kinematical corrections are taken into account in the structure and fragmentation functions. Furthermore the corrections are stable and reasonably small for/t 2 = M 2 in the range p2/4-p2. The energy dependence of R for gg--,gX is shown in fig. 2, where the results of fig. la at v / S = 0 . 6 3 TeV (full lines) are compared with the analogous one at 1.8 TeV (dashed lines). In fig. 3 we show the dependence of R on the fragmentation mass Mr, for fixed M=I.Z=PT/2.

Finally in fig. 4, we present our results for a jet algorithm it la Sterman and Weinberg, namely when the jet 470

Volume 211, number 4

PHYSICS LETTERS B gg ->

3.0

....

I ....

jet I ....

8 September 1988

+ X I ....

I ....

2.5

/

2.0 /

J

f

1.5

1.0

0.5

0.0

....

I .... 0.2

I .... 0.4

I .... 0.6

I 0.8

. . . .

Fig. 4. The ratio R for jets h la Sterman and Weinberg. Full, dashdotted and dashed lines refer to ~=0.1, 0.05 and ~=1, respectively.

p r o d u c e d at large PT is d e f i n e d by a c o n e o f s e m i a p e r t u r e ~. Full a n d d a s h - d o t t e d lines refer r e s p e c t i v e l y to ~ = 0.1 a n d ~ = 0 . 0 5 , a n d t h e l i m i t i n g case fi= 1 is p r e s e n t e d w i t h a d a s h e d line. O f course, w i t h this d e f i n i t i o n o f j e t s o n e a c c o u n t s for all possible p a r t o n s (g, q, Ct) p r o d u c e d inclusively. N o t i c e t h a t the c o r r e s p o n d i n g d i s t r i b u t i o n f u n c t i o n s do n o t e n t e r the c a l c u l a t i o n b e c a u s e the r e l a t e d 1 / e t e r m s a p p e a r i n g in the real c o n t r i b u t i o n s to the cross section are a u t o m a t i c a l l y s u b t r a c t e d by the a n a l o g o u s c o n t r i b u t i o n f r o m t w o p a r t o n s in the jet. A m o r e d e t a i l e d analysis will be p r e s e n t e d elsewhere. In c o n c l u s i o n we h a v e p r e s e n t e d the 0 ( a 3 ) c o r r e c t i o n s to g l u o n - g l u o n processes. T h e c o r r e c t i o n s are reasonably small a n d stable after t a k i n g i n t o a c c o u n t in the d e f i n i t i o n o f s t r u c t u r e a n d f r a g m e n t a t i o n f u n c t i o n s the m a i n l o g a r i t h m i c c o r r e c t i o n s o f k i n e m a t i c a l origin. We are grateful to the T h e o r y D i v i s i o n o f C E R N for the use o f the c o m p u t e r facilities.

References [ 1 ] F. Aversa, P. Chiappetta, M. Greco and J.Ph. Guillet, Phys. Lett. B 210 ( 1988 ) 225. [ 2 ] R.K. Ellis and J.C. Sexton, Nucl. Phys. B 269 ( 1986 ) 445. [3] R.K. Ellis, M.A. Furman, H.E. Haber and I. Hinchliffe, Nucl. Phys. B 173 (1980) 397. [4] M.A. Furman, Nucl. Phys. B 187 ( 1981 ) 413. [ 5 ] T. Gottschalk and D. Silvers, Phys. Rev. D 21 (1980) 102; Z. Kunszt and E. Pietarinen, Nucl. Phys. B 164 (1980) 45. [6] G. Sterman and S. Weinberg, Phys. Rev. Lett. 39 (1977) 1436. [ 7 ] G. Altarelli and G. Parisi, Nucl. Phys. B 126 ( 1977 ) 298. [8] G. Altarelli, Phys. Rep. 81 (1982) 1. [9] W.A. Bardeen, A. Buras, D.W. Duke and T. Muta, Phys. Rev. D 18 (1978) 3998; G. Altarelli, R.K. Ellis and G. Martinelli, Nucl. Phys. B 157 ( 1979 ) 461; G. Altarelli, R.K. Ellis, G. Martinelli and S.Y. Pi, Nucl. Phys. B 160 (1979) 301; R. Baier and K. Fey, Z. Phys. C 2 (1979) 339. [10] G. Curci and M. Greco, Phys. Lett. B 92 (1980) 175; D. Amati, A. Bassetto, M. Ciafaloni, G. Marchesini and G. Veneziano, Nucl. Phys. B 173 (1980) 429; A. Bassetto, M. Ciafaloni and G. Marchesini, Phys. Rep. 100 (1983) 201. [ 11 ] P. Aurenche, R. Baier, M. Fontannaz and D. Schiff, Nucl. Phys. B 286 ( 1987 ) 509, 553. [ 12 ] R.K. Ellis, P. Nason and S. Dawson, Fermilab-Pub.-87/222/T. [13] D.W. Duke and J.F. Owens, Phys. Rev. D 30 (1984) 49.

471