We present analytic results for the one-loop corrections of the helicity
amplitudes of the QCD five-parton subprocesses involving four quarks and one
gluon obtained with a standard Feynman diagram calculation using dimensional
reduction.Comment: 9 pages LaTeX, ETH-TH/94-14, KLTE-DTP/94-

Adrian Signer

Bern

Gastmans

Gunion

Kleiss

Kunszt

Mangano

van Oldenborgh

Zoltan Kunszt

Zoltán Trócsányi

Physics Letters B

English

arXiv

Kunszt, Zoltán

Signer, Adrian

Trócsányi, Zoltán

University of Debrecen Electronic Archive

ELSEVIER 
29 September 1994 
Physics Letters B 336 (1994) 529-536 
PHYSICS LETTERS B 
One-loop radiative corrections to the helicity amplitudes of QCD 
processes involving four quarks and one gluon* 
Zoltan Kunszt a, Adrian Signer a, Zolt~n Trdcsgnyi a,b 
a Theoretical Physics, ETH, Zu'rich, Switzerland 
b Department of Theoretical Physics, KLTE, Debrecen, Hungary 
Received 31 May 1994 
Editor: R. Gatto 
Abstract 
We present analytic results for the one-loop corrections of the helicity amplitudes of the QCD five-parton subprocesses 
involving four quarks and one gluon obtained with a standard Feynman diagram calculation using dimensional reduction. 
The technical developments achieved recently in perturbative QCD calculations (helicity method [ 1 - 4 ] ,  
string theory based derivations [ 5,6] ) made calculations of  one-loop corrections to helicity amplitudes up to five- 
parton and perhaps also to six-parton processes feasible. As a first result, Bern, Dixon and Kosower published 
recently the one-loop corrections to the five-gluon amplitude in QCD and in N = 1,2, 4 supersymmetric Yang- 
Mills theories. They used a string based technique and helicity method and obtained a remarkably short analytic 
answer. In a previous paper [7] we presented a simple analytic result for the singular parts of  all helicity 
amplitudes o f  all five-parton processes (in full agreement with the results of  [6] in the case o f  the five gluon 
amplitudes). In this letter we present the complete one-loop corrections for processes involving four quarks 
(equal or unequal flavors) and one gluon. We used the conventional Feynman-diagram method, however, the 
use of  the helicity technique was decisively important. Our method was tested previously by the diagrammatic 
evaluation of  the one-loop corrections of  the helicity amplitudes of  all 2 ~ 2 parton processes [ 8 ]. A systematic 
description together with the non-trivial details of  our calculation will be presented in a later publication. Here 
we mention only that the tensor integrals have been reduced to scalar integrals with a reduction method similar 
to [9] .  Calculating the integrals this way and the use of the helicity method allow us to eliminate all integrals 
which are more complicated than pentagon (box) tensor-integrals with one (two) integration momenta in the 
numerator. As a result, the Gram determinants in the denominator which blow up the size of  intermediate 
expressions were eliminated at the very beginning of  the calculation. For the sake of  simplicity we performed 
the calculation in the dimensional reduction scheme (in D = 4 - 2e dimensions). Transition rules to different 
schemes - such as conventional dimensional regularization, ' t  Hooft-Veltman - have been derived in [8] .  
* Work supported in part by the Swiss National Science Foundation, the Hungarian National Science Foundation and the Universitas 
Foundation of the Hungarian Commercial Bank. 
0370-2693/94/$07.00 (~) 1994 Elsevier Science B.V. All rights reserved 
SSD1 0370-2693 (94) 0107 1-4 
530 Z Kunszt et al./Physics Letters B 336 (1994) 529-536 
It is convenient to give our result in a crossing symmetric form for the unphysical channel where all particles 
are outgoing 0 --* 77QQqg. The momenta of the partons are labeled as 
0 --~ antiquarkl (~) + antiquark2(~)) + quarkE(Q) + quarkl (q) + gluon(g) . (1) 
The color structure of the amplitudes is the same at one loop as at tree level: 
fit (i) (77, ho; Q_., ho; Q, hQ; q, hq; g, hg) 
=g3(  g ) 2 i [  , Z  "T g" 6 (i) , -  
\-4-~.1 k )q1?12 q2q, aq,cj2(q, ho;Q, ho.;Q, hQ;q, hq;g, hg) 
(ql q2)E{q,Q} 
(ql *q2)e{q,Q} -~c ( r ) qlO,~q202a q,gll ( q, hgl; O, h~; Q, he; q, hq; g, hg) , (2) 
where i = 0 means tree level and i = 1 means one-loop approximation. 
At a given order in perturbation theory, there are only two independent color subamplitudes because we have 
the symmetry properties 
a (i)( = hc~; Q, (i) - . _ . qO,.q , O_.,ho; ha;q, hq;g, hg) =aao(Q, ho,q, hct, q, hq;Q, ha;g, hg), (3) 
~(i) (Q_, ho; 77, ho; q, hq; Q, ho; g, hg) (4) ~zqgl~(i) ktt,[ .~. ho; Q, hQ; Q, ha; q, hq; g, hg) = uOO 
Furthermore, we should consider only four helicity configurations. If we change the sign of all helicities, we 
obtain the corresponding amplitudes simply by replacing the spinor products (...) ~ - [ . . . ] ,  where the angle 
bracket and squared bracket denote spinor products with minus-plus and plus-minus helicities, 
(pq)=~O_(p)~O+(q) and [pq]=~k+(p)~k_(q). 
Due to helicity conservation along a fermion line, we have hq = - h  o and hQ = - h  O. We present our result for 
positive gluon helicity and label the remaining helicities with hq and h a, 
In order to be able to write down the result for arbitrary values of hq and ho, we introduce the helicity 
dependent momenta 
r (hq)=q if h q = -  and r(hq)=gl if hq=+,  
R(hQ)=Q if h Q = -  and R(hQ)=Q_. if hQ=+.  (5) 
We shall suppress the helicity dependence of r and R. 
For the sake of completeness, we recall the tree-level amplitudes a~ °) 
Co) (i j) (6) aij (hq, hQ, q-) = ipa(hq, hQ, q-) (ig)(gj-----~" 
We note that the helicity dependence can be absorbed completely in the factor Pa: 
(rR)2 (7) 
pa( hq, hQ, -t- ) = ( - 1 )  '~hqho (q77) (QQ) ' 
where ~ is the usual Kronecker ~. 
At one loop, the result can naturally be decomposed into soft contributions Sij(hq, hQ, +)  given in [7], 
ultraviolet renormalization terms 7~ij ( hq, h a, +) ,  terms coming from the expansion of collinear 1/e singularities 
z. Kunszt et al. / Physics Letters B 336 (1994) 529-536 531 
Cij(hq, ha, +) given by vertex and self energy integrals, finite terms composed from spinor products, dot 
products and single logarithms 79ij(hq, hQ, +),  and finite terms ~ij(hq, hQ, +) containing factors of Li2, In 2 and 
rr 2 coming from pentagon and box integrals. The labels i, j run over q, Q, q, 0 similarly to the corresponding 
labels in the color subamplitudes a~} ). The color subamplitude a[} ) has then the decomposition 
a}} ) (G -hq; Q_, -hQ; Q, hQ; q, hq; g, +) 
=$ij(hq, hQ,+) + 7P~.ij(hq, hQ,+) +Cij(hq, hQ,+) + ~ij(hq, hQ,+) +~ij(hq, hQ,+), (8) 
where we suppressed the dependence on the momenta q, q, Q, Q, g. For completeness we recall the soft contri- 
butions 
and 
SQi-l(hq, hQ,+) =-~22 {Nca(°)(hq, hQ,+) [79qg + 790_q+ 79Qg ] 
1 
uca(Q O)(hq'hQ' +) [--7900_ + 7991Q + 79glq + 790.Q + 790_q- 79Qq] 
NcaQ(2(hq, hQ, +) + 790_q - -  
1 a(O)(h 
(9) 
CF ( ., (O) l .  [7~glq ~]DQg + "pQg] SO0 - ( hq, ho, +) = - e-- ~ ~ JVcaao - ~ nq, hQ, +) + 
1 a(O) 
Nc QO_ ( hq, hQ, +) [-  ~qO. + ~qQ + ~qq + ~DO_ Q + ~DO_ q -- ~Qq] 
+Uca(Q°)(hq'ho' +) [97900- - ~qq -- 79Q.g + 79qg] 
+Nca (°) qo (hq, ho, +) [-79qq + 79qg + 79Qq-- 79Qg] }, 
where we introduced 
(lo) 
79,i = / .2\|_~___1 ~, cr = (4~r) ~r2(1 +e) 
• \ s i j /  F(1 - 2e) 
We note that the helicity dependence of the soft contribution is given by the Born factors a~ °). This holds for 
the ultraviolet renormalization contributions and the collinear (1/e) singularity as well. We record the result in 
such a form that the helicity dependence of the C terms is again completely factored into the Born terms 1. 
"]'~ij(hq, hQ, +)  = (4"~7")e 
2 eF(1 - e )  ( l l N c -  2Nf) a~?)(hq, hQ,+), (11) 
CQO(hq, hQ,+) = ¢F~(o)tt. t. + ) ( 2  3 1  ) 
~ t*OO~.t'q't'O, ~ ( U c -  N f ) ~ q  +-~l--c(~;-tq + ~i~O_.O ) 
_(O),hq, ha,+) (29 N 13 1 I0 ) 
+UO0_V ~,~-~ c + 2 Nc NT ' 
Note that the C terms are not uniquely determined since we can always shift finite contributions between 7~ and C terms. 
(12) 
532 Z. Kunszt et al. / Physics Letters B 336 (1994) 529-536 
Cra(°)rh ( 2  N 3 1 79 ) CQgl(hq, h Q , + )  = "~ Qglt q, h Q , + )  5 (  c - Nf)T2)O.O + ~ c  ( glq"~-72~O. Q) 
29 13 1 
+a(Q°)(hq'hQ'+) i-8 N c +  2 Nc Nf  . (13) 
As expected, the next-to-leading order corrections destroy some of the symmetries of the Born terms (see 
(6) ,(7)) .  However, there are several symmetry relations which remain valid even at one loop: 
a (1) - - , + ) -  ~ ( I ) t .  -j-,--~)lO.__~q,(2+__~ a
Q Q ( - - ,  -- --•Q0 t--r-, 
a(l), +, - ,  +) _ ~(~) ~ -  +, +)Io+-.q.O.-,o, QQ ~" -- --°tOO k , 
a(o1-)q ( - '  - '  +) = -a(QlO ( +' +' +)IO~Q,q~O" 
(14) 
(15) 
(16) 
We find that $0, T@, gij and Cao respect these symmetries separately. As a result the Dad terms must also 
satisfy the relations (14) and (15). In the case of CO. O, if we factor the helicity dependence into the Born terms 
as in Eq. (13), then the symmetry relation (16) is slightly violated which is compensated by a corresponding 
change in 79Q0: 
~3QO(-,-- ,+)  =-DQ~I(+,+,+)IO~Q,q,_.,O + ~ ( N c -  Nf)a(O-)q(- - , - - ,+) ln(SOQ).  17)  
\ Sg?q /I 
In order to give the results in a more compact form, we introduce the following notation 2 : 
{ ~ i } o = l n ( - S i j ' ~  _ in ( -sk t '~  {~} 
t, ) t, 2, 
{ i j ) -  1 { i j )  ..~ 1 
kl 2 (Skt Si r)2 kl 0 Skl(Skl -- Si r) 
<ijkl)-  (ik}(jl} 
<il><jk}" 
For the aQO color structure the 79 terms take the form 
Ski Sij 
{ i j )  ! { i j ) +  
kl 3- (Sk i  s i j )  3 kl 0 
( s 0 + Ski ) 
2 SijSkl(Skl -- Sij) 2' 
(o) 
79Q0(+, +, +) = aQO( +, +, +)  
~ N  2 _ - _ - SqQSqg 
X ( Nc + I <qgqQ>(qQqQ)soqSo~Q +<OgQ~>2sQg{o~:}_<glgQ~)(OQg~>SQg{oO_qQ} ' 
gtq Qg qq 
-[-Nc [~(61gQO)SQg{~g}l  ] 
- <glqQO-)<glQqO-)Sq Q ({o~qg}l + {0Q},) _ 1  - 
and 
0 ~ has - 2 n  dimensions. These functions are related to similar ones introduced in Ref, [61. 2 Note that the function kl j n mass 
18) 
19) 
(20) 
Z. Kunszt et al. / Physics Letters B 336 (1994) 529-536 533 
a (°) ( - ,  +, +) 
1 9 o 0 ( - ' + ' + )  = oO- 
x NZc + 1 ag + (qqQQ)(qOqQ)soa qO Nc (glQ_qg)(glO_qQ) SglgSTIQ -~ (Q_Qgq)2SQg #q i 1 S#qSQQ 
--(OOqg>s,~g ((glO-.qg> {q0~} 1 --(77qOg> (gQ)l) --(glO__.qQ> (1 + (glO_.qQ))s#a {~gq )l 
-(OgQq)SQg((OQgq) {~°~} 1 + (O0_qg)s#g {00}2) #q +l~(qQqQ)_- 2S24Q {00:}2] 
+ Nc [~ {,dqg}0 + 3(OOqQ)s#o {#Oqg),] + ~ Oa 
Since the results for the aQ# color structure are somewhat more complicated we make the decomposition 
7)O,~( hq, hQ, +) = a(Q°o) ( hq, h a, +) 
X I2(gc-gf)~)f~,(hq,  hQ) -~- g2-~- l~D~gl(hq, hQ)-q-gc~)lQ#(hq, hQ) Ac ~c~D~o(hq, hQ) ] (22) 
We have to record the 7)oq(hq, hQ) terms for three different helicity combinations. In the case of hq = hQ = +, 
for the flavor dependent part we get 
Df(+, +) = (gTOgQ)so_g {#O_Q }I 
1 - - OQ QQ 
- 2  (EtQqQ}(flqgQ)sqgsQg {#q }2 + (glgqQ)(glQgO)2SqgS~g {#q }3' (23) 
while the term proportional to (N~ + ! )/Nc reads 
1 QQ 
79~#(+,+) = -~(clqgQ)+ (ftqgQ)sqg((qQglO){#q }1-(~tgqQ)<glgqO) {~a}l) 
+~(gTgqQ>{OQ} 0 1 QQ - ~(glqgQ){#q }o-  2(clqQO__.>(glQqQ_.)SqQ {qga }1 
+(qgqQ) <qqgO>2Sqg {~Q }i + l(qgqQ)<qQqO)2S2qQ {0% }2 
1 - - #q 
and the leading and subleading color terms are 
1D~#(+,-q-)= (0QqO) (2 {qq_gQ }0 - {qQg }0) -1- (CIQgO)SQg (2 {#0qQ )1 -+- 3(qQq0) { qg0 }1) 
+(qqgO) (gTgqQ)sqg (2 + <qqQO>) {~ }i - (qQqO)(OqgO)gSqg {~ } , (25) 
1 0 _ Q  O.Q 
+2<qQ~lO)(qqga)sqg (<qqgO> {#07 }l- <qgqO){#OqO }i) + <qOgQ)sog {#QqQ }i" (26) 
The results for the helicity configuration hq = --, hQ = + are 
( 0q 0o) of#( -, +)= (FiQgQ)SOg {o Q }, + <YlOqQ>SQg ~sylg {QQ )2- <qOqg)s2g {#q }3 ' (27) 
5 3 4  Z. Kunszt et al. / Physics Letters B 336 (1994) 529-536 
1 ~ Oa QQ 0~'7(- '+)=2 + (gtQqQ)2s°uSoQ (#OqQ)SQg{ #q )1 3 - -- 2 (~aqa)Sglg l glq }1 S glqS OQ 
1 - - glq 
- 2 (~tQqQ)(QgQq)s°g(s°g - S°g) {OQ }2' 
3 - O.Q 3(glQqg)s°g { OOg}l' DIQo( - ,  +)= -~(QgQq)SQg {Oq }, + -~ 
s *(? - DO?(-, +)= 2(~QqQ)({ Oa }o + (QQgq)sag {gQ }, -(glQqg)sog {aa }z) " 
Finally we give the results for hq = +, hQ = - .  The flavor dependent part is 
2 (qgqQ) qg (glq }2 + (glgqQ)(glQgQ)sogSqg 
For the term proportional to (N~ + 1 )~No we have 
I9'~#(+,-) = ~ + (gl(2gO) 2 1 - (glgqQ)SOg~ (glOgQ)(glQqQ) z- sz- 
QQ Sglg f SglqS OQ 
+<glgqQ) (glqgQ)2sqg ({ o 0ff 11 + { ~qQ }1) 1 
1 + 2 (77gqQ)(g]O-'gQ)sqg ( (glqgQ)sO-` ( doff }2 -  (glQqQ)s(2q ( ctQgq }2) 
1 
+-~ ( glO_gQ ) ( glO_ qQ ) SQgS qa ( ~g }2 
+ ~(glgqQ)(glOgQ)2Sog (So_g - Sqg) {~qQ } 2 -  ~(glgqQ)(clqgQ)sqgSOQ { ~  }2' 
the leading color term reads 
~ o (  + , - )  = 5<OgqQ)<OqgQ)'q~ {08 }1 + 
and the subleading color term is 
O_e - 2(glQqQ) {Og s gig @ glq DQO( +'--) =-(77gqQ) {oq o o o 
+(gTqgQ) (2 { OOg }o + 2(OOqQ) {~qO } o ) -  (OgOQ) (glOgQ)sog (~ + 2<OOqQ)) {oQ~ }1 
+(glgQ--Q)(glO-'gQ)sog (2(O~)qQ) {~-gQ}l + 2(glgqQ){~)1) 
+(77QgQ)2SOu (2(OgqQ) + 3(QgqQ)  { ~g },- (OgqQ)(OqgQ)sqg {o O'Q }, 
-(gtgqQ)(glOqQ)sqO (2  + 2(glO-'qQ)) {odg Q }, + 2<gtOqQ)<glgqQ)(glqgQ)sqg { ~  }l " 
The g contributions can conveniently be given in terms of an auxiliary function 
(ijk). = (r(hq)ikR(hQ))"F(i.j, k). 
(28) 
(29) 
(30) 
(31) 
(32) 
(33) 
(34) 
(35) 
Z Kunszt et aL/Physics Letters B 336 (1994) 529-536 535 
where ( . . . )  has been defined in (19) and 
.U(i , j ,k ,  ) = __Li2 ( l  Smn~ _ Li2 ( l  Smn) l ( sij'] rr 2 
S i j  J Sjk / -- 2 in2 \Sjk// 6 (36) 
Li2 is the dilogarithm and m, n denote the complementary labels to (i, j, k) in the set of  labels (c7, Q, Q, q, g) 
{m,n} = {gl, Q . ,Q,q ,g} \ { i , j , k } .  (37) 
We use the convention that if both (rikR) = 0 and n = 0 then ( i jk) .  = U(i , j ,  k). Then the EQct(hq, hQ) term 
for arbitrary hq, h 0 quark helicity configuration is given as 
. ~ h q h Q  : - ~ ,  Ea#(hq, hQ, +) = --a(a°)(hq, ha, +) VQ# ~q ,~ ,Q,q ,g )  
: ~ h q h Q  : - .~. ( 0 )  ,,-~hqhQ / - +a(Q~(hq, hQ,+) 9QO tq,~d,Q,q,g) + aqc~ (hq, hQ,+) ~qCl tq,~g,Q,q,g), (38) 
where 
~he(g l ,  O , a , q , g )  = N~ [(cTgQ)o + (qqQ)2 + (qc~g)2 + (gOQ_)2 + (qQQ)2] 
1 [(qgO)o + (clgQ)o - (glgQ)o - (qgQ)o + (OQO_)2 + (glqO_)2 
Nc 
+ ( q q Q ) 2  + ( q Q a ) 2  - (qQQ)2  - (qq(2)2 - (qQO)2 - (~qQ)2] ,  (39) 
~ hqhQ / - QQ_ tq, O ,Q ,q ,g )  = Nc(gQQ)2 
l 
I-(gtgQ)o + (O.gq)o + (qqg)2 - (qqg)2 + (qQQ)2 - (qqQ)2 - (qQQ_.)2 + (qcTQ)2 Nc 
-(O.Qg)z6e. + (cTQg)2(1-,se.) - (cTQg)28e. + (qQg)2ae. - (qQg)2(l-ao.)]  , (40) 
Gqhqhe (q, Q., Q, q, g) = Nc( qOg)2 
1 [(Qgq)o - (Qgq)o - (gQQ)2 + (gQQ)2 + (q£1Q)2 - (qQQ)2  - (qqQ)2 + (qQQ)2 
Nc 
-(g~lq)2(1-&q) Jr- (gqQ)2(l-srq) - (gqO_.)2&q -q- (gqQ)2&q - (gqQ)2(l-&q)] • (41) 
As noted before, the EQO terms satisfy the symmetry relations (14) and (15) ,  therefore it is sufficient to present 
the EQO type contributions for hq = q- and arbitrary hQ: 
a(O). 
~QO(-~-,hQ,'q-) = QO( +,hQ, +) 7-[~O( gI, Q ,Q,q ,g)  
hQ - - +a~°) ( +,hQ, +)7-[h~(~l,O,Q,q,g) + a(qQ)( +,hQ, +)7-[qo(q ,Q,e ,q ,g) ,  (42) 
hQ 
where the auxiliary functions, 7-[0 have the form 
hQ _ 
~ad(q ,O . ,Q ,q ,g )  = Nc [(qgq)o + ( Q q g ) o -  (Qggl)o - ( Q g q ) o -  (QqO)o - (gqq)o 
+ (gqQ.)2 - (qcPQ)2 - (g#q):]  
1 [(cTqQ)o - (cTqQ)o + (qQg)o - (£1Qg)o + (qcTQ)2 - (qglQ)2 
Nc 
- (Q0g)2(l-,SRe) - (qO.g)2(l-am) + (qQg)26m - (0Qg)z,~m] , (43) 
536 Z. Kunszt et al. / Physics Letters B 336 (1994) 529-536 
hQ _ 
7-(QO(q, 0_, Q, q, g) = Nc [ ( q g Q ) 0 -  (gtgq)o + (Qqgt)o- (Qqg)o- (QO_.gt)o 
- (g0~))2 + (gg/q)2 + (g{)Q)2] , (44) 
7~q~(gt,h~ Q , Q , q , g )  = Nc  [ ( q g Q ) o  - (qggt)o - ( Q q g ) o  + (glqg)o 
+ (qqQ)2 - ( g q O ) 2  + ( g Q ~ ) ) 2  - (qQQ)2 + (gclq)2  - ( q q g ) 2 ]  • ( 4 5 )  
The above results are valid in the unphysical region, where the dot products are negative, therefore the 
arguments of the logarithms and dilogarithms are away from the branch cuts. To obtain the amplitudes in any 
physical channel, one has to continue analytically to the corresponding physical region and make the usual 
substitution 
Sij ---+ Sij "q'- it/. (46) 
This defines all functions in a unique way. 
As a consistency check we investigated the limiting values of our results for configurations when two of 
the external momenta, say a and b, become collinear. As expected, at most single pole terms 1/(ab) or 
1/[ab] have been found 3 . Due to color ordering some color amplitudes must remain finite for certain collinear 
configurations. Our amplitudes fulfill also these requirements. 
R e f e r e n c e s  
[1] Z. Xu, D. Zhang and L. Chang, Nucl. Phys. B 291 (1987) 392. 
[2] R. Kleiss and W.J. Stirling, Nucl. Phys. B 262 (1985) 235. 
[3] J. Gunion and Z. Kunszt, Phys. Lett. B 161 (1985) 333. 
[4] For reviews see: M.L. Mangano and S.J. Parke, Phys, Rep. 200 (1991) 301; 
R. Gastmans and T.T. Wu, International Series of Monographs on Physics, Vol. 80 (Clarendon, Oxford, 1990). 
15] Z. Bern and D.A. Kosower, Nucl. Phys. B 379 (1992) 451. 
[6] Z. Bern L. Dixon and D.A. Kosower, Phys. Rev. Lett. 70 (1993) 2677. 
[7] Z. Kunszt, A. Signer and Z. Tr6cs~inyi, Nucl. Phys. B 420 (1994) 550. 
I8] Z. Kunszt, A. Signer and Z. Tr6cstlnyi, Nucl. Phys. B 411 (1994) 397. 
[9] G. van Oldenborgh and J.A.M. Vermaseren, Zeitschr. Phys. C 46 (1990) 425. 
[ 10] Z. Bern L. Dixon D. Dunbar and D.A. Kosower, hep-ph/9403226, SLAC-PUB-6415 . 
3 The structure of the collinear limits of the ggggg and 77qggg one-loop amplitudes were studied in a recent paper by Bern et al. [ 10]. 


One-loop radiative corrections to the helicity amplitudes of QCD processes involving four quarks and one gluon

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One-loop radiative corrections to the helicity amplitudes of QCD
  processes involving four quarks and one gluon

One-loop radiative corrections to the helicity amplitudes of QCD processes involving four quarks and one gluon

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