Recently the calculations of all five-parton one-loop QCD amplitudes have
been completed. In this letter we describe how to get the corresponding
amplitudes with one gluon replaced by a photon and we give the explicit results
for the process $0 \to 2q 2Q 1\y$.Comment: 7 pages, late

Adrian Signer

Bern

Kunszt

Signer

Physics Letters B

English

arXiv

Recently the calculations of all five-parton one-loop QCD amplitudes have been completed. In this letter we describe how to get the corresponding amplitudes with one gluon replaced by a photon and we give the explicit results for the process 0 \to 2q 2Q 1\y

Signer, A

CERN Document Server

arXiv:hep-ph/9507442   28 Jul 1995
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1 Introduction
The study of photoproduction of 3+1 jets at HERA, as well as the study of direct pho-
ton production in connection with two jets at hadron colliders, requires the knowledge of
ve-parton amplitudes with external photons. Since by now all one-loop ve-parton QCD
amplitudes are known [1, 2, 3], it is natural to exploit these results in order to get the desired
amplitudes.
For amplitudes with no external quarks the photon can couple through a quark loop only.
As a result, the corresponding tree-level amplitudes vanish. We therefore concentrate on the
process with external quarks.
After the setup of the notation we describe how to get the amplitudes for 0 ! 2q2g1
and 0 ! 2q1g2 out of 0 ! 2q3g and we give the complete results for 0 ! 2q2Q1. Of
course, it is also possible to get the amplitudes for 0 ! 2q3 and the pure QED processes
out of the corresponding QCD amplitudes [3, 4], but these processes are phenomenological
less important. The construction of photonic one-loop amplitudes for one external quark
pair has also been discussed in [3].
The quarks (antiquarks) are denoted by q and Q (q and

Q), the photon by  and the
gluons are labeled from 1 to n. These labels may be used for the color index or the momentum
of the corresponding particle. The dependence on the helicities h
i
of the particles is often
suppressed
Throughout we use the normalization Tr(T
a
T
b
) = Æ
ab
, where T
a
are the SU(3) generators
in the fundamental representation. Furthermore, we use the following compact notation for
the color part:
(i : : : j)
qq
 (T
g
i
: : : T
g
j
)
qq
(1)
(i : : : j)  Tr(T
g
i
: : : T
g
j
) (2)
After replacing the structure constants f
abc
with  i=
p
2([a; b]c) and using the Fierz identity,
the color part of every Feynman diagram can be written as a linear combination of terms of
the form given in eqs. (1) and (2). The coeÆcient of an amplitude proportional to such a
color factor is called a subamplitude.
For the presentation of the (unrenormalized) amplitudes we follow closely the conventions
in ref. [2]. The amplitudes will be expressed as functions of dot products, spinor inner
products and a set of auxiliary functions (
n
ij
kl
o
x
; hijkli; P
ij
and F(i; j; k)). All these
functions as well as the ubiquitous prefactor c
 
are dened in ref. [2].
2 One external quark pair
Consider the process 0 ! 2q3g where one gluon, say g
3
, is replaced by a photon. The only
dierences of the two processes are the coupling constant and the color part. First of all we
have to replace g ! e
q
, where e
q
is the electric charge of the quark to which the photon
couples, and secondly we have to make the transition T
g
3
!
p
2.
1
1
The factor
p
2 appears because of the chosen normalization of the T
i
1
Although both steps seem to be rather trivial, they may cause some complications. The
former, because the photon may couple to quarks with dierent electric charge and the latter,
because we have to calculate additional color subamplitudes which belong to color structures
involving a trace Tr(T
g
3
). Contrary to the gluonic process this trace is not vanishing any
longer after the transition T
g
3
!
p
2.
As long as we have only one external quark and the corresponding antiquark, the diÆ-
culty concerning the replacement g ! e
q
can be handled as follows: First of all, for diagrams
without a fermion loop there is no problem, we always have g ! e
q
, where e
q
is the electric
charge of the external quark. For diagrams with a fermion loop we have to separate the dia-
grams where the photon couples to an internal fermion loop (called loop-coupled diagrams)
from those where the photon couples to the external quark (externally-coupled diagrams). In
the former case we have to replace g !
P
e
Q
 [e] where the sum extends over all fermions
Q in the loop whereas in the latter case we have the usual replacement g ! e
q
.
Since the subamplitudes 0 ! 2q3g are needed for the presentation of the results we
review the color structure of the helicity amplitudes of this process. At tree level it is given
by
A
(0)
(q; 1; 2; 3; q) = g
3
X
P (123)
(123)
qq
b
(0)
123
(q; 1; 2; 3; q) (3)
where P (1 : : : n) denotes all permutations of the elements 1 : : : n. As mentioned above,
we suppress the helicity dependence of the amplitudes in the notation. At one loop the
decomposition reads:
A
(1)
(q; 1; 2; 3; q) = g
3

g
4

2
"
X
P (1;2;3)
(123)
qq
b
(1)
123
(q; 1; 2; 3; q) (4)
+
X
i=1;2;3

(i)
qq
(kl) b
(1)
i
(q; 1; 2; 3; q) + (i) (kl)
qq
b
(1)
i;kl
(q; 1; 2; 3; q)

+ (qq) (123) b
(1)
q
(q; 1; 2; 3; q) + (qq) (321) b
(1)
q
(q; 1; 2; 3; q)
#
In this equation k and l are the two indices from the set f1; 2; 3g n i for a given i 2 f1; 2; 3g.
The subamplitudes b
(1)
i;kl
are not needed for 0! 2q3g since the corresponding color part van-
ishes. However, as we shall see, these subamplitudes have to be known for the construction
of the photonic processes. All subamplitudes can either be extracted from ref. [3] or they
are explicitly given in ref. [4].
If we replace g
3
! , the color decomposition at tree level is
A
(0)
(q; 1; 2; ; q) = g
2
p
2e
q
X
P (1;2)
(12)
qq
b
1;(0)
12
(q; 1; 2; ; q) (5)
while at one-loop we have
A
(1)
(q; 1; 2; ; q) = (6)
g
2
p
2e
q

g
4

2
2
4
X
P (1;2)
(12)
qq
b
1;(1)
12
(q; 1; 2; ; q) + (qq)(12) b
1;(1)
tr
(q; 1; 2; ; q)
3
5
2
Note that according to the discussion above the subamplitudes may contain terms propor-
tional to [e]=e
q
.
The tree-level results can be obtained immediately with
b
1;(0)
12
(q; 1; 2; ; q) = b
(0)
123
(q; 1; 2; ; q) + b
(0)
132
(q; 1; 2; ; q) + b
(0)
312
(q; 1; 2; ; q) (7)
For the subamplitude b
1;(1)
tr
the externally-coupled diagrams cancel each other, therefore,
we have no contributions proportional to e
q
N
F
. This fact enables us to get the subamplitudes
b
1;(1)
tr
out of those of the process 0! 2q3g. To do so we merely have to list all color factors of
0! 2q3g which, after the transition g
3
! , yield the color part (12). These are (123); (321)
and (3)
qq
(12). Adding up the corresponding subamplitudes yields the desired result
b
1;(1)
tr
(q; 1; 2; ; q) = (8)

b
(1)
q
(q; 1; 2; ; q) + b
(1)
q
(q; 1; 2; ; q) + b
(1)
3
(q; 1; 2; ; q)




N
F
!([e]=e
q
)
In order to present the results for the subamplitudes b
1;(1)
12
it is convenient to decompose
them further.
b
1;(1)
12
(q; 1; 2; ; q) = ic
 
B
1
12
(q; 1; 2; ; q) (9)
+ ic
 
N
F
B
1;ext
12
(q; 1; 2; ; q) + ic
 
[e]
e
q
B
1;int
12
(q; 1; 2; ; q)
In eq. (9) we put the loop-coupled diagrams in B
1;int
12
, whereas the externally-coupled dia-
grams are contained in B
1;ext
12
. Diagrams without a fermion loop are collected in B
1
12
. Note
that the subamplitude b
1;(1)
21
can be obtained from b
1;(1)
12
by simple relabeling [3, 4].
Neglecting for the moment the problems of the coupling of the photon to quarks of
dierent electric charge, the subamplitude b
1;(1)
12
(q; 1; 2; ; q) can be constructed in the same
manner as b
1;(1)
tr
. We nd
ic
 
B
1
12
(q; 1; 2; ; q) = (10)

b
(1)
123
(q; 1; 2; ; q) + b
(1)
132
(q; 1; 2; ; q) + b
(1)
312
(q; 1; 2; ; q) +N
c
b
(1)
3;12
(q; 1; 2; ; q)




N
F
!0
For the remaining needed functions B
1;ext
12
and B
1;int
12
we will give the explicit results for
the dierent helicity congurations, using the notation:
B(h
1
; h
2
; h
3
)  B(q;+; 1; h
1
; 2; h
2
; 3; h
3
; q; ) (11)
First of all we remark that it is suÆcient to give the subamplitudes for the helicity
congurations
2
( ;+;+); (+; ;+); (+;+; ) and (+;+;+). The results for all other helicity
congurations can be obtained with the help of discrete symmetries [3, 4].
For the loop-coupled diagrams we nd for all helicity congurations
B
1;int
12
(h
1
; h
2
; h
3
) =  
1
2
B
1;N
F
tr
(h
1
; h
2
; h
3
) (12)
2
Since the tree level results vanish for the helicity conguration (+;+;+), the corresponding one-loop
amplitudes are not really needed. We just give them for completeness
3
where B
1;N
F
tr
denotes the part of b
1;(1)
tr
which is proportional to [e]=e
q
. Furthermore we get
B
1;ext
12
( ;+;+) = B
1;ext
12
(+; ;+) = 0 (13)
For the other two helicity congurations the externally-coupled diagrams do not cancel and
we obtain
B
1;ext
12
(+;+; ) =
[q1][q2]
3 h12i[q3][q3]
(14)
and
B
1;ext
12
(+;+;+) =
hq1ihq2i[12]
3 hq3ihq3ih12i
2
(15)
With eq.(10) and eqs.(12 { 15) all ingredients of eq.(9) are given.
We turn now to the process 0! 2q1g2. If we start from the process 0! 2q3g and say
gluon 2 and gluon 3 become photons, there is only one color structure left.
A
(i)
(q; 1; 
2
; 
3
; q) = g

p
2e
q

2

g
4

2i
(1)
qq
b
2;(i)
(q; 1; 
2
; 
3
; q) (16)
The diagrams of this process are again decomposed into three classes. The rst contains
diagrams with no quark loop (i.e. we have to replace g
2
! e
2
q
), the second consists of
diagrams where one photon couples to the quark loop and the other photon couples to the
external quark (g
2
! [e]e
q
), and nally in the third class of diagrams both photons couple
to the quark loop (g
2
! [e
2
]). Here [e
2
] denotes
P
e
2
Q
. All diagrams of this process which
contain a quark loop and two photons coupled to the external quark line, are self-energy
insertions on external lines and vanish therefore in dimensional regularization.
Since the diagrams of the second class cancel each other for all helicity congurations it
is easy to get the helicity amplitudes.
b
2;(1)
(q; 1; 
2
; 
3
; q) =

X
P (1;2;3)
b
(1)
123
(q; 1; 
2
; 
3
; q) +N
c
b
(1)
1
(q; 1; 
2
; 
3
; q) (17)
+ N
c
b
(1)
3;12
(q; 1; 
2
; 
3
; q) +N
c
b
(1)
3;21
(q; 1; 
2
; 
3
; q)
+ N
c
b
(1)
2;13
(q; 1; 
2
; 
3
; q) +N
c
b
(1)
2;31
(q; 1; 
2
; 
3
; q)





N
F
!
[e
2
]
e
2
q
Obviously, the analytic expression obtained in this way is rather massive and further simpli-
cation is possible, but we do not pursue this any longer.
3 Two external quark pairs
For processes with external quarks which have dierent electric charge, the situation con-
cerning the replacement g ! e
q
is more complicated. Since the gluon couples in the same
way to the dierent quarks, but the photon does not, it is not possible to get the amplitudes
for the photonic process without additional work.
4
We denote the charge of q and Q by e
q
and e
Q
respectively. The color decomposition at
one loop (i = 1) and at tree level (i = 0) reads:
A
(i)
(q;

Q;Q; q; ) = (18)
g
2
p
2

g
4

2i

Æ
q

Q
Æ
Qq
e
q
u
(i)
2
(q;

Q;Q; q; ) + Æ
q

Q
Æ
Qq
e
Q
d
(i)
2
(q;

Q;Q; q; )
 
1
N
Æ
qq
Æ
Q

Q
e
q
u
(i)
1
(q;

Q;Q; q; ) 
1
N
Æ
qq
Æ
Q

Q
e
Q
d
(i)
1
(q;

Q;Q; q; )

Since the two loop-coupled diagrams cancel each other, the subamplitudes u
(1)
and d
(1)
do
not contain terms proportional to [e]=e
q
.
We write the one-loop subamplitudes in the following form:
u
(1)
1
(q;

Q;Q; q; ) = (19)
ic
 
u
l
1
(q;

Q;Q; q; ) +
ic
 
N
2
c
u
s
1
(q;

Q;Q; q; )  ic
 
N
F
N
c
u
N
F
(q;

Q;Q; q; )
u
(1)
2
(q;

Q;Q; q; ) = (20)
ic
 
N
c
u
l
2
(q;

Q;Q; q; ) +
ic
 
N
c
u
s
2
(q;

Q;Q; q; ) + ic
 
N
F
u
N
F
(q;

Q;Q; q; )
The subamplitudes d
(1)
1
and d
(1)
2
are decomposed in the same way.
Since we can use parity to change all helicities, it is suÆcient to consider only helicity
congurations with positive photon helicity. Furthermore, we have the relations
d
(i)
i
(q; h
q
;

Q;h

Q
; Q; h
Q
; q; h
q
; ; h

) = u
(i)
i
(

Q;h

Q
; q; h
q
; q; h
q
; Q; h
Q
; ; h

) (21)
For the remainder of this section we use the notation
u
(i)
(h
q
; h
Q
; h

)  u
(i)
(q; h
q
;

Q;h

Q
; Q; h
Q
; q; h
q
; ; h

) (22)
We will give the subamplitudes u
(i)
(+;+;+) and u
(i)
(+; ;+) below. The remaining sub-
amplitudes can be obtained either with parity, eq. (21) or
u
(i)
( ;+;+) =  u
(i)
(+; ;+)j
q$q;Q$

Q
(23)
u
(i)
( ; ;+) =  u
(i)
(+;+;+)j
q$q;Q$

Q
(24)
First we recall the tree level results, which can be obtained easily from the tree level
results of 0! 4q1g
u
(0)
1
(+;+;+) = u
(0)
2
(+;+;+) =  i
hq

Qi
2
h

QQihqihqi
(25)
u
(0)
1
(+; ;+) = u
(0)
2
(+; ;+) = i
hqQi
2
h

QQihqihqi
(26)
In order to present the results, it is convenient to decompose the one-loop subamplitudes
in the following way:
u
s
1
(h
q
; h
Q
; h

) = u
A
(h
q
; h
Q
; h

) + u
B
(h
q
; h
Q
; h

) (27)
u
s
2
(h
q
; h
Q
; h

) =  2 u
A
(h
q
; h
Q
; h

)  u
B
(h
q
; h
Q
; h

) (28)
u
l
2
(h
q
; h
Q
; h

) = u
A
(h
q
; h
Q
; h

)  u
l
1
(h
q
; h
Q
; h

) (29)
5
So we have to give the results for u
l
1
; u
A
; u
B
and u
N
F
for the two helicity congurations
(+;+;+) and (+; ;+). The term proportional to N
F
is the simplest.
u
N
F
(h
q
; h
Q
; h

) = iu
(0)
1
(h
q
; h
Q
; h

)
 
2
3
P

QQ
"
+
10
9
!
(30)
For the leading terms of u
1
we get
u
l
1
(+;+;+) =  iu
(0)
1
(+;+;+)
 
P
q

Q
"
2
+
P
qQ
"
2
 
2
3
P

QQ
"
 
29
18
!
(31)
+
hqihq

Qi
2
[q]
2
2 hqih

QQi
n
q

QQ
o
2
 
hq

Qihq

Qi[q]
hqih

QQi
n
q

QQ
o
1
+
hqqihq

Qi[qQ]
2 hqihqis

QQ
+
hq

Qihq

Qi[q]
2 hqih

QQis

QQ
  iu
(0)
1
(+;+;+)

F(q;

Q;Q) + F(q;Q;

Q) + F(

Q; q; ) + F(Q; q; )

and
u
l
1
(+; ;+) =  iu
(0)
1
(+; ;+)
 
P
q

Q
"
2
+
P
qQ
"
2
 
2
3
P

QQ
"
 
29
18
!
(32)
 
hqihqQi
2
[q]
2
2 hqih

QQi
n
q

QQ
o
2
 
hqqihq

QihqQi[q

Q]
hqihq

Qihqi
n
q

QQ
o
1
 
hqQi
2
hqihqih

QQi
n
q

QQ
o
0
 
hq

Qih

QQi[

Q]
hq

Qih

Qi
n
q
qQ
o
1
 
hqqihqQi
hqihq

Qihqi
n
q
qQ
o
0
+
hq

QihQi[

Q]
hqih

Qi
n
q

Q
qQ
o
1
+
hqqihqQi[q

Q]
2 hqihqis

QQ
 
hqQihqQi[q]
2 hqih

QQis

QQ
  iu
(0)
1
(+; ;+)

F(q;

Q;Q) + hqq

QQi
2
F(q;Q;

Q)
+ hq

QQi
2
F(

Q; q; ) + F(Q; q; )

With the help of the relations
u
B
(+; ;+) = u
B
(+;+;+)j
Q$

Q
(33)
u
A
(+; ;+) =  u
A
(+;+;+)j
Q$

Q
(34)
and the functions
u
B
(+;+;+) = iu
(0)
1
(+;+;+)
 
P
qq
"
2
+
P

QQ
"
2
+
3 P

QQ
"
+
13
2
!
(35)
 
hqqi
2
h

QQi[qQ]
2
2 hqihqi
n

QQ
q
o
2
 
hqqih

Qi[q][Q]
hqi
n
qq

QQ
o
2
+
hqqihq

Qih

Qi[q]
hqi
2
h

QQi
n

QQ
qq
o
1
+
n

QQ
q
o
1

+
hq

Qi
2
2 hqihqih

QQi
n

QQ
q
o
0
+
hqqi
2
[qQ]
2
2 hqihqi[

QQ]s
q
 
hq

Qi[Q]
hqis

QQ
+ iu
(0)
1
(+;+;+)

F(q; q; ) + h

Qqqi
2
F(q; q; )

6
and
u
A
(+;+;+) =  iu
(0)
1
(+;+;+)
 
P
q

Q
"
2
+
P
qQ
"
2
 
P
qQ
"
2
 
P
q

Q
"
2
!
(36)
 
hqqihq

Qi[q]
hqQihqi
n
q

QQ
o
1
 
hqQih

QQi[Q]
hqQihQi
n
q
q

Q
o
1
 
hqQih

Qi[Q]
hqihQi
n
qQ
q

Q
o
1
+
hqqihq

Qi
hqihqQihqi
n

QQ
q

Q
o
0
  iu
(0)
1
(+;+;+)

F(q;

Q;Q)  h

QQqqi
2
F(q;

Q;Q) + F(

Q; q; )
  F(

Q; q; ) + F(

Q;Q; q) F(

Q;Q; q)
+ F(Q; q; )  h

QQqi
2
F(Q; q; )

we obtain all the needed functions.
References
[1] Z. Bern L. Dixon and D. A. Kosower, Phys. Rev. Lett. 70 (1993) 2677.
[2] Z. Kunszt, A. Signer and Z. Trocsanyi, Phys. Lett. 336B (1994) 529.
[3] Z. Bern, L. Dixon and D. A. Kosower, Nucl. Phys. B437 (1995) 259.
[4] A. Signer, PhD thesis, ETH-Zurich Nr. 11143 (1995) .
7


One-loop corrections to five-parton amplitudes with external photons

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