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Factorizing one­loop contributions to two­loop Bhabha
scattering and automatization of Feynman diagram
calculations
J. Fleischer, O.V. Tarasov
Fakult˜at f˜ur Physik, Universit˜at Bielefeld
and
T. Riemann, A. Werthenbach
DESY Zeuthen
Abstract
In higher order calculations a number of new technical problems arise: one needs diagrams
in arbitrary dimension in order to obtain their needed #­expansion, zero Gram deter­
minants appear, renormalization produces diagrams with `dots' on the lines, i.e. higher
order powers of scalar propagators. All these problems cannot be accessed by the `stan­
dard' Passarino­Veltman approach: there is not available what is needed for higher loops. We
demonstrate our method of how to solve these problems.
We are moving in the direction of two­loop Bhabha scattering, which is extremely important,
in particular at higher energies, for the luminosity determination of the coming accellerators.
Factorizing one­loop contributions are obtained directly by `squaring', e.g.:
1­loop â 1­loop = 2­loop contribution
and via renormalization. As an example mass­renormalization: in any Feynman diagram we
replace
1
k 2
- (m 2 + #(m 2 )) #
1
k 2
-m 2
(1 +
#(m 2 )
k 2
-m 2
),
thus obtaining `dotted' diagrams.

1
Diagram 1; topology u1_
A
e
e
e
E
Diagram 2; topology u2_
A
A
e
Diagram 3; topology u3_
A
A
e
E
E
Diagram 4; topology u4_
A
e
e
E
e
E
Diagram 5; topology u5_
A
A
e
e
e
E
e
E
Diagram 6; topology u6_
A
e
e
A
e
E
Diagram 7; topology u7_
e
A
A
e
E
E
Diagram 8; topology u7_
A
e
A
e
E
E
Diagram 9; topology u8_
A
A
e
E E
Diagram 10; topology u9_
e
e
A
E E
Bb.ps

Calculation of 1­loop integrals
Ref.
J.F. , F. Jegerlehner and O.V. Tarasov, Nucl.Phys. B566 (2000) p.423
q \Gamma p 2
q \Gamma p 3
q \Gamma p 1
q
q \Gamma p n\Gamma1
One­loop diagram with n external legs.
Tensor integrals
I (d)
n,r = # d d q
# d/2
n
#
j=1
q µ 1 . . . q µr
c # j
j
,
= T µ1 ...µ r ({p s }, {# j }, d)I (d)
n
where # j = `index'
c j = (q - p j ) 2
-m 2
j + i# for j < n and c n = q 2
-m 2
n + i#.
Tensor operator:
T µ 1 ...µ r
({p s }, {# j }, d + ) =
1
i r
r
#
j=1
#
#a µ j
exp # i # n-1
#
k=1
(ap k )# k -
1
4
a 2 # # # # # # # # # #
a j =0
# j =i# j
#=id +
.
with
# j =
#
#m 2
j
, d + I (d) = I (d+2) ,
Only scalar integrals remain to be evaluated !

Integrals with non­zero Gram determinants
# n =
# # # # # # # # # #
Y 11 Y 12 . . . Y 1n
Y 12 Y 22 . . . Y 2n
. . .
. . . . . . . . .
Y 1n Y 2n . . . Y nn
# # # # # # # # # #
.
where
Y ij = -(p i - p j ) 2 +m 2
i +m 2
j ,
``Modified Cayley determinant'' of the diagram with internal lines 1 . . . n
() n #
# # # # # # # # # # # # #
0 1 1 . . . 1
1 Y 11 Y 12 . . . Y 1n
1 Y 12 Y 22 . . . Y 2n
. . . . . . . . . . . . . . .
1 Y 1n Y 2n . . . Y nn
# # # # # # # # # # # # #
,
``Signed minors''
# j 1 j 2 . . .
k 1 k 2 . . .
# n
will be labeled by the rows j 1 , j 2 , . . . and columns k 1 , k 2 , . . . excluded from () n . E.g. we have
# n = # 0
0
# n
Recursion Relations:
# #
0
0
# # n
# j j + I (d)
n =
n
#
k=1
# #
0j
0k
# # n
# # d -
n
#
i=1
# i (k - i + + 1) # # I (d)
n .
The operators j ± etc. shift the indices # j # # j ± 1.
This relation reduces the indices and leaves the dimension.
The following one reduces simultaneously indices and dimension:
() n # j j + I (d+2)
n = # # - # #
j
0
# # n
+
n
#
k=1
# #
j
k
# # n
k - # # I (d)
n ,

A relation reducing the space time dimension is given by:
(d -
n
#
i=1
# i + 1) () n I (d+2)
n = # # # #
0
0
# # n
-
n
#
k=1
# #
0
k
# # n
k - # # I (d)
n .
Integrals with zero kinematic determinants
(d -
n
#
i=1
# i + 1) () n I (d+2)
n = -
n
#
k=1
# #
0
k
# # n
k - I (d)
n .
To increase again the dimension d one uses relation
n
#
j=1
# j j + I (d+2)
n = -I (d)
n .
As an example we show the relation for the three­point integral C 0 with zero
Gram determinant:
(s - 4m 2
e )C 0 (m e , 0, m e , m 2
e , m 2
e , s) = -
2(d - 3)
(d - 4)
B 0 (m 2
e , m 2
e , s) -
(d - 2)
m 2
e (d - 4)
A 0 (m 2
e )

Decomposition of the diagrams into Amplitudes (diagram 7)
O 1 = U(-p 2 ) I U(p 1 ) · V (p 4 ) I V (-p 3 )
O 2 = U(-p 2 ) “
p 4 U(p 1 ) · V (p 4 ) I V (-p 3 )
O 3 = U(-p 2 ) I U(p 1 ) · V (p 4 ) “
p 2 V (-p 3 )
O 4 = U(-p 2 ) “
p 4 U(p 1 ) · V (p 4 ) “
p 2 V (-p 3 )
O 5 = U(-p 2 ) # µ U(p 1 ) · V (p 4 ) # µ V (-p 3 )
O 6 = U(-p 2 ) # µ “
p 4 U(p 1 ) · V (p 4 ) # µ V (-p 3 )
O 7 = U(-p 2 ) # µ U(p 1 ) · V (p 4 ) # µ “
p 2 V (-p 3 )
O 8 = U(-p 2 ) # µ # # U(p 1 ) · V (p 4 ) # # # µ V (-p 3 )
O 9 = U(-p 2 ) # µ # # “
p 4 U(p 1 ) · V (p 4 ) # # # µ V (-p 3 )
O 10 = U(-p 2 ) # µ # # U(p 1 ) · V (p 4 ) # # # µ “
p 2 V (-p 3 )
O 11 = U(-p 2 ) # µ # # “
p 4 U(p 1 ) · V (p 4 ) # # # µ “
p 2 V (-p 3 )
O 12 = U(-p 2 ) # µ # # # # U(p 1 ) · V (p 4 ) # # # # # µ V (-p 3 )
For diagram 8:
U(-p 2 ) · V (-p 3 ) V (p 4 ) · U(p 1 )
The on­shell diagram reads
Diagram =
12
# j=1
A j O j .
with amplitudes A j .
Crossing relations (e.g. diagram 8 ## diagram 9)
A (9)
1 = A (8)
1 - 2m e A (8)
7 - 4m e A (8)
9 + 2dA (8)
8
A (9)
2 = -A (8)
2 + 4A (8)
7 - 2(d - 4)A (8)
9 - 8m e A (8)
11
A (9)
3 = -A (8)
3 + 2A (8)
6 - 2(d - 2)A (8)
10 - 4m e A (8)
11
A (9)
4 = A (8)
4 - 2(d - 2)A (8)
11
A (9)
5 = -A (8)
5 + 2m e A (8)
6 + 4m e A (8)
7 + 8m e A (8)
9 + 4m e A (8)
10 - 12m 2
e A (8)
11 - (6d - 4)A (8)
12
A (9)
6 = -4m e A (8)
11 + A (8)
6
A (9)
7 = -4m e A (8)
11 + A (8)
7
A (9)
8 = -A (8)
8
A (9)
9 = A (8)
9
A (9)
10 = A (8)
10
A (9)
11 = -A (8)
11
A (9)
12 = A (8)
12
and exchanging t ## u. These relations also hold for crossing diagram 7 ## diagram 10 (s
## u). For the crossing 7 ## 8 (s ## t) only a change of sign occurs in the amplitudes. Thus:
we need to calculate only one SE, vertex and box.

8
1
2
3
4
1
+k­q1
2
+k­q2
3
+k­q3
4
+k
­1
­2
­3
­4
­1
+p1
­2
+p4
­3
­p2
­4
­p3
Diagram 8 topology u7_ (unique u7_) momentaset 1 (of 1)
A
e
A
e
e
E
e
E
BbInfo.ps page8

9
1
2
3
4
1
+k­Q1
2
+k­Q2
3
+k­Q3
4
+k
­1
­2
­3
­4
­1
+p1
­2
+p4
­3
­p2
­4
­p3
Diagram 9 topology u8_ (unique u8_) momentaset 1 (of 1)
A
e
A
e
e
E
e
E
BbInfo.ps page9

Further relations between amplitudes of one diagram (obtained by solving a system of
equations):
A 3 = A 2
A 7 = A 6
A 10 = A 9
A 8 =
1
4
A 1 -
m e
2
A 2 + ( A 12 )
A 9 =
1
4m e
A 1 -
1
2
A 2 -
1
2
A 6
A 11 =
1
4m 2
e
A 1 -
1
2m e
A 2
Thus: there are only 6 indpendent amplitudes.

Contribution to the di#erential cross section
d#
dcos#
=
## 2
2s
12
#
j=1
# #
B j (s, t)
4
A j (s, t) -
B j (t, s)
4
A j (t, s). # #
B 1 = -
4
t
s 2 + (2d - 8 +
24
t
m 2
e )s -
32
t
m 4
e - 8dm 2
e - 32
m 2
e
s
(t - 2m 2
e ),
B 2 =
4
t
m e (s 2 + 8m 4
e ) + (2(4 - d)m e -
24
t
m 3
e )s - 4(d + 2)m e t + 8(d - 2)m 3
e - 16
m e
s
(t - 2m 2
e ) 2 ,
B 3 = B 2 ,
B 4 = -
4
t
(m 2
e + t)s 2 + ((-4 - 2d)t + 2dm 2
e +
24
t
m 4
e )s - 2(d + 4)t 2 + 8(1 + d)m 2
e t - 8(d - 4)m 4
e
-
32
t
m 6
e -
8
s
(t - 2m 2
e ) 3 ,
B 5 = -
1
m e
B 6 + 4td +
8(st + 2t 2 + 2m 2
e s - 4m 2
e t)
s
,
B 6
m e
= -4
(d - 2)
t
(s 2
- 2tm 2
e ) + (2(d - 2)(d - 10) + 8(d - 4)
m 2
e
t
)s + 32
m 4
e
t
+ 32
m 2
e
s
(t - 2m 2
e ),
B 7 = B 6 ,
B 8 = -4
(d - 2) 2
t
s 2 + (2(d - 2)(d - 4)(d - 10) + 8(3d 2
- 8d + 8)
m 2
e
t
)s - 8(d - 2)(d - 4)t
-8(d - 2)(d 2
- 10d + 20)m 2
e - 32(d 2
- 2d + 2)
m 4
e
t - 32d
m 2
e
s
(t - 2m 2
e ),
B 9
m e
= 4(d - 2) 2 s 2
t
+ (2(2 - d)(d - 4)(d - 10) - 8(3d 2
- 8d + 8)
m 2
e
t
)s - 4(d 3 + 34d
-10d 2
- 28)t + 8(d 3
- 14d 2 + 52d - 60)m 2
e + 32(d 2
- 2d + 2)
m 4
e
t - 16d
(t - 2m 2
e ) 2
s
,
B 10 = B 9 ,
B 11 = (-4(d - 4) 2 t - 4(d - 2) 2 m 2
e )s 3 + (2(4 - d)(d 2
- 4d - 2)t 2 + 2(d 3 + 20d - 12d 2 + 32)m 2
e t
+8(3d 2
- 8d + 8)m 4
e )s 2 + ((20d 2
- 68d + 64 - 2d 3 )t 3 + 8(d - 4)(d 2
- 8d + 10)m 2
e t 2
- 8(d - 2)
(d 2
- 14d + 28)m 4
e t - 32(d 2
- 2d + 2)m 6
e )s - 8(d - 2)t 3 (t - 6m 2
e ) - 32(3d - 4)m 4
e t 2 + 64dm 6
e t,
B 12 = 4(d - 2) 3 s 3 + (-2(d - 2)(d 3
- 20d 2 + 110d - 208)t - 8(24d - 16 + d 3
- 12d 2 )m 2
e )s 2
+((12d 3
- 120d 2
- 352 + 400d)t 2
- 8(3d 3
- 42d 2 + 148d - 160)m 2
e t - 32(4 - 6d + 3d 2 )m 4
e )s
+16(3d - 2)t(t 2
- 4m 2
e t + 4m 4
e ).

Contribution to the di#erential cross section
d#
dcos#
=
## 2
2s
12
#
j=1
# #
B j (s, t)
4
A j (s, t) -
B j (t, s)
4
A j (t, s). # #
B 1 = -
4
t
(s 2
- 6m 2
e s + 8m 4
e ) - 32m 2
e - 32
m 2
e
s
(t - 2m 2
e ),
B 2 = 4
m e
t
s 2
- 24
m 3
e
t
s - 24m e t + 16m 3
e + 32
m 5
e
t - 16
m e
s
(t - 2m 2
e ) 2 ,
B 3 = B 2 ,
B 4 = -4(
m 2
e
t
+ 1)s 2 + (-12t + 8m 2
e + 24
m 4
e
t
)s - 16t 2 + 40m 2
e t - 32
m 6
e
t -
8
s
(t - 2m 2
e ) 3 ,
B 5 = 8
s 2
t
+ 24s + 24t - 32
m 4
e
t
+ 16
(t - 2m 2
e ) 2
s
,
B 6 = -8
m e s 2
t - 24m e s + 16m 3
e + 32
m 5
e
t
+ 32
m 3
e
s
(t -m 2
e ),
B 7 = B 6 ,
B 8 = -16
s 2
t
+ 192
m 2
e
t
s + 64m 2
e - 320
m 4
e
t - 128
m 2
e
s
(t - 2m 2
e ),
B 9 = 16
m e s 2
t - 192
m 3
e s
t - 48m e t - 96m 3
e + 320
m 5
e
t - 64
m e
s
(t - 2m 2
e ) 2 ,
B 10 = B 9 ,
B 11 = -16m 2
e s 3
- 32m 2
e (t - 8m 2
e )s 2
- 16(t 3
- 12m 4
e t + 0m 6
e )s - 16t 4 + 96m 2
e t 3
- 256m 4
e t(t -m 2
e ),
B 12 = 32s 3 + 96(t + 4m 2
e )s 2 + (96t 2 + 384m 2
e t - 896m 4
e )s + 160t(t - 2m 2
e ) 2 .

`Undotted' Diagrams; common factor e 2
(4#) d/2 (normalized to Born term)
Selfenergy ­ diagrams:
A 5 = A 0
4
s 2 [
1
d - 1 - 1] +B e
2
s
[
1
d - 1
(1 - z 2 ) - 1]
Vertex ­ diagrams:
A 2 = A 0
2
m e s 2 x 2 [2
1
d - 3 - (d - 4)] + B e
4m e
s 2 x 2 [1 - (d - 4)]
A 5 = 2(-A 0
1
m 2
e s
[2
1
d - 3 - x 2 z 2 ] + B e
1
s
[x 2 (1 + z 2 ) + d - 4] + F s
2
(d - 4)s
[1 + x 2 ])
`Dotted' Diagrams; common factor #(m e )/m e e 4
Selfenergy ­ diagrams:
A 5 = A 0
4
s 2 (d - 2)x 2 z 2
- B e
2
s
z 2 [1 - (d - 3)x 2 ]
Vertex ­ diagrams:
A 2 = -A 0
4
m e s 2 x 2 [
1
d - 3
x 2 + (d - 4)(1 + (d - 4)(1 - x 2 /2)) - 3 + 2x 2 ]
-B e
4m e
s 2 x 2 [(d - 4)(x 2
- (d - 4)x 2 z 2 ) + 2x 2 z 2 ] + F s
8m e
(d - 4)s 2 x 2 [1 + x 2 ]
A 5 = 2(A 0
2
s 2 x 2 [(
1
d - 3
+ 6
1
d - 5
) - (d - 4)(2x 2
- 1 + (d - 4)) + 7 - 4x 2 ]
+A 0
2
m 2
e s
[1 + 3
1
d - 5
] -B e
1
s
x 2 z 2 [(d - 4)(2x 2 + (d - 4)) + 2x 2 ] - F s
2
(d - 4)s
x 2 z 2 )

Notation:
A 0 = A 0 (m e ), B t = B 0 (m e , m e , t), B 0 = B 0 (0, 0, s), B e = B 0 (m e , m e , s)
F s = A 0 (m e )/m e
2 + B 0 (m e , m e , s), F t = A 0 (m e )/m e
2 + B 0 (m e , m e , t),
C 0 = C 0 (0, m e , 0, m 2
e , m 2
e , s), D 0 = D 0 (m e , 0, m e , 0, m 2
e , m 2
e , m 2
e , m 2
e , t, s).
F s , F t : only with factor 1
d-4 !
Further abbreviations:
r t = t/s, r u = u/s
and x 2 = 1/(1 - 4m 2
e /s), y 2 = 1/(1 - 4m 2
e /t), z 2 = 4m 2
e
s (x 2
- 1 = x 2 z 2 ).

`Undotted' Box­Diagrams; common factor e 2
(4#) d/2 r -2 t r -2 u
A 1 =
A 0
2r t
s 2 (-
1
(d - 3)
(1 + r u + 2r t r u x 2 + r u y 2
- z 2 ) - (r u + 2r t + 2r t r u x 2 + (1 + 5r u )y 2 )) -
B t r t
z 2
s
(2r t + (1 + 5r u )y 2 ) - B 0 r t
z 2
s
(1 + 2r t x 2 )r u + F t r t
z 2
s(d - 4)
(1 - 2r t - (1 + 4r u )y 2
- z 2 )
-C 0 r t z 2 (
1
(d - 3)
(r t + r u - z 2 (1 + r t + r u - z 2 )) - (r t + r u + 2r 2
t + 6r t r u - 2z 2 r t (1 - r u x 2 )))/2
+D 0 r t sz 2 (
1
(d - 3)
(r t + z 2 (r u - z 2 (1 + r t + r u - z 2 )))
+(r t (1 + 2r t + 2r u + 2r 2
t + 6r t r u + 4r 2
u ) - 2z 2 r t (1 + 2r t + 3r u - z 2 )))/4,
A 2 =
A 0
2r t
m e s 2 (-
1
(d - 3)
(1 + r t + r u + r t r u x 2
- z 2 ) - (r t + r u + r t r u x 2 + (1 + 2r u )y 2 ))
-B t
4m e r t
s 2 (r t + (1 + 2r u )y 2 ) -B 0
4m e r t
s 2 (r u + r t r u x 2 ) + F t
4m e r t
s 2 (d - 4)
(1 - (1 + 2r u )y 2
- z 2 )
-C 0
2m e r t
s
(
1
(d - 3)
((1 + r t )(r u + r t ) - z 2 (1 + r u + 2r t - z 2 )) - (r t + r 2
t
+(1 + 3r t )r u - z 2 r t (1 - x 2 r u ))) +D 0 m e r t (
1
(d - 3)
(r t + r 2
t + z 2 (r 2
t + (1 + r t )r u
-z 2 (1 + r u + 2r t - z 2 ))) + r t (1 + 3r t + 2r 2
t + 2r u (2 + 3r t + 2r u ) - z 2 (3 + 4r t + 6r u - 2z 2 ))),
A 4 =
A 0
2r t
m 2
e s 2 (-
1
(d - 3)
(1 - 2r t (1 + r t ) + r u (1 - 2r t ) + 2r t r u x 2
- r u y 2
- z 2 (1 - 2r t ))
-(r u + 2r t + 2r t r u x 2 + (1 - r u )y 2 )) -B t
4r t
s 2 (2r t + (1 - r u )y 2 ) -B 0
4r t
s 2 (r u + 2r t r u x 2 )
+F t
4r t
s 2 (d - 4)
(1 - 2r t r u - 4r t - 2r 2
t - y 2
- z 2 (1 - 2r t )) - C 0
2r t
s
(
1
(d - 3)
(r u + r t (1 - 2r t
-2r 2
t - 2r u (1 + r t )) - z 2 (1 + r u (1 - 2r t ) - r t (1 + 4r t ) - z 2 (1 - 2r t ))) - (r u + r t + 2r 2
t )
+2z 2 r t (1 - r u x 2 )) +D 0 r t (
1
(d - 3)
(r t (1 - 2r t - 2r 2
t ) + z 2 (r u (1 - 2r t - 2r 2
t ) - 2r 3
t
-z 2 (1 - r t - 4r 2
t + r u (1 - 2r t ) - z 2 (1 - 2r t )))) + r t (1 + 2r u + 2r t + 2r 2
t )
-2z 2 r t (1 + 2r t - z 2 )),

A 5 =
A 0
r t
m 2
e s
(-
1
(d - 3)
(2r t (1 - r t r u - r 2
t + r t r u x 2 ) + z 2 (r t r u - r t + 3r 2
t - r u y 2
- z 2 r t ))
-(2r t (1 + r t + 2r u + r t r u x 2 ) + 2z 2 (1 + r u )y 2 )) - B t
4r t
s
(r t (1 + r t + r u ) + z 2 (1 + r u )y 2 )
-B 0
4r t
s
(r t r u (1 + r t x 2 )) - F t
2r t
s(d - 4)
(2r t (r t + r 2
t + (2 + r t )r u ) + z 2 (r t (1 - r u - 3r t )
+(2 + 3r u )y 2 + z 2 r t )) - C 0 r 2
t (
1
(d - 3)
(2(r t - r 3
t + (1 - r 2
t )r u ) - z 2 (2 + r t + r u
-3r t r u - 5r 2
t - z 2 (1 - r u - 4r t + z 2 ))) - 2(r t + r 2
t + (1 + 2r t )r u - z 2 r t (1 - r u x 2 )))
+D 0 r 2
t s(
1
(d - 3)
(r t - r 3
t + z 2 (r t + 3r 2
t - 2r 3
t + 2(1 - r 2
t )r u - z 2 (2(1 + r t ) - 5r 2
t
+(1 - 3r t )r u - z 2 (1 - r u - 4r t + z 2 )))/2) + r t (1 + r t )(1 + r t + 2r u )
-z 2 (4r t (1 + r t ) - (r u - 4r t )r u - 2z 2 r t )/2),
A 6 =
A 0
r t
m e s 2 (
1
(d - 3)
(2(r t - r t r u x 2
- r u y 2 )) - 2(r t + r t r u x 2 + 2r u y 2 ))
-B t
4m e r t
s 2 (r t + 2r u y 2 ) - B 0
4m e r 2
t
s 2 r u x 2
- F t
2m e r t
s 2 (d - 4)
(4r t + 2r u y 2 )
+C 0
2m e r 2
t
s
(
1
(d - 3)
(r t + r u - z 2 ) + r t + 2r u - z 2 (1 - r u x 2 ))
-D 0 m e r 2
t (
1
(d - 3)
(r t + z 2 (r t + r u - z 2 )) + (r t + r u - z 2 )),
A 12 =
-A 0
r 2
t
m 2
e s(d - 3)
r u /2 + F t
r 2
t r u
s(d - 4) - C 0 r 2
t (
1
(d - 3)
((r t + r u )r u - z 2 r u ))/2
+D 0 r 2
t s(
1
(d - 3)
(r t r u + z 2 ((r t + r u )r u - z 2 r u )))/4.

`Dotted' Box­Diagrams; common factor #(me )
me
e 2
(4#) d/2 r -2 t r -2 u
A 1 =
A 0
1
s 2
((d - 4)(-2r t r u (1 + 2x 2 r t + y 2 ) - 2z 2 y 4 (1 - 3r u )) - 4r t r u (1 + 2x 2 r t + y 2 )
+2r t (1 - y 2 ) - 2z 2 (r t (1 - 2y 2 ) + y 4 (2 - 5r u ))) +B t
z 2
s
((d - 4)(-z 2 y 4 (1 - 3r u ))
+r t (1 - y 2 ) - z 2 ((1 - 2y 2 )r t + y 4 (1 - 2r u ))) +B 0
z 2
s
r t ((d - 4)((r t + y 2 )(1 - 2r u )
-r u (1 + 2x 2 r t )) + ((r t + y 2 )(1 - 2r u ) - r u (1 + 2x 2 r t ))) + F t
z 2
s(d - 4)
(r t (1 - y 2 )
-z 2 (r t (1 - 2y 2 ) + y 4 r u )) + C 0 z 2 2r t ((d - 4)(-(r t + y 2 )(1 - 2r u ) + r u (1 + 2r t )
+z 2 (2x 2 r t r u + y 2 (1 - 2r u ))) ++z 2 (2x 2 r t r u + 1 - r t + r u - z 2 ))/4
+D 0 z 2 sr t (-(d - 4)z 2 y 2 (1 - 2r u ) + 4r t r 2
u + z 2 ((y 2 (1 - 2r u ) - (1 - 2r t (1 - r u ))) + z 2 ))/4,
A 2 =
A 0
1
m e s 2
((d - 4)(-2r t (r t + r u + x 2 r t r u + y 2 (1 + r u )) + 2z 2 y 4 r u ) -
6
(d - 5)
r t (y 2 (1 + r u )
+r t ) + 2r t (1 - 3r t - 2r u - 2x 2 r t r u - y 2 (4 + 3r u )) - 2z 2 (r t - 2y 4 r u ))
+B t
4m e
s 2
(r t (1 - y 2 ) + z 2 ((d - 4)y 4 r u - r t + y 4 r u )) +B 0
4m e
s 2
r t (d - 3)(r t - r u + y 2
-2r t r u x 2 + z 2 r t r u x 2 ) + F t
m e
s 2 (d - 4)
(4r t ((1 - y 2 ) - z 2 )) + C 0
2r t m e
s
((d - 4)(2r t r u
-(r t - r u ) - y 2 + z 2 (r t + y 2 + x 2 r t r u )) + z 2 (1 + r t + (1 + x 2 r t )r u - z 2 ))
-D 0 2m e r t ((d - 4)z 2 (r t + y 2 )/2 + r t (1 + r t )r u + z 2 (1 - y 2
- r t (1 + r t + 2r u )
-z 2 (1 - r t ))/2),
A 4 =
A 0
1
m 2
e s 2 ((d - 4)(2r t (2r t (3 + r t + 2r u - x 2 r u ) + 4y 2
- r u (1 - 5y 2 )) - 2z 2 (2r t (r t + y 2 )
+(3 + 5r u )y 4 )) +
12
(d - 5)
r t (r t (3 + r t + r u ) + 2(1 + r u )y 2
- z 2 r t ) + 2r t (6r t (3 + r t )
-(3 - 10r t + 4r t x 2
- 15y 2 )r u + 12y 2 ) - 2z 2 (4r 2
t - 2r t r u y 2 + (6 + 9r u )y 4 ))
-B t
4
s 2
((d - 4)z 2 (2r t y 2 + (3 + 5r u )y 4 ) + r t r u (1 - y 2 ) - z 2 (2r t (r t + y 2 (1 + r u ))
-(3 + 4r u )y 4 )) +B 0
4
s 2
r t ((d - 4)(1 - y 2
- z 2 (1 + 2x 2 r t r u )) + (1 - y 2 )
-z 2 (1 + 2x 2 r t r u )) - F t
1
(d - 4)s 2
(4r t r u (1 - y 2 ) - 4z 2 (r t (2r t + 4y 2 + 2y 2 r u ) + r u y 4 ))
-C 0
2r t
s
((d - 4)(1 - y 2
- z 2 (1 - 2r t (2(1 + r u ) + r t - x 2 r u - z 2 ) - y 2 )) + 2z 2 r t (2 + r t
-x 2 r u - z 2 )) +D 0 r t z 2 ((d - 4)(2r t (2 + r t + 2r u - z 2 ) + y 2 ) - (1 + y 2 ) + z 2 ),

A 5 =
A 0
1
m 2
e s
((d - 4)(2r 2
t (r 2
t + 3r t + 2 + r u (2 + 2r t + 2r u - x 2 r t ))
-z 2 (r t (3r 2
t + 5r t + 2r t r u - y 2 (1 + 2r u )r u ) - z 2 (r t (r t - y 2 ) - (1 + r u )y 4 )))
+
3
(d - 5)
(2r 2
t (r 2
t + 3r t + 2 + r u (2 + 2r u + r t )) - z 2 r t (3r 2
t + 3r t + r t r u
-y 2 2(1 + r u + r 2
u ) - z 2 r t )) + 2r 2
t (3r 2
t + 9r t + 5r t r u + 6(1 + r u + r 2
u ) -
2x 2 r t r u ) + z 2 (r t (1 + r u - 22r t - 14r 2
t - 6r t r u + y 2 (3 + 17r u + 12r 2
u ))/2
+z 2 (r t (8r t - 2 - 4y 2 (1 - r u )) - 4y 4 (2 + r u ))/4)) - B t
z 2
s
(2(d - 4)(2r t (r t
+y 2 (1 + r u )) + z 2 y 2 (r t + y 2 (1 + r u ))) - r t (1 + 4r 2
t + (1 + 4r t )r u - 5y 2 (1 - r u ))
+z 2 (r t (1 + 2r t ) - 2y 2 (r t r u - y 2 ))) - B 0
1
s
(d - 3)(8r 2
t (1 + r t + r u )r u + z 2 r t (1 + 3r t
+r u (1 - 4r t ) + 2x 2 r t (1 + 2r t )r u + 2y 2 (1 + 2r u + 2r 2
u ) - z 2 (1 + 2x 2 r t r u ))) +
F t
z 2
(d - 4)s
(r t (1 + r u + 4r t (1 + r u + r t ) - y 2 (1 - 9r u )) - z 2 (r t (1 + 2r t - 2y 2 (1 + r u ))
-2y 4 r u )) + C 0 r t ((d - 4)(4r t (1 + r t + r u )r u + z 2 ((1 + r u - r t (1 + 10r u + 8r 2
u
+r t (8 + 8r u + 4r t )) + 4x 2 r 2
t r u + 2y 2 (1 + 2r u + 2r 2
u ))/2 - z 2 (1 - 4r t (1 + r u ) -
6r 2
t + y 2 (2 + 4r u + 4r 2
u ) + 2z 2 r t )/2)) - z 2 (r t (r u + 2r t (1 + r t ) - 2x 2 r t r u ) - z 2 (1 + r u
+3r t + 6r 2
t - z 2 (1 + 2r t ))/2)) +D 0 r t z 2 s((d - 4)(r t (1 + 2(r t + (1 + r t + r u )r u ) + r 2
t )
-z 2 (r t (2 + 2r u + 3r t ) - y 2 (1 + 2r u + 2r 2
u ) - z 2 r t )/2)
-r t (2 + 2r t + 2r t r u + r 2
u + 3/2r u ) + z 2 (r t (3 + 2r u ) - y 2 (1 + 2r u + 2r 2
u ))/2),
A 6 =
A 0
1
m e s 2
((d - 4)(2r t (r t - x 2 r t r u + y 2 ) - 2z 2 y 4 (1 - r u )) +
6
(d - 5)
(r t (r t
+y 2 (1 + r u ))) + 2r t (3r t - 2x 2 r t r u + 3y 2 ) + 2z 2 y 2 (2r t - y 2 (2 - r u ))) -
B t
4m e
s 2
((d - 4)z 2 y 4 (1 - r u ) + y 2 r t r u - z 2 y 2 (2r t - y 2 )) -
B 0
4m e
s 2
(d - 3)(r t (r t + y 2 )r u + z 2 x 2 r 2
t r u ) - F t
4m e
(d - 4)s 2
(r t r u y 2
- z 2 y 2 (2r t - r u y 2 )) +
C 0
2m e
s
r t ((d - 4)((r t + y 2 )r u - z 2 (r t - r u (r t x 2
- y 2 ))) + r t r u - z 2 (2r t - x 2 r t r u )) +
D 0 r t m e ((d - 4)(z 2 (r t + y 2 r u )) + r t (1 + 2r u + 2r t )r u + z 2 (r t (1 - r t - 3r u ) - y 2 r u + z 2 r t )),
A 12 =
-A 0
2
s 2
r t r u y 2
- B t
z 2
s
r t r u y 2
- F t
z 2
(d - 4)s
r t r u y 2 + C 0 2z 2 r 2
t r u /4 -D 0 z 2 sr 2
t r u /4,

One loop integrals occurring in Bhabha scattering
1. Two ­ point integrals
I (d)
G = I (d)
2 # # # # m 1 =0,m 2 =0,p 2
=
# #
(-p 2 ) (2- d
2
)
# # 2 - d
2 # # # d
2 - 1 #
2 d-3 # # d-1
2 #
I (d)
F = I (d)
2 # # # # m 1 =m,m 2 =m,p 2 =m 2
= (m 2 ) -(2- d
2
) # # 2 -
d
2
# 2 F 1
# #
1, 2 - d
2 ;
3
2 ;
p 2
4m 2
# # .
2. Three ­ point integrals
For two zero masses and final on­shell momenta we have (from J. F., F. Jegerlehner
and O.V Tarasov, unpublished)
JG
# # 2 - d
2 # =
(m 2 ) d
2 -3
2(d - 3) 2 F 1
# #
1, 1 ;
d-1
2 ;
1 -
p 2
4m 2
# # -
# ## # d-2
2 # (-p 2 ) d-4
2
2 d-2 # # d-1
2 # m 2 2 F 1
# #
1, d-2
2 ;
d-1
2 ;
1 -
p 2
4m 2
# # ,
where
JG = -C 0 (0, me, 0, m 2
e , m 2
e , p 2 ).
For the other master three­point integral with two equal masses and one zero
mass and final on­shell momenta the Gram determinant is zero. This integral
can be expressed in terms of a propagator integral as shown before. A compact
result also is
J F =
# # 2 - d
2 #
2 m 6-d 2 F 1
# #
1, 3 - d
2 ;
3
2 ;
p 2
4m 2
# # ,
where
J F = -C 0 (m e , 0, m e , m 2
e , m 2
e , p 2 ).