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November 7, 2013 281
10.8 Some techniques for one-loop diagrams
10.8.1 The ‘Feynman trick’
Consider n positive real numbers a j , j = 1..n. We can write
n∏
j=1
1
a j
=
∫ ∞
0
dz 1 dz 2 · · · dz n exp(−z 1 a 1 − z 2 a 2 − · · · − z n a n ) (10.124)
In this integral, we may define s as the sum of the z’s, and define x j as z j /s, as
follows:
n∏
j=1
1
a j
=
∫ ∞
0
dz 1 dz 2 · · · dz n ds dx 1 dx 2 · · · dx n
× exp(−z 1 a 1 − z 2 a 2 − · · · − z n a n )
× δ(z 1 + z 2 + · · · + z n − s)
(
× δ x 1 − z ) (
1
δ x 2 − z )
2
· · · δ
s
s
We can now eliminate the z’s in favor of the x’s:
n∏
j=1
1
a j
=
∫ ∞
0
dx 1 dx 2 · · · dx n ds
(
x n − z n
s
(
)
× s n−1 exp − s(x 1 a 1 + x 2 a 2 + · · · + x n a n )
)
. (10.125)
× δ(x 1 + x 2 + · · · + x n − 1) . (10.126)
A last integral over s then gives us the formula known as the Feynman trick:
n∏
j=1
1
a j
= Γ(n)
∫ 1
0
dx 1 dx 2 · · · dx n
(
x 1 a 1 + x 2 a 2 + · · · + x n a n
) −n
× δ(x 1 + x 2 + · · · + x n − 1) . (10.127)
For example,
∫
1
1
=
a 1 a 2
0
dx
1
(
xa1 + (1 − x)a 2
) 2
. (10.128)
10.8.2 A general one-loop integral
We shall compute the integral
∫
I =
d D q
(2π) D
|⃗q| n
(|⃗q| 2 + a 2 ) m (10.129)