- Source: Schwinger parametrization
Schwinger parametrization is a technique for evaluating loop integrals which arise from Feynman diagrams with one or more loops.
Using the well-known observation that
1
A
n
=
1
(
n
−
1
)
!
∫
0
∞
d
u
u
n
−
1
e
−
u
A
,
{\displaystyle {\frac {1}{A^{n}}}={\frac {1}{(n-1)!}}\int _{0}^{\infty }du\,u^{n-1}e^{-uA},}
Julian Schwinger noticed that one may simplify the integral:
∫
d
p
A
(
p
)
n
=
1
Γ
(
n
)
∫
d
p
∫
0
∞
d
u
u
n
−
1
e
−
u
A
(
p
)
=
1
Γ
(
n
)
∫
0
∞
d
u
u
n
−
1
∫
d
p
e
−
u
A
(
p
)
,
{\displaystyle \int {\frac {dp}{A(p)^{n}}}={\frac {1}{\Gamma (n)}}\int dp\int _{0}^{\infty }du\,u^{n-1}e^{-uA(p)}={\frac {1}{\Gamma (n)}}\int _{0}^{\infty }du\,u^{n-1}\int dp\,e^{-uA(p)},}
for Re(n)>0.
Another version of Schwinger parametrization is:
i
A
+
i
ϵ
=
∫
0
∞
d
u
e
i
u
(
A
+
i
ϵ
)
,
{\displaystyle {\frac {i}{A+i\epsilon }}=\int _{0}^{\infty }du\,e^{iu(A+i\epsilon )},}
which is convergent as long as
ϵ
>
0
{\displaystyle \epsilon >0}
and
A
∈
R
{\displaystyle A\in \mathbb {R} }
. It is easy to generalize this identity to n denominators.
See also
Feynman parametrization