Feynman parameter

A Feynman parameter combines propagator denominators; for example,

\frac{1}{A B} = \int_{0}^{1} \frac{d x}{[ x A + ( 1 - x ) B ] ^{2}} .

After a momentum shift, this often converts a loop integral into a rotationally symmetric one.

Table of contents
- Feynman parameter for a simple and a squared propagator Feynman parameter
- Schwinger parameterization Feynman parameter

Feynman parameter for a simple and a squared propagator ( $1/ (A B^{2}) = 2 \int_{0}^{1} (1 - x) [x A + (1 - x) B]^{- 3} d x$ )

Differentiate

1/ (A B) = \int_{0}^{1} [x A + (1 - x) B]^{- 2} d x

with respect to

B

to obtain the displayed identity. The squared denominator supplies the factor

2 (1 - x)

. Changing the parameter to

1 - x

also changes which denominator carries each weight; its use must be consistent with the shifted mass parameter.

Schwinger parameterization ( $A^{- v} = Γ (v)^{- 1} \int_{0}^{\infty} t^{v - 1} e^{- t A} d t$ )

For

Re A > 0

and

Re v > 0

, rescaling the defining gamma function integral proves this representation, with compatible complex-power branches. It turns denominator powers into exponential factors, making Gaussian integrals possible. Combining two parameters by their sum and ratio gives the usual Feynman parameter identity. Outside its convergence domain a further continuation or pole prescription is needed.

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