Schwinger parametrization

Template:Short description Script error: No such module "Unsubst". Schwinger parametrization is a technique for evaluating loop integrals which arise from Feynman diagrams with one or more loops. It is named after Julian Schwinger,^[1] who introduced the method in 1951 for quantum electrodynamics.^[2]

Description

Using the observation that

\frac{1}{A^{n}} = \frac{1}{(n - 1)!} \int_{0}^{\infty} d u u^{n - 1} e^{- u A},

one may simplify the integral:

\int \frac{d p}{A (p)^{n}} = \frac{1}{Γ (n)} \int d p \int_{0}^{\infty} d u u^{n - 1} e^{- u A (p)} = \frac{1}{Γ (n)} \int_{0}^{\infty} d u u^{n - 1} \int d p e^{- u A (p)},

for $R e (n) > 0$ .

Another version of Schwinger parametrization is:

\frac{i}{A + i ϵ} = \int_{0}^{\infty} d u e^{i u (A + i ϵ)},

which is convergent as long as $ϵ > 0$ and $A \in ℝ$ .^[3] It is easy to generalize this identity to n denominators.

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