Schwinger parametrization

Template:Short description Template:More citations needed 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

$\frac{1}{A^n}=\frac{1}{(n-1)!}\underset{0}{\overset{\mathrm{\infty }}{\int }}du{u}^{n-1}{e}^{-uA},$

Julian Schwinger noticed that one may simplify the integral:

$\int \frac{dp}{A(p{)}^n}=\frac{1}{\mathrm{\Gamma }(n)}\int dp\underset{0}{\overset{\mathrm{\infty }}{\int }}du{u}^{n-1}{e}^{-uA(p)}=\frac{1}{\mathrm{\Gamma }(n)}\underset{0}{\overset{\mathrm{\infty }}{\int }}du{u}^{n-1}\int dp{e}^{-uA(p)},$

for Re(n)>0.

Another version of Schwinger parametrization is:

$\frac{i}{A+iϵ}=\underset{0}{\overset{\mathrm{\infty }}{\int }}du{e}^{iu(A+iϵ)},$

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

See also

• Feynman parametrization

References

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