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Schwinger parametrization

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Loop integral parametrization
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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

References

  1. Schwartz, M. D. (2014). "33". Quantum Field Theory and the Standard Model (9 ed.). Cambridge University Press. p. 705. ISBN 9781107034730.


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