Feynman parametrization is a technique for evaluating loop integrals which arise from Feynman diagrams with one or more loops. However, it is sometimes useful in integration in areas of pure mathematics as well.
which is valid for any complex numbers A and B as long as 0 is not contained in the line segment connecting A and B. The formula helps to evaluate integrals like:
If A(p) and B(p) are linear functions of p, then the last integral can be evaluated using substitution.
In more general cases, derivations can be done very efficiently using the Schwinger parametrization. For example, in order to derive the Feynman parametrized form of , we first reexpress all the factors in the denominator in their Schwinger parametrized form:
and rewrite,
Then we perform the following change of integration variables,
to obtain,
where denotes integration over the region with .
The next step is to perform the integration.
where we have defined
Substituting this result, we get to the penultimate form,
and, after introducing an extra integral, we arrive at the final form of the Feynman parametrization, namely,
Similarly, in order to derive the Feynman parametrization form of the most general case, one could begin with the suitable different Schwinger parametrization form of factors in the denominator, namely,
and then proceed exactly along the lines of previous case.
Michael E. Peskin and Daniel V. Schroeder , An Introduction To Quantum Field Theory, Addison-Wesley, Reading, 1995.
Silvan S. Schweber, Feynman and the visualization of space-time processes, Rev. Mod. Phys, 58, p.449 ,1986 doi:10.1103/RevModPhys.58.449
Vladimir A. Smirnov: Evaluating Feynman Integrals, Springer, ISBN 978-3-54023933-8 (Dec.,2004).
Vladimir A. Smirnov: Feynman Integral Calculus, Springer, ISBN 978-3-54030610-8 (Aug.,2006).
Vladimir A. Smirnov: Analytic Tools for Feynman Integrals, Springer, ISBN 978-3-64234885-3 (Jan.,2013).
Johannes Blümlein and Carsten Schneider (Eds.): Anti-Differentiation and the Calculation of Feynman Amplitudes, Springer, ISBN 978-3-030-80218-9 (2021).
Stefan Weinzierl: Feynman Integrals: A Comprehensive Treatment for Students and Researchers, Springer, ISBN 978-3-030-99560-7 (Jun., 2023).