I How does the residue factorization form arise in BCFW recursion in QFT?

[Lapidus]

Below is a snipet from http://file:///C:/Users/Christian.Hollersen/Downloads/Britto_2011_2%20(1).pdf of Britto. Similar explanation can be found in the QFT books of Zee, Schwarz or the Scattering Amplitude text of Huang. Or any other text that covers BCFW recursion. My dumb question: how and why does the residue at this pole take this funny factorization form? (For clarifcation: residue is the just the word QFT people use for the numerator of a rational function with a simple pole, right?)

Thank you!

 

[OldManAndTheSeaQuark]

There are at least two ways to show the factorization of amplitudes on simple poles. An ancient proof using only properties of the S-matirx (analyticity, unitarity and cluster decomposition) can be found in Eden et al. "The Analytic S-matrix" sec. 4.5. For a more recent discussion see the nice review by Conde

http://pos.sissa.it/archive/conferences/201/005/Modave 2013_005.pdf

Alternatively there is a more local field theoretic proof given in Weinberg "The Quantum Theory of Fields Vol 1." sec. 10.2.