Understanding BCFW Recursion: Exploring the Residue Factorization Form in QFT

In summary, the conversation discusses two ways to show the factorization of amplitudes on simple poles. The first method can be found in a paper by Eden et al. titled "The Analytic S-matrix," while the second method is discussed in Weinberg's "The Quantum Theory of Fields Vol 1." The residue at the pole is referred to as the numerator of a rational function in QFT, and the question posed is about the funny form of this residue.
  • #1
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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?)

bcwf.PNG


Thank you!
 
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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.
 

Related to Understanding BCFW Recursion: Exploring the Residue Factorization Form in QFT

1. What is BCFW recursion?

BCFW recursion is a powerful mathematical technique used in theoretical physics to calculate scattering amplitudes in quantum field theory (QFT). It stands for Britto-Cachazo-Feng-Witten recursion, named after the physicists who first proposed it. It involves breaking down a complex scattering process into simpler sub-processes and using mathematical properties, such as residue factorization, to recursively calculate the final amplitude.

2. How does BCFW recursion work?

BCFW recursion works by decomposing a scattering amplitude into a sum of simpler tree-level amplitudes, where each tree-level amplitude corresponds to a specific particle exchange in the scattering process. These tree-level amplitudes can then be calculated using Feynman diagrams and the mathematical concept of residue factorization, which allows for the recursive calculation of more complex amplitudes.

3. What is the significance of BCFW recursion in QFT?

The significance of BCFW recursion lies in its ability to greatly simplify the calculation of scattering amplitudes in QFT. By breaking down a complex amplitude into simpler tree-level amplitudes, BCFW recursion allows for more efficient and accurate calculations of scattering processes. It has also led to new insights and discoveries in theoretical physics, particularly in the study of quantum gravity and supersymmetric theories.

4. What is the Residue Factorization Form in QFT?

The Residue Factorization Form is a mathematical concept used in BCFW recursion to calculate scattering amplitudes in QFT. It is based on the Cauchy's Residue Theorem, which states that the value of a complex integral is equal to the sum of the residues of its poles. In the context of BCFW recursion, the Residue Factorization Form allows for the decomposition of a complex amplitude into simpler sub-amplitudes, making the calculation process more manageable.

5. What are some applications of BCFW recursion in theoretical physics?

Aside from its significance in simplifying scattering amplitude calculations, BCFW recursion has also been applied in various areas of theoretical physics. It has been used to study the behavior of gluons in high energy collisions, as well as to explore the properties of supersymmetry and the AdS/CFT correspondence. BCFW recursion has also been crucial in the development of new theoretical frameworks, such as the scattering equations formalism and the ambitwistor string theory.

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