# BCFW recursion ## Refs - original - [[2004#Britto, Cachazo, Feng]] - [[2005#Britto, Cachazo, Feng, Witten]] - proof - [[2008#Arkani-Hamed, Kaplan]] proof using background field perturbation theory - review/learning - [[Rsc0021 Nathaniel Craig 229A Gauge theory Winter 2021]] - [[Rsc0003 ElvangHuang Scattering amplitudes]] - [Truijen notes](https://web.archive.org/web/20140221230929/http://testweb.science.uu.nl/ITF/teaching/2012/Brecht%20Truijen.pdf) - for general field theories - [[CheungShenTrnka2015]][](https://arxiv.org/pdf/1502.05057.pdf) - celestial BCFW - [[2022#Hu, Pasterski (Aug)]] ## Assumptions - usually tree-level (but can consider loops) - massless particles - locality (local Lagrangian) ## Steps 1. define shifted momenta 2. notice $A_{n}=\hat{A}_{n}(z=0)$ is just the residue of the pole at origin for $\frac{\hat{A}_{n}(z)}{z}$ 3. use Cauchy's theorem: $A_{n}=-\sum_{z_{I}} \operatorname{Res}_{z=z_{I}} \frac{\hat{A}_{n}(z)}{z}+B_{n}$ 4. notice that the poles are whenever the propagator goes on-shell - $\operatorname{Res}_{z=z_{I}} \frac{\hat{A}_{n}(z)}{z}=-\hat{A}_{\mathrm{L}}\left(z_{I}\right) \frac{1}{P_{I}^{2}} \hat{A}_{\mathrm{R}}\left(z_{I}\right)$ - used $\hat{P}_{I}^{2}=-\frac{P_{I}^{2}}{z_{I}}\left(z-z_{I}\right)$ 5. check residue at infinity ## Rules - two shifted momenta appear on two sides - otherwise the propagator will not depend on $z$ - preserve cyclic ordering ## Properties - at every stage it involves only *on-shell* particles, as opposed to virtual particles that propagate inside conventional Feynman diagrams. (the internal propagator is on-shell, and the sub-diagrams are on-shell) - nice for ==tree-level== amplitudes - no branch cuts - only simple poles (follows from locality of Feynman diagrams) ## Relation to CCFT - there is a natural notion of factorisation in CCFT, which is a nice surprise - Basically, just like a 2-gluon OPE, one could construct a multi-gluon OPE and factorize a celestial amplitude on it. It is to be expected that factorization on OPE is in 1:1 correspondence with BCFW factorization channels - see [[2014#Goncalves, Penedones, Trevisani]] - see also [[2022#Hu, Pasterski (Aug)]] ## Factorisation v.s. collinear - collinear limits are only a subset, because of signs - $k_{L}^{2}=\sum_{i \neq j \in L} k_{i} \cdot k_{j}=\sum_{i \neq j \in L} \eta_{i} \eta_{j} \omega_{i} \omega_{j} z_{i j} \bar{z}_{i j}$ - i.e. even non-collinear setup can give a pole for $1/k_L$