# Cachazo-Svrcek-Witten (CSW) relations (or MHV rules) The CSW relations allow one to construct all colour-ordered amplitudes in YM from [[0061 Maximally helicity violating amplitudes|MHV]] vertices. ## Refs - introduction in [[Rsc0003 ElvangHuang Scattering amplitudes]] - originals - [[Witten200312]]: uses the fact that all tree-level color-ordered amplitudes are related to algebraic curves in [[0330 Twistor theory|twistor space]] - [[CachazoSvrcekWitten2004]] - proof - [[Risager2005]][](https://arxiv.org/pdf/hep-th/0508206.pdf) - [[ElvangFreedmanKiermaier2008]] and more - colour-dressed version - [[2006#Duhr, Hoche, Maltoni]] ## Steps - shift - $| \hat{i}]=|i]+z c_{i} | X]$ for some arbitrary reference spinor $|X]$ - holomorphic: angular unshifted - further choice: - *Risager* shift: only $c_1$ to $c_3$ non-zero - *all-line* shift: all $c_i\ne0$ - simplification - N${}^K$MHV fall off as $1/z^K$ for large $z$ under all-line shift - so except MHV ($K=0$), all NMHV can be recursed - i.e. $MHV$ are building blocks - general result - any N${}^K$MHV tree amplitude can be written as precisely $K+1$ MHV amplitudes ## NMHV ($K=1$) - possible sub-amplitudes are anti-MHV${}_3\times$ NMHV and MHV $\times$ MHV - first one vanish - so just MHV