# Maximally helicity violating (MHV) amplitudes ## Introduction [Wiki](https://en.wikipedia.org/wiki/MHV_amplitudes): maximally helicity violating amplitudes (MHV) are amplitudes with n massless external gauge bosons, where $n–2$ gauge bosons have a particular helicity and the other two have the opposite helicity. These amplitudes are called MHV amplitudes, because at tree level, they violate helicity conservation to the maximum extent possible. The tree amplitudes in which all gauge bosons have the same helicity or all but one have the same helicity vanish. ## Related names - **MHV**: $A_{n}\left(1^{-} 2^{-} 3^{+} \ldots n^{+}\right)$ (e.g. with $n$ gluons in [[0071 Yang-Mills|YM]]) - **anti-MHV**: 2 positive and n-2 negative. obtained by flipping MHV - **Next-to-MHV**, or **NMHV**: 3 negative helicity and n-3 positive helicity gluons - **N$^K$MHV**: $K+2$ negative and $n-K-2$ positive ## Gravity - original: BGK formula in [[BerendsGieleKuijf1988]] - another form by [[MasonSkinner2008]][](http://arxiv.org/abs/0808.3907) - [[2009#Nguyen, Spradlin, Volovich, Wen]] - [[2012#Hodges]]: a simple expression using matrix determinant - 3-point: $M_{3}\left(1^{-} 2^{-} 3^{+}\right)=\frac{\langle 12\rangle^{6}}{\langle 23\rangle^{2}\langle 31\rangle^{2}}$ ## Gluons - see [[0072 Parke-Taylor n-gluon tree amplitude|Parke-Taylor formula]] ## Refs - [[Rsc0003 ElvangHuang Scattering amplitudes]] ## Related - [[0072 Parke-Taylor n-gluon tree amplitude]] - [[0058 BCFW]] - [[0352 CSW relations]] aka MHV rules