Scattering Amplitudes BCFW relation (A question)

[AT80]

I have a very trivial question to ask and it would be great if someone could
help me in this.

The statement that '3-point amplitudes' and the location of poles are sufficient to
determine any n-point amplitude at tree level is confusing to me. Don't I also need to know
4-point amlitudes, for example in YM theory ? The reason I say this is
the 4-point vertex can not be broken down. That is the residues obtained upon putting
propagators onshell will also contain 4-point functions.

What am I missing ?

Thanks for your help in advance.

 

[chrispb]

In YM, the three point vertex coefficient and the four point coefficient are related to each other; one goes like g p_mu, and the other is g^2. The second is necessary to have a gauge-invariant Lagrangian. What's neat about YM is that given the three-point vertex at tree level, one can use it to build up the n-point function from BCFW, and you get the same answer as if you'd done Feynman rules, without having the miscellaneous gauge degrees of freedom to keep track of.

In a theory where the three- and four-point vertices were uncorrelated (scalar field theory with V = g phi^3 + lambda phi^4, for example), then you could BCFW up contributions to amplitudes that contained arbitrary powers of g, but no powers of lambda, using just the three-point function, but you wouldn't get the complete answer for the amplitudes. Hope this helps!

 

[AT80]

Thanks a lot Chrispb for a fast reply. It certainly helps.