What is the essence of BCFW recursion techniques? I have recently briefly read about new methods as the Britto-Cachazo-Feng-Witten (BCFW) on-shell recursion method.


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*Can anybody please tell me about the essence of it? 

*What does it mean for the recursion to be on-shell? 

*What happens to the vacuum? 

*What is the relation to gravity?
 A: *

*The essence and roughest sketch of the recursion relation is given in equation 1.1 on page 1 in the very page you linked. The scattering amplitude $A_n$ for $n$ gluons may be written as a sum of bilinear expressions (products) involving similar amplitudes $A_m$ with $m\lt n$ which have a certain dependence on the helicity (left-handed vs right-handed) and momentum of the internal gluon that connects the two subdiagrams by a virtual propagator. So the recursion relationship is a kind of a factorization (writing $A_n$ as a product of two other amplitudes) except that we must sum many such terms.

*The recursion relation is on-shell because all the amplitudes it can calculate are on-shell which means that the external gluons have to lie on the mass shall, i.e. satisfy $p^2=0$. The terminology "on shell" comes from massive particles where it means the right relation $p^2=m^2$ and the locus of $p^\mu$ vectors in the energy-momentum space that obey it looks like a hyperboloid, a shell. The formalism only works well for on-shell amplitudes because it arose – and is probably mathematically inseparably linked to – twistor theory which is much more efficient when dealing with massless particles and conformally symmetric theories.

*Vacuum is stable and has the amplitude $1$ to evolve to itself, and no other amplitude. Maybe I don't understand what you think should be happening to the vacuum.

*Most interestingly and convincingly, in July 2012, Freddy Cachazo and David Skinner proposed and later proved somewhat analogous formula for gravity, namely the $N=8$ supergravity, see

http://arxiv.org/abs/arXiv:1207.0741

