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I have recently briefly read about new methods as the Britto-Cachazo-Feng-Witten (BCFW) on-shell recursion method.

  1. Can anybody please tell me about the essence of it?

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

  3. What happens to the vacuum?

  4. What is the relation to gravity?

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up vote 3 down vote accepted
  1. 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.

  2. 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.

  3. 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.

  4. 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

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