The Britto, Cachazo, Feng, Witten (BCFW) recursion relationship is a way in which a complex scattering process can be decomposed into tree level diagrams. The picture attached describes the process
A set of gluon momenta entering a region (we set those leaving as the negative of entering as done in the STU symmetries) may be written as the sum of products of two diagrams. To start one chooses two gluons, here the k and n lines bolded. The sum is over all cyclically ordered distributions of gluons on each sub-amplitude (one with k and the other with n mometa) and one sums further over the helicities of the internal gluon.
To formulate this requires the use of bispinors, or what are in effect twistors. BCFW recursion is a development in Witten's "twistor revolution" in string theory. The momenta for a gluon, a null momenta as it is massless, is written as p_{aa'} = λ_aω_{a'}. This exterior product is a form of twistor, and the two spinors for the inner products (λ, λ') = ε_{ab}λ^aλ^b, [ω, ω'] = ε_{a'b'}ω^{a'}ω^{b'}. (I use parentheses because carrot signs cause trouble with this blog) There is a notation convention that one spinor type has ( ) as an inner product and the other a [ ] inner product. This is the convention that has emerged and is here to stay. If we have two momenta p_{aa'} = λ_aω_{a'} and q_{aa'} = λ'_aω'_{a'} then
p•q = λ_aω_{a'}λ'_bω'_{b'}δ^a_bδ^{a'}_{b'}
= = λ_aλ'_bω_{a'}ω'_{b'}δ^a_bδ^{a'}_{b'}
= (1/2)ε^{ab}λ_aλ'_bε^{a'b'}ω_{a'}ω'_{b'} = ½(λ, λ')[ω, ω']
A tree level amplitude A(1,2,...,n-1,n) of n cyclically ordered gluons. Each gluon has momenta p_i^{aa'} = λ_i^aω_i^{a'} corresponding to the two spinors. We pick out our two gluons of interest and define a momentum
p_k(z) = λ_k(ω_k - zω_n},
p_n(z) = (λ_n + zλ_k)ω_n
which are forms of the twistor equations. The momenta of the other gluons remain unchanged p_j(z) = p_j, for j =/= k or n. This theory involves then the transformations on the two elements of the bispinor as
ω_k --- > ω_k - zω_n
λ_n --- > λ_n + zλ_k.
Now examine the amplitude under this transformation
A(z) = A(p_1, p_2, ..., p_{k-1}, p_k(z), ... p_{n-1}, p_n(z)),
Now a complex function of z. This amplitude is on shell for all z and momenta are all conserved.
Breaking up the "blob" into these two parts is then equivalent to writing this amplitude as
A_k = sum_{ij}A_{j+1}(1/P_{ij}^2)A_{k - i+1}
The momentum flowing through a tree diagram is equal to the sum of external momenta. This sum in the propagator is the sum of momenta in adjacent external lines, where here the index j stands for k and n P_{ij}(z) = p_i(z) + ...+ p_j(z) = sum j_j + p_k(z) + p_n(z). By the construction above it is clear this turns out to be independent of z. In the summation we let k lie within the range i,j and n in the range j+1 ... .
The P_{ij}(z) = P_{ij} + z_kλ_n so the square is then P_{ij}^2(z) = P_{ij}^2 - z(λ_k|P_{ij}|ω_n], here evaluated on both pairs of spinors. Thus we have
1/P_{ij}^2(z) = 1/(P_{ij}^2 - z(λ_k|P_{ij}|ω_n]) =
(1/P_{ij}^2)(1/(1+z(λ_k|P_{ij}|ω_n])/P_{ij}^2)
or as
A(z) = sum_{ij}ρ_{ij}/(z - z_{ij}), for z_{ij} = z(λ_k|P_{ij}|ω_n]/P_{ij}^2
This then has simple poles at z = z_{ij} where the residues ρ_{ij} are evaluated with ∫A(z)dz/z. The residues correspond to internal lines which are placed on shell.
This then in general corresponds to the recursion relationship, where we set
A_k = sum_{ij}sum_hA^h_{j+1}(1/P_{ij}^2)A^{-h}_{k - i+1},
where now I have included the sum over helicity states. The recursion relationship is evident where the two terms in the numerator may be further decomposed. This procedure with P_{ij}(z) = P_{ij} + z_kλ_n evaluated at the pole reduces all off-shell processes in the "blob" on the left hand side of the diagram to an on-shell process in the evaluation of residues.
