Questions on Britto-Cachazo-Feng-Witten (BCFW) recursion method

Question

I am reading about amplitude techniques, such as the spinor-helicity formalism: more specifically, the BCFW recursion relations. The lecture notes I am reading are titled as Scattering Amplitudes, and are written by Elvang and Huang. At some point in Chapter 3, the authors claim that I can shift some of the momenta of the outgoing particles, $$p_i^{\mu}\rightarrow \hat{p}_i^{\mu}=p_i^{\mu}+zr_i^{\mu}\tag{3.1}$$ such that the scattering amplitude at hand, $A_n$, becomes holomorphic, i.e. $A_n\rightarrow A_n(z)$. By applying some complex analysis techniques, we are allowed to write the physical amplitude $A_n$ (obtained by setting $z=0$) as $$A_n=-\sum_{z_{\mathcal{I}}}\text{Res}_{z=z_{\mathcal{I}}} \frac{A_n(z)}{z}+B_n\tag{3.4}$$ where $B_n$ is the residue of $A_n(z)/z$ at $z\rightarrow\infty$.

Later on, they argue that each of the residues in the same takes the following form $$\text{Res}_{z=z_{\mathcal{I}}}=-\hat{A}_L(z_{\mathcal{I}}) \frac{1}{P_I^2}\hat{A}_R(z_{\mathcal{I}})\tag{3.5}$$ where $\hat{A}_L$ and $\hat{A}_R$ are two on-shell sub-amplitudes of $A_n$, and $P_I$ is a scalar propagator connecting the Feynman diagrams of the two sub-amplitudes.

I have three questions that are related to that:

1. Why would the amplitude factorize like that?

2. Why are the sub-amplitudes on-shell? Is it guaranteed beforehand, or is it an empirical statement?

3. Later on in eq. (3.12), when they apply the BCFW recursion relations to an $n$ point amplitude $\hat{A}_L$ takes arguments $(1,\hat{P}_I^{h_I},k,...,n)$, whereas $\hat{A}_R$ takes arguments $(-\hat{P}_I^{-h_I},2,...,k-1)$. Why have the arguments corresponding to the propagator lines have been identified as the momentum carried by the scalar propagator?

Any help will be appreciated.

Answer 1

1. Because it is a connected tree-amplitude consisting of connected tree-diagrams. Cutting any line$^1$ would break each diagram apart, i.e. it would factorize into 2 connected tree-sub-amplitudes.

   It is less clear that each of the 2 connected tree-sub-amplitudes contains all connected tree-diagrams (with the pertinent external kinematics), but it is in fact true.

2. It is only the external legs of the sub-amplitudes that is on-shell, not necessarily internal lines.

   The shifted BCFW bridge momentum $\hat{P}_I(z\!=\!z_I)$ at the pole must also be on-shell, i.e. $$\hat{P}_I(z\!=\!z_I)^2=0. \tag{3.3}$$ (Here it is important that all lines are massless.) Therefore the new external legs of the sub-amplitudes are also on-shell.

   In contrast, be aware that the unshifted momenta $P_I$ above eq. (3.2) is not necessarily on-shell.

3. That essentially follows from momentum conservation within the sub-amplitudes.

References:

1. H. Elvang & Y.-t. Huang, Scattering Amplitudes, arXiv:1308.1697; section 3.1.

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$^1$ For a Dirac fermion line, the numerator of the propagator factorizes into external kinematic data for the 2 sub-amplitudes, cf. eq. (2.15).