Tree recursion under an explicit vanishing-boundary hypothesis

Let $n\ge 4$ and let $p_1,\dots ,p_n\in {\mathbb{C}}^4\setminus \{0\}$ be complex null momenta with ${\sum }_{a=1}^n{p}_a=0$. Choose spinor factors $\kappa (p_a)={\lambda }_a{\tilde{\lambda }}_a$, whose existence is established by the null-factorization result on this page, and choose distinct labels $i,j$. The spinor factors and $h$ use the convention in the spinor-helicity definition. For dimensionless $z\in \mathbb{C}$, make the two-line shift

${\lambda }_i(z)={\lambda }_i+z{\lambda }_j,{\tilde{\lambda }}_j(z)={\tilde{\lambda }}_j-z{\tilde{\lambda }}_i,$

leaving ${\tilde{\lambda }}_i$, ${\lambda }_j$ and every other spinor unchanged; set $p_a(z)={\kappa }^{-1}({\lambda }_a(z){\tilde{\lambda }}_a(z))$.

Let $A(z)=F(z)\mathrm{/}G(z)$, where $F,G\in \mathbb{C}[z]$, $G\equiv ̸0$, and common factors are canceled. A finite pole is a zero of the reduced denominator $G$; it is simple when that zero has multiplicity one. Assume $A$ is regular at $z=0$, has distinct simple finite poles $z_1,\dots ,z_m\mathrm{\ne }0$ and no other finite poles, and $A(z)\to 0$ as $z\to \mathrm{\infty }$. For a simple pole, define $r_{\alpha }={\mathrm{Res}}_{z=z_{\alpha }}A(z)$ as the coefficient of $(z-z_{\alpha }{)}^{-1}$ in its Laurent expansion. Then

$A(0)=-{\sum }_{\alpha =1}^m\frac{r_{\alpha }}{z_{\alpha }}=-{\sum }_{\alpha =1}^m{\mathrm{Res}}_{z=z_{\alpha }}\frac{A(z)}{z}\mathrm{.}$

For a factorization channel, choose a subset $S_{\alpha }\subset \{1,\dots ,n\}$ with $2\le \mathrm{\mid }{S}_{\alpha }\mathrm{\mid }\le n-2$, containing exactly one of $i,j$. Put $P_{\alpha }(z)={\sum }_{a\in S_{\alpha }}{p}_a(z)$ and $Q_{\alpha }(z)=h(P_{\alpha }(z),P_{\alpha }(z))$. Assume every pole is assigned a distinct such channel with $Q_{\alpha }(z_{\alpha })=0$ and $Q_{\alpha }^{\mathrm{\prime }}(z_{\alpha })\mathrm{\ne }0$. Since $z$ is dimensionless, $Q_{\alpha }^{\mathrm{\prime }}$ has the same inverse-length-squared units as $Q_{\alpha }$. If the residue factorization

$r_{\alpha }=\frac{{\displaystyle \sum _{s\in {\mathcal{H}}_{\alpha }}{A}_{L,s}(z_{\alpha })A_{R,-s}(z_{\alpha })}}{Q_{\alpha }^{\mathrm{\prime }}(z_{\alpha })}$

holds, where ${\mathcal{H}}_{\alpha }$ is the finite set of internal helicity labels and $A_{L,s},A_{R,-s}$ are the supplied complex values for the two lower-point channel subproblems at the shifted factorization kinematics, with the internal momenta $P_{\alpha }(z_{\alpha })$ and $-P_{\alpha }(z_{\alpha })$ null, then

$A(0)=-{\sum }_{\alpha =1}^m\frac{1}{z_{\alpha }{Q}_{\alpha }^{\mathrm{\prime }}(z_{\alpha })}{\sum }_{s\in {\mathcal{H}}_{\alpha }}{A}_{L,s}(z_{\alpha })A_{R,-s}(z_{\alpha })\mathrm{.}$

This is the BCFW recursion for the stated shift. The residue-factorization equality and the vanishing at infinity are hypotheses here; this theorem does not derive them for a chosen QFT model.

Complex null momenta and helicity weights

Null momenta admit unique spinor-helicity classes

Elvang and Huang, Scattering Amplitudes, §§3.1–3.3

Marzolla, The four-dimensional on-shell three-point amplitude in spinor-helicity formalism and BCFW recursion relations, §5

Britto, Cachazo, Feng, and Witten, Direct Proof of Tree-Level Recursion Relation in Yang-Mills Theory, §§2–3.1

The proof establishes a rational-function identity and a conditional substitution. Rationality, simple factorization poles, the residue product, and decay for a particular tree amplitude remain separate model-specific obligations. No loop integration, nonperturbative S-matrix construction, or convergence result for perturbative gravity follows.