Spinor helicity, rational factorization and tree recursion

Let $V={\mathbb{C}}^4$ with complex-bilinear form $h(p,q)=p^0{q}^0-p^1{q}^1-p^2{q}^2-p^3{q}^3$. This is $h=-\eta$ for $\eta =\mathrm{diag}(-1,1,1,1)$, the adopted $(-+++)$ coordinate convention. Coordinates $p^{\mu }$ have inverse-length units. Let ${\mathbb{C}}^2$ and ${\tilde{\mathbb{C}}}^2$ be separate two-dimensional complex vector spaces, with undotted components ${\lambda }_a$ and dotted components ${\tilde{\lambda }}_{\dot{a}}$; no complex-conjugation relation is imposed between them.

Define the linear map $\kappa :V\to M_2(\mathbb{C})$ by

$\kappa (p)=p^0{I}_2+p^1{\sigma }_1+p^2{\sigma }_2+p^3{\sigma }_3=\left(\begin{array}{cc}{p}^0+p^3& p^1-i{p}^2\\ p^1+i{p}^2& p^0-p^3\end{array}\right),$

where the dimensionless Pauli matrices are ${\sigma }_1=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$, ${\sigma }_2=\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right)$, and ${\sigma }_3=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$. Its inverse sends $M=(M_{a\dot{a}})$ to

${\kappa }^{-1}(M)=(\frac{M_{1\dot{1}}+M_{2\dot{2}}}{2},\frac{M_{1\dot{2}}+M_{2\dot{1}}}{2},\frac{M_{2\dot{1}}-M_{1\dot{2}}}{2i},\frac{M_{1\dot{1}}-M_{2\dot{2}}}{2})\mathrm{.}$

For a matrix $M$ whose image is one-dimensional (rank one), a spinor factorization is a pair $(\lambda ,\tilde{\lambda })\in ({\mathbb{C}}^2\setminus \{0\})\times ({\tilde{\mathbb{C}}}^2\setminus \{0\})$ satisfying $M_{a\dot{a}}={\lambda }_a{\tilde{\lambda }}_{\dot{a}}$. The associated momentum is $p={\kappa }^{-1}(M)$. Spinor factors have units of inverse square-root length, so their outer product has momentum units. A little-group rescaling is $(\lambda ,\tilde{\lambda })\mapsto (t\lambda ,t^{-1}\tilde{\lambda })$ for dimensionless $t\in {\mathbb{C}}^{\times }$; it leaves $p$ unchanged. On factorization pairs of a fixed $M$, identify pairs related by such a rescaling; the spinor-helicity class of $M$ is an equivalence class for this ${\mathbb{C}}^{\times }$ action.

Set ${ϵ}^{12}=1$, ${ϵ}^{21}=-1$, and ${ϵ}^{11}={ϵ}^{22}=0$, and use a second copy with dotted indices. Raise indices by ${\lambda }^a={ϵ}^{ab}{\lambda }_b$ and ${\tilde{\lambda }}^{\dot{a}}={ϵ}^{\dot{a}\dot{b}}{\tilde{\lambda }}_{\dot{b}}$. Define angle and square brackets by $⟨\lambda r⟩={ϵ}^{ab}{\lambda }_a{r}_b$ and $[\tilde{\lambda }\tilde{r}]={ϵ}^{\dot{a}\dot{b}}{\tilde{\lambda }}_{\dot{a}}{\tilde{r}}_{\dot{b}}$.

For $n\ge 1$ and assigned half-integer helicity labels $h_1,\dots ,h_n\in \frac{1}{2}\mathbb{Z}$, an outgoing helicity-$h_i$ amplitude on a domain $D$ of $n$ spinor-factorization data is a function $A_n:D\to \mathbb{C}$. Require $D$ to be invariant under independent little-group rescalings. The amplitude condition is, for each leg $i$ and each admissible $t\in {\mathbb{C}}^{\times }$,

$A_n(\dots ,t{\lambda }_i,t^{-1}{\tilde{\lambda }}_i,\dots )=t^{-2h_i}{A}_n(\dots ,{\lambda }_i,{\tilde{\lambda }}_i,\dots )\mathrm{.}$

This definition imposes no rationality, factorization, or physical scattering assumption. The next item proves that every nonzero momentum $p$ with $h(p,p)=0$ has a spinor factorization, unique up to the stated rescaling.

Dirac Clifford matrices and energy projections

Elvang and Huang, Scattering Amplitudes, §§2.2, 2.4 and 2.6

Marzolla, The four-dimensional on-shell three-point amplitude in spinor-helicity formalism and BCFW recursion relations, §§2.1, 3 and 4.1

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

For every $p\in {\mathbb{C}}^4$, $\det \kappa (p)=h(p,p)$. Thus for every nonzero $p$ with $h(p,p)=0$, $\kappa (p)$ has a spinor factorization. If $\kappa (p{)}_{a\dot{a}}={\lambda }_a{\tilde{\lambda }}_{\dot{a}}={\lambda }_a^{\mathrm{\prime }}{\tilde{\lambda }}_{\dot{a}}^{\mathrm{\prime }}$, then there is a unique $t\in {\mathbb{C}}^{\times }$ such that ${\lambda }^{\mathrm{\prime }}=t\lambda$ and ${\tilde{\lambda }}^{\mathrm{\prime }}=t^{-1}\tilde{\lambda }$. Hence the spinor factorizations of each nonzero null momentum form exactly one equivalence class under the stated rescaling.

Complex null momenta and helicity weights

Elvang and Huang, Scattering Amplitudes, §2.2

Marzolla, The four-dimensional on-shell three-point amplitude in spinor-helicity formalism and BCFW recursion relations, §3

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

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.

Let $p\mathrm{\ne }0$ be a complex null momentum and choose its spinor factorization $\kappa (p{)}_{a\dot{a}}={\lambda }_a{\tilde{\lambda }}_{\dot{a}}$, which exists by the null-factorization result on this page. Raise spinor indices and define brackets as in the spinor-helicity definition. For an undotted reference spinor $r$ with $⟨\lambda r⟩\mathrm{\ne }0$ and a dotted reference spinor $\tilde{r}$ with $[\tilde{\lambda }\tilde{r}]\mathrm{\ne }0$, define polarization tensors

$E_{+}^{a\dot{a}}(p;r)=\sqrt{2}\frac{r^a{\tilde{\lambda }}^{\dot{a}}}{⟨\lambda r⟩},E_{-}^{a\dot{a}}(p;\tilde{r})=\sqrt{2}\frac{{\lambda }^a{\tilde{r}}^{\dot{a}}}{[\tilde{\lambda }\tilde{r}]}\mathrm{.}$

The reference spinors have the same inverse-square-root length units as $\lambda$ and $\tilde{\lambda }$, so these polarization tensors are dimensionless.

Associate a vector ${\epsilon }_{\pm }^{\mu }$ to each tensor by the inverse of the raised-index Pauli map $p\mapsto {ϵ}^{ab}{ϵ}^{\dot{a}\dot{b}}\kappa (p{)}_{b\dot{b}}$. A polarization change is called pure gauge here when its vector difference is a scalar multiple of $p^{\mu }$. For any two admissible references of the corresponding chirality, the polarization-vector difference is pure gauge. Therefore, for every complex current $J\in {\mathbb{C}}^4$ satisfying the explicitly defined conservation condition $h(p,J)=0$, the contraction $h({\epsilon }_{\pm }(p;r),J)$ is independent of the reference spinor. Current conservation is a separate hypothesis; no general QFT Ward identity is asserted.

Complex null momenta and helicity weights

Null momenta admit unique spinor-helicity classes

Elvang and Huang, Scattering Amplitudes, §§2.3–2.4

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

This proposition proves reference independence only after the current-conservation condition is supplied. It does not prove that every model-specific tree current satisfies that condition.