Complex null momenta and helicity weights

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