Why split a momentum into two spinors?

The massless condition removes one degree of freedom

Through the recursion chapter we use four-dimensional spacetime, metric $(+,-,-,-)$ and natural units $\hbar=c=1$. A momentum $k^\mu=(E,k_x,k_y,k_z)$ satisfies $k^2=0$. The determinant of the matrix below is precisely that squared momentum. A nonzero matrix with zero determinant has only one independent column direction, so it factors into a column and a row.

Call these objects $\lambda_\alpha$ and $\widetilde\lambda_{\dot\alpha}$, or Weyl spinors. Each component has units of the square root of energy. They are variables for the same massless momentum, not two additional particles.

Source: Lance J. Dixon · R. Britto

An antisymmetric product retains spin information

The determinant made from two two-component columns changes sign when they are exchanged. This operation defines an angle bracket; square brackets use the corresponding dotted-spinor convention. We use $s_{ij}=\langle ij\rangle[ji]$. A numerical implementation must fix the square-bracket sign with this identity; conventions from different sources cannot be mixed.

Worked example: take $\lambda_1=\sqrt{2E}(1,0)$ and $\lambda_2=\sqrt{2E}(\cos(\vartheta/2),\sin(\vartheta/2))$. The determinant gives $\langle12\rangle=2E\sin(\vartheta/2)$. Its squared modulus is $2E^2(1-\cos\vartheta)$, twice the dot product of equal-energy future-directed null momenta. Near parallel momenta the spinor product vanishes linearly with angle.

Helicity fixes how a candidate must scale

Helicity is spin projected along the direction of motion. A massless gluon has two physical helicities, plus or minus one in units of the reduced Planck constant. An amplitude at fixed helicity may acquire a phase under changes of spinor convention, but its probability remains unchanged.

Multiply one spinor by a nonzero complex number $q$ and divide its partner by $q$. The momentum matrix is unchanged. This is little-group scaling. Physical real-state phase freedom uses $|q|=1$; general complex scaling is its analytic extension. An amplitude for helicity $h_i$ scales as $q^{-2h_i}$, providing a quick test of a proposed expression.

For a $(--++)$ four-gluon amplitude, a cyclic angle-bracket denominator contains each leg twice. A numerator $\langle12\rangle^4$ adds four powers to legs one and two. Their net weights are positive two; the remaining legs have weight negative two, as required.

Separate the color algebra

Gluons also carry color, an internal quantum label. Its symmetry is non-Abelian: the generator matrices need not commute. Normalize generators by $\operatorname{Tr}(T^aT^b)=\delta^{ab}$. A tree amplitude becomes a sum of color traces times kinematic partial amplitudes, with coupling $g^{n-2}$ stripped off. Each partial amplitude keeps a cyclic order.

For order $(1,2,3,4)$, a planar tree can have exchange channels $s_{12}$ and $s_{23}$. Cutting out legs $1$ and $3$ together breaks this cyclic order, so an independent $s_{13}$ exchange pole does not occur in this partial amplitude. The full colored amplitude recombines several orders.

All momenta are labeled outgoing. Physical incoming momenta enter with a minus sign, and incoming helicity labels are reversed under the all-outgoing convention. Momentum conservation then states that the sum is zero, making crossing comparisons easier to organize.

Source: Lance J. Dixon