Some questions about the BCFW reduction

Question

I am trying to give a fast sketch of what the BCFW reduction does and embed within it some questions at the steps which I don't seem to understand clearly. The first bullet point is sort of a very basic question about the formalism which I can't get!

Let $\{p_i\}_{i=1}^{i=n}$ be the momentum of the $n$-gluons whose scattering, $A(1,2,..,n)$ one is interested in. Let the $(n-1)^{th}$ have negative helicity and the rest be positive. So its an MHV scenario.

• For denoting the gluonic states why is it okay to use the spinor helicity formalism where for a massless Dirac particle of wave function $u(p)$ one uses the notation of, $|p> = \frac{1+\gamma^5}{2}u(p)$, $|p] =\frac{1- \gamma^5}{2}u(p)$, $<p| = \bar{u}(p)\frac{1+\gamma^5}{2}$, $[p| = \bar{u}(p)\frac{1-\gamma^5}{2}$? (..gluons are afterall not massless Dirac particles!..) What is going on? Why is this a valid description?

Then one defines analytic continuations of for the $(n-1)^{th}$ and the $n^{th}$ gluonic states as, $|p_n> \rightarrow |p_n(z)> = |p_n> + z |p_{n-1}>$ and $|p_{n-1}] \rightarrow |p_{n-1}(z)] = |p_{n-1}] - z |p_n]$.

Then the key idea is that if the amplitude as a function of $z$ tends to $0$ as $|z| \rightarrow \infty$ then one can write the analytically continued amplitude as $A(1,2,..,n,z) = \sum _{i} \frac{R_i}{(z-z_i)}$ where $z_i$ and $R_i$ are the poles and residues of $A(1,2,..,n,z)$

• Is there a quick way to see the above? (..though I have read much of the original paper..)

Answer 1

Your first question suggests to me that you should study basic references on the helicity formalism first. You might try the lecture notes by Lance Dixon or review article by Mangano and Parke.

Briefly, the idea is: given a momentum four-vector, you can express it as a matrix with spinor indices, $p_{\alpha {\dot \alpha}} = p_\mu \sigma^\mu_{\alpha {\dot \alpha}}$. If the momentum is lightlike, then $p_\mu p^\mu = 0$, which means this matrix has determinant zero. In that case, you can write it as an outer product: $p_{\alpha {\dot \alpha}} = \lambda_\alpha {\tilde \lambda}_{\dot \alpha}$. The spinors $\lambda$ and $\tilde \lambda$ are the basic objects you can express amplitudes in terms of. For instance, polarization vectors $\epsilon_\mu$ have the property $\epsilon^\mu p_\mu = 0$. Notice that, for any spinor $\mu_\alpha$, the vector $\mu_\alpha {\tilde \lambda}_{\dot \alpha}$ vanishes when dotted into $p$. In fact, a good choice of polarization vectors for positive helicity gluons is $\epsilon^+ = \frac{\mu {\tilde \lambda}}{\left<\mu~\lambda\right>}$, and for negative helicity $\epsilon^- = \frac{\lambda {\tilde \mu}}{\left[{\tilde \lambda}~{\tilde \mu}\right]}$. The "reference spinors" $\mu$ and ${\tilde \mu}$ are gauge choices, and choosing them cleverly can make calculations much easier. (They must drop out of any final amplitude.)

So, the reason you see spinors appearing in calculations with only gluons is that they're convenient ways to talk about momenta and polarization vectors for gluons with definite helicity.