# Some questions about the BCFW reduction

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..)
• One question per... question, please. So please consider splitting in into pieces (otherwise the questions are nice). – Piotr Migdal Feb 27 '12 at 8:19
• @Piotr I am not sure how to split this - since its like questions about some of the steps of a single derivation. May be you have administrative powers to split it in someway? – user6818 Feb 28 '12 at 0:00
• In the current form it is almost unanswerable (IMHO one of the main problems of TP.SE is that people asks long and multi-thread questions - it makes high cost to ask, high cost to comprehend and high cost to answer). IMHO the first question should end after the first bullet. There is no problem in asking a sequence of questions or even posting all at once. There is no problem in giving a common introduction of linking them. As I go through it, there are 4 questions which should go separate. Bear in mind that someone can know answer only to one question, or have time only to write one answer. – Piotr Migdal Feb 28 '12 at 8:30
• @Piotr Migdal Now I have split the question into two parts. Hope that helps. – user6818 Mar 6 '12 at 22:43
• @user6818: Shouldn't you have two negative helicity gluons for an MHV amplitude? – Siva May 20 '13 at 11:24

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.)
• Thanks for your reply. I think my question was ill framed. I have read about half of that review by Dixon. I am aware of this helicity formalism as much as you have written in your answer. But that doesn't help make this very clear - the point being - how is this $\lambda_\alpha$ and $\tilde{\lambda}_{\dot{\alpha}}$ chosen? Further it seems that to describe a gluon with a given polarization it seems enough to just specify its polarization either as $\epsilon^+$ or $\epsilon^-$ as you have defined – user6818 Feb 29 '12 at 19:13
• So though one needs both $\lambda_\alpha$ and $\tilde{\lambda}_\dot{\alpha}$ to define the gluon's momentum eventually what is required to completely specify it is just either one of them and another auxiliary 4-vector, $\mu$ and $\bar{\mu}$. In various calculations I have seen the convenient convention for the auxiliary vector seems to be to take as as the same auxiliary vector for all the massless gluons of a say positive helicity and let that be the momentum vector of any one of the negative helicity gluons and vice versa. – user6818 Feb 29 '12 at 19:17
• It would be great if you can make explicit as to how given the data $(p_\mu, \pm)$ about a gluon its corresponding 2-spinor $\lambda$ is chosen. – user6818 Feb 29 '12 at 19:18
• It would be great if you can make explicit as to how given the data $(p_\mu, \pm)$ about a gluon its corresponding 2-spinor $\lambda$ is chosen - This is a bit confusing since in the 4-spinor notation one would say that for gluon of momentum $k$ and the auxiliary vector being $n$ one would choose, $\epsilon_\mu^+(k,n)= \frac{<n|\gamma_\mu|k]}{\sqrt{2}<n|k>}$ and $\epsilon_\mu^-(k,n)= \frac{[n|\gamma_\mu|k>}{\sqrt{2}[k|n]}$ – user6818 Feb 29 '12 at 19:28