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..)

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Comments

One question per... question, please. So please consider splitting in into pieces (otherwise the questions are nice).

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@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?

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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.

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@Piotr Migdal Now I have split the question into two parts. Hope that helps.

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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.

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Comments

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

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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.

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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.

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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, $\epsilon_\mu^+(k,n)= \frac{}$ and $\epsilon_\mu^-(k,n)= \frac{[n|\gamma_\mu|k>}{\sqrt{2}[k|n]}$

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