• Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.


Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback


(propose a free ad)

Site Statistics

132 submissions , 111 unreviewed
3,777 questions , 1,329 unanswered
4,725 answers , 19,953 comments
1,470 users with positive rep
468 active unimported users
More ...

Some questions about the BCFW reduction

+ 3 like - 0 dislike

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..)
This post has been migrated from (A51.SE)
asked Feb 26, 2012 in Theoretical Physics by Anirbit (585 points) [ no revision ]
retagged Apr 19, 2014 by dimension10
One question per... question, please. So please consider splitting in into pieces (otherwise the questions are nice).

This post has been migrated from (A51.SE)
@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?

This post has been migrated from (A51.SE)
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.

This post has been migrated from (A51.SE)
@Piotr Migdal Now I have split the question into two parts. Hope that helps.

This post has been migrated from (A51.SE)

1 Answer

+ 3 like - 0 dislike

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.

This post has been migrated from (A51.SE)
answered Feb 29, 2012 by Matt Reece (1,630 points) [ no revision ]
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

This post has been migrated from (A51.SE)
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.

This post has been migrated from (A51.SE)
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 post has been migrated from (A51.SE)
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 $\epsilon_\mu^-(k,n)= \frac{[n|\gamma_\mu|k>}{\sqrt{2}[k|n]}$

This post has been migrated from (A51.SE)

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
To avoid this verification in future, please log in or register.

user contributions licensed under cc by-sa 3.0 with attribution required

Your rights