Question on Witten's paper: Perturbative Gauge Theory As A String Theory

[earth2]

Question on Witten's paper: "Perturbative Gauge Theory As A String Theory..."

Hi guys,

I have a question regarding formula 2.12 of Witten's paper hep-th/0312171
"Perturbative Gauge Theory As A String Theory In Twistor Space". He just states this formula but i don't really understand his 'justification' nor do I see a why to derive it myself. I tried to play around with the Pauli-Lubanski-Vector since it is in the massless limit related to helicity but with no success...

Do any of you guys have an idea?
Thanks,
earth2

 

[fzero]

It's easy to see that formula if you think about the amplitude in precisely the manner in which Witten doesn't want to, namely in terms of the momenta and polarization vectors. Expand the amplitude in a series

$$\hat{A}(\lambda_i,\bar{\lambda}_i, h_i) = \sum C_{\mu_1\cdots \mu_n \nu_1\cdots \nu_m} p_1^{\mu_1} \cdots p_n^{\mu_n} \epsilon_1^{\nu_1} \cdots \epsilon_m^{\nu_m} ,$$

where $$C_{\mu_1\cdots \mu_n \nu_1\cdots \nu_m}$$ are some coefficients that take into account however the momenta and polarizations are contracted. It is an important fact that each particle in the amplitude contributes one and only one polarization vector, so no factor of $$\epsilon_i$$ is repeated.

Now under the maps

$$p_i^\mu \rightarrow \lambda^i_a \tilde{\lambda}^i_{\dot{a}}$$
$$\epsilon_i^\nu \rightarrow \frac{\lambda^i_a \tilde{\mu}^i_{\dot{a}}}{\langle \tilde{\lambda}^i,\tilde{\mu}^i\rangle} ~\tex{or}~\frac{\mu^i_a \tilde{\lambda}^i_{\dot{a}}}{\langle \lambda,\mu\rangle},$$

we consider the action of

$$H_i = \lambda^a_i \frac{\partial}{\partial\lambda^a_i} - \tilde{\lambda}^{\dot{a}}_i \frac{\partial}{\partial\tilde{\lambda}^{\dot{a}}_i}.$$

When $$H_i$$ acts on a momentum factor, we obtain zero, since acting on $$\tilde{\lambda}$$ cancels the action on $$\lambda$$. When $$H_i$$ acts on a positive helicity polarization vector we find

$$- \frac{\mu^i_a \tilde{\lambda}^i_{\dot{a}}}{\langle \lambda,\mu\rangle^2}\langle \lambda,\mu\rangle - \frac{\mu^i_a \tilde{\lambda}^i_{\dot{a}}}{\langle \lambda,\mu\rangle} = -2(+1) \frac{\mu^i_a \tilde{\lambda}^i_{\dot{a}}}{\langle \lambda,\mu\rangle}$$

Similarly on a negative helicity polarization vector, we obtain a factor of $$2 = -2(-1)$$. So we can conclude that

$$H_i \hat{A}(\lambda_i,\bar{\lambda}_i, h_i) = - 2 h_i \hat{A}(\lambda_i,\bar{\lambda}_i, h_i) .$$

 

[earth2]

Thanks for your help! That solved my problem :)