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I am learning on-shell methods for one loop integrals from this paper: Loop amplitudes in gauge theory: modern analytic approaches by Britto. Starting with formula (18) spinor integration is explained. One first writes the loop momentum as $$(l)_{a\dot{a}}=t\lambda_a\tilde{\lambda}_\dot{a},$$ where $t\in \mathbb{R}^+$ and $\lambda$, $\tilde{\lambda}$ are spinors with $\tilde{\lambda}=\bar{\lambda}$. Why can I write $l$ in this way?

I know the decomposition $p=\lambda\tilde{\lambda}$ for momenta with $p^2=0$, but the loop momentum is not lightlike. I guess the Dirac-Delta in the integrand puts it on shell, but then where does the $t$ come from?

Next the integral over the loop momentum is turned into an integral over the spinors:

$$\int{d^4l\delta^{(+)}(l^2)f(l)}=-\int_0^\infty{\frac{t}{4}dt\int_{\tilde{\lambda}=\bar{\lambda}}{\langle\lambda \; d\lambda\rangle} [\tilde{\lambda} \; d\tilde{\lambda}] f(t,\lambda,\tilde{\lambda})}. $$

Can someone give the derivation of this formula or give me a reference where this is done?

Britto refers to the paper MHV Vertices And Tree Amplitudes In Gauge Theory by Cachazo, Svrcek and Witten. But there I can't find an explanation that I understand.

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up vote 3 down vote accepted

The decomposition $p=\lambda \tilde\lambda$ is only valid for null vectors, indeed. In loop integrals, the loop momentum may generally be off-shell but because of the delta function in these integrals, only the null values of the momentum contribute, so it's enough to deal with the null momenta and they can be factorized to the spinors in this way.

The variable $t$ is just a normalization factor that must be allowed to be arbitrary, real and positive, because $\lambda$ and $\tilde\lambda$ are going to play the role of homogeneous coordinates so one isn't allowed to distinguish them from their multiples.

The integral is just a simple change of variables on the future light cone.

The right references for these integrals are mentioned in the paper you quoted, it's [28] and [29].

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Thank you very much for your answer. This clears up my confusion about $l=t\lambda\tilde{\lambda}$. It would be very nice if you could explain how the change of variables in the integral works. That is where I am still stuck and I think it isn't explained in the references either (maybe because it is too trivial). – Johannes Mar 21 '12 at 18:54
Hi! It may be done by some straightforward algebra but it's also good to notice that both sides have the same support - they're integrals over the future light-cone - and in both cases, the delta-function imposing the support to the light cone combined with the Lorentz invariance uniquely determines the measure up to the normalization (the spinor products are Lorentz invariant much like delta of $l^2$) so they have to be equal up to the normalization one has to check (or believe). The normalization may be checked either by boring algebra or by integrating a particular example function. – Luboš Motl Mar 23 '12 at 15:37

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