I am trying to make sense of scattering amplitude recently. In some literature people say that if some number of massless particles collide together, one can theoretically express the scattering amplitude as a function of the spinor helicity variables, which transforms as $t^{-2h}$ under the little group scaling $t$ where $h$ is the helicity. As I understand, the little group is a subgroup of the Lorentz group; shouldn't the scattering amplitude be invariant under the Lorentz group action?

  • $\begingroup$ That it is a function of them does not exclude the possibility that it is invariant under the little group, even if the input variables are not (e.g. the scalar product of to vectors is invariant under rotation, even if the vectors are not), so I'm not sure what your question is. $\endgroup$ – ACuriousMind Apr 20 '15 at 15:03
  • $\begingroup$ @ACuriousMind: so you agree that the scattering amplitude should be Lorentz invariant right? But I have seen formula showing that it transform as $t^{-2h}$ under the little group, and that's what bothers me. $\endgroup$ – Daps Apr 20 '15 at 17:08
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    $\begingroup$ Some references would be helpful. $\endgroup$ – Ryan Unger Apr 22 '15 at 19:46

1) The S-matrix must be Lorentz Covariant, rather than Lorentz Invariant. That is, if $\alpha$ and $\beta$ the in and out states, they must BOTH transform as the corresponding free-particle states (free particle state $\ne$ in/out state).

$S_{\alpha,\beta} = \langle \beta | \alpha \rangle = \sum c(\alpha,\alpha') c(\beta,\beta') \ S_{\alpha',\beta'} $ (1).

If you separe the momentum labels from the spin/helicity labels: $\alpha = (p_\alpha, \sigma_\alpha)$ ($\alpha$ is a composite label for the various single particles composing the in/out state)

Then $c(\alpha,\alpha') = \delta(\Lambda p_{\alpha} - p_{\alpha'}) W(\sigma_\alpha,\sigma_\alpha')$

So you rewrite (1) as

$S(\alpha,\beta) = \sum W(\sigma_\alpha,\sigma_\alpha') W(\sigma_\beta,\sigma_\beta') S_{(\Lambda \alpha, \sigma_\alpha');(\Lambda \beta, \sigma \beta')}$

For massless particles with non zero helicity, $W$ is simply the $z \ \delta(\sigma,\sigma')$ of the litte group transformations.

For massless particles with zero helicity, $W = 1$ in agreement with the little group scaling $z^{2 h_i}$

2) All this is explained in Weinberg, Quantum Field Theory vol 1, chapter 2...the best book that was ever written.


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