Scattering Amplitude Not Invariant under Little Group?

Question

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?

Answer 1

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.