Helicity Amplitudes and Angular Decay Distributions

I discuss how to obtain angular decay distributions for sequential cascade decays using helicity methods. The angular decay distributions follow from a reasonably simple master formula involving bilinear forms of helicity amplitudes and Wigner's $d$ functions. I discuss in some detail the issue of gauge invariance for off-shell gauge bosons. As a technical exercise I calculate the linear relation between the helicity amplitudes and the invariant amplitudes of semileptonic and rare baryon decays. I discuss two explicit examples of angular decay distributions for (i) the decay $t\to b+W^+(\to \ell^+\nu_\ell)$ (which leads to the notion of the helicity fractions of the $W^+$), and (ii) the sequential decay $\Lambda_b\to\Lambda(\to p\pi^-)+J/\psi(\to \ell^+\ell^-)$.