Abstract
We investigate helicity amplitudes (HAs) of A→BC-type decays for arbitrary spin towards the kinematic endpoint. We show that HAs are proportional to product of Clebsch-Gordan coefficients (CGC) and the velocity to a non-negative power. The latter can be zero in which case the HA is non-vanishing at the endpoint. At the kinematic endpoint the explicit breaking of rotational symmetry, through the external momenta, is restored and the findings can be interpreted as a special case of the Wigner-Eckart theorem. Our findings are useful for i) checking theoretical computations and ii) the case where there is a sequence of decays, say B→B1B2 with the pair (B1B2) not interacting (significantly) with the C-particle. Angular observables, which are ratios of HAs, are given by ratios of CGC at the endpoint. We briefly discuss power corrections in the velocity to the leading order.
Highlights
We investigate helicity amplitudes (HAs) of A → BC-type decays for arbitrary spin towards the kinematic endpoint
The relations can be understood as a consequence of the restoration of the spatial rotation symmetry at the kinematic endpoint
The rotational symmetry acts like a global internal symmetry, analoguous to the isospin symmetry, and leads to simple relations amongst amplitudes in terms of Clebsch-Gordan coefficients (CGC)
Summary
One immediate consequence of (9) is that (for non-vanishing HAs the endpoint) the uniangular distribution, in the angle θB between B1 and the B-particle in A → (B → B1B2)C, is isotropic in the angle if one sums over initial state polarisation and not both decays are parity violating c.f. appendix A. This corresponds to the situation where the particles have lost any spatial reference point and can not decay into a particular direction more frequently than in any other
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