Helicity-coupling amplitudes in tensor formalism

The decay of a particle with spin to two other particles with spin is best described in the helicity formalism. It is the purpose of this paper to show that the helicity-coupling amplitudes, which appear in the angular distributions, can be expressed in terms of the covariant amplitudes in the tensor formalism. This allows for a systematic derivation for the energy dependence of the helicity-coupling amplitudes within the framework of the tensor formalism. The concept of pure intrinsic spin has been developed in the tensor formalism, for decays involving two spins in the final state, in order to bring the formalism to a form comparable to the standard $\ell S$-coupling scheme. Among several examples worked out in this paper are those involving spin-0, spin-1, or spin-2 states in the initial and the final systems and, in particular, a spin-1 state coupling to two spin-1 final states. The latter example is then specialized to the $\frac{J}{\psi }$ radiative decays into a pseudovector ($J^{\mathrm{PC}}=1^{++}$) or into a vector ($J^{\mathrm{PC}}=1^{-+}$) which is exotic, i.e., not a quarkonium. Orbital angular momenta up to $\ell =4\mathrm{}$ have been included in the examples; in particular, the decay ${\pi }_2\mathrm{}\left(1670\right)\mathrm{}\to f_2\mathrm{}\left(1270\right)+\pi$, which involves three $\ell \text{'}\mathrm{s}(\ell =0, 2, \mathrm{and} 4)$, is shown to exhibit a complicated energy-dependent form for the helicity-coupling amplitude.