# Z^0 \to 2\gamma and the Twisted Coproduct of the Poincar\'{e} Group

## TL;DR

This paper proves that the decay process $Z^0 o 2\gamma$ remains forbidden under both standard and twisted coproducts of the Poincaré group, extending Yang's theorem to noncommutative geometry frameworks.

## Contribution

It demonstrates that Yang's theorem holds even when the Poincaré coproduct is deformed by the Drinfel'd twist, independent of quantum field theory assumptions.

## Key findings

- $Z^0 o 2\gamma$ decay is forbidden for twisted coproducts.
- $Z^0 o
u +
u$ decay is forbidden for massless neutrinos with standard or twisted coproducts.
- Massive particles with odd spin cannot decay into two identical massless particles of same helicity.

## Abstract

Yang's theorem forbids the process $Z^0 \to 2\gamma$ in any Poincar\'{e} invariant theory if photons are bosons and their two-particle states transform under the Poincar\'{e} group in the standard way (under the standard coproduct of the Poincar\'{e} group). This is an important result as it does not depend on the assumptions of quantum field theory. Recent work on noncommutative geometry requires deforming the above coproduct by the Drinfel'd twist. We prove that $Z^0 \to 2\gamma$ is forbidden for the twisted coproduct as well. This result is also independent of the assumptions of quantum field theory. As an illustration of the use of our general formulae, we further show that $Z^0 \to \nu + \nu$ is forbidden for the standard or twisted coproduct of the Poincar\'{e} group if the neutrino is massless, even if lepton number is not conserved. This is a special case of our general result that a massive particle of spin $j$ cannot decay into two identical massless particles of the same helicity if $j$ is odd, regardless of the coproduct used.