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

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.