# Proof of Yang's theorem

Yang's theorem states that a massive spin-1 particle cannot decay into a pair of identical massless spin-1 particles. The proof starts by going to the rest frame of the decaying particle, and relies on process of elimination of possible amplitude structures.

Let $\vec\epsilon_V$ be the spin vector of the decaying particle in its rest frame, and let $\vec\epsilon_1$ and $\vec\epsilon_2$ be the polarization 3-vector of the massless particles with 3-momenta $\vec{k}$ and $-\vec{k}$ respectively.

In the literature, I've seen arguments saying that

$\mathcal{M_1}\sim(\vec\epsilon_1\times\vec\epsilon_2).\vec\epsilon_V$, and $\mathcal{M_2}\sim(\vec\epsilon_1.\vec\epsilon_2)(\vec\epsilon_V.\vec{k})$ don't work because they don't respect Bose symmetry of the final state spin-1 particles.

But, why is $\mathcal{M_3}\sim(\vec\epsilon_V\times\vec\epsilon_1).\epsilon_2+(\vec\epsilon_V\times\vec\epsilon_2).\epsilon_1$ excluded? Sure, it's parity violating (if parent particle is parity even), but that's not usually a problem

Thanks

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Why isn't anyone answering?! Is my question not worded nicely? Is it because nobody knows the answer? what's the matter?!?!?!?!?!?! :( :( :'( –  QuantumDot Sep 29 '12 at 20:34
Hello? is this website working? –  QuantumDot Sep 30 '12 at 20:17
I'm looking for more information on this theorem. Where did you fount it on the literature? –  Yair Nov 20 '13 at 5:09

Because $\mathcal{M}_3$ as written above actually vanishes by a simple vector identity. On the first term, write
$$(\vec{\epsilon}_V\times\vec\epsilon_1).\vec\epsilon_2=(\vec\epsilon_2\times\vec\epsilon_V).\vec\epsilon_1$$