Spin of the Pion: Neutral Decay Proves Integral Spin

[maverick280857]

Hi,

I have a question from Perkins:

For neutral pions, the existence of the decay
$$\pi^{0} \longrightarrow 2\gamma$$

proves that the pion spin must be integral (since Sy = 1) and that $s_\pi \neq 1$, from the following argument. It can be proved as a consequence of relativistic invariance that for any massless particle of spin s, there are only two possible spin substates, $S_z = \pm s$, where z is the direction of motion. Taking the common line of flight of the photons in the pion rest frame as the quantisation axis, the z-component of total photon spin in the above decay can thus have the values Sz = 0 or 2. Suppose $s_\pi = 1$; then only $S_z = 0$ is possible, and the two-photon amplitude must behave under rotation like the polynomial $P_{l}^{m}(\cos \theta)$ with m = 0, where $\theta$ is the angle of the photon relative to the z-axis.

I'm not sure I understand the line in boldface. If $s_\pi = 1$ then why is $S_z = 0$?

Thanks!

 

[xepma]

Conservation of angular momentum. two is larger than one!

 

[maverick280857]

Conservation of angular momentum. two is larger than one!

Yes. Right, but does Sz = 0 explicitly conserve angular momentum? How do I convince myself?