\eta\to\pi^0\gamma decay (Why this decay is forbidden by J?)

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Discussion Overview

The discussion centers on the decay process \(\eta \to \pi^0 \gamma\) and the reasons behind its prohibition by C parity and statistics, particularly focusing on angular momentum considerations. Participants explore the implications of spin and orbital angular momentum in this decay, comparing it to other decay processes.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant states that the decay \(\eta \to \pi^0 \gamma\) is forbidden by C parity and questions the role of statistics (J) in this prohibition.
  • Another participant notes that the initial state has spin 0, while the final state consists of a spin 0 particle and a spin 1 photon, raising questions about the total angular momentum.
  • A different participant argues that if the final state is in a J=1 configuration, the total wavefunction must be antisymmetric, which contradicts the requirement for a symmetric wavefunction due to Bose statistics.
  • One participant emphasizes that the \(\rho\) meson, which has J=1, can decay into a J=1 and J=0 particle, while the \(\eta\) meson, with J=0, cannot decay into a J=1 and J=0 particle, suggesting a fundamental difference in angular momentum conservation.
  • Another participant introduces a semiclassical perspective, arguing that a 0- state cannot decay into a 0- state plus radiation, as it would require changes in monopole moments, which are not present in this case.

Areas of Agreement / Disagreement

Participants express differing views on the implications of angular momentum and statistics in the decay process, with no consensus reached on the exact reasons for the prohibition of the decay.

Contextual Notes

Participants reference concepts such as C parity, spin, orbital angular momentum, and Bose statistics, but the discussion remains complex and unresolved regarding the specific conditions that lead to the decay's prohibition.

Alex_Rob
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Decay \eta\to\pi^0\gamma is forbidden by C parity. But why this decay is also forbidden by statistics (J)?

pi^0 spin is 0
photon spin is 1

But there are can be different situations with full orbital angular momentum of pi^0\gamma.
J=L+S and must be 0. Look like L is 0. But why? Why we can't have L=1 in this case? (unlike in \rho\to\pi^0\gamma decay)
 
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I would say that your first has spin 0, while the product has a spin 0 particle and a photon which is a spin 1 particle.
 
Again I am saying that the rho has J=1, so angular momentum conservation does not prohibit it to decay in a J=1 + J=0 particle.
The eta has J=0. And you try to make it decay into a J=1 and a J=0 particle? you can't compare these.
It's not only the CG coefficients, keep reading the conversation.

If you try to make the last configuration of pi0 and gamma to have a total angular momentum 0, then you have to set the orbital angular momentum of the two products to be L=1.
But then you are having an antisymmetric wavefunction describing your boson final states
 
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I think this is much easier to see semiclassically: you are asking if a 0- --> 0- + radiation state is possible. It is not. A 0- state can have only monopole moments, and there are no magnetic monopoles, and the electric monopole is simply charge: zero in this case. Since no moment changes, there can be no radiation.
 
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