Forbidden decays and conservation laws

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• mPlummers
In summary: Check if the total spin is equal to 13 - If yes, the decay or scattering is allowed.Yes, the steps are correct.
mPlummers
Problem Statement: Hello! I'm trying to learn how to know if a particular interaction is allowed or forbidden. I found 3 decays which i can't understand.
Relevant Equations: The decays are:
1) $\eta \rightarrow \pi ^{0}+\gamma$
2) $\phi \rightarrow \rho^{0}+\gamma$
3) $\eta \rightarrow \pi ^{0}+\pi ^{0}$
4) $\eta^{0} \rightarrow \gamma+\gamma+\gamma$

1) $\eta$ and $\pi$ have zero spin, while the photon has spin equal to 1. It should be forbidden because angular momentum is not conserved.
2) Here the angular momentum is conserved (they all have spin 1, so the total angular momentum for the final state can be 0, 1, 2, in this case has to be 1). But what about the charge conjugation? I know that they all have C = -1, so it should not be conserved because the total C for the final state is -1 x -1 = 1.
3) The angular momentum is conserved, so i would say that the decay is allowed, but i can't find it in the Particle Data Group, so I'm trying to understand if the parity is conserved. All the particles have P=-1. I know that the parity for the final state is $P_{tot} = P_{1}P_{2}(-1)^{L}$. The problem is that i don't know the value for L. Should i know that or can i set it equal to zero? If i can put L=0, the the parity is not conserved, so the decay is forbidden... but the PDG says it's possible.
4) I think this is the same case of point 3. Angular momentum is conserved, but i don't know how to add parity quantum numbers (how to choose the total L).

[Moderator's note: Moved from homework to a technical forum.]

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Up!
If the answer for this question is too long, i can isolate the most relevant problem:

If i have two particles, each one with a certain parity, how can i calculate the total parity? The formula $P_{tot} = P_{1}P_{2}(-1)^{L}$ includes the L value. How do i know its value? I usually put L=0, but as i can see that's not the case.

My guess is: is it possible that the orbital angular momentum changes between the intial (L=0) and final state (L=1)?

mPlummers said:
η\eta and π\pi have zero spin, while the photon has spin equal to 1. It should be forbidden because angular momentum is not conserved.

That's completely wrong. It could decay via P-wave, i.e. there would be one unit of orbital angular momentum in the decay. If it's forbidden, it's forbidden for some other reason.

That's completely wrong. It could decay via P-wave, i.e. there would be one unit of orbital angular momentum in the decay. If it's forbidden, it's forbidden for some other reason.

Thank you for your answer. I think i found my doubt. I have never considered before the change in L. Indeed, i think this is the only explanation also for parity conservation in these decays. Is it always possible to change the orbital angular momentum value between initial and initial state? Does it depend on the interaction?

Did you cover angular momentum in quantum mechanics? What did it say about orbital angular momentum?

Yes i did, in a previous course. I know that the total orbital angular momentum $L_{tot}$ can have values $|L_{1}-L_{2}|\leq L_{tot}\leq L_{1}+L_{2}$, and that for each value of $L_{tot}$ there are $2L_{tot}+1$ of $m_{L}$. Same rules for total angular momentum $J_{tot}$, which comes from the sum of $L$ and $S$. This means that a 2 particles state can have several values of $L_{tot}$, but i can't understand how to choose the right value.

In general all orbital angular momentum values are possible, but large values are typically unlikely. You can calculate how far apart e.g. two 100 MeV pions have to be to have L=1 in classical physics. While quantum mechanics is not classical physics you can transfer the right idea here: It is unlikely.

Tantalum-180m is a famous example from nuclear physics. As the name says it is a nuclear isomer - but so long-living that it occurs in nature and no one has observed a decay yet. It has a very high spin while the ground state or the results of a beta decay have a much lower spin.

mfb said:
In general all orbital angular momentum values are possible, but large values are typically unlikely. You can calculate how far apart e.g. two 100 MeV pions have to be to have L=1 in classical physics. While quantum mechanics is not classical physics you can transfer the right idea here: It is unlikely.

Thank you for your answer. Does it mean that, while computing the total angular momentum in the final state, i can choose different values of L? If yes, are the following steps correct to check conservation of total angular momentum?

1 - Given an interaction (decay or scattering) find the spin values of particle in the initial state and compute all the possible values of $J_{tot,i}$, putting L = 0 (collision). In the case of a decay, there's only one value of $J_{tot,i}$, depending on the spin and with L = 0.

2 - Find the spin values of the particles in the final state and compute all the possible values of $J_{tot,f}$ with $L_{tot,f}=0$. If it's not equal to $J_{tot,i}$, add a unit of L and check again. The higher the L value, the less likely the interaction.

If it's correct, is it always possible to apply these rules?

mPlummers said:
The higher the L value, the less likely the interaction.
That is a good general rule in particle physics.

What I'm trying to do is to understand if an interaction is possible in terms of conservation of total angular momentum. For example:

$\eta \rightarrow \pi ^{0}+\gamma$

- $\eta$ has spin 0
- $\pi ^{0}$ has spin 0
- $\gamma$ has spin 1

Initial state: $J _{tot,i} = 0$
To find $J _{tot,f}$ for he final state: $|L_{tot,f}-S_{tot,f}|\leq J_{tot,f} \leq L_{tot,f}+S_{tot,f}$, where $S_{tot,f} = 1$. Now, if $L_{tot,f}=0$, the total angular momentum is not conserved (it can only be 1), BUT if i put $L_{tot,f}=1$ then i can obtain $J _{tot,f}=0$. Is this reasoning correct?

Yes, see above.

What are forbidden decays?

Forbidden decays refer to particle decays that are not allowed by the fundamental conservation laws of physics. These include decays that violate charge, lepton number, baryon number, and strangeness conservation.

Why are forbidden decays important?

Forbidden decays provide important insights into the fundamental laws of physics and can help scientists understand the behavior of particles and their interactions. They also play a crucial role in testing and refining theories such as the Standard Model.

What are conservation laws?

Conservation laws are fundamental principles in physics that state that certain quantities, such as energy, momentum, and charge, cannot be created or destroyed. These laws are crucial in understanding and predicting the behavior of particles and their interactions.

How do forbidden decays violate conservation laws?

Forbidden decays violate conservation laws by allowing particles to decay into final states that have different properties than the initial state. For example, a particle may decay into a final state with a different charge, violating charge conservation.

Are there any exceptions to the conservation laws?

While conservation laws are considered fundamental principles, there are some rare exceptions that have been observed in particle physics. These exceptions are typically associated with extremely high energies or short timescales, and are still subject to ongoing research and investigation.

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