# Forbidden decay ##\rho^0\rightarrow \pi^0\pi^0##

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• Aleolomorfo
In summary, the conversation discusses the forbidden decay of ##\rho^0## into ##\pi^0\pi^0## due to Bose-Einstein statistics and the disagreement between the book's reasoning and the individual's reasoning. The individual questions whether the isospin wave function should be considered and mentions the Clebsch-Gordan coefficients as a way to understand the decay. The other person suggests that the decay is forbidden due to spin-statistic argument without considering the isospin, but acknowledges that it may also be forbidden for other reasons.
Aleolomorfo
TL;DR Summary
Reasons behind the forbidden decay ##\rho^0\rightarrow \pi^0\pi^0## and considerations about the wave function of two bosons system
Hello everybody!

I have a question regarding the forbidden decay ##\rho^0 \rightarrow \pi^0\pi^0##, but it is a general doubt.

My book states that one of the reasons why the decay is forbidden is Bose-Einstein statistics, the final state of two equal pions must be in an antisymmetric state.
My reasoning is the following. The wave function of two pions is:
$$\psi_{\pi^0\pi^0} = \psi_{space}\psi_{isospin}$$
I neglect the spin part since pions are spin-0 particles.
##\rho## is a spin-1 particle, so the orbital momentum of the two pions system is ##l=1##; analogously for isospin ##I=1##.

$$(exchange)\psi_{\pi^0\pi^0} =(exchange) \psi_{space}\psi_{isospin} = (-1)^l(-1)^I = (+1)$$

My result is not in agreement with the book. However, if I do not consider the isospin wave function everything is ok.
My question is if I should consider the isospin wavefunction. If not, why?

Look at the Clebsch-Gordan coefficients.

Look at the Clebsch-Gordan coefficients.

$$|\rho^0> = |1,0> = \frac{1}{\sqrt{2}}|1,+1>|1,-1> + 0 |1,0>|1,0> - \frac{1}{\sqrt{2}}|1,-1>|1,+1> =$$ $$=2\frac{1}{\sqrt{2}}|1,+1>|1,-1> + 0 |1,0>|1,0> = \frac{1}{\sqrt{2}}|\pi^+\pi^-> + 0 |\pi^0\pi^0>$$

From the Clebsch-Gordan coefficients I see that the decay in ##\pi^0\pi^0## is forbidden, but I do not see the link with my way of looking at the problem.

I'm not sure what to tell you. The amplitude is zero. Might it be zero for some other reason too? I guess.

The rho has Ispin 1. pi0-pi0 cannot be in an I=1 state.
The decay is Ispin forbidden, but that is not absolute.
The spin-statistic argument (without Ispin) is absolute.

vanhees71

## 1. What is a forbidden decay?

A forbidden decay is a type of particle decay that is not allowed by the laws of conservation of energy and momentum. This means that the decay cannot occur through the usual processes and must happen through a more rare or indirect pathway.

## 2. Why is the decay ##\rho^0\rightarrow \pi^0\pi^0## considered forbidden?

This decay is considered forbidden because it violates the conservation of isospin, a quantum number that is conserved in strong interactions. The rho meson (##\rho^0##) has an isospin of 1, while the two pions (##\pi^0##) each have an isospin of 0. This means that the total isospin before the decay is 1, but the total isospin after the decay would be 0, which is not allowed.

## 3. How does the forbidden decay ##\rho^0\rightarrow \pi^0\pi^0## occur?

The forbidden decay ##\rho^0\rightarrow \pi^0\pi^0## can occur through a process called annihilation, where the rho meson and one of the pions annihilate each other and produce a virtual particle that then decays into two pions. This indirect pathway allows for the decay to occur without violating the conservation laws.

## 4. What is the significance of studying forbidden decays?

Studying forbidden decays can provide valuable insights into the fundamental laws of physics and the behavior of particles. These decays are rare and difficult to observe, so studying them can help us understand the limits of our current theories and potentially discover new physics.

## 5. Are there other examples of forbidden decays?

Yes, there are many other examples of forbidden decays in particle physics. Some well-known examples include the decay of a neutron into a proton and an electron, and the decay of a muon into an electron and two neutrinos. These decays are also important for understanding the underlying laws of nature.

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