# ## \mu~\to e~ \gamma ## decay width and neutrino propagator

• I
• Safinaz
In summary: It is equal to ##m_i^2## times the product of the neutrino masses and the square of the distance between the neutrino and the nucleus.
Safinaz
Hi all,

I'm studying ## \mu \to e~ \gamma ## decay from cheng & Lie' book " gauge theory of elementary particles ". In Equation (13.84), he wrote the neutrino propagator
## \sum_i \Big ( \frac{U^{*}_{ei} U_{\mu i}}{(p+k)^2-m_i^2} \Big), ##
(where the sum taken over neutrinos flavors) in the form:

##\sum_i U^{*}_{ei} U_{\mu i} \Big ( \frac{1}{(p+k)^2} + \frac{m_i^2}{[(p+k)^2]^2} + ... \Big) ##
##= \sum_i \frac{U^{*}_{ei} U_{\mu i} m_i^2}{[(p+k)^2]^2} + ...##

Do anyone know how did he drive this ? Then he write that:

the leading term vanishes, ## \sum_i U^{*}_{ei} U_{\mu i}##, reflecting the GIM cancellation mechanism.

I can't get this statement .. Thanks,

Safinaz said:
where the sum taken over neutrinos flavors
No, the sum should be taken over the neutrino masses. The flavour eigenstates do not have definite masses.

Safinaz said:
Do anyone know how did he drive this ?

Just expand the quotient for small ##m_i##.

Safinaz said:
Then he write that:

the leading term vanishes, ## \sum_i U^{*}_{ei} U_{\mu i}##, reflecting the GIM cancellation mechanism.

I can't get this statement ..,

It follows directly from the unitarity of ##U##. It is the ##\mu##-##e## element of ##U U^\dagger = 1##.

Orodruin said:
Just expand the quotient for small ##m_i##.

Can you please write the general form, like for instance taylor expansion, it seems I'm not so good in math !

Orodruin said:
It follows directly from the unitarity of U. It is the ##\mu-e## element of ##U U^\dagger = 1##.

Actually this still not clear for me , ##\sum_i U^{*}_{ei} U_{\mu i} ## is multiplied by the first term ## \frac{1}{(p+k)^2} ## as well as the second term ## \frac{m_i^2}{[(p+k)^2]^2} ##, so why the first one which has been ignored .. also what's GIM mechanism and he says due to this mechanism this first leading term vanishes ..

Safinaz said:
Can you please write the general form, like for instance taylor expansion, it seems I'm not so good in math !

$$\frac{1}{1+x} = 1 - x + x^2 + \ldots$$

Safinaz said:
Actually this still not clear for me , ##\sum_i U^{*}_{ei} U_{\mu i} ## is multiplied by the first term ## \frac{1}{(p+k)^2} ## as well as the second term ## \frac{m_i^2}{[(p+k)^2]^2} ##, so why the first one which has been ignored .. also what's GIM mechanism and he says due to this mechanism this first leading term vanishes ..

The first term ## \frac{1}{(p+k)^2} ## is independent of ##i## and can be taken out of the sum. The second term is not independent of ##i##.

Safinaz

## 1. What is the significance of the muon to electron gamma decay width?

The muon to electron gamma decay width is an important value in particle physics, as it is a measure of the rate at which muons decay into electrons and photons. This process is governed by the weak interaction and is used to study the properties of neutrinos.

## 2. How is the muon to electron gamma decay width calculated?

The decay width is calculated using the Fermi's Golden Rule, which relates the decay rate to the square of the matrix element of the interaction and the available phase space for the decay products. In the case of muon to electron gamma decay, the matrix element is related to the coupling constants of the weak interaction and the propagator of the neutrino.

## 3. What is the role of the neutrino propagator in muon to electron gamma decay?

The neutrino propagator describes the probability amplitude for a neutrino to travel from the muon to the electron in the decay process. It is an important factor in calculating the decay width and is dependent on the mass and energy of the neutrino.

## 4. How does the muon to electron gamma decay width depend on the neutrino mass?

The decay width is directly proportional to the square of the neutrino mass. This means that a heavier neutrino would have a larger contribution to the decay width compared to a lighter neutrino.

## 5. Why is the study of muon to electron gamma decay important?

Muon to electron gamma decay is an important process in understanding the behavior of neutrinos, which are fundamental particles that have a significant role in the universe. It is also a way to probe the weak interaction and test the predictions of the Standard Model of particle physics.

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