# ##\mu##2##e\gamma## probability

• ChrisVer
In summary, the conversation discusses an exercise involving the decay of muons into electrons and photons, which violates the lepton number but is allowed due to neutrino interactions. The exercise asks to approach the problem with the HUP and find the time interval and distance traveled by the intermediate neutrinos. It also mentions the km scale of neutrino oscillations and the probability for this interaction at different energy and length scales. The conversation also mentions the branching ratio and the use of the normal derivation for the probability of this interaction. It concludes with a discussion on how to express the terms in the propagator for the neutrino transition and references to previous studies on the branching ratio.
ChrisVer
Gold Member
I was doing an exercise with the decay: $\mu \rightarrow e \gamma$
which violates the lepton number but it is (in principle) allowed due to neutrinos interactions.
The exercise asked to approach the problem with the HUP and find the time interval and "distance" traveled by the intermediate neutrinos, and compare it to the km scale of neutrino oscillations.
$L= c \Delta t = \frac{\hbar c}{2 M_W} \sim 10^{-21}~ km$
Obviously very small...

I was wondering if anyone knows a way that uses the normal derivation for the probability of this interaction. In particular I am not sure I know how to use the fermion propagator lines for the neutrinos when at some point there is a flavor change.

I think experimentally the Branching ratio has been found to be less that ##0.57 \times 10^{-12}##
http://en.wikipedia.org/wiki/Mu_to_E_Gamma

ChrisVer said:
Obviously very small...
At the length scale of the weak interaction.
What does the neutrino mixing formula tell you about the probability for this length and an energy you have to choose?

You can safely ignore all factors between .1 and 10. Just the mixing probability makes the decay impossible for all practical purposes.

Well first of all the energy of the neutrinos is also needed and $\Delta m^2$..but I think the $L$'s smallness is enough to make the sins in the probability approximately equal to their arguments.
So the probability should go like $P \sim (\Delta m^2)^2 L^2 \sim 10^{-48}$

I don't know about the energy...

ChrisVer said:
So the probability should go like $P \sim (\Delta m^2)^2 L^2 \sim 10^{-48}$
That doesn't work in terms of units. The energy is part of the formula.

As higher energies lead to less mixing, using the muon mass is probably a conservative estimate.

Or do you mean to set the probability for the transition maximum:
$\sin^2 \Big( 1.2 \frac{L}{E} \Delta m^2 \Big)=1$?
Then I guess $E(GeV) \approx 0.7 L(km) \Delta m^2$
I don't know this becomes an extremely small energy and that's why I wouldn't think of it.

Ah Ok...then

$\sin^2 \Big(1.2 \frac{L(km)}{E(GeV)} \Delta m^2 \Big) \approx \sin^2 \Big( 1.2 \frac{10^{-21}}{0.1} 10^{-3~ \text{to}~ -5}\Big) \approx 10^{-46} ~\text{to}~ 10^{-50}$

That is in the right range. A few orders of magnitude more or less do not matter, as there is absolutely no way we can find the standard model decay without advanced magic.

However I don't know how to express a diagram that looks like this:

#### Attachments

• a1.jpg
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What do you mean with "express"?
Calculate properly? That takes much more effort, and where is the point? Not even the experts bother calculating this more precise than the order of magnitude (sometimes not even that).

Well not calculate if it's so bothersome. Just trying to see how to write the terms in the propagator... (especially for the neutrino transition).

ChrisVer said:
Well not calculate if it's so bothersome. Just trying to see how to write the terms in the propagator... (especially for the neutrino transition).

Someone should correct me if I'm wrong because I'm not too familiar with this, but I think the neutrino line can be written
$$\sum_{i=1}^3 U_{\mu,i} \frac{1}{p_\mu \gamma^\mu - m_i} (U_{e,i})^*$$
where ##U## is the PMNS matrix and ##i## labels the neutrino mass eigenstates.

Note that if all the ##m_i##'s are equal, then the overall expression is proportional to ##\sum_i U_{\mu,i} (U_{e,i})^*## which is zero because it is an off-diagonal element of ##U U^\dagger= I##. So the amplitude is going to end up proportional to a difference of neutrino masses, or probably a difference of squared masses.

If you expand the above expression as a power series in the masses ##m_i##, then I think the first order term is

$$\frac{1}{p_\mu \gamma^\mu}\left(\sum_{i=1}^3 m_i U_{\mu,i} (U_{e,i})^*\right) \frac{1}{p_\nu \gamma^\nu}$$

This corresponds to how your diagram is drawn. There is a massless propagator for the electron neutrino, a massless propagator for the muon neutrino, and an insertion of the neutrino mass matrix which produces the ##\nu_\mu \to \nu_e## transition. Now we are treating the neutrino masses as perturbations; they lead to vertices like your red X at which we should insert the central expression in parentheses above.

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Actually you can't do that, because each of the vertices with the W boson and the neutrio have a $$P_L$$ projector, so if there's no masses you have

$$\gamma^{\mu} P_L \not p \not p \gamma^{\nu} P_L$$

which is zero, as the P_L ...P_L -> (...) P_R P_L

So the masses are necessary in the propagators too. Basically you have to expand it out to a higher order than what you have.

So not only is it suppressed by the mass insertion, but additionally because of the dirac structure of the currents, only mass-proportinoal terms survive. So this will go like m^2 at the amplitude level.

I think the branching ratio, compared with the eνν decay, was calculated as early as 1977 in this paper (eq. 8) by Petcov, downloadable http://ccdb5fs.kek.jp/cgi-bin/img_index?197702078 . It also appears here (eq. 10). Both are from before the actual discovery of neutrino oscillations, but it should be possible to insert current values for the mass-splittings and PMNS-elements.

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ChrisVer
tg85 said:

My neck hurts... But I was able to see the formula.

tg85 said:
It also appears here (eq. 10).

I don't have access at the moment, I will try it tomorrow.

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## 1. What is ##\mu##2##e\gamma## probability?

##\mu##2##e\gamma## probability is a measure of the likelihood of a muon decaying into an electron and a photon. It is a fundamental concept in particle physics and is used to study the interactions of muons with other particles.

## 2. How is ##\mu##2##e\gamma## probability calculated?

##\mu##2##e\gamma## probability is calculated using the Feynman diagram technique, which allows for the calculation of the probability of a particle interaction based on the quantum field theory. This involves calculating the amplitude of the interaction and then squaring it to obtain the probability.

## 3. What factors affect the ##\mu##2##e\gamma## probability?

The main factors that affect the ##\mu##2##e\gamma## probability are the energy and momentum of the particles involved, as well as the strength of the interaction between them. The type of particles and their spin also play a role in determining the probability.

## 4. What is the significance of studying ##\mu##2##e\gamma## probability?

Studying ##\mu##2##e\gamma## probability allows us to understand the fundamental interactions of particles and the laws of nature at a microscopic level. It also has practical applications, such as in medical imaging and radiation therapy.

## 5. How is ##\mu##2##e\gamma## probability related to other probabilities in particle physics?

##\mu##2##e\gamma## probability is related to other probabilities in particle physics, such as the probability of muon decay into other particles, through the conservation of energy and momentum. It is also related to the cross-section, which is a measure of the probability of a particle interaction occurring in a given area.

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