# Why is $\mu\to e \gamma$ forbidden in the SM w/o neutrino masses?

I have some questions about the next problem:

The process µ → eγ does not occur in the Standard Model in the absence of neutrino masses.

a) The muon is identical to the electron except for its mass. What is the Lagrangian density for electrons and muons coupled to electromagnetism?

Since the Lagrangian for electrons coupled to electromagnetism is $$L = - (1/4) F_{\mu \eta}F^{\mu \eta} + \bar{\psi}(i \gamma^{\mu}D_{\gamma}-m)\psi$$ where $D_{\mu} = \delta_{\mu} + ieA_{\mu}$ and $\psi$ is the electron spinor. I think the Lagrangian for electorns and muons whould just be $$L = - (1/4) F_{\mu \eta}F^{\mu \eta} + \bar{\psi}(i \gamma^{\mu}D_{\gamma}-m)\psi + \bar{\phi}(i \gamma^{\mu}D_{\gamma}-M)\phi$$ for the muon spinor $\phi$ with mass M. Is this right or am I missing an interaction term between both fermions?

b) What are the conserved charges due to the non-spacetime symmetries in this theory, and their physical interpretation? Explain, based on symmetries, why the process µ → eγ is forbidden in this theory.

I can only think on symmetries like $$\psi, \phi \rightarrow e^{i\alpha}\psi, e^{i\beta}\phi$$ Are there any others? I think I should get Lepton number and Lepton family from Noether's thoerem and conclude that lepton family (flavor) is not conserved for the process $\mu\to e\gamma$, which is why it is forbidden, is this right?

• muon number conservation? One of the reasons muon number conservation was posited ( as law from data) – anna v Apr 11 '18 at 3:54
• is there a symmetry related to EM that you've missed? are all of the symmetries global? or are there any local ones? why do you conclude lepton flavor is not conserved? – innisfree Apr 11 '18 at 7:12
• Remember, if you figure out the answers before anyone answers here, you are encouraged to answer your own question here. – innisfree Apr 11 '18 at 7:13
• I don't think there are any local symmetries. I conclude that Muon number is nos conserved and that's why this process is forbidden. – QPhysJP Apr 11 '18 at 7:53
• Is there a local symmetry associated with electromagnetism? Sorry, I think I misunderstood your comment about muon number conservation. – innisfree Apr 12 '18 at 1:51

## 1 Answer

First of all, in Standard Model Lagrangian, there are no vertex involving an electron, a muon and a photon at tree level: $$\mathcal{L}_{SM} \not\supset \; \bar \psi \, \gamma^\mu A_\mu \, \phi$$.

So, one should ask why can neutrinos with mass allow $$\mu \rightarrow e \gamma$$ eventhough it is not allowed in tree level?

In fact, muon can, in principle, decay into an electron and a photon if neutrinos oscillate because of having nonzero mass (which is the case). A muon decaying into a virtual muon neutrino and a virtual W boson can give you an electron if the muon neutrino oscillates to an electron neutrino to annihilate with the W boson. However this kind of decay is extremely rare to be observable.

• The fisrt paragraph is difficult to understand as written. Do you mean to say that there is no vertex involving $\{e,\mu,\gamma\}$, so that the proposed process does not happen at tree level? – dmckee --- ex-moderator kitten Jan 2 at 16:31
• Yes. I clarified the sentence. – Oktay Doğangün Jan 3 at 22:28