Why is the muon decay a Flavor Changing Charged Current (FCCC) process?

Question

One way of understanding why the muon decay $$\mu^-\to e^-+\bar{\nu}_e+\nu_\mu\tag{1}$$ is a Charged Current (CC) process is to write the intermediate step through which the decay (1) proceeds i.e., $$\mu^-\to \nu_\mu+\color{red}{W^-}\to \nu_\mu+\color{red}{e^-+\bar{\nu}_e}.\tag{2}$$ Now the intermediate step explains that since the process is mediated by charged current interaction or the $W$ boson, it is a Charged Current (CC) process. But what makes the process Flavor Changing (FC)? You can find this term here. As far as I can see, the process (1) or (2) conserves the total as well as individual lepton numbers $L_e,L_\mu,L_\tau$. Does one define a flavor number or something?

Answer 1

When you write the SM Lagrangian in terms of mass eigenstates, charge current couplings (W) involve generation-mixing matrices, the CKM for quarks and the PMNS for leptons. By sharp contrast, there is no mixing for neutral current couplings (Z), and generation mixing enters delicately (GIM-suppressed) at the 2-W exchange, so, then loop level—it runs on the CKM matrices of the charge current couplings!

A hadronic decay going through a W exchange $B\to \psi K$ thus mutates flavor of the same charge, a b to an s, thereby slipping down from the 3rd to the 2nd generation. Or $\Lambda \to p \pi^-$, an s to a d, from the 2nd to the first. Such flavor mutations of fermions of the same charge, all else being unremarkable, are dubbed "flavor changing", FC.

You might, ill-advisedly, marvel at the μ to e mutation as such a generational slip, and call it lepton-flavor changing, too. (But, neutrino Lepton flavor was invented to provide conservation agreement, as you note, trivially, at the interaction vertex. It has no analogy to quark flavors, and, unlike them, it is not conserved in propagation: the entire point of neutrino oscillations! This is why people repeatedly stress the "nonphysicality" of $\nu_{e,\mu,\tau}$, a shared fiction).

• Again: The $\nu_{μ,e}$ neutrinos you wrote are not mass eigenstates ($\nu_{3,2,1}$ , real, propagating particles of the fundamental SM); they are "interaction-convenience combination states", specifically invented to conserve lepton flavors in the SM interactions, $L_e,L_\mu,L_\tau$, but not in kinetic terms.

They thus straddle generations (characterized by mass stratification) through the PMNS matrix, mixing generations thoroughly, far more violently than the CKM. One has not even assigned generations to the mass eigenstates, yet, before resolution of the mass hierarchy question. The consensus is that there is no point in discussing such, yet, unless you were building recondite models. The jumble of neutrino and antineutrino components on the right hand side of your (2), then, involves all 3 generations and makes a mockery out of "flavor" and FC, since it cannot be ignored.

I hope you can see why it hardly makes sense to be discussing FCCC, at all—even though 30 years ago people were actually discussing the $\Delta S =1$ part of the effective weak hamiltonian, subdominant to the strangeness preserving part, just because CKM mixing is small...

I should contrast the above to the substantial utility of the sound FCNC concept, which usefully demarcates small loop corrections of a SM Lagrangian principle—but that would entail mission creep.

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• A teaching moment reminder, in response to comments:

Much of the popular misconceptions of flavor violations hinge on the misbegotten terms "flavor basis" and "flavor eigenstates" for $\nu_{e,\mu,\tau}$. You may see the peril of the term if you slip back to the 60s and recall the convenience superposition array coupling to the up quark in the charged weak current. That state, $d~'\equiv d \cos\theta_C+s \sin\theta_C$, where the angle is the so-called Cabbibo angle (discovered by Gell-Mann and Levy), is not a mass eigenstate. Therefore it has no well-defined flavor: it provides access to two flavors, d and s, defined through the quark mass eigenstates. It is called "in the flavor basis", since it couples to the u flavor in the charged weak vertex. (It underlies the FCCC dispatched above). It would be superflous upon sticking to the mass basis and writing down the CKM matrix in the action, which everybody does, today.

Likewise, $\nu_{e,\mu,\tau}$ are convenience superpositions which remind one that in a vast swath of reactions where neutrino oscillations have not changed their identity, $\nu_{e,\mu,\tau}$ are practical reminders that the charged lepton generation they are associated with in production will be mostly the same in their charge current absorption, usually their primary handle of observation. But, lacking a well-defined mass, they do no possess a well-defined generation, even though it is their mutation in neutrino oscillations (free propagation!) that is informally dubbed "lepton flavor change".

The all-time popular confusion is reflected in widespread poster charts of the standard model generations. These charts tabulate mass eigenstates for quarks, (u,c,t; d,s,b), on the one hand; but, instead of presenting the analog for leptons, real particles (e,μ,τ; ν₁,ν₂,ν₃), assuming the normal hierarchy prevails, they instead stick in the interaction states (ν_e,ν_μ,ν_τ), conferring a misconstrued physical significance to them and all but inviting the logical quandaries dispatched here.