Flavor-changing neutral currents (FCNC) and leptons oscillations I have some confusion regarding flavor-changing neutral currents (FCNC) processes and relative meson-antimeson oscillations. If the question has already been asked, I'm sorry, I didn't find what I was looking for.
From what I have understood, on the mass basis, the charged current sector of the Standard Model takes the form (quark)
\begin{equation}
\mathcal{L}_{CC} = -\frac{g}{\sqrt{2}} \left[ \bar{u}_{L}V_{CKM}\not\!\!W^{+}d_{L} + h.c.\right]
\end{equation}
So that at the loop level processes like $B^{0}_{d} \leftrightarrow \bar{B}^{d}_{d}$ with the following Feynman diagrams can occur (sorry for the absence of the plus sign in the W boson).


My questions are:

*

*In the lepton sector, there is the $U_{PMNS}$ matrix. So, is it also possible to have diagrams where "oscillation" with flavour violation can occur (e.g. we have internal lepton line and external lines with neutrinos)? Why is neutrino oscillation instead described quantum mechanically? Is it because we need not suppress the amplitude (unlike the GIM mechanism)?


*Why these processes are called flavour changing neutral currents? It is because there is no electric charge transfer? So what are examples of flavour changing charged currents? Are processes involving the $Z$ boson allowed? (I expect the answer is no because the weak neutral interaction couples fermions of the same type)
 A: I gather you are asking about similarities and differences between hadronic and neutrino  neutral particle oscillation. There are so many questions packed within your units, that I'll try to answer isolated facets of the jumble, apologizing, with Pascal (Je n’ai fait celle-ci plus longue que parce que je n’ai pas eu le loisir de la faire plus courte), that I lacked the time to make the answers shorter; and leave it to you to fit the facets together.
For starters, both oscillation phenomena are due to quantum-mechanical interference of complex propagation amplitudes, and are only possible due to QM. In both cases, the propagating Hamiltonian (mass) eigenstates ($K_L,~ K_S$, $B_L, ~B_S$;  $\nu_{1,2,3}$) are not the interaction combinations which production mechanisms and detectors rely on, so the mismatches in time and hence distance of the ever-shifting interaction combination states are described as an oscillation.
(There are theological discussions  about preferences in momentum vs energy eigenstates involved in the parallel propagation, but just don't go there.)
The differences between the two cases are just in the microscopic QM description of these amps. You wrote down the diagrams for Bottomness oscillations of mesons.

Why are these processes  called flavor-changing  neutral currents? It is because there is no electric charge transfer? So what are examples of flavour changing charged currents?

The standard plinth of the QFT edifice is the current, in this context a bilinear of quarks coupling to mesons, for example, and to something bosonic, like a Z or a W, or a grouping $W^+W^-$. The net charge of the object makes the difference between a charged current, versus a neutral current. In your case, the neutral currents involved are
$\bar d \Gamma b_L$ and $\bar b \Gamma d_L$ , where I've left the tensor structure of Γ vague/open, and you may construct them by pinching charged currents, as in your loop diagrams. They are not flavor neutral,  so the amp you wrote  changes bottomness by two. Because of the CKM/PMNS construction, charged currents can and do change flavor, on account of the built-in intergeneration straddling of the W couplings.

Are processes involving the $Z$ boson allowed? (I expect the answer is no because the weak neutral interaction couples fermions of the same type)

Right you are. If the neutral current couplings to the Z were not diagonal in the mass eigenstates, the unitary basis change to those would cancel itself in the couplings. But now you have exhausted this freedom, and there is nothing left to make the above bottomness-violating term flavor diagonal.

In the lepton sector, there is the $U_{PMNS}$ matrix. So, is it also possible to have diagrams where "oscillation" with flavour violation can occur (e.g. we have internal lepton line and external lines with neutrinos)?

Yes, this is the  point behind neutrino oscillations, except you got the internal particle wrong (the "oscillating"/propagating particle has to be neutral, so the neutrinos, as above), and you don't need a loop: it is a tree diagram. Here is a trail map on how to convert the snipped upper part of you second diagram to a neutrino production+oscillation+detection one.
A $\pi^+$ decays at the production vertex to a virtual $W^+$ which converts to $\nu_\mu$ and $\mu^+$, lost at production. But $\nu_\mu$ is a fiction: it is a superposition of the propagating $\nu_1,\nu_2,\nu_3$, zipping along with infinitesimally small relative delays. At the detection vertex, they suck up a virtual $W^-$ off the detector's nuclei, to turn into a negative lepton, but, this time, on account of the mismatch of velocities in the wave packet, a $e^-$, for the sake of argument. So, this presents as an oscillation ( ⇄ leptonic flavor change) of the bogus $\nu_\mu \to \nu_e$. You are not studying mesons, so you don't need two fermion lines as in B oscillations, one fermion line propagation will do.

Why is neutrino oscillation instead described quantum mechanically?

Explained above: it is all QM.

Is it because we need not suppress the amplitude (unlike the GIM mechanism)?

Actually yes, in a way, if by this you mean "what makes the respective oscillations rare/slow enough to be detectable at macroscopic scales?" Indeed, (do the estimate calculation!), in the hadronic case, the doubly week, GIM suppressed, transition amp is sufficiently slow to give you a macroscopic (lab) effect for the high masses involved; while, for neutrinos, the tree amp suffices, on account of the smallness of neutrino masses: The relevant parameter for the distance of oscillation is L ~ E/m², long baseline.
