That’s indeed correct about the Z! An electron and positron can annihilate into a neutrino and antineutrino.

I do not think I wrote a very detailed post about virtual particles, I think in the early Feynman diagram posts I referred to them hand wavily as quantum magic. (The lecture notes linked above probably have a little more detail, but not much.) The point is that as long as they’re not directly observable (i.e. purely virtual) they can get away with a lot of things, including energy non-conservation.

Thanks for the interest!
F

I really love your posts, but I’m a little stuck on the last lump of exercises. Can you post the solutions, for my future reference.

Also, can you ‘turn’ the Z interractions like you can with QED interractions, to make two leptons turn into two neutrinos? It doesn’t intuitively feel like you could, but I wanted to check.

Also also, have I missed a link to a discussion of how virtual particles work (like, for example, how they get the energy to work, and why they so quickly break down), or was there never one?

(This is the way that Feynman rules are usually defined, so I forgot to point this out. I’ll add an update to the main post.)

With that piece of information it should be clear how to “derive” the other rules: If you want the diagram for a photon turning into a W+/W- pair, then you swap the in-going W+ into an out-going W- and the in-going W- into an outgoing W+. Similarly, if you want the diagram for a W+ emitting a photon, you swap the in-going photon for an out-going photon and the in-going W- for an out-going W+.

(It should also be clear now that charge is indeed conserved.)

Great point. Thanks.
-F

Thanks for the response. I can’t think of any easy way around this. If you said the neutrino leg of the Feynmann diagram had finite length then I suppose the nu_tau coming out of an e-W interaction would have nonzero probability (via oscillation)

On the other hand, I think your tutorial still works if you replace nu_e by nu_1, etc. throughout, and maybe would be no more confusing to the beginner. It’s the older physicists who would be asking “what is this nu_1?!”

I was hoping to sweep this nuance under the rug for a while until I figured out how to best explain flavor physics without going into any mathematical formalism (i.e. writing matrices).

As you said, we assume that the quark states are already mass eigenstates and so we don’t write u_1, u_2, u_3 but rather u, c, t. I’ve been writing neutrinos in the same way.

As I suspect you already know, this is slightly misleading since the e and mu lepton flavors have a large mixing in the 1 and 2 mass eigenstates so it’s a little silly to call one mass eigenstates an ‘electron neutrino’ rather than a ‘muon neutrino.’

At any rate, these are all excellent points which I’ll think of a good way to address down the road.

I hope you agree that the small liberties I’ve taken here are reasonable given the targeted public audience, though I’ll be careful in the future to be more technically precise or at least point out subtleties for physicists.

Thanks!
Flip

This seems to be a case of knowing a little too much! Here I am introducing the “post-1998″ Standard Model, or the Standard Model with Dirac neutrino masses (i.e. with RH gauge singlet neutrinos). In this model the leptons and neutrinos have analogous weak interactions as the up and down-type quarks. Thus you have the same misalignment in flavor and mass basis: the CKM matrix in the quark sector is analogous to the PMNS matrix in the lepton sector, and one has tree-level lepton-flavor violation which is suppressed by the right-handed neutrino mass scale. (In this last statement we are assuming some kind of see-saw mass mechanism.)

Indeed, you are both correct that in the minimal Standard Model (pre neutrino-oscillation) there were only left-handed neutrinos and hence one could go into a basis where both flavor and masses are diagonalized simultaneously. And indeed, this is usually what people refer to as the ‘Standard Model.’ (Certainly before 1998 when neutrino oscillations were discovered.) However, I chose to introduce the SM + Dirac Neutrino model because it makes it much simpler to understand the analogous structure in the quark sector, emphasizing the parallel weak representations of the leptons and quarks without having to mention any group theory. Further, the see-saw model, while technically an extension of the Standard Model, is ‘natural’ in light of grand unification since (1) it fits exactly into SO(10) reps and (2) the see-saw mechanism “points” to a large scale on the order of the GUT scale.

Anyway, you are both right that few (if any) textbooks would show flavor-changing W vertices in the lepton sector. Part of this is because books written before 1998 were written at a time when neutrino-oscillation data was unclear. However, I think the SM + Dirac Neutrino model is the ‘minimal’ model that accommodates neutrino oscillations. (I haven’t thought too deeply about the structure of models with Majorana neutrino masses—but these seem unsatisfying since the relevant flavor structure would come from a higher dimension operator.)

At any rate, I accept your comments but hope you appreciate the particular pedagogical choices I made. I will address this issue in a future post where I discuss neutrinoless double beta decay.

Cheers,
Flip