# Higgs Mechanism Confusion

I find myself confused about the Higgs mechanism, but more specifically about how the leptons get their mass.

The intuitive picture I have in my head is that since the leptons are interacting with the Higgs field, when the Higgs moves to a new VEV, the leptons now need to interact with this field as they move through space, which gives the mass.

But this does not really seem to hold water. The massless leptons are two component Weyl spinors. In order to get a massive lepton, you need to actually combine two different Weyl spinors into a 4-component Dirac spinor. This means that the new massive lepton is actually a new particle that did not exist before, or a condensate of two different old particles. But it doesn't seem right to think of it as the same particle as before (the massless lepton) just with a mass.

Am I missing something here?

In the standard model, fermions distinctly do not get their mass from the Higgs mechanism. They get it from the spontaneous breaking of the SU(2)×U(1) chiral-type symmetries through the asymmetric vacuum of the Higgs field. The right-and-left chiral fermions couple together in a Yukawa coupling, and the v.e.v. of the Higgs provides a piece in that coupling that serves as a fermion mass term.

(The Higgs mechanism describes how three degrees of freedom in the Higgs field are actually necessary components of massive gauge fields, instead.)

Your fantasy pictures you are providing do not make much sense to me, and, frankly, if they came from popular science reporting, hell has a special place for science reporters providing such.

I would not presume to contribute to the mental cacophony of such pictures. Formally, a fermion mass amounts to a linkage between the right and the left chiral pieces of the fermion, which is only achievable through the subtle couplings of both components to the peculiarly configured Higgs field vacuum. Something quite analogous happens in QCD, the gluon-driven strong interactions, where the dynamical vacuum links lefters and righters to break chiral symmetry and generates much much much more mass for quarks and hadrons (not leptons).

• I actually heard the intuitive picture from John Ellis, in this video: youtube.com/watch?v=YL96kNBHQ00 (which happens to be in french). Jun 28, 2021 at 13:50
• Avec tout le respect que nous devons à John, il ne mentionne jamais les fermions dans son rêve. Son fantasme pourrait avoir plus de sens pour le mécanisme de Higgs, concernant les masses de bosons... Jun 28, 2021 at 13:59
• En effet, vous avez raison, il ne mentionne pas les fermions. Il devait vouloir dire les bosons, comme vous dites. Mon erreur! Jun 28, 2021 at 14:08

Mass terms for fermions just require that both left and right handed states for them exist in the theory. If these left and right Weyl spinors happened to transform in the same representation of the gauge group with the same couplings then it would be a valid choice for us to treat them as a single Dirac spinor but that's not what happens in the standard model.

Perhaps you are thinking of the simplest toy model where there is only a single scalar field which turns the Yukawa term $$$$\lambda \psi_L \phi \psi_R$$$$ into a mass term after expanding around $$\left < \phi \right > = v$$. In this case, Lorentz invariance would require all $$SU(2) \times U(1)$$ indices of $$\psi_L$$ to be contracted with those of $$\psi_R$$ and the parity violation we observe would be impossible. However, the standard model allows $$\psi_L$$ and $$\psi_R$$ to transform differently because it posits that the Higgs is not a singlet but transforms under $$SU(2) \times U(1)$$ as well.