Reconciling mass due to the Higgs mechanism with a particle being asymptotic

Question

I have a basic understanding of how the Higgs mechanism works, and how it can be used to explain, or perhaps rather accommodate, masses of elementary particles like fermions, without breaking gauge invariance.

In this mechanism, the mass of an elementary particle like an electron, is due to the interaction between the electron and the Higgs field. When working out the interaction rate of say an electron and photon in Compton scattering, we would take the incoming and outgoing electron (and the photon) to be external/asymptotic states. By definition, this means that they do not interact further and propagate freely according to their respective equations of motion. How can we reconcile this assumption with the fact that the mass of the electron is due to interaction with the Higgs field? This would seem to imply that we cannot really think of the electron as external/asymptotic. What am I missing here?

The buzzword you might base your search on is "SM complementarity": how to make gauge singlet observables out of composite operators, in your case the electron and the Higgs fields. Highly technical, and active in the early 80s (Susskind et al.). — Cosmas Zachos, Mar 11 '23 at 16:02
Thanks! I did not really find anything when I searched for "standard model complementarity", but I saw the paper you linked in the linked post. — Étienne Bézout, Mar 11 '23 at 16:13
Yes, this should be helpful. Might track down the refs of this one. — Cosmas Zachos, Mar 11 '23 at 16:16
Thanks! I case someone else is interested, a cursory reading suggests that the following could also be useful: https://arxiv.org/abs/1912.08680. — Étienne Bézout, Mar 11 '23 at 16:19

score 3 · Answer 1 · answered Mar 12 '23 at 13:16

While in popular science it is common to say that the mass of elementary particles is due to the interaction with the Higgs, there are a few caveats that should be mentioned.

The mechanism of spontaneous symmetry breaking in the standard model does lead to the occurrence of the mass terms for fermions and gauge bosons. In this sense, the presence of the Higgs does generate the masses. Furthermore, the same mechanism leads to interaction vertices between the massive particles and the Higgs boson where the interaction coupling constants are proportional to the particle's mass. In other words, the heavier the particle, the stronger its interaction with the Higgs field.

Nevertheless, the terms in the action generating the interaction with the Higgs and generating the particles' masses are different. They are both given by the SSB mechanism and their couplings are related, but they are different terms. Hence, in the asymptotic limit in which you are treating the particles as free, you are dropping the interaction with the Higgs, but the masses come from other terms and you still keep them. You can drop the interactions but keep the mass terms because you are explicitly assuming the particle is not interacting with anything else in this asymptotic limit.

Shortly, the statement "the mass of an electron is due to its interaction with the Higgs field" is not incorrect, but should be interpreted carefully. The mass term is generated by the SSB and the same mechanism generates an interaction with a coupling constant proportional to the electron's mass, but the mass term for the electron is not literally the interaction term with the Higgs boson.

Reconciling mass due to the Higgs mechanism with a particle being asymptotic

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