# Why has the Higgs-field inertia?

The Higgs field is responsible for the masses of all elementary particles. Including the Higgs particle. But doesn't this transfer the question of mass, or inertia, to the Higgs field itself? Why has the Higgs field inertia which can be transmitted to the massless elementary particles?

• Are you aware of the non-popular view on the Higgs (i.e. the spontaneous symmetry breaking mechanism)? – Nihar Karve Apr 1 at 14:42
• @NiharKarve Yes, I am. Do you mean that the field has zero energy while being in a non-zero state? – Deschele Schilder Apr 1 at 14:44

The mass of the Higgs is not due to the Higgs mechanism. There simply is a mass-term in the Higgs field Lagrangian.

The whole idea that Higgs is the originator of mass of otherwise massless particles is quite mistaken, at least as it stands in the collective psyche of the popular culture. ;) The correct statement is that coupling to Higgs gives mass to fermions of the Standard Model and they would have been massless otherwise. As such, in the framework of quantum field theory, a field can simply have a mass term -- there is nothing wrong with it. However, when a theory has certain symmetries, it can forbid the field from having the mass because the mass term might violate the symmetry. This is precisely what happens with the fermions of the Standard Model, the $$SU(2)_L$$ gauge symmetry forbids the mass term. However, due to the coupling of these fermions with the Higgs field, via the mechanism of spontaneous symmetry breaking, they acquire a mass.

In any case, the moral is that there is nothing that says that you need to somehow have a source of mass. The Lagrangian of a field can simply have a mass term. It is the particular case of some field theories where certain particles might not have mass due to symmetries but they nonetheless acquire mass due to spontaneous symmetry breaking. However, this need not be the case. Ironically, as I mentioned earlier, the mass of the Higgs does not come from its coupling with itself. It is massive regardless of spontaneous symmetry breaking.

• Aren't elementary particles the ones of the SM? – Deschele Schilder Apr 1 at 15:11
• @DescheleSchilder I am not sure I understand your question but to the extent I do, I would like to point out two things: $1$. The Higgs boson is not a fermion (see, I wrote "fermions of the Standard Model"). $2$. The point of pointing out as to whether something is a fundamental aspect of the framework of quantum field theory or a rather accidental aspect of the standard model is to distinguish what is in principle forbidden and what just happens to be forbidden in a particular model. – Dvij D.C. Apr 1 at 15:21
• Well, the quarks and leptons (and the intermediate gauge bosons) "acquire" their mass by interacting with the Higgs field. From where does the inertia of the Higgs field come? – Deschele Schilder Apr 1 at 15:32
• @DescheleSchilder What do you mean by intermediate gauge bosons? Gauge bosons are massless and they do not acquire their mass by interacting with the Higgs field -- they are simply massless. And again, the mass of the Higgs field comes from the mass term in its Lagrangian. As I tried to stress, a field can simply have a mass term, it is not a human rights violation. ;) – Dvij D.C. Apr 1 at 15:36
• @DescheleSchilder Yes, sorry, my bad for forgetting the WZ bosons. But yes, the mass of the Higgs is not explained via spontaneous symmetry breaking because it does not need spontaneous symmetry breaking to acquire mass. If it helps, nobody (as far as I know) feels the need to somehow explain the mass of a field if the field can simply have a mass term (there is no need for an explanation, a field can simply have a mass term). It is when the mass term is forbidden due to symmetry, one needs to look for where the mass might come from if the said particle is known to be massive. – Dvij D.C. Apr 1 at 15:47

The two are correlated, obviously, but they're not the same thing.

Example: the mass of each of the elements of the periodic system. The mass of a Helium nucleus is close to two times the mass of a proton plus two times the mass of a neutron, but not quite.

The inertial mass of a nucleus involves the energy state of that nucleus. A higher energy state comes with a correspondingly larger inertial mass.

The inertia of energy is unrelated to the Higgs mechanism.

My understanding is that the Higgs mechanism doesn't even address the question of inertia.

One author phrased this as follows (I'm quoting from memory): coupling to the Higgs field imposes an energy cost. The mass acquired due to coupling to the Higgs field is the inertial mass that corresponds to that energy.

• Do you think energy has inertia? – Deschele Schilder Apr 2 at 7:31
• @DescheleSchilder Special relativity leads to attributing inertial mass to energy content. This was first discussed in 1905 by Einstein in an article titled: "Ist die Trägheit eines Körpers von seinem Energieabhalt abhängig?" (Title of translated version: "Does the inertia of a body depend on its energy content?" I recommend the discussion Einstein on the inertia of energy by Kevin Brown. Using 'm' for inertial mass the expression for the amount of inertial mass corresponding to energy content E is $$m = \frac{E}{c^2}$$ – Cleonis Apr 2 at 8:40
• The famous equation only says that mass is equivalent to energy. Their values can be known from each other's values. Has a photon inertia? It can push other particles (when they absorb the photon). But can we push the photon? With other photons, yes (photon-photon scattering), but there is no three-photon vertex. There are always other virtual particles involved in the Feynman diagram – Deschele Schilder Apr 2 at 9:12
• @DescheleSchilder I recommend that you turn this into the subject of a physics.stackexchange question. For instance, according to the principle of equivalence a spinning flywheel has a larger gravitational mass than that same flyweel in non-spinning state. According to relativistic physics the rotational kinetic energy has inertial mass, and the PoE asserts equivalence of inertial and gravitational mass. For a practical experiment the mass difference is below detection limit, but the outcome of the thought experiment is unambiguious. – Cleonis Apr 2 at 9:36
• I was already typing the question, in fact! But then I saw the question was already asked: physics.stackexchange.com/questions/587347/… – Deschele Schilder Apr 2 at 9:42