# Masses of Fermions, string theory, Higgs mechanism

• I
Gold Member
Take the first family of fundamental fermions, u, d, e-, and ν. The u and d are more massive than the e- and the e- is more massive than the ν. The u and d interact via 4 forces, the e- interacts via 3 forces, and the ν interacts via 2 forces. The fermions that interact via the most forces are the more massive.

Does string theory suggest a connection between the masses of the fundamental fermions and the number of forces a fermion interacts with?

Thanks!

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First let's just think in terms of field theory. I don't know if you thought of this yourself, but this kind of idea has been around for a while. It can be expressed in terms of self-energy, e.g. the neutrino mass as arising from weak-interaction self-energy, the electron mass as arising from electromagnetic self-energy, the up and down quark masses as arising from chromodynamic as well as electromagnetic self-energy.

What standard physics says, is that the mass of these particles has a nonzero contribution from other factors (e.g. seesaw mechanism for the neutrino, Higgs mechanism for the others), but does indeed have a substantial component coming from the self-energy. Here's a brief discussion. I quote Polchinski as saying that roughly 20% of the electron mass comes from self-energy; in the next sentence he says, "for quarks the effect is larger due to the larger SU(3) coupling, so that the self-energy is of order the mass itself".

But in our current understanding, the mass scales of electron, up, and down are set fundamentally by yukawa couplings to the Higgs (and in the case of the neutrino, by unknown physics such as a seesaw with a heavy right-handed neutrino). The number of forces does not determine the mass even qualitatively, and even in the realm of the self-energy contribution, the strength of the individual forces counts for more than the overall number.

Nonetheless: in another thread, we have extensively discussed @arivero's 2011 generalization of the Koide formula, and it comes in two forms, an "unperturbed" version in which electron and up masses are zero, and a slightly "perturbed" version which is the realistic one. Could the "perturbation" come from the self-energy? It's a logical idea, but lacking a genuine model, it's just an idea.

As for string theory, the string theory account of mass is exactly as in field theory: as free particles, the chiral fermions are massless, they pick up a mass by yukawa coupling to the Higgs, and then the mass runs due to the self-energy. The only difference is that string theory has to provide extra details. Field theory just says it's "massless fermions interacting with a Higgs vev", with yukawa couplings and gauge couplings as free parameters. In string theory, this will correspond to strings and branes interacting in some specific way (there are many ways to do it), and the values of the couplings will be determined by the geometric details rather than being free parameters.

So in string phenomenology, one normally supposes that even in the first generation of fermions, there's a nonzero yukawa coupling that contributes to the mass, and the self-energy is just a correction rather than the whole contribution. It's certainly mathematically possible to obtain "rank one mass matrices" in which only the third-generation yukawas are nonzero. There are somewhat contrived field-theory models in which the first and second generations then get their masses from virtual top quarks. The possibility that string theory can do it better is at least worth some attention.

• ohwilleke and Spinnor
ohwilleke
Gold Member
Does string theory suggest a connection between the masses of the fundamental fermions and the number of forces a fermion interacts with?

I would be a bit less charitable and more direct than @mitchell porter in answering this part of the question.

String theory has no meaningful phenomenology that tells us anything about how it would give rise to the masses of the fundamental fermions. The scientific community has not even identified any version of string theory that produces the right mix of fundamental particles at electro-weak energies and below to reproduce what we see in the Standard Model, let alone any of the experimentally measured constants of those particles in the Standard Model.

Basically, string theory has nothing meaningful to say about the subject.

String theory may have something meaningful to say at some point in the distant future, but it isn't at all close to being sufficiently developed to say anything of the kind right now. String theorists are struggling with whether there are even an universes that could arise consistent with String theory that could have the same overall topology as our universe. There are some general strategies from getting from a vague theoretical construct to a predictive theory that have been suggested, and string theory was instrumental in predicting the existence of a spin-2 massless graviton (a quantum gravity hypothetical particle which has not yet been proven to exist), but that is about it.

ohwilleke
Gold Member
Take the first family of fundamental fermions, u, d, e-, and ν. The u and d are more massive than the e- and the e- is more massive than the ν. The u and d interact via 4 forces, the e- interacts via 3 forces, and the ν interacts via 2 forces. The fermions that interact via the most forces are the more massive.

This is an intriguing observation which has been made in the past many times independently. And, if there were just one generation of fermions, the notion that their masses arise from their self-energy would seem plausible.

For example, the ratio of the electron mass to the neutrino mass is on the same order of magnitude as the ratio of the electromagnetic force (which impacts electrons but not neutrinos) to the weak force (which impacts both electrons and neutrinos), and the up and down quarks, which are subject to the strong force in addition to the electromagnetic and weak forces, are heavier than either of the leptons, which are not.

But, there are problems with the concept even at the first generation level.

First, the self-energy of the up quark and the down quark due to the strong force ought to be identical, they have the same weak force charge, and the up quark has a stronger electromagnetic charge in magnitude than the down quark. But, the down quark is more massive than the up quark, not the other way around. In fact, the down quark is almost twice as massive, despite having half the electromagnetic charge and despite the fact that self-energy from electromagnetism should be much smaller than self-energy from the strong force.

Indeed, even though the relative self-energy of the top quark to the bottom quark, in the third generation, and the up quark to the down quark, in the first generation, ought to be about the same, the top quark is more than 40 times as massive as the bottom quark, while the up quark has roughly half the mass of the down quark.

Second, the ratio of the up and down quark masses relative to the electron mass is much smaller than the ratio of the strength of the strong force to the strength of the electromagnetic force (the strong force is on the order of 137 times stronger than EM, but the quarks are on the order of 5 times more massive than the electron).

Furthermore, a mechanism that is based upon mass generation through self-energy does nothing to explain the masses of second and third generation particles, which have approximately the same self-energy as their less massive first generation counterparts, unlike the Higgs mechanism which explains the masses of all of the charged fermions in the SM. We also know that a fundamental mass that does not arise solely from interactions with fields is mathematically inconsistent with the SM (which doesn't itself mean that mass arising from self-energy is prohibited in the SM).

Basically, all of the best theories for figuring out why the fundamental particles have the masses that they do, and why, more generally, the SM constants have the values that they do are phenomenological and "numerological", i.e. observations like yours about relationships and patterns without any really solid theoretical basis.

This isn't to say that the fundamental particle masses in the SM appear to be simply random. There are some phenomenological relationships that seem to hold reasonably well (which are beyond the scope of this particular thread), and probably reflect an order with a source in a deeper theory, which is currently unknown to us. But, self-energy alone probably isn't the answer.

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• ftr and Spinnor