Why should the mass of leptons to be near of proton and QCD/chiral scales?

The mystery of the mass of the top being in the electroweak scale can be justified by the Higgs mechanism itself; in some sense the top mass is the only "natural" mass, the other masses of fermions being protected by some unknown mechanism so that they are "zero" relative to the electroweak: the typical values of the Yukawas are in the $10^{-3}$ range and even smaller. The masses are still "natural" relative to GUT or Planck Scale cutoffs, because then their corrections are logarithmic and about the same order of magnitude that the mass itself, in fact about a 30%.

My question is about the mass of tau and mu. Is there some reason for them to be in the GeV range, where QCD masses -proton and pion, if you wish, or glue and chiral scales- are?

I am asking for theories justifying this. For instance, Alejandro Cabo tries to produce first the quark masses from the one of the top quark via QCD, very much as Georgi-Glashow electron-muon in the early seventies, and then he expects that the leptons should have masses similar to the quarks via the electromagnetic/electroweak coupling (albeit some group theoretical reason could be enough).

Hierarchically, and from the point of view that any near zero value for a coupling is a hint for a broken symmetry (so that in the limit where this symmetry is unbroken, the coupling is exactly zero), what I would expect is a symmetry protecting all the yukawas except the top, and then still a subgroup protecting the first generation.

• Note that besides $M_\mu$ near of the pion mass and $M_\tau$ near the glueball, the Koide basic mass $(\frac 13 \sum_l \sqrt M_l)^2$ is about 313 MeV, similar to the constituient mass. – arivero Feb 17 '11 at 11:28
• The self-energy contributions to the electron mass is only logarithmic in $\Lambda$. This doesn't explain why the mass of the electron is small. – QGR Feb 17 '11 at 15:04
• The electron is massless in the SM before gauge symmetry breaking and then acquires a mass proportional to a Yukawa coupling and the Higgs vev. You can't just ignore this and start talking about E&M self-energy. You're discussing the QFT of the 1930's. – pho Feb 17 '11 at 16:09
• @Jeff and @QGR, you are right, the argument is irrelevant, because $\delta m \approx \alpha m \ln {1 \over m \Lambda_P}$ is more or less the same percentage of $m$, about a 30%, around all the interesting range of values of $m$ (the masses are natural in this sense, jut because it only varies logarithmic, as @QGR says). I mentioned it because Nottale postulates a fixed value for $\delta m /m$ and uses it to "predict" electron mass but it misleads, so let me to edit it. – arivero Feb 17 '11 at 17:57

The anthropic principle might explain it.

• The difference in mass of the neutron and proton is roughly the difference in mass between that of the down and up quarks, which is comparable to that of the electron mass. If the proton were significantly heavier, they would decay into neutrons — which are electrically neutral — leading to no atoms or chemistry.

• The instability of neutrons with a half life of 15 minutes is necessary, as otherwise, most of the hydrogen and neutrons would fuse together into deuterium during the big bang nucleosynthesis epoch. This means the mass of the neutron should be slightly larger than the sum of proton and electron masses.

• For chemistry as we know it to be possible, molecules ought to have definite shapes, as opposed to a delocalized quantum superposition of nuclei positions. This requires the nucleons to be at least 1000 times heavier than the electron, fixing the QCD scale.

• The weakness of the weak interaction is due to the smallness of the Yukawa couplings for the first generation of fermions. If the weak interaction were significantly stronger than it is in our universe, the neutrinos generated during the explosive fusion when the core of a massive star collapses will interact too strongly with the stellar matter that they would be absorbed before reaching the outer regions of the star. In our universe, the weak interaction is sufficiently weak that enough neutrinos can reach the outer regions, but yet strong enough to interact sufficiently there to blow the outer region away in a supernova. Without supernovae, we won't have planets with significant quantities of heavier elements needed for life.

However, I can't think of any anthropic reason why the Yukawa coupling for the tau lepton isn't of order unity.

• But the main puzzle is about tau and muon. Of course, if you say that the electron scale fixes the other two, it could be an argument. – arivero Feb 17 '11 at 15:25
• The third point can't be right--- why wouldn't you be able to do muonic chemistry with nuclei in superpositions? It doesn't make much difference. The other arguments are fine. – Ron Maimon Dec 31 '11 at 12:16