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If you look at the masses of leptons and quarks in the standard model, you will see that the masses are not very similar but are orders of magnitude apart.

e.g. Top quark mass is about 40 times bottom quark mass which is about 40 times strange quark mass.

Likewise the lepton masses are orders of magnitude apart in each generation.

If, for example, the known fermions were made of smaller particles you would expect to see the masses (squared) follow more of an arithmetic progression. So this probably rules out this idea.

In a theory like string theory, the masses depend on how certain extra dimensions are curled up which in turn affects how much the Higgs boson interacts with each fermion. But would we expect to see orders of magnitude mass differences in most cases?

Likewise should we expect the neutrino masses to be orders of magnitude different from each other. And WHY would we expect this? (What physical principle?)

Qmechanic
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  • Lepton and quark masses are tunable parameters in the Standard Model, aren't they? As such how can there be any theoretical "reason" for the discrepancy (beyond a possible invocation of the anthropic principle)? – Nihar Karve Mar 17 '21 at 07:40
  • @NiharKarve The masses can be constrained or even determined in beyond-standard-model theories – Mitchell Porter Mar 17 '21 at 08:24
  • Well, I thought there might be even some mathematical principle, such as distributions of eigenvalues of random matrices, or some such. If the top quark is a tunable parameter and so is bottom quark and stange, it is weird that there is about a 40x difference in masses of these 3 particles and not say 2x or 1000000000000000x. Most particles in the SM in generations seem to be about 1-3 orders of magnitude apart for no aparent reason. –  Mar 17 '21 at 14:09
  • Related: https://physics.stackexchange.com/q/470376 – Nihar Karve Mar 18 '21 at 05:43

2 Answers2

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The masses come from, and are proportional to, the yukawa couplings to the Higgs field. So part of the question is, how to obtain yukawas that differ by so many orders of magnitude?

I may have overlooked something, but all the examples I can think of, fall into one of two categories. Either the yukawa is an exponential function of something else - in which case the "something else" only has to vary by a small factor, in order to produce large variation among the yukawas; or, the yukawa is obtained from Feynman diagrams in which different numbers of some vertex appear, giving rise to different powers of the associated coupling.

The examples of exponential dependence best known to me, involve localization of the SM fermions at different points in compact extra dimensions. The distance apart in the extra dimensions is the small quantity, on which the yukawa is then exponentially dependent.

As for the other option, the Wikipedia page on family symmetries gives several examples. Typically, the three generations have a different "family charge" under some new U(1) symmetry, which then determines the number of vertices in the underlying process which gives rise to the yukawa coupling. For example see figure 6, page 33, in these lectures.

Something else to remember is that the yukawas actually come in 3x3 matrices. So in order to obtain the hierarchy, one will consider different properties for these matrices - their "texture" (number and location of zero elements), their "rank" (number of linearly independent columns) - that approximate the observed spectrum of masses, and then look for symmetries etc that will give the yukawa matrices that structure.

I will add that this is my favorite paper on the mass hierarchy, but the extra structure it observes is apparently not something that any known mechanism can naturally generate.

Mitchell Porter
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  • That is interesting, so it might be the case that the Higgs is acting via a double or triple vertex Feynman diagram on some fermions, making them approximate a coupling constant of $\alpha^3$ for example. That would make some sense. Not sure it would completely explain the big mass range but interesting. –  Mar 19 '21 at 17:23
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Here is a long comment on masses in the standard model from a experimental point of view:

All masses were supposed to be equal before symmetry breaking as the available energy in the expanding cosmos was distributed to larger volumes. There were only the group structures, example ., which remained unchanged.

After symmetry breaking and the coming to existence of the Higgs field, the particles acquire mass, but the group structure is unchanged.

Thus it is the way that the Higgs field is functionally implemented that gives mathematically the experimental different masses to different particles in the group structure, to fit the data. At present the vacuum expectation value and the Higgs mass are fitted from the data.

It may be that in a different future theory these could be theoretically predicted.

anna v
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  • While this is true, the question is more about why the particles acquire masses that are more like a geometric than an arithmetic progression. –  Mar 18 '21 at 14:02
  • @zooby I answer : the way that the Higgs field is functionally implemented That is why I say it is a comment. I do not know the maths, but I expect that the group structure, the quantum numbers,.at symmetry breaking, enter in the calculation of the mass of the fermions in the table. – anna v Mar 18 '21 at 14:52