Why do elementary particles have such strange looking mass ratios?

Why do elementary particles have such strange looking mass ratios?

Elementary fermions of four different values of electric charge are observed in nature. In units of the charge of the positron, these values are 0, 1/3, 2/3, 1. The positron has charge 1, the up quark has charge 2/3, the anti-down quark has charge 1/3, and the anti-neutrino has charge 0. Their respective anti-particles have same absolute value of the electric charge, but of the opposite sign [electron, anti-up, down, neutrino]. We hence say that charge is quantised in these discrete units? But why these and only these, and why is it quantised in the first place?

Each of these four particles has a corresponding second generation particle, and a third generation particle. The second and third generation copies have the same electric charge as its first generation relative. Thus the counterparts of the electron are the muon and the tau-lepton, each of them having charge minus one. The down quark has as its counterparts the strange quark and bottom quark, all having charge 1/3. The up quark's copies are called charm and top, and they all have charge 2/3. We see that the value of electric charge does not change across generations.

The only known difference between the three copies of  a particle is their mass. The electron has a mass of 0.5 MeV. The muon at about 105 MeV is some 200 times heavier than the electron, The tau lepton at 1777 MeV is about 3500 times heavier than the electron. 

The up quark, charm and top have respective masses of 2.3 MeV, 1275 MeV and 173210 MeV. The down, strange and bottom quark have respective masses of 4.7 MeV, 95 MeV and 4180 MeV.

Neutrinos are known to have a non-zero mass, but much smaller than the mass of the charged fermions, and the actual value of their masses is unknown. Let us leave them out of the discussion for now.

It is obvious that unlike electric charge, mass ratios appear strange and random, and show no apparent pattern. If we decide say to compare the various masses with respect to the up quark, and for simplicity take the square root of the ratio, we get the numbers 1, 1.4, 0.47, 6.4, 6.8, 23.5, 42, 274.3, 27.8 The elegant simplicity of quantised values of electric charge is lost. Is there a simple pattern to these mass ratios, or not, and how are we to find the pattern, in case there is one? This profound question has remained unanswered for decades after all these particles were found and their masses measured experimentally.

An answer may now have been found, and it comes from a very surprising and unexpected quarter. We started in a very different realm: addressing a foundational problem in quantum theory. Quantum mechanics is formulated on an external classical space-time but such classical elements should strictly not be a part of quantum theory. We should be able to describe quantum phenomena without referring to classical time. In one specific approach to finding such a description, it has been found that the theory must be  formulated in eight dimensions, which are labelled not by real numbers, but by eight dimensional numbers known as the octonions.

When we try to place fermions in such an octonionic space-time, we are in for a surprise. We are not allowed to assign arbitrary properties and quantum numbers to these particles. The space-time dictates that the electric charge must be quantised, precisely in the units 0, 1/3, 2/3 and 1, as observed. And the space-time also dictates that particles have anti-particles of opposite charge, and come in three generations. 

If the space-time dictates there are three generations, and if the only difference between the three generations is mass, the octonionic space-time must also determine mass ratios, just as it determines ratios of electric charge. Happily, it does. Just as there is a charge value associated with every particle, there is a mass number associated with every particle, called its Jordan eigenvalue. These Jordan mass numbers are shown in the table below (Ignore the numerical entry in the first column). The entries in the next three columns are the respective mass numbers.

These mass numbers are very simple and pretty. For the three generational copies of a particle, the middle value is its electric charge, and the other two values are symmetrically placed about the middle value, always departing from the middle by a factor of square root of 3/8. 

The theory dictates that mass ratios will be determined by these Jordan values, and these mass ratios are shown in the table below, scaled with respect to the down quark mass which is set as one. It can be easily verified that these simple fractions reproduce the strange pattern of mass ratios observed in nature! Mass is quantised, just as electric charge is, and the ratios are quite simply given, though not as simple as the charge ratios 1/3, 2/3 and 1. 

We believe that the mystery of the strange mass ratios has finally been solved. At the heart of the resolution lies the realisation that elementary particles are described by the octonions, as long suspected by several physicists.

Also:

Mass ratios and Majorana neutrinos:

Getting the correct mass ratios from the octonion algebra requires us to assume that the neutrino is a Majorana particle [i.e. its own anti-particle]. Assuming the neutrino to be a Dirac particle gives wrong mass ratios. We hence predict that Neutrinoless Double Beta Decay does occur in nature and will indeed be observed once adequate sensitivity is achieved in experiments.

Reference: https://www.preprints.org/manuscript/202101.0474/v4