The Standard Model (SM) of particle physics

The SM describes all particles and fields observed until now, except the gravitational field. These are fields of three types: fermions (quarks, leptons, neutrinos), gauge fields (field of the electromagnetic, weak and strong interactions). Moreover, it contains some not yet observed Higgs field(s).


The SM contains 24 fermions, which appear in some well-defined structured way. Why they appear in such a way is unknown. This is one of the puzzles of the SM, and our approach allows to solve it.

Thus, the structure of the 24 fermions of the SM can be described by the following table:

red quarks green quarks blue quarks leptonic sector
1. generation (down',up) (down',up) (down',up) (electron e, neutrino νe)
2. generation (strange', charmed) (strange', charmed) (strange', charmed) (myon μ, neutrino νμ)
3. generation (bottom',top) (bottom',top) (bottom',top) (τ-meson τ, neutrino ντ)

Only three generations?

If there are three generations, why not more? Maybe some more generations will be found in future experiments? The probability is low, because there are good empirical arguments against the existence of more generations. Of course, if the mass of all particles of the next generation would be larger than the heaviest particle we are able to create in our accelerators now, there would be no disagreement with observation. For most of the particles of the next generation this would be a quite natural assumption, with one exception: the next neutrino. Even if it would be much larger than the heaviest neutrino, by some factor which is of order of the mass relations between different quarks or leptons, it would be nonetheless light in comparison with the particles we can observe today, that means, light enough to be observable today. People have looked for some effects, which would be caused by such a particle, and havn't found them.

Thus, it seems likely that the known three generations are not simply the beginning of some infinite series of generations, but that there are exactly three generations.

The natural question is, why three? Why not four, or five? The answer given in our approach is, that we have three generation because we live in a three-dimensional space.


The mass terms make the whole picture of the SM much less beautiful. You may have noticed that in the table we have assigned an apostrophe to the down, strange and bottom quarks. This is because the mass terms appear quite strange. In the three-dimensional space of the upper quarks (up, charmed, top) as well as in those of the lower quarks (down, strange, bottom) the quarks themself have well-defined and different masses, are so-called "mass eigenstates". But from point of view of the weak interaction, the picture looks different. The weak interaction transforms an up quark not into a down quark, but into a linear combination of all three lower quarks, a state which has no well-defined mass and is denoted in the table by down'. Similarly, the states which interact with the charmed and the top quark are also such strange linear combinations denoted here by strange' and bottom'.

A similar mixing happens between the neutrinos.

Gauge fields

The gauge group of the standard model is SU(3) x SU(2) x U(1), and it consists of following three parts:

These are already all particles of the SM which have been observed, up to now, in particle accelerators.

The weak gauge fields (W and Z bosons) have masses. Instead, the other gauge particles – the gluons of strong interaction and the photon – are massless.

The Higgs sector

There is, yet, some other part of the SM, known as the Higgs sector. The Higgs particle has not been observed until now. Moreover, there are lots of different theoretical variants for the Higgs particle(s). There is only some agreement that some Higgs sector is necessary. So that, if it will not be observed in near future, this would be considered to be problematic.

There is, yet, not much what can be said about the Higgs sector in our lattice model, because this has to be, together with the mass terms, left to future research.

The point is that the Higgs sector is the mechanicsm which gives, during some symmetry breaking, the SM particles it's masses. In our lattice model, we also need some sort of symmetry breaking, to give the particles mass. But this symmetry breaking may be very different from the mechanism given by the Higgs mechanism. So, it is not clear at all, if there has to be some Higgs particle in our approach too.