The maximal gauge group possible in our model (as a Wilson gauge field or a field describing lattice deformations), which is compatible with a few general principles (preservation of symplectic structure, of Euclidean symmetry, of neutrality of the vacuum), is the group

G_{max} ≅ S(U(3)_{c}×U(2)_{L} × U(1)_{R})

It contains the whole SM gauge group

G_{SM} ≅ SU(3)_{c} x SU(2)_{L} x U(1)_{Y}

and only a single additional gauge field U(1)_{U}, which I have named ** upper axial boson**. It's charge is

I_{U} = γ^{5}(I_{3}- ^{1}⁄_{2}).

That means, it acts on all upper particles — particles with isospin I_{3}= - ^{1}⁄_{2} — the up quark, charmed quark, and top quark, as well as the electron, myon, and the τ-particle. Instead, all lower particles (those with isospin I_{3}= ^{1}⁄_{2}) — the down quark, strange quark, and bottom quark, as well as the three neutrinos — have charge 0.

The factor γ^{5} is what makes it an "axial boson". Left-handed and right-handed parts of the same particle have different charges.

Thus, our axioms allow not only the observable gauge fields of the SM, but also an additional field which has not been observed yet. Now, it would be possible to say that this additional field is only allowed, not required. There is no necessity for this additional field, thus, using Ockham's razor, we should not introduce it. But in effective field theory, a different principle holds (which can be also understood as a principle of simplicity of another type): The principle is "what is possible, compatible with the basic symmetries, will appear". Following this principle, the upper axial boson will exist.

There are two ways to handle such a situation: First, to hope that this additional particle will be observed in some future. The alternative is to search for an explanation why the additional field is not observable.

The author of these pages hopes for the first possibility — that the upper axial boson really exists, but simply has not been observed until now. This seems possible: First, it is a diagonal field, thus, does not lead to any particle decays which would be impossible otherwise. Then, it's mass and interaction constant are unknown. If its mass wil be sufficiently large, it will be impossible to observe it until a large enough collider will be build. As well, if the interaction constant is sufficiently small, it becomes hard to observe it as well.

Of course, observing such an additional particle field would be the best, ultimate support for this model. Nonetheless, it seems necessary to think about other ways too. And there is an important open problem which allows, in principle, to explain why we do not observe an upper axial boson. This problem is the gauge anomaly.

From point of view of the standard (BRST) approach to gauge field quantization, not every gauge action can be quantized. There is an important restriction — there should be no ABJ (Adler-Bell-Jackiw) anomaly in the action of the gauge group on fermions.

The condition is a quite technical one. We have to look for the "axial part" of the trace of the product of three (possibly, but not necessarily different) generators τ_{i} of the gauge action. The axial part of an expression A we obtain as the expression A_{1} in the decomposition A_{0} + γ^{5}A_{1}, where nor A_{0}, nor A_{1} contain γ^{5}. Thus, a theory is anomaly free, if, whatever the choice of the generators τ_{i}, there should be

Tr ({ τ_{1L} τ_{2L}} τ_{3L}) = Tr ({ τ_{1R} τ_{2R}} τ_{3R})

The SM gauge group is free of anomalies. Instead, for τ_{1} = τ_{2}= τ_{3} = I_{U} (the charge of the upper axial gauge group U(1)_{U}) violates this condition.

That means, according to the standard procedure for the quantization of gauge fields, our maximal gauge group G_{max} would be forbidden as anomalous. We would have, instead, to look for a non-anomalous subgroup of G_{max}, which could be quantized.

Now, **G _{SM} is the maximal anomaly-free subgroup of G_{max}**. Thus, this seems to be a nice way to obtain the SM gauge group.

Unfortunately for this solution of the problem, in our approach it is not that simple to forbid anomalous gauge groups. The point is that the standard approach requires, for the quantization of gauge fields, exact gauge symmetry. The anomaly is forbidden because it violates gauge symmetry.

But in our approach, the gauge degrees of freedom are physical. We have no exact gauge symmetry on the lattice, and do not need it. Thus, there seems no reason to forbid anomalous gauge fields as well.

Does that mean that the idea to use the gauge anomaly to explain why U(1)_{U} is unobservable has failed? Not necessarily. The gauge anomaly exists, and it will lead to nontrivial physical effects in our approach too. Which effects? This is yet unknown. This is an open problem for future research.

And it is quite possible that, at the end, the result will be similar — that the final theory (more accurate, the large distance limit) has to be an anomaly-free group. Therefore, the observation that the SM gauge group is the maximal anomaly-free subgroup of G_{max} is not completely useless. It may become a key for the solution of the problem.