The Standard Model of Particle Physics is built out of gauge groups and the Dirac algebra. The Dirac algebra describes the coupling of elementary particles to the spacetime background. Ideally, we want a model that is independent of the background, so that it applies universally at all times and in all places. The Standard Model is not background independent, although a lot of physicists assume it is. The reason it is not background independent is because it is Lorentz-invariant. Lorentz invariance tells you how things appear differently if you move fast. But if you move fast, then the background changes. Since you cannot edit out the background (quantum gravity), Lorentz invariance is inconsistent with background independence.
That is, in a nutshell, the inconsistency between gravity and quantum mechanics. The Standard Model is built to be Lorentz invariant, not background independent. Consistency with General Relativity requires background independence. My SU(7,2) model, expounded in recent posts, explains how you can choose between the Lorentz invariant Standard Model with a single generation of fermions, and a background independent model with three generations of fermions. The background independent model contains a discrete generation symmetry, that changes the generations of the electrons. It therefore cannot contain any mass, because mass breaks this symmetry. Mass therefore comes from the Dirac equation that couples the quantum numbers (weak isospin, weak hypercharge, lepton number, baryon number, flavours, etc.) to the gravitational background.
It follows that all the elementary particle masses (and I mean all) are background dependent. The mass of the electron is a completely meaningless concept unless you have a gravitational background in which to measure this mass. I know this is heresy, but it is obvious. You do not need any fancy mathematical model to deduce this obvious fact. I realised the truth of this obvious fact some ten years ago, long before I started developing any mathematical models at all. But physicists won’t listen to arguments like that, so I have been forced to develop the mathematical models in order to try and blind them with mathematics.
Physicists think that masses of elementary particles are intrinsic to the particles. This is impossible for the simple reason that elementary particles contain only a finite amount of information. Moreover, if masses were intrinsic to the particles, then we would not need the Dirac equation to couple the masses to the particles, or couple the particles to spacetime. In that case the entire foundation of quantum mechanics would be redundant. This is absurd.
It appears from the work I have done, that a background independent model is not only massless, but is also discrete. So it doesn’t have mixing angles either. Apart possibly from the Weinberg angle, which is half the larger angle in a discrete right-angled triangle with sides of lengths 2 and 3. All the other “unexplained” parameters of the Standard Model are background dependent. Every single one. Even the fine structure constant. Gasp! Is nothing sacred? No, nothing is sacred.
The nonsense that is talked about neutrino masses arises from the same confusion between background independence and Lorentz invariance. The flavour of a neutrino is a quantum number, and is therefore background independent. Therefore it is not Lorentz invariant. Particle physicists try to deal with this by defining a Lorentz invariant neutrino eigenstate – which means a mass eigenstate. But they can’t measure this (rest) mass because it is too small. I would say that it is actually exactly zero, but it would be more accurate to say it is meaningless, rather than zero. It is meaningless (or zero) in the same way that the (rest) mass of the photon is meaningless (or zero). A neutrino can never be at rest, just as a photon can never be at rest. The concept of “rest mass” does not apply to them.
If you insist, as my collaborators do, that your model of particle physics must be Lorentz invariant, then you cannot do better than the Standard Model. This is a maximal Lorentz-invariant model and contains everything that actually is Lorentz invariant. But it is not background independent, and therefore when the background changes even a tiny bit (say when you rotate the gravitational field by a millionth of a radian) you will get “anomalies” creeping in to your experimental results. These anomalies are ten a penny, they’re all over the place once you start looking for them. The only way to resolve these anomalies is to abandon the idea that you want a Lorentz-invariant model, and seize the idea that you want a background-independent model. And it you want a background independent model, then you first have to take out the 20 or so background dependent parameters from the Standard Model, by explaining exactly how they depend on the background.
That’s what I started to do ten years ago. I think I managed to take out about 15 of them, but no-one listened so I gave up.