The Standard Model has a mixture of broken and unbroken symmetries, but I don’t understand why. In the real world, all symmetries are broken, always. At best, symmetry can be approximate. So I don’t understand why the strong force is described by an unbroken colour symmetry group SU(3), that implies the mediators are massless gluons, when there is a fundamental principle that says massless particles mediate long-range forces, and short-range forces like the strong nuclear force are mediated by massive particles. This is a fundamental contradiction that no physicist has ever been able (or willing) to explain to my satisfaction. In the real world, the strong force SU(3) symmetry must be broken, just as the weak isospin symmetry SU(2) and weak hypercharge symmetry U(1) are in the Standard Model.
To understand how symmetries break, let us go back to first principles, and consider how a sphere behaves in Special Relativity. In Newtonian mechanics, a sphere is a sphere is a sphere, and has spherical symmetry described by the rotation group SO(3). In Special Relativity, a moving sphere contracts in its direction of motion, so that protons in the LHC, for example, are as flat as pancakes (or even more so). This is an effective breaking of symmetry, due to the motion of the observer relative to the object being studied. Of course, in the proton’s own (or “proper”) reference frame, the symmetry is not broken. It is only broken in the observer’s reference frame. This is a fundamental physical principle: broken symmetries arise from the “peculiar” (in the technical sense used in astronomy) motion of the observer. No “observer” means no observation, no context, no symmetry-breaking. But the “observer” does not have to be human, or conscious, the “observer” can equally well be the Solar System itself.
So, the symmetries of a sphere in Special Relativity can be broken by Lorentz contractions in three directions independently, so that we add 3 degrees of freedom to SO(3) to get SO(3,1), which has 6 degrees of freedom altogether. The three “rotation” symmetries are “internal” symmetries, but what we observe are the other three symmetries – they correspond to velocities in the observer’s frame of reference. We cannot observe the 3 “internal” symmetries directly, only the other 3. But we might be able to deduce the internal symmetry from the way a particle interacts with the outside world.
Now let’s look at the weak force, with gauge group SU(2). Since SU(2) is just a double cover of SO(3), the mathematics of the symmetry-breaking (but not the physics) is essentially the same as in the Lorentz group. There is a “Lorentz transformation” in one direction that increases the mass of the Z boson relative to the mass of the W bosons in the two perpendicular directions. This is mathematically equivalent to the way that the inertia of a moving particle increases in Special Relativity. It’s the same formula, in (almost) the same group. Again, we do not observe the “internal” massless symmetries, i.e. the theoretical massless gauge bosons, directly, what we observe are the other three degrees of freedom, identified as the Z, W+ and W- bosons.
Why doesn’t the Standard Model treat the strong force in the same way? I have no idea. Why don’t we treat the strong force in the same way? No reason – let’s do it! We have a group SU(3) that contains 8 independent unobservable “colour” symmetries, and we have a corresponding set of 8 (I guess?) observable symmetries, which I guess we want to identify as massive bosons – for example the “meson octet” consisting of three pions, four kaons and the eta meson. Or I might prefer to count the five kaons that are observed in experiments, rather than the four that exist in the old theoretical model, in which case I identify the pions with the subgroup SO(3), rather than the subgroup SU(2) that Gell-Mann et al. used in the old days. Or there might be some other way of doing it.
In any case, we have established the principle that unbroken symmetries are represented by compact “gauge groups”, and that broken symmetries are observed when we extend to non-compact groups, so that we actually have something to measure. We also know that the unbroken symmetries consist of U(1) weak hypercharge, SU(2) weak isospin, and SU(3) colour, making 12 degrees of freedom for these internal symmetries. Now we follow Georgi and Glashow, who noticed that the smallest simple Lie group that contains a product of these three groups is SU(5). But we don’t make the mistake that they made, of assuming this SU(5) is compact. We take the other possibility, that is SU(3,2), instead.
Then we find that “the” compact subgroup of SU(3,2) is exactly the gauge group of the Standard Model. But it isn’t unique, so what do we mean by “the”? To see what this question means, go back to the Lorentz group SO(3,1) in Special Relativity. If we want “the” rotation subgroup, then we have to specify the observer: the internal or “proper” rotation symmetry of a proton is quite different from the external or “peculiar” rotation symmetry of the observer. The “proper” rotation symmetry is (I assume) exact, but the “peculiar” rotation symmetry is broken. Both are described by subgroups SO(3) of SO(3,1), but they are completely different copies of SO(3). In a linear collider, the two copies intersect in SO(2) rotation symmetry around the direction of motion, but in a circular collider even this symmetry is broken, and the two copies of SO(3) are completely disjoint.
Much the same thing happens in SU(3,2). We can take a copy of “the” compact subgroup that is completely internal and completely unbroken, or we can take a copy that is completely external and completely broken, or we can do what the Standard Model does, which is a half-hearted attempt to break the symmetries of the SU(2) part and leave everything else alone. Indeed, when physicists tell me about the symmetries, they tell me that SU(2) symmetry “is” broken, and that SU(3) symmetry “is not” broken, as though these are physical facts. But obviously they are not facts, they are points of view. Both broken and unbroken symmetries exist in the SU(3,2) model, so you just have to specify which ones you are talking about. OK, so the Standard Model talks about “broken” SU(2) and “unbroken” SU(3), which is perfectly fine as far as it goes, but what experiment actually observes is the broken SU(3), so we might as well talk about that instead.
The first question to ask is, how much symmetry-breaking is available in SU(3,2)? Well, the unbroken symmetries are 12-dimensional, and the total dimension is 24, so there are 12 dimensions of symmetry-breaking available. That is exactly the number we need in order to break all the symmetries, and have nothing left over. We can then choose two disjoint copies of the gauge group, one internal and completely unbroken, one external and completely broken. Together, these two copies of the gauge group cover the whole of the 24 dimensions. To get a complete picture of the broken symmetries, you need 12 parameters to get into the right copy of the gauge group, and another 12 to move around inside that copy once you get there. In other words, there are 24 parameters altogether, that you need to measure by experiment. This is roughly the number we expect from the Standard Model, but the precise number depends on how exactly we treat “the” Lorentz group(s).
Let’s first look at the 12 parameters inside the broken copy of the gauge group. These are all “rotations”, because the gauge group is compact, and in principle they split into 1+3+8 parameters, but of course there are many ways of expressing the information, so this splitting is not necessarily evident in the Standard Model. Most of these “rotations” must be essentially the 9 mixing angles of the SM, which leaves 3 for ordinary common or garden rotations of space, or perhaps for coupling constants for the three forces. Some work needs to be done on the details, but the principles are clear enough.
Now look at the 12 parameters outside the broken copy of the gauge group. These are all “boosts”, that is non-compact degrees of freedom, so are real numbers, mostly masses, I presume. If they are all masses, they neatly cover the 12 masses of the 12 fundamental fermions. But perhaps neutrino masses, which cannot be directly detected by experiment, are better taken to be effectively zero, leaving these three degrees of freedom for Lorentz transformations, or something else. Again we have some details to sort out, but the basic principle that the 24 degrees of freedom split into 6 for the Lorentz group, 9 mixing angles and 9 detectable masses seems well-established.
So far, so good. What about gravity? I am glad you asked that question, because if we include the Lorentz group in our scheme, then we have to break the Lorentz symmetry as well as everything else. And it is gravity that breaks the Lorentz symmetry. The way it does this is to distinguish between an “internal” copy of the Lorentz group, describing how electromagnetism works in the laboratory, and an “external” copy of the Lorentz group, describing the motion of the Solar System as a whole. In other words, we distinguish between a “local” and a “global” definition of inertial frame. These definitions, by the way, use SO(3,1), not SL(2,C), so take the Lorentz group to be inside “the” “real” subgroup SO(3,2) of the “complex” group SU(3,2). You can also use SO(3,2) to define the Dirac gamma matrices, and hence write down a Dirac equation acting on a 5-component “spinor” (compared to the Dirac 4-component spinor).
This is quite remarkable, because you get a Dirac equation without having to use SL(2,C). And now we know which is the real Lorentz group, and which is the impostor! We’re well into Act 3 of the opera, and the almighty confusion of Act 2 is beginning to sort itself out. Deus ex machina has arrived in the form of gravity, and is going to take control of the situation. First of all, the impostor SL(2,C) “Lorentz group” has been unmasked, and banished to Hades. We have already noted that the “unbroken” symmetries live only in Heaven, and not on Earth. The stage is set for a completely unexpected denouement, ending in a quartet for soprano (electro-magnetism), contralto (the weak force), tenor (the strong force) and bass (gravity), all united and living happily in harmony ever after. Massive chords of C major for the whole orchestra, and the curtain finally falls on the whole extraordinary tale.
Gravity reveals that he knows how to explain the 24 unexplained parameters, and proceeds to do so, one by one. It all comes down to the confusion between the real Lorentz group and the impostor. And the relationship between the Lorentz group of the laboratory, and the Lorentz group of the Solar System. There are a number of parameters that relate the two, including the number of days in a year, the tilt of the Earth’s axis, the inclination of the Moon’s orbit and the number of days in a month. Gravity tells the assembled cast that the electron/proton mass ratio is half the sine of the angle of tilt divided by the number of days in a year. At least it was back in 1973, when the real Lorentz group was sent to prison, and the impostor started walking around in his place. The impostor failed to maintain this relationship between gravitational and inertial mass, but now the formula has been revealed, everyone can see the deception for themselves. But Act 3 is not yet finished, and we leave gravity in the middle of his magnificent aria, explaining the unexplained in the stylised manner of Hercule Poirot.