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Why does the $1/2$ spin of fundamental fermions (electrons, quarks, and neutrinos) split them into three variants that differ only in mass, while the integer spins of massless fundamental bosons (e.g. photons and gluons) causes no such splitting?

If the bosons had mass, would they also split into three generations?

Terry Bollinger
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    Some bosons do have mass, e.g., the Higgs. And there is a real possibility that the Higgs actually is a multiplet with multiple masses. A more general issue is what framework one could use to answer this type of question. You used the string-theory tag -- do you want a string-theory answer? One could also try to answer this based on the anthropic principle, in which case I suspect the answer is that there is no answer, since I doubt it matters anthropically that we have the tau lepton. –  Oct 13 '14 at 01:57
  • +BenCrowell, if string theory has possible insights on that one I'd be delighted to hear them! Also, your point about Higgs is fascinating — has anyone actually postulated a Higgs triplet? – Terry Bollinger Oct 13 '14 at 02:06
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    There are some good articles about the Higgs at this blog: profmattstrassler.com –  Oct 13 '14 at 02:08
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    Heuristically, I think of it like this: The fermions are matter particles, and must obey no particular restrictions. The fundamental bosons arise through gauge symmetries, and there's only "one" (Lie-algebra valued) gauge field for each such symmetry. Having three generations of photons would mean having three different unbroken $\mathrm{U}(1)$ symmetries, for example. – ACuriousMind Oct 13 '14 at 12:06
  • Regarding string theory and @Heterotic's intriguing comment that it does (or can) predict generations: Can anyone elaborate on that specific point? Is there a simple (heh!) example somewhere that shows how e.g. a vibrating string might lead to some fixed number of harmonics? Ben Crowell, any pointers on that? To me, the generations issue is still one of the top outstanding and most fascinating mysteries still remaining in physics. It's just so "not obvious" that it has to be telling us something hugely important. – Terry Bollinger Oct 15 '14 at 14:08
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    @TerryBollinger : The "simplest" example are orbifold models. They have some number of fixed points and states are localized around these points, so the discrete number of generations arises this way. Another example is free-fermionic models. Have a look at figure 1 in this technical paper: http://arxiv.org/abs/arXiv:1403.4107 to see that they can get thousands of models with a given number of generations within their framework. – Heterotic Oct 19 '14 at 09:41
  • Heterotic, thanks, that's great info and much appreciated! – Terry Bollinger Oct 19 '14 at 16:28

2 Answers2

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The number of gauge bosons is restricted by symmetry: a given theory with a certain gauge invariance admits as many gauge bosons as there are generators of the corresponding gauge group. For example, there is one generator for $\mathrm{U}(1)$, resulting in the existence of a photon. $\mathrm{SU}(3)$ admits eight generators, which yield eight gluons. This is true without any reference to generations. Regarding the Higgs, it is not entirely clear that there exists only one particle.

The number of fermionic (matter) particles in the Standard Model is not dictated by gauge symmetry and in principle one could construct a model with more or fewer generations. The question as to why there are exactly three generations is still an open problem, even in string theory.

Frederic Brünner
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  • If supersymmetry exists, then we have a whole bunch of other bosons out there. How does that fit with this argument? Would there be no contradiction with this argument because things like selectrons and squarks aren't gauge bosons? –  Oct 13 '14 at 15:50
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    The symmetry argument holds true for gauge bosons. To my understanding, there would be as many generations of superpartners as there are for quarks and leptons. – Frederic Brünner Oct 13 '14 at 16:30
  • If I could just suggest: Perhaps the more interesting facet of the symmetries question might be why, if they do exist, they are so badly broken. The breaks are like windows (or perhaps detectors) into some broader reality where things aren't as smooth as simple symmetries would suggest... and that has to make them interesting and insightful, assuming we can ever figure out the root causes of the breaks. – Terry Bollinger Oct 15 '14 at 14:12
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There are two things that define a particle physics model (at low energies). The first one is the gauge group G we want the model to be symmetric under. For the Standard Model (SM) we set this to $G=SU(3)\times SU(2)\times U(1)$ (for good experimental reasons!). This will uniquely determine the number of gauge bosons needed to make the model consistent.

The second ingredient is the matter content, i.e. how many fermions and scalar fields we want the model to include. In the SM we choose 3 generations of fermions (this includes the leptons, quarks and their CP partners) and 1 scalar field (the Higgs).

From a low energy point of view, the two choices are completely independent and we can choose anything we like. For example it is conceivable to have a model with $SU(5)$ gauge symmetry and no matter at all, etc...

However, the OP is not alone in not being satisfied with this apparent lack of symmetry in treating fermions and bosons. Most physicists would indeed agree that we should try and find a more unified description and hope that nature has such a bigger symmetry at a deeper level. Theories beyond the Standard Model are indeed following such an approach. The best studied example are supersymmetric theories in which every boson/fermion has a superpartner that is a fermion/boson. In this theories, the symmetry of numbers is restored!

All in all, to answer questions like the number of generations in the SM, the masses of the fermions, etc you need a candidate for a complete theory at high energies and the answers will depend on this candidate. String theory is considered to be the most successful framework for this job and string models do indeed make concrete predictions about (among other things) the number of generations we should be observing. Unfortunately, there are too many models (vaccua) to choose from and no obvious way to make the choice...

Heterotic
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  • Your comment about string theory being capable of predicting generations, but also leading to too many models, is at the heart of why I get very frustrated with it. A formal framework is also necessarily a language in which assertions can be made, and string theory is no exception. A predictively interesting language, such as the symmetries Dirac used to predict antimatter, should only be capable of making a limited number of such assertions. The number of vacuua in string theory in contrast implies that its fundamental semantic units— string vibrations — are too rich too predict anything. – Terry Bollinger Oct 14 '14 at 11:13