Should the Higgs be found at the LHC, but no supersymmetry (assuming for the sake of argument that the LHC be capable of eliminating all versions of SUSY that are motivated by solving the hierarchy problem), what then? Are there alternatives for stablising the Higgs scale that don't involve SUSY (other than Randall-Sundrum)?
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Randall Sundrum doesn't really work to stabilize the Higgs scale. In those cases where people claim it does, it is a low planck scale theory and it is trivially experimentally excluded. But this is not your question.
The most natural mechanism for stabilizing the Higgs is technicolor. This is the idea that the Higgs is an emergent boson, like the pion, of a strong color force.
One motivation for this is the following: suppose you got rid of the Higgs completely, so that you consider the Higgsless standard model. Do all the Fermions become massless, and the weak interactions become long range?
The answer is no. In the absence of the Higgs, QCD vacuum condensates pick up the slack. QCD with massless quarks is not so different from QCD as we know it, except for the pions being close to massless. With more flavors, there are more chiral condensates, and there are more pions, so you break flavor SU(6)xSU(6) to SU(6) (the six massless flavors), and get a full SU(6) worth of pions. You have an udsctb condensate, instead of the ud condensate with a smaller s component, as in our world.
But the up-down condensate breaks chiral symmetry, meaning it is a condensate of left-right quarks which means it breaks chiral SU(2). The chiral condensates are Higgslike, they define a direction in SU(2) space and they have a U(1) charge. Their expectation value breaks weak SU(2)xU(1) to an "electromagnetic" U(1).
But this breaking is SU(6) symmetric, meaning it cannot give a different mass to the different quarks, and flavor isospin SU(6) is exact. Further, the leptons only interact with the condensate through the U(1) interaction, and I am not sure what their masses will be.
The idea of Technicolor is to repeat the QCD story at the Higgs scale, by producing a Higgs mechnanism by a new gauge symmetry group. The technical problem is to introduce interactions with the existing standard model fermions which give the standard model fermions a proper mass. There are technical difficulties here, which lead many people to believe that the theory must be slow running to work, this is called "walking" technicolor.
Such theories stabilize the Higgs automatically, because the Higgs is produced nonperturbatively at the analog of the QCD scale for the new confining gauge field. This also explains the QCD-Higgs coincidence problem (this doesn't have a name in the literature), the 2-order of magnitude coincidence between the QCD scale which determines the mass of the proton and the Higgs scale, which determines the mass of everything else. This is strange, because a-priori, there is no relation even in SUSY models.
The Wikipedia page on this is surprisingly detailed. The space of renormalizable theories is enormously rich, and it should be possible to make the idea work. But this is probably will be much easier once there is more LHC data to give clues.
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$\begingroup$ Thanks (I didn't really mean that RS theories "stabilise" the Higgs, I just wanted to head off answers that might go in that direction). With regard to technicolour, I am told that an SM Higgs should be distinguishable from technicolour at the LHC (indeed, shouldn't we already be seeing technipions and their ilk appearing if it were true?), so I wanted to know if there were models with a genuinely elementary scalar Higgs that isn't troubled by hierarchy. $\endgroup$– JamesCommented Oct 12, 2011 at 14:17
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$\begingroup$ @RonMaimon: Speaking as someone not in HEP, I've always wondered about resolutions to the Higgs fine tuning without SUSY. Do you have any comments on things like top-condensate? It seems a bit too coincidental that top-higgs (phenomenological) coupling is about unity. $\endgroup$– gennethCommented Oct 12, 2011 at 14:53
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$\begingroup$ @genneth: I don't know top-condensate models, I never took them seriously enough to read the papers, but you need a new force to bind the tops together and condense them, and then why doesn't it act on other generations? It seems inelegant to break generation symmetries. I think it is best to wait a few months--- we will definitely know the answer. $\endgroup$ Commented Oct 13, 2011 at 6:29
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1$\begingroup$ @James: So you want to protect a scalar to be massless without SUSY? This is difficult--- the only way I can think of is to make the Higgs a Goldstone boson. If it is a Goldstone boson of a broken chiral symmetry, you're back in technicolor land. If you break something else at high energies, maybe you can get a good Higgs. I believe this is the subject of "little Higgs" theories, but these are usually in the context of large extra dimensions, and I do not read large-extra-dimension papers on principle, but perhaps one should make an exception for the purpose of answering these questions. $\endgroup$ Commented Oct 13, 2011 at 6:33
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$\begingroup$ Thanks, that's what I suspected. It was on my mind, since the LHC seems to be putting the squeeze on vanilla SUSY. Now I need to go off and find out what it's telling us about technicolour. $\endgroup$– JamesCommented Oct 13, 2011 at 7:43
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Yes, there is a mechanism. It's the strong anthropic principle. For there to be consciousness, intelligent life has to evolve in a hospitable environment. With a slightly smaller hierarchy between the Planck scale and the electroweak breaking scale, massive astronomical bodies like galactic superclusters/clusters, galaxies or stars would probably collapse into black holes. Gravity becomes too strong.
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$\begingroup$ you mean like a sort of cosmological-scale Zeno effect, where by simply existing we are continuously measuring a hospitable environment, and hence making a unstable vacuum to become effectively stable? $\endgroup$– lurscherCommented Oct 11, 2011 at 16:48
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$\begingroup$ @lurscher: The Higgs instability is not a vacuum instability, it is just a way to keep the mass low compared to the Planck scale without fine tuning. $\endgroup$ Commented Oct 12, 2011 at 4:55