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What is the current thinking on the little hierarchy problem in light of a potential Higgs mass above 120 GeV? A few years ago, at least, I remember various phenomenologists saying that this at least makes life rather difficult for the MSSM.

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The newest paper with "little hierarchy problem" in the title is one from May 2011 by Feldman, Kane, Kuflik, Lu: - those gravitinos and moduli around 30 TeV do solve it, aside from other problems. Another 2011 paper: – Luboš Motl Dec 12 '11 at 19:08
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I don't think there is a little hierarchy problem in general (this is why I never regarded it as a real problem; it is a somewhat mild issue only in the most minimal SUSY theory).

The origin of the little hierarchy is that in the MSSM, at tree-level, the Higgs cannot be heavier than 90 GeV or so (see hep-ph/9709356 for a still relevant review). So in light of the LEP bound (115 GeV) one has to get a pretty decent contribution (to the quartic coupling) from loops. To make the loops sufficiently large one needs the stop to be somewhat heavier than the top. But that comes back to hunt you since it destabilizes the Higgs mass. Typically, for a Higgs around 120-130 I believe one would have to live with tuning at the per mil level (more optimistically, one could perhaps get 10^{-2}).

However, innocuous additions to the MSSM allow for heavier Higgs particles at tree-level. For instance, one can add a singlet (this leads to the so called NMSSM) or one could add an additional gauge group that is broken at the TeV scale (this enhances the quartic coupling through new D-terms). One can cook models where there is virtually no tuning whatsoever.

Note that there is no model independent bound on the stop mass, it could easily be 300-400 GeV in some scenarios. Therefore, in the presence of additional tree-level terms in the Higgs potential, there is no tuning.

(It should be added, though, that in many models beyond the MSSM one looses gauge coupling unification...)

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I remember the NMSSM. But, well, ick, no? And your last comment (if you want various charged stuff floating around) sort of spoils the point of the whole thing. But now I guess I'm getting more opinionated :). – Aaron Dec 14 '11 at 1:35
In fact, the stop mass can be 200 GeV or perhaps even lower; as you say, it's model-dependent. Hopefully the story of stop mass limits (or signals?) will change very quickly over the next year. – Matt Reece Dec 14 '11 at 3:40
Matt: Yes I agree, there is also the famous small window for the stop mass below 200 GeV. Aaron: I don't know why you say the NMSSM is more ugly than the MSSM. They look comparable in complexity to me. Also the NMSSM does not introduce any new charged fields. It jeopardizes unification for a different reason, a Landau pole for a Yukawa coupling.... – Zohar Ko Dec 14 '11 at 11:20
I guess there's two ways of looking at it -- you could say you're doubling the number of fields out there, why not throw in another two? Or you could say, oh my god, you're doubling the number of fields, and you want to throw in another chiral, too, because you're still not fitting the real world? I never was a good phenomenologist.... – Aaron Dec 14 '11 at 13:26
I am also not a true phenomenologist, although I spend a lot of time thinking about pheno. To me all these models look equally complex, I would not judge something just because it has one more field... – Zohar Ko Dec 14 '11 at 14:18

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