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As far I as I know, and from naturalness considerations, a 125 GeV Higgs mass is rather large for the MSSM. This is because in the MSSM $$m_h^2 \lesssim M_z^2 \cos^22\beta + \Delta$$ where $\Delta$ represents top/stop loop corrections to the Higgs. It takes the form $$\Delta \sim \ln(\frac{m_\text{stop}}{m_\text{top}}) + \text{mixing}$$

Moreover, the Z mass is determined by the Higgs mass parameter, $m_{H_u},$ and $\mu$ (in the large $\tan\beta$ limit. i.e. $\tan\beta \geq 10$). And $m_{H_u} \sim - m_\text{stop}$, so the larger $m_\text{stop}$ the larger the fine-tuning in the MSSM. Now, only by large $m_\text{stop}$ (and mixing) one can reach a Higgs mass greater than LEP/LHC bounds. But this is already associated with large fine-tuning as I said.

So, in this view, a 125 GeV Higgs mass is large for the MSSM. (At least for the cMSSM.)

However, I realised that some people think 125 GeV is much less than what the MSSM predicts! Now this I don't understand at all. So could someone please explain what this view is based on?

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I recommend you an excellent December 2011 paper on this very question:

http://arxiv.org/abs/arXiv:1112.3068

Matt Reece, a co-author of this paper and a contributor at this very forum, just accepted a job at Harvard which is a great choice, I think.

The value 125 GeV in MSSM is indeed high if you want to have light superpartners. It requires either near-maximal mixing of the stops; or very heavy stops. The latter choice makes the "little hierarchy problem" of the Higgs lighter than the relevant superpartners worse.

The paper concludes that the surviving regions are corners of the MSSM parameter space; or new physics of the Higgs sector is needed.

But of course, there are many flexible things here – some people may tell you it's natural for superpartners to be much heavier, and so on (the 135 GeV bound would refer to very high or very mixed superpartners), and the mass becomes totally OK in some not-so-minimal SSMs with extra matter.

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Thanks for the reference, Luboš. Actually I know this paper. Having read it is one reason why I was surprised to read the following statement (here: t.co/Nprg3NLi): "...However, the simplest version of SUSY, called the minimally supersymmetric standard model (MSSM), predicts a mass range for the Higgs boson that is much greater than 125 GeV." He says MSSM, so he's not talking about the NMSSM or something else. So, I'm still not sure what is he talking about, especially when he says: "...much greater than 125"! –  stupidity May 23 '12 at 18:59
    
Right, I would probably join those who guess that the particular sentence in the article is just wrong. MSSM surely prefers "naturally" smaller values than 125 GeV, such as those "slightly above" the Z-boson mass only. –  Luboš Motl May 23 '12 at 19:10
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I'm glad to hear that. I asked him on twitter but he didn't respond. So I thought maybe I'm missing something obvious (to him), hence my inquiry here. Anyways, many thanks, Luboš. –  stupidity May 23 '12 at 19:17

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