In that vein, this post is about a complicated relationship between the 125 GeV Higgs and supersymmetry. There is this lore that SUSY predicts the Higgs mass below 130 GeV, and you might have heard people saying that the recent almost-discovery of the Higgs is an incredible success of supersymmetry. Well, strictly speaking, the number 130 GeV is taken out of my ass. Instead, with some degree of rigor, one can make the following 3 statements:
- Minimal SUSY without fine-tuning predicts the Higgs mass close to the Z boson mass, that is about 90 GeV.
- Minimal SUSY ignoring fine-tuning predicts the Higgs boson lighter than 160 GeV.
- Non-minimal SUSY in general makes no predictions about the Higgs mass.
The statement #1 on my list boils down to the fact that in the MSSM the quartic term in the Higgs potential (which fixes the Higgs mass, given its vacuum expectation value) is not a free parameter. Instead, supersymmetry ties the quartic coupling to the electroweak gauge couplings.
Up to 1-loop precision the Higgs mass is given by the formula:
(for vanishing A-terms, a large tanβ, and universal stop masses, and setting yt=1). In the first approximation one gets the famous bound m_Higgs ≤ m_Z. Thus, if the MSSM were for real, the Higgs should have been seen at LEP.
Only when supersymmetry is badly broken, that is when the top mass is much smaller than the mass of its scalar partner the stop, the one-loop logarithmic term can be large enough to raise the Higgs mass considerably above the Z boson mass. In particular, for the 125 GeV Higgs the tree-level and loop contributions must be, amusingly, almost exactly equal. The price for making the stop mass large goes under the name of fine-tuning. Since vacuum equations in the MSSM generically marry the SUSY scale to the weak scale, m_stop ~ m_Z , as soon m_stop >> m_top one needs to carefully tune the parameters of the theory so as to cancel various excessive contributions to the Z boson mass. This goes against the original motivation for supersymmetry which was precisely to exorcise fine-tuning.
This brings us the statement #2 on my list. When the fine-tuning issue is ignored, the scenario known as split supersymmetry (SS), the Higgs mass in the MSSM can be much larger than the Z boson mass. In the plot on the right (from this paper), you can see that the Higgs mass can reach 155 GeV for scalar SUSY partner masses at the GUT scale. From the same plot, one finds that the 125 GeV mass correspond to roughly 10 TeV squark masses. Thus, the almost-discovery of the 125 GeV Higgs at the LHC clearly points to Somewhat Split Supersymmetry (SSS) ;-)
All in all, the story of Higgs and SUSY is getting less like a Hollywood romance and more like a Ken Loach movie of hardship and misery. Of course, it is well known that 10 TeV squark masses are not an inevitable consequence of the MSSM and 125 GeV Higgs. Playing with another SUSY breaking parameter, the so-called A-term, the Higgs mass can be dialed to any desired value. When the A-term is judiciously chosen, the scalar top partners could even be at a few hundred GeV, well within the reach of last year's LHC run. See the violet band in the plot on the right. Thus a happy ending cannot be completely excluded at this point. However, more and more theorists are beginning to prepare an exit strategy, like ...nobody said SUSY had to show up at the LHC, maybe fine-tuning 1:1000 is not so bad, maybe SUSY is really at 10 TeV, etc... In a sense, this is right: from the theory point of view there is no fundamental difference between 1 in 100 and 1 in 1000 fine-tuning. Only a practical one, for LHC experimentalists :-)