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 :-)

To wrap up this inflammatory post: the point I was trying to make is that 125 GeV Higgs is not a successful prediction but rather a serious setback from the point of view of SUSY. In non-minimal SUSY any Higgs mass is possible. Minimal SUSY can accommodate any mass up to almost 160 GeV, depending on how much fine-tuning you're willing to accept; 125 GeV Higgs points to 10 TeV squarks, outside the LHC reach.

## 23 comments:

I think many of the statements you make in this post are taken "out of your ass". The fact is that there is an upper bound of 130 GeV on the Higgs mass in the MSSM, not 160 GeV as you imply in your misleading post. Why don't you stick with crap like technicolor and leave SUSY to real physicists?

You probably might know it already, but the "split" SUSY scenario is also the one favored from the compactified stringy perspective. :]

Despite what Anonymous @ 4:46 says, the MSSM in the split SUSY limit can reach a Higgs mass of ~155 GeV, as confirmed in the recent Giudice/Strumia paper with current top mass value, etc, updating the older calculations. In the supersplit limit, the bound is about 140 GeV. So any claimed bound of 130 GeV relies on an assumption that all superpartners are relatively light (below 2 TeV, perhaps, depending on the specific assumptions that were made about A-terms and so on). Not sure why the previous Anonymous feels the need to get so angry when he or she is just wrong.

Hi Jester,

A question. Do you think that it would be accurate to say that expecting a large tan(beta) (say, >=10) constitutes fine-tuning in and of itself? It seems to me that, since tan(beta) is the ratio of VEVs of the two Higgs doublets, and there's no a priori reason to expect that they should be different from each other or from the SUSY breaking scale, one should expect tan(beta) on the order of unity. It also seems to me from looking at old articles that theoretical physicists used to expect values like 1 or 2 until the late 90's, when LEP ruled out light Higgs and made it very difficult to harmonize new limits with small tan(beta). Any thoughts?

Anonymous@4:46 Nobody who thinks that supersymmetry describes the real world is doing real physics, as Martin Veltman always says. You will see how wrong you are in a year, when the LHC will have destroyed the last bit of hope that proponents of supersymmetry might still have.

Hi Jester,

as a relative outsider i always wonder about the negative side-effects of shifting the superpartners up. so e.g. in a scenario where the stop is almost at the GUT scale, assuming that the other superpartners lie in approximately the same region, the gauge couplings would run essentially like in the SM up to there. so this would screw one of the favored SUSY arguments - gauge coupling unification. it also would probably deprive us of a LSP as dark matter candidate.

if you put all that back into the mixture - i.e. if SUSY is actually required to solve some of the problems it was always argued to solve, even ignoring the fine tuning problem - i guess the reasonable Higgs mass range is much smaller than the strictly allowed one.

Hi Jester,

the dominant one-loop correction to the light-Higgs mass is proportional to mt^4.

Ptr, I set yt = 1 to reduce clutter.

Chris, there's no contradiction having very heavy squarks and leptons and much lighter gauginos/higgsinos, in which case gauge coupling unification and dark matter can be accommodated while Higgs gets heavier. That's exactly the idea of split susy.

I don't think a moderate tanbeta is fine-tuning. In the MSSM it's basically a free parameter set by the Bmu term (the soft mass term mixing the 2 Higgs doublet). It's straightforward to build a susy breaking scenario explaining any value in the range 0-100. I guess in the early days they assumed tanbeta \sim 1 because for particle physicists any parameter is order 1 until proven otherwise ;-)

Dear Jester, I noticed that before these vitriolic and misleading texts, you co-wrote a SUSY phenomenology paper. Isn't it going to be cool for an author to SUSY phenomenology paper to lose $10,000 in a bet where he claims that SUSY doesn't exist? ;-) LM

Far more than 1 paper, actually. Hatred, dear Lubos, is often born of unrequited love :-)

Dear Jester, SUSY loves you and once it's discovered and you pay, and be sure I will not allow you not to pay when it's found (and my $100 is waiting for you if it's not found), I will love you, too! ;-)

Because my fellow SUSY fans who remained anonymous were not too specific about the details, let me add some complaints.

Your first description that SUSY implies the light Higgs mass around the Z mass - at the tree level, below the tree mass - is the tree level approximation so it's manifestly an inaccurate starting point to discuss the exact value of the Higgs mass.

But it is already telling us a qualitative lesson: the Higgs mass should be close to the Z mass. On the other hand, non-SUSY theories without any other solution to the hierarchy problem really predict the Higgs mass to be O(1) in the Planck units: this prediction is wrong by more than 10 orders of magnitude.

This is the kind of tuning and problem you should compare your perceived SUSY problems with. You don't do it fairly. You only look for flaws of SUSY and you overlook much bigger but fully analogous problems of non-SUSY models.

Even if you assume that there's no Landau pole in the Higgs self-coupling, which is really a sort of a wishful thinking if you don't offer a mechanism or symmetry that takes care of it, the Higgs mass should naturally be in many hundreds of GeVs. That's already in a mild conflict with the 125 GeV value, too. Much larger conflict than your (invented) conflict between 125 and 160 GeV.

Second, your "calculated" figure of 160 GeV is silly, too. If you naively substitute the GUT mass for the stop, you could get the illusion that it may get this high. But in that case, one safely violates gauge coupling unification and gets serious problems with cosmology where the multi-TeV scale for the particles is damn useful. Various stringy models have independent reasons for the generation of the multi-10-TeV scale etc.

Moreover, the stop really can't be at the GUT scale for the 125 GeV Higgs because a lighter stop is needed to prevent the Higgs potential instability which kicks in earlier than that.

There are subtleties but the reasonable numerical conclusions are that the MSSM allows the light Higgs to go up to 130 or 135 or 137 GeV; 160 GeV is simply a "5-sigma wrong" result and you should get an F for that result.

Indeed, for some totally indiscriminate models, one could perhaps design models where the Higgs mass is much higher. Well, experiments are already suggesting that this is not likely to be the right path. The garden-variety SUSY models fine-tuned to produce stuff "behind the corner" have been refuted by the LHC; but models that sent everything way too far about the current energy frontier are already in a mild conflict with the observations, too.

Cheers

LM

Hi Jester,

What program do you use to do your graphs?

The plots are not mine, they're from the papers I linked to. Draper et al. use FeynHiggs, while Binger uses his own code, I suppose.

Shut up, Lubos. Nobody wants you here. Get out!

Anonymous commenter Nr.16 (above me), on a physics stack exchange site I would flag your trolling post (as offensive) for moderator attention such that it gets deleted ...

Thanks, Jester! I just watched "The 2012 Apocalypse" from the Discovery Channel (see my blog, it's about the end of the world) which made me appreciate how relatively (and this is an important word) rational, accurate, and balanced even your description of supersymmetry is! ;-)

I also think Lubos vs Jester makes interesting reading. Not that I understand much of it to be honest!

max respect to Jester for showing the strong correlation between suggesting that softly broken susy is unpredictive, and a sharp increase in the amount of troll activity.

once this has been achieved, however, I'd suggest moving the talk about bets and/or religious thinking to Dorigo's, Motl's, Woit's and the like... it'd be a pity to lose the focus on physics here.

thanks for the expanation!

The problem is the Higgs at 125 seems to force some amount of fine tuning no matter where we go. There is no natural model available. This is slightly troubling, since it seems that the standard model by itself might be meta stable on cosmological timescales by the vacuum stability bounds.

The Nmssm or some Zprime variants are currently sitting at 1 in 100, it won't take long to go up to 1 in 1000.

I'm not quite sure what this means for fundamental physics

Whether at 125GeV the vacuum is stable in the SM assuming GUT scale new physics only is a question that i think can never be answered conclusively without knowing the exact form of the new physics. it just is within the 4pi ambiguity of the effective couplings of these new terms. At 140 GeV or 119 GeV things would have been clear, but this case is absolutely borderline.

if you think about it in terms of funding of new colliders, this is about as bad as can be: some vague hints that the theory is not complete at the 10-100 TeV scale but no strong argument to expect anything up to the GUT scale either.

As far as predictions go, looks to me like both SUSY and non-SUSY SM predict some upper bound on the Higgs mass -- SUSY's bound is about 10 times better -- without SUSY the Higgs would have to be below a TeV or so.

Fine-tuning problem is addressed by SUSY to a large extent, but this does not imply that Higgs mass prediction without SUSY is in the Planck Scale.

If we take precision measurements into account the non-SUSY prediction for Higgs mass is almost as good as with SUSY. And these precision results have nothing to do with SUSY -- in fact they push the SUSY scale higher up by limiting flavour changing processes etc.

Ravi K

Ravi K: "If we take precision measurements into account the non-SUSY prediction for Higgs mass is almost as good as with SUSY."

Thanks, that restored the Jester-Motl balance!

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