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March 25, 2004

Fine-tuned

One-loop contribution to quartic Higgs coupling
Corrections to quartic Higgs self-coupling from stop and top loops.

We all learned on our grandfather’s knee that supersymmetry required a light Higgs. Back then, this was a cheering thought, for it meant that we would not have to wait too long for the Higgs to be discovered. The years passed, and the experimental lower bound on the mass of the Higgs crept slowly upwards. We now know that it must be heavier than 114 GeV or so.

Scott Thomas was in town the other week, and gave a very nice colloquium, explaining how serious the situation has become for the MSSM.

At tree level, m h<m Zcos(2β) m_h \lt m_Z \cos(2\beta) where tan(β)=H/H˜\tan(\beta)= \langle H\rangle/\langle\tilde{H}\rangle, and HH & H˜\tilde{H} give masses, respectively, to the up and down type quarks. The inequality becomes an equality in the limit that the mass of one of the other neutral scalars in the Higgs sector, m Am_A\to\infty.

With m Z=91m_Z = 91 GeV, and m h>114m_h\gt 114 GeV, this bound is clearly violated. Fortunately, the one-loop corrections to the quartic self-coupling, depicted above tend to push this number up. m h 2=m Z 2cos 2(2β)+6|λ t| 2m t 24π 2log(m t˜/m t) m_h^2 = m_Z^2 \cos^2(2\beta)+\frac{6|\lambda_t|^2 m_t^2}{4\pi^2}\log(m_{\tilde{t}}/m_t) Note that the supersymmetric cancellation between the two diagrams means that the result depends only logarithmically on the stop mass. To fit the current lower bound on m hm_h, the stop must be heavy m t˜>850GeV m_{\tilde{t}} \gt 850\, \text{GeV} And each time we push up the lower bound on the Higgs mass, the lower bound on the stop mass goes up exponentially.

One-loop contribution to quadratic Higgs coupling
Corrections to the quadratic Higgs self-coupling dominated by stop loops.

While the corrections to the quartic terms in the Higgs potential depend only logarithmically on the stop mass, the corrections to the quadratic terms are proportional to m t˜ 2m_{\tilde{t}}^2. m 2|μ| 23|λ t| 2m t˜ 28π 2log(m t˜/M) m^2 \sim |\mu|^2 - \frac{3 |\lambda_t|^2 m_{\tilde{t}}^2}{8\pi^2} \log (m_{\tilde{t}}/M) where MM is a messenger mass, at which the loop-momentum integral is effectively cut off. (It’s precisely these radiative corrections that drive this term negative, and lead to the electroweak symmetry-breaking.)

To end up with an electroweak symmetry-breaking scale around (100GeV) 2(100\, \text{GeV})^2, one needs the μ\mu parameter (the coefficient of HH˜H\tilde{H} in the superpotential) to be in the TeV range, and its value must be tuned to within a few percent.

Personally, I can live with a fine-tuning in the 1% range. But you would not have to push the Higgs mass up too much further to make even me nervous.

Posted by distler at March 25, 2004 12:30 AM

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6 Comments & 5 Trackbacks

Re: Fine-tuned

I can’t help but suggest that the title of this post really needs to be:

Stop! In the name of love!

Posted by: Aaron on March 25, 2004 12:35 AM | Permalink | Reply to this

Supremes

I definitely need a Motown-themed blog. Thank you for helping me realize this.

Posted by: Jacques Distler on March 25, 2004 12:42 AM | Permalink | PGP Sig | Reply to this

Re: Fine-tuned

Finally! A physics post that I understand…

Gordy Kane’s view was that fine-tuning was in the eye of the beholder: he was (maybe still is) dreaming of a string model that would *predict* a particular ratio between mu and the soft masses, which would turn out to be the right one.

Others less sanguine are busy constructing non-minimal SSM’s with heavier Higgses. (You mentioned Murayama’s programme for this some time ago.)

Curiously, in contrast to the period a couple of years ago when the experimentalists were depressed because the LEP Higgs ‘signal’ went away on reanalysis, a more recent summing-up appears to be a good bit more optimistic. See here.

Posted by: Thomas Dent on March 25, 2004 1:11 PM | Permalink | Reply to this

Published in Nature?!

Hmmm. That BBC article was remarkably content-free. Is there a hep-ex eprint we might read?

Yes, I know people are looking at alternatives. One has to say, though, that most of the cures look worse than the disease.

One of the main points of Scott’s talk was that most of the obvious things you might try don’t actually help.

Posted by: Jacques Distler on March 25, 2004 10:10 PM | Permalink | PGP Sig | Reply to this

Re: Published in Nature?!

It wasn’t a research article in Nature, just a sort of review of the current status. No eprint.

The bottom line is that the excess at 115 GeV has a 9% probability of being background. So, not even 2 sigma.

The ‘significance’ is that the BBC is correcting its previous terrible reporting which implied that the experimentalists think that the Higgs probably doesn’t exist. Of course, one would prefer them not to exaggerate in either direction in the first place.

Myself, I wish the mass bound were going up. Because that would imply that the Tevatron was working properly!!!

Posted by: Thomas Dent on March 26, 2004 8:11 AM | Permalink | Reply to this

BBC, Nature

Peter Renton article is not so optimistic about the Higgs. Exactly, he says “either” the Higgs or some other physics mimicking the Higgs at 155 GeV. In his analysis he speaks of indirect calculations as the ones that zeroed towards the Top.

But one should also add -speaking randomly; I have not studied the ALEPH equipment- that there is the possibility of noise from detectors. For instance, nuclear physics hints a source of noise at 115 GeV. Perhaps the same source causing the minima in yields of antiproton bombarding of Au at CERN.

Posted by: Alejandro Rivero on May 15, 2004 2:22 PM | Permalink | Reply to this
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