# 115 GeV, 170 GeV, and the noncommutative standard model

Several years ago, noncommutative geometry was used to describe the standard model, somehow yielding a prediction of 170 GeV for the mass of the Higgs boson, a prediction which was falsified a few years later.

Meanwhile, for a long time there have been indications of a Higgs boson at 115 GeV. As it happens, 115 GeV and 170 GeV form somewhat natural theoretical bounds on the mass of the Higgs: above 170 GeV, the theory would develop a Landau pole, and below 115 GeV, the electroweak vacuum would become unstable.

So I'm wondering whether the mechanism or the logic behind the original prediction can be reversed, so as to drive the Higgs mass to the lower bound, rather than to the upper bound, in an altered version of the noncommutative standard model.

EDIT: I am nowhere near decoding how the prediction of 170 GeV was made, but simple algebra shows that it is amazingly easy to get very close to 115 GeV instead, by adjusting some of the penultimate quantities appearing in the calculation.

A comment at Resonaances points out that $\sqrt{2} m_W$ is close to 115 GeV (it's a little over 113 GeV). In equation 5.15 of hep-th/0610241, we see this formula:

$$m_H = \sqrt{2\lambda}\frac{2M}{g}$$

On page 36 (section 4.1), we read that $M = m_W$ and $g = \sqrt{4\pi\alpha}$. So for $m_H$ to be approximately 115 GeV, we need $\lambda = \pi\alpha$.

In equation 5.10, we have that lambda-tilde is approximately $4/3 \pi\alpha_3$, and in remark 5.1 we read that "the factor of 4/3 in (5.10) should be corrected to 1". So we have $\tilde\lambda = \pi\alpha_3$. Almost what we need - it's just a different lambda and a different alpha! :-)

And I'll point out again that 115 GeV is a special value theoretically: it's in a narrow range of values for m_H for which, given the measured value of m_top, the vacuum of the minimal standard model is metastable (see figure 13, arxiv:0704.2232). So I can't believe any of this is coincidence.

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Dear Mitchell, I think that you overestimate a bit - 170 GeV is not the highest possible Higgs mass avoiding the Landau pole. You can go higher than that and Alain Connes thought that he could literally calculate 170 GeV - from a formula, not just by trying to saturate an inequality. I don't think he has any "NC" model compatible with a 115 GeV Higgs. – Luboš Motl Apr 22 '11 at 13:03
The 170 GeV calculation was assuming that there were no new particles in some large mass range above the Higgs. If there are new particles there, then his calculations don't work, and I believe you can get a variety of different masses for the Higgs, but there are too many possibilities to make a prediction (and it is very hard to figure out what all the possibilities are).. So if the LHC finds only the Higgs, then the NCG model is falisified. If it finds a number of other particles, then we'll have to see whether there's an NCG model that encompasses them all. – Peter Shor Apr 22 '11 at 16:43
+1 for asking a well phrased and well formatted question! (Not bad content-wise either ;-) – David Zaslavsky Apr 22 '11 at 22:34
I've rendered the math in your edit in LaTeX (we have MathJax active on the site, so we can write decent math), but not being familiar with this work I didn't know if you wanted (for instance) $\lambda$ or $\Lambda$. – dmckee Aug 31 '11 at 15:53