# 115 GeV, 170 GeV, and the non-commutative 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 non-commutative 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.

• 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. Apr 22, 2011 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. Apr 22, 2011 at 16:43
• 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 $\Lambda$. Aug 31, 2011 at 15:53

The question about the 170 GeV prediction is obsolete now.

But the issue:

to drive the Higgs mass to the lower bound, rather than to the upper bound, in an altered version of the noncommutative standard model

remains relevant and has been addressed indeed by an updated almost-commutative spectral extension of the standard model. This extension according to http://arxiv.org/abs/1304.8050 claims basically that:

[the] obstruction to lower [the Higgs mass] was overcome in http://arxiv.org/abs/1208.1030 simply by taking into account a scalar field which was already present in the full model which [was] computed previously in http://arxiv.org/abs/1008.3980. One lesson which [was] learned on that occasion is that [one has] to take all the fields of the noncommutative spectral model seriously, without making assumptions not backed up by valid analysis, especially because of the almost uniqueness of the Standard Model in the noncommutative setting.

Incidently this new neutral singlet scalar ﬁeld (σ) comes originally from some Majorana term in the spectral action responsible for a type I see-saw mechanism implying existence of heavy right-handed neutrinos (http://arxiv.org/abs/hep-th/0610241). It is worth mentioning that this σ field can purpotedly stabilize the Higgs coupling and prevent it from becoming negative at higher energies thus make it consistent with its mass of 126 Gev, providing a vev for σ of the order of $10^{11}$ GeV compatible with the Majorana mass that could explain the actual neutrino phenomenology. This last choice of parameter can be interpreted as some fine-tuning ...

• In my question, I was looking for a physical mechanism which would force the Higgs mass to lie at the edge of its allowed range. Estrada and Marcolli arxiv.org/abs/1208.5023 achieved something like this, by adding the assumption of asymptotic safety arxiv.org/abs/0912.0208 to the noncommutative SM... Jun 28, 2013 at 6:06
• In the extension with the second scalar, the range of plausible mass values is shifted lower, so the experimental value is somewhere in the middle of the new range. It deserves to be mentioned, but Estrada and Marcolli is closer to what I wanted... Jun 28, 2013 at 6:13
• One issue which deserves to be examined, is whether the Higgs mass is technically natural, in the extension. I will think about this, in connection with the other question physics.stackexchange.com/q/69176/1486 Jun 28, 2013 at 6:15

I just noticed that this question was still unanswered. As far as I can recall, the logic in Chamseddine-Connes-Marcolli was very straight: to get values for the SM parameters at GUT scale, and then run down. So it is not easy to find a way to retort the prediction. On other hand, time ago a different bunch of papers, on the topic of "infrared fixed point" for the top yukawa coupling, shown that the values of order unity were favoured, so the CCM result is not surprise.

I think to remember that the extra ingredient of CCM was an equation relating -at GUT scale- the mass of the W, or the sum of masses of the gauge bosons, to the sum of masses of fermions. So as the top was the most massive fermion, this was really a prediction of the mass of the top. It was very amusing because such kind of sums are typical of susy, but here no hint of susy appears in the argument.

really the prediction of top mass was just an exersice to show that the model was realistic. The main point of the work was the peculiar dimensionality of the new spectral triple.