## Tuesday, October 24, 2006

### Alain Connes et al.: predictions for masses

Ali H. Chamseddine, Alain Connes, and Matilde Marcolli argue in their paper
that they can essentially predict the Higgs mass and the top quark mass. These interesting authors always attract attention. The reader is supposed to have seen a previous paper or a blog comment about it,
The new paper again shares the gauge coupling unification with GUT theories - which I think is still put in - but it also argues that they can relate the top quark mass to the W mass. More precisely, they argue on page 4 that at the unification scale,
• sum (3 generations) m_e^2 + m_nu^2 + 3 m_d^2 + 3 m_u^2 = 8 M_W^2.
I suppose that this is some tree-level formula although their treatment of the loop effects and renormalization is not transparent to me (partly because they argue that their model also has gravity in it). At any rate, this kind of a relation seems new to me - and my guess is that it will look new to all particle physicists and string theorists - and if it is not bogus, it would be really something. ;-)

Expert readers should try to decode the mathematically rich formalism and understand where their relation comes from.

The left-hand side is originally constructed from the squares of the Yukawa couplings. To have an idea what it means, realize that the left-hand side is dominated by 3 m_{top}^2, so you roughly get
• m_{top} = sqrt(8/3) M_W
which is marginally compatible with observations. After some arguments culminating on page 55, they also claim to predict the Higgs mass around 170 GeV. I am sure people should look at these statements in detail to see whether there is something materially new about them which would be fascinating.

Update: In 2008, Alain Connes became the most hapless scientist who ever tried to predict the Higgs mass. The CDF and D0 combined data at the Tevatron were able to falsify one particular possible value of the Higgs mass and it happened to be 170 GeV. See the article about this ironic story involving Connes. ;-)

If you're interested in skeptical results of a Harvard discussion:
1. It is not so difficult to predict reasonable values of the Higgs boson: All models that are well-defined at high energies inevitably lead to a Higgs mass in a reasonable window 115-170 GeV. If the mass were below 115 GeV (the same as the observational lower bound), the quartic coupling would run negative at some scale between a TeV and the GUT scale and destabilize the vacuum. On the other hand, if the mass were above 170 GeV, we would encounter a Landau pole below the GUT scale. The Connes et al. value is close to developing the Landau pole at high energies.
2. The relations between the masses of the fermions and W bosons can't hold generally at low energies because the equation is not invariant under the RG flow. If they mean something, they describe physics at high energies. But at high energies, one could probably construct many models that would lead to very different predictions - e.g. GUT with all the extra gauge bosons. With this viewpoint in mind, the prediction can't be viewed as a natural consequence of the simple low-energy limit but rather a randomly chosen extrapolation of it to high energies.
3. I was trying to conjecture that the relations of these kinds could hold at the string scale for masses in a broad class of models extracted from quantum gravity. This conjecture can be quickly falsified. In braneworlds and probably even in heterotic strings, one can obtain exponentially small Yukawa couplings as long as the Weyl fermions are localized at distinct brane intersections and/or different orbifold singularities so that the couplings are dominated by worldsheet instantons. On the other hand, the W boson masses are unsuppressed because they live in the "bulk" as opposed to intersections or singularities.
4. A general relation between the squares of the Yukawa couplings and the gauge couplings is not an unexpected feature because the models based on deconstruction typically generate the Yukawa couplings from the same parameters as the gauge couplings - from some higher-dimensional gauge couplings. But the spectrum of possibilities how to do that is probably large and Connes et al. just proposed one possibility from a large landscape of possibilities.
To summarize, I would bet that the derived relations are guesses that depend on so many arbitrary choices rather than a robust or unique theory that they can't quite be trusted. Also, I don't think that it is correct to describe this theory as a "unification" because it doesn't unify the forces. The different gauge forces still correspond to independent parts of the structure and the only sense in which they are unified with gravity is the terminology that calls the gauge bundles and corresponding Dirac's operators a "form of non-commutative geometry".