How Kane et al get the Higgs mass

[mitchell porter]

Kane, Lu and Zheng have a paper today summarizing how they obtain a Higgs of about 125 GeV within the G2-MSSM (M-theory compactified on a "G2 manifold" so as to give the supersymmetric standard model). I'm not a big fan of this prediction, the machinery behind it is very complicated and I would prefer something like Shaposhnikov-Wetterich; but I do still want to see how it works.

The final stages of the calculation can be seen symbolically in the figure on page 6 and the mass matrix at the bottom of page 7. The need for a working cosmology is used to obtain a value of about 50 TeV for the masses of MSSM up and down Higgses, stop squark, etc, at the GUT scale (see the right of the figure); then renormalization group equations are used to run these masses down to low energies. These quantities enter into the mass matrix on page 7, at low energies the smaller eigenvalue of this matrix is about 125 GeV, and this object will behave like a SM Higgs.

That's about as much as I understand so far. The paper lists the starting assumptions on page 4, but the text only describes the argument informally. I would like to get the argument into deductive form, with links to papers containing the detailed calculations, so that a total novice could follow the logic of it. But that might take a while.

 

[mitchell porter]

Some general remarks on the framework employed by Kane et al:

They work in M theory. So there are ten dimensions of space and one dimension of time, with a metric field, a fermionic field called the gravitino, and an electromagnetism-like "three-form" field called the C-field. There may also be "twobranes" and "fivebranes" flying around, as sources of the C-field.

They work in M theory compactified on a seven-dimensional space called a singular G2 manifold. So at each apparent point in our three macroscopic dimensions of space, there is in fact a Planck-scale seven-dimensional space. As well as being a sort of complicated hyper-donut with about 100 size and shape parameters, this 7-manifold contains "singular 3-surfaces" which might be conceptualized as "creases" or "ridges" in the 7-manifold where the metric becomes a little singular. There will also be points on these 3-surfaces, conical singularities, where the metric becomes even more singular.

There are nonabelian gauge superfields on the 3-surfaces, and chiral superfields (in representations of the gauge group) at the conical singularities. So for each singular 3-surface in the 7-manifold, there is a supersymmetric GUT. The super-GUTs on different 3-surfaces also interact gravitationally through the bulk of the 7-manifold.

These 3-surfaces are really the M-theoretic counterpart of a "braneworld". So returning to the macroscopic perspective, if you were to zoom in on a point in space, you wouldn't just discover a 7-manifold, you'd find a 7-manifold with a number of 3-surfaces embedded in it. Since the same 3-surfaces are found at every "macroscopic point", they each really define a volume of six space dimensions, three large, three small; like a 6+1-dimensional Kaluza-Klein world embedded in the larger 10+1-dimensional Kaluza-Klein world of M-theory.

In the G2-MSSM scenario, there are always at least three of these "M-theory braneworlds" coexisting in the larger space of "macroscopic Minkowski space times 7-manifold". Each of them contains a separate super-GUT. The visible world is contained in just one of them.

For example, there might be SO(10) gauge superfields, three conical singularities each of which contains a SM generation (a 16-dimensional representation), and a fourth point which contains a Higgs superfield. All those fields would be contained in just one of the 3-surfaces, and then this SO(10) GUT would be broken to the MSSM by a Wilson line of flux within the 3-surface.

The other 3-surfaces contain "hidden sectors". One of them will be a strongly interacting super-GUT which dynamically breaks supersymmetry; susy-breaking is then transmitted by gravity to the other 3-surfaces, which is how susy gets broken in the visible-sector MSSM. The reason one wants at least two hidden sectors is to stabilize the shape and size of the 7-manifold, in a way that makes it rigid at energies below 50 TeV or so (two hidden sectors are required for algebraic reasons I haven't tried to understand). This rigidity means that you won't get physical effects resulting from tremors in the shape of the 7-manifold, at energies that we can measure.