A GUT Model That Makes Some Progress On SM Constants

Grant Unified Theories are notorious for trying to find a unified structure that encompasses the Standard Model particles, while not providing any meaningful insight into the origins of the fundamental mass and mixing parameters of the Standard Model.

An exception is the pre-print Feruglio, Patel and Vicinoa, "A realistic patter of fermion masses from a five-dimensional SO(10) model" (July 2, 2015).  It's abstract states that:

We provide a unified description of fermion masses and mixing angles in the framework of a supersymmetric grand unified SO(10) model with anarchic Yukawa couplings of order unity. The space-time is five dimensional and the extra flat spatial dimension is compactified on the orbifold S¹/(Z₂ x Z'₂), leading to Pati-Salam gauge symmetry on the boundary where Yukawa interactions are localised. The gauge symmetry breaking is completed by means of a rather economic scalar sector, avoiding the doublet-triplet splitting problem. The matter fields live in the bulk and their massless modes get exponential profiles, which naturally explain the mass hierarchy of the different fermion generations. Quarks and leptons properties are naturally reproduced by a mechanism, first proposed by Kitano and Li, that lifts the SO(10) degeneracy of bulk masses in terms of a single parameter. 

The model provides a realistic pattern of fermion masses and mixing angles for large values of tan β. It favours normally ordered neutrino mass spectrum with the lightest neutrino mass below 0.01 eV and no preference for leptonic CP violating phases. The right handed neutrino mass spectrum is very hierarchical and does not allow for thermal leptogenesis. We analyse several variants of the basic framework and find that the results concerning the fermion spectrum are remarkably stable

As the authors explain further:

Below the GUT scale the theory looks like the MSSM and we expect standard SUSY gauge coupling unification. In order to suppress the higher order corrections in Eq. (2.8), we take c ≡ ΛπR ≈ O(100) so that the cut-off of the theory, Λ can be lifted up to the Planck scale . . . . The higher order corrections are at the percent level and remain smaller than experimental uncertainty in the fermion mass data we adopt. The theory provides a predictive framework for fermion masses and mixing angles[.]

This model does not reproduce the Standard Model constants exactly (it assumes that a group of constants relevant to determining the observable values are random numbers on the order of 1), is not terribly original, and has some rough spots that mark it as an incomplete work.  But, it at least takes a serious stab at developing a model that reproduces the hierarchy of Standard Model mass and mixing constants up to an order of magnitude level, and even risks displaying calculations made with it. This makes it quite notable relative to the ordinary flurry of toy models that make no concrete predictions at all about the Standard Model constants.

They renormalize the observable parameters up to the GUT scale using the results for this in the Minimal Supersymmetric Standard Model (MSSM) from prior literature and derive their predicted values at the GUT scale.

By choosing a "normal" neutrino hierarchy, setting tan β = 50, and testing random values of the nine free parameters of the theory within the range 0.5-1.5 (i.e. of order 1), they manage to get results that are within two sigma of the observed values of the parameters about 0.05% of the time, i.e. one time in 2000 (compared to less than one in 100,000 trials in an inverted hierarchy, and a poor fit indeed with a smaller value of tan β).

In the MSSM, tan β is the ratio of the vacuum expectation values of the two Higgs doublets, so there is one Higgs field at the electroweak scale of about 246 GeV, and another Higgs field at a scale of about 12.3 TeV, which is very roughly speaking the scale where supersymmetric effects would be more visible.

The model predicts, which fitted to known experimental values, a lightest neutrino mass of about 3.9 meV, a neutrinoless double beta decay Majorana mass of 4.96 meV, and a spectrum of right handed neutrino masses of 190 GeV, 802 TeV, and 1.42*10¹¹ TeV.

I personally seriously doubt that there are right handed neutrinos at all, suspect that neutrinos do not have any Majorana mass, and expect that the lightest neutrino mass is on the order of 1 meV or less.  In other words, I think all of these predictions are wrong.  But, they are at least not yet contradicted by direct experimental evidence.

There are a host of good reasons to believe that the universe is not described by the MSSM or something quite similar to it, particularly after the first run of the LHC's data is out, but the MSSM does provide a more mathematically tractable way to deal some general concepts related to grand unified theories generally that may be pertinent to one that actually does explain the Standard Model.  So, while these results must be taken with the heap of salt, it is nice to see something working on a model they can do some calculations with in order to get some intuition about which corner of MSSM parameter space is the best fit to the real world, from which the nature of a GUT that actually does describe the universe might be more easily conceived.