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Checking a quadrillion of heterotic Pati-Salam models

Benjamin Assel, Kyriakos Christodoulides, Alon E. Faraggi, Costas Kounnas, and John Rizos (currently in the U.K., France, and Greece) decided to look at a gang of million of billions (2^{51}) of heterotic models in the free fermionic formulation,

Classification of heterotic Pati-Salam models (arXiv),
that are higgsed down from an SO(10) to the Pati-Salam SO(6) x SO(4) group. They draw lots of histograms that show the distribution of desirable qualitative "traits" among these models.

The qualitative conclusions, if imposed one by one, are reducing the number of selected models by the following percentages:
  • no gauge group enhancement: 79% of all
  • complete families: 28% of the above
  • three generations: 1.3% of the above
  • Pati-Salam-breaking Higgs: 8.0% of the above
  • Standard-Model-breaking Higgs: 8.1% of the above
  • no massless exotics: 0.64% of the above
  • minimal Pati-Salam Higgs: 26% of the above
When you multiply all the factors, you will find out that 0.32 ppm (parts per million) of the models have all the desirable qualitative traits. So even in this pretty realistic and promising subclass of the string vacua, it would surely be wrong to say that the observed qualitative properties are "almost generic".

On the other hand, they're not impossibly unlikely, either. One vacuum among 3 million vacua satisfies the qualitative requirements they have imposed - it means that the "fine-tuning" needed to reproduce the detailed qualitative properties is not insane. On the other hand, extra selection or fine-tuning has to work to get realistic values of all the continuous parameters that are predicted for the low-energy effective field theory.

The right number of generations and the absence of massless exotics seem to be the most severe constraints. Each of them reduces the models by two orders of magnitudes or so. I still do think that there should be a more robust reason - not just an anthropic selection that picks 1 guy among 3 million - that explains why it is natural for Nature to satisfy the qualitative constraints above. But in the absence of a robust argument, I am not quite sure about it.

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