Asymptotic Gravity Doesn't Play Well With Other Beyond The Standard Model Theories

One possible way to integrate gravity and the Standard Model of Particle Physics is called 'asymptotic gravity", in which the gravitational coupling constant runs with energy scale to an ultraviolet fixed point.

It turns out that this theory is inconsistent with many beyond the Standard Model theories, because the structure of the theory limits the number of extra-dimensions and fundamental particles that a theory can have while remaining consistent with the theory. 

For what it is worth, the author really make you work to see in the body text of their paper that their conclusion holds in ways that surely could have been explained much more clearly. But, the conclusion is still buried in there.

It is consistent with the minimum particle content, the particle content of the Standard Model, in either four or five dimensions, but not more. (We know that there are at least four dimensions.)

We conclude that, within the truncation specified above, and for a fixed number of vectors, there is an upper bound on the number of fermions and scalars that are compatible with asymptotic safety. The standard model has 12 vectors (1 photon, 8 gluons and 3 weak bosons), 4 scalars (one Higgs and three Goldstone modes that are “eaten up” by the weak bosons and become their longitudinal degrees of freedom) and 45/2 Dirac fermions. The non-integer number of fermions arises, since the standard model is chiral, i.e., it is constructed from Weyl fermions. In the gravitational β functions, a Dirac fermion essentially equals two Weyl fermions. As the standard model does not include right-handed neutrinos, it contains 45 Weyl fermions. . . . Generally, the allowed region in the NS, ND-plane for fixed NV shrinks rapidly, as the number of extra dimension is increased. We conclude that an experimental discovery of universal extra dimension could pose a challenge for the asymptotic safety scenario. Note that scenarios in which only gravity can propagate into the extra dimensions are not restricted by these results.

In four dimensions, it could handle up to three more spin-1/2 fermions (such as right handed neutrinos) and up to 2 more scalars. Basically, two scalars equals one Dirac fermion that is both left and right handed, for purposes of the matter content budget of the theory.

Asymptotic gravity is not compatible with the particle content of the Minimal Supersymmetric Standard Model (MSSM), an SU(5) GUT, or an SU(10) GUT theory. This is because: "a large number of scalars and fermions cannot be accommodated, unless a large number of vectors is postulated at the same time." In particular:

We now turn to supersymmetric models, which must contain a gravitino. As it contributes to the beta function for G with the same sign as the graviton, it allows to extend the number of matter fields slightly in comparison to the gravitino-less case. In particular, simple SUGRA still admits a viable gravitational fixed point. The MSSM+SUGRA, however, contains too many fermions and lies in the excluded region. 

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In this work, we have added the contribution of spin- 3/2 gravitinos. We find that a model of pure supergravity admits an asymptotically safe gravitational fixed point. If we add matter, bounds on the number of allowed matter fields persists. In particular, the matter content of the MSSM is not compatible with a viable fixed point within a model with one graviton and one gravitino.

But, if the fermion superpartners in a supersymmetric theory were vector bosons instead of scalar bosons, these bounds wouldn't be a barrier to merging asymptotic gravity and supersymmetry, however.