Finite Theories after the discovery of a Higgs-like boson at the LHCThese theories are finite – i.e. cancelling all UV divergences – to all orders in perturbation theories. One may guarantee that supersymmetric grand unified theories are finite to all orders if he cancels the beta-functions for gauge couplings and anomalous dimensions of the Yukawa couplings up to one-loop and/or two-loop level; and if he also relates the Yukawa couplings with the gauge couplings in certain ways. Linear relationships for the squared masses – certain sum rules – are natural conditions that arise.

The authors analyze all the available experimental constraints and show that the Higgs mass around \(125\GeV\) is very naturally predicted by those models. However, their selected example models, FUTA and FUTB, usually have the spectrum of superpartners at the level of several \(\TeV\)'s. The lightest neutralino and/or both staus are the only particles that manage to be lighter than \(1\TeV\) in their example model in Table 1.

It's surely interesting that such models are "positively correlated" with the experiments, if I have to describe their vague compatibility with the known measurements modestly. These models are surely attractive from an aesthetic viewpoint. In the future, people may understand some justification that actually makes the finiteness "necessary" or "implied by a deeper theory".

So far, it doesn't look like that the finiteness of an effective theory is a "must". I think that effective low-energy theories that one extracts from string theory are not finite in general – even though string theory, including all the stringy stuff, is of course finite. Even if you work at the level of field theories – supersymmetric GUTs – the authors remind us that when the grand unified gauge symmetry is broken, the finiteness disappears.

Intuitively, I find it natural to believe that all divergences in effective field theories should be due to some symmetry breaking and the finiteness should be restored rather soon after you restore various symmetries and other things as you study the short-distance origin of our effective field theories. Needless to say, the finiteness condition is a huge constraint on the effective field theories which strengthens the predictive power of these theories considerably.

Still, at this moment, we don't really know for sure whether the finiteness of the effective field theory is a good guide or not.

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