This condition doesn't seem to be a "must" – the divergences may very well be taken care of by the high-energy phenomena (string theory ultimately takes care of all divergences so its approximations don't have to be finite by themselves) – but it is an aesthetically intriguing condition, anyway. Now, the same authors released a new paper
Finite Theories Before and After the Discovery of a Higgs Boson at the LHC (S. Heinemeyer, M. Mondragon, G. Zoupanos)where they calculate some new predictions and intriguing details.
They focus on the third-generation fermions and their superpartners, the Higgs sector, and the gauginos. The nicest FUTs they consider boast names such as FUTA and FUTB – the latter seem particularly attractive. They also take some LHCb results into account. In these models, \(\tan\beta\) is typically rather large, \(\mu\) is almost necessarily negative.
The spectra seem very intriguing and consistent with everything we know. Unfortunately, they're inaccessible to the LHC – or marginally accessible – and perhaps even inaccessible to ILC/CLIC. I like the representative table of a FUTB model here:\[
\begin{array}{|l|l||l|l|}
\hline
m_b(M_Z) & 2.74 &&
m_t & 174.1 \\ \hline
m_h & 125.0 &&
m_A & 1517 \\ \hline
m_H & 1515&&
m_{H^\pm} & 1518 \\ \hline
m_{\tilde t_1} & 2483 &&
m_{\tilde t_2} & 2808 \\ \hline
m_{\tilde b_1} & 2403 &&
m_{\tilde b_2} & 2786 \\ \hline
m_{\tilde \tau_1} & 892 &&
m_{\tilde \tau_2} & 1089 \\ \hline
m_{\tilde\chi_1^\pm} & 1453 &&
m_{\tilde\chi_2^\pm} & 2127 \\ \hline
m_{\tilde\chi_1^0} & 790 &&
m_{\tilde\chi_2^0} & 1453 \\ \hline
m_{\tilde\chi_3^0} & 2123 &&
m_{\tilde\chi_4^0} & 2127 \\ \hline
m_{\tilde g} & 3632 && {\rm masses}& {\rm in}\,\GeV
\\ \hline
\end{array}
\] You see that the LSP is the lightest neutralino below \(800\GeV\). Staus are just somewhat heavier, \(900\GeV\) and \(1100\GeV\). Both sbottoms and stops fit the pattern that the lightest and heaviest one is at \(2500\GeV\) and \(2800\GeV\), respectively. The second lightest neutralino and the lightest chargino sit at \(1450\GeV\), the remaining four faces of the God particle find themselves above \(1500\GeV\) while the heavier chargino and the heaviest two neutralinos are above \(2100\GeV\). Finally, the gluino is above \(3600\GeV\).
Particularly the last figure is rather high (in a broader ensemble of models they analyze, the masses may go up to \(10\TeV\) or so). We would have trouble to see such a gluino for years. But this model or at least similar models may be right. From a theoretical viewpoint, I see absolutely no preference when I compare models with gluinos at \(1200\GeV\) and \(3600\GeV\). Some people become very emotional and start to say that one of them has to be right or wrong or its rightness or wrongness means something a priori. Well, it just doesn't. Nature doesn't give a damn whether it's easy or hard for us to observe the superpartners. Once we observe them, many new things start to be clear. If we don't observe them, we are still extremely far from ruling out supersymmetry – and nice special supersymmetric models such as FUTB in this paper.
Its not my – or other humans' – job to rate the beauty of the values of particle physics parameters that emerge from Nature's decisions. It's Her job. Nevertheless, I must say that I would find a spectrum like the table above – or many other tables – elegant. It would probably mean that all these obnoxious idiots who like to say bad things about SUSY could remain loud for many more years. That's an annoying vision from a personal viewpoint but it can't change anything about the reality and it is less important than the actual beauty and physical near-inevitability that is carried by supersymmetry at some scale. If the known – mostly theoretical – evidence makes two models equally plausible and elegant, then one is obliged to love both of them equally, regardless of the fact that one of them may be much more accessible to the experiments. I view this commandment as a part of the scientific integrity.
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