Finite naturalness of a scalar singlet extension of the standard model ?

Exploring an "ugly" model with very small  couplings to the Standard Model which does not care about quadratically divergent quantum corrections...
... like the "good" supersymmetric universe but evades the "bad" multiverse solution:
The naturalness principle strongly influenced high-energy physics in the past decades [1], leading to the belief that physics beyond the Standard Model (SM) must exist at a scale ΛNP  such that quadratically divergent quantum corrections to the Higgs squared mass are made finite (presumably up to a log divergence) and not much larger than the Higgs mass Mh itself. This ideology started to conflict with data after TeVatron measured the top mass (which implies a sizeable order-one top Yukawa coupling λt) and after LEP excluded new charged particles below 100 GeV [2]. Indeed, imposing that the SM one loop correction to M2h... is smaller than M2h×∆, where ∆ is the amount of allowed fine-tuning (ideally ∆<∼1), implies (computing the fine tuning  with respect to high-scale parameters) ΛNP <∼√∆× 50GeV . 
The most plausible new physics motivated by naturalness is supersymmetry. [...] 
However, no new physics has been so far seen at LHC with √s = 8 TeV, such that, in models that aim to be valid up to high energies, the unit of measure for ∆ presently is the kilo-fine-tuning. While this is not conclusive evidence, while special models that minimise fine-tuning are being considered, while naturalness arguments can be weakened by allowing for a finer tuning, while various searches have not yet been performed, while LHC will run at higher energy, etc, it is fair to say that the most straightforward interpretation of present data is that the naturalness ideology is wrong. 
This situation leads to consider the opposite extremum: the Higgs is light due to huge cancellations [4] because ‘anthropic selection’ destroyed naturalness. 
Here we explore an intermediate possibility, that sometimes appeared in the literature, more or less implicitly. We name it ‘finite naturalness’. The idea is that we should ignore the uncomputable quadratic divergences, so that the Higgs mass is naturally small as long as there are no heavier particles that give large finite contributions to the Higgs mass.[...] 
Anyhow, the goal of this paper is not advocating for the ‘finite naturalness’ scenario. Instead, we want to explore how experiments can test if it satisfied in nature. [...] 
Another experimental result which might suggest the presence of physics beyond the SM is the fact that the SM potential (for the currently favored values of the Higgs and top masses) develops an instability at field values above about 108 GeV, leading to vacuum decay with a rate much longer than the age of the universe [7]. 
There are many ways to avoid this instability, which employ loop corrections from new particles with sizeable couplings to the Higgs [...]. Thereby, in the context of finite naturalness, this kind of new physics is expected to be around the weak scale. 
This is however not a general conclusion. Indeed there is one special model where the instability is avoided by a tree level effect with small couplings. Adding to the SM a scalar singlet S with interactions to the Higgs described by the potential [38] 
 V = λH(HH −v2)2 + λS(SS −w2)2 + 2λHS(HH −v2)( SS −w2)   
the low-energy Higgs quartic coupling is given by λ = λH −HS2/λS at tree level. This model allows to stabilize the SM vacuum compatibly with ‘finite naturalness’ even if the singlet is much above the weak scale, provided that the couplings λHS and λS are small. A singlet with this kind of couplings is present within an attempt of deriving the SM from the framework of non commutative geometry [39]. 
Finally, observations of cosmological inhomogeneities suggest that the full theory incorporates some mechanism for inflation. At the moment the connection with the SM is unknown, even at a speculative level. A successful inflaton must have a flat potential, which is difficult to achieve in models; at quantum level flatness usually demands small couplings of the inflaton to SM particles. An inflaton decoupled from the SM would satisfy ‘finite naturalness’. A free scalar S with mass M≈1013 GeV is the simplest inflaton candidate; it satisfies finite naturalness provided that its couplings to the Higgs λHS is smaller than about 10-10. It is interesting to notice that this roughly is the maximal mass compatible with ‘finite naturalness’: at three loops gravity gives a finite correction to the Higgs mass, δm2 ∼y2M6/M4PL(4π)6 [we thank A. Arvinataki, S. Dimopoulos and S. Dubovsky for having noticed and pointed to us such effect]. 
(Submitted on 28 Mar 2013 (v1), last revised 29 Apr 2014 (this version, v3))

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