Which minimal Standard Model with a singlet scalar extension...

... compatible with LHC1?
In the Standard Model (SM), the Higgs mass around 125 GeV implies that the electroweak vacuum is metastable since the quartic Higgs coupling turns negative at high energies. I point out that a tiny mixing of the Higgs with a heavy singlet can make the electroweak vacuum completely stable. This is due to a tree level correction to the Higgs mass-coupling relation, which survives in the zero-mixing/heavy-singlet limit. Such a situation is experimentally indistinguishable from the SM, unless the Higgs self-coupling can be measured. As a result, Higgs inflation and its variants can still be viable.

(Submitted on 1 Mar 2012 (v1), last revised 22 May 2012 (this version, v3))

... in agreement with Planck+Bicep2?
We have studied the possibility of embedding the inflationary sector into the SM through a Z2 Higgs portal, in a model that we dubbed “SMS” [for SM with a real singlet]...
Since a heavy scalar, S, coupling to the Higgs through a portal (and taking a large VEV) can provide a stabilization mechanism via a tree-level threshold
[26, 27], it is important to determine whether this scalar could drive inflation as well. We have found that successful inflation can indeed occur in this model, but the stabilization of the potential cannot be provided by the inflaton itself, as we explain below.
We have first studied the potential energy valley that provides an attractor trajectory for inflation, focusing in the limit of small Higgs portal coupling, λSH. We have shown that inflation can be described in a very good approximation with a single-field effective theory along this valley. The inflaton field corresponds to a combination of the heavy singlet and the Higgs, and represents the length travelled along the valley. However, in this limit inflation takes place mostly in the direction of S, which makes reasonable identifying this field as the inflaton. This identification is exact in the decoupling limit, i.e. when λSH is sent to zero. The variation of the potential energy along the floor of the valley is described by a Mexican hat potential, which provides a good fit to current CMB data [34] if the inflation takes a large VEV of the order of 15 MP . This is one of the reasons why inflation has to proceed mostly in the direction of the singlet, since the extension of the valley in the Higgs direction for small λSH is much smaller than the VEV of the singlet. In addition, a small effective quartic coupling is required to fit the amplitude of primordial perturbations, which also impedes inflation from going substantially along the Higgs direction. During inflation, the field S interpolates between a hilltop-like behaviour and a quadratic potential with positive curvature. In particular, we find that one may have a tensor-scalar ratio as small as r ≥ 0.04 for values of AS and nS within their 95% confidence levels. Using these results, we are able to estimate the associated mass scale of the (very heavy) singlet, which turns to be of the order of 1013GeV, as well as its self-coupling λS10-13. In the limit of small coupling between the SM and the singlet, λSH cannot be determined with CMB measurements, but it may be possible to do it for larger values, thanks to the deformations that the inflationary valley would undergo in that case. Assuming that the SM is indeed unstable and using a purely classical argument, we have estimated that λSH has to be at most 10-16 if the Higgs is to be safe during inflation. On top of this, we stress the relevance of quantum fluctuations, which are important even in the extreme case of exact decoupling limit [12]. 
If the heavy singlet of the SMS plays the role of the inflaton, the tree-level threshold stabilization mechanism mentioned above does not apply. There are two reasons for this. First, the singlet mass required for successful inflation, ∼1013GeV, lies above the SM instability scale ΛI (which is of the order of 1012 GeV) for the central values of the Higgs and top masses. And second, because even if the masses where such that the instability would be pushed beyond the mass scale of the singlet, the conditions of applicability for the mechanism are largely incompatible with the values of the couplings needed for inflation. We find that the tree-level stabilization only has a chance of being successful for a very restricted narrow band of top masses, which is most likely negligible. 
Furthermore, we have shown that regardless of any consideration related to inflation, the mechanism fails in general for a sufficiently small portal coupling λSH. This is due to the appearance of a relevant scale, which was not identified in previous works. This scale can grow above the SM instability scale as λSH decreases, eventually becoming unbounded in the decoupling limit. The actual scale that has to be compared to ΛI, in order to determine whether the threshold effect can cure the instability, is the largest one between two competing scales ... that have opposite behaviours under variations of λSH. The need of the new scale [2|m2S|/ λSH ] that we have identified in this work can be understood intuitively by realizing that the value of the quartic threshold in the decoupling limit can be kept unchanged if the self-coupling of the singlet is modified accordingly. This implies that the stabilization cannot depend on the value of the quartic threshold alone, since it is clear that the mechanism should not work if the SM and the singlet are completely disconnected from each other. The two competing scales, [|m2SλSH λλ ] and [2|m2S|/ λSH ], can be identified by following the potential along the directions given by the lines of minima with respect to the Higgs and the heavy singlet. 
Coming back to inflation, the inapplicability of the threshold stabilization is worrisome given the large Higgs fluctuations sourced by inflation. Stabilization, however, can be simply achieved for models with m< 172.15 GeV, which are still allowed by current experimental results from ATLAS and CMS. Other possibilities to stabilize the potential can also be imagined. One that we have considered here consists in including a second singlet stabilized at the origin, which does not change the shape of the potential energy valleys at tree-level. We have checked that whenever the potential is stabilized in either of these ways, the predictions for inflation for small λSH including loop corrections are essentially identical to the ones obtained at tree-level...
(Submitted on 27 May 2015)
... satisfying finite naturalness?
There are many ways to avoid [the SM potential] 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 ... 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 ≈ 1013GeV 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’...
(Submitted on 28 Mar 2013 (v1), last revised 29 Apr 2014 (this version, v3))

//added on 3 June 2015
...possibly motivated by the spectral noncommutative geometrization program? 
I have already quoted the article from Lebedev et al. here and the last one from Farina et al. there. I  have chosen to do it again in this post to contextualize the new article by Ballesteros and Tamarit. Their recent work is particularly interesting for very clearly and pedagogically addressing the consistency of the interplay between the Standard Model of particle physics and the inflationary sector of the standard model of cosmology. After the success of LHC1 and the Planck mission and meanwhile Bicep program, LUX, Xenon, IceCube, AMS-02... and now LHC2 are struggling to extract signal from noise, it could be interesting for phenomenologists from zepto to zetta space (astroparticle physicists?) not only to wait and see the Wimp Miracle but also to contemplate the Standard Model Marvel : a SM that could be extended in an economical way up to a Planck scale possibly derived from a Higgs sector and merging then with the standard model of cosmology... Advocating such and idea is of course a personal idiosyncra(z)y derived from my personal readings of the science literature :-) which this blog is a netographical guide so to speak. I let the reader build her-is opinion and to inform us of any comment. 
To end with, let me motivate the fact that the statement by Farina et al. according to whom "at the moment the connection [between inflation and]... the SM is unknown, even at a speculative level" could be qualified somehow:

... when we apply our formalism to the NonCommutative Geometry (NCG) traditionally used to describe the standard model of particle physics, we find a new U(1)B-L gauge symmetry (and, correspondingly, a new B − L gauge boson). This, in turn, implies the existence of a new complex Higgs field σ that is a singlet under SU(3)C×SU(2)L×U(1)Y but transforms with charge +2 under U(1)B-L, allowing it to form a majorana-like Yukawa coupling σνRνR with two right-handed neutrinos (so that, if it obtains a large VEV, it induces seesaw masses for the neutrinos). It is striking, on the one hand, that this precise extension of the standard model has been previously considered in the literature [22, 23] on the basis of its cosmological advantages; and, on the other hand, that the new field σ can resolve a previous discrepancy between the observed Higgs mass and the NCG prediction [18–21]. Note that in the previous works [18, 19 ,20, 21] which introduced the σ field for this purpose, it was a real field, and a gauge singlet. By contrast, from the perspective presented here, the fact that σ is complex, and transforms under U(1)B-L, is the key to its existence: had it been real, it would not have been induced by the covariance argument of the previous section. It is important to carefully reconsider the phenomenological and cosmological implications of the standard model extension which we have landed on here, especially in light of the extra constraints imposed by the spectral action. This is an exciting topic for future work.
(Submitted on 22 Aug 2014 (v1), last revised 14 Jan 2015 (this version, v2))

...we have uncovered a higher analogue of the Heisenberg commutation relation whose irreducible representations provide a tentative picture for quanta of geometry. We have shown that 4-dimensional Spin geometries with quantized volume give such irreducible representations of the two-sided relation involving the Dirac operator and the Feynman slash of scalar fields and the two possibilities for the Clifford algebras which provide the gamma matrices with which the scalar fields are contracted. These instantonic fields provide maps Y, Y' from the four-dimensional manifold M4 to S4 . The intuitive picture using the two maps from M4 to S4 is that the four-manifold is built out of a very large number of the two kinds of spheres of Planckian volume. The volume of space-time is quantized in terms of the sum of the two winding numbers of the two maps. More suggestively the Euclidean space-time history unfolds to macroscopic dimension from the product of two 4-spheres of Planckian volume as a butterfly unfolds from its chrysalis. Moreover, amazingly, in dimension 4 the algebras of Clifford valued functions which appear naturally from the Feynman slash of scalar fields coincide exactly with the algebras that were singled out in our algebraic understanding of the standard model using noncommutative geometry thus yielding the natural guess that the spectral action will give the unification of gravity with the Standard Model (more precisely of its asymptotically free extension as a Pati-Salam model as explained in [5]).
(Submitted on 4 Nov 2014)