### A series of motivated phenomenological leaps to explain the smallness of the Higgs mass?

**An intriguing chain of hypotheses to fish out the Planck scale from the Higgs completed Standard Model**

Still following the Higgs compass (and looking for a better answer to a personal haunting question) the blogger proposes to her-his reader some excerpts of a nicely written article from A. Gorsky

*et al.*about a hypothetical conformal symmetry hidden in the Standard Model to save it asymptotically (remark : the title of the post is naively inspired by another article on naturalness by J. D. Wells discussed here and which deals with several*realistic intellectual leaps*that could have been made to understand the smallness of the electron mass...):Motivated Phenomenological Leap (MPL) #1:

It is an experimental fact that the Higgs mass, the top-quark mass and v [the Higgs ﬁeld vacuum expectation value] satisfy, with miraculous accuracy, the relations

4m^{2}_{H}= 2m^{2}_{top}= v^{2}

i.e.there seems to be a clear conspiracy between the Higgs, the top-quark and the W/Z-boson masses... one obtains for the Yukawa (y) and Higgs self-couplings (λ) [the values]

y = 1 and λ = 1/8(1)

... it [could be] possible that these special values of the couplings and the corresponding mass relations point to some hidden symmetry underlying the SM, which should further enhance a conformal-like symmetry at the Planck scale, that is strongly suggested by W. Bardeen...

MPL #2 :

The values λ(0), g(0), y(0) are ﬁne-tuned, so that at the Planck scale[M_{Pl}]... one obtains simultaneouslyλ' (M_{Pl}) = 0andλ(M(2)_{Pl}) = 0

Nature started at the Planck level at a very distinguished point, where λ is stable and vanishing (free scalar theory), and after that the RG evolution, mainly due to the evolution of the gauge couplings, which were not stable at M_{Pl}brought the scalar ﬁeld to its present state with a very concrete potential. Reverting the statement,the λφ...^{4}sector of the standard model [could be] ﬁne-tuned to be “asymptotically secure”, instead of exhibiting unhealthy Landau pole behavior

(2)The differenceξ = |1/8y^{2}−λ| < 0.05remains small all along the RG-evolution region... Stability of the above difference gets especially well pronounced ... if the parameters of the SM are chosen so thatare exact... Assuming [this adjustment really takes place, when improved by higher-loop corrections and more precise measurements,] this choice of the SM parameters leads to another interesting bonus, namely to the fact[that]...for the values of the parameters of the Standard Model which lead to the relations(2), the 1-loop effective potential [for the scalar field] has a second almost degenerate minimum at a ﬁeld value practically equal to the Planck scale...

MPL #3 :

... Thus,the Planck scale, which is not present in the lagrangian of the Standard Model, is nevertheless hidden in the actual values of its parameters and the conjectured property (1) of the fundamental theory atM_{Pl}.All these puzzling facts seem to imply that the parameters of the Standard Model are not at all accidental. Instead, they may be fully determined by an assumption, that(1)the Standard Model is a low energy limit of a very special fundamental theory deﬁned naturally at the Planck scale, which is the next fundamental threshold in particle physics. Moreover, these relations imply that there is some additional symmetry, which underlies the Standard Model and the deeper fundamental theory. This symmetry should automatically protect the vacuum expectation value of the Higgs ﬁeld(in order to protect relations like) and, hence, solve the hierarchy problem (in the spirit of [11,12a,12b]).

(Submitted on 1 Sep 2014 (v1), last revised 2 Oct 2014 (this version, v2))

**To conclude with a bold but may be not unrealistic intellectual leap**

we reviewed the evidence that the Standard Modellies at a very special point of the parameter space. Namely, that itis connected by the Renormalisation Group evolution to a theory with enhanced (conformal-like) symmetry at Planck energies, where it is supposed to be mixed with quantum gravity and, perhaps, string theory.If true, this implies the exciting possibility that the actual values of couplings, which may seem ﬁne tuned at our energies, may just reﬂect the fact that we are looking at the low energy limit of an UV-healthy theory, thus providing a kind of reﬁnement of the renormalisability principle. In other words, it is possible thatthe low-energy theory is not only necessarily a gauge theory, but in addition its scalar sector should be very special, just as a consequence of being a low-energy effective theory. This option, if actually realized, would resolve at once many puzzles about the Standard Model.

Id.

Needless to say that the blogger would be very interested to know if the spectral noncommutative extension of the standard model with its very specific scalar sector points to the same kind of UV-healthy theory. As far as a possible scale invariance is concerned here is what the spectral action teaches us:

The Dirac operator being a differential operator has the dimensions of mass.The spectral action in noncommutative geometry is deﬁned as a function of a dimensionless operator which is taken to be the Dirac operator divided by some arbitrary large mass scale. The arbitrariness of the mass scale naturally suggests to make this scale dynamical by introducing a dilaton ﬁeld in the Dirac operator of the noncommutative space deﬁned by the standard model. To understand the appearance of the mass scales of the spectral action, we evaluated all interactions of the dilaton with the matter sector in the standard model.We found the remarkable result that the low-energy action, when evaluated in the Einstein frame, is scale invariant except for the Einstein-Hilbert term and the dilaton kinetic term. The resulting model is almost identical to the one proposed in the literature [3],[2],[4]. The main motivation in these works is the observation thatthe standard model is classically almost scale invariant, with the symmetry only broken by the mass term in the Higgs potential. The symmetry is restored by the use of a dilaton ﬁeld. When coupled to gravity, neither the dilaton kinetic energy nor the scalar curvature are scale invariant, leading to a Jordan-Brans-Dicke theory of gravity. The vacuum expectation value of the Higgs ﬁeld is then dependent on the dilaton and is classically undetermined.Quantum corrections break the scale invariance of the scalar potential and change the vacuum expectation value of the Higgs ﬁeld. The dilaton acquires a large negative expectation value given by −m and a small mass. The hierarchy in mass scales is due to the large Yukawa coupling of the top quark. The dilaton expectation value can range between the GUT scale of 10^{15}Gev to the Planck scale of 2.4·10^{18}Gev. The hierarchy in mass scales is not possible if the dilaton kinetic energy and the gravitational action were scale invariant.It is remarkable that all the essential features of building a scale invariant standard model interactions to generate a mass hierarchy and predict the Higgs mass are naturally included in the spectral action without any ﬁne tuning. It is worth mentioning that the scalar potential of exactly the same model considered here was shown to admit extended inﬂation and a metastable ground state...

(Submitted on 14 Dec 2005 (v1), last revised 16 Mar 2006 (this version, v3))

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