Le boson de Higgs en sait peut-être plus qu'on ne croit sur la hierarchie des masses (Chapitre 6)

Qu'est-ce que le boson de Higgs sait que nous ne sav(i)ons pas? 

La masse disparue
On a vu hier que la version minimale la plus populaire de l'extension supersymétrique du modèle standard était pour le moment sinon mise en défaut expérimentalement du moins soumise à de fortes contraintes empiriques de sorte que la nécessité théorique d'une supersymétrie pour la physique de basse énergie se fait moins sentir. Dès lors, il est légitime de revenir en arrière et de voir où la supersymétrie, vue comme extension naturelle du modèle standard, a commencé. On (re)tombe du coup sur un célèbre théorème d'interdiction en théorie quantique des champs, celui de Coleman-Mandula, dont la supersymétrie est un échappatoire bien connu. Mais n'y en a-t-il pas d'autres? Si bien sûr, et le second plus connu dans ce registre est peut-être bien la symétrie conforme :

Coleman-Mandula theorem allows the Poincare group to be generalized to two global groups, one is the super-Poincare group and the other is the conformal group. It is remarkable to notice that these two groups might yield resolution of the gauge hierarchy problem by a completely different idea, and they also yield a natural generalization of local gauge group, the former gives rise to the local super-Poincare group leading to supergravity whereas the latter does the local conformal group leading to conformal gravity. 
According to recent results by the LHC, supersymmetry on the basis of the super-Poincare group seems not to be taken by nature as resolution of the gauge hierarchy problem. Then, it is natural to ask ourselves if the conformal group, the other extension of the Poincare group, gives us resolution of the gauge hierarchy problem. Indeed, inspired by an interesting idea by Bardeen, there has appeared to pursue the possibility of replacing the supersymmetry with the conformal symmetry near the TeV scale in an attempt to solve the hierarchy problem. It is worth noting that the principle of conformal invariance is more rigid than the supersymmetry in the sense that in many examples the conformal symmetry predicts the number of generations as well as a rich structure for the Yukawa couplings among various families. This inter-family rigidity is a welcome feature of the conformal approach to particle phenomenology. In the conformal approach, it is thought that the electro-weak scale and the QCD scale as well as the masses of observed quarks and leptons are all so small compared to the Planck scale that it is reasonable to believe that in some approximation they are exactly massless. If so, then the quantum field theory which would be describing the massless fields should be a conformal theory as it has no mass scale. In this scenario, the fact that there are no large mass corrections follows from the condition of conformal invariance. In other words, the ’t Hooft naturalness condition is satisfied in the conformal approach, namely in the absence of masses there is an enhanced symmetry which is the conformal symmetry. 
... in the present context, it seems to be of interest to consider the issue of renormalizability. Usually, in quantum field theories, the condition of renormalizability is imposed on a theory as if it were a basic principle to make the perturbation method to be meaningful, but its real meaning is unclear since there might exist a theory for which only the non-perturbative approach could be applied without relying on the perturbation method at all. To put differently, the concept of renormalizability means that even if one is unfamiliar with true physics beyond some higher energy scale, one can construct an effective theory by confining its ignorance to some parameters such as coupling constants and masses below the energy scale. Thus, from this point of view, it is unclear to require the renormalizability to theories holding at the highest energy scale, the Planck scale, such as quantum gravity and superstring theory. On the other hand, given a scale invariance in a theory, all the coupling constants must be dimensionless and operators in an action are marginal ones whose coefficient is independent of a certain scale, which ensures that the theory is manifestly renormalizable. In this world, all masses of particles must be then generated by spontaneous symmetry breakdown. 
In previous works, we have shown that without resort to the Coleman-Weinberg mechanism, by coupling the non-minimal term of gravity, the U(1) B-L gauge symmetry ... is spontaneously broken in the process of spontaneous symmetry breakdown of global or local scale symmetry at the tree level and as a result the U(1) B-L gauge field becomes massive via the Higgs mechanism. One of advantages in this mechanism is that we do not have to introduce the Higgs potential in a theory. 
Ichiro Oda, Higgs Mechanism in Scale-Invariant Gravity, 08/2013

Scale invariance is a well-known symmetry that has been studied in many physical contexts. A strong physical motivation for incorporating scale symmetry in fundamental physics comes from low energy particle physics. Namely, the classical action of the standard model is already consistent with scale symmetry if the Higgs mass term is dropped. This invites the idea, which many have considered, that the mass term may emerge from the vacuum expectation value of an additional scalar field φ (x) in a fully scale invariant theory. Another striking hint of scale symmetry occurs on cosmic scales: the (nearly) scale invariant spectrum of primordial fluctuations, as measured by WMAP and the Planck satellite. This amazing simplicity seems to cry out for an explanation in terms of a fundamental symmetry in nature, rather than as just the outcome of a scalar field evolving along some particular potential given some particular initial condition... 

For example consider the usual standard model with all the usual fields, including the doublet Higgs field H(x) coupled to gauge bosons and fermions, but add also an SU(2)×U(1) singlet φ(x) (plus right handed neutrinos and dark matter candidates), and take the following purely quartic renormalizable potential involving only the minimal set of scalar fields ... This model ... is the minimal extension of the standard model that is fully scale invariant at the classical level, globally. ... The field φ(x), which we call the “dilaton”, absorbs the scale transformations and is analogous to the dilaton in string theory. In the current context, it has a number of interesting features: Due to SU(3)×SU(2)×U(1) gauge symmetry, the singlet φ is prevented from coupling to all other fields of the Standard model - except for the additional right handed singlet neutrinos or dark matter candidates. These features of φ, that prevent it from interacting substantially with standard visible matter except via the Higgs ... suggest naturally that φ itself could be a candidate for dark matter. .. the vacuum expectation value of the Higgs may fluctuate throughout spacetime, depending on the dynamics of φ(x), without breaking the scale symmetry. However, if for some reason (e.g. driven by quantum fluctuations or gravitational interactions) φ develops a vacuum expectation value φ0 which is constant in some region of spacetime, then the Higgs is dominated by a constant vacuum expectation value ... fixed by observation to be approximately 246 GeV. The Higgs vacuum ... provides the source of mass for all known elementary forms of matter, quarks, leptons, and gauge bosons (while φ0 may be the source of Majorana mass for neutrinos). The observation at the LHC of the Higgs particle, which is just the small fluctuation on top of the vacuum value v, has by now solidified the view that this is how nature works in our region of the universe, at least up to the energy scales of the LHC. 

Itzhak Bars, Paul Steinhardt, Neil Turok, Local Conformal Symmetry in Physics and Cosmology, 07/07/2013