### La masse du boson de Higgs est une boussole (1er épisode)

**Déconstruire le problème de la hiérarchie dans le secteur de Higgs?**

Ce billet se veut le premier d'une série où on essaiera de collecter des informations sur les solutions possibles au problème de la hiérarchie et de la naturalité du boson de Higgs (merci encore à Jonathan Butterworth de nous avoir indiqué des références intéressantes :-).

Frequently, the hierarchy problem is formulated in a strong form, saying that if the Higgs mass is not protected by a symmetry and hence we expect MH = O( ) and supposing all couplings are O(1) then M_{H}= O(Λ) implies Mi = O(Λ) (i = H,W, Z, t, · · ·) for all masses. In other words, why is the electroweak scale v not just ? However, this type of argument is rather formal. According to this kind of interpretation, the term “spontaneous symmetry breaking” would become quite meaningless if the breaking would be naturally at the “hard” scale and not at a much lower “soft” one, as it is anticipated usually. It would mean that the symmetric phase is not recovered at the hard scale. In effective theories one has to distinguish between short range (e.g. lattice spacing a ∼ Λ^{−1}) order and long range order quantities, the latter emerging from collective behavior as encountered in phase transitions. The fact that criticality requires the temperature to be tuned to its critical value does not mean that the critical temperature is Tc = O(1/a). Note that, in field theory language, the reduced temperature (T − Tc)/Tc is proportional to the renormalized mass square m^{2}_{ren}= m^{2}_{bare}−m^{2}_{c bare}, where m^{2}_{c bare}is the critical bare mass for which the renormalized mass is zero. The key point is that a limit Λ → ∞ need not exist as is a given physical quantity. The critical “fine tuning” T ∼ Tc is not a fine tuning problem giving an answer to why Tc ≪ 1/a. In typical cutoff systems encountered in condensed matter physics an order parameter associated with a first order phase transition, like the Higgs VEV v in our case, is by no means O(&Lamda;). Rather it is a matter of a collective phenomenon of the system with infinitely many degrees of freedom. Below the critical temperature, on the bare level, depending on the given effective short range interaction between the intrinsic degrees of freedom, the system is building up long range order and domain structures. The critical temperature Tc as well as an order parameter like the magnetization M are macroscopic quantities. Long range effective quantities emerging in critical phenomena, are effects we seewhen looking at a system from far away and do not simply reflect the microscopic structure. The emergence of long range collective patterns is what I called self-tuning or self-protection above. It is the natural case in critical or quasi-critical condensed matter systems. So it is natural to have v ≪ Λ and unnatural to expect v ∼ Λ. In the SM, in addition, stetting v = 0 enhances the symmetry in any case (the gauge- and chiral-one), in spite the Higgs mass square persists getting corrections O(Λ^{2}), which in the symmetric phase boosts up the physical mH to an O(Λ) quantity.

Fred Jegerlehner,The hierarchy problem of the electroweak Standard Model revisited, 14/09/2013

**Faire des divergences quadratiques le moteur de l'inflation?**

Faire d'un obstacle l'axe d'un levier pour sinon soulever le monde du moins essayer de résoudre une grande énigme c'est ce que propose Fred jegerlehner dans un article tout récent qui prolonge les idées développées précédemment.

... the SM in the broken phase has no hierarchy problem. In contrast, in the unbroken phase (in the early universe) the quadratic enhancement of the mass term in the Higgs potential is what promotes the Higgsto be the inﬂaton scalar ﬁeld. Thus the “quadratic divergences” provide the necessary condition for the explanation of the inﬂation proﬁle as extracted from Cosmic Microwave Background (CMB) data [9].Note that in the unbroken phase, which exists from the Planck scale down to the Higgs transition not very far below the Planck scale, the bare theory is the physical one and a hierarchy or ﬁne-tuning problem is not an issue there. Standard Model Higgs vacuum stability bounds have been studied some time ago in Ref. [10,11], for example. Surprisingly, the Higgs mass determined by the LHC experiments revealed a value which just matched or very closely matched expectations from vacuum stability bounds.

Fred Jegerlehner, Higgs inflation and the cosmological constant, 16/02/2014

**Ajouter un grain de sel non commutatif à l'équation Higgs = inflaton?**

La spécificité de ce blog réside dans sa quête obstinée d'une trace de non commutatif dans le monde physique or cet aspect n'a pas encore été mentionné dans ce billet. Le texte qui suit a pour but de pallier ce défaut en présentant des travaux sur le lien entre boson de Higgs et inflaton dans la perspective de la physique spectrale non commutative. Il n'y a par contre a priori aucun lien entre les idées développées par Nelson et Sakellariadou ci dessous et celles de Jegerlehner.

Considering the product of ordinary Euclidean spacetime (i.e., space-time but with imaginary time) by a finitespace (with the properties discussed above), a geometric interpretation of the experimentally confirmed effective low energy model of particle physics was given in Ref. [7]. Investigating cosmological consequnecs of this proposal, we have concluded that the Higgs field can play the rˆole of the inflaton field within the noncommutative approach to the standard model, provided inflation will take place at a scale higher than the strong weak unification scheme, 1017GeV. In order to find the precise value of this scale, a detailed analysis of the running of the couplings above unification would be required. However, let us emphasise that the aim of this paper is simply to note that within the noncommutative geometry approach to unifying gravity and the Standard Model, it is possible to have an epoch of inflation sourced by the dynamics of the Higgs field. In addition, this type of noncommutative inflation could have specific consequences that would discriminateit from alternative models. In particular, since the theory contains all of the Standard Model fields, along withtheir couplings to the Higgs field, which in this scenario plays the rôle of the inflaton, a quantitative investigation of reheating should be possible. More significantly, the cosmological evolution equations for inhomogeneous perturbations differs from those of the standard Friedmann-Lemaître-Robertson-Walker cosmology [10]. This raises the possibility that signatures of this noncommutative inflation could be contained within the cosmic microwave background power spectrum.

William Nelson et Mairi Sakellariadou,Natural inflation mechanism in asymptotic noncommutative geometry,09/03/2009

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