### Bossons sur le (grand frère du) boson scalaire (de Higgs à 125GeV)

**Naturalité du boson de Higgs associée à un pseudo-dilaton dans un contexte de théorique quantique des champs**

En ce temps estival, le blogueur se plonge dans le flot des prépublications scientifiques récentes qui tournent autour de l'extension du secteur scalaire des théories quantiques des champs, toujours guidé par son outil méta-heuristique favori (la géométrie non commutative et le principe d'action spectrale).

On commence aujourd'hui avec un article qui fait singulièrement écho à la problématique évoquée dans un précédent billet intitulé Naturalité des extensions minimales du secteur scalaire du Modèle Standard

The Higgs boson presents several well-known puzzles associated with the problem of the naturalness of the existence of a low mass fundamental 0+ ﬁeld in quantum ﬁeld theory. The naturalness issue is associated with how scale symmetry is implemented (or not) for the Higgs boson, andthere has been a recent upsurge of interest in models that attempt to maintain a classical scale invariance which is broken only by scale anomalies[1–4].Here we explore this idea in the context of an extension of the Standard Model (SM) that includes a new gauge singlet scalar ﬁeld σ coupled to the Higgs sector via ultra-weak couplings.In particular,we assume that the Higgs couples to the singlet ﬁeld σ through a portal interactionζ_{1}σ^{2}H.^{†}H. Electroweak breaking is induced when σ acquires a VEV by quantum loops, i.e., through Coleman- Weinberg (CW) symmetry breaking [5], and thus yields a mass for σ and for the Higgs boson. We consider the case that the σ ﬁeld VEV f is much larger than the weak scale,f ≫ v_{weak}, in which case the couplingζ_{1}must be ultra-weak,|ζ_{1}| = m^{2}_{H }/ f^{2}≪ 1.At ﬁrst sight, constructing a model with ultra-weak scalar couplings would seem to be a foolish thing to do since most SM couplings are either technically naturally small (e.g., the electron or up and down quark Higgs-Yukawa couplings) or are of order the gauge couplings, such as gtop ∼ g3. For example, the Higgs quartic coupling λ receives additive contributions from the large O(1) couplings gtop, g2 and g1, and thus λ is not ultra-weak. Thereforewe must ask ifζ_{1}can be technically naturally small. The answer is yes: there exists a custodial symmetry for ultra-weak couplings amongst singlet ﬁelds.This is a “shift symmetry” and it has a Noether current whose divergence is small, ∝ ζi. This is the reason why ultra-weak couplings can remain ultra-weak in the renormalization group (RG) evolution; the ’t Hooft naturalness of ultra-weak couplings is the exact shift symmetry in the limit ζi→ 0.We have seen shift symmetry in another guise before. Shift symmetry naturally casts σ as a pseudo-dilaton...

Given that the scale of gauge couplings in the SM is O(1), the shift symmetry limit can exist only if the σi are gauge singlet ﬁelds.Indeed, it is not meaningful to talk about shift symmetries for ﬁelds that carry gauge charges such as the Higgs boson (unless one is interested in the consequences of dynamics in the limit that gauge couplings can be ignored). The couplings λi of ﬁelds such as the Higgs boson will receive additive corrections from gauge couplings and will not be multiplicatively renormalized. They will run according to the RG and become comparable in size to the gauge couplings. Of course, our argument is subject to gravitational effects.All ﬁelds including σ couple to gravity, which is a gauge theory, so the condition of ultra-weak ζi couplings is subject to whether or not the shift symmetry can be maintained in the context of gravity. This can be done if the contributions to the RG equations from conformal couplings ξi, which appear in terms like1/2ξσ^{2}R, can remain ultra-weak. These, in turn, will involve eﬀective gravitational couplings, an example of which is the recent “Agravity” model of Salvio and Strumia [10]...

Hence,the shift symmetry may be a powerful constraint that admits a natural sector of ultra-weakly coupled physics...

Up to now we have assumed that the theory obeys classical scale invariance in the sense that scale invariance is broken only through the trace anomaly.This assumes, as is the case in dimensional regularization, that the radiative corrections to scalar masses that are quadratically dependent on the cut-oﬀ scale are cancelled by the bare mass terms, leaving the scalars massless before spontaneous symmetry breaking. This makes sense in a pure ﬁeld theory because only the renormalized masses are physical.However, new physics at a high scale can spoil this by introducing contributions to the scalar masses that are proportional to the high scale.This is the case if there is a stage of Grand Uniﬁcation, for which the contributions are proportional to the mass scale of the heavy GUT states, but can also happen even if there are no massive states, for example when the new scale is generated by the CW mechanism. In the model presented here, such corrections would aﬀect the Higgs mass and give rise to the usual hierarchy problem, but they also aﬀect the singlet state, despite its ultra-weak couplings, because a contribution to the σ mass squared of O(ζiΛ^{2}) will dominate over the CW potential for Λ > O(TeV).To avoid this we envisage two possibilities.

The ﬁrst is that there are no high scales of the type discussed above. Of course this cannot be true if gravity is included, but, as discussed above, it may be that gravity respects the shift symmetry and the gravitational corrections to the dilaton mass are small.However, one would still expect an unacceptably large contribution to the Higgs mass, thereby reintroducing the hierarchy problem. Alternatively,if the model is UV complete so that it does not have Landau poles, gravity may not contribute to the scalar masses at all [11]. This case is analogous to that of a pure ﬁeld theory with classical scale invariance and guarantees that the scalar sector remains massless in the absence of spontaneous symmetry breaking.The second possibility is to super-symmetrize the modelso that the quadratic mass terms have a low SUSY scale cut-off. In this case, one can have a stage of Grand Uniﬁcation without introducing unacceptably large scalar mass contributions. A supersymmetric version ofthe model requires an additional Higgs doublet that somewhat complicates the model.

Kyle Allison, Christopher T. Hill, et Graham G. Ross,Ultra-weak sector, Higgs boson mass, and the dilaton,24 avril 2014

**A la recherche d'une symétrie cachée qui rendrait naturel le (couplage ultra-faible du) boson de Higgs (à un scalaire réel singulet?)**

The ﬁeld σ with ultra-weak couplings is formally analogous to a dilaton, as occurs in a spontaneous breaking of scale symmetry.Let us examine this relationship.Spontaneous scale symmetry breaking can be viewed in two ways. The conventional description is to start with a scale invariant theory, containing a dilaton with a shift-invariant potential, and matter ﬁelds. The dilaton’s shift symmetry is broken by the coupling to matter, e.g., as in Yukawa couplings.The stress-tensor is traceless. The dilaton can then acquire a nonzero VEV, and the matter ﬁelds then acquire mass, but the stress tensor remains traceless. Hence,we end up with a scale invariant theory, massive matter, and a massless dilaton as the Nambu-Goldstone boson.Alternatively, we can start with massive matter ﬁelds, and we include a dilaton with a shift-invariant potential, but with couplings to matter that again break the shift symmetry.Now we compute the stress tensor and ﬁnd that it is not traceless, i.e., the scale current is not conserved. However,we can ﬁnd a linear combination of the scale current and the dilaton shift current that is conserved; the theory has a hidden symmetry after all.

Id.

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