Sérénade à trois (scalaires) / Design for living (up to the Planck scale)

In search of a LHC-testable minimal conformal extension of the standard model / A la recherche de l'extension conforme minimale du modèle standard 

The gauge hierarchy problem continues to be one of the most pressing questions of modern theoretical physics. It is a naturalness problem which, at its core, asks the question why the electroweak scale can be light in spite of a high-energy embedding of the Standard Model (SM) into a more complex theory with other heavy scalar degrees of freedom. One approach to solve the hierarchy problem is the systematic cancellation of bosonic and fermionic loop contributions to the Higgs mass within supersymmetry. However, due to the fact that no supersymmertic particle has been observed yet, alternative approaches are appealing.

A radical way of addressing this problem is the assumption that the fundamental theory describing Nature does not have any scale. In such a conformal model, the symmetry can be realized non-linearly and explicit scales can appear. This approach has inspired a number of papers in the recent years, see for example [1–28] and references therein. A common feature of those works is the need for additional bosonic degrees of freedom, as in the SM alone the large top mass does not permit radiative breaking of the electroweak symmetry. The conceptual difficulty in the conformal model building is the nature of the symmetry, which is sometimes misleadingly called classical scale invariance. This symmetry is anomalous, since generically the renormalization-group (RG) running of the parameters leads to a non-vanishing trace of the energy-momentum tensor, which enters the divergence of the scale current.

We now argue that, if conformal invariance is a fundamental symmetry of Nature, then the quantum field theory must have a vanishing trace anomaly at some scale. In the absence of explicit mass parameters, the trace of the energy-momentum tensor is given by a weighted sum of the beta functions. The anomalous Ward identity thus allows only logarithmic dependence of physical quantities on the renormalization scale. Any quadratically divergent contributions to the Higgs mass must therefore be purely technical and are typically introduced by explicitly breaking the conformal invariance by regulators. The formal divergences can be absorbed by appropriate counterterms.

Our analysis changes the perspective under which the hierarchy problem is viewed. The question is not why in a given model the Higgs mass is light, but rather whether a quantum field theory with a given set of fields and parameters is stable under renormalization group translations. This RG stability will be our essential criterion to distinguish models and to analyze whether a particular parameter configuration is allowed. This criterion selects certain representations which can be added to the SM. We find that only the interplay of scalars, fermions and gauge bosons can lead to the desired RG stability. 

In this paper we revisit several classically scale-invariant models and investigate whether they can be low-energy realizations of a conformal theory. Including all relevant effects we find that in contrast to previous studies, for example [20], the SM extension by one real scalar field is not consistent with this requirement. Eventually, we identify the minimal conformal extension of the SM Higgs sector to consist of the usual complex Higgs doublet supplemented by two real scalar gauge singlets, one of which develops a non-zero vacuum expectation value (vev). In this context, minimality implies that the SM gauge group is not altered and the additional number of representations is minimal. We find that the scalar field without the vev can be a viable dark matter candidate. Furthermore, small neutrino masses can be easily accommodated in this model. Another important result of our work is that the physical Higgs will have sizable admixtures of one of the singlet scalars which can be used to constrain our model’s parameter space.

...

Let us now ... concentrate on the conformal SM extended by two real scalar gauge singlets (CSM2S), one of which (say S) acquires a finite vev during electroweak symmetry breaking (EWSB), i.e. S = v_S + σ. The most general scalar potential which is consistent with the SM gauge symmetries and classical scale invariance can be written as

V = λ_φ(φ^†φ)² + λ_SS⁴ + λ_RR⁴  + κ_φS(φ^†φ)S² + κ_φR(φ^†φ)R² +κ_SRS²R² 

  + κ₄SR(φ^†φ)²+ κ₅S³R + κ₆SR³.                                                (28)

In order to reduce the number of free parameters, we impose an additional global ℤ₂ symmetry in the following way R→ −R , (29) with all other fields in the theory left invariant. The three terms in the last line of Eq.(28) are odd under the above transformation and are thus forbidden. Note, furthermore, that the definition in Eq.(29) implies absolute stability of R which therefore might be a viable dark matter candidate...  

Here, we concentrate on the case in which the physical Higgs boson ... is not identified with the pseudo Goldstone boson (PGB) of broken scale invariance... For given portal coupling  κ_φS(Λ_GW), we can therefore directly calculate the value of λφ at the Gildener-Weinberg (GW) scale...  

Our calculation for positive scalar mixing angle gives the plot shown in {the figure below}. As discussed above, we vary one of the portal couplings, κφS, and the PGB mass. In accordance with the discussion ...{motivating the assumption ln(Λ_GW) ∼ ln(m_Higgs)}, we immediately discard those parameter points, which imply a large separation between Λ_GW and the electroweak scale v (grey area on the left). Since the effective potential’s perturbative expansion is no longer reliable if ln(<ϕ>/Λ_GW) is too large, we additionally exclude points, for which the hierarchy between the GW scale and the condensate <ϕ> becomes sizable (grey area on the top). For small portal couplings |κ_φS| and sufficiently low PGB masses, m_PGB≤15 GeV, we then find a viable region of parameter space (red area). In this regime, a fully consistent extrapolation of the model up to the Planck scale is possible, while reproducing the correct low-energy phenomenology. 

Largest possible UV scale in {a conformal SM extended by two real scalar gauge singlets}.
(Case in which the physical Higgs is not the pseudo Goldstone boson ...)
 The white arrow marks the example point {m_PGB=3 GeV,  κ_φS= −0.0018}.

The available parameter space can be further narrowed down by noting that the mixing in the Higgs sector will effect the signal strength of Higgs events observed at the LHC. The currently measured signal strength constrains the scalar mixing angle to sin β ≤ 0.37 [34]. By including this limit in {the} Figure {above} we can rule out all points below the dashed black curve. An other interesting phenomenological aspect is the existence of exotic Higgs decays. The Higgs boson can decay into two PGBs, which then further decay to SM particles. In this decay chain possible final states contain, H → 4 jets, H → 4 leptons, H → 4γ. H → 2 jets 2γ, H → 2 jets 2 leptons, H → 2 leptons 2γ. While the hadronic decays have a large background at the LHC, the final states containing leptons can be well distinguished. In particular the leptons are pairwise boosted in contrast to a decay mediated by the electro-weak gauge bosons. Furthermore, the H → 4γ can provide a very clean signature and only has the background coming from highly suppressed Higgs self interactions. This opens a window of opportunity to test a symmetry implemented close to the Planck scale, directly at the TeV scale... 

... separating the wheat from the chaff? / séparer le bon grain de l'ivraie ?

The present study contains the analysis of simple conformal extensions of the Higgs sector in which radiative symmetry breaking within the Coleman-Weinberg mechanism can take place. As a consequence of nonlinearly realized conformal symmetry implemented at a much higher scale, the usual gauge hierarchy problem is avoided. For this scenario to be consistent, the vanishing of the trace anomaly at the high scale is necessary. As simple extensions of the Standard Model (SM), we consider theories with the same gauge group. Hence, there is always the beta function of the Abelian gauge coupling which can only vanish in the UV once gravity contributions become significant. Thus, our necessary condition is that the renormalization-group (RG) running remains stable and does not develop Landau poles below the Planck scale. We have used the Gildener-Weinberg formalism, ensuring the perturbative nature of our expansion, and have taken into account the complete one-loop RG equations. In particular, we include contributions from field renormalization. 

We find that none of the conformal extensions of the Higgs sector by one scalar SU(2)L multiplet meets the stability criteria. The additional scalar can be either real or complex and acquire a vacuum expectation value or not. In all cases the models develop a Landau pole far below the Planck scale...  

In particular, the simple model discussed in [20], in which the SM is extended by one real SU(2)L-singlet scalar and right-handed sterile neutrinos, turns out to be unstable. Indeed, even though a Yukawa coupling y gives a negative contribution proportional to −y⁴ to the beta function of the Higgs self-coupling, the scalar field wave function renormalization unavoidably introduces positive terms scaling as +λy² . Therefore, it is obvious that, with growing λ, the fermionic contributions always destabilize the system even more.  

Other extensions of the Higgs sector by one SU(2)L scalar representation turn out to be unstable as well, as for example the conformal inert doublet model [23]... 

Having excluded those simplest theories, we find the minimal model, which leads to correct radiative breaking of electroweak symmetry and is RG stable, among the extensions of the Higgs sector by two scalars... 

Within the minimal model, we find that one of the scalar singlets is an excellent dark matter candidate, since it does not develop a vacuum expectation value. Its effective phenomenology is similar to the Higgs portal model, see for example [38,39,40] and references therein. We observe the dark matter mass to be confined to a rather small region between 300 GeV and 370 GeV. Furthermore, we checked that the parameter space considered by us is consistent with cosmological observations, i.e. the scalar field abundance does not overclose the universe. However, a detailed study of the dark matter phenomenology goes beyond the scope of this article... 

Furthermore, the points of the parameter space in which we find stable RG running predict sizable singlet scalar admixtures to the physical Higgs state with sines of the mixing angle between 0.12 and 0.48. The mixing can be compared to the SM prediction which leads to a constraint on the mixing angle. The current LHC upper limit of sinβ<0.37 [34] therefore already rules out a large fraction of the parameter space. The complete model might be tested by the LHC in the ongoing run.  

Finally, we would like to remark that in the minimal conformal extension of the SM neutrino masses can easily be accommodated. Once we introduce right-handed neutrino fields as SM gauge singlets they naturally possess a conformal and gauge-invariant, Majorana-type Yukawa coupling to the scalar singlet S. 8 Additionally, we obtain Dirac-type Yukawa couplings with the SM lepton and Higgs doublet. After electroweak symmetry breaking the Yukawa couplings lead to a neutrino mass matrix that realizes a type-I seesaw mechanism [10]. Of course, it remains to be checked whether including the Majorana Yukawa coupling negatively influences the RG running. Based on our observations regarding the effects of the top quark Yukawa coupling on the RGEs, we expect changes due to y_M to be controllable.

Minimal Conformal Extensions of the Higgs Sector

Alexander J. Helmboldt, Pascal Humbert, Manfred Lindner, Juri Smirnov

(Submitted on 11 Mar 2016)

Another trio of scalars from a noncommutative perspective (much more challenging to test) / Un autre trio de scalaire dans une perspective noncommutative (beaucoup plus difficile à tester)

The noncommutative approach predicts all the fermionic and bosonic spectrum of the standard model, and the correct representations. One can also take as a prediction that there are no other particles to be discovered, except for the three scalar fields: the Higgs field, the singlet field and the dilaton field {see the quoted article below}. The dynamics of the fields are governed by the interactions obtained from the spectral action principle, which is based on using a function of the Dirac operator defining the metric of the noncommutative space... In fact the singlet responsible for the right-handed neutrino mass gets mixed in a non-trivial way with the Higgs field. The potential was derived and given in full detail in our earlier work [2]. Recently and in more than one work [3], [4], [5], [6], it was shown that adding a singlet (real or complex) scalar field, whose potential mixes with the Higgs field, has important consequences. It turned out that the RG equations of the combined Higgs-singlet system solves the stabilization problem faced with a light Higgs field of the order of 125 Gev avoiding making the Higgs quartic coupling negative at very high energies. Remarkably, the form of the Higgs-singlet potential proposed recently agree with the one we derived before from the spectral action [2]. The quartic couplings are determined at unification scale in terms of the gauge and Yukawa couplings. Running these relations down with the scale, give values consistent with the present data for the Higgs and top quark mass.

Resilience of the Spectral Standard Model
Ali H. Chamseddine, Alain Connes(Submitted on 5 Aug 2012 (v1), last revised 27 Aug 2012 (this version, v2))

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 that the 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 [constrained to be smaller than 10^-6 eV]. 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 10¹⁵ Gev to the Planck scale of 2.4·10¹⁸ 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. It also evades the problems of the original version of extended inflation.

Scale Invariance in the Spectral Action

Ali H. Chamseddine, Alain Connes

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