Higgs boson at LHC Run 2 : our workhorse for probing effective dimension-6 operators of the Standard Model fields

The particle physicist without low energy SUSY but a Higgs boson is a modern sisyphus
If one suppose following Adam Falkowski (aka Jester) that no slightest hint for a particular scenario beyond the Standard Model has emerged so far (I will venture to nuance this statement another day) any good physicist has to resort to effective field theory. This is what is reported below, following a model-independent framework. 

The standard model (SM) of particle physics was proposed back in the 60s as a theory of quarks and leptons interacting via strong, weak, and electromagnetic forces [1]. It is build on the following principles
  • #1 The basic framework is that of a relativistic quantum field theory, with interactions between particles described by a local Lagrangian. 
  • #2 The Lagrangian is invariant under the linearly realized local SU(3)×SU(2)×U(1) symmetry.
  • #3 The vacuum state of the theory preserves only SU(3)×U(1) local symmetry, as a result of the Brout-Englert-Higgs mechanism [2, 3, 4]. The spontaneous breaking of the SU(2)×U(1) symmetry down to U(1) arises due to a vacuum expectation value (VEV) of a scalar field transforming as (1, 2)1/2 under the local symmetry
  • #4 Interactions are renormalizable, which means that only interactions up to the canonical mass dimension 4 are allowed in the Lagrangian...
The SM passed an incredible number of experimental tests. It correctly describes the rates and differential distributions of particles produced in high-energy collisions; a robust deviation from the SM predictions has never been observed. It predicts very accurately many properties of elementary particles, such as the magnetic and electric dipole moments, as well as certain properties of simple enough composite particles, such as atomic energy levels. The discovery of a 125 GeV boson at the Large Hadron Collider (LHC) [89] nails down the last propagating degree of freedom predicted by the SM. Measurements of its production and decay rates vindicates the simplest realization of the Brout-Englert-Higgs mechanism, in which a VEV of a single SU(2) doublet field spontaneously breaks the electroweak symmetry. Last not least, the SM is a consistent quantum theory, whose validity range extends to energies all the way up to the Planck scale (at which point the gravitational interactions become strong and can no longer be neglected).  
Yet we know that the SM is not the ultimate theory. It cannot account for dark matter, neutrino masses, matter/anti-matter asymmetry, and cosmic inflation, which are all experimental facts. In addition, some theoretical or esthetic arguments (the strong CP problem, flavor hierarchies, unification, the naturalness problem) suggest that the SM should be extended. This justifies the ongoing searches for new physics, that is particles or interactions not predicted by the SM. 
In spite of good arguments for the existence of new physics, a growing body of evidence suggests that, at least up to energies of a few hundred GeV, the fundamental degrees of freedom are those of the SM. Given the absence of any direct or indirect collider signal of new physics, it is reasonable to assume that new particles from beyond the SM are much heavier than the SM particles. If that is correct, physics at the weak scale can be adequately described using effective field theory (EFT) methods. 
In the EFT framework adopted here the assumptions #1 . . . #3 above continue to be valid... Thus, much as in the SM, the Lagrangian is constructed from gauge invariant operators involving the SM fermion, gauge, and Higgs fields. The difference is that the assumption #4 is dropped and interactions with arbitrary large mass dimension D are allowed. These interactions can be organized in a systematic expansion in D. The leading order term in this expansion is the SM Lagrangian with operators up to D = 4. All possible effects of heavy new physics are encoded in operators with D > 4, which are suppressed in the Lagrangian by appropriate powers of the mass scale Λ. Since all D=5 operators violate lepton number and are thus stringently constrained by experiment, the leading corrections to the Higgs observables are expected from D = 6 operators suppressed by Λ2 [14]. I will assume that the operators with D > 6 can be ignored, which is always true for v≪Λ... 
Using the dependence of the signal strength on EFT parameters worked out in Section 4 and the LHC data in Table 2 one can constrain all CP-even independent Higgs couplings in Eq. (3.2)... In the Gaussian approximation near the best fit point I find the following constraints:
 where the uncertainties correspond to 1σ...
The Higgs boson has been discovered, and for the remainder of this century we will study its properties. Precision measurements of Higgs couplings and determination of their tensor structure is an important part of the physics program at the LHC and future colliders... it is important to (also) perform these studies in a model-independent framework. The EFT approach described here, with the SM extended by dimension six operators, provides a perfect tool to this end. One should be aware that Higgs precision measurements cannot probe new physics at very high scales. For example, LHC Higgs measurements are sensitive to new physics at Λ∼1TeV at the most. This is not too impressive, especially compared to the new physics reach of flavor observables or even electroweak precision tests. However, Higgs physics probes a subset of operators that are often not accessible by other searches. For example, for most of the 9 parameters in Eq. (5.1) the only experimental constraints come from Higgs physics. It is certainly conceivable that new physics talks to the SM via the Higgs portal, and it will first manifest itself within this particular class of D = 6 operators. If this is the case, we must not miss it.
(Submitted on 30 Apr 2015 (v1), last revised 12 May 2015 (this version, v2))

In the following I chose to focus on two specific model-dependant frameworks for the sake of argument.

Scalar fields are the last hype in  high-energy physics...
As a start, we assume that new physics does not violate known gauge and Lorentz symmetries in the SM so that the higher dimensional operators obtained by integrating out the heavy degrees of freedom also satisfy the same symmetries. There is only one dimension-5 operator (for one family of fermions) consistent with this, i.e., the Weinberg operator that gives rise to Majorana mass for neutrinos [1]. This operator violates the lepton number by two units. In the case of dimension-6 operators, the original attempt to compile a complete basis [2] was later found to be redundant [3, 45], leaving 64 independent operators (also for one family of fermions) [6] with five of them violating either baryon or lepton number [1, 7, 8]...
There are some attractive motivations to consider models with an extended scalar sector. For example, new scalar bosons in these models may facilitate a strong first-order phase transition for successful electroweak baryogenesis, provide Majorana mass for neutrinos, and/or have a connection with a hidden sector that houses dark matter candidates. Even though it may not be possible to directly probe this sector due to the heavy masses of new scalar bosons and/or their feeble interactions with SM particles, they can nevertheless leave imprints in some electroweak precision observables...  
Although it is widely believed that the standard model (SM) is at best a good effective theory at low energies, the fact that the observed 125-GeV Higgs boson has properties very close to that in the SM suggests that the new physics scale is high and the new degrees of freedom are likely to be in the decoupling limit. Therefore, it is useful to work out an effective field theory (EFT) in terms of operators up to dimension 6 and composed of only the SM fields...
In this paper, we have analysed the EFT of the SM Higgs field for a wide class of weakly coupled renormalizable new physics models extended by one type of scalar fields and respecting CP symmetry, concentrating on the dimension-6 operators that have corrections to the electroweak oblique parameters and current Higgs observables. We have shown that for the new scalar field of specific representations (SU(2)L singlet, doublet, and triplet), there are “accidental” interactions between the scalar and the SM Higgs fields that lead to dimension-6 operators at both tree and one-loop level. For the scalar field of a general representation under the SM gauge groups, we have pointed out that there are only two generic quartic interactions that will lead to dimension-6 operators only at one-loop level... 
(Submitted on 23 May 2015)

Spectral action principle could make phenomenologist's life easier as it talks naturally to scalars
The coupling constants of the three gauge interactions run with energy [1]. The ones relating to the non-abelian symmetries are relatively strong at low energy, but decrease, while the abelian interaction increases. At an energy comprised between 1013−1017GeV their values are very similar, around 0.52, but, in view of present data, and in absence of new physics, they fail to meet at a single scale. Here by absence of new physics we mean extra terms in the Lagrangian of the model. The extra terms may be due for example to the presence of new particles, or new interaction. A possibility could be supersymmetric models which can alter the running and cause the presence of the unification point [2].
The standard model of particle interaction coupled with gravity may be explained to some extent as a particular for of Noncommutative, or spectral geometry, see for example [3] for a recent introduction. The principles of noncommutative geometry are rigid enough to restrict gauge groups and their fermionic representations, as well as to produce a lot of relations between bosonic couplings when applied on (almost) commutative spaces. All these restrictions and relations are surprisingly well compatible with the Standard Model, except that the Higgs field comes out too heavy, and that the unification point of gauge couplings is not exactly found. We have nothing new to say about the first problem, which has been solved in [4, 5, 6, 78] with the introduction of a new scalar field σ suitably coupled to the Higgs field, but we shall address the second one.
Some years ago the data were compatible with the presence of a single unification point Λ. This was one of the motivations behind the building of grand unified theories. Such a feature is however desirable even without the presence of a larger gauge symmetry group which breaks to the standard model with the usual mechanisms. In particular, the approach to field theory, based on noncommutative geometry and spectral physics [10], needs a scale to regularize the theory. In this respect, the finite mode regularization [11, 12, 13] is ideally suited. In this case Λ is also the field theory cutoff. In fact using this regularization it is possible to generate the bosonic action starting form the fermionic one [14, 15, 16], or describe induced gravity on an equal footing with the anomaly-induced effective action [17].
The aim of this paper is to investigate whether the presence of higher dimensional terms in the standard model action − dimension six in particular − may cause the unification of the coupling constants. The paper may be read in two contexts: as an application of the spectral action, or independently on it, from a purely phenomenologically point of view. 
From the spectral point of view, the spectral action [10] is solved as a heath kernel expansion in powers in the inverse of an energy scale. The terms up to dimension four reproduce the standard model qualitatively, but the theory is valid at a scale in which the couplings are equal. The expansion gives, however, also higher dimensional terms, suppressed by the power of the scale, and depending on the details of the cutoff. This fixes relations among the coefficients of the new terms. The analysis of this paper gives the conditions under which the spectral action can predict the unification of the three gauge coupling constants...  
In this paper we have calculated the sixth order terms appearing in the spectral action Lagrangian. We have then verified that the presence of these terms, with a proper choice of the free parameters, could cause the unification of the three constants at a high energy scale. Although the motivation for this investigation lies in the spectral noncommutative geometry approach to the standard model, the result can be read independently on it, showing that if the current Lagrangian describes an effective theory valid below the unification point, then the dimension six operator would play the proper role of facilitating the unification. In order for the new terms to have an effect it is however necessary to introduce a scale of the order of the TeV, which for the spectral action results in a very large second momentum of the cutoff function
We note that we did not require a modification of the standard model spectral triple, although such a modification, and in particular the presence of the scale field σ, could actually improve the analysis. From the spectral action point of view the next challenge is to include the ideas currently come form the extensions of the standard model currently being investigated.
(Submitted on 24 Oct 2014)

The phenomenological requirement of a new TeV scale physics is exciting for the experimentalist. Could-it be related to expections from some noncommutative geometricaly compatible TeV-scale Left-Right models with superconnections? Anyhow the extremely large value asked for the second momentum of the cutoff function in the spectral action functional seems awkward in the perspective of the canonical noncommutative geometer educated physicist. Future will say if further studies involving an effective field theory survey of the full spectral standard model with its Majorana right-handed neutrinos and real or complex SM-singlet scalar fields (not to mention a Pati-Salam extension) will show the same kind of trend or follow the orthodox motto up to high energy partial unification scales ...

last edit 27 June


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