## Abstract

An elementary, weakly coupled and solitary Higgs boson allows one to extend the validity of the Standard Model up to very high energy, maybe as high as the Planck scale. Nonetheless, this scenario fails to fill the universe with dark matter and does not explain the matter–antimatter asymmetry. However, amending the Standard Model tends to destabilize the weak scale by large quantum corrections to the Higgs potential. New degrees of freedom, new forces, new organizing principles are required to provide a consistent and natural description of physics beyond the standard Higgs.

## 1. The Standard Model and the mass problem

The strong, weak and electromagnetic interactions of elementary particles are described by gauge interactions based on a symmetry group SU(3)_{c}×*SU*(2)_{L}×*U*(1)_{Y}. The structure of this gauge symmetry followed from the discovery by the Gargamelle experiment (http://home.web.cern.ch/about/experiments/gargamelle) of the neutral current responsible for the scattering *e*^{−}*ν*_{μ}→*e*^{−}*ν*_{μ} that could not be mediated by a photon exchange. Gauge theory is not only a way to classify particles and assign quantum numbers to them but it is also a dynamical principle that predicts particular couplings among particles. And the structure of these interactions has been well tested at the Large Electron–Positron Collider, for instance in the process *e*^{+}*e*^{−}→*W*^{+}*W*^{−}. While this is certainly true at least for the 3-point functions, namely the interactions involving at least three particles, the gauge structure is actually badly violated at the level of the 2-point functions, namely in the mass spectrum: the observed mass terms for the leptons and the gauge bosons are not gauge invariant since the gauge group is chiral, and also acts nonlinearly on the gauge fields. This apparent clash calls for a spontaneous breaking of the gauge symmetry.

In the broken phase, a (massive) spin one particle describes three different polarizations: two transverse ones plus an extra longitudinal one which is known to decouple in the massless limit. Precision electroweak (EW) measurements have established the Brout–Englert–Higgs mechanism [1,2] and the longitudinal *W*^{±} and *Z* certainly correspond to the eaten Nambu–Goldstone bosons [3,4] associated with the breaking of the approximate global chiral symmetry SU(2)_{L}×SU(2)_{R} to its diagonal subgroup. The gauge field masses are conveniently rewritten as the kinetic terms for these Goldstones (*σ*^{a}, *a*=1,2,3, are the usual Pauli matrices and ):
1.1which indeed exactly reproduces the standard mass Lagrangian in the unitary gauge (*Σ*=1). However, this is a description that is not self-consistent at very high energy since it leads to scattering amplitudes growing with the energy as follows from the Goldstone equivalence theorem by expanding the Lagrangian (1.1) (*V* denotes *W*^{±} or *Z*):
1.2with
1.3In the absence of any new weakly coupled elementary degrees of freedom cancelling this growth, perturbative unitarity will be lost around 4*πv*≈3 TeV (or , depending on the exact criterion used to define strong coupling) and new strong dynamics will kick in and soften the UV behaviour of the amplitude, for instance via the exchange of massive bound states similar to the *ρ* meson of quantum chromodynamics. The prime incarnation of these models of dynamical EW symmetry breaking is technicolour [5,6], which after being under siege by the EW precision data, had to surrender with the discovery [7,8] of a particle that behaves remarkably like the Standard Model (SM) Higgs boson.

## 2. The Standard Model Higgs boson

The simplest example of new dynamics that can restore perturbative unitarity consists of a single (canonically normalized) scalar field, *h*, coupled to the longitudinal *V* 's and to the SM fermions, *f*, as [9,10]
2.1The scalar self-interactions can be parametrized as
2.2For *a*=1, the scalar exchange cancels the growing piece (1.3) of the *V* _{L}*V* _{L} amplitude at high energy. Furthermore for *b*=*a*^{2}, the inelastic *V* _{L}*V* _{L}→*hh* amplitude also remains finite at high energy, while for *ac*=1 the amplitude does not grow either. The point *a*=*b*=*c*=1, and also *d*_{3}=*d*_{4}=1, *c*_{2}=*b*_{3}=⋯=0, defines the SM Higgs boson and the scalar resonance then combines with the EW Goldstones to form a doublet transforming *linearly* under *SU*(2)_{L}×SU(2)_{R}. It is an elementary, weakly coupled and solitary scalar field that unitarizes all the scattering amplitudes [11–14]. The SM Higgs boson has been the subject of intense research both from the theoretical and experimental sides (see for instance the study of Carena *et al*. [15] for a short presentation of the Higgs boson physics) and has driven the efforts of the high-energy community over many decades.

## 3. The hierarchy problem and the need for new physics

### (a) The UV sensitivity of the Higgs mass

With the addition of the Higgs boson, the SM is now theoretically consistent at the perturbative level *a priori* up to very high scale, possibly the Planck scale of quantum gravity. With the discovery of the Higgs boson by the ATLAS and CMS experiments [7,8], the picture would be perfect if it were not for the fact that the quantum corrections to the Higgs potential reveal a dramatic sensitivity to the details of the physics at very high energy, as if Newton would have realized that the exact value of the top quark mass plays a crucial role in the motion of the Moon around the Earth. This property goes against our intuition that physical phenomena at different scales decouple from each other. Concretely, the one-loop corrections to *M*_{h}, the mass parameter in the Higgs potential, are depicted in figure 1 and amount to
3.1As an example, for a 10 TeV cutoff, the gauge, top and Higgs contributions to the Higgs mass square corrections are, respectively, of the order of (600 GeV)^{2}, −(1.5 TeV)^{2} and (800 GeV)^{2}, all quite far from what the Higgs mass should be. The SM particles give unnaturally large corrections to the Higgs mass: they destabilize the Higgs vacuum expectation value (vev) and tend to push it towards the UV cutoff of the SM. Some precise adjustment (fine-tuning) between the bare mass and the one-loop correction is needed to maintain the vev of the Higgs around the weak scale: given two large numbers, their sum/difference will naturally be of the same order unless these numbers are almost equal up to several significant digits. This is the so-called hierarchy problem [16–18]. It is a generic technical problem in any theory involving some elementary light scalar fields.

It is often argued that the quadratic divergences in the Higgs mass corrections have no meaning since they can be set to 0 in dimensional regularization. Hence the belief that there is no hierarchy problem. This is actually true in the SM (at least when gravity is ignored) which involves a single scale. The hierarchy problem exists only when multiple scales are present. The hierarchy problem can be seen when dealing with the renormalized running Higgs mass, see Barbieri [19] for a recent narrative, and results from *finite* threshold corrections generated by new particles coupled to the Higgs boson: these corrections are proportional to the masses of these new particles and potentially drive the value of the Higgs mass itself to the largest energy scale unless they are precisely counter-balanced by the boundary value of the Higgs running mass at high energy. See figure 2 for illustration.

### (b) Stabilization of the Higgs potential by symmetries

Extra structures, particles and/or symmetries, are needed to stabilize the Higgs potential and to screen the radiative corrections:

— The

*spin trick*[20]; 21: in general, a particle of spin*s*has 2*s*+1 degrees of polarization with the only exception of a particle moving at the speed of light, in which case fewer polarizations may be physical. And conversely if a symmetry decouples some polarization states then the particle will necessarily propagate at the speed of light and thus remain massless. For instance, gauge invariance ensures that the longitudinal polarization of a vector field is non-physical, and chiral symmetry keeps only one fermion chirality: both spin-1 and spin- particles are protected from dangerous radiative corrections. Unfortunately, this spin trick cannot be used for a spin-0 particle such as the SM Higgs scalar boson.— The

*Goldstone theorem*[4,22]: when a global symmetry is spontaneously broken, the spectrum contains a*massless*spin-0 particle. However, here again, it seems difficult to invoke this trick to protect the SM Higgs boson from radiative corrections since a Nambu–Goldstone boson can only have some derivative couplings, unlike the Higgs field. Little Higgs models [23] have been constructed to circumvent these difficulties and they provide realistic examples of Higgs as a (pseudo-)Nambu–Goldstone boson.

In the late 1960s, the Coleman–Mandula and Haag–Lopuszanski–Sohnius theorems [24,25] taught model-builders how to apply the spin trick to spin-0 particles: the four-dimensional Poincaré symmetry has to be enlarged. The first construction of this type consists of embedding the four-dimensional Poincaré algebra into a super-algebra. Then the supersymmetry between fermion and boson extends the spin trick to scalar particles: spin-0 particles acquire the chirality of their spin- superpartners. Actually, there exists an even simpler way of enlarging the Poincaré symmetry, which is going into extra dimension(s): the five-dimensional Poincaré algebra obviously contains the four-dimensional Poincaré algebra as a sub-algebra. After compactification of the extra dimensions, from a four-dimensional point of view, the higher dimensional gauge field decomposes into a four-dimensional gauge field (the components along our four-dimensional world) and four-dimensional scalar fields (the components along the extra dimensions). The symmetry between vectors and scalars allows one to extend the spin trick to spin-0 particles.

Neither supersymmetry nor higher dimensional Poincaré symmetry are exact symmetries of nature. Therefore, if they ever have a role to play, they have to be broken. In order not to lose any of their benefits, this breaking has to proceed without reintroducing any strong UV dependence into the renormalized scalar mass square: a *soft breaking* of these symmetries is needed.

### (c) Composite Higgs models

The hierarchy problem and the absence of any apparent new physics that could stabilize the mass of an elementary Higgs boson make it tantalizing to consider the Higgs boson as a composite bound state emerging from a strongly interacting sector. In order to maintain a good agreement with EW data, it is sufficient that a mass gap separates the Higgs resonance from the other resonances of strong sector (the resonances that will ultimately enforce good behaviour of the *WW* scattering amplitudes). Such a mass gap can naturally follow from dynamics if the strongly interacting sector possesses a global symmetry, *G*, spontaneously broken at a scale *f* to a subgroup *H*, such that the coset *G*/*H* contains a fourth Nambu–Goldstone boson that can be identified with the Higgs boson. Simple examples of such a coset are SU(3)/SU(2) or SO(5)/SO(4) [26,27], the latter being favoured since it is invariant under the custodial symmetry (some non-minimal models with extra Nambu–Goldstone bosons have also been constructed [28]). Attempts to construct composite Higgs models in four dimensions were made in the early 1980s (e.g. [29–34]) and modern incarnations have been recently investigated in the framework of five-dimensional warped models where, according to the principles of the AdS/CFT correspondence, the holographic composite Higgs boson now originates from a component of a gauge field along the fifth dimension with appropriate boundary conditions [35].

The composite Higgs models offer a nice and continuous interpolation between the SM and technicolour-type models. The dynamical scale *f* defines the compositeness scale of the Higgs boson: when *ξ*=*v*^{2}/*f*^{2}→0, the Higgs boson appears essentially as a light elementary particle (and its couplings approach the ones predicted by the SM) while the other resonances of the strong sector become heavier and heavier and decouple; on the other hand, when *ξ*→1, the couplings of the Higgs boson to the *W*_{L}'s go to zero and unitarity in gauge boson scattering is ensured by the exchange of the heavy resonances. Below its compositeness scale, a composite Higgs is described by the effective Lagrangian (2.1) where the couplings generically differ from their SM values by an amount of order *ξ*, e.g. for the SO(5)/SO(4) models, the couplings are given by
3.2

A composite Higgs boson fails to fully unitarize *V* *V* scattering amplitudes and other heavy resonances, maybe vector ones, will be needed to restore perturbative unitarity at high energy after experiencing an enhancement of the scattering amplitude compared with the SM. A growth of the *V* *V* →*hh* cross section with the energy is also expected and could be a good probe of the composite nature of the Higgs boson [10,36].

## 4. Conclusion: the need for a Higgs precision programme

‘With great power comes great responsibility’ say a good book and a bad movie. And ‘with great discoveries come great measurements’. The Higgs boson is definitively one of the greatest discoveries of recent years [15]. And a dedicated study of its properties and couplings is offering a way to infer what the structure of physics beyond the SM can be. Natural models aiming at offering a rationale of why the Higgs mass is screened from high-energy corrections at the quantum level generically predict some deviations in the Higgs couplings compared to the SM predictions of the order of 1–100%. The current Higgs data accumulated at the Large Hadron Collider (LHC) by the ATLAS and CMS collaboration already constrain the Higgs couplings to massive gauge bosons and to fermions not to deviate by more than 20–30% from the SM predictions (figure 3). Identifying the expected deviations in the Higgs couplings should be one priority for the next run of the LHC and might call for a dedicated programme at future colliders.

## Funding statements.

This work has been partially supported by the Spanish Ministry MICINN under contract FPA2010-17747 and by the European Commission under the ERC Advanced Grant 226371 *MassTeV*.

## Footnotes

One contribution of 12 to a Discussion Meeting Issue ‘Before, behind and beyond the discovery of the Higgs boson’.

- © 2014 The Author(s) Published by the Royal Society. All rights reserved.