Abstract
The mechanism extending spontaneous symmetry breaking to gauge fields had a considerable impact on both theoretical and experimental elementary particle physics. It is corroborated by the discovery of the Z and the W, and by the precision electroweak tests. A detection of its Scalar boson(s) would not only constitute a direct verification of the mechanism, but knowledge of its couplings to known particles could pave the way to the world hitherto hidden beyond the Standard Model. These topics are discussed with emphasis on conceptual issues.

1 Spontaneous breaking of a global symmetry
Chiral symmetry breaking
Spontaneous symmetry breaking was introduced in relativistic quantum field theory by Nambu in analogy with the BCS theory of superconductivity [1] . The problem studied by Nambu [2] and Nambu and Jona-Lasinio [3] is the spontaneous breaking of the chiral U (1) symmetry of massless fermions resulting from the arbitrary relative (chiral) phase between their decoupled right and left constituent neutrinos. They then generalize to include isospin. Fermion mass cannot be generated from a chiral invariant interaction in a perturbation expansion but may arise through a (non-perturbative) self-consistent fermion condensate: this breaks the chiral symmetry spontaneously. Nambu [2] showed that such spontaneous symmetry breaking (SSB) is accompanied by a massless pseudoscalar. This is interpreted as the chiral limit of the (tiny on the hadron scale) pion mass. Such interpretation of the pion constituted a breakthrough in our understanding of strong interaction physics. In the model of reference [3] , it is shown that SSB also generates a massive Scalar boson, which I denote by a capital to emphasize its role in what follows.
The simple Goldstone U(1) model
The significance of the massless boson(s) and of the massive Scalar boson(s) occurring in SSB is well illustrated in a simple model devised by Goldstone [4] . A complex scalar field φ experiences a potential V (φ * φ). The Lagrangian density, Writing φ = φ + ϕ and φ = (φ 1 + iφ 2 )/ √ 2, the U (1) symmetry breaking is revealed by selecting the expectation value φ to lie in some direction, say φ 1 , of the (φ 1 , φ 2 ) plane. The quadratic terms in ϕ 1 and ϕ 2 yield the mass squared of their respective fields, namely, using the condition φ 1 2 = µ 2 /2λ at the minimum,
The simple Goldstone U(1) model
Thus ϕ 2 describes a massless boson, ϕ 1 a massive one, and the "order parameter" φ 1 may be viewed as a condensate of ϕ 1 bosons. Their significance is brought to light in Fig.2 and Fig.3 depicting respectively classical ϕ 1 and ϕ 2 waves on the background φ 1 . Nambu's discovery of the massless pion resulting from spontaneous chiral symmetry breaking.
The simple Goldstone U(1) model
They will be labeled massless Nambu-Goldstone (NG) bosons. Formal proofs corroborating the above simple analysis can be found in the literature [5] . These excitations in the φ 1 direction describe fluctuations of the order parameter φ 1 . They are volume effects and their energy does not vanish when the wavelength becomes increasingly large. They correspond in Fig.1 to a climbing of the potential. The quantum excitations ϕ 1 are thus now massive, in agreement with Eq.(1.2). These considerations can be again extended to more general SSB (even to discrete ones) to account for order parameter fluctuations. Lorentz invariance imposes that such massive excitations are necessarily scalar particles. They were also already present in reference [3] and will be denoted in general as massive Scalar bosons. The above considerations are restricted to spontaneous symmetry breaking of global continuous symmetries. Global means that the symmetry acts everywhere in space-time: for instance in the U (1) Goldstone model the parameter α in φ → e iα φ is independent of the space-time point x. We now discuss the extension from global to local symmetries.
The simple Goldstone U(1) model
2 The symmetry breaking mechanism for gauge fields
From global to local symmetry
The global U (1) symmetry in Eq.(1.1) is extended to a local one φ(x) → e iα(x) φ(x) by introducing a vector "gauge field" A µ (x) transforming under such local "gauge transformations" as
From global to local symmetry
The Lagrangian density becomes 
From global to local symmetry
Here, (D µ φ) A are covariant derivatives, F a µν are field strengths and φ A belongs to the representation of G generated by T a AB . The potential V is invariant under G.
From global to local symmetry
The local abelian or non-abelian gauge invariance of Yang-Mills theory hinges apparently upon the massless character of the gauge fields A µ , hence on the long-range character of the forces they transmit, as the addition of a mass term for A µ in the Lagrangian Eq.(2.1) or (2.2) destroys gauge invariance. But short-range forces such as the weak interaction forces seem to be as fundamental as the electromagnetic ones. To reach a basic description of such forces one is tempted to link this fact to gauge fields masses arising from spontaneous broken symmetry.
From global to local symmetry
However the problem of SSB is very different for global and for local symmetries.
The mechanism
This Section is based on the field-theoretic approach of reference [6] . In view of slips often made about the content and the dates of the 1964 papers quoted in this Section 2.2, references to these papers are detailed [17] .
Breaking by Scalars
Let us first examine the abelian case U (1) as realized by the complex scalar field φ exemplified in Eq.(2.1). The interaction between the complex scalar field φ and the gauge field A µ is
Breaking by Scalars
As in the Goldstone model of Section 1.2, the SSB Yang-Mills phase is realized by a non vanishing expectation value for φ = (φ 1 + iφ 2 )/ √ 2, which we choose to be in the φ 1 -direction.
Breaking by Scalars
with φ 1 = φ 1 + ϕ 1 and φ 2 = ϕ 2 , where as previously ϕ 2 and ϕ 1 are respectively the NG massless boson and the massive Scalar boson.
Breaking by Scalars
In the covariant gauges, the free propagator of the field A µ is
Breaking by Scalars
where η is a gauge parameter. In what follows, we shall choose the Landau gauge defined by
Breaking by Scalars
The polarization tensor Π µν of the gauge field in lowest order perturbation theory around the self-consistent vacuum is given by the tadpole graphs of Fig.4 , We see that, as a consequence of the contribution from the NG boson, the polarization tensor is transverse
Breaking by Scalars
and yields a singular polarization scalar Π(q 2 ) at q 2 = 0,
Breaking by Scalars
From Eqs.(2.6), (2.7) and (2.8), the dressed gauge field propagator becomes 9) which shows that the A µ -field gets a mass M V ,
Breaking by Scalars
The transversality of the polarization tensor Eq.(2.7) results from the contribution of the NG boson and agrees with a Ward identity which guarantees that gauge invariance is preserved.
Breaking by Scalars
Thus the mass of the gauge field A µ acquired through the absorption of the NG boson is gauge invariant.
Breaking by Scalars
The generalization of these results to the non-abelian case described by the action Eq. (2.2) is straightforward. Writing the generators in terms of the real components of the fields, one gets the mass matrix 
The fate of the massless NG boson
The diagrams of Fig.4 show that the NG boson is absorbed in the gauge field propagator. This yields the required longitudinal polarization of the massive gauge field encoded in the numerator of Eq.(2.9) on the mass shell q 2 = M 2 V . The massless NG boson actually disappears entirely from the physical spectrum. This is an immediate consequence of gauge invariance. Consider indeed Fig.2 . As explained in Section 1.2, the massless NG mode originates in global SSB from the vacuum degeneracy: the energy of the excitations depicted in Fig.2b and 2c tend to zero in the limit of infinite wavelength because they generate in that limit a vacuum equivalent to the original one under a symmetry operation. But local symmetry means that the configurations of Fig.2b and 2c carry no energy at all ! They are thus simply redundant description of the same gauge invariant vacuum, a redundancy not unexpected when fields are described by potentials A µ . Therefore there is no vacuum degeneracy, no spontaneous symmetry breaking and thus no massless NG boson ! 1 An apparent symmetry breaking, akin to the Goldstone model U (1) SSB, appears when one chooses a fixed orientation of the average Scalar field φ , e.g. φ = φ 1 . But this description is only a convenient gauge choice. It allows for the conventional assignment of group quantum numbers (such as isospin) to particles in perturbation theory. I shall therefore qualify also as SSB the mechanism generating mass for gauge fields but one should keep in mind that the symmetry is not intrinsically broken, a fact that renders the disappearance of massless NG bosons obvious. Their degrees of freedom are recovered in the longitudinal polarization of the massive gauge fields 2 .
The fate of the massive Scalar boson
A lee97@example.com shows that the stretching of (classical) Scalar fields are independent of local rotations of the φ-field in the (φ 1 , φ 2 ) plane. This translates the fact that the modulus of the φ-field is gauge invariant. Hence the Scalar bosons survives the gauging and their classical analysis is identical to the one given for the Goldstone model in Section 1.2. The coupling of the Scalar boson to the massive gauge bosons follows from the graphs in Fig.4 . Using Eq.(2.5) one gets the two tree-level vertices of Fig.5 where the heavy wiggly lines 1 A more detailed description of the distinctive features of global and local SSB can be found in reference [7] . Formal proofs for the absence of massless NG bosons were given by Higgs [8] , and then by Guralnik, Hagen and Kibble [9] . These proofs do not make use explicitly of the unicity of the gauge invariant vacuum.
The fate of the massive Scalar boson
2 A non relativistic precursor of this effect was found by Anderson [10] in condensed matter physics. Namely in superconductivity the massless mode of the broken U (1) symmetry disappears by being absorbed by electron density oscillations, namely by the longitudinal "massive" plasma mode.
The fate of the massive Scalar boson
on the right hand side represent (tree-level) dressed massive gauge propagators. The vertex couplings follow from Eq.(2.10).
Dynamical symmetry breaking
The symmetry breaking giving mass to gauge vector bosons may also arise from a fermion condensate. If a spontaneously broken global symmetry is extended to a local one by introducing gauge fields, the massless NG bosons disappear as previously from the physical spectrum and their absorption by gauge fields renders these massive.
The renormalization issue
The interest in the symmetry breaking mechanism stems from the fact that it provides, as does quantum electrodynamics, a taming of quantum fluctuations. This allows the computation of the quantum effects necessary to cope with precision experiments. In other words, the theory is "renormalizable", in contradistinction to the theory of genuine non-abelian massive vector fields.
The renormalization issue
The massive vector propagator Eq.(2.9), which is also valid in the non-abelian case by diagonalizing the mass matrix Eq.(2.11), differs from a conventional free massive vector propagator.
The renormalization issue
The numerator of the former is transverse for all momenta while the numerator of the latter, However there is a catch. The pole at q 2 = 0 in Eq.(2.9) has a negative residue and therefore is potentially violating unitarity. A glimpse into the solution of the problem appears from comparing our approach to the one of Higgs [12] . Higgs obtained most of our results from the classical equations of motion. In addition, he showed how to eliminate all contributions of the massless NG boson in that limit by the following field transformation 12) where B µ satisfies the conventional classical equations of motion of a massive vector field. In terms of propagators Eq.(2.12) becomes the identity Although these arguments suggest that the mechanism can be consistent, it is a highly non trivial affair to show that the fully interacting theory is renormalizable and unitary. This was proven by 't Hooft and Veltman [13] , who thereby established the quantum consistency of the SSB mechanism 3 .
The electroweak theory
In the electroweak theory for weak and electromagnetic interactions [15] , the gauge group is taken to be the chiral group SU (2)×U (1) 
The electroweak theory
The discovery of the Z and W bosons in 1983 and the precision experiments testing the quantum consistency of the theory establish the validity of the mechanism. The Scalar boson itself is presently search for and would provide a direct proof of it and would also characterize its precise realization. This will be further discussed in the following Section. The couplings of the Scalar to the massive W and Z bosons follow from 
Perspectives
Hopefully a Scalar boson will be found. Will it be identifiable to the electroweak Scalar of the Standard model, at least up to energies presently available at the LHC ? and what would be hidden beyond the Standard Model at possible higher energies? Although the answer must obviously be deferred to experiment, one may try, assuming that some Scalar will be found, some educated guess based on the belief that the organization of nature in scientific terms should be based on a minimal number of disjoint hypothesis. The usefulness of this exercise is that properties of known particles may perhaps pave the way to the unknown.
Perspectives
To try to make sense of this type of Occam's razor approach, I will select and compare some perspectives. I consider two perspectives which I find appealing. The first one I label historical : it takes at its starting point the SSB mechanism. The second one evolves out of the requirement that elementary fermions be massive; I will label it logical. In both perspectives, now that the chiral character of weak interactions is well established for massive fermions, the backbone of the electroweak theory is taken to be its group-theoretical content: the chiral group SU (2) × U (1).
Perspectives
-a) A "historical" perspective
Perspectives
The Fermi theory of weak interactions, formulated in terms of a four Fermi point-like current-current interaction, was well-defined in lowest order perturbation theory and successfully confronted many experimental data. However, it is clearly inconsistent in higher orders because of uncontrollable divergent quantum fluctuations. In order words, in contradistinction to quantum electrodynamics, the Fermi theory is not renormalizable. This difficulty could not be solved by smoothing the point-like interaction by a massive, and therefore short-range, charged vector particle exchange (the W + and W − bosons): theories with massive charged vector bosons are not renormalizable either. It is the electroweak theory, based on the renormalizable SSB mechanism applied to the group SU (2) × U (1), that provided a suitable tool for testing experiment. Its Scalar boson has yet to be confirmed.
Perspectives
The Scalar boson was introduced there through a field experiencing a Goldstone-type potential. In absence of a yet definite experimental answer, should this be expected to be an elementary particle (at least to testable scales) or is it a phenomenological description of a composite object?
Perspectives
The SSB mechanism described in Section 2 could be realized by an elementary Scalar condensate (Section 2.2.1) or dynamically (Section 2.2.4) in which case the Scalar boson would, at best, be a bound state. The absence of known elementary scalar particle, and the fact that a neutral scalar condensate could be, as is often the case in condensed matter physics, only a phenomenological description of a more complex dynamics, may suggest a dynamical realization of the mechanism. This is comforted by the fact that simple dynamical models, such as Technicolor, can be constructed. Technicolor generate gauge vector boson masses, but its extension to produce elementary fermion masses is more problematic. Giving mass to the fermions dynamically, which is natural in this perspective, might require additional groups which have then to confront many experimental constraints. As a rule, full dynamical symmetry breaking is very laborious and the corresponding phenomenological Scalar(s) may have higher masses 4 .
Perspectives
The "historical" perspective suggests that at low energies, the Scalar decays of Fig.6a (but not 6b) and Fig.7 could perhaps be presently detectable except for the last diagram involving fermion loops. Of course similar "fermiophobic" constraints would apply to the creation process.
Perspectives
At higher energies one would expect the emergence of a complex spectrum beyond the standard model one. To eliminate them one has to extend the symmetry to a local one and thus the gauge fields of the group must be introduced. It is natural to require that the elementary Scalar giving mass to the fermions acts then on the gauge fields. Of course one just recovers the electroweak theory and all decays of Fig.6 and Fig.7 should occur. The simplicity of the "logical" perspective, as compared to the "historical" one, is marred by the introduction of an elementary Scalar disconnected from the fermionic content of the theory and submitted to the ad hoc Goldstone-4 For a review on dynamical electroweak symmetry breaking see reference [16] .
Perspectives
type potential. These drawbacks could largely be avoided if some hidden supersymmetry, broken at larger energy scales, would be present. It would ensure, independent of the usual rather weak argument of "naturalness", that elementary scalars do appear and are accompanied by fermions at different masses.
Perspectives
The "logical" perspective suggests that at low energies, the electroweak theory provides the correct description. At higher energies one would expect some kind of hidden supersymmetry.
Perspectives
In the "historical" perspective there appear to be no particular reason for postulating supersymmetry.
Perspectives
If there is some sense in selecting these two perspectives, it would mean that if the Scalar is indeed discovered but is "fermiophobic", one would expect at higher energy scales no supersymmetry but instead a very complex structure with many new particles. If however the Scalar has the properties predicted by the electroweak theory, supersymmetry broken at high energy would be expected but the complexity of group and dynamical structures at high energy might be tamed. Clearly, these considerations stem from some kind of aesthetic prejudices, whatever that means, and only experiment will tell. They just convey a touch of personal feelings about the problem: I became increasingly impressed by the coherence out of the very few hypothesis embedded in the "logical" perspective and I took this occasion to stress it.

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