Some alternatives to the standard Higgs field

Hopefully within a couple of years LHC will tell us something about the origin of Electroweak Symmetry Breaking (EWSB). While there is no strong argument against the standard picture based on a fundamental scalar doublet, it is entertaining to consider alternative scenarios. Some of these do contrast with widely held beliefs about quantum field theories and therefore would require some rather radical rethinking if found to be true. I will describe some of them here, without any pretence at completeness.

Technicolor

By far the most developed alternative to a fundamental scalar Higgs is Technicolor. The simplest version of Technicolor would be just a scaled up version of QCD. This idea originates from the observation that QCD by itself already provides a source of EWSB. In the approximation in which heavy quarks can be ignored and the masses of the u and d quark can be neglected, QCD is invariant under the group SU(2)_L x SU(2)_R of independent unitary transformations of the left and right quark doublets. But the spectrum of hadronic states does not exhibit this symmetry, so this (approximate) symmetry must be spontaneously broken. What breaks it is a quark condensate and by Goldstone’s theorem the fluctuations around this nontrivial vacuum are massless. Now in the real world the quarks are not massless, but this picture remains approximately correct also if the quarks have small masses, and the corresponding (pseudo) Goldstone bosons can be identified with the pions. When we bring the EW interactions into the picture, the group SU(2)_L is gauged. But then the chiral condensate “dynamically breaks” SU(2)_L. If there was not a Higgs field (or something similar to it), then the W would “eat up the pions” and acquire a mass of the order of 1MeV.

Since the mass of the W is much higher than 1MeV, QCD cannot be the origin of EWSB. However, this suggests that there may exists a new, as yet unobserved, interaction called technicolor, with the same structure of QCD but with a chiral condensate at a scale of the order of a few hundred GeV. In this picture the VEV of the scalar is replaced by this Technicolor condensate and the Higgs particle – if present - would be a composite of the techniquarks. The W would get its mass by “eating” the technipions, which disappear from the spectrum (while the ordinary pions obviously remain).

This picture is very attractive, because it avoids the naturalness problems that haunt a fundamental scalar. In fact, the so called hierarchy problem would be solved essentially in the same way in which it is solved in QCD. There are particles, such as the proton, whose mass is to a large extent independent of the EWSB mechanism. The EWSB gives mass to the constituents of the proton, but these masses only contribute a small fraction of the mass of the proton. The rest comes from the internal motion of the quarks and from the gluon fields, and is entirely governed by QCD. One can ask: why is this mass so small compared to the Planck scale, or the grand unification scale? This question is independent of the analogous question for EWSB, and it has a convincing explanation within QCD: the strong coupling constant starts out relatively weak at the GUT scale and since it runs only logarithmically, one has to run down many orders of magnitude in energy before it becomes sufficiently strong to form bound states, or a condensate. So in Technicolor theories also the hierarchy between the GUT scale and the mass of the W would be explained by this simple dynamical mechanism.

This particular realization of Technicolor as a scaled up version of QCD is unfortunately in contrast with electroweak precision data, but there is a community of committed physicists that keep this idea alive and have produced more elaborate models that can get around such restrictions. For an overview of current research in this field, see e.g. here.

Aside from Technicolor there are other less known and/or less developed scenarios which predict that a Higgs particle will not be seen or at least will be hard to detect at LHC.

Truly Higgsless models

The most radical, absolutely Higgsless scenario would consist in having simply a mass term for the gauge fields. The mass term breaks gauge invariance and renders the theory perturbatively nonrenormalizable, so this possibility is usually not taken into consideration. But let us consider it from a slightly different point of view. It is possible to rewrite the mass term in a gauge invariant way by introducing a field U with values in the group SU(2) and transforming by left multiplication under gauge transformations. One can write a gauge invariant kinetic term for this gauge field, of the form f² tr DU^-1DU, where DU=dU+WU and f is a coupling with dimension of mass. One can choose the gauge so that U=1 and in this gauge the kinetic term of U becomes the mass term of W. This is a nonabelian version of a trick first introduced in electrodynamics by Stuckelberg. So by using this trick we can recover gauge invariance. The problem with renormalizability remains, however, because the field U is a nonlinear sigma model and the nonlinear sigma model is perturbatively nonrenormalizable. This particular Higgsless theory based on a gauged sigma model has been known since long (early works were Appelquist and Bernard and Longhitano, see also Herrero and Morales). Because of the renormalizability issue this model is generally considered as an effective field theory with a cutoff somewhere under the TeV scale.

This gauged sigma model approach to EWSB can also be seen as a special limit of the standard model Higgs. Consider a potential of the form λ(φ²-υ²)². If one sends the quartic self-coupling λ to infinity keeping υ fixed, the Higgs potential becomes sharply peaked around its minima, and in the limit becomes a constraint, forcing the Higgs to lie in a three-sphere. This three-sphere can be identified with the group SU(2), parametrizing the gauge degrees of freedom of the Higgs field (the Goldstone bosons). The fourth degree of freedom, the physical Higgs particle, becomes infinitely massive in this limit and disappears from the spectrum. So the gauged nonlinear sigma model can be seen as the strong coupling limit of the standard model. It is clear that perturbative methods are not well suited to study this kind of limit. However, this issue has been studied on the lattice and the conclusion of much work is that such a strongly coupled theory could not exist because of the triviality problem. An oversimplified and pedestrian way of understanding this is as follows. The quartic self coupling λ has a beta function such that it grows towards the UV and decreases towards the IR. Actually, if one starts from a finite value of λ at some finite energy, then going towards the UV λ blows up at a finite scale (this is known as the Landau pole). If we tried to move this Landau pole out to infinity, then at any finite energy the renormalized coupling λ would be zero, so the theory would be free and in particular it could not drive the EWSB. So we come again to the conclusion that this must be regarded as an effective field theory with a cutoff in the TeV range.

There is still one way in which the Higgsless theory might work beyond te TeV scale, and this is if the nonlinear sigma model was asymptotically safe. This would mean that the coupling f has an UV attractive fixed point, and more generally also all other couplings that multiply higher derivative terms have a fixed point, with a finite number of relevant directions. If this was the case, then by placing ourselves in the subspace that is attracted to the fixed point when going towards higher energies, we would be assured that the theory has a good ultraviolet limit. It is not necessary to assume that the theory follows such a trajectory up to infinite energy. At some high scale “new physics” may appear, but in this scenario the scale of new physics could be shifted to arbitrarily high energies, in principle up to the Planck scale. This possibility has not been studied in detail so far, but some evidence for asymptotic safety of some nonlinear sigma models has been given recently here. It is worth noting that the nonlinear sigma models have a great deal in common with theories of gravity and that this work was actually stimulated by work on the asymptotic safety of gravity.

Another way in which the Higgs particle may disappear has been discussed by Pawlowski and Raczka in the 1990’s. In some sense this is the exact opposite of the scenario described above. Suppose that instead of letting λ to infinity we let λ go to zero. The potential then vanishes in the limit, and the Higgs particle becomes massless. (This is sometimes called the Prasad-Sommerfield limit, in the context of soliton theory.) In this limit the theory is scale invariant, and if one further adds a suitable gravitational term to the action, the theory becomes locally scale invariant (i.e. Weyl invariant). Weyl transformations act multiplicatively on the Higgs field, so one can fix the Weyl gauge by requiring that the Higgs field be constant. In this kind of “unitary Weyl gauge” the Higgs field disappears from the spectrum. The Goldstone bosons remain and do their job as usual, but the scale of the VEV is not set by the minimum of the potential. All we can say is that it is not zero, it remains otherwise unspecified.

This model could be seen also from another point of view. A mass for the gauge fields breaks, in addition to gauge invariance, also scale invariance. One can make a non-scale-invariant theory invariant even under local scale transformations by another version of the Stuckelberg trick. One introduces a new scalar field that scales multiplicatively under scale transformations, called a dilaton, and replaces every mass by the dilaton, times a dimensionless coupling. More generally, any dimensionful parameter in the action is replaced by the dilaton raised to the canonical dimension of the coupling. The result is a theory that is Weyl invariant. The Pawlowski-Raczka theory can be seen as a massive gauge theory that has been written in a gauge invariant and scale invariant form by introducing Stuckelberg fields both for gauge and scale transformations. One can recover the gauged nonlinear sigma model picture by fixing the Weyl gauge where the dilaton is a constant. Thus one can see this as a theory where scale invariance is “spontaneously broken”.

Connections with gravity and role of conformal invariance

Interestingly, this picture has some features in common with an analogous picture in the theory of gravity. In the case of gravity, linear gauge invariance (which contains scale transformations as a subgroup) is “spontaneously broken” at the Planck scale by the VEV of the metric. Up to linear transformations, the metric can be in only a finite number of states, characterized by its rank and signature (here we are assuming homogeneity in spacetime). Similarly, in this Weyl invariant Higgs theory, the Higgs field could be in only two states, up to scale transformations: the state where it is zero and the state where it is nonzero. Local scale invariance eliminates the continuous degree of freedom associated to the Higgs scale, leaving just a discrete degree of freedom. As a consequence, the choice between broken and unbroken phase cannot be made by tuning a potential, as in the usual formulation.

Let us summarize the discussion of the previous section as follows. Of the four components of the standard Higgs field, three are gauge degrees of freedom. The fourth one could disappear from the spectrum in two different ways: either by having infinite mass (which is the same as saying that it does not exist) or by having zero mass and being itself a gauge degree of freedom under Weyl transformations. In the latter case, all components of the Higgs field would be pure gauge and could be eliminated by gauge fixing. In this process they would generate masses for gauge fields. Three components of the Higgs (the angular variables) become the longitudinal components of the W and Z, and in this sense have already been observed. In the Pawlowski-Raczka picture also the fourth component (the radial variable) would suffer a similar fate, giving mass to the gauge field for scale transformations. Now, in the real world we do not know of any particle that can be interpreted as such a gauge field. So what is the meaning of this gauge field? Since scale transformations are spacetime transformations, there is here an intriguing possibility of a direct link between Higgs physics and gravitational physics. So let us digress briefly to discuss the role of this gauge field in gravity.

In the theory of gravity there is a linear connection on spacetime, an object that in particle physics would be called a gauge field for the group GL(4). The trace part of this gauge field (in the algebra indices) is a gauge field for local scale transformations. (In Weyl’s original proposal, this field was identified with the electromagnetic field, but it was soon understood that this interpretation is physically incorrect.) In the standard formulation of Einstein’s theory the connection is not treated as an independent degree of freedom: instead, it is a function of the metric. It is required that the covariant derivative of the metric be zero, and this freezes all the components of the gauge field that are not in the subalgebra of SO(1,3). The condition of vanishing torsion freezes the remaining components. Instead of imposing these conditions on the connection as a priori constraints, it is more instructive to think of the connection as a dynamical variable with a large mass. The mass for the components of the connection which are not in the Lorentz subalgebra can be generated for example by a term quadratic in the covariant derivative of the metric. Among these components is the gauge field for scale transformations, and in Einstein’s theory we can say that its mass originates from the VEV of the metric. But the preceding discussion shows that even if such gravitational terms were absent from the action, the VEV of the Higgs field, in the Pawlowski-Raczka scenario, would also give a mass to this gauge field. (If the Higgs potential did not vanish, scale invariance would be broken but not spontaneously broken.)

There are therefore two independent possible origins for the mass of this particular gauge field: a gravitational origin, driven by the VEV of the metric, and an EWSB origin, driven by the VEV of the Higgs field. (In GUTs, there could be a third origin, driven by the VEV of the scalars that break the GUT group.) It is perfectly possible that these different mechanisms are simultaneously present. The dominant contribution to the mass would then come from the VEV of the metric and the standard model Higgs would only contribute a tiny correction. This would be entirely similar to the standard picture for the mass of the W and Z, whose dominant part comes from the VEV of the scalar Higgs with a small correction from the QCD condensate. But there have also been speculations that there is really only one fundamental scale in Nature, which could be either the electroweak scale or the Planck scale. In the first case the scale is set by the VEV of the Higgs and the Planck scale would arise from an extremely strong coupling of the curvature scalar to this VEV (see e.g. here). In the second case the scale is set by the VEV of the metric, and the electroweak scale would arise via a very weak coupling of the electroweak sector to this VEV.

To conclude this section let me mention that in a somewhat similar vein also Faddeev has suggested that the Higgs field be identified in some way with the conformal factor of the metric.

Other unconventional ideas

In the last few years, there has been some revived interest in the idea that conformal invariance may stabilize the electroweak scale against quantum corrections, and therefore provide a solution to the fine tuning problems of the standard model.

Meissner and Nicolai, propose that the electroweak scale is generated by quantum effects in a classically scale invariant theory, containing in addition to the standard model fields also right handed neutrinos and an additional scalar that enters into the right handed neutrino’s Majorana-type mass. Using dimensional regularization, the (Coleman-Weinberg-type) effective potential develops a nonzero VEV due essentially to the conformal anomaly. They show that without much fine tuning the model can hold up to the Planck scale (assuming that there are no other intermediate scales). In this model the Higgs field is present, though with some peculiar phenomenological signatures. Similar ideas have also been explored by Foot, Kobakhidze and Volkas.

There have been several suggestions that interactions of a conventional Higgs particle with a hidden sector may make its detection very hard. See Van der Bij or the general discussion by Wilczek. In particular, people have discussed the interactions of the Higgs with a conformally invariant “unparticle” sector. A more recent proposal in this general class is that the Higgs field itself is an “unparticle” (hence the name “unhiggs field”). There are various versions of this story. The original suggestion by Stancato and Terning is phrased in terms of a field with a nonlocal kinetic term, containing the momenta to power 2(2-d). The usual Higgs field would have d=1, but now d is allowed to lie somewhere between 1 and 2. The claim by Stancato and Terning is that such a field would still be able to drive EWSB and furthermore that it could also unitarize W-W scattering. This paper has been followed by several others. Calmet, Deshpande, He and Hsu describe the Higgs unparticle as a particular convolution of infinitely many degrees of freedom possessing a continuous mass distribution. They claim that in this description the fermion mass generation, unitarization and radiative corrections can be computed in a simpler way. Falkowski and Perez-Victoria describe the nonlocal Unhiggs action as the boundary effective action of a local field in a five dimensional world.

Last update june 24, 2010