**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 f^{2}
tr DU^{-1}DU, 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(φ^{2}-υ^{2})^{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