4.1. Instabilities in Scalar Theory
We start with the Lagrangian of a scalar
model with a single real field
,
Distinct from (8), we have no background field here, and we use momentum representation. As before, the metric is Euclidean.
If , the quantization is around , and after renormalization, m is the physical mass, which does not follow, such as the renormalized coupling , from the quantum theory, and one may calculate all quantum corrections in perturbation theory, at least in principle.
If
, if quantizing around
, the ground state (vacuum) energy acquires an imaginary part already from the first radiative correction, and the system becomes unstable. For a constant field
, the potential belonging to Lagrangian (103) has the ‘Mexican hat’ shape. The system will leave the ‘top of the hill’ and roll down to the bottom. Mathematically, this is realized by a shift,
of the field, and quantization around
, which has the meaning of the vacuum expectation value of the field,
, or the condensate. In general, the condensate may depend on coordinates. However, at least in the beginning, one considers a homogeneous condensate.
With the substitution (104), and
, Lagrangian (103) turns into
with a new mass,
This Lagrangian has a linear contribution, which must vanish (we quantize after substitution (104) again around
), i.e.,
must hold on the tree level (this relation will be modified by the quantum corrections). At once, as can be seen easily, this is the condition for the minimum in
of the tree level part.
The first quantum correction is the ‘tr ln’-term in (54). Using (56), we obtain
with
This function is calculated in
Appendix A. It has an ultraviolet divergence, and one must apply some regularization. In the
Appendix A, we represent two of them, the zeta functional one and the momentum cut-off. The first one is more elegant while the second provides more physical insight. To demonstrate the ideas behind the renormalization, in this subsection, we use the second one.
Inserting
, (106), into (A4),
we reordered
by powers of the condensate
and a contribution, which is not a power of
since it contains the logarithm of
. The renormalization starts by considering the powers of
as counterterms and by absorbing them into the tree-level Lagrangian by the substitutions
The second line in (110) does not depend on
and can be added as a constant term to the Lagrangian. We do not show this explicitly. Moreover, the specific form of substitution such as (111) is unimportant, and we will not show them in the following. It is only important that such substitutions are possible. We mention that the substitutions (111) become infinite ones in the limit of removing the regularization,
. When performing the corresponding substitutions with the zeta functional regularization, i.e., with
as given by (A2), these will be finite.
As known, regularization introduces a new, arbitrary dimensional factor. In the cut-off regularization, it is
that has the dimension of momentum, and in the zeta functional regularization, it is
, as mentioned in
Section 3.2. This circumstance opens the possibility of finite renormalization. Usually, such arbitrariness is fixed by so-called normalization conditions, which need a physical interpretation or justification. We mention that changes in these conditions, or in the regularization parameters, can be described in terms of the renormalization group. For instance, the factor in front of the logarithm in the upper line in (111) is the first coefficient in the so-called
function. The renormalization group plays an important role in quantum field theory, but it is also outside the scope of this review.
Using the renormalization freedom, we take the ‘tr ln’ contribution in the form
It is motivated by the tree level minimum, which remains unchanged when including this
since it was chosen such that the normalization conditions
hold.
Now we can discuss the interpretation of these results. With (57), we obtain an effective potential, including the first radiative correction, in the form
with
. Considered as a function of
, it has a minimum at
, and the mass
M takes in this minimum the value
. It is real. This way, one has a consistent theory with a condensate. For our needs, this is sufficient. We mention that this approach allows for the inclusion of temperature (via the Matsubara representation) and the study of symmetry restoration with a rising temperature. In doing so, infrared problems occur that require the inclusion of an infinite number of higher radiative corrections (graphs). All this is outside the given review. The interested reader may consult [
22] for a general introduction and motivation, Ref. [
23] for the summation formalism, and the most recent paper [
24] for the actual state and for further literature.
Finally, we discuss the massless case,
, which was the original motivation for [
8]. In that case, model (103) does not have any dimensional parameter. As already mentioned, such a parameter comes in with regularization. All previous discussions and formulas are applicable until (112), where formally putting
would produce an infinity. There is no reasonable normalization condition opposite to the massive case. In connection with the Casimir effect, this point was discussed in detail in [
7] (
Section 4.3). Here, we take
in the form
where
is arbitrary. The complete one-loop effective potential reads
Following [
8], we took
as the normalization condition (note, their coupling
is our
).
This function has the interesting property that from the logarithm for small
negative contributions appear, which result for
in a minimum,
Formally, this is to the right from the tree level minimum at
. However, in the perturbative approach, this is exponentially small and must be considered as zero. For the minimum to exit, the logarithm must be negative and of the same order as the tree contribution in (116), i.e., perturbatively small. This can be achieved only by a very small condensate
.
In [
8], one finds another, even more convincing explanation. By improving (117) by the renormalization group equation, or by letting the coupling constant run, negative values of the effective potential occur only for
in a region beyond the Landau pole, hence in a nonphysical region.
In summary, in this section, we have seen that in a pure scalar theory, an instability may occur when the initial mass has a negative, nonzero square. This instability results in a condensate. This phenomenon can be viewed as the scalar part of the Higgs phenomenon. In case the initial mass is zero, no realistic minimum and no condensate appear.
4.2. Scalar QED and Abelian Higgs Model
In this section, we consider the massless scalar QED. Its Lagrangian reads
where
is the field strength of the electromagnetic field, and
is a complex scalar field. With the covariant derivative
this Lagrangian is invariant under the group U(1). Regarding (8), the electromagnetic field is now a dynamic one, and we added the self-interaction of the scalar field. This model is a generalization of the pure scalar model considered in the preceding subsection. It is also called the Abelian Higgs model for the spontaneous symmetry breaking that it allows.
Following the ideas of [
8], we turn to real fields
,
and rewrite the Lagrangian in the form
It has, in addition to the gauge symmetry, O(2) symmetry, and its scalar part is a special case of an O(2N) model.
Like in the scalar case, we imagine a mass term added with a negative mass square. This would result in an instability and force us to consider the possibility of having a condensate. Therefore, we make a shift
of the scalar fields. Thereby, we follow [
8], slightly changing the notations. In general, any shift away from the origin in the (
)-plane would be as good as 4.2.5.
With (123), the Lagrangian turns into
where the dots stand for interaction terms that we do not need to consider (after performing the shift). In addition, we did not show the terms that are linear in the fields since these must vanish in the extremum of the Lagrangian. Equation (124) shows the constant and the quadratic parts of the Lagrangian. However, it has two cross-terms. The first one disappears when using the symmetry
between the scalar fields. The second can be diagonalized by the substitution
,
, which results in
After the shift, the electromagnetic field acquires a mass
, and the scalar fields acquires masses
and
.
Now we calculate the effective potential. For the ‘tr ln’ contributions, we use Formula (115) for all modes,
and take into account that the electromagnetic field now has three polarizations.
We look for a minimum of this effective potential and discuss, such as in the preceding subsection, the tree level and the quantum correction, which should be of the same order. Demanding that the logarithm is of order one, this requires
With such a relation, we have
, and (36) simplifies,
For the arbitrary constant
, we take the value of the effective potential,
, and from the first derivative of (127), we arrive at
which confirmed (127). The potential in this minimum is
The interpretation of
is that it is the vacuum expectation of the field,
, so that with (129), the effective potential finally takes the form
We started this subsection with Lagrangian (119). It has two parameters, the pair (
) of dimensionless couplings. We ended with the effective potential (41), which has the parameter pair (
). Now
has a dimension. It has an arbitrary parameter coming in with regularization and for which we take
. In [
8], the change from the dimensionless
to the dimensional
is called
dimensional transmutation. In addition, they showed that in this case, application of the renormalization group does not remove the minimum.
The outcome from this subsection is the understanding that by adding an Abelian vector field to the scalar sector, a condensate and symmetry breaking occur, as shown in
Figure 4. Thereby, the initial coupling
, which should obey relation (127), drops out from the final result (131), and, unlike the pure scalar model, renormalization group improvement does not destroy this picture.
It is interesting to look at the fate of the modes under the symmetry breaking. Initially, we have two photon modes and two scalar modes. After the symmetry breaking, one scalar mode goes into massive photon mode, and the other remains. This can be seen, for instance, using a module-phase representation,
. The field
is massless and represents the Goldstone boson. However, together with the electromagnetic sector, it appears as a pure gauge and can be eliminated. A similar discussion can be found in [
25] as well as in the discussion of further developments.
Of course, there are further developments of this model. The most important is the inclusion of finite temperature. One needs to perform resummation to eliminate infrared problems. In the result, as shown in [
26], a first-order phase transition shows up, and the symmetry is restored when raising the temperature. A physically important extension is the electroweak phase transition (or crossover as the lattice calculations suggest), which is still a topic under actual discussion [
27].
4.3. Instabilities in Non-Abelian Gauge Theories
In this subsection, we consider instabilities in non-Abelian gauge theories. The most relevant are QCD and the electroweak theory. Both involve non-Abelian vector fields, which have in a magnetic background a spectrum such as in (44) or (83). Its peculiarity results from their spin,
, which causes a coupling to a magnetic background field with negative energy. Frequently, this property is discussed as anti-screening or paramagnetism. In QCD, which is a massless theory, the result is a tachyonic state that is present for any strength of the background field making the ground state, which is considered a Savvidy vacuum, unstable. This case was discussed in
Section 3.3.3. The depths of the minimum (97), of the effective potential are as shallow as in the massless scalar case in
Section 4.1, Equation (118), namely exponentially decreasing for small coupling. However, unlike the scalar case, it is not washed out by renormalization group improvement due to asymptotic freedom.
In the electroweak theory, the mass of the W-boson postpones the instability until the field strength reaches the critical value , i.e., until the background field becomes very strong.
In the literature, the instability was discussed mainly in the context of an applied magnetic field, which would cause pair production and could have cosmological implications. The spontaneous generation of a magnetic field is not considered due to the large mass threshold. The topic was intensively discussed at the beginning of the 1980s, i.e., long before the discovery of the Higgs boson, for instance in [
28,
29,
30,
31]. Therefore, all parameter regions were of interest. A much-discussed question, for instance in [
29], was the gauge fixing to be applied in the background field, unitary gauge or the
-gauge. Both are possible and deliver, finally, the same result.
As an example of these developments, we consider [
30]. The model consists of an SU(2)-field coupled to a triplet of Higgs fields. Combining (31) and (119), the Lagrangian reads
where the covariant derivative,
provides the coupling to both, the background field
and gauge potential
. Following (41), we rotate the fields,
and keep the third component. Thereby, the derivative term of the Higgs field turns into
the mass and self-interaction terms, accordingly, into
The spontaneous symmetry breaks sets after changing the sign of the mass square,
, such as in
Section 4.1, with a shift of the third component of the Higgs field,
From (135), the W-field acquires a mass,
, in addition to what we have seen in
Section 4.2.
Now we have a charged vector field with mass and a charged scalar with mass in the magnetic background, with as a neutral vector field and as a neutral scalar field. The first two contribute with their spectrum in analogy to (83) and (64), as well as two modes from and one mode from , such as (109), to the one-loop effective potential. On the given level, all contributions are additive.
We demonstrate as an example the result obtained in [
30] in this situation. The following expression for the effective Lagrangian was obtained
with
and where
In their notations,
, and
is the scalar condensate. For the numerical evaluation, it is meaningful to use the relation
The corresponding effective potential, normalized to
, is shown in
Figure 5 as a function of the condensate
for two values of the magnetic field. The solid line represents the real part, and the dashed line is the imaginary part. It is clearly seen that for small
B, the effective potential is real and shows the minimum known from the Abelian Higgs model. For a larger magnetic field, an additional minimum appears near the origin together with an imaginary part. This minimum corresponds to that known from the Savvidy vacuum, and it becomes the deeper one. In addition, it is seen that the magnetic minimum is much shallower than the scalar one. For the picture, quite a large value of the coupling was chosen to obtain both minima shown in one picture.
Similar pictures were obtained in [
32], where, however, the focus was on the formation of a lattice of magnetic flux lines. At the beginning of the 1980s, there were a number of investigations of the effective potential. For instance, in [
31], similar pictures were obtained, but there was a question about the symmetry restoration by a large magnetic field, partly by early discussions in [
33]. Similar discussions can be found in [
28,
34]. In [
35], it was argued that this is not possible as long as the ground state is not stable. The magnetic vacuum remains, as seen, for example, in the right panel in
Figure 5, as well as the scalar condensate with it, in this minimum for all values of the magnetic field. Before the discovery of the Higgs boson, investigation of the effective potential of the electroweak model in a magnetic field was used to obtain bounds on the Higgs mass; for a review see [
36].
The question of symmetry restoration was also investigated at finite temperatures. The general expectation, according to Refs. [
37,
38], was that broken symmetry, the spontaneous creation of a magnetic field in our case, will be restored at a sufficiently high temperature. However, as found in [
39,
40], this is not the case.
4.4. Electroweak Magnetism
Beyond the instabilities caused by the one-loop corrections, it is also interesting to study non-Abelian fields on the classical level. As an especially intriguing feature, in quite a number of papers, the analogy to a (dual) superconductor was considered. In this spirit, the question arises of how the field responds to an applied homogeneous magnetic field. Will this response be a lattice of flux lines or something else? The first discussion of this question can be found in [
41]. It was assumed that domains will form in the plane perpendicular to the applied field with a size smaller than that necessary for the formation of an unstable mode. Thereby, formation of the unstable mode is considered an infrared, long-range effect. These domains are formed by magnetic flux tubes. In order to restore Lorentz invariance of the model, in [
41], a model was developed with a fluid-like superposition of the flux tubes. However, this model, which is known as the “Copenhagen vacuum”, was criticized independently in [
40,
42] and was finally abandoned.
The formation of the lattice of flux lines is of interest in the electroweak theory. In [
43], a mechanism was discussed that goes beyond the quadratic approximation and involves the interaction with a longitudinal mode of the W-field. In [
44], the condensate was discussed in the Georgi–Glashow model. In a linearized approximation, the existence of a vortex-like condensate was confirmed, and the relation to a type II superconductor was discussed. In [
45], these results were discussed in the electroweak model near the Bogomolny bound
where
is the Higgs coupling constant, and
is the Weinberg angle. In [
35], similar results were reported using a perturbative expansion near the bound. As a result, it was concluded that beyond the critical field strength, a periodic lattice of flux lines will be formed.
To demonstrate these ideas, we give a representation of the basic methods used. First, we consider the derivation of exact (classical) equations of motion for the tachyonic modes, which are a breakdown of the general Yang–Mills equations for this specific component. Second, we represent the attempts to lower the effective potential by some superposition of tachyonic modes.
4.4.1. Equations of Motion for the Tachyonic Mode
In [
44], a restricted model was discussed. It has an O(3) symmetry, and it is similar to the SU(2)-model considered in
Section 2.4 with an additional mass term for the W-field,
The fields are a charged vector field
and a neutral vector field
, related to the initial fields
by the relation (41). The W-field stands for the W-boson and the A-field for the electromagnetic field. Restricting the mode carries the instability
which are the only non-vanishing components. In this case, and assuming dependence on
only, the field equations can be simplified. This calculation is shown in
Appendix C. Finally, the equations for mode (143) read
The first line is a Liouville-type equation for the W-field, and the second line gives the expression for the magnetic field in terms of the W-field. The energy of the solutions can be expressed in the form
where
is the critical field strength. This way, the classical energy is always non-negative, as one should expect. We also mention that the system can be rewritten in terms of first-order equations using (A25) and the field strength
in addition to the second line in (144). The system of equations, generalizing the above ones to the case of the bosonic part of the electroweak theory, was derived in [
45]. A review and further discussions in [
46] focused especially on the interpretation of the second line in (144). The plus sign on the second term, which is an enhancement of the applied magnetic field by the system, is interpreted as anti-screening caused by the asymptotic freedom.
4.4.2. A Lattice of Flux Lines
In this subsection, we consider the formation of a lattice of magnetic flux lines such as that in a type II superconductor. We follow the first attempt of this kind [
47]. The authors start with a background field in the Landau gauge such as in (11). The equation for the tachyonic mode follows as a special case from
(42). The general solution is a superposition of harmonic oscillator ground state functions,
where
is an arbitrary coefficient function. All these states have the same energy and realize the degeneracy of the ground state.
In [
47], two ‘naive’ choices for
are discussed,
In the first case, the solution describes a magnetic field with a Gaussian shape on the plane
= 0. In the second case, performing the Gaussian integration over
, a similar picture appears on the
plane. In either case, the field is of finite size in one direction. The classical energy has translational invariance in all three spatial directions and is proportional to the corresponding lengths, ∼
, whereas the energy of the naive solutions is proportional only to two, say ∼
. Thus, one needs to look for a superposition that would be proportional to the classical energy. With this motivation, the following ansatz was made,
where
and
c are some constants. For
, we obtain an infinite set of leaves each with a Gaussian magnetic field. For
, an interference sets in, which will be constructive for
. Inserting (148) into (146) results in
With the notations,
,
, one comes to a representation in terms of a Theta function,
This solution forms a regular lattice of magnetic flux lines. Further, in an ingenious calculation, Ref. [
47] calculated the classical energy of this configuration, using an expression for the energy density derived earlier in [
48], and came to the result
with
. In hindsight, this formula should be a special case of 4.4.21.
Finally, the quantum contribution was added. For this, the one-loop vacuum energy in a homogeneous background was taken. The corresponding expression is, with the reversed sign, the second term in (96). This is, of course, a crude approximation. For the magnetic field, some average of (150) can be taken, which will not have an essential influence on the final result.
For the thus-obtained total energy, a minimum was found by a variation of all parameters entering, for instance, the coupling parameter
e. As a result, an effective potential in the minimum was found (
),
With
, this effective potential is negative. The depth of this minimum is not decreasing, unlike in (97), and for small coupling, it will certainly become the deeper one.
The above ideas were also applied to the electroweak theory, for instance in [
32]. Here, a decisive parameter is the ratio
, where
is the Higgs mass and
is the W-boson mass. For
(these masses were not known at that time), an instability and a first-order phase transition similar to the Abelian Higgs model was found. In the opposite case, the magnetic instability sets in when the magnetic background field exceeds the critical value
set by the W-boson mass. Making a perturbative expansion for small
, the formation of a lattice of magnetic flux lines was derived, which is similar to the Abrikosov lattice in a superconductor. The formation of such a lattice was confirmed in [
45] for the case of
, where
is the Z-boson mass, using the Bogomolny method, i.e., by reducing the equations of motion to first-order equations that are easier to solve.
The above results have been reconsidered several times, the latest in [
49]. The ansatz (146) had been modified using the symmetric gauge so that the initial solution describes a flux line (in place of a flux plane). The superposition of such solutions is then directly in terms of the center position. In addition, a regular lattice of these flux lines was chosen. It is argued that the time evolution of an initially homogeneous field would result in such a lattice. Then, the energy of such a configuration was minimized with respect to the scale parameter
of the solutions. Numerically, a minimum was found that is appreciably below the energy of the homogeneous field (the authors call it ‘classical field’).
In summary, there is a variety of instabilities in theories involving non-Abelian fields. In electroweak theory, due to the W-boson mass, the threshold for these is far beyond reachable energies (at least on Earth). In all cases, a spontaneous generation of a magnetic field is energetically favorable. However, these states are all unstable. A solution for the problem caused by this instability has not yet been found, despite numerous attempts, the latest being [
50] in the massless case.
4.5. On the Fate of a Homogeneous Background Field
Being interested in non-perturbative, infrared effects, homogeneous fields are a prime candidate for a background field. Not only do such fields allow for quite explicit calculations, inhomogeneities tend to increase the classical energy. The general properties of background fields in an SU(2) theory were discussed in detail in [
51]. It was shown that there are two types of translational invariant fields, of non-Abelian type, where the potential
is constant in a suitable gauge, and an Abelian type field that may be taken in the form
, where
is a constant field strength. For the first type, it was shown that any constant potential with a non-vanishing field strength yields an unstable action, and for that from the second type, only Euclidean self-dual fields may be stable. These have infinitely many zero-mode excitations whose influence on the stability was left open. In addition, one should remember that self-duality in the Euclidean region implies an imaginary electric field in the Minkowski region.
The topic of a homogeneous Abelian background field was reconsidered independently a few years later in [
52] from a variational point of view. The variations were taken in a class of fields being the unstable perturbations of a constant Abelian background, thereby including their quartic self-interaction. It was assumed that the stable fluctuations may be handled subsequently as perturbations. However, carrying out this program, it was observed that inclusion of the quartic terms resulted in a scaling of the coupling, which is different from the usual one that follows from the ultraviolet properties. In addition, it was mentioned that the inevitable mixing of the stable and the unstable modes could produce contributions of the same order as that from the unstable modes. Soon after, in [
53], it was shown that the mixing of unstable and stable modes makes this procedure unreliable. More precisely, it was shown ‘that the minimum of the restricted action,
, is of no help’ (
is the homogeneous background field and
are the unstable fluctuations modes). Their discussion rests on the observation that there is no solution of the Yang–Mills equations of the form
for non-vanishing
. This way, all attempts to consider a homogeneous background field are put under question.
After submitting this review, a (re-)derivation of the effective potential for a homogeneous (anti)self-dual background field appeared [
54]. In generalizing the initial work [
55], the author succeeded in summing up all zero modes explicitly, i.e., to solve a problem left open in [
51]. The resulting effective Lagrangian is the same as in the one-loop approximation but without the imaginary part. Further, in [
54], this result was generalized by deforming the electric field,
with
, breaking the self-duality. In this case, one has in the quadratic approximation, the known unstable modes. However, taking the full interaction, i.e., including the quartic terms, in this case, the now unstable modes could be summed up, such as the zero modes in the self-dual case. At the end, the one-loop result without the imaginary part was confirmed, including the case
, i.e., a pure gluomagnetic background. The relation of this result to [
53] was not discussed.