Higgs effect without lunch

Reduction in effective space–time dimensionality can occur in field-theory models more general than the widely studied dimensional reductions based on technically consistent truncations. Situations where wave function factors depend non-trivially on coordinates transverse to the effective lower dimension can give rise to unusual patterns of gauge symmetry breaking. Leading-order gauge modes can be left massless, but naturally occurring Stueckelberg modes can couple importantly at quartic order and higher, thus generating a ‘covert’ pattern of gauge symmetry breaking. Such a situation is illustrated in a five-dimensional model of scalar electrodynamics in which one spatial dimension is taken to be an interval with Dirichlet/Robin boundary conditions on opposing ends. The Stueckelberg mode remains in the theory as a propagating scalar degree of freedom from a dimensionally reduced perspective, but it is not ‘eaten’ in a mass-generating mechanism. At leading order, it also makes no contribution to the conserved energy; for this reason, it may be called a (non-ghost) ‘phantom’. This simple model illuminates a mechanism which also has been found in gravitational braneworld scenarios. This article is part of the theme issue ‘The future of mathematical cosmology, Volume 2’.

Reduction in effective space-time dimensionality can occur in field-theory models more general than the widely studied dimensional reductions based on technically consistent truncations. Situations where wave function factors depend non-trivially on coordinates transverse to the effective lower dimension can give rise to unusual patterns of gauge symmetry breaking. Leading-order gauge modes can be left massless, but naturally occurring Stueckelberg modes can couple importantly at quartic order and higher, thus generating a 'covert' pattern of gauge symmetry breaking. Such a situation is illustrated in a five-dimensional model of scalar electrodynamics in which one spatial dimension is taken to be an interval with Dirichlet/Robin boundary conditions on opposing ends. The Stueckelberg mode remains in the theory as a propagating scalar degree of freedom from a dimensionally reduced perspective, but it is not 'eaten' in a mass-generating mechanism. At leading order, it also makes no contribution to the conserved energy; for this reason, it may be called a (nonghost) 'phantom'. This simple model illuminates a mechanism which also has been found in gravitational braneworld scenarios.
This article is part of the theme issue 'The future of mathematical cosmology, Volume 2'.

Vacuum structure and the Higgs mechanism
This article is about some of the unusual symmetrybreaking effects that can occur in systems whose lowenergy effective dynamics is best described in a lowered space-time dimensionality.
The phenomenon of symmetry breaking triggered by a non-symmetric vacuum in a theory with a continuous the D = 4 effective theory has the massless spectrum of a supergravity theory, beginning in the action at quadratic order with a Fierz-Pauli spin-two Lagrangian, the interaction coefficients at fourth order and higher in fields turn out not to have the values one expects for a generally covariant theory in an order-by-order field expansion [13,14].
This 'funny numbers' phenomenon is the defining feature of covert symmetry breaking in an effective theory where the lower-dimensional gauge or gravitational fields remain otherwise massless: a Higgs effect without lunch. A full exploration of how this works in the Type IIA model of [9,10,12] becomes quite involved. However, simpler toy models of the phenomenon can be constructed by declaring non-standard boundary conditions for fields in the transverse space dimensions. This was done for a simple illustrative Maxwell-plus-scalar model in [15], which shows the essential features of such a covert symmetry breaking scenario, and to which we next turn. In reviewing [15], we will expand on and clarify a key issue surrounding the number of propagating degrees of freedom in the effective theory. In particular, we will shine a light on the existence of a 'phantom'-a non-ghostly scalar that only contributes to the energy at the interacting level. Instead of the technically rather involved Type IIA supergravity embedding of the Salam-Sezgin model, one can create an illustrative dimensional reduction model in which fluctuation fields need to have a non-trivial dependence on an extra 'transverse' coordinate. Since this will not generate a consistent reduction in the technical sense, the dimensional reduction will need to be understood in the sense of a low-energy effective theory where lower-dimensional dynamics is dominant. Start with Maxwell theory in a five-dimensional space-time with structure in which the intention will be to choose non-standard boundary conditions on fields at the z = 1 end of the I = [0, 1] interval. The D = 5 space-time metric is denoting the reduction coordinate by z and the d = 4 reduced space-time coordinates by x μ . The Maxwell gauge fields A μ (x, z), A z (x, z) will satisfy standard D = 5 equations of motion and the A μ components will be assigned standard Dirichlet boundary conditions at the z = 0 end of the I = [0, 1] interval, but at the z = 1 end they will be assigned Robin boundary conditions. The Dirichlet/Robin boundary conditions chosen for A μ are accordingly For the A z component of the D = 5 gauge field, it will prove to be necessary for its transversespace derivative ∂ z A z to satisfy the boundary conditions (2.3). In the d = 4 Minkowski subspace directions, all fields and their associated derivatives will be required to fall off as usual at spatial infinity. The corresponding D = 5 Maxwell action also needs to include a boundary term at the z = 1 end and where 4 = ∂ μ ∂ μ . Including the boundary term S BT in the action (2.4) is necessary in order to allow the Robin condition for the field A μ to be incorporated into a well-posed variational problem generating the field equations (2.6) and (2.7).
In order for the boundary conditions (2.3) on A μ to be gauge invariant, one requires the following restrictions on the form of the local gauge parameter Λ: where c 1 and c 2 are constants. In order for the gauge parameter to vanish at asymptotic infinity, we require c 1 = c 2 = 0. In addition to the local gauge symmetry (2.5), the Dirichlet/Robin boundary conditions (2.3) for A μ allow an additional but restricted 'harmonic' gauge symmetry where we will again take the constants c 3 and c 4 to vanish.

(b) Mode expansions
Now let us expand the D = 5 gauge fields in a mode expansion so that we can derive an effective theory for the leading-order d = 4 modes. For A μ satisfying the boundary conditions (2.3), one needs an expansion basis satisfying the same boundary conditions. In order to solve equation (2.6) by separation of variables, one requires a complete set of basis functions ξ , = 0, 1, . . . , ∞, satisfying the transverse wave function problem where ξ (z) = ∂ z ξ (z). The L 2 normalized solutions to this transverse problem are and tan ω = ω , ω > 0. (2.14) When it comes to A z , the situation is somewhat different. No specific boundary conditions are required in deriving (2.7) from the variation of the action (2.4) because the only term containing δA z on the boundaries of I already vanishes when the equations of motion are satisfied. One learns the behaviour of A z on the boundaries instead directly from compatibility with equation (2.6). Substituting the expansion (2.11) into (2.6), one obtains where ω 0 = 0. Accordingly, one sees that ∂ z A z must lie within the span of the ξ (z) basis, i.e.
Integrating this equation and noting that for > 0 the indefinite integral of ξ (z) is proportional to its derivative, one has the expansion (c) Leading-order d = 4 effective field theory: no lunch Using the mode expansions (2.11) and (2.17) in the D = 5 field equations (2.6) and (2.7), one obtains an equivalent formulation of the theory in terms of d = 4 expansion fields. Our main interest here will be the structure of the leading-order theory. This may be done either by working directly with the D = 5 action (2.4) and using orthonormality properties of §2(b) mode expansions to separate the leading-order terms or by making the mode expansions in the D = 5 field equations (2.6) and (2.7), separating out the leading-order modes and then reconstructing the leading effective action. Either way, one obtains simply where f μν = ∂ μ a ν − ∂ ν a μ and a μ (x) and g(x) are just the = 0 modes in the expansions (2.11) and (2.17) with the = 0 index dropped. The higher ≥ 1 modes excluded from the leading-order action (2.18) are all massive, as in an ordinary Kaluza-Klein expansion. Thus, one expects (2.18) to accurately describe the leading gauge-field dynamics of the system at energies well below the mass of the lightest ( = 1) such higher mode, even when interactions with sources are introduced. What does the action (2.18) describe? A key point to note is that expanding in terms of the ξ i (z) and ζ (z) transverse wave function basis has not produced a leading-order d = 4 system with a massive vector field. So this reduction has not generated a standard Higgs mechanism-the leading-order vector field a μ (x) = a 0 μ (x) remains massless. So there is no lunch-feast going on here. The next question is: what are the degrees of freedom described by the effective action (2.18)?
The answer involves the U(1) gauge symmetry (2.5) and the harmonic symmetry (2.9). For the leading-order modes, the U(1) gauge transformations are, expanding Λ(x, z) also in the ξ i (z) basis with λ(x) = λ 0 (x), Note the role of the g(x) field in (2.18): it accompanies the otherwise non-gauge-invariant ∂ μ ha μ term to make the U(1) gauge-invariant combination ∂ μ h(∂ μ g − a μ ). This is a place where reduction with a non-standard transverse wave function zero mode has had an effect: the g(x) field is behaving like a Stueckelberg field [16 -18] repairing an otherwise non-gauge-invariant term. Unlike the original Stueckelberg implementation of such a field in a vector field mass term, however, the g(x) field in (2.18) does not participate in the generation of an effective-theory mass. To analyse the dynamical radiative degrees of freedom of the system, we employ Fourier analysis on the source-free field equations following from (2.18): Fourier transforming the fields and gives the momentum-space equations and condition at spatial infinity, a 0 → 0 as |x i | → ∞, then implies a 0 = 0. Consequently, there remain only the standard two dynamical degrees of freedom in the a μ Maxwell field. The momentumspace Lorentz condition is also obtained: p μ a μ = −p o a 0 + p i a i = 0.
The remaining degree of freedom for the system (2.18) is then seen in equation (2.25): p 2 g(p) = 0. Thus, in Coulomb gauge, there is an additional candidate scalar degree of freedom residing in the Stueckelberg field. This would be entirely expected in a standard Kaluza-Klein reduction where dependence on the z reduction coordinate is simply suppressed, in which case the A z component of the D = 5 Maxwell field becomes a massless scalar in d = 4. In the present Dirichlet/Robin system, however, the remaining g field has a more tenuous existence. As we have seen in (2.9), there is one more transformation which we have not yet exploited: D = 5 pure Maxwell theory with Dirichlet/Robin boundary conditions on the transverse interval I also has the harmonic symmetry. By combining the harmonic transformation (2.9) with a U(1) gauge transformation (2.5), one can trade the δa μ = ∂γ harmonic symmetry transformation of a μ for a transformation δg = −γ , leaving a μ unchanged. The harmonic condition on γ corresponds precisely to the support set of g, so one can use the reformulated harmonic transformation to send g → 0.
One can summarize the degree-of-freedom count for the reduced effective-theory system described by the action (2.18) as follows: there is no massive lunch, and even the remaining scalar g has a kind of 'phantom' (but not ghost!) existence-barely a crumb on the side. Only the two d = 4 Maxwell degrees of freedom are unambiguously dynamical. We will investigate this structure further by looking at the Hamiltonian, in order to see what impact the presence of g would have on the conserved energy.

(d) Effective-theory energy
From the leading-order effective action (2.18), one passes to a canonical form of the action in the standard way, defining a canonical momentum π a (x) = δI/δψ a conjugate to each field ψ a in (2.18). This yields and π g = −ḣ, (2.38) whereψ = ∂ψ/∂t. The corresponding canonical action has the Hamiltonian density One may verify that Hamilton's equations following from variation of the canonical action (2.39) are a first-order system that is fully equivalent to the Lagrangian field equations (2.20)-(2.22). The system's total energy is the value of E = d 3 xH for solutions to the field equations, as discussed above in §2(c). From the Coulomb-gauge analysis there, one has h = constant, so from (2.38), one also has π g = 0. Moreover, the field a 0 has no conjugate momentum and its variation yields the constraint ∂ i π i = 0 just as in standard d = 4 Maxwell theory. Consequently, for solutions to the effective-theory field equations arising from the action (2.18), one has the conserved total energy which is simply equal to the standard, positive semidefinite, energy of Maxwell theory alone. Although allowed by the field equations, the g field does not contribute to the total energy. Clearly, the fact that g may be removed by the residual harmonic symmetry is the reason for its nonappearance in the total energy (2.41). At least in the source-free theory we have considered so far, the g field may be considered redundant.

Covert symmetry breaking (a) Scalar electrodynamics
In §2, we have seen that the unambiguous leading-order dynamics of the D = 5 system (2.4) is just d = 4 Maxwell theory. The non-standard Dirichlet/Robin boundary conditions on the interval I do give rise to the h and g fields in the leading-order effective action (2.18), but the analysis of §2 shows that h has no dynamics while g may be considered redundant, giving no contribution to the total energy. Following [15], we now extend the model by U(1) gauge-covariantly coupling the system (2.4) also to a D = 5 complex scalar field Φ and where e is the charge of the complex scalar field Φ. For simplicity, one may take the D = 5 scalar field Φ to satisfy Dirichlet/Dirichlet boundary conditions on the interval I. One may expand Φ in a transverse basis appropriate to the Dirichlet/Dirichlet boundary conditions, where {θ n (z) = √ 2 sin(m n z)} with n = 1, 2, . . . and m n = nπ . Under the U(1) gauge symmetry (2.5), the φ (n) complex scalar modes transform in a way that mixes the various n levels where the matrix (I i ) nm is One may expand the system (3.1) into d = 4 modes either by substituting the expansions of A μ , A z and Φ into the higher-dimensional equations of motion or by inserting these expansions into the higher-dimensional action in order to obtain an action for the d = 4 formulation of the system. The two procedures give equivalent results. Keeping all expansion modes for A μ , A z and Φ, the U(1) gauge symmetry is maintained to all orders, albeit in a rather complicated way. Now consider just the leading-order effective theory, keeping just the = 0 massless gauge sector as described in §2 and the lightest n = 1 complex scalar mode. The main point of this illustrative D = 5 scalar electrodynamics model now comes into focus: it is simple enough that the various mode integrals governing expansions beyond the free theory (i.e. for action terms cubic and higher in fields) can straightforwardly be done. In addition to the matrix I nm given in (3.5) Noting that terms Φ∂ μ Φ and ΦΦA μ obey Dirichlet/Robin conditions, and thus can be expanded in the {ξ (z)} basis, one needs the product rules θ n (z)θ m (z) = I nm ξ (z) and θ n (z)θ m (z)ξ k (z) = I nm k ξ (z). (3.6) The covariant derivative operator for the scalar field is then Keeping just the leading = 0 gauge modes and the n = 1 modes of the scalar, writing simply φ = φ 1 and keeping terms up to cubic order in the interacting theory, one has where I 11 0 = √ 3/2. One can accordingly identify the d = 4 effective-theory charge for the leading gauge-scalar coupling to be q Eff = √ 3/2e in terms of the charge e of the D = 5 theory.

(b) An unanticipated seagull coefficient
Continuing on to quartic order in fields, one encounters a key peculiarity of this scalar electrodynamics construction with non-standard boundary conditions. As always in a model with a U(1) gauge field coupled to a complex scalar field, one expects to have d 4 xa μ a μ φφ 'seagull' terms occurring at quartic order. 1 From the effective-theory charge q Eff identified at cubic order, one would expect the gauge-field-coupled scalar kinetic term to become − d 4 x(D μ φ)D μ φ where is the usual gauge covariant derivative. Were that the case, the quartic-order d 4 xa μ a μ φφ seagull term coefficient would be just q 2 Eff . That is not what arises from the transverse-space integrals, however. Instead, what one finds involves I 11 00 = 1 − (3/2π 2 ) = (I 11 0 ) 2 . This is the phenomenon of covert symmetry breaking: no mass generation occurs as in a standard Higgs mechanism, but symmetry breaking in higher couplings, starting at quartic order in fields, does.
Of course, if one keeps all gauge and scalar modes, the full D = 5 theory is retained and no symmetry breaking occurs. But this D = 5 U(1) symmetry mixes the various gauge and scalar field mode levels in a complicated way. Covert symmetry breaking is intrinsically a phenomenon related to a low-energy approximation, in which the massive higher modes are not independently excited, but are integrated out using the leading-order results of their field equations.
That such a phenomenon can occur starting at quartic order may be expected on general grounds, as well as the fact that when it does occur, it cannot be fully remedied by field redefinitions [8]. What can be altered by field redefinitions, however, is the presentation of the phenomenon. The leading-order effective theory is obtained by integrating out all higher massive modes. For the gauge-field sector, that process is unambiguous when one restricts attention to the leading = 0 modes. For the complex scalar, however, some rearrangement is possible. Integrating out massive modes at leading order basically involves discarding derivatives of such a mode in its field equation, but keeping the mass term in what then becomes an algebraic equation at leading order, solving for the mode in question. Field redefinitions involving the Stueckelberg modes g ( ) change the way the D = 5 U(1) symmetry acts on the various modes. If one defines [15] ϕ (n) = exp(ieg ( ) I nn ) exp(−ieg (k) I k ) nm φ (m) , (3.10) then instead of the D = 5 U(1) transformations mixing between n modes of the scalar field, the ϕ (n) transform diagonally and canonically, ϕ (n) → exp(ieλ ( ) I nn )ϕ (n) , (3.11) noting that exp(ieλ ( ) I nn ) is a phase and not a matrix.
Integrating out the n > 1 massive modes and restricting to the = 0 gauge modes and the ϕ = ϕ (1) scalar field, the result through cubic terms remains the same as in (3.8), but when keeping terms up to quartic order, one has where D μ is the usual covariant derivative (3.9) and while X is a more complicated but explicitly calculable positive coefficient. In the effective-action form (3.12), one sees both the preservation of the U(1) symmetry by Stueckelberg fields but also its breaking. The top line of (3.12) has the normal d = 4 gauge-covariant coupling of a μ to the complex scalar field ϕ, while the second line shows the difference seagull vertex (a μ − ∂ μ g)(a μ − ∂ μ g)ϕϕ with a coefficient involving theĨ combination which gives the deviation from the standard seagull coefficient. Since this structure involves the U(1) gauge invariant combination (a μ − ∂ μ g), it makes the preservation of the D = 5 U(1) symmetry manifest, but since this is only achieved via the presence of the Stueckelberg field g, it also makes the covert symmetry breaking phenomenon manifest.
The alternative between the original φ (n) expansion and the transformed ϕ (n) expansion may be likened to the alternative between the gauge basis and the mass basis in the CKM mechanism [19,20] in the Standard Model. In the φ (n) basis, there are no terms directly mixing the a ( ) μ gauge modes and the g ( ) Stueckelberg modes, but the transformations of the φ (n) scalar modes mix between different n levels. In the ϕ (n) basis, on the other hand, the scalar modes transform diagonally and canonically but then there are terms involving products of the a ( ) μ and the g ( ) Stueckelberg modes together with scalar modes, as one sees in (3.12).

Outlook
The key issues that we have considered in this article concern the low-energy effective-theory dynamics of a higher-dimensional theory which has a natural effective-theory interpretation in a lower dimension, but in which transverse wave function modes have a non-trivial dependence on a transverse coordinate. Such effective reductions can occur in a wide variety of situations, including the Type IIA supergravity lift of the Salam-Sezgin model as considered in [9] as well as in the simpler Dirichlet/Robin scalar electrodynamics considered here and in [15]. A take-home message is that there are interesting wider varieties of symmetry breaking mechanisms than the standard BEHGHK mechanism. Moreover, reductions in effective space-time dimensionality can occur without requiring technically consistent dimensional reductions.
The D = 5 Dirichlet/Robin scalar electrodynamics model illustrated here has leading-order field dynamics in which massless Maxwell theory is unambiguously part of the d = 4 effective theory, but there is also the g scalar mode, which, however, makes no contribution to the conserved energy at leading order and may be considered redundant at that level. The full story of that mode in the interacting theory remains to be clarified, as well as the roles of analogous modes in the IIA supergravity/Salam-Sezgin gravitational model. Another question going beyond what has been considered here is what happens when the Dirichlet/Robin scalar electrodynamics system generates a localized source in five dimensions for the gauge sector. One may expect a transition between near-field D = 5 behaviour near the source and far-field d = 4 behaviour away from it. Such a near-field/far-field transition is found when a delta-function source localized in the higher dimension is coupled to the Type IIA/Salam-Sezgin model [21]; one might expect similar behaviour in the D = 5 scalar electrodynamics case.
Models that come close to gauge-symmetric models but which display gauge-symmetry breaking without generating masses for the leading-order gauge modes can have subleading corrections which could be of interest from a variety of different perspectives. One such might be whether these corrections can serve as hints of hitherto unrecognized higher-dimensional space-time structure.
All authors gave final approval for publication and agreed to be held accountable for the work performed therein.
Conflict of interest declaration. We declare we have no competing interests.