Vector-Field Domain Walls
aa r X i v : . [ h e p - t h ] A ug Vector-Field Domain Walls
J.L. Chkareuli a,b , Archil Kobakhidze a,c and Raymond R. Volkas ca E. Andronikashvili Institute of Physics, 0177 Tbilisi, Georgia b I. Chavchavadze State University, 0162 Tbilisi, Georgia c School of Physics, The University of Melbourne,Victoria 3010, Australia
E-mail: [email protected] , [email protected], [email protected]
Abstract
We argue that spontaneous Lorentz violation may generally lead to metastabledomain walls related to the simultaneous violation of some accompanying discretesymmetries. Remarkably, such domain wall solutions exist for space-like Lorentzviolation and do not exist for the time-like violation. Because a preferred spacedirection is spontaneously induced, these domain walls have no planar symmetryand produce a peculiar static gravitational field at small distances, while their long-distance gravity appears the same as for regular scalar-field walls. Some possibleapplications of vector-field domain walls are briefly discussed.
Gauge fields as vector Nambu-Goldstone bosons
Relativistic invariance and local gauge symmetries are cornerstones of our currentunderstanding of elementary particles and their interactions. Elementary particles asquanta of their corresponding fields are classified through irreducible representationsof the Poincar´e group, while the local gauge symmetries prescribe their dynamics.This theoretical picture is most successfully realized within the celebrated StandardModel of quarks and leptons and their fundamental strong, weak and electromagneticinteractions.Nevertheless, it is conceivable that local gauge symmetries and the associatedmasslessness of gauge bosons might have a completely different origin, being inessence dynamical rather than due to a fundamental principle. This point of view ispartially motivated by the peculiarities of local gauge symmetries themselves, which,unlike global symmetries, represent redundancies of the description of a theory ratherthan being “true” symmetries. In addition, the dynamical origin of massless par-ticle excitations is very well understood in terms of spontaneously broken globalsymmetries. Based on these observations, the origin of massless gauge fields as thevector Nambu-Goldstone bosons appearing due to spontaneous Lorentz invarianceviolation (SLIV) [1] has gained new impetus [2]-[7] in recent years. While the first models realizing the SLIV conjecture were based on the fourfermion interaction (where the photon was expected to appear as a fermion-antifermioncomposite state [1]), the simplest model for SLIV is in fact given by a conventionalQED type Lagrangian extended by an arbitrary vector field potential energy. Forthe minimal polynomial containing only the vector field bilinear and quadrilinearterms one comes to the Lagrangian L = − F ab F ab − λ (cid:0) A a A a − n v (cid:1) , (1)where n a ( a = 0 , , ,
3) is a properly-oriented unit Lorentz vector, n = n a n a = ±
1, while λ and v are, respectively, dimensionless and mass-squared dimensionalpositive parameters. This potential means in fact that the vector field A a develops aconstant background value h A a i = n a v and Lorentz symmetry SO (1 ,
3) breaks downto SO (3) or SO (1 ,
2) depending on whether n a is time-like ( n a >
0) or space-like( n a < A a ( x ) = n a ( v + φ ) + A a ( x ) , n a A a = 0 (2)one finds that the A a field components, which are orthogonal to the Lorentz violatingdirection n a , describe a massless vector Nambu-Goldstone boson, while the φ fieldcorresponds to a Higgs mode. Independently of the problem of the origin of local symmetries, Lorentz violation in itself hasattracted considerable attention as an interesting phenomenological possibility which may be probedin direct Lorentz non-invariant extensions of quantum electrodynamics (QED) and the StandardModel [8]-[10]. A a = n v (which appears virtuallyin the limit λ → ∞ from the potential (1)) was also intensively discussed in theliterature (see paper [7] and references therein). Actually, both of these modelsare equivalent in the infrared energy domain, where the Higgs mode is consideredinfinitely massive, and amount to gauge invariant QED taken in the axial gauge, aswas shown in tree [11] and one-loop [6] approximations. This axial gauge, n a A a = 0,singles out the pure Goldstonic modes in the vector field as per Eq. (2). We show here that systems described by Lagrangians that are extensions of type(1) may possess topologically stable domain wall configurations. Remarkably, suchdomain wall solutions exist for space-like Lorentz violation and do not exist fortime-like violation.One may at first suspect that the topological stability of such domain wall solu-tions may stem from the fact that the discrete Z symmetry A a ( x ) → − A a ( x ) of theLagrangian (1) lies outside the connected part of the SO (1 ,
3) Lorentz symmetrygroup and, when minimal couplings to matter fields ψ ( x ) are included, is nothingbut the charge conjugation invariance C : A a ( x ) → − A a ( x ) , ψ ( x ) → ψ c ( x ) . (3)The nonzero VEVs h A a i = ± n a v spontaneously break this symmetry. We maythus search for a vector-field domain wall solution which asymptotes to the two Z degenerate vacua, + vn a and − vn a , along some spatial direction.However, the model, as it stands in Eq.(1), does not yet have topologicallystable walls. The point is that the potential terms in the model (1) have an extraaccidental symmetry SO (1 , ′ which rotates vector fields like the Lorentz symmetrywhile leaving the space-time coordinates untransformed. This SO (1 , ′ is in fact ageneric internal symmetry of the potential in the model (1) or in any other QEDpolynomial extension. As a result, the vacuum state four-vector of type + vn a canbe transformed into the vector − vn a by a continuous SO (1 , ′ rotation, so thesystem does not possess disconnected vacua. For pure geometrical reasons, thisargument holds just for space-like domain wall solutions ( n = −
1) and would not We shall discuss what we mean by “ topologically stable” in this context in more detail below.But in short: it is well-known that SLIV theories do not have stable “vacua”, since the Hamilto-nians are not bounded from below. Since “vacua”, which are more accurately to be termed “localminima”, are used as the boundary conditions for domain wall solutions, those solutions can neverbe absolutely stable. Nevertheless, local minima will be metastable. Topological (meta-)stabilitythen means that the local minima used as boundary conditions for domain wall solutions form adisconnected manifold, just as in the usual case of absolutely-stable topological domain walls inscalar field theories. See also [12] for a discussion of topological defects in scalar field theories withexplicit Lorentz invariance violation.
2e applicable to time-like domain walls ( n = 1), if such solutions might appearthrough spontaneous Lorentz violation. However, as we show below, only the space-like domain wall solutions specifically appear in the model of type (1) so one mightthink that they all are unstable.To achieve stability, the model can be extended, as we shall see shortly, byintroducing a second vector field B a into the model (1). There naturally appears,apart from the discrete Z symmetry ( A a → − A a , B a → − B a ), also the interchangesymmetry A ↔ B in the properly arranged Lagrangian model of the A and B fields. We shall show that the spontaneous breakdown of this interchange discretesymmetry may provide stability for vector-field domain walls.We consider first the one-vector field case to illustrate some generic features ofvector-field domain walls, and then proceed to the two-vector field model. The general domain wall solution in flat spacetime can be searched for using thesimple ansatz, A a ( x ) = n a V ( n · x ) , (4)where n b is another constant unit four-vector, n · x = n b x b . The equation of motionfollowing from (1) then reads (cid:2) n n a − ( n · n ) n a (cid:3) V ′′ ( n · x ) − λn n a (cid:0) V ( n · x ) − v (cid:1) V ( n · x ) = 0 , (5)where primes denote the differentiation with respect to the argument of a func-tion. As one can immediately see multiplying the equation (5) by vector n a thatstatic domain wall solutions appear only for the pure space-like spontaneous Lorentzviolation with orthogonal n and n directions, n = n = − , n a n a = 0 . (6)Then the equation (5) for V reduces to the familiar equation for the scalar domainwall: V ′′ − λV (cid:0) V − v (cid:1) = 0 . (7)The solution for the above equation is well known: V ( n · x ) = v tanh [ m ( n · x )] , (8)where m = p λ/ v .Consider for definiteness a domain wall extending along the z -direction, i.e. n a =(0 , , , z = 0 with the Lorentz violating vacua taken in the y -direction, i.e. n a = (0 , , , A = v tanh ( mz ) , A = A = A = 0 . (9)Observe that due to the Lorentz non-invariant VEVs, this configuration does nothave an x − y planar symmetry, contrary to the case of the ordinary scalar field3omain wall [13], [14], [15]. This important difference is reflected in the energy-momentum tensor of our domain wall configuration (9): T ab ≡ − F ac F bc − λA a A b (cid:0) A c A c − n v (cid:1) + 14 η ab h F cd F cd + λ (cid:0) A c A c − n v (cid:1) i = m v cosh ( mz ) diag (cid:2) , − , − ( mz ) , (cid:3) . (10)Note the fact that the preferred Lorentz-violating direction in the plane of the vectordomain wall implies that T = T in contrast to a standard scalar domain wall,though its surface energy density, σ = Z + ∞−∞ T dz = 43 mv , (11)is the same as for a scalar wall, as directly follows from Eq.(10).However, as was argued above, this vector domain wall solution is unstable sincethe vacuum (0 , , + v,
0) in this model can be continuously changed to the vacuum(0 , , − v,
0) by a proper rotation in the x − y plane. So, one has to think about someextensions of the model among which the two-vector field system seems to be thesimplest possibility. The most general Lagrangian of two vector fields A and B possessing the independentdiscrete symmetries, A → − A, B → B,A → A, B → − B, (12)together with the interchange symmetry I , A ↔ B, (13)may be written in the following simple form, L ( A,B ) = − F A − F B − µ A + B ) − α (cid:0) A + B (cid:1) − β A B ) − γ AB ) , (14)in a self-evident notation for contractions of A and B fields and their field-strengthtensors ( A = A a A a , AB = A a B a , A = ( A a A a ) , F A = F Aab F abA etc.) and positivedimensionless coupling constants α , β and γ . The sign of the mass parameter, µ / L ( A,B ) also possessesan accidental symmetry SO (1 , ′ A × SO (1 , ′ B which is broken to a “diagonal”subgroup by the γ -term in Eq.(14). Also, when α = β , these two terms combineinto ( A + B ) which possess an internal SO (2) also exhibited by the mass andkinetic terms. But this continuous symmetry is explicitly broken by the γ -term, andalso if α = β . The non-existence of this SO (2) will be important for establishingdomain wall topological stability.One can readily confirm that this potential has the following extrema. The firstextremum has both the A and B fields identically equal to zero, but it is arrangedto be a local maximum, providing µ >
0. Second, there is the “symmetrical”extremum given by the following expectation values of the fields: h A a i = h B a i = n a s µ α + β + γ , (15)where n a is a spacelike unit vector, n a = −
1. If this were the vacuum, it wouldspontaneously break the discrete symmetries of Eq.(12) while leaving the interchangesymmetry I of Eq.(13) exact.There are also “asymmetric” extrema. One class is given by h A a i = n a v, h B a i = 0 with v ≡ p µ /α, (16)which is degenerate with the I -related configuration h A a i = 0 , h B a i = n a v. (17)A vacuum of this type breaks I and one of the Z symmetries of Eq.(12). The otherclass is given by h A a i = n a v, h B a i = n b v with v ≡ s µ α + β , (18)where n a = n b = − n a · n b = 0. It is degenerate with its I -symmetry correlate.Note that the vacua of Eqs.(15) and (18) are unsuitable for domain-wall stabilitysince all configurations of those types form connected manifolds. For example, thereare two-field analogues of the solution we derived in the one-vector field model. Onesuch configuration is A a ( x ) = n a v tanh[ m ( n · x )] , B a ( x ) = 0 (19)and another is its I -symmetry partner. However, for the parameter space regionwhere Eqs.(16) and (17) are the vacua, a qualitatively different wall solution exists,related to the spontaneous breaking of the interchange symmetry I rather than the Recall that by “vacuum” we really mean metastable local minimum. < α ≤ β + γ , α ≤ β. (20)All the other extrema discussed above are local maxima when the inequalities (20)hold. Note that in this range of parameters the potential in (14) is not boundedfrom below. The reason is that ( A B ) term is neither positive nor negative definite,and thus for 0 < α < β one always finds a direction in ( A, B ) space along whichthe potential runs to −∞ . However, it is unreasonable to insist on the boundness ofthe potential since, as mentioned earlier, the total Hamiltonian is known to be un-bounded [17], [18]. The corresponding instabilities are caused by sufficiently strongfluctuations of the Higgs modes of vector fields around a vacuum. For small (linear)perturbations (e.g., in the low energy regime of the theory), however, stability can bemaintained [18]. Thus we only demand that vacua are realized as local minima, andthey are stable under the linear perturbations. We will confirm shortly below thatour domain wall solutions are also stable perturbatively against linear perturbationsof the Higgs modes.A prototype for a domain wall solution has the form A a ( x ) = n a v (1 + tanh[ m ( n · x )]) / , B a ( x ) = n a v (1 − tanh[ m ( n · x )]) / I -degenerate vacua, h ( A a , B a ) i = ( vn a ,
0) and (0 , vn a ) (22)along some spatial direction n · x ∈ ( −∞ , + ∞ ). One can easily see that these vacuacannot be rotated to each other in principle. The only continuous symmetry thatmight do it is the SO (2) acting in the ( A, B ) space, but this symmetry is explicitlybroken in the Lagrangian, as discussed earlier.The prototype solution above is the simplest analytic configuration of this typeone can write down using the standard hyperbolic tangent function. In theoriesinvolving multiple fields, other kinds of kink-like functions can also be solutions,though they can usually only be obtained numerically. It is often the case thatthe analytic prototype corresponds to a particular relationship holding amongst theparameters in the potential, and if that relationship does not hold then wall solutionsstill exist but must be computed numerically.To establish that wall solutions of the type we want exist, it suffices to con-strain ourselves to the analytic prototype and show that it indeed can be a solution.Putting the ansatze (21) into the equations of motion of the vector fields A a ( x ) and B a ( x ) one finds that such a solution indeed exists, with m = p λ/ v = µ/ √
2, inthe parameter plane β + γ = 3 α (23)This relation is consistent with the potential stability conditions (20), providing γ ≤ β . As emphasized already, for other values of these constants a numericalintegration of the equations of motion is required.6onsider, again as in the one-vector field case, a domain wall extending alongthe z -direction, i.e. n a = (0 , , , z = 0 with the Lorentz violatingvacua taken both in the y -direction, i.e. n a = (0 , , , A ( z ) = v [1 + tanh( mz )] / , B ( z ) = v [1 − tanh( mz )]) / A = A = A = 0 and B = B = B = 0. Equally, they could be taken inthe orthogonal x - and y -directions, respectively. In any case, due to the preferredspace directions in these vacua, our domain wall configuration does not possess an x − y planar symmetry and has, as it can easily be confirmed , the following energy-momentum tensor T ab = m v ( mz ) diag (cid:2) , − , − ( mz ) , (cid:3) (25)being exactly half of the corresponding tensor in the one-vector field case (10).To conclude, we found vacuum configurations in the model where only one ofthe two vector fields A or B is condensed, thus producing three Goldstone modescollected into a photon-like multiplet, while its counterpart has a mass of order theLorentz violation scale and decouples from the low-energy physics. Actually, themass-squared of the non-condensed field being from the outset negative (tachyonic)becomes now positive. If, say, A is condensed (while B is not) the mass of the B -fieldexitations along n a direction is given by M B k = µ p ( β + γ ) /α − µ √ , (26)while the excitations orthogonal to n a have the mass, M B ⊥ = µ p β/α − . (27)So, on one side of the wall one has massless A photons and massive B bosons, whileon the other side obe has massive A bosons and massless B photons. In this sense,the model contains in fact only one massless vector field in all possible observationalmanifestations. Before discussing the gravitational properties of our two-field domain wall solutionswe would like to show that the wall solution is stable against linear perturbations inthe Higgs modes, φ A and φ B : A µ = n a ( V A + φ A ) + a a and B a = n µ ( V B + φ B ) + b a .Here V A = v (1 + tanh[ m ( n · x )]) / V B = v (1 − tanh[ m ( n · x )]) / V A and V B and keeping only the terms linear in φ A and φ B , we obtain the linearized equations of motion for the Higgs modes: (cid:2) (cid:3) + ( n · ∂ ) (cid:3) φ A = ( µ − α ( V A + V B )) φ A − αV A V B φ B , (28) (cid:2) (cid:3) + ( n · ∂ ) (cid:3) φ B = ( µ − α ( V A + V B )) φ B − αV A V B φ A , (29)7he stability can be easily established by looking at linear combinations: φ + = φ A + φ B and φ − = φ A − φ B . Namely, it can be straightforwardly obtained from(27) and (28) that the mode φ + is simply a massive ( m + = √ µ ) free field with the(Lorentz-violating) dispersion relation ω = 2 µ + ~k − ( ~n · ~k ) >
0. The solution forthe the mode φ − can be written as φ − = N − e − iω + ik x x + ik y y f ( z ). For f ( z ) we obtain, f ′′ ( z ) + (cid:2) − ω + k x + µ (1 + 3 tanh ( mz )) (cid:3) f ( z ) = 0 (30)This equation is essentially the same as the corresponding equation obtained in thecase of the the usual one-field scalar domain wall. Hence, the domain wall solutionsare perturbatively stable against linear Higgs mode perturbations, despite the factthat the total Hamiltonian is not bounded from below.A more generic analysis of the linear perturbations (including those of Goldstonicmodes a a and b a ) is a complicated problem and can not be handled analytically.However, we expect that the Goldstonic perturbations do not induce instabilities dueto the topological reasons. Indeed, due to the topological charge conservation, thedomain wall configuration (21) can ”decay” by emitting a a and b a quanta only into aconfiguration with the same topological charge and lower surface energy. However,it is not difficult to check that any domain wall configuration which interpolatesbetween I -degenerate vacua, ( n ′ a v,
0) and (0 , n ′ a v ), have the same energy density T (25). Therefore, the only source of the instability of domain walls is the metastabilityof the vacua (16,17), and this instability shows up at non-linear level. Thus we arguethat the wall solutions are indeed metastable, though we acknowledge that this hasnot been explicitly checked through linear stability analysis. Next we are interested in the gravitational properties of vector-field domain walls.Let us start by considering the weak gravity approximation, g µν ( x ) = η µν + h µν ( x ) . (31)(where the tensors now have “curved” indices µ, ν... rather than the previous “flat”indices a, b, ... ). In this approximation we can use the flat spacetime solution for thevector domain wall (24) and the associated energy-momentum tensor (25). In theharmonic gauge, ∂ µ ( h µν − / η µν h ) = 0, the Einstein equations are, h ′′ µν ( z ) = 2 M ( T µν − / η µν T ) , (32)8here T = T µµ is the trace of the energy-momentum tensor. The reflection ( z → − z )symmetric solution to the above equations is h = − h = − v M (cid:2) mx )) + cosh − ( mz ) (cid:3) ,h = v M (cid:2) ln (cosh( mz )) − cosh − ( mz ) (cid:3) (33) h = v M [ln (cosh( mz ))] . One can see a peculiar static gravitational field of the vector wall at small dis-tances. Interestingly, one of the h components, namely h , is vanishing at smalldistances mz ≪
1. At the same time all h components grow linearly with | z | awayfrom the wall’s center. Thus, strictly speaking, the weak field approximation is notvalid at large | z | , so one has to find a solution in the full nonlinear theory. As iswell known [14], [15], planar symmetric spacetime metrics (in the case of scalar-fielddomain walls) in Einstein gravity without a cosmological constant are necessarilytime-dependent. We find below that the same appears in our case as well despitethe planar symmetry being broken for vector-field domain walls.In principle, we have to simultaneously solve the Einstein equations and theequations of motion for the vector field in the spacetime given by an a priori unknownmetric. Analytic solutions are difficult, if not impossible, to obtain. To proceedfurther we will thus consider the thin wall approximation which is sufficient for ourpurpose because we are interested here in how the metric behaves away from thewall (where the above linear approximation is not valid). Indeed, the finite-thicknessdomain-wall solutions must approach thin-wall-limit solution in regions distant fromthe wall, where the“microscopic” details of the wall structure are not essential. Inthe thin-wall limit , m → ∞ such that mv = const . , the energy-momentum tensor(25) takes the form, T µν = σδ ( z )diag [1 , − , − , , (34)which coincides with the energy-momentum tensor for the scalar field domain wall.Therefore, we eventually come to the same solution[14], [15] for metric ds ≡ g µν dx µ dx ν = (1 − k | z | ) [ dt − e kt ( dx + dy ) − dz ] . (35)where k = 2 πG N σ. We would like to conclude this section with the following remark concerning thethin wall approximation, comparing a limiting procedure from polynomial theory(which causes Lorentz violation) to the case of standard QED. Consider for simplicitythe one-vector model given by the Lagrangian (1). As we have pointed out in theend of the Section 1, in the limit λ → ∞ such that v remains finite, which isdifferent from the thin-wall limit considered immediately above, the potential in (1)reduces to the gauge field constraint A µ = n v , and this model is in fact equivalentto a gauge invariant U (1) theory which may be identified with QED taken in the9onlinear gauge. This is also a type of thin-wall limit, since m → ∞ , but in thiscase the energy density σ diverges and thus the wall, even if it might be stablein this generic model, would be infinitely massive and decoupled from low-energyphysics. Therefore, standard QED, though it may be considered as a low-energyapproximation for a general polynomial vector field theory (1), is fundamentallyfree from domain-wall type solutions. The vector-field domain walls described above can be produced in the early universethrough the Kibble mechanism [19]. Ordinary (scalar field) domain walls evolvecosmologically as per ρ ∼ /a in the non-relativistic limit, where a is the cosmologicalscale factor. They therefore dominate the energy density of the universe from veryearly times unless the corresponding discrete symmetry breaking scale v is less than ∼
100 MeV. In fact, it must be less even than ∼ M eV if the anisotropy induced bythe walls on the cosmic microwave background radiation is to be below experimentallimits .The cosmological evolution of the vector-field domain walls seems to largelyfollow to the same scenario as the regular scalar field walls. However, there may besome difference as well. A full analysis of this problem is beyond the scope of thispaper, but we do wish to make some simple observations.One reason for a difference between scalar and vector domain walls is the vi-olation of Lorentz invariance on the domain wall surface. Heuristically, this canbe understood as follows. Because of Lorentz symmetry breaking on the surfaceof a vector-field wall [ SO (1 , → SO (1 , In the context of the inflationary scenario, the domain wall problem exists if the universe getsreheated enough to allow the production of domain walls after inflation, i.e. T reheat . > v . Otherwise,the domain walls are either inflated away from the visible universe, or they simply are not produced.
10e differetn from those for ordinary walls, and may be weaker. Detailed numericalmodelling of a network of vector-field domain walls would probably be necessary tocompute the precise bounds, including from the generation of microwave backgroundanisotropy.
We have argued that spontaneous Lorentz-invariance violation leads to domain wallsolutions related to the simultaneous violation of the accompanying discrete symme-tries, which may be charge-conjugation and/or interchange symmetry. Dependingon the specific model, such a configuration may be unstable or metastable.As was illustrated by the example of the simple one-vector field model, suchdomain wall solutions can exist for space-like Lorentz violation and do not exist fortime-like violations. Though the one-vector field model leads to an unstable wall,the two-vector field extension seems to also be of physical interest. There naturallyappears, apart from the discrete Z symmetries ( A a → − A a , B a → − B a ), the in-terchange symmetry A ↔ B in an appropriate Lagrangian model of the A and B fields. This additional discrete symmetry provides a mechanism to achieve metasta-bility for vector-field domain walls. We found a metastable vacuum configurationin the model where only one of the two vector fields condenses on each side of thewall, thus producing Goldstone modes collected into a photon-like multiplet while itscounterpart acquires the Lorentz violation scale order mass and decouples from thelow-energy physics. Because a preferred space direction is spontaneously induced,these domain walls have no planar symmetry and produce a peculiar static gravi-tational field at small distances, while their long-distance gravity appears the sameas for regular scalar-field walls. Cosmological bounds on these domain walls maybe weaker than for the usual scalar walls, but a more detailed analysis is requiredbefore this can be quantified.The vector-field domain walls are especially interesting if this QED type model isfurther extended to the Standard Model (where the Lorentz-violating vector field isthen taken to be coupled to the hypercharge current rather than the electromagneticone), and also to grand unified theories. Because the discrete symmetry that isspontaneously broken may be the charge conjugation, one application could be tobaryogenesis, since particles and antiparticles are expected to behave differently dueto their interactions with a wall. Another application might concern the extensionof the model to higher dimensions and the possibility of trapping of gauge fields(both Goldstonic and non-Goldstonic) to a 4-dimensional vector-field domain wallappearing in the higher dimensional bulk. We may return to these interesting pointselsewhere. Acknowledgments
One of us (J.L.C.) appreciates the warm hospitality shownto him during his visit to the Theoretical Particle Physics Group at the Universityof Melbourne where part of this work was carried out. The visiting scholar award11eceived from the University of Melbourne is also gratefully acknowledged. The workof A.K. and R.R.V. was supported in part by the Australian Research Council. Wethank Alexander Vilenkin for interesting comments.
References [1] J. D. Bjorken, “A Dynamical Origin For The Electromagnetic Field,” AnnalsPhys. (1963) 174.[2] J. L. Chkareuli, C. D. Froggatt and H. B. Nielsen, “Lorentz invariance and originof symmetries,” Phys. Rev. Lett. (2001) 091601 [arXiv:hep-ph/0106036].[3] P. Kraus and E. T. Tomboulis, “Photons and gravitons as Goldstonebosons, and the cosmological constant,” Phys. Rev. D (2002) 045015[arXiv:hep-th/0203221].[4] A. Jenkins, “Spontaneous breaking of Lorentz invariance,” Phys. Rev. D (2004) 105007 [arXiv:hep-th/0311127].[5] S R. Bluhm and V. A. Kostelecky, “Spontaneous Lorentz violation,Nambu-Goldstone modes, and gravity,” Phys. Rev. D (2005) 065008[arXiv:hep-th/0412320].[6] A. T. Azatov and J. L. Chkareuli, “Nonlinear QED and physical Lorentz in-variance,” Phys. Rev. D (2006) 065026 [arXiv:hep-th/0511178].[7] J. L. Chkareuli, C. D. Froggatt, J. G. Jejelava and H. B. Nielsen, “ConstrainedGauge Fields from Spontaneous Lorentz Violation,” Nucl. Phys. B (2008)211 [arXiv:0710.3479 [hep-th]].[8] S. M. Carroll, G. B. Field and R. Jackiw, “Limits on a Lorentz and ParityViolating Modification of Electrodynamics,” Phys. Rev. D (1990) 1231.[9] D. Colladay and V. A. Kostelecky, “Lorentz-violating extension of the standardmodel,” Phys. Rev. D , 116002 (1998) [arXiv:hep-ph/9809521].[10] S. R. Coleman and S. L. Glashow, “High-Energy Tests of Lorentz Invariance,”Phys. Rev. D , 116008 (1999) [arXiv:hep-ph/9812418].[11] Y. Nambu, “Quantum electrodynamics in nonlinear gauge,” Progr. Theor.Phys. Suppl. Extra (1968) 190.[12] M. N. Barreto, D. Bazeia and R. Menezes, “Defect Structures in Lorentz andCPT Violating Scenarios,” Phys. Rev. D , 065015 (2006)[13] A. Vilenkin, “Gravitational Field Of Vacuum Domain Walls And Strings,”Phys. Rev. D (1981) 852. 1214] A. Vilenkin, “Gravitational Field Of Vacuum Domain Walls,” Phys. Lett. B (1983) 177.[15] J. Ipser and P. Sikivie, “The Gravitationally Repulsive Domain Wall,” Phys.Rev. D (1984) 712.[16] S. R. Coleman, “The Fate Of The False Vacuum. 1. Semiclassical Theory,”Phys. Rev. D (1977) 2929 [Erratum-ibid. D (1977) 1248].[17] R. Bluhm, N. L. Gagne, R. Potting and A. Vrublevskis, “Constraints and Sta-bility in Vector Theories with Spontaneous Lorentz Violation,” Phys. Rev. D (2008) 125007 [Erratum-ibid. D (2009) 029902] [arXiv:0802.4071 [hep-th]].[18] S. M. Carroll, T. R. Dulaney, M. I. Gresham and H. Tam, “Instabilities in theAether,” Phys. Rev. D (2009) 065011 [arXiv:0812.1049 [hep-th]].[19] T. W. B. Kibble, “Topology Of Cosmic Domains And Strings,” J. Phys. A9