Born-Infeld Axion-Dilaton Electrodynamics and Electromagnetic Confinement

A generalization of Born-Infeld non-linear vacuum electrodynamics involving axion and dilaton fields is constructed with couplings dictated by electromagnetic duality and SL(2,R) symmetries in the weak field limit. Besides the Newtonian gravitational constant the model contains a single fundamental coupling parameter b0. In the absence of axion and dilaton interactions it reduces, in the limit as b0 tends to infinity, to Maxwell's linear vacuum theory while for finite b0 it reduces to the original Born-Infeld model. The spherically symmetric static sector of the theory is explored in a background Minkowski spacetime where numerical evidence suggests the existence of axion-dilaton bound states possessing confined electric flux.


Introduction
The existence of new forms of matter that interact only with gravitation has been recently advocated in order to account for a number of puzzles in modern cosmology. However the experimental detection of such states remains elusive. Unified models of the basic interactions also predict a large class of undetected states that may induce experimental signatures predicted by low energy effective string models. Phenomenological models of the strong interactions (QCD) also demand "axionic" states to ameliorate anomalies in the presence of the observed lepton families and account for the observed imbalance of matter over anti-matter [1][2][3]. Furthermore the simplest generalization of Einsteinean gravitation involves a gravitational scalar field that modifies certain predictions of Einstein's theory [4,5]. Perhaps the coupling of hypothetical axions and dilaton scalar fields to the electromagnetic field offers the most promising mechanism leading to their experimental detection [6]. It is therefore worth analyzing new effective field theories involving such interactions [7]. Although a number of traditional searches for axion particles are based on natural modifications to the linear Maxwell theory in vacuo, this may be a weak-field approximation to a more general non-linear vacuum electrodynamics. Indeed, in the absence of axions and dilatons, such a theory was first formulated by Born and Infeld [8] in 1934. This theory has acquired a modern impetus from the observation that it emerges naturally in certain string-inspired quantum field theories [9] and it is perhaps unique among a large class of non-linear electrodynamic models in its causal properties in background spacetimes [10,11]. String theories also naturally include candidates for axion and dilaton states that at the Planck scale have prescribed couplings among themselves and the Maxwell field. In low-energy effective string models these couplings give rise to particular symmetries in the weakfield limit. Such models have been extensively studied by Gibbons et al [12][13][14] with particular reference to the preservation of linear realizations of SL(2, R) symmetry [13] and non-linear realizations of electromagnetic duality in the context of Born-Infeld electrodynamics with a dilaton [14]. In this letter we report on a new model that naturally incorporates both axion and dilaton fields in the context of Born-Infeld vacuum non-linear electrodynamics.

Axion-dilaton Born-Infeld Electrodynamics
If {R a b } denotes the curvature 2-forms of the Levi-Civita connection, g = η ab e a ⊗ e b the spacetime metric with η ab = diag{−1, 1, 1, 1} and {e a } a local g-orthonormal co-frame, F the Maxwell 2-form, ϕ the dilaton scalar and A the axion scalar, the model that arises from string theory in a weak-field limit [13] follows by a variation of the action S[g, A, ϕ, A] = M Λ 0 where the 4-form Λ 0 on spacetime M is with F = d A and * is the Hodge map associated with g. In (1) the constants are in terms of the Planck length the Newtonian gravitational constant G N and the permittivity of free space ǫ 0 1 . The term involving p 4 in (1) denotes the traditional coupling of the axion field to the electromagnetic field while the term involving p 5 is a natural dilaton coupling. One of the original aims given by Born and Infeld in generalizing the vacuum Maxwell theory was to construct a theory possessing bounded spherically symmetric static electric fields. Their theory invoked a new fundamental constant b 0 with the physical dimensions of an electric field strength. They demonstrated that their field equations admitted such solutions. Furthermore, by assuming that the finite mass of such an electromagnetic field configuration could be identified with the electron Born and Infeld were able to estimate the magnitude of b 0 . While such an argument is suspect in the context of subsequent developments, the idea of ameliorating the Coulomb singularity in the electric field using a non-linear electromagnetic self-coupling remains attractive.
The generalization considered here also reduces to the model defined by (1) in a weak-field limit. Furthermore in the absence of axion and dilaton contributions it reduces in the limit b 0 → ∞ to Maxwell's linear vacuum theory while for finite b 0 it reduces to the original Born-Infeld model. It involves the Newtonian gravitational coupling constant G N in addition to b 0 and a parameter τ = ±1 and is obtained by varying the action: with The structure of the argument of the square root in (8) follows from the (non-trivial) identity where exp(ϕ) = α 2 − β 2 , A = −2αβ, F = 1 2 F ab e a ∧ e b and * F = 1 2F ab e a ∧ e b . The 4-form Λ τ in (7) is expressed in the so-called Einstein "frame". However, it turns out that the importance of the dilaton and axion in our model is more readily exposed by moving to the Jordan "frame" by making the Weyl transformation g → g = ψ −2 g where ψ = exp(ϕ). After the field equations have been obtained in the following by varying Λ τ , we will choose g to be a flat background metric thereby endowing the metric g in the Einstein frame with non-zero curvature where the dilaton is non-constant.
Introducing e a = ψ −1 e a where {e a } is a g-ortho-normal co-frame, it follows that with ⋆ the Hodge map associated with g, {R a b } the curvature 2-forms of the Levi-Civita connection of g, In the following attention is restricted to the case where τ = 1.
The non-linear vacuum Maxwell equations follow as d F = 0 (since F = d A) and (by varying A) and and from ψ variations Using g-ortho-normal co-frame variations one obtains the gravitational field equations in the Jordan frame where Solving the full field systems in the Einstein and Jordan frames should lead to the same conclusions (up to identification of the spacetime metric) but finding exact solutions to such systems is non-trivial. A natural approximation is to neglect the couplings of dynamic gravitation to the other fields by neglecting the gravitational field equations. However the neglect of these equations in the Einstein frame will, in general, lead to solutions for the axion, dilaton and electromagnetic fields in a g-flat background with different behaviours from those calculated in a g-flat background in the Jordan frame.
To probe the consequences of adopting a g-flat background metric in the presence of the explicit coupling of the dilaton ψ to the curvature R a b in (15) we develop two approximation schemes in the Jordan frame. The first approach neglects (16) from the outset with g set to a background g-flat metric. The second approach employs (16) to express the Ricci scalar in terms of the dilaton field. The result is used to eliminate R ab ∧ ⋆(e a ∧ e b ) from (15), yielding We then set g to the background g-flat metric (thereby ignoring the gravitational field equation (16) for the remainder of the analysis). In the following, we refer to the field system (12), (13), (14), (15) as System 1 and the field system (12), (13), (14), (18) as System 2.
In the static spherically symmetric sector an ortho-normal co-frame field for a background metric g is e 0 = c d t, e 1 = d r, e 2 = r d θ, e 3 = r sin θ d φ in spherical polar coordinates. In terms of the dimensionless radial coordinate ρ = r/L we write: with ψ = ψ(ρ), A = A(ρ). Equations (14), (12) reduce to for some integration constant Γ 0 . The dimensionless constantλ is defined bŷ and The ODE for ψ in System 1 follows directly from adopting a g-flat metric in (15), whereas (18) yields: as the ODE for ψ in System 2 If S 2 is any 2-sphere of radius ρL centered at ρ = 0 the electric flux of any state crossing the surface of this sphere is 4πq = S 2 ⋆G. The value of q will be interpreted as the total electric charge within this sphere. Hence for the above spherically symmetric static field configuration determined by any constant Γ 0 such charge is q = 2ǫ 0 b 0 L 2 Γ 0 .
Clearly, analytic solutions to Systems 1 and 2 are unlikely; both are however amenable to numerical analysis. The simplest approach is to differentiate (21) with respect to ρ and treat each coupled system as an initial value problem specified by a choice of ψ(ρ 0 ), The initial field conditions should be consistent with a real Γ 0 . Starting from ρ = ρ 0 each system can be readily integrated numerically to the regions ρ > ρ 0 and ρ < ρ 0 and the solution monitored to check that Γ 0 remains constant. Solutions possessing the elementary charge q = e were investigated and the constantλ = 10 −80 was chosen, which yields a value for b 0 commensurate with Born and Infeld's model of the electron 2 . The quartic equation (26) for A(ρ 0 ) was used to fix the electric charge q of the solution. In particular, the choice q = e was implemented by algebraically solving (26) for A(ρ 0 ) and using the prescribed values of ψ(ρ 0 ), Λ(ρ 0 ) and Γ 0 = e/(2ǫ 0 b 0 L 2 ).           figure 5, and a schematic diagram of the normalized Λ 2 determining the electric field is shown in figure 1a. The axion field A is mostly negative over the range in ρ shown above and the plot shows A normalized with its sign reversed for convenience. This procedure yielded A(ρ 0 ) = 0 to within numerical precision. A consistent picture that emerges from extensive numerical analysis of both systems of ODEs is the existence of confined solutions, i.e. fields that are zero for ρ greater than some real positive non-zero constant.
For the same initial data ψ(ρ 0 ), A(ρ 0 ), ϕ ′ (ρ 0 ), A ′ (ρ 0 ), Λ(ρ 0 ) the electric field of solutions to Systems 1 and 2 appears to be regular throughout all space (see figures 1b and 2; the subscript "max" indicates the maximum value of a field over the range of ρ shown). Other solutions diverge at the origin but terminate at finite ρ (see figures 3 to 5). Finally, some solutions to System 2 are finite, non-zero and continuous on a subset ρ a < ρ < ρ b for positive non-zero ρ a , ρ b and zero elsewhere (see figures 1a and 6); however, we were unable to replicate this behaviour using System 1. This implies that the gravitational field equation (16) plays a significant role in determining the dilaton interaction with the electromagnetic field and axion despite the neglect of curvature in favour of a g-flat metric in the Jordan frame.
The existence of states where electric, axion and dilaton fields have finite support in space is interesting and unexpected. The role of the axion and dilaton is critical since no such configurations can occur in the spherical symmetric static sector of the original Born-Infeld theory. The field cut-offs arise when the trajectory {ψ(ρ), A(ρ), Λ(ρ)} approaches the boundary of the domain ∆(ϕ, Λ, A) ≥ 0 in field space.

Conclusion
An extension of the original Born-Infeld model has been developed to include axion and dilaton fields. Motivated by low-energy effective string actions and their symmetries the model reduces in weak field or weak coupling limits to the original Born-Infeld model, SL(2, R) covariant axion-dilaton models or linear Maxwell electrodynamics. In the absence of axion and dilaton couplings it contains only one dimensionless coupling constantλ and is thereby analogous to the original Born-Infeld model regarding its spherically symmetric static gravity-free sector. Two approximation schemes have been developed to explore the significance of dilaton couplings to curvature in the Jordan frame. Both entail working with a g-flat metric although the treatment of the gravitational field equations differ in the two schemes. Clearly it would be valuable to find an exact static spherically symmetric solution with non-zero curvature for the model in either the Einstein or Jordan frame in order to assess the validity of these schemes. If such a solution (generalizing the Reissner-Nordström solution to the Einstein-Maxwell system) exists that substantiates the approximations leading to system 2 then numerical evidence suggests the existence of both finite mass electrically charged and neutral states in this sector. The latter are novel and are composed of mutually coupled electric, axionic and dilatonic fields that exist in a bounded region of space. Some states are bound by a single sphere; others are bound by two concentric spheres, much as a static electric field is confined in a spherical capacitor in Maxwell theory. The electric charge sources for such states reside in induced surface charge densities on the bounding spheres.
The model would then provide a mechanism for confined static abelian fields via their mutual interaction. Since U(1) ⊂ SU(2) ⊂ SU(3) it would be of interest to explore whether such a mechanism arises in a non-abelian generalization. At the abelian level it suggests the possibility of new types of electrically neutral axion-dilaton bound states with no direct interaction with external electromagnetic fields.