Born-Infeld Axion-Dilaton Electrodynamics and Electromagnetic Confinement

Abstract

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.

Full text

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arXiv:1012.4995v1 [hep-th] 22 Dec 2010
Born-Infeld Axion-Dilaton Electrodynamics and Electromagnetic
Confinement
D.A. Burton∗†
T. Dereli‡
R.W. Tucker†∗
December 23, 2010
Abstract
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 pa-
rameter b0. In the absence of axion and dilaton interactions it reduces in the limit b0 ?→ ∞ to Maxwell’s
linear vacuum theory while for finite b0 it reduces to the original Born-Infeld model. The spherically sym-
metric 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.
1 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 [1–3]. 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]. 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 weak-
field limit. Such models have been extensively studied by Gibbons et al [12–14] 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.
∗Physics Department, Lancaster University, Lancaster LA1 4YB, UK
†The Cockcroft Institute of Accelerator Science and Technology, Daresbury WA4 4AD, UK
‡Department of Physics, Ko¸ c University, 34450 Istanbul, Turkey
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2 Axion-dilaton Born-Infeld Electrodynamics
If Rab denotes the curvature 2-forms associated with the spacetime metric g = ηabea⊗ ebwhere ηab =
diag{−1,1,1,1} in a local coframe {ea}, 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 varying the action S[g,A,ϕ,A] =?
Λ0=p1Rab∧ ⋆eab+ p2dϕ ∧ ⋆dϕ + p3exp(−2ϕ)dA ∧ ⋆dA
+ p4AF ∧ F + p5exp(ϕ)F ∧ ⋆F
and F = dA. In this expression the constants are
MΛ0
where the 4-form Λ0on spacetime M is
(1)
p1=
c3
8πGN
=
?
L2, (2)
p2= p3= −2?
p4= p5=ε0
L2, (3)
2c,
(4)
in terms of the Planck length
L =
?
8π?GN
c3
,(5)
the Newtonian gravitational constant GN and the permittivity of free space ǫ0
(1) denotes the traditional coupling of the axion field to the electromagnetic field while the term involving p5
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 b0with 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 b0. 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 b0?→ ∞ to Maxwell’s linear vacuum
theory while for finite b0it reduces to the original Born-Infeld model. It involves the Newtonian gravitational
coupling constant GN in addition to b0and a parameter τ = ±1 and is obtained by varying the action:
1. The term involving p4 in
Sτ[g,A,ϕ,A] =
?
M
Λτ
(6)
where
Λτ=p1Rab∧ ⋆eab+ p2[dϕ ∧ ⋆dϕ + exp(−2ϕ)dA ∧ dA]
+ fτ(X,Y,ϕ,A) ⋆ 1, (7)
with
fτ(X,Y,ϕ,A) = τε0b2
0
c
?
1 −
?
1 − τeϕX
b2
0
− τAY
b2
0
−[eϕY − AX]2
4b4
0
?
,(8)
X = ⋆(F ∧ ⋆F) and Y = ⋆(F ∧ F).
1All tensor fields in this article have dimensions constructed from the SI dimensions [M],[L],[T],[Q] where [Q] has the unit of
the Coulomb in the MKS system. We adopt [g] = [L2],[ϕ] = [A] = 1,[G] = [Q], [F] = [Q]/[ǫ0] where the permittivity of free space
ǫ0has the dimensions [Q2T2M−1L−3] and c =
α in 4 dimensions one has [⋆α] = [α][L4−2r] where ⋆ denotes the Hodge map associated with g.
1
√ǫ0µ0denotes the speed of light in vacuo. Note that, with [g] = [L2], for r−forms
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The structure of the argument of the square root in (8) follows from the (non-trivial) identity
√τ
b0
= 1 − τ exp(ϕ)X
b2
−det
?
ηab+
αFab+
√τ
b0
− τ AY
β˜Fab
?
−[exp(ϕ)Y − AX]2
4b4
0
b2
00
(9)
where exp(ϕ) = α2− β2, A = −2αβ, F =1
In the following attention is restricted to the case where τ = 1. The non-linear vacuum Maxwell equations
follow as dF = 0 (since F = dA) and (by varying A)
2Fabea∧ eband ⋆F =1
2˜Fabea∧ eb.
d ⋆ G = 0 (10)
where
⋆ G = 2cfX⋆ F + 2cfYF (11)
and fX= ∂Xf etc. From A variations one has
− 2p2d[exp(−2ϕ) ⋆ dA] + fA⋆ 1 = 0 (12)
and from ϕ variations
− 2p2d ⋆ dϕ − 2p2exp(−2ϕ)dA ∧ ⋆dA + fϕ⋆ 1 = 0. (13)
Finally from ortho-normal coframe variations one obtains the Einstein equations
p1Rbc∧ ⋆eabc= τa[F,ϕ,A,g] (14)
where
τa[F,ϕ,A,g] =p2(iadϕ ∧ ⋆dϕ + dϕ ∧ ia⋆ dϕ)
+ p2exp(−2ϕ)(iadA ∧ ⋆dA + dA ∧ ia⋆ dA)
− (f − X fX− Y fY) ⋆ ea− fX(iaF ∧ ⋆F − F ∧ ia⋆ F).
Although we shall consider the system in a background Minkowski spacetime the above stress-energy-
momentum forms that act as a source of gravitation can be used to calculate the gravitational mass of the
field configurations satisfying the above field equations for F,ϕ and A.
In the static spherically symmetric sector the Minkowski metric is written as above with the coframe field
e0= cdt,e1= dr,e2= rdθ,e3= r sinθdφ in spherical polar coordinates. In terms of the dimensionless radial
coordinate ρ = r/L we write:
F = b0Λ(ρ)dr ∧ cdt
with ϕ = ϕ(ρ),A = A(ρ). From the above field equations one finds the ordinary system of coupled equations:
ˆλ
2
(15)
(16)
∂2
ρϕ +2
ρ∂ρϕ = −
?
ϕ3−(φ7−5
?∆(ϕ,Λ,A)
AΛ4
?∆(ϕ,Λ,A),
= Γ0
8)ϕ4Λ2
?
, (17)
∂2
ρA +2
ρ∂ρϕ = −
ˆλ
8
(18)
ρ2(2φ5Λ + Λ3A2)
4?∆(ϕ,Λ,A)
(19)
for some integration constant Γ0. The dimensionless constantˆλ is defined by
ˆλ =
?8π?GN
c3
?2ǫ0b2
0
2?c
(20)
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(a) Electric axion-dilaton bound state with
confined fields.
(b) Electric axion-dilaton bound state with
un-confined fields.
Figure 1: Electric field lines arising from a numerical analysis of (17), (18), (19). Figure 1(a) illustrates a state
where the electric field lines emanate and terminate on spheres of finite radius in space. The state in figure 1(b)
with unconfined electric flux generalizes the static spherically symmetric solution in the original Born-Infeld
electron model.
and
∆(ϕ,Λ,A) = ϕ8− ϕ5Λ2− cA2ϕ4
4
(21)
If S2is any 2-sphere of radius ρL centered at ρ = 0 the electric flux of any state crossing the surface of this
sphere is 4πq =?
q =1
Clearly an analytic solution to the system (17), (18), (19) is unlikely. It is however amenable to numerical
analysis. The simplest approach is to differentiate (19) with respect to ρ and treat the coupled system as an
initial value problem specified by a choice of ϕ(ρ0),A(ρ0),ϕ′(ρ0),A′(ρ0),Λ(ρ0) with
?
?∆(ϕ(ρ0),Λ(ρ0),A(ρ0))
The initial field conditions should be consistent with a real Γ0. Starting from ρ = ρ0the system can be readily
integrated numerically to the regions ρ > ρ0and ρ < ρ0.
If the constant q determined by Γ0is non-zero for a sphere of infinite radius, the static field configuration
will be deemed to have electric charge q. The time-like Killing 1-form K = dt of the Minkowski metric is used
to define the total gravitational mass-energy Mc2of the field configuration:
S2⋆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 Γ0 such charge is
2ǫ0b0L2Γ0.
Γ0=ρ2
0
4
2ϕ5(ρ0)Λ(ρ0) + Λ3(ρ0)A2(ρ0)
?
. (22)
Mc2=
?
Σ
τ0
(23)
where Σ is any constant t space-like hyper-surface in spacetime. This contains contributions from F,A and ϕ.
An interesting feature that emerges from a numerical analysis of the system above is the existence of bounded,
piecewise continuous solutions on a subset of 0 < ρ < ∞ (see figure 1). Depending on the choice of initial data
ϕ(ρ0), A(ρ0), ϕ′(ρ0), A′(ρ0), Λ(ρ0) some solutions are finite, non-zero and continuous on a subset ρa< ρ < ρb
for positive non-zero ρa,ρb and zero elsewhere (see figure 1(a)). Other solutions appear finite, non-zero and
continuous throughout all space (see figure 1(b)). Solutions with fields finite for ρa< ρ < ρb can have finite
gravitational mass while states with total non-zero finite flux of ⋆G out of a large sphere at ∞ will have non-zero
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electric charge. Numerical evidence suggests that both types of solutions exist in this model. The precise values
of the mass and charge depend on the values chosen forˆλ and Γ0and the accuracy of the numerical analysis.
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
{ϕ(ρ),A(ρ),Λ(ρ)} approaches the boundary of the domain
limiting the values of the fields is analogous to the mechanism that limits the speed of a massive particle to be
less than c in relativistic dynamics.
?∆(ϕ,Λ,A) ≥ 0 in field space. The mechanism
3 Conclusion
An extension of the original Born-Infeld model has been developed to include axion and dilaton fields. Mo-
tivated 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 a background Minkowski spacetime it contains only one dimensionless coupling constant
ˆλ and is thereby analogous to the original Born-Infeld model regarding its spherically symmetric static sector.
Numerical evidence suggests the existence of both finite mass electrically charged and neutral states in this
sector. The latter are novel since they are composed of mutually coupled electric, axionic and dilatonic fields
that exist in a region of space bounded by a pair of 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 surface charge
densities on the bounding spheres. The model provides 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.
4 Acknowledgments
DAB and RWT are grateful for the hospitality provided by the Department of Physics, Ko¸ c University and for
financial support as part of the ALPHA-X collaboration (EPSRC grant EP/E001831/1). TD is grateful for the
hospitality provided by the Department of Physics, Lancaster University, and the Cockcroft Institute, UK. All
authors thank TUBA (The Turkish Academy of Sciences) for financial support.
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