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Eur. Phys. J. C (2017) 77:519

DOI 10.1140/epjc/s10052-017-5073-5

Regular Article - Theoretical Physics

Einstein–Maxwell-axion theory: dyon solution with regular

electric ﬁeld

Alexander B. Balakin

, Alexei E. Zayats

Department of General Relativity and Gravitation, Institute of Physics, Kazan Federal University, Kremlevskaya Street 18, Kazan 420008, Russia

Received: 29 March 2017 / Accepted: 17 July 2017 / Published online: 4 August 2017

Abstract In the framework of the Einstein–Maxwell-axion

theory we consider static spherically symmetric solutions

which describe a magnetic monopole in the axionic environ-

ment. These solutions are interpreted as the solutions for an

axionic dyon, the electric charge of which is composite, i.e.

in addition to the standard central electric charge it includes

an effective electric charge induced by the axion–photon cou-

pling. We focus on the analysis of those solutions which are

characterized by the electric ﬁeld regular at the center. Special

attention is paid to the solutions with the electric ﬁeld that is

vanishing at the center, and that has the Coulombian asymp-

tote, and thus displays an extremum at some distant sphere.

Constraints on the electric and effective scalar charges of

such an object are discussed.

1 Introduction

In 1987 Wilczek has formulated the idea that for a distant

observer the magnetic monopole in an axionic environment

looks like a dyon with a magnetic and effective electric charge

[1]. This idea was based on the prediction of the axion electro-

dynamics where the interaction between the radial magnetic

ﬁeld attributed to the monopole and the surrounding pseu-

doscalar (axion) ﬁeld produces the radial electric ﬁeld with-

out real electric charge at the center. That is why it is said that

Wilczek in 1987 presented the ﬁrst example of the so-called

axionic dyon. The axion electrodynamics on which this result

was based have been established and developed in the decade

1977–1987, being inspired by the theoretical discovery of

Peccei and Quinn of the CP-invariance conservation [2] and

by discussions about a new light pseudo-Goldstone boson

introduced by Weinberg [3] and Wilczek [4]. The model of

e-mail: Alexander.Balakin@kpfu.ru

e-mail: Alexei.Zayats@kpfu.ru

the coupling of the pseudoscalar and electromagnetic ﬁelds

was formulated in a covariant form by Ni in [5]; the axion

electrodynamics written in the 3-dimensional form was used

by many authors (see, e.g., the work of Sikivie [6]). Since

the axions are considered to be candidates to the dark mat-

ter particles [7–15] the physics of axions had become one of

the key elements of numerous applications to cosmology and

astrophysics. These applications take into consideration var-

ious models of interaction of gravitational, electromagnetic,

scalar and pseudoscalar ﬁelds which are nowadays called

the Einstein–Maxwell-axion and Einstein–Maxwell-axion–

dilaton models (see, e.g., [16–18]). Also, these applications

focus the attention on the models which belong to the class

of theories associated with extended axion electrodynamics

[19–26].

In 1991 Lee and Weinberg [27] studied spherically sym-

metric solutions for static black holes with a massless axion-

like scalar ﬁeld; in fact it was a realization of the Wilczek

idea in the framework of the Einstein–Maxwell-axion the-

ory. Lee and Weinberg have obtained self-consistent master

equations for the axion ﬁeld and metric coefﬁcients, ana-

lyzed the asymptotic properties of the solutions and stud-

ied the analytic and numeric solutions for the cases of large

and small values of the constant of the axion–photon cou-

pling. If we omit the initial electric charge at the center of the

object described in [27] we ﬁnd the solution for the axionic

dyon, which was obtained in the framework of the Einstein–

Maxwell-axion model and was predicted in [1]usingthe

simple Maxwell-axion model. In this sense it can be said

that in [1,27] the authors presented the ﬁrst (static) exam-

ple of the so-called longitudinal magneto-electric cluster in

which the magnetic and axionically induced electric ﬁelds

are parallel to one another. Later the solutions describing the

Longitudinal Clusters were found in the systems with the

pp-wave symmetry [28] and in the context of the search for

ﬁngerprints of relic axions in the terrestrial magnetosphere

[29].

123

519 Page 2 of 13 Eur. Phys. J. C (2017) 77 :519

Now we are interested to ﬁnd a regular solution for

the axionic dyon. What does this mean? In 1968 Bardeen

[30] attracted the attention to solutions of the ﬁeld equa-

tions which are regular in the center. The ﬁrst idea was to

modify the equations for the electric ﬁeld so that it will

be ﬁnite at r = 0; for instance it might be the func-

tion E(r) =

with E(0) =

and the Coulombian

asymptote E(r) →

. In the framework of the Einstein–

Maxwell theory the regularity in the center assumes that

not only the electric ﬁeld is ﬁnite but the metric coefﬁ-

cients and all curvature invariants are ﬁnite as well. The

story of the search for regular solutions is worthy to be

subject of special review; we would like to mention only

three details in this context. First, the nonminimal coupling

of electromagnetic and gauge ﬁelds can provide the grav-

itational ﬁeld to be regular (see, e.g., [31–33]). Second,

the nonminimal coupling can provide the electric ﬁeld to

be ﬁnite in the center (see, e.g., [34–36]). Third, for solu-

tions with a magnetic monopole ﬁeld the situation is not

perfect; for the mentioned solutions the ﬁrst invariant of

the electromagnetic (or gauge) ﬁeld, B

− E

is not reg-

ular in the center since the magnetic ﬁeld, B(r) =

in contrast to the electric one, cannot be ﬁnite there. As

for the second (pseudo)invariant (B · E) it is possible for

it to be ﬁnite in the center, when the electric ﬁeld is

not only ﬁnite, but tends to zero not more slowly than

. On the other hand, if the electric ﬁeld strength E is

ﬁnite at the origin but does not vanish, the vector ﬁeld E

has a hedgehog-like singularity. Therefore if we expect to

ﬁnd a solution which is characterized by the electric ﬁeld

being regular in the center in the strict sense of the word,

we should require the condition E(0) = 0tobesatis-

ﬁed.

Thus, searching for the regular axionic dyons we are

faced with the problem to ﬁnd an exact solution for the ﬁeld

equations for which the electric component vanishes both at

r → 0 and r →∞. Below we intend to show that this is

possible for a magnetic monopole surrounded by the pseu-

doscalar (axion) ﬁeld when the guiding parameters of the

model are speciﬁcally coupled.

The paper is organized as follows. In Sect. 2 basic details

of the Einstein–Maxwell-axion theory are revived and well-

known solutions with the vanishing constant of the axion–

photon coupling (γ = 0) are recovered, using the har-

monic spacetime coordinates. In Sect. 3 we analyze the

solutions with nonvanishing γ ; in Sect. 3.1 we discuss an

example of an exact solution for the axionic dyon singu-

lar at the center; in Sect. 3.2 we study (analytically) the

regular solutions of the axion electrostatics in the back-

ground of magnetic monopole; the results of the numer-

ical study are presented in Sect. 3.3. Section 4 contains

conclusions.

2 Einstein–Maxwell-axion model

2.1 Basic formalism

The action functional of the Einstein–Maxwell-axion model

takes the form

EMa



√

−g d

L =

2κ

γ F

∗

−

∇

φ∇

φ +

. (1)

Here R is the Ricci scalar; g is the determinant of the metric

tensor g

; κ is the Einstein constant, F

is the Maxwell ten-

sor,

∗

denotes its dual tensor, φ stands for the pseudoscalar

(axion) ﬁeld; γ is the constant of the axion–photon coupling;

and m

is the axion mass.

The variation of the action (1) with respect to potentials of

the electromagnetic ﬁeld A

, to the axion ﬁeld φ, to the space-

time metric g

gives, respectively, the equations of axion

electrodynamics

∇

+ γ

∗

∇

φ = 0, (2)

the equation for the axion ﬁeld φ

∇

φ + m

φ +

∗

= 0, (3)

and the equations for the gravitational ﬁeld

−

= κ(T

(M)

+ T

(a)

). (4)

Here T

(M)

and T

(a)

are the energy-momentum tensors for

the electromagnetic and axion ﬁelds, respectively, which are

deﬁned as follows:

(M)

− F

, (5)

(a)

=∇

φ∇

φ −

∇

φ∇

φ +

. (6)

The dual Maxwell tensor satisﬁes the equation ∇

∗

= 0

which is free from information on the axion ﬁeld.

2.2 Static spherically symmetric spacetime

Let us consider a static spherically symmetric spacetime with

the metric

= e

−2β(u)

− e

2β(u)

−4ρ(u)

+ e

−2ρ(u)

d

(7)

123

Eur. Phys. J. C (2017) 77 :519 Page 3 of 13 519

We use the harmonic coordinate system [37] in which the

variable u plays the role of a radial coordinate; the spatial

inﬁnity corresponds to u = 0. We assume that the axion ﬁeld

depends on the radial coordinate only, i.e. φ = φ(u).This

system is more convenient to analyze scalar ﬁeld models.

However, when we will need to revive the usual spherical

coordinate notation we put

r = e

β(u)−ρ(u)

. (8)

At the spatial inﬁnity, i.e at u = 0, asymptotic behavior of the

spacetime metric is supposed to be Minkowskian. It means

that

β(0) = 0,ρ(u)|

u→0

∼ ln u. (9)

In this paper we focus on the study of conﬁgurations with a

magnetic monopole located at the center; the Maxwell tensor

components are chosen to be equal to

= q(u)e

−2β(u)

, F

θϕ

= μ sin θ, (10)

where the constant μ relates to the magnetic charge, q(u) is

a function to be found. To characterize the electric ﬁeld it is

convenient to introduce also the scalar quantity E deﬁned as

follows:

E ≡

= q e

2ρ−2β

. (11)

In fact, the scalar E is the tetrad component of the Maxwell

tensor E =

√

−F

. In these terms the equations of axion

electrodynamics (2) reduce to one equation

(

q − γμφ

)



= 0, (12)

yielding the solution

q = Q + γμ

(

φ − φ

)

. (13)

The constant of integration φ

is the value of the axion ﬁeld

at the inﬁnity, i.e. φ(u = 0) = φ

; similarly we deﬁne Q =

q(0).

The axion ﬁeld equation (3) takes now the form



− γμq e

−2β

− m

φ e

2β−4ρ

= 0. (14)

Using (13) this equation can be rewritten as follows:



−γ

q e

−2β

−m

(

q + γμφ

− Q

)

2β−4ρ

= 0. (15)

There are four nontrivial equations of the gravitational

ﬁeld. For the metric (7) four nonvanishing components of

the Einstein tensor G

= R

−

R are

= e

−2β+4ρ

(β



− ρ



+ e

−2ρ

), (16)

= G

= e

−2β+4ρ

(−β



+ ρ



+ ρ



), (17)

= e

−2β+4ρ

(−β



+ ρ



− 2β



+ 2ρ



+ e

−2ρ

). (18)

The corresponding four nonvanishing components of the

energy-momentum tensor T

= T

k(M)

k(a)

take the form

(see (5) and (6))

−2β+4ρ

[(μ

+ q

−2β

− φ



, (19)

= T

−2β+4ρ

[−(μ

+ q

−2β

+φ



(20)

−2β+4ρ

[(μ

+ q

−2β

+ φ



. (21)

If we assume (as in [27]) that the axion ﬁeld is massless,

= 0, three independent equations for gravity ﬁeld can be

rewritten



+ e

−2ρ

= 0, (22)



(μ

+ q

−2β

= 0, (23)



− e

−2ρ

= β



−

(μ

+ q

−2β

. (24)

Clearly the ﬁrst equation is decoupled from other ones and

can be immediately resolved. Indeed, the ﬁrst integral of (22)



− e

−2ρ

= C, (25)

and the solution satisfying the condition (9) takes the form

ρ = ln (C, u),

(C, u) ≡

⎧

⎪

⎨

⎪

⎩

sinh νu

, when C = ν

> 0,

u, when C = 0,

sin νu

, when C =−ν

< 0.

(26)

Also one can check directly that, with (26), Eq. (23)isadif-

ferential consequence of (24) with (13). Thus, the key sub-

system of master equations consists of the following pair of

equations:



2γ



−

(μ

+ q

−2β

= C,



− γ

q e

−2β

= 0. (27)

When the quantities β(u) and q(u) are found the axion ﬁeld

and the electric ﬁeld can be reconstructed,

φ = φ

q − Q

γμ

, E(u) = q(u) e

−2β



(C, u). (28)

123

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