2838 Page 4 of 17 Eur. Phys. J. C (2014) 74:2838
the scalar functions
c
(k) being for its four eigenvalues,
4
(k) = 0. The eigenvalues
c
(k) depend on F and on
any two of the three momentum-containing Lorentz invari-
ants k
2
= k
2
− k
2
0
, kF
2
k, k
˜
F
2
k, subject to one relation,
k
˜
F
2
k
2F
− k
2
=
kF
2
k
2F
, where F is taken on the external field,
2F = B
2
. The solution of (12) has an arbitrary longitudinal
part:
D
μτ
(k) =
4
c=1
D
(c)
(k)
(c)
μ
(c)
τ
(
(c)
)
2
,
D
(c)
(k) =
(k
2
−
c
(k))
−1
, c = 1, 2, 3
arbitrary, c = 4
. (16)
It also has a diagonal form in the same terms as (13). This
propagator has three components, corresponding to separate
eigenmodes. Each of them has a pole in the 4-momentum
plane, where solutions of the corresponding dispersion equa-
tions lie, i.e. on t he photon mass shell, defined by the equation
k
2
−
c
(k) = 0.
Now, the solutions (9), (10) may be, respectively, written
as
a
lin
μ
(k) =
4
c=1
1
(k
2
−
c
(k))
(c)
μ
(
(c)
)
2
( j
τ
(
k
)
(c)
τ
), (17)
a
nl
μ
(k) =
4
c=1
1
(k
2
−
c
(k))
(c)
μ
(
(c)
)
2
( j
τ
nl
(
k
)
(c)
τ
). (18)
2 Response of magnetized vacuum to static electric field
in the infrared approximation
In the rest of the paper we shall be treating the equations
of Sect. 2.1 in the low-momentum–low-frequency (infrared)
approximation, k
μ
∼ 0, which stems from the assumption
that the effective action [A] is a local functional of the
field strengths F
μν
in the sense that it does not contain their
space and time derivatives. Examples of such an action are
the Heisenberg–Euler action available in the one-loop [39]
and two-loop [51] approximations in QED, the Born–Infeld
[52] action, etc. Within the local limit the second- and third-
rank polarization tensors (3) were calculated in [1]togive
the result
2
μν
x, x
=
L
F
∂
2
∂x
μ
∂x
ν
− η
μν
x
−
L
FF
F
μα
F
νβ
+ L
GG
˜
F
μα
˜
F
νβ
∂
2
∂x
α
∂x
β
×δ
(
4
)
x − x
,
(19)
2
The fourth-rank tensor in the same approximation is also available
[2].
νρσ
x, x
, x
=−O
νρσ αβγ
∂
∂x
α
∂
∂x
β
δ
(
4
)
x − x
×
∂
∂x
γ
δ
(
4
)
x
− x
,
(20)
where
O
μτ σ αβγ
= L
GG
F
γσ
αμβτ
+
F
αμ
βτγσ
+
F
βτ
αμγ σ
+ L
FF
η
μτ
η
αβ
− η
μβ
η
ατ
F
γσ
+F
αμ
η
τσ
η
γβ
− η
βσ
η
γτ
+ F
βτ
η
μσ
η
γα
− η
ασ
η
γμ
+ L
FGG
F
αμ
F
βτ
F
γσ
+
F
αμ
F
βτ
F
γσ
+
F
αμ
F
βτ
F
γσ
+ L
FFF
F
αμ
F
βτ
F
γσ
, (21)
which expresses them in terms of the derivatives of the effec-
tive Lagrangian taken at the constant external field value,
F
μν
= const.,
L
F
=
dL(F, 0))
dF
F=F
, L
FF
=
d
2
L(F, 0)
dF
2
F=F
,
L
GG
=
∂
2
L(F, G)
∂G
2
F=F,G=0
, (22)
L
FFF
=
d
3
L(F, 0)
dF
3
F=F
, L
FGG
=
d
dF
∂
2
L(F, G)
∂G
2
F=F,G=0
.
(23)
It is taken into account that once the external field is purely
magnetic in a certain Lorentz frame, the invariant G for it is
zero while F is positive.
It was established in [53 ] that the second-rank polariza-
tion operator (19) has indeed the structure of (13) with the
eigenvalues
a
(k) in the infrared regime being
1
(k
2
, kF
2
k, F)
k→0
= k
2
L
F
, (24)
2
(k
2
, kF
2
k, F)
k→0
= k
2
L
F
− (k
˜
F
2
k)L
GG
, (25)
3
(k
2
, kF
2
k, F)
k→0
= k
2
L
F
− (kF
2
k)L
FF
. (26)
Henceforth, we shall be dealing only with sources that
are static in a reference frame, where the external field is
magnetic, i.e. time-independent charges at rest in a magnetic
field,
j
μ
(x) = δ
0μ
q(x),
˜
j
μ
(k) = (2π)δ
0μ
δ(k
0
) ˜q(k), (27)
where the tilde marks the Fourier-transformed function.
2.1 Linear response—modified Coulomb law at large
distances
Employing the source (27)in(17) and taking into account
that at k
0
= 0 out of all the three ( nontrivial) eigenvectors
123