210 Brazilian Journal of Physics, vol. 38, no. 2, June, 2008

Self-Energies for Interacting Fields with a Compactified Spatial Dimension

F. A. Barone∗
ICE - Universidade Federal de Itajubá, Av. BPS 1303, Caixa Postal 50, 37500-903, Itajubá, MG, Brazil.

M. Hott†
Departamento de Fı́sica e Quı́mica, UNESP-Campus de Guaratinguetá, 12500-000, Guaratinguetá, SP, Brazil.

Received on 4 November, 2007

We study the self-energy of a model with a tri-linear coupling among spinless fields on a manifold with one
spatial dimension compactified. The model is considered to be composed by a quantized scalar field interacting
with a classical scalar one. In order to improve the compactification we take the quantum field satisfying quasi
periodic boundary conditions, which interpolates continuously the periodic and anti-periodic conditions, whilst
the background field satisfies periodic boundary conditions. All the calculations are performed in Euclidean
coordinates.

Keywords: Quantum fields; Compactified dimensions

I. INTRODUCTION

Since the seminal papers of T. Kaluza [1] and O. Klein
[2], there have been many studies of models with extra di-
mensions in the context of strings theory and development of
works investigating models with compactified spatial dimen-
sions in the context of Quantum Field Theory. In this scenario
we can cite, for instance, studies about the proton stability
in six dimensions [3], grand unified theories in extra dimen-
sions [4], the descriptions of the Higgs particle from models
with extra dimensions [5], a justification of the Electroweak
symmetry breaking with extra spatial dimensions compacti-
fied and without the introductions of the Higgs field [6, 7], the
development of spontaneous broken symmetry of spaces with
compactified dimensions [8], the role of the Casimir effect in
the radius stabilization of extra dimensions [9], the gauge in-
variance of massive modes of quantum fields in theories with
extra dimensions compactified [10], thermal effects of fields
with spatial dimensions compactified [11], the existence of an
unbroken subgroup of the six-dimensional Lorentz symmetry
for models with chiral compactification [12]. Other examples
are the studies of the influence of boundary conditions in the
cosmological scenario [13], in the corrections to the masses of
fields [14, 16], in symmetry breaks [17, 18] and in the renor-
malization of quantum field theories in non-simply connected
space-time [19].

In the context of Quantum Field Theory it is relevant
to investigate if the existence of extra compactified dimen-
sions could produce some different characteristics on systems
(which could be controlled in the laboratory) in comparison
to the situation where extra dimensions do not exist. A good
scenario which can be of some inspiration for such kinds of
phenomena is the Quantum Field Theory at Finite Tempera-
ture (QFTFT), once in the imaginary time formalism we have
one of the dimensions, the time dimension, compactified. In

∗e-mail: fbarone@unifei.edu.br
†e-mail: hott@feg.unesp.br

this sense, models of QFTFT can exhibit narrow similarities
with corresponding ones in Quantum Field Theory in 3 + 1
dimensions at zero temperature but with one of the spatial di-
mension compactified.

It is well known that in quantum field theory at finite tem-
perature (QFTFT), some Feynman diagrams of the theory can
be non-analytical [20, 21], in the sense that, the zero limit of
the external momentum is not well defined. In other words,
the limit Π(k0 = 0,k → 0) does not commute with the limit
Π(k = 0,k0 → 0), where Π(k0,k) designates a Feynman am-
plitude and k0 (k) stands for the time (space) component of
external momentum. Such non-analyticity is due to a branch
cut along the real axis of the complex plane of the time-
component of the external momentum, with branch point at
the origin, if the internal propagators of the loop carries the
same mass [22]. The branch point is the threshold above
which the amplitude for the decay of real particles into “ther-
mal particles”, and vice versa, is different from zero. The non-
analyticity can be seen in any one of the formalisms used to
treat QFTFT, but the doubling of degrees of freedom is man-
ifested naturally in the real-time formalisms. The real-time
formalisms are also appropriate settings to analyze effective
actions at finite temperature, since the derivatives of any field,
in terms of which one express the effective Lagrangian, are
assumed to be continuous and there is no need to resort to
analytic continuation from discrete to continuous momenta.
Unfortunately, even the derivative expansion of the effective
Lagrangian is plagued with the non-analyticity that renders it
essentially non-local [23], unless the external field is static.
The imaginary-time formalism in quantum field theory at fi-
nite temperature is also thoroughly used to analyze effective
potentials and thermal corrections to the masses of particles.

A natural question that can be raised is: in analogy with
QFTFT, how the non-commutativity is manifested and which
kind of phenomena it can produce when one considers quan-
tum field theory at zero temperature but with one spatial di-
mension compactified. This could be an interesting subject
in the study of systems defined on manifolds with one spatial
dimension compactified, like nanoubes, and also in Kaluza-
Klein theories.



F. A. Barone and M. Hott 211

Due these considerations, in this work we analyze the influ-
ence of boundary conditions on the quantum correction to the
mass of a real classical scalar field interacting with a quan-
tized complex scalar field [24]. The quantum field satisfies
quasi-periodic boundary conditions, which interpolates con-
tinuously the periodic and anti-periodic cases in one of the
spatial dimensions. The classical field is taken to be satisfy-
ing periodic boundary condition along the same spatial direc-
tion. To keep an analogy with QFTFT the calculations are
performed in Euclidean time in 3+1 dimensions. We analyze
the behavior of the self-energy diagram as a function of the
modulus and direction of the external momentum, and point
out the possible origin for the non-commutativity. We show
that the correction to the masses for directions of propagation
perpendicular and parallel to the compactified axis can be the
same depending on how the limit of the external momentum
going to zero is taken. In other words, it is shown that the lim-
its Π(k‖ = 0,k⊥→ 0) and Π(k⊥ = 0,k‖→ 0), where k‖ (k⊥)
is the component of the external momentum parallel (trans-
verse) to the compactified dimension [27], commute. The first
(second) of the limits is considered when the field propagates
in the direction transverse (parallel) to the compactified axis.
We also show that this commutativity does not occur for an
otherwise arbitrary direction of propagation.

We consider this specific model because it was studied
previously in the context QFTFT in the analysis of non-
commutativity and effective action [23]. The graphics for
the self-energy in this model is simpler than the photon self-
energy in the QED, for instance, which exhibits a tensor struc-
ture. Furthermore, the chosen boundary conditions interpolate
continuously the periodic and anti-periodic cases and its role
in the non-commutativity of self-energy graphics were not
studied in the literature previously, although similar bound-
ary conditions in QFTFT in 0+1 dimensions were studied in
the context of interpolating statistics [25, 26].

II. THE SELF-ENERGY

We shall consider a model which is described, in Euclidean
coordinates, by the Lagrangian density

L = (∂µφ)(∂µφ∗)+m2|φ|2 +
1
2

(∂µB)2 +
1
2

M2B2 +gB|φ|2,
(1)

with φ≡ φ(x) being a quantized complex scalar field, B≡B(x)
a real classical scalar field (not quantized) and g a coupling
constant. We also consider that the B(x) classical field satisfies
periodic boundary condition in the third spatial coordinate x3

B(x0,x1,x2,x3) = B(x0,x1,x2,x3 +a) , (2)

and the φ(x) satisfies the so called quasi-periodic condition

φ(x0,x1,x2,x3) = ei2πθφ(x0,x1,x2,x3 +a). (3)

In the condition above θ is a parameter whose value is in the
interval 0 ≤ θ < 1, and a is the compactifying length. Note
that for θ = 0 and θ = 1/2 we recover the periodic and anti-
periodic boundary conditions, respectively. Similar boundary
conditions have been imposed in the imaginary time to present
systems that obey an interpolating statistics [25] and also to
analyze the equation of state of a gas of particles obeying an
interpolating statistics [26].

In the appendix we use perturbation theory with fields satis-
fying finite boundary conditions, in order to write the effective
Lagrangian density for the classical field B(x), up to second
order in the coupling constant g, as

Le f f =
1
2

(∂µB)2 +
1
2

M2B2 +
1
2

B(x)
(

Π(∂)
)

B(x), (4)

where the self-energy Π(∂) is given by the differential opera-
tor

Π(∂) = g2
∫ d3p⊥

(2π)3
1
a

∞

∑
`=−∞

{
1

p2
⊥+

[ 2π
a (`+θ)

]2 +m2

1

(p⊥− i∂⊥)2 +
[ 2π

a (`+θ)− i∂3
]2 +m2

}
, (5)

with ∂⊥ = (∂0,∂1,∂2) the derivative with respect to the coordinates transverse to the compactified axis, ∂3 the derivative with
respect to the compactified coordinate x3 and p⊥ = (p0, p1, p2). The above expression is written in terms of derivatives of the
external field B(x) and the derivatives in the equation (4) act to the right. The derivative expansion, which constitutes in a Taylor
series of Π(a,θ,m,∂3) in powers of ∂/m, is useful to obtain a local effective Lagrangian for that field. Later we are going to see
that, due to the non-analyticity of the self-energy in the origin of the external momenta, a derivative expansion of the effective
lagrangian is not well-defined.

In the Appendix we show that in the momenta space the effective action integral for B(x) becomes

Se f f = −
∫

d4x Le f f =

= −1
2

∫ d3k⊥
(2π)3

∞

∑
n=−∞

B̃−n(−k⊥)

[
k2
⊥+

(
2nπ

a

)2

+M2 + Π̃(k⊥,n)

]
B̃n(k⊥) , (6)

where k⊥ = (k0,k1,k2) is the component of the momentum of the field B(x) in the direction transverse to the x3 axis, B̃n(k⊥) is



212 Brazilian Journal of Physics, vol. 38, no. 2, June, 2008

the Fourier transform of the field B(x) and

Π̃(k⊥,n) = g2
∫ d3p⊥

(2π)3
1
a

∞

∑
`=−∞

{
1

p2
⊥+

[ 2π
a (`+θ)

]2 +m2

1

(p⊥+k⊥)2 +
[ 2π

a (`+θ)+ 2πn
a

]2 +m2

}
(7)

is the self-energy in the momenta space. The set of vectors
(p⊥, p3 = 2π

a (`+θ)) and (k⊥,2πn/a) are the momenta of the
fields φ(x) and B(x) respectively.

The summation which appears in (7) is over integer values
of l and can be calculated using the expression

∞

∑
`=−∞

f (`) =−π ∑
poles
f (z)

Res
[

f (z)cot(πz)
]

, (8)

which is valid for any function f (z) that does not have poles
at z = 0,±1,±2, ... and has the limit lim|z|→∞ f (z) = 0 [21].
In expression (8) the summation on the right hand side is over
the poles of f (z), with z a complex variable. By using the fact
that

coth(z) = 1+2nB(2z) , (9)

where nB(z) is the Bose-Einstein distribution function (B-E
factor),

nB(z) =
1

exp(z)−1
, (10)

the equation (7) can be split into two contributions

Π̃(k⊥,n) = Π̃(k2)+ Π̃(a,θ,k⊥,k3), (11)

where the first term is the sole contribution to the self-energy
without boundary conditions once it depends on neither the
length a nor the interpolating parameter θ; the second one is
the contribution due to the boundary conditions. From now on
we concentrate on this second part of the self-energy which is
written as

Π̃(a,θ,k⊥,k3) =
1
2

∫ d3p⊥
(2π)3

[
1

ωp⊥

1
k2 +2(k⊥ ·p⊥+ ik3ωp⊥)

nB
(
aωp⊥ +2πiθ

)
+

+
1

ωp⊥+k⊥

1
k2−2(k⊥ ·p⊥+ ik3ωp⊥+k⊥)

nB
(
aωp⊥+k⊥ +2πiθ+ iak3

)
]

+C.C. , (12)

with

k2 = k3
2 +k⊥2, k3 =

2π
a

n, , ωp⊥ =
√

p2
⊥+m2 and ωp⊥+k⊥ =

√
p2
⊥+m2. (13)

The C.C. designates the complex conjugate of its preceding terms.
In the equation (12) we make the change of variables p⊥→−(p⊥+k⊥) in the integrals which contains k3 in the argument of

the B-E factor nB, and integrate in the angular variables to obtain

Π̃(a,θ,k⊥,k3) =
1

(2π)2
1

4k⊥

∫ ∞

0
d p⊥

p⊥
ωp⊥

[
ln

(
k2 +2(k⊥p⊥+ ik3ωp⊥)
k2−2(k⊥p⊥− ik3ωp⊥)

)
nB

(
aωp⊥ +2πiθ

)
+

+ ln
(

k2 +2(k⊥p⊥− ik3ωp⊥)
k2−2(k⊥p⊥+ ik3ωp⊥)

)
nB

(
aωp⊥ +2πiθ+ iak3

)
]

+C.C. . (14)

By rotating back to Minkowskian space-time and expand-
ing (14) in powers of the external momenta, one can verify
from the expression for the effective action, equation (6), that
the zero order term in the external momenta gives the zero
order correction, in powers of the external momenta, to the
mass of the B(x) field, up to second order in the coupling con-
stant g. One can also see that the components of the external

momenta, k3 and k⊥, are not in the same foot in (14) and the
zero-momentum limit depends on the path in the momenta
space along which it is taken. We say that based on similar
results in QFTFT where a similar picture is present and two
masses come out, the plasmon mass due to time oscillations
and the Debye mass related to the screening of the external
field. Since here the boundary conditions are imposed on one



F. A. Barone and M. Hott 213

of the space coordinates, we associate the zero-momentum
limit Π̃(a,θ,k3 = 0,k⊥→ 0) with the correction to the mass
when the field B(x) propagates in the direction transverse to
the x3, whilst the limit Π̃(a,θ,k⊥→ 0,k3 → 0) is the correc-
tion to the mass related to the propagation of the field parallel
to the compactified dimension. The first of the limits corre-
sponds to set k3 = 0 first and then to take the limit k⊥ → 0.
On the other hand, the second limit corresponds to take the
limit k⊥ → 0 first and only after the contribution of n = 0 is
evaluated. It worths to mention that the time dimension plays
no role in this scenario; as a matter of fact we could suppose
the real field to be a static one and, in such a case, we might
speak of different Debye screening lengths that are given by
the inverse of the “masses”, depending on which direction we
approach the origin of the momentum space. In the next sec-
tion the propagation directions perpendicular and parallel to

the x3 are analyzed in some detail.

III. THE PARALLEL AND PERPENDICULAR
PROPAGATIONS

In this section we calculate the quantum corrections to the
mass of the B(x) field for propagations parallel and perpen-
dicular to the x3 axis. For this task, it is convenient to define
the dimensionless variables

u =
p⊥
m

, v =
k⊥
m

, z =
k3

m
, s = ma , (15)

in order to write equation (14) in the form

Π̃(a,θ,k⊥,k3) = g2 1
(2π)2

1
4

∫ ∞

0
du

u√
1+u2

I (z,v,u,s,θ) , (16)

where we have defined

I (z,v,u,s,θ) =
1
v

[
ln

(
v2 + z2 +2(vu+ iz

√
1+u2)

v2 + z2−2(vu− iz
√

1+u2)

)

×
(

nB(s
√

1+u2 +2πiθ)+nB(s
√

1+u2−2πiθ− isz)
)]

+C.C. . (17)

Notice that the variable z defined in (15) is discrete once k3, defined in (13), is a discrete variable.
Let us, first, study the correction to the mass when the propagation is perpendicular to the x3 axis. In this case the external

momentum is given by kµ = (k⊥,0), that is, k3 = 0. This amounts to z = 0 in (17), what leads to

I (z = 0,v,u,s,θ) =
1
v

[
ln

(
(v+2u)2

(v−2u)2

)(
nB(s

√
1+u2 +2πiθ)+nB(s

√
1+u2−2πiθ)

)]
. (18)

By substituting (18) into (16) we have the self-energy for a propagation perpendicular to the x3 axis, whose zero-momentum
limit, k⊥→ 0, gives the zero-order correction, in powers of v2 = k⊥2/m2, to the mass for the B(x) field. This limit is equivalent
to take v→ 0, that is

lim
v→0

I (z = 0,v,u,s,θ) =
2
u

(
nB(s

√
1+u2 +2πiθ)+nB(s

√
1+u2−2πiθ)

)
, (19)

By using (16), (19) and the definitions (15) we have

lim
k⊥→0

Π̃′(a,θ,k⊥ = 0,k3) = g2 1
(2π)2

1
2

∫ ∞

0
du

1√
1+u2

nB(ma
√

1+u2 +2πiθ)+C.C . (20)

We would have obtained the same result if we had set the external momenta equal zero from the very beginning, that is, if we
had assumed k⊥ = 0 and n = 0 in the expression (7), but this procedure is not reliable since the variables z and v are not treated
in the same foot and the integrand of (16) is not analytic in the origin of the (z,v) plane. This can be seen by taking the limit
v→ 0 and then set z = 0.

Now, let us consider the correction to the mass when the propagation is parallel to the x3 axis. In this situation the external
momentum is given by kµ = (0,k3), that is, k⊥ = 0, or equivalently v = 0. So, by taking v = 0 in the expression (17) we obtain

I (z,v = 0,u,s,θ) =
4u

z(z+2i
√

1+u2)

(
nB(s

√
1+u2 +2πiθ)+nB(s

√
1+u2−2πiθ− isz)

)
+C.C. (21)

The above expression evaluated at z = 0 furnishes us the
zero-order correction, in powers of z2 = k3

2/m2, to the mass
of the external field when the propagation is parallel to the



214 Brazilian Journal of Physics, vol. 38, no. 2, June, 2008

compactified axis. One can see that the above expression is
not well defined for n = 0, so we need a different prescription
for I (z,v = 0,u,s,θ) which can obtained by constructing a
function of the continuous variable z defined in such a way
that, when z = 2π/(ma)n, it coincides with the values given
by the right hand side of (21). In addition, this new function
must have a well defined limit when z = 0.

In order to do that it is important to notice that the last term
in the argument of the second B-E factor of the expression
(21) (the same factor is also present in the C.C. terms) can be
rewritten as isz = 2πn. Then, from (10), one can conclude that
they contribute a factor exp(±i2πn) = 1 for any n integer and

as such, does not influence the result of I (z,v = 0,u,s,θ)||z=0.
As a matter of fact this argument is correct only for n 6= 0, but
not for n = 0, since each term of I (z,v = 0,u,s,θ) exhibits
divergences in n = 0. In other words, once expression (21) is
well defined for n 6= 0 one can put exp(±i2πn) = 1, for n = 0
this expression is ill defined, so we need another prescription
for I (z,v = 0,u,s,θ) in this case.

The presence of the the term exp(±i2πn) in equation (21)
plays a central role in the final results, as shall be exposed. If
one takes exp(±i2πn) = 1 in the expression (21) for any value
of n (including n = 0) one obtains

I ′(z,v = 0,u,s,θ) =
4u

z(z+2i
√

1+u2)

(
nB(s

√
1+u2 +2πiθ)+nB(s

√
1+u2−2πiθ)

)
+C.C. =

=
8u

z2 +4(1+u2)

(
nB(s

√
1+u2 +2πiθ)+nB(s

√
1+u2−2πiθ)

)
(22)

which has a well defined value for z = 0, namely

I ′(z,v = 0,u,s,θ)
∣∣
z=0 =

2u
(1+u2)

(
nB(s

√
1+u2 +2πiθ)+nB(s

√
1+u2−2πiθ)

)
. (23)

By substituting this result in the expression (16), one can deduce that the screening mass for the field propagating along the x3
direction is given by

Π̃′(a,θ,m,k⊥,k3)
∣∣
k3=0 = g2 1

(2π)2
1
2

∫ ∞

0
du

u2

1+u2 nB(ma
√

1+u2 +2πiθ)+C.C . (24)

It is worth mentioning that the prime in equations (23) and (24) means that these quantities where obtained by taking
exp(±i2πn) = 1 in equation (21) and setting z = 0 after that.

One can see that the corrections to the mass given by expressions (20) and (24) do not coincide with each other. We say that
the limit k3 = 0,k⊥→ 0 does not commute with the limit k⊥→ 0,k3 = 0.

Note that in obtaining the result (24) we are assuming that in the expression (22) z is a continuous variable and the expression
(23) is the analytic behavior of the limit limz→0I (z,v = 0,u,s,θ). On the other hand, by assuming that the behavior of I (z,v =
0,u,s,θ) for small values of k3 (k3 << 2π/a) must be analytically continued by taking into account the contribution of exp(±isz),
we have that

lim
z→0

I (z,v = 0,u,s,θ) =
2u√

1+u2
×

[
1√

1+u2
nB(s

√
1+u2 +2πiθ)

+ s exp
(

s
√

1+u2 +2iπθ
)[

nB(s
√

1+u2 +2πiθ)
]2

]
+C.C. , (25)

which differs from the result in (23) by the term involving the square of the B-E factor.
By substituting (25) into (16) we have

lim
k3→0

Π̃(a,θ,m,k⊥ = 0,k3) = g2 1
(2π)2

1
4

∫ ∞

−∞
du u

[
u

(1+u2)3/2 nB(s
√

1+u2 +2πiθ)

+
su

1+u2 exp
(

s
√

1+u2 +2iπθ
)[

nB(s
√

1+u2 +2πiθ)
]2

]
+C.C. , (26)

which can also be written in the form

lim
k3→0

Π̃(a,θ,m,k⊥ = 0,k3) =−g2 1
(2π)2

1
4

∫ ∞

−∞
du u

d
du

[
1√

1+u2
nB(s

√
1+u2 +2πiθ)

]
+C.C. . (27)



F. A. Barone and M. Hott 215

By performing an integration by parts and using (15) and (20), it can be shown that

lim
k3→0

Π̃(a,θ,m,k⊥ = 0,k3) = g2 1
(2π)2

1
2

∫ ∞

0
du

1√
1+u2

nB(ma
√

1+u2 +2πiθ)+C.C.

= lim
k⊥→0

Π̃(a,θ,m,k⊥,k3 = 0) . (28)

That is, the correction to the mass generated when the propa-
gation is parallel to the x3 axis is the same as the one generated
in a propagation perpendicular to the x3 axis.

At this point we would like to mention that in the reference
[17] a gauge invariant mass term for real photons propagating
in the direction perpendicular to the compactified axis is also
found due to vacuum polarization. By working with Feyn-
man parametrization to calculate the effects of twisted and
untwisted spinors L. Ford found an expression for the vacuum
polarization tensor that is analytic in the origin of momenta
space and, as a result, the photon mass generated dynamically
is unique. In this section we have shown the origin of the non-
analyticity on the self-energy but have also shown how to cir-
cumvent this problem by considering an analytic continuation
for the self-energy for small values of the external momenta
and, as a consequence, we have also found a unique result for
the correction to the mass of the real scalar field.

We notice that a derivative expansion of the effective ac-
tion is still put in jeopardy, because the next coefficients in
the Taylor series exhibits a more severe non-analyticity. Once
each of those coefficients is obtained from derivatives of the
self-energy evaluated at zero-momentum, and each derivative

results in one extra power of k in the numerator, the zero-
momentum limit of these higher power of momenta in the nu-
merator dominate over the zero-momentum limit of the B-E
factors.

IV. SELF-ENERGY FOR AN ARBITRARY PROPAGATION
DIRECTION

In this section we study the behavior of the zero-momentum
limit of the self-energy when it is taken in an arbitrary direc-
tion close to the origin of the external momenta plane. The
approach used here is the same developed in the first of the
references in [20] to analyze the non-analyticity in the context
of QFTFT. We are interested in the correction to the mass of
the real B(x) field when this is propagating along an arbitrary
direction. We consider the analytic extension of expression
(17) for small values of z by taking into account the factors
exp(±isz) . This all means that, for small values of momenta,
z is a continuous variable and we can write the components of
the external momentum kµ in polar coordinates, as follows

k⊥ = k cosϕ , k3 = k sinϕ with − (π/2)≤ ϕ≤ (π/2) , (29)

in such a way that expression (17) becomes

I (z,v,u,s,θ) = I (ξ,ϕ,u,s,θ) =
1

ξcosϕ

[
ln

(
ξ+2(ucosϕ+ i

√
1+u2 sinϕ)

ξ−2(ucosϕ− i
√

1+u2 sinϕ)

)

×
(

nB(s
√

1+u2 +2πiθ)+nB(s
√

1+u2−2πiθ− isξsinϕ)
)]

+C.C. , (30)

where we have defined the dimensionless quantity ξ = k/m.
The zero-momentum limit (k → 0) is taken by maintaining ϕ, such that we can control the direction we approach the origin

of the momentum plane. In this way we have

lim
ξ→0

I (ξ,ϕ,u,s,θ) =
2u

u2 + sin2 ϕ
nB(s

√
1+u2 +2πiθ)+

+

[
i ln

(
+2ucosϕ+2i

√
1+u2 sinϕ

−2ucosϕ2i
√

1+u2 sinϕ

)
tanϕ sexp(s

√
1+u2 +2πiθ)

(
nB(s

√
1+u2 +2πiθ)

)2
]
+C.C. . (31)

Note that for ϕ = 0, which corresponds to k3 = 0 and a propagation perpendicular to the x3 axis, expressions (31) and (19) are
identical to each other, as they might be. Similarly, for a propagation parallel to the x3 axis we take ϕ =±π/2 and the expression
(31) becomes equivalent to (25). This last case can be obtained by using the fact that

lim
ϕ→±π/2

[
tanϕ ln

(
2ucosϕ+2i

√
1+u2 sinϕ

−2ucosϕ+2i
√

1+u2 sinϕ

)]
=−2i

u√
1+u2

. (32)



216 Brazilian Journal of Physics, vol. 38, no. 2, June, 2008

We would like to point out that the logarithm that appears in equation (31) is pure imaginary, namely

ln

(
+2ucosϕ+2i

√
1+u2 sinϕ

−2ucosϕ+2i
√

1+u2 sinϕ)

)
= i

[
2arctan

(√
1+u2

u
tanϕ

)
−π sgn

(√
1+u2

u
tanϕ

)]
, (33)

where

sgn(x) =





1 if x > 0
0 if x = 0
−1 if x < 0

. (34)

By substituting (31) into (16) and using the definition of s (15), we have that the correction to the mass when the B(x) field
propagates along an arbitrary direction in fact depends on the direction we approach the origin of the external momenta plane
since the result depends on ϕ, namely

Π̃(a,θ,m,k⊥ = 0,k3 = 0) = g2 1
(2π)2

1
4

∫ ∞

0
du

u√
1+u2

[
2u

u2 + sin2 ϕ
nB(ma

√
1+u2 +2πiθ)+

+

(
i ln

(
+2ucosϕ+2i

√
1+u2 sinϕ

−2ucosϕ+2i
√

1+u2 sinϕ

)
(tanϕ) maexp(ma

√
1+u2 +2πiθ)

(
nB(ma

√
1+u2 +2πiθ)

)2
)]

. (35)

For θ = 0 and θ = 1/2 we have the cases of periodic and
anti-periodic boundary conditions, respectively. We could
take θ = 0 or θ = 1/2 from the beginning of the calculations,
such that the results would be the same as the ones obtained
from (35).

V. CONCLUSIONS AND FINAL REMARKS

In this paper we have found the correction to the mass for
a real scalar field, in 3 + 1 dimensions and in the static limit,
due to quantum fluctuations of a complex scalar field with one
spatial dimension compactified. Specifically, we have submit-
ted the real scalar field to periodic conditions and the complex
scalar field to quasi-periodic conditions. The results interpo-
late continuously the periodic and anti-periodic cases for the
complex field.

We have shown that, despite the fact that the correction to
the mass exhibits a dependence on the direction we approach
the origin of the external momenta plane, the results for prop-
agations parallel to and perpendicular to the compactified axis
are equal to each other, that is, we have the same contribution
to the mass for the cases where ϕ = 0 and ϕ =±π/2. As can
be seen from (35), this curious fact does not happen for any
other propagation direction.

It would be interesting to extend these results to the same
model but in 4 + 1 dimensions, in order to investigate if it is
possible to have a mechanism of mass generation for fields
due to the existence of an extra spatial dimensions compacti-
fied.

Appendix

The generating functional for the Lagrangian (1) is given
by the path integral

Z[J] =
∫

Dφθ

∫
DBP exp

[
−

∫
d4x [L + J(x)B(x)]

]
, (36)

where Dφθ and DBP means that the integrals must be taken
only over field configurations that satisfies the conditions (2)
and (3).

By following the same procedure used in quantum field the-
ory without boundary conditions, we write

φ =
1√
2
(φ1 + iφ2) , (37)

what allows us to integrate (36) on the fields φ1 and φ2

Z[J] =
∫

DBP exp
[−tr ln(1−gĜB)

]
exp

[
−

∫
d4x

1
2

(
(∂µB)(∂µB)+M2B2)+ JB

]
, (38)

where we have discarded a normalization factor, Ĝ is the propagator operator of the scalar field φθ(x) under the boundary
condition (3) and we have used the definition (1).

Up to second order in the coupling constant g, we can write expression (38) in the form

Z[J] =
∫

DBP exp
[
−

∫
d4x

1
2

(
(∂µB)(∂µB)+M2B2)+(J−gG(0))B

]



F. A. Barone and M. Hott 217

exp
[
−g2

2

∫ ∫
d4xd4y B(y)G(y,x)G(x,y)B(x)

]
, (39)

where G(x,y) = 〈x|Ĝ|y〉 is the Green’s function of the scalar field with the conditions (3) and G(0) = G(x,x). From the above
expression one can define a new external current J′ = J−gG(0) and identify the effective action for the B field

Se f f =
∫

d4x
1
2

(
(∂µB)(∂µB)+M2B2)+

∫ ∫
d4xd4y

(
g2

2
B(y)G(y,x)G(x,y)B(x)

)
. (40)

It is convenient to write the Se f f in momentum space. For
this task we consider that, due to the conditions (2), the B field
can be written in the form

B(y) =
∫ d3k⊥

(2π)3

∞

∑
n=−∞

B̃n(k⊥)eik⊥·y⊥ψn(y3) , (41)

where quantities with subscript⊥ are to be read as those trans-
verse to the compactified axis and

ψn(y3) =
1√
a

exp
(

2nπi
a

)
. (42)

Due to (3), the Green’s function G(x,y) is written as

G(x,y) =
∫ d3k⊥

(2π)3 eip⊥·(x⊥−y⊥)
∞

∑
`=−∞

φ`(x3)φ∗`(y
3)

p2
⊥+q2

` +m2 , (43)

where

φ`(x3) =
1√
a

exp
(

i
2`π+θ

a

)
(44)

and

q` =
2`π+θ

a
. (45)

By substituting (41), (42), (43), (44) and (45) in (40) and
using the identities

∫
dy⊥e−i(k⊥+p⊥−p′⊥)·y⊥ = (2π)3δ3(k⊥+ p⊥− p′⊥) (46)

and

∫
dy3ψn(y3)φ∗`′(y

3)φ`(y3) =
1√
a

δn+`−`′,0 , (47)

where δn+`−`′,0 is the Kronecker delta, we can rewrite the ef-
fective action (40) in the form (6) with the corresponding ef-
fective Lagrangian (4).

Acknowledgments

MH thanks to Conselho Nacional de Desenvolvimento
Cientı́fico e Tecnológico (CNPq-Brasil) for the financial sup-
port. FAB thanks to Fundação de Amparo à Pesquisa do Es-
tado de São Paulo (FAPESP) for the financial support.

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