arXiv:0704.1990v2  [hep-th]  3 Apr 2008
Vacuum Polarization by a Magnetic Flux Tube at
Finite Temperature in the Cosmic String Spacetime
J. Spinelly1 ∗and E. R. Bezerra de Mello2 †
1.Departamento de F́ısica-CCT
Universidade Estadual da Paráıba
Juvêncio Arruda S/N, C. Grande, PB
2.Departamento de F́ısica-CCEN
Universidade Federal da Paráıba
58.059-970, J. Pessoa, PB
C. Postal 5.008
Brazil
November 11, 2018
Abstract
In this paper we analyse the effect produced by the temperature in the vacuum
polarization associated with charged massless scalar field in the presence of magnetic
flux tube in the cosmic string spacetime. Three different configurations of magnetic
fields are taken into account: (i) a homogeneous field inside the tube, (ii) a field
proportional to 1/r and (iii) a cylindrical shell with δ-function. In these three
cases, the axis of the infinitely long tube of radius R coincides with the cosmic
string. Because the complexity of this analysis in the region inside the tube, we
consider the thermal effect in the region outside. In order to develop this analysis,
we construct the thermal Green function associated with this system for the three
above mentioned situations considering points in the region outside the tube. We
explicitly calculate in the high-temperature limit, the thermal average of the field
square and the energy-momentum tensor.
PACS numbers: 98.80.Cq, 11.27. + d, 04.62. + v
1 Introduction
It is well known that different types of topological defects may have been created in the
early Universe after the Planck time by the vacuum phase transition [1, 2]. These include
domain walls, cosmic strings and monopoles. Among them cosmic string and monopole
seem to be the best candidates to be detected.
∗E-mail: jspinelly@uepb.edu.br
†E-mail: emello@fisica.ufpb.br
http://arxiv.org/abs/0704.1990v2
Many years ago Nielsen and Olesen proposed a theoretical model comprised by Higgs
and gauge fields, that by a spontaneous broken of gauge symmetry produces linear topo-
logical defect carrying a magnetic flux named vortex [3]. A few years later, Garfinkle
investigated the influence of this topological object on the geometry of the spacetime
[4]. Coupling the energy-momentum tensor associated with the system to the Einstein
equations, he found static cylindrically symmetric solutions. The author also shown that
asymptotically the spacetime around the vortex is a Minkowiski one minus a wedge. The
core of the vortex has a non-zero thickness and the magnetic flux inside. Two years later
Linet [5] obtained, under some specific condition, exact solutions for the complete set of
differential equation. He was able to show that the structure of the respective spacetime
corresponds to a conical one, with the conicity parameter being expressed in terms of the
energy per unity length of the vortex.1
In recent papers, we have investigated the vacuum polarization effects associated with
massless scalar [7, 8] and fermionic [9, 10] fields, in the presence of a magnetic flux tubes
of finite radius in the cosmic string spacetime at zero temperature. In these analysis
we considered that the magnetic fields are confined into an infinitely long tube of radius
R around the cosmic string. Three different configurations of magnetic field, H(r), are
taking into account in our analysis:
i) H(r) =
Θ(R− r) , homogeneous field inside, (1)
ii) H(r) =
2παRr
Θ(R− r) , field proportional to 1/r inside, (2)
iii) H(r) =
δ(r − R) , cylindrical shell, (3)
where R is the radius extent of the tube, Θ is the Heaviside’s function and Φ is the
total magnetic flux.2 The ratio of the flux to the quantum flux Φo, can be expressed by
δ = Φ/Φ0 = N + γ, where N is the integer part and 0 < γ < 1.
In the framework of the quantum field theory at finite temperature, a fundamental
quantity is the thermal Green function, Gβ(x, x
′). For the scalar field it should be periodic
in the imaginary time with period β, which is proportional to the inverse of the temper-
ature. Because we are interested to obtain the thermal Green function, it is convenient
to work in the Euclidean analytic continuation of the Green function performing a Wick
rotation on the time coordinate, t → iτ . So, we shall work on the Euclidean version of
the idealized cosmic string spacetime, which in cylindrical coordinates, can be described
by the line element below:
ds2 = dτ 2 + dr2 + α2r2dθ2 + dz2 , (4)
where α is a parameter smaller than unity which codify the presence of a conical two-
surface (r, θ) .3
1The complete analysis about the behavior of the gauge and matter fields near the cosmic string’s core
can only be obtained numerically. Some recent numerical analysis [6] about the structure of supermassive
cosmic strings show that two different kind of solutions for the metric tensor exist.
2The configurations for magnetic flux provide the same magnetic flux on the two-surface perpendicular
to the z−axis in coordinate system defined in (4).
3For a typical Grand Unified Theory, α = 1−O(10−6).
The vacuum polarization effects associated with a charged scalar field due to a mag-
netic field confined in a tube of finite radius in Minkowski spacetime has been first analysed
by Serebryanyi [11]. A few years later this analysis, for an idealized cosmic string space-
time, has been considered by Guimarães and Linet [12]; however the magnetic flux was
considered as being a line running through the string. The effect of the temperature on
this vacuum polarization was also investigated by Guimarães in [13]. In this context,
inspired by our previous work [7, 8], we decided to investigate the effect of the temper-
ature on the vacuum polarization effects associated with charged massless scalar field in
presence of magnetic flux tube in the cosmic string spacetime, considering the three dif-
ferent configurations of magnetic field given before. The standard procedure to develop
this analysis is by calculating the respective thermal Euclidean Green function. This can
be done for an ultrastatic spacetime4 by knowing the Green function at zero tempera-
ture. The analysis of the thermal effects on the vacuum polarization effects associated
with massless bosonic and fermionic fields in the global monopole spacetime have been
considered in [14, 15, 16] few years ago.
This paper is organized as follows: In the section 2 we calculate the thermal Euclidean
Green function associated with the system for the three different models of magnetic
fields. Using the results obtained, we calculate in the sections 3 and 4, the thermal
renormalized vacuum expectation value of the field square and the energy-momentum
tensor, respectively. We leave for the section 4 our conclusions.
2 The Euclidean thermal Green function
The Green function associated with the charged massless scalar field at zero temperature
in the presence of a electromagnetic field, must obey the following non-homogeneous
second-order differential equation
−ggµνDν
G(x, x
) = −δ(4)(x− x′) , (5)
where Dµ = ∂µ − ieAµ, Aµ being the four-vector potential.
In order to reproduce the configurations of magnetic fields along the z−direction given
by (1)-(3), we write the the vector potential by Aµ = (0, 0, A(r), 0), with
A(r) =
a(r) . (6)
For the two first models considered, we can represent the radial function a(r) by:
a(r) = f(r)Θ(R− r) + Θ(r −R) , (7)
f(r) =
r2/R2, for the model (i) and
r/R, for the model (ii).
For the third model,
a(r) = Θ(R− r) . (9)
4An ultrastatic spacetime admits a globally defined coordinate system in which the components of the
metric tensor are time independent and the conditions g00 = 1 and g0i = 0 hold
As we have mentioned, in this work we shall continue in the same line of investigation
started in [7, 8], calculating at this time, the thermal contribution on the vacuum polar-
ization effects in the region outside the magnetic tube for the three magnetic fields under
consideration.
The Euclidean Green functions for points outside the magnetic flux tube at zero-
temperature are given below:
• For the models (i) and (ii),
T=0(x, x
eiN∆θ
8π2αrr′ sinh u0
ei∆θ sinh(γu0/α) + sinh[(1− γ)u0/α]
cosh(u0/α)− cos∆θ
dωωJ0
(∆τ)2 + (∆z)2
ein∆θDjn(ωR)K|ν|(ωr)K|ν|(ωr
′), j = 1, 2 , (10)
where
n−N − γ
, (11)
cosh uo =
r2 + r
+ (∆τ)2 + (∆z)2
Djn(ωR) =
H ′j(R)I|ν|(ωR)−Hj(R)I ′|ν|(ωR)
Hj(R)K
|ν|(ωR)−H ′j(R)K|ν|(ωR)
. (13)
In the above equations the functions Hj(r) are given by:
H1(r) =
Mσ1,λ1
, (14)
with σ1 = (
− ω2R2α
)/2 and λ1 = n/2α, and
H2(r) =
Mσ2,λ2 (ζr) , (15)
with σ2 =
(δ2 + ω2R2α2)−1/2, λ2 = n/α and ζ =
(δ2 + ω2R2α2)1/2. Moreover,
Mσ,λ is the Whittaker function, while I|ν| e K|ν| are the modified Bessel functions
[17].
• For the model (iii), we have
T=0(x, x
8π2αrr′ sinh u0
sinh(u0/α)
cosh(uo/α)− cos∆θ
dωωJ0
(∆τ)2 + (∆z)2
ein∆θDn(ωR)K|n/α|(ωr)K|n/α|(ωr
′) . (16)
where
Dn(ωR) =
|ν|(ωR)I|n|/ω(ωR)− I|ν|(ωR)I
|n|/α(ωR)
I|ν|(ωR)K
|n|/α(ωR)− I
|ν|(ωR)K|n|/α(ωR)
. (17)
We can observe that the first term on the right hand side of (10) is, up to a gauge
transformation, equivalent to the result presented by Guimarães and Linet [12] for a
charged massless scalar field in the presence of a magnetic flux running through the
cosmic string; and as to (16), its first term corresponds the results found by Smith [18]
and by Linet in [19] for a massless scalar field without charge. However, the seconds
terms of booths expressions represent corrections on the respective Green functions due
to a non-vanishing radius R attributed to the magnetic flux; off course these corrections
vanish when we take R → 0.
Following the prescription given in the papers by Braden [20] and Page [21], the
thermal Green function, GT (x, x
′), can be expressed in terms of the sum
GT (x, x
GT=0(x, x
′ − lλβ) , (18)
where λ is the ”Euclidean” unitary time-like vector and β = 1/kBT , being kB the Boltz-
mann constant and T the absolute temperature.
In agreement with the equations (10), (16) and (18), the thermal Green functions
associated with the massless scalar field in the cosmic string spacetime and in the presence
of magnetics field, are given by:
• For the models (i) and (ii),
T (x, x
′) = G
T (x, x
dωωJ0
(∆τ + lβ)2 + (∆z)2
ein∆θDjn(ωR)K|ν|(ωr)K|ν|(ωr
′) , j = 1, 2 . (19)
• For the model (iii),
T (x, x
′) = GT (x, x
dωωJ0
(∆τ + lβ)2 + (∆z)2
ein∆θDn(ωR)K|n|/α(ωr)K|n|/α(ωr
′) . (20)
The thermal Green functions G
T (x, x
′) and GT (x, x
′) which appear in (19) and (20)
have being obtained a few years ago by Guimarães [13] and Linet [22], respectively. So, we
shall not repeat them. In fact what we are really interested in this paper is to analyze the
seconds terms in these Green functions, which correspond to the thermal contributions
due to the finite thickness admitted for the magnetic flux tube.
3 The Computation of 〈φ̂∗(x)φ̂(x)〉Ren. at Non-zero Tem-
perature
The main objective of this section is to investigate the effects produced by the temperature
in the renormalized vacuum expectation value of the square of the charged massless scalar
field, 〈φ̂∗(x)φ̂(x)〉, in the presence of a magnetic flux tube of finite radius. Formally this
quantity is given by taking the coincidence limit of the Green function:
〈φ̂∗(x)φ̂(x)〉T = lim
GT (x, x
′) . (21)
However, this procedure provides a divergent result and the divergence comes exclusively
from the first terms of the right hand side of (19) and (20)5. In order to obtain a finite and
well defined result, we must apply some renormalization procedure. Here we shall adopt
the point-splitting renormalization one. It has been observed that the singular behavior
of the Green function has the same structure as given by the Hadamard one, which on
the other hand can be written in terms of the square of the geodesic distance between two
points. So, here we shall adopt the following prescription: we subtract from the Green
function the Hadamard one before applying the coincidence limit as shown below:
〈φ̂∗(x)φ̂(x)〉T,Ren. = lim
[GT (x, x
′)−GH(x, x′)] . (22)
We can write the result as:
〈φ̂∗(x)φ̂(x)〉T,Ren. = 〈φ̂∗(x)φ̂(x)〉T,Reg. + 〈φ̂∗(x)φ̂(x)〉CT=0 + 〈φ̂∗(x)φ̂(x)〉CT . (23)
The first term on the right hand side of the above expression, represents, for the
models (i) and (ii), the thermal contribution coming from the interaction between charged
massless scalar field with a magnetic flux considered as a line running along the cosmic
string. From the Guimarães’s paper [13], this term is:
〈φ̂∗(x)φ̂(x)〉T,Reg. =
16π2αβr
r cosh u/2
cosh u/2
F (γ)α (u, 0) du , (24)
where
F (γ)α (u, 0) = −2
sin [πγ/α] cos [u (1− γ) /α] + sin [u (1− γ) /α] cos [πγ/α]
cosh u/α− cosπ/α
. (25)
For the model (iii) an analogous expression can be obtained from the previous one, by
taking γ = 06. (An interesting aspect of these results is that the vacuum polarizations
depend only on the fractional part of the ration of the magnetic flux by the quantum one,
In the high-temperature limit (β → 0), Guimarães showed that,
〈φ̂∗(x)φ̂(x)〉T,Reg. ≈
M (γ)
, (26)
where the constant M (γ) is given by
M (γ) =
16π2α
F (γ)α (u, 0)
cosh(u/2)
du , (27)
5A special feature of these Green functions is that the correction due to the magnetic tube’s radius is
finite in the coincidence limit.
6See paper [22].
and in the zero-temperature limit (β → ∞),
〈φ̂∗(x)φ̂(x)〉T,Reg. ≈
, (28)
with ω(γ) being given by
ω(γ) = − 1
γ − 1
. (29)
The two last terms in (23) are corrections on the vacuum polarization due to fi-
nite thickness of the radius of the magnetic tube and non-zero temperature. The term
〈φ̂2(x)〉CT=0, corresponds to correction due to finite thickness of the radius of tube only, it
was given in [7, 8]. For the two firsts models the corrections are similar. They are given
by the component l = 0 in (19), and read
(2π)2α
Din(ωR)K
|ν|(ωr) . (30)
Because we are mostly interested to study the vacuum polarization for points very far
from the cosmic string, we shall consider R/r ≪ 1. Comparing the behavior of Din(ωR)
with the behavior of K2|ν|(ωr), we have shown that we may approximate the integrand of
(30) assuming to the coefficient Djn its first order expansion in ωR , which reads:
Djn(ωR) = −
Γ(|ν| + 1)Γ(|ν|)
wnj − |ν|znj
wnj + |ν|znj
)2|ν|
, (31)
where
wn1 =
δ − n
Mλ1,λ2(δ/α) +
Mγ1,λ1(δ/α), (32)
zn1 = Mλ1,λ2(δ/α), (33)
wn2 =
δ − n
Mλ2,λ1(2δ/α) +
Mγ2,λ2(2δ/α) , (34)
zn2 = Mλ2,λ1(2δ/α) , (35)
being γ1 = (n+2α)/2α, γ2 = (n+α)/α and ν =
n−N−γ
. The most important contributions
in (30) comes from the component n = N in the summation. In this way, considering
only the dominant term, the corrections are given by:
〈φ̂∗(x)φ̂(x)〉CT=0 = −
(2π)2r2
α(2γ + α)
wNj − (γ/α)zNj
wNj + (γ/α)z
)2γ/α
. (36)
As we can see the corrections present extra dependence on the radial coordinate, conse-
quently they are appreciable only in the region close to the tube.
As to the third model there happens a very interesting phenomenon. For n 6= 0, the
coefficient Dn(ωR) in (16) vanishes at least as fast as (ωR)
2|n|/α when ωR → 0. On the
other hand, D0(ωR) vanishes only with the inverse of the logarithm, so quite slowly. In
this way the most relevant contribution to the summation comes from n = 0. For this
D0(ωR) =
+ C − α/δ
, (37)
where C is the Euler constant. In [8] we have explicitly shown that the correction on the
vacuum polarization effect for the third model, consequence of a non-vanishing radius of
the magnetic flux, is mainly given by:
〈φ̂∗(x)φ̂(x)〉CT=0 = −
8π2αr2 ln
e−C+α/δ
) . (38)
We also have shown that for α = 0.99 and δ = 0.2 this term is of the same order of the
standard vacuum expectation value of the field square in the absence of magnetic flux, up
to the distance r which exceeds the radius of the observable Universe.
As we have already mentioned, the last term in (23), 〈φ̂∗(x)φ̂(x)〉CT , is consequence
of combined effects of the nonvanishing thickness of the magnetic tube and temperature.
This is a new contribution and we shall focus on it. For the first two models, this term
can be expressed by
〈φ̂∗(x)φ̂(x)〉CT =
2π2αr2
dvvJ0 (vζl)
Djn(vR/r)K
|ν|(v) , j = 1, 2 , (39)
with ζ = β/r. In the above expression we have introduced a dimensionless variable
v = ωr.
As in the zero temperature analysis, because we are considering R/r ≪ 1, the most
important contribution for the above summation comes from the n = N component. So,
on basis what we have already discussed for points very far from the string, equation (39)
can be approximate to
〈φ̂∗(x)φ̂(x)〉CT = −
(2π)2αr2
Γ(γ/α + 1)Γ(γ/α)
wNj − (γ/α)zNj
wNj + (γ/α)z
)2γ/α
dvv1+2γ/αJ0 (vζl)K
|γ/α|(v) , j = 1, 2 . (40)
From the above expression we can observe that the thermal content of this correction
is in the summation S given below:
S(ζ) =
dvv1+2γ/αJ0 (vζl)K
|γ/α|(v) . (41)
Unfortunately it is not possible to obtain a closed expression to this term, and provide a
complete information about the thermal behavior of (40); on the other hand, it is possible
to give its main information. In order to do that we shall divide the integral above in
two parts: from [0, 2π/ζ ] and from [2π/ζ,∞). Due the strong exponential decay of the
modified Bessel function for large argument, the contribution in the interval [2π/ζ,∞)
can be neglected in the high-temperature regime, i.e., for ζ ≪ 1; moreover using the series
properties for the Bessel function [17],
J0(vζl) = −
, 0 < v < 2π/ζ , (42)
(41) can be approximated to
S(ζ) = −1
∫ 2π/ζ
dvv1+2γ/αK2|γ/α|(v) +
∫ 2π/ζ
dvv2γ/αK2|γ/α|(v) . (43)
In the high-temperature limit the most relevant contribution is given by the second
term. Adopting the same approximation criterion to discard the integral in the inter-
val [2π/ζ,∞), we may evaluate the integral by taking its upper limit going to infinity. In
this way the most relevant contribution to the thermal vacuum polarization effect is [17]7:
S(ζ) =
22−2γ/α
Γ2 (1/2 + γ/α) Γ (1/2 + 2γ/α)
Γ (1 + 2γ/α)
. (44)
Consequently, in the high-temperature regime the correction term, 〈φ̂∗(x)φ̂(x)〉CT , is dom-
inated by:
〈φ̂∗(x)φ̂(x)〉CT = −
(γ,α,R)
)2γ/α
, j = 1, 2 , (45)
(γ,α,R)
4π3/2α
Γ2 (1/2 + γ/α) Γ (1/2 + 2γ/α)
Γ (1 + 2γ/α) Γ(1 + γ/α)Γ(γ/α)
wNj − (γ/α)zNj
wNj + (γ/α)z
Hence, the expression for renormalized vacuum expectation value of the square of the
charged massless scalar field, in the high-temperature limit, is given by
〈φ̂∗(x)φ̂(x)〉T,Ren. ≈
M (γ)
(γ,α,R)
)2γ/α
. (46)
From this expression we can observe that the two sub-dominant contributions are of the
same order of magnitude for points near the tube.
For the third model, 〈φ̂∗(x)φ̂(x)〉CT , is given by the expression below:
〈φ̂∗(x)φ̂(x)〉CT =
2π2αr2
dvvJ0 (vζl)
Dn(vR/r)K
|n|/α(v) . (47)
Again, as in the zero temperature analysis of the vacuum polarization effect, the
most relevant contribution comes from the n = 0 component of the above summation for
R/r ≪ 1. The respective coefficient is:
D0(vR/r) =
ln (v/qr)
, (48)
where
e−C+α/δ . (49)
In this way (47) can be written by:
〈φ̂∗(x)φ̂(x)〉CT =
2π2αr2
S̄(ζ) , (50)
7By numerical analysis, in [24] we have reached to S(ζ), the same behavior found here, i.e., S(ζ) ≈ 1/ζ.
where
S̄(ζ) =
dvvJ0 (vζl)D0(vR/r)K
0(v) . (51)
Applying the previous summation property to the Bessel functions, in the high-
temperature regime, (51) can be given by
S̄(ζ) = −
∫ 2π/ζ
dvvD0(vR/r)K
0(v) +
∫ 2π/ζ
dvD0(vR/r)K
0(v) . (52)
For ζ << 1 the most relevant contribution to S̄ is given by second term; moreover in
this limit we can obtain an approximated expressions for it by taking the upper limit of
the integral going to infinite. In this way (52) can be evaluated by
S̄(ζ) ∼=
K20 (v)
ln(v)− ln(qr)
. (53)
Unfortunately this expression presents a pole for v = qr. However, this pole is con-
sequence of the approximation adopted and it is located in the region where where the
approximation is no more valid8. In fact the full expression to D0(vR/r) has no singu-
larity. D0 is a slowly increasing function of v. As we have mentioned, the full expression
to the coefficient D0 vanishes with the inverse of the logarithm for v ≈ 0, and grows less
slower than e2vR/r for large v. The square of the modified Bessel function, K20 , provides
an integrable logarithmic divergence for small argument, and decays with e−2v for large
argument. Except for very small value of v, where the dominant contribution to D0 is
given by 1/ ln(v), D0 can be well approximated by
9 −1/ ln(qr). In figure 1, this argument
is also numerically justified. There, it is exhibited the behavior of the exact integrand of
S(ζ), and its approximated expression discarding the factor ln(v) in the denominator, for
specific values of the parameters.
Accepting the above arguments, we can represent the main contribution to (53) by:
S̄(ζ) = − 1
ζ ln(qr)
dvK20(v) , (54)
Consequently we obtain
〈φ̂∗(x)φ̂(x)〉CT = −
8αβr ln
e−C+α/δ
] . (55)
Now for this model, the expression to the renormalized vacuum polarization effect in
high-temperature limit reads
〈φ̂∗(x)φ̂(x)〉T,Ren. ≈
M (γ)
8αβr ln
e−C+α/δ
] . (56)
8For γ = 0.2, N = 0 and α = 0.99, qr = 2r
eC−α/ζ is of order 105 for r/R = 103, so vR/r = 102
consequently bigger than unity.
9In [23], B. Allen at all adopted a similar procedure to provide an approximate expression to the
vacuum expectation value of the field square in the cosmic string spacetime considering a general structure
to its core; in [8] this procedure has is also been adopted to calculate the 〈φ∗φ〉.
0.5 1 1.5 2 2.5 3 3.5 4
Figure 1: The dashed curve corresponds the exact integrand of the second term of the
right hand side of (52), and the solid line the approximated expression discarding ln(v)
in the denominator of (53). In the numerical analysis we have used α = 0.99, γ = 0.2,
N = 0 and r/R = 103.
On basis of previous discussion about the vacuum polarization effect at zero temper-
ature, we can conclude that although being sub-dominant, the thermal correction due to
the non vanishing radius of the magnetic flux tube, is so relevant as the standard term pro-
portional to 1/βr, for points at very large distance from the cosmic string. Consequently,
it can be considered as a long-range effect.
4 The Computation of 〈T̂ νµ (x)〉Ren. at Non-zero Tem-
perature
The energy-momentum tensor, Tµν(x), is a bilinear function of the fields, so we can
evaluate its vacuum expectation value, 〈Tµν(x)〉, by the standard method using the Green
function [25]. The thermal vacuum average, consequently, can also be obtained by using
the thermal Green function.
The renormalized vacuum expectation value of the energy-momentum tensor at non-
zero temperature for the system adopted here can be calculated by:
〈T̂ νµ 〉T,Ren. = lim
µ(α,φ)GT (x, x
′)−Dν′µ(1,0)GH(x, x′)] , (57)
where Dν
µ(α,φ) stands a second order non-local differential operator in presence of a mag-
netic field and cosmic string, defined by
µ(α,φ) = (1− 2ξ)DµD
ν′ − ξ(DµDν +Dµ′D
2ξ − 1
δνµDσD
, (58)
ξ being the non-minimal curvature coupling. In the above expression we consider that
Dσ = ∇σ−ieAσ, with∇σ being the covariant gravitational derivative and the bar denoting
complex conjugate. Again we shall consider the magnetic field configuration given by (7)
and (8) for the two first models, and by (9) for the third model. The respective thermal
Green function, GT (x
′, x), are given by (19) and (20), respectively.
Due to the form of the Green function, we can express (57) by
〈T̂ νµ (x)〉T,Ren. = 〈T̂ νµ (x)〉T,Reg. + 〈T̂ νµ (x)〉CT=0 + 〈T̂ νµ (x)〉CT . (59)
The first and second term of the right hand side of the above expression have already
been computed in [13] and [8], respectively. As to the first term, Guimarães has shown
〈T̂ νµ (x)〉T,Reg. =
diag(−3, 1, 1, 1) + 〈T̃ νµ (x)〉T , (60)
where the second term on the right hand side is given in terms of five integrals. However,
the author was able to show that in the high-temperature limit (β → 0), this term
is proportional to 1/βr3, and in the zero-temperature limit (β → ∞) 〈T̂ νµ (x)〉T,Ren. is
proportional to 1/r4. The second term in (59), 〈T̂ νµ (x)〉CT=0, is consequence of a non-
vanishing radius to the magnetic flux. For the two first models, its most important
contribution is proportional to (1/r4)(R/r)2γ/α, consequently is relevant only in the region
near the magnetic tube; on the other hand, for the third model this term presents a long-
range effect, similar to what happens in the vacuum expectation value of the field square.
The new contribution, 〈T̂ νµ (x)〉CT , is consequence of a non-vanishing radius attributed to
the magnetic flux tube and the temperature.
For the two first models this term is given by
〈T̂ νµ 〉CT = lim
µ(α,φ)G
C,(j)
T (x, x
′) , (61)
C,(j)
T (x, x
l 6=0
dωωJ0
(∆τ + lβ)2 + (∆z)2
ein∆θDjn(ωR)K|ν|(ωr)K|ν|(ωr
′) , j = 1, 2 . (62)
Following the same procedure adopted in the last section, we shall consider only the
n = N component in the summation. Substituting (62) into (61), and using the approx-
imated expression to the coefficient D
N , after long calculation we arrive at the following
result:
〈T̂ νµ 〉CT = −
4π2αr4
Γ (γ/α) Γ (1 + γ/α)
wNj − (γ/α)zNj
wNj + (γ/α)z
)2γ/α
0 , a
1 , a
2 , a
where
0 = (4ξ − 1)
3 + I
1 − 2(γ/α)I
2 + 2 (γ/α)
1 = I
1 − 2 (γ/α + 2ξ) I
2 + 4ξ(γ/α)I
4 − I
2 = 2(γ/α) (γ/α− 2ξ) I
4 + 4ξI
(4ξ − 1)
3 + I
1 − 2(γ/α)I
2 + 2 (γ/α)
3 = 2I
5 + (4ξ − 1)
3 + I
1 − 2(γ/α)I
2 + 2 (γ/α)
dvv3+2γ/αJ0 (vζl)K
γ/α+1(v) , (64)
dvv2+2γ/αJ0 (vζl)Kγ/α+1Kγ/α(v) , (65)
dvv3+2γ/αJ0 (vζl)K
γ/α(v) , (66)
dvv1+2γ/αJ0 (vζl)K
γ/α+1(v) (67)
dvv2+2γ/α
J1 (vζl)
K2γ/α(v) . (68)
In order to provide the most relevant contribution to (64)-(67), we adopt the same pro-
cedure adopted in last section: we divide the interval of integration in two parts, from
[0, 2π/ζ ] and from [2π/ζ,∞). Again in the high-temperature regime, ζ ≪ 1, the integral
in the second interval can be neglected due the exponential decay of the modified Bessel
function. Finally using the series properties for the Bessel function [17], and after some
intermediate steps, we obtain:
〈T̂ νµ 〉CT = −
γ,α,R
)2γ/α
0 , b
1 , b
2 , b
, j = 1, 2 , (69)
where
γ,α,R
(1 + 2γ/α)
2π3/2α
Γ2 (1/2 + γ/α) Γ (1/2 + 2γ/α)
Γ (3 + 2γ/α) Γ(1 + γ/α)Γ(γ/α)
wNj − (γ/α)zNj
wNj + (γ/α)z
being
ξ − 1
1 + 5(γ/α) + 8 (γ/α)
+ 4 (γ/α)
1 + 4(γ/α)− 8ξ
1 + 3(γ/α) + 2 (γ/α)
2 = −
4 + 5(γ/α)− 4 (γ/α)2 − 8ξ
1 + 4(γ/α) + 5 (γ/α)
+ 2 (γ/α)
3 = −
1 + 4(γ/α) + 8 (γ/α)
+ 8 (γ/α)
1 + 5(γ/α) + 8 (γ/α)
+ 4 (γ/α)
Analysing the result we can observe that 〈T̂ νµ 〉CT becomes relevant in the region near the
magnetic tube. In this region, it is of the same order of magnitude as the sub-dominant
contribution obtained in [13] in the high-temperature limit. Also it is possible to check
that for the conformal factor ξ = 1/6, the trace of the correction vanishes, i.e., 〈T̂ µµ 〉CT = 0
For the third model the new contribution is given by:
〈T̂ νµ 〉CT = lim
µ(α,φ)G
C,(3)
T (x, x
′) , (70)
where
C,(3)
T (x, x
l 6=0
dωωJ0
(∆τ + lβ)2 + (∆z)2
ein∆θDn(ωR)K|n|/α(ωr)K|n|/α(ωr
′) . (71)
Using the same procedure adopted to calculate the thermal average of the field square,
we shall consider only the component n = 0 in the thermal Green function above. Substi-
tuting (71) into (70), using the approximated expression to the coefficient D0, and taking
into account the same considerations to overcome the integral problem found in (53), we
obtain:
〈T̂ νµ 〉T,Ren. = −
4π2αr4 ln (q/r)
0 , a
1 , a
2 , a
, (72)
where the coefficients a(0)µ are given by the expressions found in the precedent analysis by
taking γ = 0. Moreover in the high-temperature regime (ζ ≪ 1), we obtain:
〈T̂ νµ 〉T,Ren. = −
8αβr3 ln
e−C+α/δ
]diag(b
0 , b
1 , b
2 , b
3 ) . (73)
Here we also observe that a long-range effect appears, consequently it is so relevant as
the sub-dominant contribution proportional to 1/βr3 for points at large distance from the
tube.
5 Concluding Remarks
In this paper we have analysed the thermal effects on the renormalized vacuum expectation
value of the square of a charged massless scalar field, 〈φ̂∗φ̂〉, and in the energy-momentum
tensor, 〈T̂ νµ 〉, in the cosmic string spacetime considering the presence of a magnetic flux
of finite radius. Three specific configurations of magnetic filed have been considered. For
them we could express these vacuum polarization effects as the sum of three different
terms as follows:
• The first one represents the thermal contribution coming from the interaction be-
tween the charged field with a magnetic flux considered as a line running along the
cosmic string. This contribution has been exactly calculated in [13].
• The second term is the zero temperature contribution on the vacuum polarization
due to the non vanishing radius of the magnetic flux tube. This contribution has
been analysed in detail in [8].
• The third contribution is the new one. It comes from the combination of the non
vanishing radius of the magnetic flux tube and temperature. It goes to zero for
R → 0 and for T → 0.
Unfortunately this new contribution cannot be expressed in terms of any analytical
function. In order to provide the most important quantitative information about the ther-
mal behavior of this contribution, we adopted an approximated procedure. We consider
this term in the high-temperature regime. Doing this it was possible, by using the series
property for the Bessel function, to obtain an analytical expression to this correction.
Apart from the homogeneous thermal contribution to the vacuum polarizations, there
appears in calculations of the thermal average of the field square and energy-momentum
tensor, corrections due to the geometry of the spacetime, and due to the non-vanishing
magnetic flux. The latter presents two parts: one is given as the magnetic flux be a line
running along the string, the standard contribution, and the other consequence of a finite
transversal radius. These corrections are sub-dominant and depend on the distance to
the string. For the two first models for the magnetic fields, the corrections on the thermal
average due to the finite radius, are very similar and only relevant in the regions near
the tube; as to the third model, it becomes so important as the standard one for large
distance.
Although the structure of the magnetic field produced by a U(1)−gauge cosmic string
cannot be presented by any analytical function, its influence on the thermal vacuum
polarization effects of charged matter fields takes place for sure. So in this case, the
geometric and magnetic interactions provide contributions. The relevant physical question
is how important they are. Trying to clarify this important question, we adopted specific
fields configurations which allow us to develop an analytical procedure: we assumed that
the magnetic field extent R is much bigger that the cosmic string’s radius considered here
equal to zero.10 So our main conclusion is that: although the structure of the magnetic
field cannot be very well understood, its influence on the thermal vacuum polarization
effect can be so relevant as the influence of the gravitational filed itself.
A real cosmic string would be immersed in a bath of primordial heat radiation. The
behavior of particle detectors near straight strings immersed in thermal radiation, in
free space or passing through black holes, has been analyzed in [26]. The influence of
a magnetic field surrounding the string on this detector, can be evaluated by using the
corresponding thermal Green functions calculated in this paper. In addition, some results
obtained here may shed light upon the vacuum polarization effects induced by a realistic
vortex configuration in early cosmology, where the temperature of the Universe was really
high. By these results, we can see that thermal contributions to the vacuum expectation
10In fact for the U(1)−gauge cosmic string the magnetic field extent is approximately δ ≈ 1
while the cosmic string radius ξ ≈ 1
. So we are considering the case where
>> 1.
value of the field square and the energy-momentum tensor, modify the zero-temperature
quantities in that epoch. Because of this, they should be taken into account in, for ex-
ample, the theory of structure formation.
Acknowledgment
One of us (ERBM) wants to thanks Conselho Nacional de Desenvolvimento Cient́ıfico
e Tecnológico (CNPq.) for partial financial support, FAPESQ-PB/CNPq. (PRONEX)
and FAPES-ES/CNPq. (PRONEX).
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  In this paper we analyse the effect produced by the temperature in the vacuum
polarization associated with charged massless scalar field in the presence of
magnetic flux tube in the cosmic string spacetime. Three different
configurations of magnetic fields are taken into account: $(i)$ a homogeneous
field inside the tube, $(ii)$ a field proportional to $1/r$ and $(iii)$ a
cylindrical shell with $\delta$-function. In these three cases, the axis of the
infinitely long tube of radius $R$ coincides with the cosmic string. Because
the complexity of this analysis in the region inside the tube, we consider the
thermal effect in the region outside. In order to develop this analysis, we
construct the thermal Green function associated with this system for the three
above mentioned situations considering points in the region outside the tube.
We explicitly calculate in the high-temperature limit, the thermal average of
the field square and the energy-momentum tensor.

Introduction
It is well known that different types of topological defects may have been created in the
early Universe after the Planck time by the vacuum phase transition [1, 2]. These include
domain walls, cosmic strings and monopoles. Among them cosmic string and monopole
seem to be the best candidates to be detected.
∗E-mail: jspinelly@uepb.edu.br
†E-mail: emello@fisica.ufpb.br
http://arxiv.org/abs/0704.1990v2
Many years ago Nielsen and Olesen proposed a theoretical model comprised by Higgs
and gauge fields, that by a spontaneous broken of gauge symmetry produces linear topo-
logical defect carrying a magnetic flux named vortex [3]. A few years later, Garfinkle
investigated the influence of this topological object on the geometry of the spacetime
[4]. Coupling the energy-momentum tensor associated with the system to the Einstein
equations, he found static cylindrically symmetric solutions. The author also shown that
asymptotically the spacetime around the vortex is a Minkowiski one minus a wedge. The
core of the vortex has a non-zero thickness and the magnetic flux inside. Two years later
Linet [5] obtained, under some specific condition, exact solutions for the complete set of
differential equation. He was able to show that the structure of the respective spacetime
corresponds to a conical one, with the conicity parameter being expressed in terms of the
energy per unity length of the vortex.1
In recent papers, we have investigated the vacuum polarization effects associated with
massless scalar [7, 8] and fermionic [9, 10] fields, in the presence of a magnetic flux tubes
of finite radius in the cosmic string spacetime at zero temperature. In these analysis
we considered that the magnetic fields are confined into an infinitely long tube of radius
R around the cosmic string. Three different configurations of magnetic field, H(r), are
taking into account in our analysis:
i) H(r) =
Θ(R− r) , homogeneous field inside, (1)
ii) H(r) =
2παRr
Θ(R− r) , field proportional to 1/r inside, (2)
iii) H(r) =
δ(r − R) , cylindrical shell, (3)
where R is the radius extent of the tube, Θ is the Heaviside’s function and Φ is the
total magnetic flux.2 The ratio of the flux to the quantum flux Φo, can be expressed by
δ = Φ/Φ0 = N + γ, where N is the integer part and 0 < γ < 1.
In the framework of the quantum field theory at finite temperature, a fundamental
quantity is the thermal Green function, Gβ(x, x
′). For the scalar field it should be periodic
in the imaginary time with period β, which is proportional to the inverse of the temper-
ature. Because we are interested to obtain the thermal Green function, it is convenient
to work in the Euclidean analytic continuation of the Green function performing a Wick
rotation on the time coordinate, t → iτ . So, we shall work on the Euclidean version of
the idealized cosmic string spacetime, which in cylindrical coordinates, can be described
by the line element below:
ds2 = dτ 2 + dr2 + α2r2dθ2 + dz2 , (4)
where α is a parameter smaller than unity which codify the presence of a conical two-
surface (r, θ) .3
1The complete analysis about the behavior of the gauge and matter fields near the cosmic string’s core
can only be obtained numerically. Some recent numerical analysis [6] about the structure of supermassive
cosmic strings show that two different kind of solutions for the metric tensor exist.
2The configurations for magnetic flux provide the same magnetic flux on the two-surface perpendicular
to the z−axis in coordinate system defined in (4).
3For a typical Grand Unified Theory, α = 1−O(10−6).
The vacuum polarization effects associated with a charged scalar field due to a mag-
netic field confined in a tube of finite radius in Minkowski spacetime has been first analysed
by Serebryanyi [11]. A few years later this analysis, for an idealized cosmic string space-
time, has been considered by Guimarães and Linet [12]; however the magnetic flux was
considered as being a line running through the string. The effect of the temperature on
this vacuum polarization was also investigated by Guimarães in [13]. In this context,
inspired by our previous work [7, 8], we decided to investigate the effect of the temper-
ature on the vacuum polarization effects associated with charged massless scalar field in
presence of magnetic flux tube in the cosmic string spacetime, considering the three dif-
ferent configurations of magnetic field given before. The standard procedure to develop
this analysis is by calculating the respective thermal Euclidean Green function. This can
be done for an ultrastatic spacetime4 by knowing the Green function at zero tempera-
ture. The analysis of the thermal effects on the vacuum polarization effects associated
with massless bosonic and fermionic fields in the global monopole spacetime have been
considered in [14, 15, 16] few years ago.
This paper is organized as follows: In the section 2 we calculate the thermal Euclidean
Green function associated with the system for the three different models of magnetic
fields. Using the results obtained, we calculate in the sections 3 and 4, the thermal
renormalized vacuum expectation value of the field square and the energy-momentum
tensor, respectively. We leave for the section 4 our conclusions.
2 The Euclidean thermal Green function
The Green function associated with the charged massless scalar field at zero temperature
in the presence of a electromagnetic field, must obey the following non-homogeneous
second-order differential equation
−ggµνDν
G(x, x
) = −δ(4)(x− x′) , (5)
where Dµ = ∂µ − ieAµ, Aµ being the four-vector potential.
In order to reproduce the configurations of magnetic fields along the z−direction given
by (1)-(3), we write the the vector potential by Aµ = (0, 0, A(r), 0), with
A(r) =
a(r) . (6)
For the two first models considered, we can represent the radial function a(r) by:
a(r) = f(r)Θ(R− r) + Θ(r −R) , (7)
f(r) =
r2/R2, for the model (i) and
r/R, for the model (ii).
For the third model,
a(r) = Θ(R− r) . (9)
4An ultrastatic spacetime admits a globally defined coordinate system in which the components of the
metric tensor are time independent and the conditions g00 = 1 and g0i = 0 hold
As we have mentioned, in this work we shall continue in the same line of investigation
started in [7, 8], calculating at this time, the thermal contribution on the vacuum polar-
ization effects in the region outside the magnetic tube for the three magnetic fields under
consideration.
The Euclidean Green functions for points outside the magnetic flux tube at zero-
temperature are given below:
• For the models (i) and (ii),
T=0(x, x
eiN∆θ
8π2αrr′ sinh u0
ei∆θ sinh(γu0/α) + sinh[(1− γ)u0/α]
cosh(u0/α)− cos∆θ
dωωJ0
(∆τ)2 + (∆z)2
ein∆θDjn(ωR)K|ν|(ωr)K|ν|(ωr
′), j = 1, 2 , (10)
where
n−N − γ
, (11)
cosh uo =
r2 + r
+ (∆τ)2 + (∆z)2
Djn(ωR) =
H ′j(R)I|ν|(ωR)−Hj(R)I ′|ν|(ωR)
Hj(R)K
|ν|(ωR)−H ′j(R)K|ν|(ωR)
. (13)
In the above equations the functions Hj(r) are given by:
H1(r) =
Mσ1,λ1
, (14)
with σ1 = (
− ω2R2α
)/2 and λ1 = n/2α, and
H2(r) =
Mσ2,λ2 (ζr) , (15)
with σ2 =
(δ2 + ω2R2α2)−1/2, λ2 = n/α and ζ =
(δ2 + ω2R2α2)1/2. Moreover,
Mσ,λ is the Whittaker function, while I|ν| e K|ν| are the modified Bessel functions
[17].
• For the model (iii), we have
T=0(x, x
8π2αrr′ sinh u0
sinh(u0/α)
cosh(uo/α)− cos∆θ
dωωJ0
(∆τ)2 + (∆z)2
ein∆θDn(ωR)K|n/α|(ωr)K|n/α|(ωr
′) . (16)
where
Dn(ωR) =
|ν|(ωR)I|n|/ω(ωR)− I|ν|(ωR)I
|n|/α(ωR)
I|ν|(ωR)K
|n|/α(ωR)− I
|ν|(ωR)K|n|/α(ωR)
. (17)
We can observe that the first term on the right hand side of (10) is, up to a gauge
transformation, equivalent to the result presented by Guimarães and Linet [12] for a
charged massless scalar field in the presence of a magnetic flux running through the
cosmic string; and as to (16), its first term corresponds the results found by Smith [18]
and by Linet in [19] for a massless scalar field without charge. However, the seconds
terms of booths expressions represent corrections on the respective Green functions due
to a non-vanishing radius R attributed to the magnetic flux; off course these corrections
vanish when we take R → 0.
Following the prescription given in the papers by Braden [20] and Page [21], the
thermal Green function, GT (x, x
′), can be expressed in terms of the sum
GT (x, x
GT=0(x, x
′ − lλβ) , (18)
where λ is the ”Euclidean” unitary time-like vector and β = 1/kBT , being kB the Boltz-
mann constant and T the absolute temperature.
In agreement with the equations (10), (16) and (18), the thermal Green functions
associated with the massless scalar field in the cosmic string spacetime and in the presence
of magnetics field, are given by:
• For the models (i) and (ii),
T (x, x
′) = G
T (x, x
dωωJ0
(∆τ + lβ)2 + (∆z)2
ein∆θDjn(ωR)K|ν|(ωr)K|ν|(ωr
′) , j = 1, 2 . (19)
• For the model (iii),
T (x, x
′) = GT (x, x
dωωJ0
(∆τ + lβ)2 + (∆z)2
ein∆θDn(ωR)K|n|/α(ωr)K|n|/α(ωr
′) . (20)
The thermal Green functions G
T (x, x
′) and GT (x, x
′) which appear in (19) and (20)
have being obtained a few years ago by Guimarães [13] and Linet [22], respectively. So, we
shall not repeat them. In fact what we are really interested in this paper is to analyze the
seconds terms in these Green functions, which correspond to the thermal contributions
due to the finite thickness admitted for the magnetic flux tube.
3 The Computation of 〈φ̂∗(x)φ̂(x)〉Ren. at Non-zero Tem-
perature
The main objective of this section is to investigate the effects produced by the temperature
in the renormalized vacuum expectation value of the square of the charged massless scalar
field, 〈φ̂∗(x)φ̂(x)〉, in the presence of a magnetic flux tube of finite radius. Formally this
quantity is given by taking the coincidence limit of the Green function:
〈φ̂∗(x)φ̂(x)〉T = lim
GT (x, x
′) . (21)
However, this procedure provides a divergent result and the divergence comes exclusively
from the first terms of the right hand side of (19) and (20)5. In order to obtain a finite and
well defined result, we must apply some renormalization procedure. Here we shall adopt
the point-splitting renormalization one. It has been observed that the singular behavior
of the Green function has the same structure as given by the Hadamard one, which on
the other hand can be written in terms of the square of the geodesic distance between two
points. So, here we shall adopt the following prescription: we subtract from the Green
function the Hadamard one before applying the coincidence limit as shown below:
〈φ̂∗(x)φ̂(x)〉T,Ren. = lim
[GT (x, x
′)−GH(x, x′)] . (22)
We can write the result as:
〈φ̂∗(x)φ̂(x)〉T,Ren. = 〈φ̂∗(x)φ̂(x)〉T,Reg. + 〈φ̂∗(x)φ̂(x)〉CT=0 + 〈φ̂∗(x)φ̂(x)〉CT . (23)
The first term on the right hand side of the above expression, represents, for the
models (i) and (ii), the thermal contribution coming from the interaction between charged
massless scalar field with a magnetic flux considered as a line running along the cosmic
string. From the Guimarães’s paper [13], this term is:
〈φ̂∗(x)φ̂(x)〉T,Reg. =
16π2αβr
r cosh u/2
cosh u/2
F (γ)α (u, 0) du , (24)
where
F (γ)α (u, 0) = −2
sin [πγ/α] cos [u (1− γ) /α] + sin [u (1− γ) /α] cos [πγ/α]
cosh u/α− cosπ/α
. (25)
For the model (iii) an analogous expression can be obtained from the previous one, by
taking γ = 06. (An interesting aspect of these results is that the vacuum polarizations
depend only on the fractional part of the ration of the magnetic flux by the quantum one,
In the high-temperature limit (β → 0), Guimarães showed that,
〈φ̂∗(x)φ̂(x)〉T,Reg. ≈
M (γ)
, (26)
where the constant M (γ) is given by
M (γ) =
16π2α
F (γ)α (u, 0)
cosh(u/2)
du , (27)
5A special feature of these Green functions is that the correction due to the magnetic tube’s radius is
finite in the coincidence limit.
6See paper [22].
and in the zero-temperature limit (β → ∞),
〈φ̂∗(x)φ̂(x)〉T,Reg. ≈
, (28)
with ω(γ) being given by
ω(γ) = − 1
γ − 1
. (29)
The two last terms in (23) are corrections on the vacuum polarization due to fi-
nite thickness of the radius of the magnetic tube and non-zero temperature. The term
〈φ̂2(x)〉CT=0, corresponds to correction due to finite thickness of the radius of tube only, it
was given in [7, 8]. For the two firsts models the corrections are similar. They are given
by the component l = 0 in (19), and read
(2π)2α
Din(ωR)K
|ν|(ωr) . (30)
Because we are mostly interested to study the vacuum polarization for points very far
from the cosmic string, we shall consider R/r ≪ 1. Comparing the behavior of Din(ωR)
with the behavior of K2|ν|(ωr), we have shown that we may approximate the integrand of
(30) assuming to the coefficient Djn its first order expansion in ωR , which reads:
Djn(ωR) = −
Γ(|ν| + 1)Γ(|ν|)
wnj − |ν|znj
wnj + |ν|znj
)2|ν|
, (31)
where
wn1 =
δ − n
Mλ1,λ2(δ/α) +
Mγ1,λ1(δ/α), (32)
zn1 = Mλ1,λ2(δ/α), (33)
wn2 =
δ − n
Mλ2,λ1(2δ/α) +
Mγ2,λ2(2δ/α) , (34)
zn2 = Mλ2,λ1(2δ/α) , (35)
being γ1 = (n+2α)/2α, γ2 = (n+α)/α and ν =
n−N−γ
. The most important contributions
in (30) comes from the component n = N in the summation. In this way, considering
only the dominant term, the corrections are given by:
〈φ̂∗(x)φ̂(x)〉CT=0 = −
(2π)2r2
α(2γ + α)
wNj − (γ/α)zNj
wNj + (γ/α)z
)2γ/α
. (36)
As we can see the corrections present extra dependence on the radial coordinate, conse-
quently they are appreciable only in the region close to the tube.
As to the third model there happens a very interesting phenomenon. For n 6= 0, the
coefficient Dn(ωR) in (16) vanishes at least as fast as (ωR)
2|n|/α when ωR → 0. On the
other hand, D0(ωR) vanishes only with the inverse of the logarithm, so quite slowly. In
this way the most relevant contribution to the summation comes from n = 0. For this
D0(ωR) =
+ C − α/δ
, (37)
where C is the Euler constant. In [8] we have explicitly shown that the correction on the
vacuum polarization effect for the third model, consequence of a non-vanishing radius of
the magnetic flux, is mainly given by:
〈φ̂∗(x)φ̂(x)〉CT=0 = −
8π2αr2 ln
e−C+α/δ
) . (38)
We also have shown that for α = 0.99 and δ = 0.2 this term is of the same order of the
standard vacuum expectation value of the field square in the absence of magnetic flux, up
to the distance r which exceeds the radius of the observable Universe.
As we have already mentioned, the last term in (23), 〈φ̂∗(x)φ̂(x)〉CT , is consequence
of combined effects of the nonvanishing thickness of the magnetic tube and temperature.
This is a new contribution and we shall focus on it. For the first two models, this term
can be expressed by
〈φ̂∗(x)φ̂(x)〉CT =
2π2αr2
dvvJ0 (vζl)
Djn(vR/r)K
|ν|(v) , j = 1, 2 , (39)
with ζ = β/r. In the above expression we have introduced a dimensionless variable
v = ωr.
As in the zero temperature analysis, because we are considering R/r ≪ 1, the most
important contribution for the above summation comes from the n = N component. So,
on basis what we have already discussed for points very far from the string, equation (39)
can be approximate to
〈φ̂∗(x)φ̂(x)〉CT = −
(2π)2αr2
Γ(γ/α + 1)Γ(γ/α)
wNj − (γ/α)zNj
wNj + (γ/α)z
)2γ/α
dvv1+2γ/αJ0 (vζl)K
|γ/α|(v) , j = 1, 2 . (40)
From the above expression we can observe that the thermal content of this correction
is in the summation S given below:
S(ζ) =
dvv1+2γ/αJ0 (vζl)K
|γ/α|(v) . (41)
Unfortunately it is not possible to obtain a closed expression to this term, and provide a
complete information about the thermal behavior of (40); on the other hand, it is possible
to give its main information. In order to do that we shall divide the integral above in
two parts: from [0, 2π/ζ ] and from [2π/ζ,∞). Due the strong exponential decay of the
modified Bessel function for large argument, the contribution in the interval [2π/ζ,∞)
can be neglected in the high-temperature regime, i.e., for ζ ≪ 1; moreover using the series
properties for the Bessel function [17],
J0(vζl) = −
, 0 < v < 2π/ζ , (42)
(41) can be approximated to
S(ζ) = −1
∫ 2π/ζ
dvv1+2γ/αK2|γ/α|(v) +
∫ 2π/ζ
dvv2γ/αK2|γ/α|(v) . (43)
In the high-temperature limit the most relevant contribution is given by the second
term. Adopting the same approximation criterion to discard the integral in the inter-
val [2π/ζ,∞), we may evaluate the integral by taking its upper limit going to infinity. In
this way the most relevant contribution to the thermal vacuum polarization effect is [17]7:
S(ζ) =
22−2γ/α
Γ2 (1/2 + γ/α) Γ (1/2 + 2γ/α)
Γ (1 + 2γ/α)
. (44)
Consequently, in the high-temperature regime the correction term, 〈φ̂∗(x)φ̂(x)〉CT , is dom-
inated by:
〈φ̂∗(x)φ̂(x)〉CT = −
(γ,α,R)
)2γ/α
, j = 1, 2 , (45)
(γ,α,R)
4π3/2α
Γ2 (1/2 + γ/α) Γ (1/2 + 2γ/α)
Γ (1 + 2γ/α) Γ(1 + γ/α)Γ(γ/α)
wNj − (γ/α)zNj
wNj + (γ/α)z
Hence, the expression for renormalized vacuum expectation value of the square of the
charged massless scalar field, in the high-temperature limit, is given by
〈φ̂∗(x)φ̂(x)〉T,Ren. ≈
M (γ)
(γ,α,R)
)2γ/α
. (46)
From this expression we can observe that the two sub-dominant contributions are of the
same order of magnitude for points near the tube.
For the third model, 〈φ̂∗(x)φ̂(x)〉CT , is given by the expression below:
〈φ̂∗(x)φ̂(x)〉CT =
2π2αr2
dvvJ0 (vζl)
Dn(vR/r)K
|n|/α(v) . (47)
Again, as in the zero temperature analysis of the vacuum polarization effect, the
most relevant contribution comes from the n = 0 component of the above summation for
R/r ≪ 1. The respective coefficient is:
D0(vR/r) =
ln (v/qr)
, (48)
where
e−C+α/δ . (49)
In this way (47) can be written by:
〈φ̂∗(x)φ̂(x)〉CT =
2π2αr2
S̄(ζ) , (50)
7By numerical analysis, in [24] we have reached to S(ζ), the same behavior found here, i.e., S(ζ) ≈ 1/ζ.
where
S̄(ζ) =
dvvJ0 (vζl)D0(vR/r)K
0(v) . (51)
Applying the previous summation property to the Bessel functions, in the high-
temperature regime, (51) can be given by
S̄(ζ) = −
∫ 2π/ζ
dvvD0(vR/r)K
0(v) +
∫ 2π/ζ
dvD0(vR/r)K
0(v) . (52)
For ζ << 1 the most relevant contribution to S̄ is given by second term; moreover in
this limit we can obtain an approximated expressions for it by taking the upper limit of
the integral going to infinite. In this way (52) can be evaluated by
S̄(ζ) ∼=
K20 (v)
ln(v)− ln(qr)
. (53)
Unfortunately this expression presents a pole for v = qr. However, this pole is con-
sequence of the approximation adopted and it is located in the region where where the
approximation is no more valid8. In fact the full expression to D0(vR/r) has no singu-
larity. D0 is a slowly increasing function of v. As we have mentioned, the full expression
to the coefficient D0 vanishes with the inverse of the logarithm for v ≈ 0, and grows less
slower than e2vR/r for large v. The square of the modified Bessel function, K20 , provides
an integrable logarithmic divergence for small argument, and decays with e−2v for large
argument. Except for very small value of v, where the dominant contribution to D0 is
given by 1/ ln(v), D0 can be well approximated by
9 −1/ ln(qr). In figure 1, this argument
is also numerically justified. There, it is exhibited the behavior of the exact integrand of
S(ζ), and its approximated expression discarding the factor ln(v) in the denominator, for
specific values of the parameters.
Accepting the above arguments, we can represent the main contribution to (53) by:
S̄(ζ) = − 1
ζ ln(qr)
dvK20(v) , (54)
Consequently we obtain
〈φ̂∗(x)φ̂(x)〉CT = −
8αβr ln
e−C+α/δ
] . (55)
Now for this model, the expression to the renormalized vacuum polarization effect in
high-temperature limit reads
〈φ̂∗(x)φ̂(x)〉T,Ren. ≈
M (γ)
8αβr ln
e−C+α/δ
] . (56)
8For γ = 0.2, N = 0 and α = 0.99, qr = 2r
eC−α/ζ is of order 105 for r/R = 103, so vR/r = 102
consequently bigger than unity.
9In [23], B. Allen at all adopted a similar procedure to provide an approximate expression to the
vacuum expectation value of the field square in the cosmic string spacetime considering a general structure
to its core; in [8] this procedure has is also been adopted to calculate the 〈φ∗φ〉.
0.5 1 1.5 2 2.5 3 3.5 4
Figure 1: The dashed curve corresponds the exact integrand of the second term of the
right hand side of (52), and the solid line the approximated expression discarding ln(v)
in the denominator of (53). In the numerical analysis we have used α = 0.99, γ = 0.2,
N = 0 and r/R = 103.
On basis of previous discussion about the vacuum polarization effect at zero temper-
ature, we can conclude that although being sub-dominant, the thermal correction due to
the non vanishing radius of the magnetic flux tube, is so relevant as the standard term pro-
portional to 1/βr, for points at very large distance from the cosmic string. Consequently,
it can be considered as a long-range effect.
4 The Computation of 〈T̂ νµ (x)〉Ren. at Non-zero Tem-
perature
The energy-momentum tensor, Tµν(x), is a bilinear function of the fields, so we can
evaluate its vacuum expectation value, 〈Tµν(x)〉, by the standard method using the Green
function [25]. The thermal vacuum average, consequently, can also be obtained by using
the thermal Green function.
The renormalized vacuum expectation value of the energy-momentum tensor at non-
zero temperature for the system adopted here can be calculated by:
〈T̂ νµ 〉T,Ren. = lim
µ(α,φ)GT (x, x
′)−Dν′µ(1,0)GH(x, x′)] , (57)
where Dν
µ(α,φ) stands a second order non-local differential operator in presence of a mag-
netic field and cosmic string, defined by
µ(α,φ) = (1− 2ξ)DµD
ν′ − ξ(DµDν +Dµ′D
2ξ − 1
δνµDσD
, (58)
ξ being the non-minimal curvature coupling. In the above expression we consider that
Dσ = ∇σ−ieAσ, with∇σ being the covariant gravitational derivative and the bar denoting
complex conjugate. Again we shall consider the magnetic field configuration given by (7)
and (8) for the two first models, and by (9) for the third model. The respective thermal
Green function, GT (x
′, x), are given by (19) and (20), respectively.
Due to the form of the Green function, we can express (57) by
〈T̂ νµ (x)〉T,Ren. = 〈T̂ νµ (x)〉T,Reg. + 〈T̂ νµ (x)〉CT=0 + 〈T̂ νµ (x)〉CT . (59)
The first and second term of the right hand side of the above expression have already
been computed in [13] and [8], respectively. As to the first term, Guimarães has shown
〈T̂ νµ (x)〉T,Reg. =
diag(−3, 1, 1, 1) + 〈T̃ νµ (x)〉T , (60)
where the second term on the right hand side is given in terms of five integrals. However,
the author was able to show that in the high-temperature limit (β → 0), this term
is proportional to 1/βr3, and in the zero-temperature limit (β → ∞) 〈T̂ νµ (x)〉T,Ren. is
proportional to 1/r4. The second term in (59), 〈T̂ νµ (x)〉CT=0, is consequence of a non-
vanishing radius to the magnetic flux. For the two first models, its most important
contribution is proportional to (1/r4)(R/r)2γ/α, consequently is relevant only in the region
near the magnetic tube; on the other hand, for the third model this term presents a long-
range effect, similar to what happens in the vacuum expectation value of the field square.
The new contribution, 〈T̂ νµ (x)〉CT , is consequence of a non-vanishing radius attributed to
the magnetic flux tube and the temperature.
For the two first models this term is given by
〈T̂ νµ 〉CT = lim
µ(α,φ)G
C,(j)
T (x, x
′) , (61)
C,(j)
T (x, x
l 6=0
dωωJ0
(∆τ + lβ)2 + (∆z)2
ein∆θDjn(ωR)K|ν|(ωr)K|ν|(ωr
′) , j = 1, 2 . (62)
Following the same procedure adopted in the last section, we shall consider only the
n = N component in the summation. Substituting (62) into (61), and using the approx-
imated expression to the coefficient D
N , after long calculation we arrive at the following
result:
〈T̂ νµ 〉CT = −
4π2αr4
Γ (γ/α) Γ (1 + γ/α)
wNj − (γ/α)zNj
wNj + (γ/α)z
)2γ/α
0 , a
1 , a
2 , a
where
0 = (4ξ − 1)
3 + I
1 − 2(γ/α)I
2 + 2 (γ/α)
1 = I
1 − 2 (γ/α + 2ξ) I
2 + 4ξ(γ/α)I
4 − I
2 = 2(γ/α) (γ/α− 2ξ) I
4 + 4ξI
(4ξ − 1)
3 + I
1 − 2(γ/α)I
2 + 2 (γ/α)
3 = 2I
5 + (4ξ − 1)
3 + I
1 − 2(γ/α)I
2 + 2 (γ/α)
dvv3+2γ/αJ0 (vζl)K
γ/α+1(v) , (64)
dvv2+2γ/αJ0 (vζl)Kγ/α+1Kγ/α(v) , (65)
dvv3+2γ/αJ0 (vζl)K
γ/α(v) , (66)
dvv1+2γ/αJ0 (vζl)K
γ/α+1(v) (67)
dvv2+2γ/α
J1 (vζl)
K2γ/α(v) . (68)
In order to provide the most relevant contribution to (64)-(67), we adopt the same pro-
cedure adopted in last section: we divide the interval of integration in two parts, from
[0, 2π/ζ ] and from [2π/ζ,∞). Again in the high-temperature regime, ζ ≪ 1, the integral
in the second interval can be neglected due the exponential decay of the modified Bessel
function. Finally using the series properties for the Bessel function [17], and after some
intermediate steps, we obtain:
〈T̂ νµ 〉CT = −
γ,α,R
)2γ/α
0 , b
1 , b
2 , b
, j = 1, 2 , (69)
where
γ,α,R
(1 + 2γ/α)
2π3/2α
Γ2 (1/2 + γ/α) Γ (1/2 + 2γ/α)
Γ (3 + 2γ/α) Γ(1 + γ/α)Γ(γ/α)
wNj − (γ/α)zNj
wNj + (γ/α)z
being
ξ − 1
1 + 5(γ/α) + 8 (γ/α)
+ 4 (γ/α)
1 + 4(γ/α)− 8ξ
1 + 3(γ/α) + 2 (γ/α)
2 = −
4 + 5(γ/α)− 4 (γ/α)2 − 8ξ
1 + 4(γ/α) + 5 (γ/α)
+ 2 (γ/α)
3 = −
1 + 4(γ/α) + 8 (γ/α)
+ 8 (γ/α)
1 + 5(γ/α) + 8 (γ/α)
+ 4 (γ/α)
Analysing the result we can observe that 〈T̂ νµ 〉CT becomes relevant in the region near the
magnetic tube. In this region, it is of the same order of magnitude as the sub-dominant
contribution obtained in [13] in the high-temperature limit. Also it is possible to check
that for the conformal factor ξ = 1/6, the trace of the correction vanishes, i.e., 〈T̂ µµ 〉CT = 0
For the third model the new contribution is given by:
〈T̂ νµ 〉CT = lim
µ(α,φ)G
C,(3)
T (x, x
′) , (70)
where
C,(3)
T (x, x
l 6=0
dωωJ0
(∆τ + lβ)2 + (∆z)2
ein∆θDn(ωR)K|n|/α(ωr)K|n|/α(ωr
′) . (71)
Using the same procedure adopted to calculate the thermal average of the field square,
we shall consider only the component n = 0 in the thermal Green function above. Substi-
tuting (71) into (70), using the approximated expression to the coefficient D0, and taking
into account the same considerations to overcome the integral problem found in (53), we
obtain:
〈T̂ νµ 〉T,Ren. = −
4π2αr4 ln (q/r)
0 , a
1 , a
2 , a
, (72)
where the coefficients a(0)µ are given by the expressions found in the precedent analysis by
taking γ = 0. Moreover in the high-temperature regime (ζ ≪ 1), we obtain:
〈T̂ νµ 〉T,Ren. = −
8αβr3 ln
e−C+α/δ
]diag(b
0 , b
1 , b
2 , b
3 ) . (73)
Here we also observe that a long-range effect appears, consequently it is so relevant as
the sub-dominant contribution proportional to 1/βr3 for points at large distance from the
tube.
5 Concluding Remarks
In this paper we have analysed the thermal effects on the renormalized vacuum expectation
value of the square of a charged massless scalar field, 〈φ̂∗φ̂〉, and in the energy-momentum
tensor, 〈T̂ νµ 〉, in the cosmic string spacetime considering the presence of a magnetic flux
of finite radius. Three specific configurations of magnetic filed have been considered. For
them we could express these vacuum polarization effects as the sum of three different
terms as follows:
• The first one represents the thermal contribution coming from the interaction be-
tween the charged field with a magnetic flux considered as a line running along the
cosmic string. This contribution has been exactly calculated in [13].
• The second term is the zero temperature contribution on the vacuum polarization
due to the non vanishing radius of the magnetic flux tube. This contribution has
been analysed in detail in [8].
• The third contribution is the new one. It comes from the combination of the non
vanishing radius of the magnetic flux tube and temperature. It goes to zero for
R → 0 and for T → 0.
Unfortunately this new contribution cannot be expressed in terms of any analytical
function. In order to provide the most important quantitative information about the ther-
mal behavior of this contribution, we adopted an approximated procedure. We consider
this term in the high-temperature regime. Doing this it was possible, by using the series
property for the Bessel function, to obtain an analytical expression to this correction.
Apart from the homogeneous thermal contribution to the vacuum polarizations, there
appears in calculations of the thermal average of the field square and energy-momentum
tensor, corrections due to the geometry of the spacetime, and due to the non-vanishing
magnetic flux. The latter presents two parts: one is given as the magnetic flux be a line
running along the string, the standard contribution, and the other consequence of a finite
transversal radius. These corrections are sub-dominant and depend on the distance to
the string. For the two first models for the magnetic fields, the corrections on the thermal
average due to the finite radius, are very similar and only relevant in the regions near
the tube; as to the third model, it becomes so important as the standard one for large
distance.
Although the structure of the magnetic field produced by a U(1)−gauge cosmic string
cannot be presented by any analytical function, its influence on the thermal vacuum
polarization effects of charged matter fields takes place for sure. So in this case, the
geometric and magnetic interactions provide contributions. The relevant physical question
is how important they are. Trying to clarify this important question, we adopted specific
fields configurations which allow us to develop an analytical procedure: we assumed that
the magnetic field extent R is much bigger that the cosmic string’s radius considered here
equal to zero.10 So our main conclusion is that: although the structure of the magnetic
field cannot be very well understood, its influence on the thermal vacuum polarization
effect can be so relevant as the influence of the gravitational filed itself.
A real cosmic string would be immersed in a bath of primordial heat radiation. The
behavior of particle detectors near straight strings immersed in thermal radiation, in
free space or passing through black holes, has been analyzed in [26]. The influence of
a magnetic field surrounding the string on this detector, can be evaluated by using the
corresponding thermal Green functions calculated in this paper. In addition, some results
obtained here may shed light upon the vacuum polarization effects induced by a realistic
vortex configuration in early cosmology, where the temperature of the Universe was really
high. By these results, we can see that thermal contributions to the vacuum expectation
10In fact for the U(1)−gauge cosmic string the magnetic field extent is approximately δ ≈ 1
while the cosmic string radius ξ ≈ 1
. So we are considering the case where
>> 1.
value of the field square and the energy-momentum tensor, modify the zero-temperature
quantities in that epoch. Because of this, they should be taken into account in, for ex-
ample, the theory of structure formation.
Acknowledgment
One of us (ERBM) wants to thanks Conselho Nacional de Desenvolvimento Cient́ıfico
e Tecnológico (CNPq.) for partial financial support, FAPESQ-PB/CNPq. (PRONEX)
and FAPES-ES/CNPq. (PRONEX).
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