Brazilian Journal of Physics, vol. 37, no. 1B, March, 2007 1
Condensation of Vortex-Strings: Effective Potential Contribution Through Dual
Actions
Rudnei O. Ramos,∗ Daniel G. Barci,† and Cesar A. Linhares‡
Departamento de Fı́sica Teórica, Universidade do Estado do Rio de Janeiro, 20550-013 Rio de Janeiro, RJ, Brazil
J. F. Medeiros Neto§
Instituto de Fı́sica, Universidade Federal do Pará, 66075-110 Belem, Pará, Brazil
Topological excitations are believed to play an important role in different areas of physics. For example, one
case of topical interest is the use of dual models of quantum cromodynamics to understand properties of its
vacuum and confinement through the condensation of magnetic monopoles and vortices. Other applications are
related to the role of these topological excitations, nonhomogeneous solutions of the field equations, in phase
transitions associated to spontaneous symmetry breaking in gauge theories, whose study is of importance in
phase transitions in the early universe, for instance. Here we show a derivation of a model dual to the scalar
Abelian Higgs model where its topological excitations, namely vortex-strings, become manifest and can be
treated in a quantum field theory way. The derivation of the nontrivial contribution of these vacuum excitations
to phase transitions and its analogy with superconductivity is then made possible and they are studied here.
PACS numbers: 11.10.Wx, 98.80.Cq
Keyword: dual models, vortices, phase transitions
I. INTRODUCTION
Topological excitations, or defects, are nonhomogeneous
solutions of the field equations of motion in many types of
field theory models [1, 2, 3]. They are finite energy and stable
configurations that emerge as a consequence of a spontaneous
symmetry breaking process. Mathematically, defects are pre-
dicted to appear whenever some larger group of symmetry G
breaks into a smaller one H such that there are a nontrivial
homotopy group πk(G/H) of the vacuum manifold different
from the identity. Well known examples are kinks, or do-
main walls (k= 0) [2], that originate from a discrete symmetry
breaking, strings, or vortices (k = 1), for example originating
from a continuous gauge symmetry breaking U(1) → 1 [4]
and magnetic monopoles (k = 2), e.g. from a SO(3)→ U(1)
symmetry breaking [5, 6]. Since many phase transitions in na-
ture are associated to symmetry breakings, topological excita-
tions are a common feature in these processes and are in fact
observed in many systems in the laboratory, like in ferromag-
netism, helium superfluidity, superconductivity and in many
other condensed matter system and they are also expected to
have appeared in phase transitions in the early universe as well
(for a general review, please see [7]).
In the study of phase transitions in quantum field theory
one basic quantity usually computed is the effective potential,
which is an important tool in the study of phase transitions
in scalar and gauge field theories [8]. It is equivalent to the
homogeneous coarse-grained free-energy density functional
of statistical physics, with its minima giving the stable and,
when applicable, metastable states of the system. For inter-
acting field theories the effective potential is evaluated per-
∗Electronic address: rudnei@uerj.br
†Electronic address: barci@uerj.br
‡Electronic address: linhares@dft.if.uerj.br
§Electronic address: jfmn@ufpa.br
turbatively, with an expansion in loops being equivalent to an
expansion in powers of ~ [9]. The one-loop approximation is
then equivalent to incorporating the first quantum corrections
to the classical potential. Recall that the effective potential,
taking a scalar field theory as an example, is obtained from
the effective action Γ[φc], where it is defined in terms of the
connected generating functional W [J] as
Γ[φc] =W [J]−
d4xJ(x)φc(x) , (1)
with the classical field φc(~x, t) defined by
φc(~x, t)≡ δW [J]/δJ(x), and
W [J] =−i~ln
Dφ exp
S[φ,J]
. (2)
In order to evaluate Γ[φc] perturbatively, one writes the
field as φ(~x, t) → φ0(~x, t) + η(~x, t), where φ0(~x, t) is a field
configuration which extremizes the classical action S[φ,J],
δS[φ,J]
δφ |φ=φ0 = 0, and η(~x, t) is a small perturbation about that
extremum configuration. The action S[φ,J] can then be ex-
panded about φ0(~x, t) and, up to quadratic order in η(~x, t), we
can use a saddle-point approximation to the path integral to
obtain for the connected generating functional,
W [J] = S[φ0]+~
d4xφ0(x)J(x)+ i
∂µ∂µ+V ′′(φ0)
In order to obtain the one-loop expression for Γ[φc], we first
note that writing φ0 = φc−η we get to first order in ~, S[φ0] =
S[φc]−~
d4xη(x)J(x)+O(~2). Using this result and Eq. (3)
into Eq. (1) we find, as J → 0,
Γ[φc] = S[φc]+ i
∂µ∂µ+V ′′(φc)
. (4)
http://arxiv.org/abs/0704.1117v1
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mailto:jfmn@ufpa.br
2 Rudnei O. Ramos et al.
The effective action can also be computed as a derivative ex-
pansion about φc(~x, t),
Γ[φc] =
−Veff(φc(x))+
(∂µφc)
2 Z(φc(x))+ . . .
The function Veff(φc) is the effective potential. For a constant
field configuration φc(~x, t) = φc we obtain
Γ[φc] =−ΩVeff(φc) , (6)
where Ω is the total volume of space-time. Comparing Eqs.
(4) and (6) we obtain for the one-loop effective potential,
Veff(φc) =V (φc)− i
Ω−1Trln
∂µ∂µ+V ′′(φc)
. (7)
When working at non-vanishing temperature, the same
functional techniques can be used. In this case one is in-
terested in evaluating the generating functional (the partition
function) Zβ[J] which is given by the path integral [10]
Zβ[J] = N
Dφexp
d3x(LE − Jφ)
, (8)
where the integration is restricted to paths periodic in τ with
φ(0,~x) = φ(β,~x), LE is the Euclidean Lagrangian, and N is
a normalization constant. Again one expands about an ex-
tremum of the Euclidean action and calculates the partition
function by a saddle-point evaluation of the path integral. The
result for the one-loop approximation to the effective potential
Veff(φc,T ) =Veff(φc)
2π2β4
dx x2ln
1− exp
x2 +β2V ′′(φc)
As the effective potential is equivalent to the free energy func-
tional (for a constant field configuration), all thermodynamics
functions follow from it. In particular the different phases,
critical temperature of phase transition and temperature de-
pendence of the field vacuum expectation value can be ob-
tained from (9).
From the above discussion it is clear that the one-loop ap-
proximation to the effective action, Eq. (4), works best when
the classical field does not differ much from the configuration
that extremizes the classical action, φc = φ0 +η ∼ φ0, since in
this case the saddle-point evaluation to the path integral is ad-
equate. Also, φc(~x, t) must be nearly constant so that the effec-
tive potential can be obtained from Eq. (6). As J → 0, φc(~x, t)
is identified with 〈φ〉, the vacuum expectation value. One ma-
jor problem we see in this whole approach of studying the
phase structure of field theory models from the effective po-
tential is when the action functional, determined from Eq. (8)
is dominated not by homogeneous, constant field configura-
tions but by nonhomogeneous ones. In those situations when
other stable, finite energy solutions to the field equations of
motion exist, we expect these configurations to dominate the
partition function over the homogenous solutions for instance
close to the critical temperature [11]. Under these circum-
stances the effective potential, which includes only contri-
butions to the partition function from constant background
field configurations becomes inappropriate to study the phase
transition and we must rely in other approaches, for example
studying the phase transition directly from the effective action
or free energy for the topological configurations [11, 12], or
taking directly a field theoretic description for the topological
excitations [13]. In either case we are faced with the problem
of accounting for nonlocal contributions in the perturbative
expansion, which is only amenable of analysis up to the low-
est order leading order. To circumvent these difficulties we
make use of the techniques of dualization in field theory, from
which the degrees of freedom of the topological excitations
are explicitly realized in the functional action. This method
is described in the following section, Sec. II, where we spe-
cialize to the case of the scalar Abelian Higgs model, whose
topological solutions are vortex-strings. In Sec. III we evalu-
ated the dual action for the model, making explicit the vortex-
strings degrees of freedom and how they couple to the matter
fields. In Sec. IV we show how an effective potential calcula-
tion for an averaged vortex-string field can be computed and
the results and interpretation of the phase transition obtained
from this quantity. Finally, we give our conclusions in Sec. V.
II. THE STRING SOLUTIONS IN THE SCALAR ABELIAN
HIGGS MODEL
In this work we will use the Abelian Higgs model, with La-
grangian density for a complex scalar field φ and gauge field
Aµ given by
L =−1
µν + |Dµφ|2 −V(φ) , (10)
where, Fµν = ∂µAν−∂νAµ, Dµ = ∂µ− ieAµ and V (φ) is a sym-
metry breaking potential, for example given by
V (φ) =−m2φ |φ|
, (11)
with positive parameters m2φ and λ. The symmetry breaking
U(1) → 1 with homotopy group π1 6= 1 indicates the exis-
tence of string-like topological excitations in the system, or
Nielsen-Olesen strings [4] (for an extended introduction and
review see e.g. Ref. [3]). For example, for a unit winding
string solution along the z axis, the classical field equations
of motion obtained from the Lagrangian density (10) admit a
stable finite energy configuration describing the string given
by (using the cylindrical coordinates r,θ,z)
φstring =
ρ(r)√
eiθ , (12)
Aµ,string =
A(r) ∂µθ , (13)
Brazilian Journal of Physics, vol. 37, no. 1B, March, 2007 3
where the functions ρ(r) and A(r) vanish at the origin and
have the asymptotic behavior
φ(r → ∞) → ρv ≡
A(r → ∞) → 1 . (14)
The functions ρ(r) and A(r) can be obtained numerically by
solving the classical field equations for φ and Aµ. If we write
the field φ as φ = ρexp(iχ)/
2, then from (12) and (13) for
the string, at spatial infinity ρ goes to the vacuum ρv and Aµ
becomes a pure gauge. This also gives, in order to get a finite
energy for the string configuration, that ∂µχ = eAµ at r → ∞,
so Dµφ = 0. This leads then that, by taking some contour C
surrounding the symmetry axis, and using Stokes’ theorem, to
the nonvanishing magnetic flux
∂µχdxµ = 2π/e . (15)
Since φ must be single-valued, the Eq. (15) implies that on
the string χ must be singular. Therefore, the phase χ can be
separated into two parts: in a regular part and in a singular
one, due to the string configuration,
χ(x) = χreg(x)+χsing(x) , (16)
where the singular (multivalued) part χsing(x) can be related
to a closed world-sheet of an vortex-string [14],
εµνλρ∂λ∂ρχsing(x) = n
dσµν(x)δ4[x− y(ξ)] = ωµν , (17)
where n is a topological quantum number, the winding num-
ber, which we here restrict to the lowest values, n = ±1, cor-
responding to the energetically dominant configurations. The
element of area on the world sheet swept by the string is given
dσµν(x) =
d2ξ (18)
and yµ(ξ) represents a point on the world sheet S of the vortex-
string, with internal coordinates ξ0 and ξ1. As usual, we con-
sider that ξ1 is a periodic variable, since we work with closed
strings, whereas ξ0 will be proportional to the time variable
(at zero temperature), in such a way that ξ1 parameterizes a
closed string at a given instant ξ0. Eq. (17) is known as the
vorticity. Eqs. (17) and (18) entails the vortex-string degrees
of freedom and then can be used to identify the topological
vortex string contributions to the partition function.
Let us briefly recall two main previous methods to take into
account the effect of topological strings in phase transitions.
The first attempt to do so made use of semiclassical methods
[2]. In the semiclassical method we use directly the nonho-
mogeneous string solutions, Eqs. (12) and (13), when evalu-
ating the effective action. In this case the effective action is
evaluated after taking fluctuations around the string vacuum
solutions, φ→ φstring+φ′ and Aµ →Aµ,string+A′µ and the func-
tional integration performed over the fluctuation fields φ′ and
A′µ. In the one-loop approximation, this gives the analogous to
Eq. (4), with the constant background field now replaced by
the scalar string background configuration plus those analo-
gous loop corrections for the gauge field string configuration.
But from Eqs. (4) and (8), we see that the effective action at
finite temperature is just associated with the free energy of the
system, where, here is the free energy in the presence of the
string field configurations. This is the procedure used for in-
stance in the papers in Ref. [11]. The free energy relevant for
the study is written as [11]
Fstring =−
Zstring
, (19)
where Zstring is the partition function evaluated in the presence
(imposing the appropriate boundary conditions for) of strings,
Zstring is the partition function for the trivial (constant) vac-
uum sector of the model (and then Eq. (19) is actuals the free
energy difference between the string and trivial vacuum sec-
tors). β, as always, is the inverse of the temperature (we use
throughout this work, unless explicitly noted, with the natural
units ~,c,kB = 1) and L is the size of the system.
The difficult with the approach given by (19), which be-
comes evident from Eq. (4) when we are dealing with noncon-
stant background fields, is the nonlocal terms that appears in
higher order perturbation terms when expanding the effective
action (in this case, the effective action for the string back-
ground configurations). The only terms amenable of analysis
are the one-loop leading order terms. Analogous approach
based on the semiclassical method, is the direct evaluation of
the classical partition function taking into account the string
degrees of freedom, as performed by the authors in Ref. [12]
and where the contribution and interpretation of the phase
transition based on the picture of string condensation is an-
alyzed using known statistical physics results.
Another approach that has been used is to define a field
creation operator for vortex-string excitations and then work
directly with the correlation functions in terms of these opera-
tors. This is the approach for instance taken in Refs. [13, 15].
However, also in this approach the evaluation of correlation
functions already at tree-level order is involved and results are
lacking beyond that order (though in the first reference of [13]
results for the asymptotic behavior of the two-point correla-
tion function for vortex operators were obtained at one-loop
order, but only for the 2+ 1 dimensions case).
III. THE DUAL ACTION FOR VORTEX-STRINGS
Let us start by writing the partition function for the Abelian
Higgs model (10), which, in Euclidean space-time is given by
4 Rudnei O. Ramos et al.
DADφDφ∗ exp
d3xLE − SGF
, (20)
where in the above expression LE denotes the Lagrangian den-
sity (10) in Euclidian space-time and SGF is some appropriate
gauge-fixing and ghost term that must be added to the action
to perform the functional integral over the relevant degrees
of freedom. A dual action to the original one is obtained
from (20) by appropriately performing Hubbard-Stratonovich
transformations on the original field in such a way to become
explicitly the strings degrees of freedom, like in the form of
Eq. (16) and (17). For that, we first write the complex Higgs
field φ in the polar parameterization form φ = ρeiχ/
2. Then,
the scalar phase field χ is split in its regular and singular terms,
like in Eq. (16).
Lets for now, for convenience, omit the gauge fixing term
SGF in Eq. (20) and re-introduce it again in the final
transformed action. Following e.g. the procedure of Refs.
[16, 17, 18, 19, 20], the functional integral over χ in Eq. (20)
can then be rewritten as
Dχ exp
ρ2 (∂µχ+ eAµ)
Dχsing DχregDCµ
C2µ − iCµ
∂µχreg
− iCµ
∂µχsing + eAµ
Dχsing
DWµν exp
V 2µ + eκAµVµ+ iπκWµνωµν
, (21)
where we have performed the functional integral over χreg
in the second line of Eq. (21). This gives a constraint on
the functional integral measure, δ(∂µCµ), which can be repre-
sented in a unique way by expressing the Cµ in terms of an
antisymmetric field, Cµ = −i κ2 εµνλρ∂νWλρ ≡ κVµ, which then
leads to the last expression in Eq. (21). κ is some arbitrary pa-
rameter with mass dimension and ωµν is the vorticity, defined
by Eq. (17) for the singular phase part of χ. Next, in order to
linearize the dependence on the gauge field in the action we
introduce a new antisymmetric tensor field Gµν through the
identity
d4xF2µν
DGµν exp
G2µν −
G̃µνFµν
,(22)
with G̃µν ≡ 12 εµνλρGλρ.
Substituting Eqs. (21) and (22) back into the partition
function, we can immediately perform the functional in-
tegral over the Aµ field. This leads to the constraint
εµναβ∂µ
Gαβ −Wαβ
= 0 can be solved by setting Gµν =
Wµν − 1µW (∂µBν − ∂νBµ), where we defined, for convenience,
eκ = µW and Bµ is an arbitrary gauge field. Using these ex-
pressions back in the partition function (and re-introducing the
gauge fixing term) we then finally obtain the result
DWµνDχsing DBµDρ
× exp
−Sdual
Wµν,Bµ,ρ,χsing
− SGF
, (23)
where the dual action is given by
Sdual =
2e2ρ2
V 2µ +
(µWWµν − ∂µBν + ∂νBµ)2 +
(∂µρ)
ρ4 + iπ
Wµνωµν
. (24)
The model described by Sdual is completely equivalent to the
original Abelian Higgs model, in the polar representation ob-
tained from Eq. (20) and so, any calculations done using (23)
must lead to the same results as those done with the original
action. The advantage of this dual formulation (24) is that it
explicitly exhibits the dependence on the singular configura-
tion of the Higgs field, making it appropriate to study phase
transitions driven by topological defects. At the same time
it also show, from the last term in Eq. (24), that the vortex-
string’s degrees of freedom coupled to the matter field through
the antisymmetric (or Kalb-Ramond) field. Now, if we come
to the part concerning the gauge fixing term SGF in (23), we
Brazilian Journal of Physics, vol. 37, no. 1B, March, 2007 5
see from Eq. (24) that the dual action exhibits invariance un-
der the double gauge transformation: the hypergauge transfor-
mation
δWµν(x) = ∂µξν(x)− ∂νξµ(x) ,
δBµ = µW ξµ(x) , (25)
and the usual gauge transformation
δBµ = ∂µθ(x) , (26)
where ξµ(x) and θ(x) are arbitrary vector and scalar functions,
respectively. Choosing ξµ = Bµ in the first transformation is
equivalent to fix the gauge through the condition Bµ = 0 [17]
and this is equivalent to choose the unitary gauge in Eq. (23).
The complete form for the gauge fixing action accounting for
the gauge invariances (25) and (26) was obtained in Ref. [21],
which, besides an overall normalization factor independent of
the action fields (and the background Higgs field) give for the
quantum partition function the complete result [21]
Z = N
DWµν DρDBµ DηDη exp
2e2ρ2
V 2µ +
(µWWµν − ∂µBν + ∂νBµ)2
(∂µρ)
ρ4 −ηρ−3η− 1
(∂µWµν)
µWWµν (∂µBν − ∂νBµ)+
(∂µBµ)
. (27)
where η, η are the ghost fields used to exponentiate the Jaco-
bian ρ−3 in the functional integration measure in Eq. (23) and
θ,u and ξ are gauge parameters.
IV. THE EFFECTIVE POTENTIAL FOR LOCAL
VORTEX-STRINGS AVERAGED FIELDS
Let us turn now to the study of the problem of vortex-strings
condensation during a phase transition. Thus, to proceed fur-
ther with the evaluation of the string contribution to the par-
tition function we introduce a (nonlocal) field associated to
the string. Quantizing the vortex–strings as nonlocal objects
and associating to them a wave function Ψ[C], a functional
field, where C is the closed vortex–string curve in Euclidean
space-time, and noting that the interaction term of the vortex-
string with the antisymmetric field in Eq. (24) is in the form of
a current coupled to the antisymmetric field, following Refs.
[22, 23] we can define the string action term in the form
Sstring(Ψ[C],Wµν) =
|DσµνΨ[C]|2 −M40|Ψ[C]|2
where Dσµν is a covariant derivative term defined by [24]
Dσµν(x) =
δσµν(x)
− i2πµW
Wµν(x) , (29)
where δσµν(x) is to be considered as an infinitesimal rectan-
gular deformation on the string’s worldsheet. It can be eas-
ily checked that Eq. (28) is invariant under the combined
gauge invariances (25) and (30) if the hypergauge transfor-
mation (25) is now supplemented by the vortex–string field
transformation
Ψ[C]→ exp
−i2πµW
dxµξµ(x)
Ψ[C] . (30)
M40 in Eq. (28) is a dynamical mass for the strings,
M40 ≡
with τs is the string tension (the total energy per unit length
of the vortex-string) [22, 23], which, in terms of the pa-
rameters of the Abelian Higgs model, it is given by [25]
τs = πρ2c ε(λ/e
2), where ε(λ/e2) is a function that increases
monotonically with the ratio of coupling constants. a in Eq.
(31) can approximately be given by the string typical radius
can be expressed as [12])
1/a ∼ mφ
, (32)
where Tc is the mean-field critical temperature, Tc =
12m2φ/(3e
2 + 2λ/3) [26]. ρc, the Higgs vacuum expecta-
tion value, can likewise be expressed as
. (33)
By defining a local string field as [22]
ψ̂C ≡ 4
|Ψ[C]|2 , (34)
6 Rudnei O. Ramos et al.
where l is the length of a curve C, and Cx,t represents a curve
passing through a point x in a fixed direction t. The vacuum
expectation value of ψ̂C is denoted by ψC and represents the
sum of existence probabilities of vortices in Cx,t . In terms of
ψ̂C, it can be shown that the contribution of the vortices to the
quantum partition function, indicated by the last term in Eq.
(24) and the integration over χsing, can be written as [22]
DΨ[C]DΨ∗[C]e
M40 ψ̂C+
µνψ̂C
. (35)
Eq. (35) implies, together with Eq. (24), that an im-
mediate consequence of ψC 6= 0 is the increase of the Wµν
mass. This is directly associated with a shift in the mass of
the original gauge field in the broken phase, MA = eρc, as
M2A → M2A(1+ψC). Since the field ψC, defined by Eq. (34),
works just like a local field for the vortex-strings, we are al-
lowed to define an effective potential for its vacuum expec-
tation value ψC in just the same way as we do for a constant
Higgs field. Since this vortex-string field only couples directly
to Wµν, at the one-loop level the effective potential for ψC will
only involve internal propagators of the antisymmetric tensor
field. This effective potential, at one-loop order and at T = 0,
was actually computed in Ref. [22] in the Landau gauge for
the antisymmetric tensor field propagator and it is given by (in
Euclidean momentum space and at finite temperatures)
1-loop
eff (ψC) =
M40ψC
(2π)3
ω2n +k
2 +M2A(1+ψC)
, (36)
where ωn = 2πn/β are the Matsubara frequencies for bosons.
When ψC = 0, in the absence of string vacuum contribu-
tions to the partition function, we re-obtain the standard result
for the one-loop contribution to the Higgs effective potential
coming from the gauge field loops. The sum over the Mat-
subara frequencies in (36) is easily performed [26]. We can
also work with the resulting expression by expanding it in the
high-temperature limit MA
1+ψC/T ≪ 1 and for e2/λ≪ 1,
which corresponds to deep in the second order regime of
phase transition for the scalar abelian Higgs model. This
is analogous to the phenomenology of the Landau-Ginzburg
theory for superconductors, where the parameter (e2/λ)−1
(also called the Ginzburg parameter), measuring the ratio of
the penetration depth and the coherent length, controls the
regimes called Type II and Type I superconductors. In our
case, the coherent length is governed by a ∼ 1/MH , where
MH is here the temperature dependent Higgs mass, while the
penetration depth is proportional to 1/MA, where MA is the
(temperature dependent) gauge field mass. This way we find
a manageable expression for the finite temperature effective
potential given by [21]
eff,string(ψC)≃
M40 +
3e2ρ2c
16π2a2
e2ρ2c
(1+ψC)
3/2 T − 3e
4ρ4c ln [2/aT ]
ψ2C , (37)
where M0, a and ρc are given by Eqs. (31), (32) and (33).
We can then see that the quantum and thermal corrections
in the effective potential for strings, Eq. (37), are naturally
ordered in powers of α = e2/λ. Therefore, in the regime
α ≪ 1 the leading order correction to the tree-level potential
in Eq. (37) is linear in ψC, while the second and the third
correction terms are O(α3/2) and O(α2), respectively. Thus,
the linear term in ψC controls the transition in the deep sec-
ond order regime since the other terms are all subleading in
α. Thus, near criticality, determined by some temperature Ts
where the linear term in Eq. (37) vanishes, V
eff,string(ψC) ∼ 0
in the α ≪ 1 regime. Ts is interpreted as the temperature of
transition from the normal vacuum to the state of condensed
strings, is then determined by the temperature where the lin-
ear term in ψC in Eq. (37) vanishes and it is found to be re-
lated to the mean field critical temperature, for which the ef-
fective mass term of the Higgs field, obtained from V
eff (ρc),
vanishes. Using again Eqs. (32) and (33), with the result
τsa2 ∼ O(1/λ) and in the perturbative regime e2 ≪ λ ≪ 1,
after some straightforward algebra, we find the relation
Tc −Ts
e−1/λ
[1+O(α)] , (38)
with next order corrections to the critical temperatures dif-
ference being of order O(α). This result for Ts allows us to
identify it with the Ginzburg temperature TG for which the
contribution of the gauge field fluctuations become important.
These results are also found to be in agreement with the calcu-
lations done by the authors in Ref. [12], who analyzed an anal-
ogous problem using the partition function for strings config-
urations, in the same regime of deep second order transition.
For the case where the gauge fluctuations are stronger, i.e.,
for α = e2/λ & 1, the second term in Eq. (37) of order α3/2
induces a cubic term ρ3c to the effective potential, favoring the
appearance of a first order phase transition instead of a second
order one. Here we see that the non-trivial vacuum ψc 6= 0
above the critical temperature Ts enhance the first order phase
transition by an amount (1+ψc)3/2. Hence, since Ts ∼ Tc,
we see that the driven mechanism of the first order transition
can be interpreted as a melting of topological defects. This
mechanism is very well known in condensed matter physics
[20] and always leads to a first order phase transition (except
in two dimensions).
V. CONCLUSIONS
We have interpreted here the phase transition in the scalar
Abelian Higgs model as a process of condensation of vortex-
Brazilian Journal of Physics, vol. 37, no. 1B, March, 2007 7
strings condensation. Our analysis was based on a dual real-
ization of the original model in such a way to make explicitly
the vortex-strings degrees of freedom of the nontrivial vacuum
of the model. This way, by constructing a field theory model
for string fields, the finite temperature effective potential for a
local expectation value for the string field was obtained. The
transition temperature obtained from this effective potential,
the temperature of transition from the normal vacuum to the
state of condensed strings, was then obtained and identified
with the Ginzburg temperature for which gauge field fluctua-
tions become important.
Possible extensions of this work could, for example, to in-
clude magnetic monopoles, like in the context of the compact
Abelian Higgs model [27], in which case monopoles could be
added as external fields in the dual transformations. The study
of finite temperature effects and possible consequences for the
confinement picture in the dual superconductor model, should
be possible in the context of the study performed in this work.
Acknowledgments
The authors would like to thank Conselho Nacional de
Desenvolvimento Cientı́fico e Tecnológico (CNPq-Brazil),
Fundação de Amparo à Pesquisa do Estado do Rio de Janeiro
(FAPERJ) for the financial support. R.O.R. would like to
thank the organizers of the conference Infrared QCD in Rio
for the invitation to talk about this work at the conference.
[1] S. Coleman, Aspects of Symmetry (Cambridge University Press,
Cambridge, 1985).
[2] R. Rajaraman, Solitons and Instantons (North Holland, Ams-
terdam, 1989).
[3] A. Vilenkin and E. P. S. Shellard, Cosmic Strings and Other
Topological Defects (Cambridge University Press, Cambridge,
2000).
[4] H. B. Nielsen and P. Olesen, Nuc. Phys. B 61, 45 (1973).
[5] G. ’t Hooft, Nucl. Phys. B 79, 276 (1974).
[6] A. M. Polyakov, JETP Lett. 20, 194 (1974); Soviet Phys. JETP
41, 988 (1976).
[7] T. W. B. Kibble, Symmetry breaking and defects, in T.W.B. Kib-
ble (Imperial Coll., London) . IMPERIAL-02-3-5, Nov 2002.
35pp. Lectures given at NATO Advanced Study Institute and
COSLAB School: Patterns of Symmetry Breaking, Crakow,
Poland, 15-29 Sep 2002. ArXive: cond-mat/0211110.
[8] L. Dolan and R. Jackiw, Phys. Rev. D 9, 3320 (1974); S. Wein-
berg, Phys. Rev. D 9, 3357 (1974).
[9] S. Coleman and E. Weinberg, Phys. Rev. D 3, 1888 (1973); P.
Ramond, Field Theory: A Modern Primer, (Addison-Wesley,
1990).
[10] J. Kapusta, Finite Temperature Field Theory, (Cambridge Uni-
versity Press, 1989).
[11] C. A. de Carvalho, D. Bazeia, O. J. P. Eboli and G. C. Marques,
Phys. Rev. D 32, 3256 (1985); J. M. Guerra Jr., G. C. Marques
and S. J. Rodrigues, Phys. Rev. D 42, 2022 (1990).
[12] E. Copeland, D. Haws and R. Rivers, Nuc. Phys. B 319, 687
(1989).
[13] E. C. Marino, G. C. Marques, R. O. Ramos and J. S. Ruiz, Phys.
Rev. D 45, 3690 (1992); E. C. Marino and R. O. Ramos, Phys.
Rev. D 49, 1093 (1994).
[14] P. A. M. Dirac, Phys. Rev. D 74, 817 (1948).
[15] E. C. Marino, Phys. Rev. D 53, 1001 (1996).
[16] K. Lee, Phys. Rev. D 48, 2493 (1993).
[17] P. Orland, Nuc. Phys. B 428, 221 (1994).
[18] E. T. Akhmedov, M. N. Chernodub, M. I. Polikarpov and M. A.
Zubkov, Phys. Rev. D 53, 2087 (1996).
[19] D. Antonov and D. Ebert, Eur. Phys. J. C 12, 349 (2000); D.
Antonov, Int. J. Mod. Phys. A 14, 4347 (1999).
[20] H. Kleinert, Lett. Nuovo Cimento 35, 405 (1982); Phys. Lett.
B 293, 168 (1992); see also, Chapter 13 in the second volume
of H. Kleinert, Gauge fields in Condensed Matter, Vol I: Su-
perflow and Vortex lines (World Scientific, Singapore, 1989);
Gauge fields in Condensed Matter, Vol II: Stresses and Defects,
Differential geometry, Crystal defects (World Scientific, Singa-
pore, 1989).
[21] R. O. Ramos, J. F. Medeiros Neto, D. G. Barci, C. A. Linhares,
Phys. Rev. D 72, 103524 (2005).
[22] K. Seo and A. Sugamoto, Phys. Rev. D 24, 1630 (1981).
[23] H. Kawai, Progr. Theor. Phys. 65, 351 (1981).
[24] Y. Nambu, in Broken Symmetry (World Scientific, Hong Kong,
1995).
[25] M. B. Hindmarsh and T. W. B. Kibble, Rept. Progr. Phys. 48,
477 (1995).
[26] L. Dolan and R. Jackiw, Phys. Rev. D 9, 2904 (1974).
[27] M. N. Chernodub, Phys. Rev. D 69 094504 (2004).
http://arxiv.org/abs/cond-mat/0211110

  Topological excitations are believed to play an important role in different
areas of physics. For example, one case of topical interest is the use of dual
models of quantum cromodynamics to understand properties of its vacuum and
confinement through the condensation of magnetic monopoles and vortices. Other
applications are related to the role of these topological excitations,
nonhomogeneous solutions of the field equations, in phase transitions
associated to spontaneous symmetry breaking in gauge theories, whose study is
of importance in phase transitions in the early universe, for instance. Here we
show a derivation of a model dual to the scalar Abelian Higgs model where its
topological excitations, namely vortex-strings, become manifest and can be
treated in a quantum field theory way. The derivation of the nontrivial
contribution of these vacuum excitations to phase transitions and its analogy
with superconductivity is then made possible and they are studied here.

Brazilian Journal of Physics, vol. 37, no. 1B, March, 2007 1
Condensation of Vortex-Strings: Effective Potential Contribution Through Dual
Actions
Rudnei O. Ramos,∗ Daniel G. Barci,† and Cesar A. Linhares‡
Departamento de Fı́sica Teórica, Universidade do Estado do Rio de Janeiro, 20550-013 Rio de Janeiro, RJ, Brazil
J. F. Medeiros Neto§
Instituto de Fı́sica, Universidade Federal do Pará, 66075-110 Belem, Pará, Brazil
Topological excitations are believed to play an important role in different areas of physics. For example, one
case of topical interest is the use of dual models of quantum cromodynamics to understand properties of its
vacuum and confinement through the condensation of magnetic monopoles and vortices. Other applications are
related to the role of these topological excitations, nonhomogeneous solutions of the field equations, in phase
transitions associated to spontaneous symmetry breaking in gauge theories, whose study is of importance in
phase transitions in the early universe, for instance. Here we show a derivation of a model dual to the scalar
Abelian Higgs model where its topological excitations, namely vortex-strings, become manifest and can be
treated in a quantum field theory way. The derivation of the nontrivial contribution of these vacuum excitations
to phase transitions and its analogy with superconductivity is then made possible and they are studied here.
PACS numbers: 11.10.Wx, 98.80.Cq
Keyword: dual models, vortices, phase transitions
I. INTRODUCTION
Topological excitations, or defects, are nonhomogeneous
solutions of the field equations of motion in many types of
field theory models [1, 2, 3]. They are finite energy and stable
configurations that emerge as a consequence of a spontaneous
symmetry breaking process. Mathematically, defects are pre-
dicted to appear whenever some larger group of symmetry G
breaks into a smaller one H such that there are a nontrivial
homotopy group πk(G/H) of the vacuum manifold different
from the identity. Well known examples are kinks, or do-
main walls (k= 0) [2], that originate from a discrete symmetry
breaking, strings, or vortices (k = 1), for example originating
from a continuous gauge symmetry breaking U(1) → 1 [4]
and magnetic monopoles (k = 2), e.g. from a SO(3)→ U(1)
symmetry breaking [5, 6]. Since many phase transitions in na-
ture are associated to symmetry breakings, topological excita-
tions are a common feature in these processes and are in fact
observed in many systems in the laboratory, like in ferromag-
netism, helium superfluidity, superconductivity and in many
other condensed matter system and they are also expected to
have appeared in phase transitions in the early universe as well
(for a general review, please see [7]).
In the study of phase transitions in quantum field theory
one basic quantity usually computed is the effective potential,
which is an important tool in the study of phase transitions
in scalar and gauge field theories [8]. It is equivalent to the
homogeneous coarse-grained free-energy density functional
of statistical physics, with its minima giving the stable and,
when applicable, metastable states of the system. For inter-
acting field theories the effective potential is evaluated per-
∗Electronic address: rudnei@uerj.br
†Electronic address: barci@uerj.br
‡Electronic address: linhares@dft.if.uerj.br
§Electronic address: jfmn@ufpa.br
turbatively, with an expansion in loops being equivalent to an
expansion in powers of ~ [9]. The one-loop approximation is
then equivalent to incorporating the first quantum corrections
to the classical potential. Recall that the effective potential,
taking a scalar field theory as an example, is obtained from
the effective action Γ[φc], where it is defined in terms of the
connected generating functional W [J] as
Γ[φc] =W [J]−
d4xJ(x)φc(x) , (1)
with the classical field φc(~x, t) defined by
φc(~x, t)≡ δW [J]/δJ(x), and
W [J] =−i~ln
Dφ exp
S[φ,J]
. (2)
In order to evaluate Γ[φc] perturbatively, one writes the
field as φ(~x, t) → φ0(~x, t) + η(~x, t), where φ0(~x, t) is a field
configuration which extremizes the classical action S[φ,J],
δS[φ,J]
δφ |φ=φ0 = 0, and η(~x, t) is a small perturbation about that
extremum configuration. The action S[φ,J] can then be ex-
panded about φ0(~x, t) and, up to quadratic order in η(~x, t), we
can use a saddle-point approximation to the path integral to
obtain for the connected generating functional,
W [J] = S[φ0]+~
d4xφ0(x)J(x)+ i
∂µ∂µ+V ′′(φ0)
In order to obtain the one-loop expression for Γ[φc], we first
note that writing φ0 = φc−η we get to first order in ~, S[φ0] =
S[φc]−~
d4xη(x)J(x)+O(~2). Using this result and Eq. (3)
into Eq. (1) we find, as J → 0,
Γ[φc] = S[φc]+ i
∂µ∂µ+V ′′(φc)
. (4)
http://arxiv.org/abs/0704.1117v1
mailto:rudnei@uerj.br
mailto:barci@uerj.br
mailto:linhares@dft.if.uerj.br
mailto:jfmn@ufpa.br
2 Rudnei O. Ramos et al.
The effective action can also be computed as a derivative ex-
pansion about φc(~x, t),
Γ[φc] =
−Veff(φc(x))+
(∂µφc)
2 Z(φc(x))+ . . .
The function Veff(φc) is the effective potential. For a constant
field configuration φc(~x, t) = φc we obtain
Γ[φc] =−ΩVeff(φc) , (6)
where Ω is the total volume of space-time. Comparing Eqs.
(4) and (6) we obtain for the one-loop effective potential,
Veff(φc) =V (φc)− i
Ω−1Trln
∂µ∂µ+V ′′(φc)
. (7)
When working at non-vanishing temperature, the same
functional techniques can be used. In this case one is in-
terested in evaluating the generating functional (the partition
function) Zβ[J] which is given by the path integral [10]
Zβ[J] = N
Dφexp
d3x(LE − Jφ)
, (8)
where the integration is restricted to paths periodic in τ with
φ(0,~x) = φ(β,~x), LE is the Euclidean Lagrangian, and N is
a normalization constant. Again one expands about an ex-
tremum of the Euclidean action and calculates the partition
function by a saddle-point evaluation of the path integral. The
result for the one-loop approximation to the effective potential
Veff(φc,T ) =Veff(φc)
2π2β4
dx x2ln
1− exp
x2 +β2V ′′(φc)
As the effective potential is equivalent to the free energy func-
tional (for a constant field configuration), all thermodynamics
functions follow from it. In particular the different phases,
critical temperature of phase transition and temperature de-
pendence of the field vacuum expectation value can be ob-
tained from (9).
From the above discussion it is clear that the one-loop ap-
proximation to the effective action, Eq. (4), works best when
the classical field does not differ much from the configuration
that extremizes the classical action, φc = φ0 +η ∼ φ0, since in
this case the saddle-point evaluation to the path integral is ad-
equate. Also, φc(~x, t) must be nearly constant so that the effec-
tive potential can be obtained from Eq. (6). As J → 0, φc(~x, t)
is identified with 〈φ〉, the vacuum expectation value. One ma-
jor problem we see in this whole approach of studying the
phase structure of field theory models from the effective po-
tential is when the action functional, determined from Eq. (8)
is dominated not by homogeneous, constant field configura-
tions but by nonhomogeneous ones. In those situations when
other stable, finite energy solutions to the field equations of
motion exist, we expect these configurations to dominate the
partition function over the homogenous solutions for instance
close to the critical temperature [11]. Under these circum-
stances the effective potential, which includes only contri-
butions to the partition function from constant background
field configurations becomes inappropriate to study the phase
transition and we must rely in other approaches, for example
studying the phase transition directly from the effective action
or free energy for the topological configurations [11, 12], or
taking directly a field theoretic description for the topological
excitations [13]. In either case we are faced with the problem
of accounting for nonlocal contributions in the perturbative
expansion, which is only amenable of analysis up to the low-
est order leading order. To circumvent these difficulties we
make use of the techniques of dualization in field theory, from
which the degrees of freedom of the topological excitations
are explicitly realized in the functional action. This method
is described in the following section, Sec. II, where we spe-
cialize to the case of the scalar Abelian Higgs model, whose
topological solutions are vortex-strings. In Sec. III we evalu-
ated the dual action for the model, making explicit the vortex-
strings degrees of freedom and how they couple to the matter
fields. In Sec. IV we show how an effective potential calcula-
tion for an averaged vortex-string field can be computed and
the results and interpretation of the phase transition obtained
from this quantity. Finally, we give our conclusions in Sec. V.
II. THE STRING SOLUTIONS IN THE SCALAR ABELIAN
HIGGS MODEL
In this work we will use the Abelian Higgs model, with La-
grangian density for a complex scalar field φ and gauge field
Aµ given by
L =−1
µν + |Dµφ|2 −V(φ) , (10)
where, Fµν = ∂µAν−∂νAµ, Dµ = ∂µ− ieAµ and V (φ) is a sym-
metry breaking potential, for example given by
V (φ) =−m2φ |φ|
, (11)
with positive parameters m2φ and λ. The symmetry breaking
U(1) → 1 with homotopy group π1 6= 1 indicates the exis-
tence of string-like topological excitations in the system, or
Nielsen-Olesen strings [4] (for an extended introduction and
review see e.g. Ref. [3]). For example, for a unit winding
string solution along the z axis, the classical field equations
of motion obtained from the Lagrangian density (10) admit a
stable finite energy configuration describing the string given
by (using the cylindrical coordinates r,θ,z)
φstring =
ρ(r)√
eiθ , (12)
Aµ,string =
A(r) ∂µθ , (13)
Brazilian Journal of Physics, vol. 37, no. 1B, March, 2007 3
where the functions ρ(r) and A(r) vanish at the origin and
have the asymptotic behavior
φ(r → ∞) → ρv ≡
A(r → ∞) → 1 . (14)
The functions ρ(r) and A(r) can be obtained numerically by
solving the classical field equations for φ and Aµ. If we write
the field φ as φ = ρexp(iχ)/
2, then from (12) and (13) for
the string, at spatial infinity ρ goes to the vacuum ρv and Aµ
becomes a pure gauge. This also gives, in order to get a finite
energy for the string configuration, that ∂µχ = eAµ at r → ∞,
so Dµφ = 0. This leads then that, by taking some contour C
surrounding the symmetry axis, and using Stokes’ theorem, to
the nonvanishing magnetic flux
∂µχdxµ = 2π/e . (15)
Since φ must be single-valued, the Eq. (15) implies that on
the string χ must be singular. Therefore, the phase χ can be
separated into two parts: in a regular part and in a singular
one, due to the string configuration,
χ(x) = χreg(x)+χsing(x) , (16)
where the singular (multivalued) part χsing(x) can be related
to a closed world-sheet of an vortex-string [14],
εµνλρ∂λ∂ρχsing(x) = n
dσµν(x)δ4[x− y(ξ)] = ωµν , (17)
where n is a topological quantum number, the winding num-
ber, which we here restrict to the lowest values, n = ±1, cor-
responding to the energetically dominant configurations. The
element of area on the world sheet swept by the string is given
dσµν(x) =
d2ξ (18)
and yµ(ξ) represents a point on the world sheet S of the vortex-
string, with internal coordinates ξ0 and ξ1. As usual, we con-
sider that ξ1 is a periodic variable, since we work with closed
strings, whereas ξ0 will be proportional to the time variable
(at zero temperature), in such a way that ξ1 parameterizes a
closed string at a given instant ξ0. Eq. (17) is known as the
vorticity. Eqs. (17) and (18) entails the vortex-string degrees
of freedom and then can be used to identify the topological
vortex string contributions to the partition function.
Let us briefly recall two main previous methods to take into
account the effect of topological strings in phase transitions.
The first attempt to do so made use of semiclassical methods
[2]. In the semiclassical method we use directly the nonho-
mogeneous string solutions, Eqs. (12) and (13), when evalu-
ating the effective action. In this case the effective action is
evaluated after taking fluctuations around the string vacuum
solutions, φ→ φstring+φ′ and Aµ →Aµ,string+A′µ and the func-
tional integration performed over the fluctuation fields φ′ and
A′µ. In the one-loop approximation, this gives the analogous to
Eq. (4), with the constant background field now replaced by
the scalar string background configuration plus those analo-
gous loop corrections for the gauge field string configuration.
But from Eqs. (4) and (8), we see that the effective action at
finite temperature is just associated with the free energy of the
system, where, here is the free energy in the presence of the
string field configurations. This is the procedure used for in-
stance in the papers in Ref. [11]. The free energy relevant for
the study is written as [11]
Fstring =−
Zstring
, (19)
where Zstring is the partition function evaluated in the presence
(imposing the appropriate boundary conditions for) of strings,
Zstring is the partition function for the trivial (constant) vac-
uum sector of the model (and then Eq. (19) is actuals the free
energy difference between the string and trivial vacuum sec-
tors). β, as always, is the inverse of the temperature (we use
throughout this work, unless explicitly noted, with the natural
units ~,c,kB = 1) and L is the size of the system.
The difficult with the approach given by (19), which be-
comes evident from Eq. (4) when we are dealing with noncon-
stant background fields, is the nonlocal terms that appears in
higher order perturbation terms when expanding the effective
action (in this case, the effective action for the string back-
ground configurations). The only terms amenable of analysis
are the one-loop leading order terms. Analogous approach
based on the semiclassical method, is the direct evaluation of
the classical partition function taking into account the string
degrees of freedom, as performed by the authors in Ref. [12]
and where the contribution and interpretation of the phase
transition based on the picture of string condensation is an-
alyzed using known statistical physics results.
Another approach that has been used is to define a field
creation operator for vortex-string excitations and then work
directly with the correlation functions in terms of these opera-
tors. This is the approach for instance taken in Refs. [13, 15].
However, also in this approach the evaluation of correlation
functions already at tree-level order is involved and results are
lacking beyond that order (though in the first reference of [13]
results for the asymptotic behavior of the two-point correla-
tion function for vortex operators were obtained at one-loop
order, but only for the 2+ 1 dimensions case).
III. THE DUAL ACTION FOR VORTEX-STRINGS
Let us start by writing the partition function for the Abelian
Higgs model (10), which, in Euclidean space-time is given by
4 Rudnei O. Ramos et al.
DADφDφ∗ exp
d3xLE − SGF
, (20)
where in the above expression LE denotes the Lagrangian den-
sity (10) in Euclidian space-time and SGF is some appropriate
gauge-fixing and ghost term that must be added to the action
to perform the functional integral over the relevant degrees
of freedom. A dual action to the original one is obtained
from (20) by appropriately performing Hubbard-Stratonovich
transformations on the original field in such a way to become
explicitly the strings degrees of freedom, like in the form of
Eq. (16) and (17). For that, we first write the complex Higgs
field φ in the polar parameterization form φ = ρeiχ/
2. Then,
the scalar phase field χ is split in its regular and singular terms,
like in Eq. (16).
Lets for now, for convenience, omit the gauge fixing term
SGF in Eq. (20) and re-introduce it again in the final
transformed action. Following e.g. the procedure of Refs.
[16, 17, 18, 19, 20], the functional integral over χ in Eq. (20)
can then be rewritten as
Dχ exp
ρ2 (∂µχ+ eAµ)
Dχsing DχregDCµ
C2µ − iCµ
∂µχreg
− iCµ
∂µχsing + eAµ
Dχsing
DWµν exp
V 2µ + eκAµVµ+ iπκWµνωµν
, (21)
where we have performed the functional integral over χreg
in the second line of Eq. (21). This gives a constraint on
the functional integral measure, δ(∂µCµ), which can be repre-
sented in a unique way by expressing the Cµ in terms of an
antisymmetric field, Cµ = −i κ2 εµνλρ∂νWλρ ≡ κVµ, which then
leads to the last expression in Eq. (21). κ is some arbitrary pa-
rameter with mass dimension and ωµν is the vorticity, defined
by Eq. (17) for the singular phase part of χ. Next, in order to
linearize the dependence on the gauge field in the action we
introduce a new antisymmetric tensor field Gµν through the
identity
d4xF2µν
DGµν exp
G2µν −
G̃µνFµν
,(22)
with G̃µν ≡ 12 εµνλρGλρ.
Substituting Eqs. (21) and (22) back into the partition
function, we can immediately perform the functional in-
tegral over the Aµ field. This leads to the constraint
εµναβ∂µ
Gαβ −Wαβ
= 0 can be solved by setting Gµν =
Wµν − 1µW (∂µBν − ∂νBµ), where we defined, for convenience,
eκ = µW and Bµ is an arbitrary gauge field. Using these ex-
pressions back in the partition function (and re-introducing the
gauge fixing term) we then finally obtain the result
DWµνDχsing DBµDρ
× exp
−Sdual
Wµν,Bµ,ρ,χsing
− SGF
, (23)
where the dual action is given by
Sdual =
2e2ρ2
V 2µ +
(µWWµν − ∂µBν + ∂νBµ)2 +
(∂µρ)
ρ4 + iπ
Wµνωµν
. (24)
The model described by Sdual is completely equivalent to the
original Abelian Higgs model, in the polar representation ob-
tained from Eq. (20) and so, any calculations done using (23)
must lead to the same results as those done with the original
action. The advantage of this dual formulation (24) is that it
explicitly exhibits the dependence on the singular configura-
tion of the Higgs field, making it appropriate to study phase
transitions driven by topological defects. At the same time
it also show, from the last term in Eq. (24), that the vortex-
string’s degrees of freedom coupled to the matter field through
the antisymmetric (or Kalb-Ramond) field. Now, if we come
to the part concerning the gauge fixing term SGF in (23), we
Brazilian Journal of Physics, vol. 37, no. 1B, March, 2007 5
see from Eq. (24) that the dual action exhibits invariance un-
der the double gauge transformation: the hypergauge transfor-
mation
δWµν(x) = ∂µξν(x)− ∂νξµ(x) ,
δBµ = µW ξµ(x) , (25)
and the usual gauge transformation
δBµ = ∂µθ(x) , (26)
where ξµ(x) and θ(x) are arbitrary vector and scalar functions,
respectively. Choosing ξµ = Bµ in the first transformation is
equivalent to fix the gauge through the condition Bµ = 0 [17]
and this is equivalent to choose the unitary gauge in Eq. (23).
The complete form for the gauge fixing action accounting for
the gauge invariances (25) and (26) was obtained in Ref. [21],
which, besides an overall normalization factor independent of
the action fields (and the background Higgs field) give for the
quantum partition function the complete result [21]
Z = N
DWµν DρDBµ DηDη exp
2e2ρ2
V 2µ +
(µWWµν − ∂µBν + ∂νBµ)2
(∂µρ)
ρ4 −ηρ−3η− 1
(∂µWµν)
µWWµν (∂µBν − ∂νBµ)+
(∂µBµ)
. (27)
where η, η are the ghost fields used to exponentiate the Jaco-
bian ρ−3 in the functional integration measure in Eq. (23) and
θ,u and ξ are gauge parameters.
IV. THE EFFECTIVE POTENTIAL FOR LOCAL
VORTEX-STRINGS AVERAGED FIELDS
Let us turn now to the study of the problem of vortex-strings
condensation during a phase transition. Thus, to proceed fur-
ther with the evaluation of the string contribution to the par-
tition function we introduce a (nonlocal) field associated to
the string. Quantizing the vortex–strings as nonlocal objects
and associating to them a wave function Ψ[C], a functional
field, where C is the closed vortex–string curve in Euclidean
space-time, and noting that the interaction term of the vortex-
string with the antisymmetric field in Eq. (24) is in the form of
a current coupled to the antisymmetric field, following Refs.
[22, 23] we can define the string action term in the form
Sstring(Ψ[C],Wµν) =
|DσµνΨ[C]|2 −M40|Ψ[C]|2
where Dσµν is a covariant derivative term defined by [24]
Dσµν(x) =
δσµν(x)
− i2πµW
Wµν(x) , (29)
where δσµν(x) is to be considered as an infinitesimal rectan-
gular deformation on the string’s worldsheet. It can be eas-
ily checked that Eq. (28) is invariant under the combined
gauge invariances (25) and (30) if the hypergauge transfor-
mation (25) is now supplemented by the vortex–string field
transformation
Ψ[C]→ exp
−i2πµW
dxµξµ(x)
Ψ[C] . (30)
M40 in Eq. (28) is a dynamical mass for the strings,
M40 ≡
with τs is the string tension (the total energy per unit length
of the vortex-string) [22, 23], which, in terms of the pa-
rameters of the Abelian Higgs model, it is given by [25]
τs = πρ2c ε(λ/e
2), where ε(λ/e2) is a function that increases
monotonically with the ratio of coupling constants. a in Eq.
(31) can approximately be given by the string typical radius
can be expressed as [12])
1/a ∼ mφ
, (32)
where Tc is the mean-field critical temperature, Tc =
12m2φ/(3e
2 + 2λ/3) [26]. ρc, the Higgs vacuum expecta-
tion value, can likewise be expressed as
. (33)
By defining a local string field as [22]
ψ̂C ≡ 4
|Ψ[C]|2 , (34)
6 Rudnei O. Ramos et al.
where l is the length of a curve C, and Cx,t represents a curve
passing through a point x in a fixed direction t. The vacuum
expectation value of ψ̂C is denoted by ψC and represents the
sum of existence probabilities of vortices in Cx,t . In terms of
ψ̂C, it can be shown that the contribution of the vortices to the
quantum partition function, indicated by the last term in Eq.
(24) and the integration over χsing, can be written as [22]
DΨ[C]DΨ∗[C]e
M40 ψ̂C+
µνψ̂C
. (35)
Eq. (35) implies, together with Eq. (24), that an im-
mediate consequence of ψC 6= 0 is the increase of the Wµν
mass. This is directly associated with a shift in the mass of
the original gauge field in the broken phase, MA = eρc, as
M2A → M2A(1+ψC). Since the field ψC, defined by Eq. (34),
works just like a local field for the vortex-strings, we are al-
lowed to define an effective potential for its vacuum expec-
tation value ψC in just the same way as we do for a constant
Higgs field. Since this vortex-string field only couples directly
to Wµν, at the one-loop level the effective potential for ψC will
only involve internal propagators of the antisymmetric tensor
field. This effective potential, at one-loop order and at T = 0,
was actually computed in Ref. [22] in the Landau gauge for
the antisymmetric tensor field propagator and it is given by (in
Euclidean momentum space and at finite temperatures)
1-loop
eff (ψC) =
M40ψC
(2π)3
ω2n +k
2 +M2A(1+ψC)
, (36)
where ωn = 2πn/β are the Matsubara frequencies for bosons.
When ψC = 0, in the absence of string vacuum contribu-
tions to the partition function, we re-obtain the standard result
for the one-loop contribution to the Higgs effective potential
coming from the gauge field loops. The sum over the Mat-
subara frequencies in (36) is easily performed [26]. We can
also work with the resulting expression by expanding it in the
high-temperature limit MA
1+ψC/T ≪ 1 and for e2/λ≪ 1,
which corresponds to deep in the second order regime of
phase transition for the scalar abelian Higgs model. This
is analogous to the phenomenology of the Landau-Ginzburg
theory for superconductors, where the parameter (e2/λ)−1
(also called the Ginzburg parameter), measuring the ratio of
the penetration depth and the coherent length, controls the
regimes called Type II and Type I superconductors. In our
case, the coherent length is governed by a ∼ 1/MH , where
MH is here the temperature dependent Higgs mass, while the
penetration depth is proportional to 1/MA, where MA is the
(temperature dependent) gauge field mass. This way we find
a manageable expression for the finite temperature effective
potential given by [21]
eff,string(ψC)≃
M40 +
3e2ρ2c
16π2a2
e2ρ2c
(1+ψC)
3/2 T − 3e
4ρ4c ln [2/aT ]
ψ2C , (37)
where M0, a and ρc are given by Eqs. (31), (32) and (33).
We can then see that the quantum and thermal corrections
in the effective potential for strings, Eq. (37), are naturally
ordered in powers of α = e2/λ. Therefore, in the regime
α ≪ 1 the leading order correction to the tree-level potential
in Eq. (37) is linear in ψC, while the second and the third
correction terms are O(α3/2) and O(α2), respectively. Thus,
the linear term in ψC controls the transition in the deep sec-
ond order regime since the other terms are all subleading in
α. Thus, near criticality, determined by some temperature Ts
where the linear term in Eq. (37) vanishes, V
eff,string(ψC) ∼ 0
in the α ≪ 1 regime. Ts is interpreted as the temperature of
transition from the normal vacuum to the state of condensed
strings, is then determined by the temperature where the lin-
ear term in ψC in Eq. (37) vanishes and it is found to be re-
lated to the mean field critical temperature, for which the ef-
fective mass term of the Higgs field, obtained from V
eff (ρc),
vanishes. Using again Eqs. (32) and (33), with the result
τsa2 ∼ O(1/λ) and in the perturbative regime e2 ≪ λ ≪ 1,
after some straightforward algebra, we find the relation
Tc −Ts
e−1/λ
[1+O(α)] , (38)
with next order corrections to the critical temperatures dif-
ference being of order O(α). This result for Ts allows us to
identify it with the Ginzburg temperature TG for which the
contribution of the gauge field fluctuations become important.
These results are also found to be in agreement with the calcu-
lations done by the authors in Ref. [12], who analyzed an anal-
ogous problem using the partition function for strings config-
urations, in the same regime of deep second order transition.
For the case where the gauge fluctuations are stronger, i.e.,
for α = e2/λ & 1, the second term in Eq. (37) of order α3/2
induces a cubic term ρ3c to the effective potential, favoring the
appearance of a first order phase transition instead of a second
order one. Here we see that the non-trivial vacuum ψc 6= 0
above the critical temperature Ts enhance the first order phase
transition by an amount (1+ψc)3/2. Hence, since Ts ∼ Tc,
we see that the driven mechanism of the first order transition
can be interpreted as a melting of topological defects. This
mechanism is very well known in condensed matter physics
[20] and always leads to a first order phase transition (except
in two dimensions).
V. CONCLUSIONS
We have interpreted here the phase transition in the scalar
Abelian Higgs model as a process of condensation of vortex-
Brazilian Journal of Physics, vol. 37, no. 1B, March, 2007 7
strings condensation. Our analysis was based on a dual real-
ization of the original model in such a way to make explicitly
the vortex-strings degrees of freedom of the nontrivial vacuum
of the model. This way, by constructing a field theory model
for string fields, the finite temperature effective potential for a
local expectation value for the string field was obtained. The
transition temperature obtained from this effective potential,
the temperature of transition from the normal vacuum to the
state of condensed strings, was then obtained and identified
with the Ginzburg temperature for which gauge field fluctua-
tions become important.
Possible extensions of this work could, for example, to in-
clude magnetic monopoles, like in the context of the compact
Abelian Higgs model [27], in which case monopoles could be
added as external fields in the dual transformations. The study
of finite temperature effects and possible consequences for the
confinement picture in the dual superconductor model, should
be possible in the context of the study performed in this work.
Acknowledgments
The authors would like to thank Conselho Nacional de
Desenvolvimento Cientı́fico e Tecnológico (CNPq-Brazil),
Fundação de Amparo à Pesquisa do Estado do Rio de Janeiro
(FAPERJ) for the financial support. R.O.R. would like to
thank the organizers of the conference Infrared QCD in Rio
for the invitation to talk about this work at the conference.
[1] S. Coleman, Aspects of Symmetry (Cambridge University Press,
Cambridge, 1985).
[2] R. Rajaraman, Solitons and Instantons (North Holland, Ams-
terdam, 1989).
[3] A. Vilenkin and E. P. S. Shellard, Cosmic Strings and Other
Topological Defects (Cambridge University Press, Cambridge,
2000).
[4] H. B. Nielsen and P. Olesen, Nuc. Phys. B 61, 45 (1973).
[5] G. ’t Hooft, Nucl. Phys. B 79, 276 (1974).
[6] A. M. Polyakov, JETP Lett. 20, 194 (1974); Soviet Phys. JETP
41, 988 (1976).
[7] T. W. B. Kibble, Symmetry breaking and defects, in T.W.B. Kib-
ble (Imperial Coll., London) . IMPERIAL-02-3-5, Nov 2002.
35pp. Lectures given at NATO Advanced Study Institute and
COSLAB School: Patterns of Symmetry Breaking, Crakow,
Poland, 15-29 Sep 2002. ArXive: cond-mat/0211110.
[8] L. Dolan and R. Jackiw, Phys. Rev. D 9, 3320 (1974); S. Wein-
berg, Phys. Rev. D 9, 3357 (1974).
[9] S. Coleman and E. Weinberg, Phys. Rev. D 3, 1888 (1973); P.
Ramond, Field Theory: A Modern Primer, (Addison-Wesley,
1990).
[10] J. Kapusta, Finite Temperature Field Theory, (Cambridge Uni-
versity Press, 1989).
[11] C. A. de Carvalho, D. Bazeia, O. J. P. Eboli and G. C. Marques,
Phys. Rev. D 32, 3256 (1985); J. M. Guerra Jr., G. C. Marques
and S. J. Rodrigues, Phys. Rev. D 42, 2022 (1990).
[12] E. Copeland, D. Haws and R. Rivers, Nuc. Phys. B 319, 687
(1989).
[13] E. C. Marino, G. C. Marques, R. O. Ramos and J. S. Ruiz, Phys.
Rev. D 45, 3690 (1992); E. C. Marino and R. O. Ramos, Phys.
Rev. D 49, 1093 (1994).
[14] P. A. M. Dirac, Phys. Rev. D 74, 817 (1948).
[15] E. C. Marino, Phys. Rev. D 53, 1001 (1996).
[16] K. Lee, Phys. Rev. D 48, 2493 (1993).
[17] P. Orland, Nuc. Phys. B 428, 221 (1994).
[18] E. T. Akhmedov, M. N. Chernodub, M. I. Polikarpov and M. A.
Zubkov, Phys. Rev. D 53, 2087 (1996).
[19] D. Antonov and D. Ebert, Eur. Phys. J. C 12, 349 (2000); D.
Antonov, Int. J. Mod. Phys. A 14, 4347 (1999).
[20] H. Kleinert, Lett. Nuovo Cimento 35, 405 (1982); Phys. Lett.
B 293, 168 (1992); see also, Chapter 13 in the second volume
of H. Kleinert, Gauge fields in Condensed Matter, Vol I: Su-
perflow and Vortex lines (World Scientific, Singapore, 1989);
Gauge fields in Condensed Matter, Vol II: Stresses and Defects,
Differential geometry, Crystal defects (World Scientific, Singa-
pore, 1989).
[21] R. O. Ramos, J. F. Medeiros Neto, D. G. Barci, C. A. Linhares,
Phys. Rev. D 72, 103524 (2005).
[22] K. Seo and A. Sugamoto, Phys. Rev. D 24, 1630 (1981).
[23] H. Kawai, Progr. Theor. Phys. 65, 351 (1981).
[24] Y. Nambu, in Broken Symmetry (World Scientific, Hong Kong,
1995).
[25] M. B. Hindmarsh and T. W. B. Kibble, Rept. Progr. Phys. 48,
477 (1995).
[26] L. Dolan and R. Jackiw, Phys. Rev. D 9, 2904 (1974).
[27] M. N. Chernodub, Phys. Rev. D 69 094504 (2004).
http://arxiv.org/abs/cond-mat/0211110
