Global-String and Vortex Superfluids in a Supersymmetric Scenario
Global-String and Vortex Superfluids in a Supersymmetric Scenario
C.N. Ferreira 1,∗ J. A. Helayël-Neto 2,† and W.G. Ney 1‡
Núcleo de Estudos em F́ısica,
Centro Federal de Educação Tecnológica de Campos
Rua Dr. Siqueira, 273, Campos dos Goytacazes,
Rio de Janeiro, Brazil, CEP 28030-130
Centro Brasileiro de Pesquisas F́ısicas,
Rua Dr. Xavier Sigaud 150, Urca,
Rio de Janeiro, Brazil, CEP 22290-180
(Dated: November 1, 2018)
The main goal of this work is to investigate the possibility of finding the supersymmetric version
of the U(1)-global string model which behaves as a vortex-superfluid. To describe the superfluid
phase, we introduce a Lorentz-symmetry breaking background that, in an approach based on su-
persymmetry, leads to a discussion on the relation between the violation of Lorentz symmetry and
explicit soft supersymmetry breakings. We also study the relation between the string configuration
and the vortex-superfluid phase. In the framework we settle down in terms of superspace and su-
perfields, we actually establish a duality between the vortex degrees of freedom and the component
fields of the Kalb-Ramond superfield. We make also considerations about the fermionic excitations
that may appear in connection with the vortex formation.
PACS numbers: 12.60.Jv,11.27.+d
I. INTRODUCTION
Stable vortex states may appear as an interesting man-
ifestation of superfluidity. As one of the motivations,
these vortices have been observed in bosonic or fermionic
diluted gases[1, 2]. In the case of the bosonic vortices,
the detection has been confirmed by analyzing the den-
sity variations in an expanding Bose-Einstein Condensate
(BEC)[2, 3]. In Fermi systems, we do not expect signif-
icant density variations[4] but, under certain conditions,
the density variations may be induced by the presence
of one or more vortices that can be present in nuclear
matter[5]. The interior of a neutron star, that is the only
known system close to nuclear and neutron matter, con-
stitutes the appropriate scenario for the vortex formation
induced by the rotational state of the star.
There are important observations of astrophysical rele-
vance that might be influenced by the presence of vortices
in the interior of neutron stars; for instance, the pulsar
glitches. The glitching events represent a direct manifes-
tation of the presence of superfluid vortices in the inte-
rior of the star, the triggering event being an unbalance
between the hydrodynamical forces acting on the vor-
tex and the force of interaction of the vortex with nuclei
presence in the crust, pinning force [6, 7]; but, there are
doubts about the value of the pinning force.
One is related to the value of the energy gap in uni-
form neutron matter whereas the second problem is due
to the very outlined way of treating vortex states in neu-
∗Electronic address: crisnfer@pq.cnpq.br
†Electronic address: helayel@pq.cnpq.br
‡Electronic address: wander@cefetcampos.br
tron matter. Global strings, which behave as a vortex
superfluidity states, appear when a discrete symmetry is
broken. These strings, as the local strings [9, 10, 11], were
most likely produced during phase transitions [12], and
appear in some Grand-Unified Gauge Theories. They
carry a large energy density [10]. Both global and lo-
cal strings were mainly studied as a possible mechanism
for the seed density perturbation which has become a
structure of large scale of the Universe we observe today
[13, 14].
Nowadays, the approach considering cosmic string
configurations has been revisited in connection with
string theory[12, 15, 16] and the Wilkinson Microwave
Anisotropy Probe (WMAP)[17]. The importance of this
context is related to a possible measurement at the level
of string theory, which has supersymmetry (SUSY) as
one of its main characteristics. SUSY is also related
to cosmic strings in other contexts, where one contem-
plates the possibility that the boson-fermion symmetry
was manifest in the early Universe, but it was broken
approximately at the same time when these topological
defects were formed.
Many recent works investigate local strings by adopt-
ing a supersymmetric framework [18, 19, 20, 21, 22, 23].
In the context of the star formation [24], SUSY appears
as one of the most interesting mechanisms to describe
cold dark matter[25, 26]. Both, local and global strings,
are also important for their contribution to the gravita-
tional radiation background[27]; in the case of the global
symmetry, instead of radiating gravitationally, the dom-
inant radiation mechanism for these strings is the emis-
sion of massless Nambu-Goldstone bosons [28]. Global
strings which behave as a vortex superfluidity states are
connected with a Kalb-Ramond field. In some publica-
tions, it has been shown that, in the low-energy regime,
http://arxiv.org/abs/0704.1935v3
mailto:crisnfer@pq.cnpq.br
mailto:helayel@pq.cnpq.br
mailto:wander@cefetcampos.br
the effective action that presents the Kalb-Ramond fields,
that also appear in string theory, provides an accurate
description of the dynamics of global strings[29].
The Kalb-Ramond field[30, 31] is an antisymmetric
tensor. This tensor, whenever interacting with a mas-
sive Higgs field, gives us a source. The system may have
applications to superfluid helium and axion cosmology. A
global vortex behaves as a superfluid if the Kalb-Ramond
field breaks Lorentz symmetry in the background. The
Kalb-Ramond fields in context of the topological defects
can be studied in [32], with SUSY framework [20, 23]
and associated with Lorentz-symmetry breaking can be
studied in [21, 22]. For these implications, in this work,
we analyze the equivalence of the vortex-superfluids to
global strings in a supersymmetric context. This analogy
is important to propose alternative models to be consid-
ered; the vortex stability and the fermionic and bosonic
behaviors of the matter can in the future enlighten us
how to understand the vortex states in fermionic matter.
The outline of this paper is as follows: in Section 2,
we present some considerations about vortex superfluid
models.
In Section II, we devote our attention to showing how
the Lorentz-symmetry violation by the Kalb-Ramond
background induces explicit SUSY breaking terms. In
Section III, we focus on the general properties of the su-
persymmetric model for the vortex and treats some spe-
cific properties of the supersymmetric superfluid phase
in the model we study. In this Section, we also carry
out the superfield identifications at zero temperature and
start a discussion on the fermionic excitations. Section
IV Fermionic excitations are the main issue presented.
In Section V, we propose a discussion on the non-zero
temperature treatment of the model. Finally, in Section
VI, we draw our General Conclusions.
II. THE IMPLICATIONS OF THE
LORENTZ-VIOLATING BACKGROUND FOR
SOFT SUSY BREAKING
In this Section, we discuss the implications of the pres-
ence of the Lorentz-violating background for a soft SUSY
breaking. The idea here is to understand how the explicit
soft SUSY breaking works to yield mass to some of the
field of the model.
We consider the possibility to get the scalar masses
in SUSY theories by working with a Lorentz-symmetry
violating background. This framework is important for
a better understanding of the relation between explicit
SUSY breaking and Lorentz-symmetry violation. It is
important to stress that the problem of the SUSY break-
ing is a very important matter, related with the hiearchy
problem and the mass constraints on the supersymmet-
ric particles. In this section, let us start off with the
following supersymmetric Lagrangian:
LK = Φ
†e4gGΦ|θθθ̄θ̄. (1)
where the ingredient superfields of the model are: a chiral
scalar supermultiplet, Φ(φ, χ, F ), that contains a com-
plex scalar field, φ, a spinor, χa, and an auxiliary com-
plex scalar field, F . The chiral scalar supermultiplet Φ
can be θ-expanded according to the following expression
Φ = e−iθσ
µθ̄∂µ [φ(x) +
2θaχa(x) + θ
2F (x)], (2)
G is the Kalb-Ramond field-strength superfield defined in
terms of the chiral spinor superfield as
(DaΣa − D̄ȧΣ̄ȧ) (3)
where
Σa = ψa(x) + θ
bΩba(x) + θ
ξa(x) + iσ
aȧ∂µψ̄
ȧ(x)
−iθσµθ̄∂µψ̄a(x)− iθσµθ̄θȧ∂µΩȧa(x)
θ2θ̄2✷ψa(x)
The chirality condition for this field is D̄ȧΣa = 0.
The Kalb-Ramond field accommodated in Ωȧb(x) is
given by
Ωab = −ǫabρ(x) + (σµν )abBµν(x). (5)
with ρ(x) and Bµν(x) being complex fields,
ρ(x) = P (x) + iM(x),
Bµν(x) = 14
Bµν − iB̃µν(x)
B̃µν(x) =
ǫµναβB
αβ(x) (7)
The components P and ψa are compensating fields and
are not present in the θ-expanssion of G, as it shall be
explicitly given below.
The superfield G(M, ξ, G̃µ), which plays a central role
in connection with local vortices ([20]), accomodates the
real scalar, M, the fermion ξ and the dual of the Kalb-
Ramond field strength G̃µ. It can be θ-expanded accord-
ing to the following expression:
G = −
ξ̄ȧ +
aȧθ̄
aȧθ̄
2∂µξ̄
ȧ − 1
aȧθ̄
ȧ∂µξ
θ2θ̄2✷M ; (8)
Now, we have all the elements to illustrate how the
Lorentz-symmetry violation, signaled by the background
of the Kalb-Ramond field, is intimately connected to the
appearance of explicit (soft) SUSY breaking terms.
We can notice that the superfield G carries only some
degrees of freedom of Σa, the fermionic field ψa does not
appear, ρ appears only through M , and as G̃µ is related
to the 2-form, the Kalb-Ramond field, Bµν
G̃µ =
ǫµναβG
ναβ . (9)
Gµνκ = ∂µBνκ + ∂νBκµ + ∂κBµν (10)
∂µχ+ g
2|φ|2G̃µG̃µ
+gG̃µ
χ̄γµχ−
φ̄∂µφ+
φ∂µφ̄
where in this discussion we consider Φ → ΦegM . The
Lagrangian L
is the interaction Lagrangian, and its
explicit form is not important to show the relation be-
tween the Lorentz and SUSY breakings.
Let us consider the split Gµνλ as
Gµνλ = G
(self)
(ext)
. (12)
The external Lorentz-symmetry breaking background
is given by
(ext)
ρǫ0ijk =
ρǫijk. (13)
The crucial point here is the justification of why the
background value of Gµνκ in (13) yields an explicit soft
breaking of SUSY. The whole idea here is that the back-
ground for Gµνλ given in (13) lies on a θ-component of
G, actually, the θσµθ̄G̃µ in (8), which necessarily signals
an explicit SUSY breaking. If the first component ( the θ
-independent one) set up a non-trivial background then
SUSY may not background, then SUSY may not be bro-
ken; however, whenever the background value sits on a
non-trivial θ-component, SUSY is necessarily explicitly
broken down, and this is the case here.
The relevant bosonic part of the (11) important to ana-
lyzed the Lorentz-Breaking relation with SUSY breaking
in the background is
L = ∂µφ
∗∂µφ+ g2|φ|2G̃µG̃µ −
φ∗∂µφ− φ∂µφ∗
By splitting the Kalb-Ramond fields as (12) and by
adopting the ansatz of a Lorentz-breaking background,
(13) there emerges a mass term for the bosons,
LL−SUSY−B = g
ρ|φ|2, (15)
Terms like that may appear as a result of sponta-
neous breaking of SUSY[34]. Soft explicit SUSY breaking
terms are very important in connection with the physics
derived from the Minimally Supersymmetric Standard
Model (MSSM). In view of that, we try to stress here
on the connection between a Lorentz-symmetry violating
background and the appearance of explicit SUSY break-
ing terms.
III. THE SUPERSYMMETRIC VERSION FOR
A GLOBAL VORTEX AND THE SUPERFLUID
BEHAVIOR
In the present section, we study the supersymmetric
framework setting up the general formalism that gives us
the terms to construct the global vortex and study the su-
perfluid behavior. The action studied in the previous Sec-
tion (1) helps us in understanding the consequences of the
Lorentz-breaking background in connection with SUSY.
Actually, we have to see how the presence of a back-
ground yielding Lorentz symmetry violation also leads
to an explicit SUSY breaking. In the present Section, let
us adopt the action to study the vortex configuration as
in the sequel:
LK = Φ
†e4gGΦ|θ2θ̄2 + S†S|θ2θ̄2
+W |θ2 + W̄ |θ̄2 . (16)
These superfields satisfy a chirality constraint, given
by the condition D̄ȧΦ = 0 and D̄ȧS = 0. The superfield
Φ is defined in (2) and G in (8); the superfield S has the
same properties as Φ and can be θ- expanded according
S = e−iθσ
µ θ̄∂µ [S(x) +
2θaζa(x) + θ
2H(x)]. (17)
In the expression (16), W is the superpotential whose
general form is
W = aiΦi + bijΦiΦj + cijkΦiΦjΦk. (18)
The scalar-field potential is given by
ĀiAi =
|2 (19)
where Ai is the auxiliary component of the φi-superfield.
Let us study the possibility to obtain the supersym-
metric version of the global vortex potential according to
the model discussed in [33].
In the case of a global gauge symmetry, the chiral
superfield Φi transforms as a phase under the U(1)-
symmetry:
Φ′i = e
−iqiΛΦi (20)
where qi are U(1) global charges and Λ is the rigid U(1)
rotation angle. The qi and Λ are real constants.
It is possible to build up a potential with spontaneously
broken gauge symmetry using three chiral superfields.
For cosmic strings with a local gauge symmetry, one usu-
ally needs two charged fields φ±, with respective U(1)
charges q±, and a neutral field, Φ0. This mechanism re-
mains the same if we have a global transformation (20);
in this case, the superpotential takes the form:
W (Φi) = µΦ0
Φ+Φ− − η2
In this approach, the neutral field Φ0 is important to
give us the term responsible for the mass of the scalar
field of the theory, but, in a global vortex superfluid con-
figuration, we have a Lorentz breaking background, as
discussed in[33], and we have proven, in the previous Sec-
tion, that the Lorentz breaking introduces masses for the
scalars; then, we can adopt a simpler potential, with one
superfield Φ, with charge qΦ, and another superfield, S,
with charge qS , satisfying the constraint qS = −2qΦ, so
W = hSΦ2 (22)
This form of the superpotential, in connection with the
Lorentz-symmetry breaking (15) of the previous Section,
leads to the Mexican hat configuration, responsible for
the global vortex behavior that characterises superfluid-
The SUSY transformations read as below:
ǭȧξ̄
ȧ − i
ǫaξa, (23)
δξa = 2σ
aȧǭ
∂µM − iG̃µ
, (24)
δG̃µ =
ǫb(σµν)ab∂νξa +
ǭḃ(σ̄
µν)ḃȧ∂ν ξ̄
ȧ (25)
It is important to point out here that the soft super-
symmetry breaking terms do not invalidate the super-
symmetric transformations; actually, Lorentz symmetry
and SUSY are broken down by the background, but they
are both symmetries of the action. So, SUSY transfor-
mations as translations in superspace are not lost.
Global strings appear whenever a U(1) global symme-
try is spontaneously broken. After the breaking of the
U(1) symmetry, a massless Goldstone boson emerges that
yields a long-range force. The bosonic Lagrangian result-
ing from this supersymmetric model and that is relevant
for the superfluid can be written as
LB = ∂µφ
φ+∂µS
|φ|2GµνρGµνρ−V ′+G̃µJµ.
for simplicity, we also adopt the redefinition Φ′ → ΦegM .
The current jµ is given by
φ∗∂µφ− φ∂µφ∗
The bosonic potential V ′ , that comes from the eq.(22),
is given by
′ = g2|φ|4 + 4g2|S|2|φ|2. (28)
We perform the background splitting (12) of (49), so
that the full potential is
V = h2|φ|4 − (g2ρ− 4h2|S|2)|φ|2. (29)
This potential shows us that the U(1)-breaking gives
mass to the moduli field |φ| while the phase of the scalar
field remains massless. In the low-energy limit, the com-
plex scalar field of the Goldstone model can be repre-
sented as below:
φ = ϕ(r)eiα, (30)
where r is the radial coordinate. The boundary condi-
tions are given by
ϕ(r) = 0 to r = δ
ϕ(r) = η to r → ∞. (31)
The configuration (30)-(31) is the same as the one
of the local vortices, but the long-range interactions of
global strings, happen due to their coupling to a massless
Goldstone field, cause their dynamics to be substantially
different from those of the local strings. Spontaneous
symmetry breaking requires that ϕ(r) have mass and α
be a massless Goldstone boson. This breaking triggered
by the soft SUSY breaking term we introduce, takes place
whenever
(g2ρ− 4h2|S|2) > 0. (32)
This relationship is crucial to ensure the stability of the
potential, for it guarantees the vortex is formed around
the right ground state. This justifies our claim, stated in
the previous Section, on the importance of the term that
breaks SUSY explicitly (and softly) for the stability of
the global vortex. A solution to the vortex configuration
exists if g2 ≥ 4h2|S|2. In this configuration, we consider
the boundary conditions to ϕ, given by (31) and, for the
S-field, we consider the ansatz S = s(r)eiΛ. Outside
the string, we consider the field 〈S〉 = 0. The global
vortex presents a minimum roll and a central maximum
characterizes the Mexican hat potential. By analyzing
the potential minimum outside the string, with 〈φ〉 = η
and 〈S〉 = 0, we have η = gρ
. The ansatz in the core of
the string allows us to analyze it in comparison with the
fermionic Yukawa potential that are the subject of the
section IV.
The effective Lagrangian
LB = ∂µϕ∂
ϕ+ ϕ2∂µα∂
α+ ∂µs∂
s+ s2∂µΛ∂
ϕ2G(self)µνρ G
(self)
ρεijkϕ2G
(self)
µν − V. (33)
Now, let us write, the current Jµν it in terms of the
Kalb-Ramond field, according to the functional relation
below:
µd4x =
ǫµαβγ∂
αBβγJµd4x
ǫαµβγB
µν (34)
where
Jµν =
ǫµναβ∂
φ− φ∂βφ∗
The configuration in the core of the string, where the
commutator is not zero, [∂µ, ∂ν ]α 6= 0, in the presence of
a vortex. We can see this clearly by considering a straight
vortex along the z- axis, the azimuthal angle, and inte-
grating over a two-surface orthogonal to the string yields,
[∂x, ∂y]αdxdy = 2π, or [∂x, ∂y]α =
δ(x)δ(y)
, then, in the
presence of the vortex the α is a multi-valued function of
the coordinates and Jµν 6= 0 on the vortex core.
Outside the string core, φ can be represented as φ ∼
η exp(iα(x)) and 〈S〉 = 0; the effective Lagrangian for the
Goldstone mode (in the presence of the global strings at
large distances of the core, which are non-massive exci-
tations) can be written as
L = η2∂µα∂
(self)
µνρ G
(self)
εijkG
(self)
ijk +
µν . (36)
We use the fact that a real massless scalar field in four-
dimensional Minkowski space is equivalent to a rank-2
anti-symmetric tensor , Bµν [30, 31]; the nature of this
equivalence in the case of the global strings can be found
in [28]. In SUSY, this duality property can be understand
by superfield identification
Φ†Φ ∼ G. (37)
In fact, the left part that contains the vortex superfield
gives us the term ϕ2∂µα and the right side gives us a term
related with the dual field, G̃µ. The identification (37)
gives us other contributions, related to the scalar field
ϕ2 = 2ηM (38)
The fermionic part is
∗ = −iηξa (39)
2χ̄ȧϕ = iηξ̄ȧ (40)
and the vortex identification part
|ϕ2|σµ∂µα+ χ̄χ =
ηǫµνλρσ
µ∂νBλρ. (41)
We can notice that the fermionic part modifies the
usual vortex duality relation [33]. If we neglect the
fermionic contribution, eq.(41) can be written as
ϕ2∂µα =
ηǫµνλρ∂
νBλρ. (42)
The identification of the bosonic part given by (42) has
the same form as the [33] for the global vortex configu-
ration, but, with the supersymmetric invariance, we can
always have a fermionic part.
The only remaining dynamical degree of freedom is the
scalar (Goldstone boson) field, α. In this approach, we
have the action for the (global) static string:
(Self)
µνβ G
(Self)
ρεijkG
(self)
ijk +
µνd4x (43)
where β = 1 + g2η2.
At this point, it is advisable to remind that an ex-
plicit Lorentz-symmetry breaking, as stated above, may
be rephrased in terms of a softly explicit SUSY breaking
term as the one we consider here [8]. Now, let us study
the solution at long distances compared to the string
core, when the interaction of the vortex with the clas-
sical Goldstone-boson field is described by an effective
Lagrangian. The stress tensor in the background consid-
ers a string at rest point in û direction we have
T 00 = T ii = ρ = p (44)
T 0i = β
ρG0jk
ǫijk. (45)
The equation of the motion for the Kalb-Ramond field
µαβ =
Jαβ (46)
We obtain the solution
ûirj − ûjri
It yields the stress tensor interaction part given by
T 0i = 2
(û× r)i
A single straight global string has a logarithmically di-
vergent energy per unit of length. We can think that
these strings could be ignored because they appear to be
unphysical. However, following cosmological phase tran-
sitions, global strings may form loops with finite total
energy or open strings with finite energy per horizon.
An interesting application that some authors have been
envisaging is the possibility that radiative decay of closed
loops be connected with density fluctuation in the process
of structure formation. This approach, considering the
data basis, has been ruled out alone, but together with in-
flationary models and considering the noise of the exper-
imental data, we can still consider them [17]. In the ap-
proach of [33], that is considered here, the vortex configu-
ration is stable in the presence of the special background
that breaks the Lorentz invariance[33]. The fact that the
superfluid vortex is immersed in a Lorentz-noninvariant
fluid suggests that the correct model for a superfluid vor-
tex involves the choice of a special background. The rela-
tivistic force law for the response of a vortex to the local
field Gµνρ is analogous to the Lorentz force law in Elec-
trodynamics. The external force due the background field
interaction is given as F i = JjkG
jki =
ρǫjkiJjk . The
bosonic part of the solution has the same form as in the
non-supersymmetric model, but, in our construction, the
solution presents the explicit dependence on the param-
eters h and g and on the effects of the fermions. In the
supersymmetric version, the introduction of a Lorentz-
symmetry violating background gives us important impli-
cations on the fermionic background that we shall discuss
in the next section, when we study the supersymmetric
superfluid.
IV. THE ANALYSIS OF THE FERMIONS AND
THE SUPERFLUIDITY BEHAVIOR
In a supersymmetric framework, besides the bosonic
degrees of freedom, there are fermionic partners in the
theory. In this section, let us analyse the behavior of
the fermions that accompany the bosonic fields. The
fermionic action can be written as:
χ̄σµ∂µχ+
BµνJ µν + LFKR + Lint, (49)
where Jµν is the fermionic current of the vorticity. The
latter can also be expressed as follows:
Jµν =
ǫµναβ∂
αχ̄σβχ, (50)
where
χ̄σµχ. (51)
The Lagrangian LFKR contains the fermionic Kalb-
Ramond couplings and reads as below:
LFKR =
χ̄σµG̃µχ. (52)
This Lagrangian (52) amounts to a mass contribution
given by the Lorentz-breaking parameter present in (13),
that is,
LmassFKR =
χ̄χ. (53)
Lint is the interacting Lagrangian, where we include
the Yukawa terms which induce masses to the fermions
that couple to the vortex. These Yukawa couplings are
collected in:
Y = g
2φζaχa + 2φ
ζ̄a + Sχ
χa + S
. (54)
From (54) and (53), there follows an interesting pos-
sibility. If we choose 〈S〉 = 0 to be zero in the core of
the string, the mass (53) does not appear in the core,
the fermionic interaction term vanishes in the core and
the fermions and χ and ξ become massless. In this case,
where the fermions is not have mass, the fermionic zero-
modes propagate with the speed of light in the z-direction
and particles can be ejected from the vortex. Outside
the vortex, these particles have masses induced by the
φ-interaction Yukawa term and by (53), as induced by
the Lorentz-symmetry breaking [35].
V. SECOND-ORDER PHASE TRANSITIONS
AND THE RELATION OF THE S-FIELD WITH
THE TEMPERATURE
In this Section, let us study a physical interpretation
of the field S. Up to now, we know that the field S is
important for the zero-modes. Now, let us study another
interpretation, possibly related with second-order phase
transitions. The potential (29) can represent a high-
temperature effective potential[28], that can be written
V (φ, T ) = m2(T )|φ|2 + h2|φ|4 (55)
where we identify S with the temperature, T . We actu-
ally consider |S|2 = T 2, then
m2(T ) = h2(4T 2 − η2). (56)
The term m(T ) is the mass for the φ-field, whenever
the state is symmetric, 〈|φ|〉 = 0. This mass vanishes
when T = Tc, and
. (57)
Another important case occurs for T > Tc; the effec-
tive mass m2(T ) is positive and the minimum of V is at
φ = 0. The physical interpretation of this result is that
the expectation value of φ vanishes. This means that the
symmetry is restored at high temperature. The symmet-
ric vacuum becomes unstable and φ develops a non-zero
expectation value. Minimizing V , as in (55), we obtain,
for T < Tc,
|φ| =
T 2c − T 2
An important realization of the second phase transition
is the fact that |φ| grows continuously from zero, as the
temperature decreases from the critical temperature, Tc.
The cosmological point of view, when the supersym-
metric Universe cools through the critical temperature,
is that the field φ develops an expectation value of mag-
nitude (58). The evolution of the phases α of φ and
Λ of S with the temperature is not determined only by
local physics; their values outside depends on random
fluctuations and α and Λ take different values in differ-
ent regions of space during the evolution. But, since the
free energy is minimized, these phases after the Universe
expansion can become precedent sections with the tem-
perature T = 0. We can define the correlation length,
Π(t), to be the length scale above which the values of α
and Λ are uncorrelated. The evolution of Π(t) depends
on details of the relaxation processes. Indeed, Π(t) has
to satisfy the causality bound. The correlation length
cannot establish scales greater than the causal horizons
related with the distance travelled by the light during the
life-time of the Universe. For T < Tc, the scalar field de-
velops an expectation value corresponding to some point
in the manifold M of the minima of the effective poten-
tial V . We can see in (58) that the term that in our model
is given by the soft SUSY breaking, was presented in the
high temperature state, but, in the case T ≫ Tc, SUSY
breaking can be neglected, and we can consider the Uni-
verse as being in a supersymmetric phase. To understand
this fact, we need String Theory arguments, not contem-
plated in this work. This analysis only gives us knowledge
about the vortex formation, and it is not able to provide
us with information on the Lorentz breaking. But, if we
consider that, when the temperature becomes low vortex
formation may take place, then Lorentz symmetry may
be violated and there occurs a vortex-superfluid forma-
tion, as we have analysed throughout this paper.
VI. GENERAL CONCLUSIONS
In this work, we have shown that it is possible to build
up a string vortex by modelling the vortex superfluid in a
supersymmetric context. We have analyzed the potential
that gives us the correct string vortex configuration, in
zero temperature, it presents a soft SUSY-breaking in-
duced by the hidden sector. We have also analyzed the
bosonic aspects of the duality representation of the vor-
tex. We also analysed the physical interpretation of the
extra field S related to the presence of the zero-mode and
we show that we can relate it to the temperature. It is
advisable to comment here that the violation of Lorentz
symmetry introduced is independent of the soft SUSY
explicit breaking terms. The latter has been considered
to be correct taking into account stability aspects of the
potential. It would be very interesting to eventually un-
derstand if the violation of Lorentz symmetry and the
explicit SUSY breaking could be related to one another.
This would render our proposal more interesting, in that
we would be dealing with less arbitrary parameters. Also,
it would clarify the interplay between Lorentz-symmetry
violation (in the sense of particle transformations) and
SUSY explicit breaking that describes mass splittings
among bosons and fermions that belong to the same su-
permultiplet. This is an issue that could be investigated
better in a future work. The interesting phenomenolog-
ical aspect of this discussion would be checking whether
properties like the masses of the SUSY particles, such as
the photino and the higgsino, would necessarily signal to
some type of Lorentz-symmetry breaking. In section V,
we have proposed a discussion in a cosmological evolution
context, but, as already pointed out, these vortices also
appear in star cores. In the case of the neutron stars, as
presented in the Introduction, the force that dictates the
vortex stability is induced by the nuclear matter; but,
we do not eliminate the possibility of the presence of a
Lorentz-symmetry breaking to have an important role in-
side the star. Nothing guarantees that the matter inside
the star has a Lorentz-invariant behavior, because the
high energy envolves, in analogy with high energy γ-rays
from extragalatic sources[36]. Another point is if we con-
sider that dark matter is mostly composed by supersym-
metric particles, the relation between the Lorentz and
SUSY breaking may become important to understand
the parameters of the model. Then, the possibility of
the Lorentz-symmetry breaking in supersymmetric mat-
ter becomes relevant for the dark matter stability around
the stars [25] and particles can then be ejected out of
these astrophysical structures. In this work, we do not
have an application for these objects, but we understand
that our model can be an alternative possibility to under-
stand some phenomena involving high energies. The next
step is studying the fermionic implication of the Lorentz-
symmetry breaking Kalb-Ramond background and how
we could find out a mechanism to justify its appearance,
the relation between Lorentz and SUSY breaking and the
origin of the hidden sector represented by soft breaking
of global SUSY.
Acknowledgments:
W. Bietenholz is acknowledged for a critical read-
ing and for many helpful suggestions on an original
manuscript. The authors would also like to thank
(CNPq-Brasil) for the invaluable financial support.
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