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Eur. Phys. J. C (2014) 74:3002

DOI 10.1140/epjc/s10052-014-3002-4

Regular Article - Theoretical Physics

Self-dual vortices in Abelian Higgs models with dielectric function

on the noncommutative plane

W. García Fuertes

1,a

, J. Mateos Guilarte

2,b

Departamento de Física, Facultad de Ciencias, Universidad de Oviedo, 33007 Oviedo, Spain

Departamento de Física Fundamental and IUFFyM, Universidad de Salamanca, 37008 Salamanca, Spain

Received: 9 May 2014 / Accepted: 23 July 2014 / Published online: 21 August 2014

Abstract We show that Abelian Higgs Models with a

dielectric function deﬁned on the noncommutative plane

enjoy self-dual vorticial solutions. By choosing a partic-

ular form of the dielectric function, we provide a family

of solutions whose Higgs and magnetic ﬁelds interpolate

between the proﬁles of the noncommutative Nielsen–Olesen

and Chern–Simons vortices. This is done both for the usual

U(1) model and for the SU(2) ×U(1) semilocal model with

a doublet of complex scalar ﬁelds. The variety of known

noncommutative self-dual vortices which display a regular

behavior when the noncommutativity parameter tends to zero

results in this way considerably enlarged.

1 Introduction

Although local quantum ﬁeld theory has had an impressive

success as a framework for describing the dynamics of ele-

mentary particles at the current accessible energies, there are

indications that, at some stage in the route toward a more

fundamental theory, the idea of locality as a basic assump-

tion of physics should be given up. The exact way in which

nonlocality would arise in that underlying theory is not clear,

but a possibility that has often been considered by theorists

is that, for lengths below some scale

√

θ, spacetime has to

be replaced by a different, blurred entity, in which the coor-

dinates x

become noncommuting quantities ˆx

with com-

mutators among them of order θ. The reasons for consider-

ing noncommutative quantum ﬁeld theories formulated on

this arena are diverse. Originally, noncommutative QFT’s

appeared in an attempt to use the scale

√

θ as a cutoff for

ultraviolet divergences, but later they were seen as effective

theories on the spacetime foam resulting from the modiﬁed

uncertainty principle arising in quantum gravity, as some

e-mail: wifredo@uniovi.es

e-mail: guilarte@usal.es

low-energy limits of the theory of open strings propagat-

ing on a constant Kalb–Ramond ﬁeld or as describing the

low-energy quantum ﬂuctuations of stacks of D-branes in

the context of the IIB matrix model. The noncommutativ-

ity of spatial coordinates emerges also in condensed matter

contexts, such as the motion of very light charged particles in

strong magnetic ﬁelds as happens in the quantum Hall effect.

For reviews of the formalism of noncommutative quantum

ﬁeld theory and some of its motivations and uses or their

possible role in phenomenology; see [1–4].

The study of the different classes of solitons appearing in

ﬁeld and string theory is an important topic, both because

they are stable objects with interesting dynamical behavior

and because their conserved charges allow an interpretation

of the solitons as supersymmetric BPS states, which can give

signiﬁcant information on the nonperturbative regime of the

theory. In this respect, noncommutative QFT are especially

appealing, because they can accomodate regular solitonic

solutions in situations in which the usual commutative ﬁeld

theory would give singularities. This happens because Der-

rick’s theorem, which is based on the scaling properties of the

lagrangian kinetic- and potential-energy terms under dilata-

tions of the coordinates, ceases to be valid in the noncom-

mutative case due to the presence of the f undamental length

√

θ. As a consequence, it is possible to ﬁnd noncommuta-

tive scalar solitons even in theories without kinetic terms,

and there is even a so-called solution generating technique

which can be used to construct scalar and gauge solitons

starting from trivial vacuum solutions [5]. This kind of soli-

tons, however, become singular when the noncommutativity

parameter θ is driven to zero.

In this paper we are going to study the self-dual vor-

tices arising in a class of noncommutative Abelian models in

which the kinetic Maxwell term incorporates a dielectric fac-

tor which is a function of the Higgs ﬁeld. This dielectric con-

tributionto the action, which spoils renormalizability, is how-

ever, a common occurrence in the effective truncation to low

123

3002 Page 2 of 16 Eur. Phys. J. C (2014) 74:3002

energy of supersymmetric theories. In the commutative case,

self-dual vortices in Abelian models with dielectric function

have been studied in [6–8]or[9], and other related Higgs

models which arise from effective supersymmetric theories

are dealt with in [10,11] and [12 ]. Here, moving to the non-

commutative plane, we will consider two variants among this

kind of Abelian systems. First, we will pay attention to the

case where there is only one complex scalar ﬁeld and the

local symmetry group is U(1); this is the simplest paradigm

for the Higgs mechanism and, from the phenomenological

side, has interest as a Ginzburg–Landau model for super-

conductivity (the scalar ﬁeld is the order parameter between

type I and II superconductivities). Then we will extend the

treatment to consider a model with a doublet of scalar ﬁelds

enjoying a mixture of global SU(2) and local U(1) symme-

tries; this semilocal situation is a quite interesting limit of

the electroweak theory and has been a subject of research

in the ﬁeld of cosmic strings. In both cases, we will focus

on self-dual solutions which continue to be regular when θ

goes to zero. For that, we follow closely the treatment given

in the articles [13] and [14] by Lozano, Moreno and Scha-

posnik. In these references, the authors solve the self-duality

equations for, respectively, noncommutative Nielsen–Olesen

and Chern–Simons–Higgs U(1) vortices by means of a very

convenient ansatz which leads to some discrete recurrence

relations. On the other hand, in [7] a speciﬁc form of the

dielectric function which interpolates between the commu-

tative Nielsen–Olesen and Chern–Simons energy densities

was proposed. We use this function (with a slightly dif-

ferent parametrization) to ﬁnd the noncommutative vortices

interpolating between those found in [13] and [14], and also

between their semilocal counterparts. The main theme of this

paper is thus to combine the ﬂexibility provided by a dielec-

tric function with the techniques to deal with the noncommu-

tative self-dual equations developed by the authors of [13,14]

to show how the spectrum of self-dual noncommutative vor-

tices with good behavior for θ → 0 can be considerably

enlarged.

2 The Abelian Higgs model with dielectric function and

its self-duality equations

We are working on a three-dimensional spacetime with coor-

dinates (x

, x

) and metric η

μν

= diag(1, −1, −1),but

the spatial coordinates x

, x

are not real numbers but fuzzy

variables with uncertainty relation

Δx

≥

(1)

where θ is some positive real number. In this setup, we shall

consider a dynamical modelcontaining a complexscalar ﬁeld

φ and a gauge ﬁeld A

interacting through the action

S =





−

G ∗ F

μν

∗ G ∗ F

μν

+ D

φ ∗ D

−

W ∗ W



where the star stands for the Groenewold–Moyal product

f (x) ∗ g(x) = exp



∂

∂x

∂

∂x

j



f (x)g(x



)



x=x



. (2)

The formalism of noncommutative gauge ﬁeld theories is

explained, for instance, in [15]or[16]. In this particular

model, the scalar ﬁeld transforms with the fundamental rep-

resentation of the U

∗

(1) gauge group:

φ −→ Λ ∗ φ

φ −→

φ ∗ Λ

†

while A

is a U

∗

(1) connection

−→ Λ ∗ A

∗ Λ

†

Λ ∗ ∂

†

such that the covariant derivative of the scalar ﬁeld and the

gauge ﬁeld strength are

φ = ∂

φ − ieA

∗ φ

μν

= ∂

− ∂

− ie( A

∗ A

− A

∗ A

The ﬁeld φ is self-interacting through a potential quadratic

in W, a function of the star productof φ and

φ, W = W(φ∗

φ).

Also, we allow for a non-minimal scalar-gauge interaction

driven by the dielectric function G = G(φ ∗

φ). In this way,

G and W transforms under the adjoint representation of the

gauge group

G −→ Λ ∗ G ∗ Λ

†

W −→ Λ ∗ W ∗ Λ

†

exactly as F

μν

does, so that the gauge invariance of the action

is guaranteed. In the following, we will also assume that G is

positive deﬁnite and that W vanishes only when the product

φ ∗

φ takes its vacuum expectation value, denoted v

Going to the temporal gauge A

= 0 and after some con-

venient rescalings

→

φ →

φv→

the energy E of the static ﬁeld conﬁgurations takes the form

E =





G ∗ B ∗ G ∗ B + D

φ ∗ D

W ∗ W



where B is the magnetic ﬁeld

B = F

= ∂

− ∂

− i

(

∗ A

− A

∗ A

)

and the spatial covariant derivatives are now D

φ = ∂

φ −

∗ φ, k = 1, 2. This form of the energy functional is

123

Eur. Phys. J. C (2014) 74:3002 Page 3 of 16 3002

amenable to a Bogomolny splitting. The quadratic term in

the covariant derivatives of the Higgs ﬁeld is written as [17]



φ ∗ D



(

φ + iD

)

∗



φ − i D



+ φ ∗

φ ∗ B}

(3)

where an irrelevant contour term has been discarded, and the

other two terms can be arranged as





G ∗ B ∗ G ∗ B +

W ∗ W







(

G ∗ B + W

)

− W ∗ G ∗ B



, (4)

where the square is in the sense of the ∗-operation and the

cyclic property



xf(x) ∗ g(x) ∗ h(x) =



xh(x) ∗ f (x) ∗ g(x)

of the Groenewold–Moyal product has been used. By com-

bining (3) and (4), we see that, if W is chosen in such a way

that

W ∗ G = φ ∗

φ − v

, (5)

the energy of the ﬁeld conﬁgurations which satisfy the self-

duality equations

G ∗ B =−W, (6)

φ + iD

φ = 0(7)

is proportional to the magnetic ﬂux

E = v



xB,

which is indeed a boundary term by virtue of



(x) ∗ A

(x) =



(x) ∗ A

(x).

For ﬁnite-energy conﬁgurations, the ﬁelds at inﬁnity depend

only on the polar angle. The derivatives entering in (2)are

therefore proportional to inverse powers of distance and then,

in the asymptotic region of the noncommutative plane, the

star product of ﬁelds converges to the ordinary product. This

means that the classiﬁcation in topological sectors can be

directly taken over from the well-known results valid in the

commutative plane. In particular, the magnetic ﬂux is quan-

tized. Hence, the solutions of (6)–(7) minimize the energy

in each topological sector and are, therefore, bona ﬁde solu-

tions of the Euler–Lagrange equations. If we now denote by

the inverse of G according to the star product and take into

account that W and G commute between themselves because

both are functions of φ ∗

φ, the use of the constraint (5) turns

the ﬁrst self-duality equation into the more convenient form

B =





∗ (v

− φ ∗

φ), (8)

to be used in what follows.

Functions on the noncommutative plane can be traded by

operators on the Hilbert space H = L

) by means of the

Weyl map

f (x

, x

) −→

( ˆx

, ˆx

) =

(2π)



Δ(k)

f (k),

where the Weyl kernel is

Δ(k) = exp



−i(k

ˆx

+ k

ˆx

)



and

f (k) =



ik·x

f (x) is the Fourier transform of f (x);see

[5,15]. The transformation is consistent in the sense that the

star products are mapped to ordinary operator products on the

Hilbert space. The use of the operator side of the Weyl map is

very convenient for dealing with the self-duality equations,

especially if we express them in holomorphic coordinates

z =

+ ix

√

¯z =

− ix

√

and introduce the harmonic oscillator ladder operators

ˆa =

ˆx

+ i ˆx

√

2θ

ˆa

†

ˆx

+ i ˆx

√

2θ

with commutator



ˆa, ˆa

†



= 1 consistent with the uncertainty

relation (1). One can check [13,14] that, in terms of these

operators, the self-duality equations have the form

−

√



†

, A

¯z



−

√



a, A



− i



, A

¯z







− φ

φ),

√

[

a,φ

]

− iA

¯z

φ = 0

with φ, A

and A

¯z

representing here the operators

and

¯z

arising by applying the Weyl map to the Higgs and

gauge ﬁelds of the original theory, but all hats have been

suppressed to alleviate notational cluttering. Also, the vortex

energy can be now computed as the trace

E = 2πθv

on the Hilbert space.

3 The interpolating model: noncommutative vortices

By choosing the dielectric function in different forms it

is possible to ﬁnd self-dual noncommutative vortices with

gauge and scalar ﬁelds displaying a wide variety of proﬁles.

In particular, an interesting option proposed in [7]istoﬁx

123

3002 Page 4 of 16 Eur. Phys. J. C (2014) 74:3002

G(φ ∗

φ) in such a way that it can accommodate the proﬁles

of the two most prominent types of vortices from a phys-

ical point of view: the Nielsen–Olesen and Chern–Simons

vortices. This can be achieved by using

G =



(1 − λ) + λβ φ ∗

where the square root should be understood in the sense of the

star product, λ is a non-dimensional parameter with values in

the interval [0, 1] and β is an arbitrary constant with inverse

mass squared dimension. Thus, the self-dual equations for

this model are

−

√



†

, A

¯z



−

√



a, A



− i



, A

¯z



= i



(1 − λ) + λβ φ



− φ

φ),

√

[

a,φ

]

− iA

¯z

φ = 0.

For λ = 0, these equations are precisely the self-dual equa-

tions of the ordinary Abelian Higgs Model [13], while for

λ = 1 they coincide with those of the relativistic Chern–

Simons–Higgs model [14], with the Chern–Simons κ cou-

pling given by κ

2β

. Thus, by continuously varying λ

between 0 and 1 we can ﬁnd vortices with ﬁeld proﬁles which

interpolate between the solutions arising in these two theo-

ries.

3.1 Solving the noncommutative vortex equation

Let us ﬁrst consider, following [17] where more details can

be found, the case of very large noncommutative parameter

θ. By expanding in inverse powers of θ

φ = φ

∞

−1

+···

¯z

√



¯z

)

∞

¯z

)

−1

+···



the self-dual equations for general λ are, to leading order,

exactly the same that arise for Nielsen–Olesen vortices:

∞

= v

i(A

¯z

)

∞

[

a,φ

∞

]

As is well known [17 ,18], these equations have a solution

for each positive integer n which can be expressed in terms

of the shift operators | kk + n| for the harmonic oscillator:

∞

= v

∞



k=0

| kk + n |, (9)

¯z

)

∞

= i

∞



k=0



√

k + 1 + n −

√

k + 1



| kk + 1| (10)

Because

| k + n=



(k + n)(k + n − 1) ···(k + 1) | k,

the scalar ﬁeld operator can be recast as

∞



†

)

and, in this way, the vorticial character of the solution is

apparent through the factor a

(which is the noncommuta-

tive guise of the familiar angular dependence of type z

for

commutative vortices). This character can be corroborated

by computing the magnetic ﬁeld, which is proportional to

the projector onto the | 0 state,

∞

=−iF

z¯z



†

,(A

¯z

)

∞





a,(A

)

∞



−



)

∞

,(A

¯z)

∞



| 00 |,

and thus checking that the solution contains n quanta of the

magnetic ﬂux

= 2πθTr

∞

= 2π n,

as is appropriate for a vortex.

However, the presence of θ in the denominator of the mag-

netic ﬁeld shows that these solutions will become singular if

we try to extend them to the commutative θ = 0 case. In

order to obtain a solution valid for all values of θ, it is natural

to modify the solution (9)–(10)fortheθ =∞case by trying

an ansatz with a different coefﬁcient for each shift operator,

φ = v

∞



k=0

| kk +n|, (11)

¯z

=−

√

∞



k=0

| kk + 1|, (12)

which wasproposed for the Abelian HiggsModels in [13] and

for the Chern–Simons–Higgs Model in [14]. By substitution

in the self-dual equations, one ﬁnds a system of algebraic

equations for the f

and d

coefﬁcients,

k+1

√

k + 1 f

k+1

−

√

k + n + 1 f

√

k−1

− 2

√

k + 1d

+ d

− d

k−1

= θv

(1 − λ + λβ v

)(1 − f

)

which can be solved along the lines explained in these ref-

erences. By writing d

as d

√

k + 1 −

√

k + n + 1 + e

the ﬁrst equation gives the new coefﬁcient e

in terms of the

coefﬁcients as

√

k + n + 1



1 −

k+1



(13)

123

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