# The 1+1 Dimensional Abelian Higgs Model Revisited: Physical Sector and Solitons

**ABSTRACT** In this paper the two dimensional abelian Higgs model is revisited. We show that in the physical sector, the solutions to the Euler-Lagrange equations include solitons.

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arXiv:0812.0732v1 [hep-th] 3 Dec 2008

The 1 + 1 Dimensional Abelian Higgs Model Revisited:

Physical Sector and Solitons1

Laure GOUBA†,‡, Jan GOVAERTS∗,⋆,♦and M. Norbert HOUNKONNOU♦

†National Institute for Theoretical Physics (NITheP),

Stellenbosch Institute for Advanced Study (STIAS),

Private Bag X1, Matieland 7602, Republic of South Africa

E-Mail: gouba@sun.ac.za

‡African Institute for Mathematical Sciences (AIMS),

6 Melrose Road, 7945 Muizenberg, Republic of South Africa

E-Mail: laure@aims.ac.za

∗Center for Particle Physics and Phenomenology (CP3),

Institut de Physique Nucl´ eaire, Universit´ e catholique de Louvain (U.C.L.),

2, Chemin du Cyclotron, B-1348 Louvain-la-Neuve, Belgium

E-Mail: Jan.Govaerts@uclouvain.be

⋆Fellow, Stellenbosch Institute for Advanced Study (STIAS),

7600 Stellenbosch, Republic of South Africa

♦International Chair in Mathematical Physics and Applications (ICMPA-UNESCO Chair),

University of Abomey–Calavi, 072 B. P. 50, Cotonou, Republic of Benin

E-Mail: norbert.hounkonnou@cipma.uac.bj

In this paper the two dimensional abelian Higgs model is revisited. We show that in the physical

sector, the solutions to the Euler–Lagrange equations include solitons.

1 Introduction

The two dimensional abelian Higgs model is revisited. Using the Dirac formalism for constrained systems,

it has been established [1] that this model, in its gauge invariant physical sector, corresponds to the coupling

of a pseudoscalar field, namely the electric field, with a real scalar field, as a matter of fact the Higgs field

which is the radial component of the original complex scalar field in field configuration space. At the

classical level, the Euler–Lagrange equations of motion lead to a system of coupled non-linear equations of

which the spectrum of solutions includes solitons. The linearised spectrum of fluctuations of these solitonic

solutions has been identified in order to ascertain their classical stability.

This contribution is organised as follows. Section 2 discusses the Euler–Lagrange equations of

motion. Upon compactification of the spatial dimension into a circle, in Section 3 static solutions to these

equations are constructed in closed analytic form in terms of the Jacobi elliptic functions. In Section 4 the

linearised spectrum of fluctuations for these classical solutions is identified. Some concluding remarks are

provided in Section 5.

1Contribution to the Proceedings of the Fifth International Workshop on Contemporary Problems in Mathematical

Physics, Cotonou, Republic of Benin, October 27–November 2, 2007, eds.

(International Chair in Mathematical Physics and Applications, ICMPA-UNESCO, Cotonou, Republic of Benin, 2008),

pp. 164–169.

Jan Govaerts and M. Norbert Hounkonnou

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2 The Euler–Lagrange Equations of Motion

The 1 + 1 dimensional abelian Higgs model is described by the Lagrangian density

L = −1

4[∂µAν− ∂νAµ][∂µAν− ∂νAµ] + |(∂µ+ ieAµ)φ|2− V (|φ|),

where Aµis the gauge field with gauge coupling constant e and φ a complex scalar field with self-interactions

described by the U(1) gauge invariant potential V (|φ|). Choosing a parametrisation of the complex field

as φ = ρeiϕ/√2 and through the Dirac formalism for constrained systems [2] the model in its physical

sector is described by [1]

Lphys=1

2

1

ρ2(∂µB)2−1

2e2B2+1

2(∂µρ)2− V (ρ) − ∂0[1

ρ2B∂0B] + ∂1[B(∂0ϕ + eA0)], (1)

where B = −1

respectively,

eE and E is the electric field. The Euler–Lagrange equations of motion for ρ and B are,

∂2

µρ +1

ρ3(∂µB)2+∂V

∂ρ= 0,∂µ

?1

ρ2∂µB

?

+ e2B = 0.(2)

These equations are non-linear and coupled, and not all their solutions can be analytically found. However

it might be possible to find analytical solutions for static states of finite energy. In order to reduce

the difficulties of solving these equations, we consider configurations where the electric field vanishes,

E(t,x) = 0. In fact the total energy is Ephys= Ek+ Ep, where Ekis the total kinetic energy and Epthe

total potential energy of the fields, with,

Ek=

?L

−L

dx

?

1

2ρ2(∂

∂tB)2+1

2(∂

∂tρ)2

?

,

and

Ep=

?L

−L

dx

?1

2

1

ρ2(∂

∂xB)2+1

2e2B2+1

2(∂

∂xρ)2+ V (ρ)

?

.

From these expressions, one notices that if B is non zero but ρ vanishes for some value of x, we have a

singularity in the equations of motion and furthermore the energy may become infinite, unless the electric

field vanishes faster than ρ in such a manner that the quotient of these zeros remains finite. But such

behaviour induces large spatial gradients in both ρ and B, implying a large value for the total energy. Hence

in order to minimise the energy, one needs to consider configurations with B = 0. Another advantage of

this restriction is that the equations are no longer coupled and we simply have to solve

B = 0,∂2

µρ +∂V

∂ρ= 0. (3)

A further assumption is required though, since it is still difficult to construct all the time dependent

solutions to the above equation (except for those obtained by Lorentz boosts from a static solution).

Moreover, any time dependence for a solution increases its total energy. Consequently we restrict further

to static configurations in which ρ only depends on the space variable x, leading to the single non trivial

equation,

d2ρ(x)

dx2

B(t,x) = 0,

−dV (ρ(x))

dρ(x)

= 0, (4)

where for the potential energy henceforth the Higgs choice will be made, V (ρ) = (1/8)M2?ρ2− ρ2

0

?2, with

M > 0 a mass scale and ρ0the expectation value of the scalar field.

3Static Solutions

In order to solve the above equation, let us choose to compactify space into a circle of length 2L with

−L ≤ x ≤ L. For any constant values of ρ associated to the minima of the potential, where the potential

also vanishes given our choice of subtraction constant, the non-linear equation in (3) is satisfied. These

constant solutions B(t,x) = 0, ρ(t,x) = ±ρ0 correspond to vacuum configurations, of which the total

energy values are minimal and vanish.

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Besides these configurations, we also have non constant static solutions which behave like solitons.

Given the equation for ρ,

ρ′′−∂V

∂ρ= 0, (5)

where ρ′′(x) stands for the double derivative with respect to x, there exists a conservation law expressed

as

1

2((ρ′(x))2= V (ρ) + C0, (6)

C0being some integration constant. In this form the solution is readily obtained by quadrature. On the

circle of length 2L one finds,

ρs(x) = ±ρ0

?

2k2

1 + k2sn

?

1

2Mρ0

?

2

1 + k2x

?

.

it being understood that the solutions obey either periodic or anti-periodic boundary conditions given the

remaining freedom in the sign of ρ existing for the choice of polar parametrisation of the complex scalar

field φ [1]. These solutions are thus expressed in terms of the Jacobi elliptic functions [3]. According to

whether a periodic or an anti-periodic solution is obtained, the remaining single integration constant is

related to the elliptic modulus k through the condition,

LMρ0

= 4(n + r)K(k)

?

periodic,

antiperiodic.

1 + k2

2

,n ∈ N,

r=

?

0 :

1/2 :

Their total energy is represented through the expression

Ephys=

?+∞

−∞

dx

?1

4M2?ρ2

s− ρ2

0

?2+ C0

?

.

By substituting the above explicit expression, one finds

Ephys=

LM2ρ4

2(n + r)K(k)

0

?2(n+r)K(k)

0

dy

??1

4+

k2

(1 + k2)2

?

−

2k2

(1 + k2)sn2y +

2k4

(1 + k2)2sn4y

?

.

This quantity is finite since 0 ≤ k2≤ 1 et 0 ≤ sn2y ≤ 1. This very fact together with the spatial profile

ρ(x) of these configurations justifies their interpretation as solitons. In particular the periodic solution

with n = 1 in the decompactification limit L → ∞ reduces to the celebrated kink solutions of the φ4real

scalar field theory in 1+1 dimensions.

4 Spectrum of Fluctuations

Given the explicit solutions of the previous Section, in the present Section we address the issue of the

classical stability under linearised fluctuations in field configuration space. This requires the computation

of the spectrum of fluctuation eigenvalues, to ascertain that none of these is negative, which otherwise

would establish that some modes have an unbounded above exponentially growing amplitude, spelling

disaster for the corresponding solution.

Let us consider arbitrary time- and space-dependent fluctuations around the identified solutions,

B(t,x) = δB(t,x), ρ(t,x) = ρs(x) + δρ(t,x). The corresponding linearised Lagrangian density, expanded

to second order in these fluctuations, is

L =

1

2ρ2

s(∂µδB)(∂µδB) −1

2e2(δB)2+1

2(∂µρs)(∂µρs) + (∂µρs)(∂µδρ)

+

1

2(∂µδρ)(∂µδρ) − V (ρs) − δρV′(ρs) −1

2(δρ)2V′′(ρs) .

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Applying the variational principle in the fields δB and δρ, the linearised Euler–Lagrange equations for

these fields read

∂µ∂µδρ + V′′(ρs)δρ = 0,∂µ

?1

ρ2

s

∂µδB

?

+ e2δB = 0. (7)

These equations being linear in the field variables, a normal mode expansion is warranted, with a spectrum

of eigenfrequencies to be identified through the following ansatz for time dependence (the general solution

then being constructed through linear combinations),

δρ(t,x) = f(x)e−iω1t+ f∗(x)eiω1t, δB(x,t) = g(x)e−iω2t+ g∗(x)eiω2t. (8)

If there exist solutions for which either of the eigenfrequencies ω1or ω2is pure imaginary, namely such

that ω2

i< 0 (i = 1,2), this would imply an exponential run-away time dependence for at least one of

the linearised fluctuation modes, hence instability of the corresponding classical solution. In the above

parametrisation of the normal modes, f(x) and g(x) stand for complex-valued functions on the circle, with

g(x) periodic and f(x) periodic or anti-periodic according to the periodicity properties of the classical

solution of which the stability needs to be established.

By direct substitution of the above ansatz into (8), the following eigenvalue equations are derived,

?

−d2

1

ρ2

s(x)

dx2+ V′′(ρs)

d

dx+ e2ρ2

?

f(x) = ω2

1f(x) ,(9)

?

−ρ2

s(x)d

dx

s(x)

?

g(x) = ω2

2g(x) . (10)

In the case of the vacuum configuration, B = 0, ρ = ±ρ0, the solutions to (9) and (10) are readily

constructed through a Fourier series analysis on the circle for the unknown functions g and f, with

g(x) = 1/(2L)?+∞

n=−∞eiπn

Lxgn and f(x) = 1/(2L)?+∞

n=−∞eiπn

Lxgn. The spectrum is then found to be

given as

fn :ω2

1

=

?πn

?πn

L

?2

?2

+ M2ρ2

0,

gn :ω2

2

=

L

+ e2ρ2

0,

which is indeed positive definite. As it should, the vacuum configuration is indeed stable against all possible

fluctuations in the fields.

For the non-trivial soliton configurations, the eigenvalue problem reads,

?d2

dy2− 6k2sn2y

?

f(y) = −

?

2ω2

M2ρ2

1

0

+ 1

?

(1 + k2)f(y), (11)

?

−d2

dy2+ WB(y)

?

ψ(y) =2(1 + k2)ω2

2

M2ρ2

0

ψ(y),(12)

where

WB(y) = 2

?2e2k2

M2

sn2y −1 + k2

2

+

1

sn2y

?

. (13)

In order to obtain (12), the following change of variable was introduced in (10), g(x) = ρs(x)ψ(x). As

a matter of fact, (11) is a Lam´ e equation, of which the solutions have been discussed and classified in

Ref. [3], with the results listed in the Table below.

As may be seen from those results, the eigenfunctions f(y) = sny · cny and f(y) = sny · dny both

have a positive eigenspectrum of ω2

1values and are both antiperiodic. However the last three solutions

in the Table correspond to periodic functions, one of which is the zero mode associated to infinitesimal

spatial translations, while among the other two there always exists one with a strictly negative eigenvalue

ω2

1. Consequently the solitons possessing a periodic boundary condition on the circle are unstable against

fluctuations corresponding to that specific mode.

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f(y)ω2

1

sny · cny

1

2M2ρ2

0

3

k2+1

sny · dny

1

2M2ρ2

0

3k2

1+k2

cny · dny0

sn2y −

1

3k2(1 + k2±?1 − k2(1 − k2)

1

2M2ρ2

0

?

1 ∓ 2

√

1−k2(1−k2)

1+k2

?

For what concerns the function g(x) and its fluctuation spectral equation (12), the latter eigenvalue

problem being equivalent to solving the Schr¨ odinger equation for the potential WB(y), the issue may also

be addressed from that point of view. Even though the Schr¨ odinger equation may not be explicitly solve

for that choice of potential, using the fact that the potential (13) is positive, implies in any case that the

spectrum of ω2

2eigenvalues is likewise positive. No instability may arise in the g(x), namely the B sector

of fluctuations.

5Concluding Remarks

The vacuum configuration has thus been confirmed to be stable against all fluctuations. Non-trivial static

soliton configurations are also stable, but only in the sector of anti-periodic solitons and thus an anti-

periodic boundary condition on the function f(x). Periodic solitons, though, are always unstable.

Even though this work has some overlap with some previous studies, its originality lies with the

fact that it considers specifically only the physical sector of the 1+1 dimensional abelian U(1) Higgs

model without applying any gauge fixing procedure whatsoever. In particular the factorisation between

the actual physical sector and the decoupled gauge variant sector offers two advantages. First, that no

artefacts due to gauge fixing are introduced. Second, that potential spurious instabilities lying solely within

the gauge variant and unphysical sector are avoided from the outset, a feature any other approach having

been developed so far cannot achieve. All these approaches have until now relied on some gauge fixing

procedure rather than applying such a gauge invariant and physical factorisation. Hence these approaches

runs the risk of identifying instabilities which in fact are pure gauge and thus unphysical. Such difficulties

are avoided from the outset within our approach.

When the potential V (ρ) is taken to be the Higgs potential, as we did, the static solutions in ρ with

B = 0 are in fact those of Refs. [4,5]. These authors also solve the model of a single complex scalar field

on the circle, while in Ref. [6], the authors consider the 1 + 1 abelian Higgs Model on the circle, but in

contradistinction to what we have done, they apply a gauge fixing procedure in terms of the gauge field

components A0and A1. We recover the same solutions on the circle. Our work has established that the

soliton configurations with anti-periodic boundary conditions are stable. In Refs. [5,6], the static soliton

solutions with periodic boundary conditions are also found to be unstable.

A possible continuation of the present study would be the computation of the complete spectrum

of fluctuation eigenfrequencies for the stable and unstable classical soliton configurations, inclusive of

the electromagnetic sector contributions but retaining only the physical degrees of freedom, in order to

determine the quantum corrections of the mass spectrum of these solitonic field configurations of the 1+1

dimensional abelian U(1) Higgs model, whether on the circle or on the real line.

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Acknowledgements

This work, part of Laure Gouba’s Ph.D. thesis, was revisited during her stay at the African Institute for

Mathematical Sciences (AIMS) as a Postdoctoral Fellow and Teaching Assistant. L.G. would like to thank

Profs. Neil Turok, founder of AIMS, and Fritz Hahne, Director of AIMS, as well as the AIMS family for

their support and hospitality.

J.G. is grateful to Profs. Hendrik Geyer, Bernard Lategan and Frederik Scholtz for the support

and the hospitality of the Stellenbosch Institute for Advanced Study (STIAS) with the grant of a Special

STIAS Fellowship which made a stay at STIAS and NITheP possible in April-May 2008. He acknowledges

the Abdus Salam International Centre for Theoretical Physics (ICTP, Trieste, Italy) Visiting Scholar

Programme in support of a Visiting Professorship at the ICMPA-UNESCO (Republic of Benin). J.G.’s

work is also supported by the Institut Interuniversitaire des Sciences Nucl´ eaires, and by the Belgian

Federal Office for Scientific, Technical and Cultural Affairs through the Interuniversity Attraction Poles

(IAP) P6/11.

References

[1] L. Gouba, Th´ eories de jauge ab´ eliennes scalaire et spinorielle ` a 1 + 1 dimensions: Une ´ etude non

perturbative, Ph.D. Thesis (University of Abomey–Calavi, 2005), unpublished; see also the preceding

contribution in this Volume, which includes further details with regards to our notations.

[2] J. Govaerts, Hamiltonian Quantisation and Constrained Dynamics (Leuven University Press, Leuven,

1991).

[3] F. M. Arscott, Periodic Differential Equations, An Introduction to Mathieu, Lam´ e and Allied Func-

tions (Pergamon Press, New York, 1964).

[4] N. S. Manton and T. M. Samol, Phys. Lett. B 207, 179 (1988).

[5] Y. Brihaye and T. N. Tomaras, The Goldstone Model Static Solutions on S1,

e-print arXiv:hep-th/9810061 (October 1998).

[6] Y. Brihaye, S. Giller, P. Kosinski and J. Kunz, Phys. Lett. B 293, 383 (1992).

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