University of Cincinnati

UNIVERSITY OF CINCINNATI

______, 20 _____

I,______, hereby submit this as part of the requirements for the degree of:

______in: ______It is entitled: ______

Approved by: ______CURVATURE DEPENDENCE OF CLASSICAL SOLUTIONS EXTENDED TO HIGHER DIMENSIONS

A dissertation submitted to the

Division of Research and Advanced Studies of the University of Cincinnati

in partial fulfillment of the requirements for the degree of

DOCTORATE OF PHILOSOPHY (PH.D.)

in the Department of Physics of the College of Arts and Sciences

2003

Athula Herat

B.Sc., University of Colombo, Sri Lanka, 1994 M.S., University of Cincinnati, 1996

Committee Chair: Dr. Peter Surnayi Abstract

I study the curvature dependence of extended classical solutions of interacting ﬁeld theories. The main motivation behind my study of classical solutions is conﬁnement.

It has become apparent that classical objects (monopoles and vortices) are responsible for color conﬁnement in QCD. Therefore, studying the properties of these objects is

extremely important. The main goal of this investigation is to accurately determine

the energy of extended classical solutions. In most applications it is assumed that the

energy depends simply on the length of the object, irrespective of the shape. I show

that the curvature eﬀects are highly non-trivial.

I start with the kink solutions of the (2+1) dimensional linear sigma model. In

particular, I show that the curvature energy of a kink in two spatial dimensions, as

a prototype of extended classical solutions, is always negative. Assuming that the

deviations of the kink from the straight line are small, I derive a closed form for

the curvature energy. This investigation clearly demonstrates that the energy of the

(2+1) dimensional kink has a positive length term and a negative curvature term.

Next I look at extended vortex solutions. Vortices are (2+1) dimensional soliton

solutions of gauge theories coupled to complex scalar Higgs ﬁeld. In the trivially

extended vortices the core of the vortex forms a straight line. Recent lattice studies

show that vortices appear not be straight. Therefore, it is important to study the

curvature eﬀects of vortex solutions. However, unlike in the case of the kink, no

analytical vortex solutions exist, which makes the study of curved vortices an ex-

i tremely complex one. This problem can be made less daunting if circular vortices are considered. Therefore, I study the circular vortex of the U(1) gauge theory. I use numerical techniques to obtain the circular vortex conﬁgurations that minimize the Hamiltonian. I show that the curvature energy of the circular vortex is negative, which clearly demonstrates that the total energy of the (3+1) dimensional circular vortex is less than the length energy. I.e. the extended abelian vortex prefers the curved state to the straight one.

ii iii Acknowledgments

Throughout my graduate student life at the Department of Physics, University of

Cincinnati, I have have been extremely lucky to have had the guidance, support, and the friendship of many people. I would like to take this opportunity to extend my gratitude to all of them.

First and foremost, I would like to thank my advisor Dr. Peter Suranyi for giving me the opportunity to work with him. His underlying support through the years, academically, professionally, as well as personally, has made it possible for me to achieve this goal. It has been a pleasure to work under his guidance and I am indebted him for his patient help throughout the course of my research. I am extremely grateful to Dr. Rohana Wijewardhana for being both a mentor and a friend. His guidance and assistance from the very ﬁrst day I set foot in Cincinnati, especially during times when the going was tough, has been invaluable to me. I would like to extend a special thanks to Dr. Frank Pinski for his invaluable advice and assistance with the numerical aspects of the research. I thank Dr. Brian Meadows and Dr. Mark Jarrell as well as all of the above for serving in my dissertation committee.

It has been a great privilege to work and interact with my talented and gifted colleagues and friends. I thank them all for their cooperation and their entertaining company. I am grateful to David Will for helping me solve every single computer issue that came up. I would also like to thank Dr. Mike Sokoloﬀ for his help in various aspects of my stay at the Department of Physics.

iv Without the dedication and the support of my family, I would not have had the courage to undertake this endeavor. I owe a special debt of gratitude to my wonderful

wife, Manori, for her loving support and for the countless sacriﬁces she has made over

the years so that I could achieve my dream. To my beautiful daughter Hiruni, I

am so glad that you came into my life during this entire process. I could not have

completed this eﬀort without the both of you. I am also grateful to my sister and

brother-in-law (Tamara and Sampath Jayawardane) and my parents-in-law (Walter

and Indrani Singhabahu) for their love and support.

I would like to dedicate this work to my parents (Samson and Ramani Herat) who

have been a constant source of encouragement and inspiration to me throughout my

life. I am grateful to them for their warmth, support and guidance.

v Contents

1 Introduction 5

1.1 Conﬁnement ...... 6

1.2 Models of Conﬁnement ...... 7

1.2.1 Dual superconductor model of conﬁnement ...... 8

1.2.2 Abelian Projection ...... 11

1.2.3 Abelian Projection on a Lattice ...... 14

1.2.4 Center vortex model of conﬁnement ...... 18

1.2.5 Monopoles vs. Vortices ...... 23

2 The Undulating Kink 25

2.1 Introduction ...... 25

2.2 Kink solution ...... 27

2.3 Extension to 2+1 dimensions ...... 29

2.4 Exact solutions for small deviations from the straight line ...... 32

2.5 Non-inﬁnitesimal undulation ...... 36

1 3 The Circular Vortex 44

3.1 Introduction ...... 44

3.2 The Straight Vortex ...... 46

3.2.1 Solutions of the straight Vortex ...... 52

3.2.2 Results ...... 55

3.3 The Circular Vortex ...... 57

3.4 Numerical Treatment ...... 62

3.4.1 Variational Method ...... 63

3.4.2 Jacobian Iteration Method ...... 64

3.5 Results ...... 65

4 Conclusions 68

4.1 Classical Objects and Conﬁnement ...... 69

4.1.1 ’t Hooft’s dual superconductivity model ...... 69

4.1.2 Abelian Projection ...... 70

4.1.3 Center Vortex Model of Conﬁnement ...... 72

4.2 The Undulating Kink ...... 75

4.3 The Circular Vortex ...... 77

2 List of Figures

1.1 (a) The Abrikosov string between the monopoles in the superconductor and

(b) an analogue of the Abrikosov string between the chromo-electrically

charged particles in the dual superconductor...... 9

1.2 Abelian and nonabelian potentials (with self energy V0 subtracted), Ref. [17]. 16

1.3 The abelian potential (diamonds) in comparison with the photon contribu-

tion (squares), the monopole contribution (crosses) and the sum of these

two parts (triangles), Ref. [17]...... 17

1.4 (a) The ”Spaghetti vaccum” of condensed vortices (b) The ”monopole vacumm”

appears as a result of abelian projection [25] ...... 24

2.1 (a) A plot of the static kink(anti-kink) solution (2.4) (b) The energy density

of the kink (2.5). It is localized...... 29

3.1 Plot of the straight vortex field configurations for different values of the

coupling constant. (a) β = βc = 1/4, critical coupling. (b) β = 3βc. (c)

β = 2βc. (d) β = βc/2. (e) β = βc/3...... 56

3 3.2 Plot of the variation method approximate circular vortex ﬁeld conﬁgurations

for critical coupling, β = βc = 1/4 ...... 64

3.3 Plot of the circular vortex field configurations for different values of the

coupling constant. (a) β = βc = 1/4, critical coupling. (b) β = 3βc. (c)

β = 2βc. (d) β = βc/2. (e) β = βc/3...... 66

4 Chapter 1

Introduction

One of the major developments of twentieth century physics has been recognition that all the known interactions can be described in terms of gauge theories [1]. Gauge theories in general exhibit a rich spectrum of finite energy classical solutions. Of these vortices, monopoles, and instantons are the best known topological solutions in 2, 3, and 4 dimensions. These classical objects play a major role in describing a wide variety of phenomena such as magnetic field lines in type II superconductors [3], cosmic strings in grand unified theories [4], and objects responsible for confinement in dynamical models of nonabelian gauge theories [5][6], to name a few. Our investigations of classical solutions extended to higher dimensions have been manily motivated by the recent success of these dynamical models of confinement.

5 1.1 Conﬁnement

Ample experimental evidence exists to support the fact that the strongly interacting particles, hadrons (i.e. protons, neutrons,etc.), are made of more elementary con- stituents called quarks. There is further evidence that, in addition to their electric charge, quarks carry an additional quantum number called color. In fact, quarks come in three colors. A non-abelian gauge theory, Quantum ChromoDynamics(QCD) explains the interactions between quarks as due to the exchange of a set of particles called the gluons. This theory is similar to Quantum Electrodynamics (QED) that explains the Electromagnetic interactions between electrically charged particles as due to the exchange of a particle, the photon. The main diﬀerence is that gluons interact among themselves while the photons do not. The self interactions between gluons play an important role in determining the nature of the color force.

The strength of the strong force between quarks is given by the gauge coupling strength. Perturbation theory, which is the mathematical technique used in solving quantum theories, can only be applied in the regime where the coupling strength is relatively week. Perturbative calculations in Quantum Chromodynamics indicates that the gauge coupling becomes weak at short distance scales. At zero separation, quarks act like a set of free particles. This behavior is termed asymptotic freedom.

QCD is well understood in this perturbative regime and agrees with all known ex- periments. On the other hand, at long distances the coupling becomes large and

6 perturbation theory fails. Understanding the long distance behavior of QCD is extremely important if we want to consider it as the theory of strong interactions.

Experimental evidence indicates that any viable theory of strong interactions should also contain one important feature, conﬁnement. Free quarks or any other objects carrying the quantum number color have never been observed outside hadrons.

It seems that quarks are conﬁned inside hadrons at a distance scale of about one

15 Fermi, i.e, 10− meters. This feature is referred to as confinement. There is still no analytical proof of confinement in QCD. Confinement is a long distance phenomena

and perturbation theory cannot be used to explore it as the gauge coupling becomes

strong in this regime. Therefore, it is extremely important to understand how QCD

could be analyzed in its non perturbative regime.

Although an analytical proof is absent it is evident from results of numerical

simulations that conﬁnement is indeed present in QCD. Wilson [2] in his elegent

formulation of QCD on a space-time lattice illustrated the existence of conﬁnement

in the strong coupling limit.

1.2 Models of Conﬁnement

As mentioned previously, conﬁnement is one of the most important, yet unsolved,

aspects of non-abelian gauge theories. Since no analytical proof of conﬁnement ex-

ist, it becomes essential to look at models that capture the dynamical degrees of

freedom that are responsible for conﬁnement in order to understand the physical un-

7 derpinnings. Perhaps the most popular of such theories of quark conﬁnement is the dual-superconductor picture of ’t Hooft.

1.2.1 Dual superconductor model of conﬁnement

In mid 1970’s, ’t Hooft, Nambu and Mandelstam proposed an intriguing idea that confinement in non-abelian gauge theories could be due to the dual version of superconductivity [7, 8, 6]. The BCS theory of superconductivity requires the existence of phonon mediated electron-electron pairs, namely the Cooper pairs. These spin zero bosons undergo Bose-Einstein condensation at low temperatures, and the condensate exhibits infinite conductivity. In the superconducting phase photons acquire a mass, thus, external magnetic field lines are either completely expelled from the superconductor (type I) or squeezed into thin flux tubes (type II). This phenomenon is known as the Meissner effect. The magnetic flux between a pair of imaginary monopole- antimonopole pair inside the superconductor, will then be squeezed into a flux tube

(Abrikosov string) [3] and will give rise to a linear attractive potential between them

(see Figure 1.1a). Therefore, we can say that the monople-antimonopole pair is con-

ﬁned.

This is very similar to what we desire in QCD, i.e the existence of a linear, attractive potential between a pair of chromo-electric charged objects. However, the role of the chromo-electric and chromo-magnetic ﬁelds are reversed compared to the superconductor, and hence referred to as a dual-superconductor model. Consider the QCD

8 (a) (b)

Figure 1.1: (a) The Abrikosov string between the monopoles in the superconductor and

(b) an analogue of the Abrikosov string between the chromo-electrically charged particles in the dual superconductor. vacuum to be a condensate of monopoles. Then, in this state, gluons would become

massive, giving rise to a dual version of the Meissner eﬀect. Suppose a pair of objects

with opposite chromo-electric charge (quarks or gluons) is placed in this medium. The

chromo-electric ﬁeld between the pair would get squeezed into ﬂux tubes (see Figure

1.1 b) and they would experience a linear, attractive potential. This is conﬁnement.

However, this picture has two problems with it:

1) The dual-superconductor model is based on a theory, in which electro-magnetic

duality is manifest, while QCD is a non-abelian theroy.

2) The existence of monopoles is essential in this scenario, where as in non-abelian

gauge theories (without Higgs bosons), there are no monopole solutions.

9 As a solution to these two problems, ’t Hooft proposed the concept of abelian gauge

fixing [9]. By definition, abelian gauge fixing reduces QCD to an abelian gauge theroy, where one fixes the non-Abelian part of the gauge, such that the Cartan subgroup remains an exact symmetry. In the case of SU(2), U(1) remains exact, and this left-

over symmetry corrresponds to magnetic ﬁelds. As a remarkable fact, in this abelian

gauge, magnetic monopoles appear as topological objects corresponding to the non-

N 1 N 1 trivial homotopy group Π2(SU(N)/U(1) − ) = Z − . By abelian gauge ﬁxing, QCD

is reduced to an abelian gauge theory with monopoles, and thus provide a theoretical

basis for the dual superconductor model of conﬁnement.

Diﬀerent abelian projections of non-abelian theories lead to diﬀerent abelian theo-

ries, and the most popular of these is the maximal abelian projection (MA) [10, 11].

The MA gauge is the abelian gauge in which the diagonal components of the gauge

ﬁelds are maximized by gauge transformations. In this gauge, the physical informa-

tion of the gauge conﬁguration is concentrated in the diagonal components. Lattice

studies of theories in the MA gauge give a strong indication of abelian dominance,

which means that the string tension and the chiral condensate are mostly given by

the abelian variables [12, 13]. There is also lattice evidence that in the MA gauge

only the monopole part of the abelian variables contribute to non-perturbative QCD,

and this feature is refered to as monopole dominance [17, 16]. The presence of abelian

10 and monopole dominance is considered to be strong evidense in favour of the dual-

superconductor model of conﬁnement [14, 15].

1.2.2 Abelian Projection

The abelian monopoles arise from non-abelian gauge ﬁelds as a result of the abelian

projection suggested by ’t Hooft [9]. The abelian projection is a partial gauge ﬁxing under which the abelian degrees of freedom remain unﬁxed. Since the original SU(N)

gauge symmetry group is compact, the remaining abelian gauge group is also compact. However it is known that abelian gauge theories with compact gauge symmetry group contain abelian monopoles. Therefore SU(N) gauge theory in the abelian gauge has monopoles.

As we have mentioned before, there are inﬁnite number of abelian projections. How- ever, to demonstrate the method of abelian projection let us ﬁrst consider the simplest example, i.e. the case of F12 abelian projection for SU(2) gauge theory.

Under the gauge transformation the component of the ﬁeld strength tensor, Fˆ12,

transform as

Fˆ (x) Fˆ0 (x) = Ω†Fˆ (x)Ω. (1.1) 12 → 12 12

The F12 gauge is deﬁned as:

Fˆ (x) Diagonal Matrix. (1.2) 12 ≡

11 Then in this gauge, the ﬁeld strength tensor is invarient under U(1) gauge transformations. i.e.,

ˆ ˆ F12(x) = ΩU† (1)F12(x)ΩU(1). (1.3) where,

eiα(x) 0 ΩU(1)(x) =   (1.4) iα(x)  0 e−      Therefore, the gauge condition (1.2) ﬁxes the SU(2) group up to the diagonal U(1) subgroup.

The SU(2) gauge ﬁeld transforms under the gauge transformation as:

i Aˆ Ω†Aˆ Ω Ω†∂ Ω. (1.5) µ → µ − g µ

where the SU(2) gauge ﬁeld is given by,

A3 A+ ˆ µ µ 1 2 Aµ =   , where Aµ± = Aµ Aµ. (1.6) 3 ±  Aµ− Aµ   −    If we use the F12 gauge condition, then under the residual U(1) gauge transformation

the components of the nonabelian gauge ﬁeld Aˆµ will transform as:

1 A3 A3 ∂ α, and A+ e2iαA+. (1.7) µ → µ − g µ µ → µ

3 + Here, we can see that Aµ transforms like an abelian gauge ﬁeld, where as the ﬁeld Aµ

plays the role of a charge 2 abelian vector matter ﬁeld.

12 Thus, we obtain abelian ﬁelds from a non-abelian theory. Now, let us look at the

appearence of monopoles. For simplicity let us use a = A3 . Since, H~ = ~ ~a, if µ µ ∇ × a is regular, then ~ H~ = 0. i.e. there are no monopoles. However, Ω, the gauge ∇ · transformation matrix may contain singularities. In that case the abelian ﬁeld a

may be non-regular. The non-abelian ﬁeld strength tensor Fˆµν is not invariant under

singular gauge transformations. i.e.:

Fˆ Fˆ0 = Ω†Fˆ Ω + singular part, (1.8) µν → µν µν where,

i singular part = Ω†(x)[∂ ∂ ∂ ∂ ]Ω(x). (1.9) −g µ ν − ν µ

If, we deﬁne the abelian ﬁeld strength tensor as that we obtain from singular gauge

transformation (1.5), i.e. f = ∂ a ∂ a , then f may contain singularities (Dirac µν µ ν − ν ν µν strings). i.e.

f = f r + f s , where ε ∂ f r = 0 and ε ∂ f s = 0. (1.10) µν µν µν µναβ ν αβ µναβ ν αβ 6

= ~ H~ = 0. Therefore, we can conclude that the abelian projected theory can ⇒ ∇ · 6 contain monopoles. To ﬁnd the charge of the monopole, consider a monopole and

surround it by an inﬁnitesimally small sphere S. Then the charge is given by the

integral,

1 m = H~ d~σ (1.11) 4π I

13 It has been shown that monopole charge can take values, m = 0, 1 , 1 ,...... [10, 11]. ± 2g ± g The quantization of the monopole charge is due to topological reasons. i.e. the surface integral shown in equation (11) is equal to the topological winding number of

SU(2) over the sphere S surrounding the monopole. And, the quantization comes

about due to the electric charge being ﬁxed by equation (1.5), and consequently the

magnetic charge having to obey the Dirac quantization condition.

According to the dual superconductor model, the abelian gauge freedom remain-

ing after the gauge transformation, together with the monopoles, are responsible for

confinement. Therefore, once the gauge fixing is done all the off-diagonal elements of

the gauge ﬁeld are discarded. This process is referred to as Abelian projection.

To solve the problem of conﬁnement it is insuﬃcient to prove the existance of monopoles,

but one needs to further establish that the monopoles condense. However, the only

known method of investigating this feature is through lattice calculations.

1.2.3 Abelian Projection on a Lattice

In the previous section the Fˆ12 gauge was used to illustrate the process of abelian

gauge ﬁxing. However, there are an inﬁnite number of abelian projections. Instead

of diagonalizing the component Fˆ12, one can choose to diagonalize any hermitian

operator G, that transforms as the adjoint representation of the gauge group, i.e. G

Ω†GΩ. Each operator G deﬁnes an abelian projection. →

14 Recent lattice studies have shown that infrared phenomena such as conﬁnement and chiral symmetry breaking are to a very good extent reproduced in the Maxmial

Abelian(MA) gauge [18, 15]. This method was ﬁrst devised by Kronfeld, Laursen,

Schierholtz, and Wiese [10]. The SU(2) gauge ﬁelds, Ul, on the lattice, are deﬁned

by SU(2) matrices attached to the liks l. Under gauge transformation, the ﬁeld, Ul,

transforms as Uµ0 (x) = Ω†(x)Uµ(x)Ω(x + aµˆ). In the SU(2) lattice formalism, the MA gauge is deﬁned so as to maximize the functional,

1 F [U] = T r U (x)σ U †(x)σ (1.12) 2 µ 3 µ 3 x,µ X

Using the standard parametrization for the link matrix Ul,

iθl iχl cosϕle sinϕle Ul =   (1.13) iχl iθl  sinϕle− cosϕle−   −    where θ, χ [ π, +π] and ϕ [0, π] ∈ − ∈

In this parametrization,

R [Ul] = cos2ϕl. (1.14) Xl Here it can be observed that the maximization of R corresponds to the maximization of the diagonal elements of the link matrix. Once the gauge ﬁxing is done one is free to perform the abelian projection. This is done be replacing the full gauge variables U = a I + i~a ~σ that are deﬁned on the links in the lattice by the abelian 0 ·

15 links A given by;

a I + ia σ A = 0 3 3 (1.15) 2 2 a0 + a3 p

In order to test this theory one has to check whether the abelian projected theory

0.6 V-V0 0.5 ab ab V -V0 0.4

0.3

0.2 V(R) 0.1

-0.1

-0.2 2 4 6 8 10 12 14 16 R

Figure 1.2: Abelian and nonabelian potentials (with self energy V0 subtracted), Ref. [17]. is indeed equivalent to the original non-abelian theory. This is acheived by looking for abelian dominance in the projected theory. The notion of the abelian dominance means that the expectation value of the physical physical quantity < χ > in the non-abelian theory coincides with the corresponding expectation value in the abelian theory obtained by the abelian projection. Among the well-studied problems is the

abelian dominence for string tension σ [12, 13, 17]. In this case, χ(Uˆnonabelian) = σSU(2)

and χ(Uˆabelian) = σU(1). Both string tensions σSU(2) and σU(1) are calculated by means

16 0.8 Vab 0.7 Vph Vmon,fs 0.6 Vmon,fs+Vph

0.5

0.4 V(R) 0.3

0.2

0.1

0 2 4 6 8 10 12 14 16 R

Figure 1.3: The abelian potential (diamonds) in comparison with the photon contribution

(squares), the monopole contribution (crosses) and the sum of these two parts (triangles),

Ref. [17]. of Wilson loops. In ref. [17] the authors have carried out an accurate study of SU(2) gluodynamics on a 344 lattice. The abelian and non abelian potentials are shown in

Figure (1.2), while the photon and monopole contributions are shown in Figure (1.3)

[17]. From the slopes of Figures (1.2) and (1.3) one can derive the following results:

σU(1) = 92%σSU(2), σj = 95%σU(1), where σj refers to the monopole contribution to the string tension. From the ﬁrst result one sees clear evidence of abelian dominence while the later indicates monopole dominance, which means that the monopoles are responsible for most of the SU(2) string tension.

Other calculations performed have shown condensation of monopoles in the conﬁning

17 phase. At the same time mesearments have also revealed that chiral symmetry is also

broken int the abelian projected theory and the size of the chiral condensate takes

the desired values. The main conclusion is that the QCD vacuum behave like a dual superconductor, where the monopole currents are condensed and are responsible for conﬁnement.

1.2.4 Center vortex model of conﬁnement

In the preceding section I looked at the dual superconductor model of conﬁnement, where the QCD vacuum is considered to be a monopole condensate, and as a con- sequence, the abelian charges conﬁned. Evidence of abelian dominance in lattice calculations is indicative of the successfulness of this theory.

The idea that extended vortex conﬁgurations are responsible for conﬁnement in SU(N) gauge theories was recognized by several people, including Nielsen and Olesen [20], ’t

Hooft [21], Cornwall [22], and Mack [23]. This picture is referred to as the spaghetti vacuum. Here, the QCD vacuum is understood to be a condensate of vortices of some

finite thickness, carrying magnetic flux in the center of the gauge group, hence refered to as center vortices. Recently, results of numerous lattice studies have indicated that the center vortex picture to be a strong candidate for describing confinement.

In center vortex model of SU(N) gauge theories, the gauge ﬁxing conditions is choosen to reduce the full SU(N) gauge symmetry to the discrete center group, ZN . This is

18 reﬀered to as center gauge ﬁxing. The most popular center gauges include, two ver- sions of the Maximial Center (MC) gauge [24, 25, 26], and the Laplacian Center(LC) gauge [27, 28].

I will ﬁrst describe the ”indirect” maximal center gauge. One starts by ﬁxing to the maximal abelian gauge [10]. As shown in the privious section (equation 1.12), for

SU(2) this amounts to maximizing the quantity,

T r Uµ(x)σ3Uµ†(x)σ3 (1.16) x,µ X Then the abelian projection is carried out where the full link variables are replaced by

the abelian link variables according to equation (1.15). These link variables transform

under the residual symmetry as U(1) gauge ﬁelds. The indirect maximal center gauge

uses this residual symmetry to bring the abelian link variables as close as possible to

SU(2) center elements I. This is acheived by writting ±

eiθ A =   (1.17) iθ  e−      and using the residual U(1) symmetry to maximize

2 cos (θµ(x)) (1.18) x,µ X

This leaves a remnant Z2 symmetry. This is the (indirect) maximal center gauge.

Then Center Projection is carried out by replacing the full link variable U by the

19 center element ZI, where

Z sign(cosθ) = 1 (1.19) ≡ ±

This is refered to as the ’indirect’ maximal center gauge because the center is maximized in the abelian link variable, rather than directly in the full link variable. Lattice

results show evidence of center dominance in this gauge. However, the agreement of the string tension extracted from this theory compare with the original string tension

before gauge ﬁxing is not as good as one would desire. Better results have been re-

ported in the ”direct” center gauge.

The direct maximal center gauge ﬁxing is accomplished by maximizing the functional

F [U] over the gauge transformation V(x), where

3 1 V 2 1 V V F [U] = T rU (x) = T r σ U (x)σ U † . (1.20) 4 µ 6 i µ i µ x,µ x,µ " i # X X X 1 3 = T r σ V (x)U (x)V †(x + aµˆ)σ V (x + aµˆ)U †(x)V †(x) . 6 i µ i µ x,µ " i # X X re-arranging this,

1 3 F [U] = T r V †(x)σ V (x)U (x)V †(x + aµˆ)σ V (x + aµˆ)U †(x) 6 i µ i µ x,µ " i # X X this can be written in the form

1 3 F [U] = g (x)g (x + aµˆ) T r σ U (x)σ U †(x) 6 i i i µ i µ x,µ i X X Write F [U] in terms of the adjoint representation quantities,

1 3 F [U] = g (x)U ab(x)g (x + aµˆ) (1.21) 3 ia µ ia x,µ iab X X 20 Consider the standard form of the gauge ﬁeld,

U (x) = a + i~a ~σ, where a2 + ~a2 = 1 µ 0 · 0

2 then F [U] = a0. Therefore by maximizing a0, ~a is minimized, and results in the gauge ﬁeld becomingP as close as possible to the elements of the center, U 1. µ ∼ ± This is the direct Maximal Center (MC) gauge. Once the gauge is ﬁxed, the center projection is carried out by substituting, a + i~a ~σ sign a . 0 · → 0

Recently, lattice studies of MC theories have investigated whether the ”projection vortices” that appear in direct MC theories due to the excitations of Z2 gauge fields have anything to do with the existence of center vortices in the full, unprojected lattice configurations. L. Del Debbio et al.[26] have shown that indeed theses projection vortices locate center vortices in the full theory. In their calculations the string tension of the vortex contribution to the Wilson loops is found to match, quite accurately, the string tension extracted from the full Wilson loops. This is center dominence. These results seem to support the view that center vortices are responsible for confinement.

However, recently there has been criticism of the methods used to fix the gauge on the lattice in MC projection models [28]. The relationship between the ZN excitations which are located on a plaquette and the center vortices of the original SU(N) theory proceeds through the gauge fixing scheme. It is the gauge fixing technique used and not the maximal center projection that has raised doubts about the valid-

21 ity of this model. Gauge ﬁxing is accomplished by iterative local maximization of

a gauge functional; and this process stops once a local maximum is reached. There

is no guatrantee that a gobal maximum has been reached, and as a result the local

maximum achieved depends on the starting gauge conﬁguration. This situation has

been shown to have harmful eﬀects in the MC gauge [30].

As a solution to this problem, Alexandrou, d’Eliya, and de Forcrand introduced the

Laplacian Center (LC) gauge [31, 27]. Consider the adjoint form of the functional

F [U]. 1 3 F [U] = g (x)U ab(x)g (x + aµˆ) (1.22) 3 ia µ ia x,µ iab X X One only needs two orthogonal vectors (say g1a and g2a) in order to uniquely deﬁne the adjoint representation gauge transformation. Therefore, it is suﬃecient to keep only two terms i = 2 and i = 2 in the sum over i. g1a and g2a are normalized and are orthogonal to each other. In the LC gauge the normalization and the orthogonality

conditions on the vectors gia are relaxed. At certain points ’x’, g1a is parallel to g3a,

and these points form codimension 2 set. Then at such points the gauge freedom

is locally enlarged from Z2 to U(1), and the gauge trnsformation become singular.

These deﬁne the location of vortices. At points where the gauge freedome is further increaded to SU(2) for codimension 3 sets that can be identiﬁed with monopoles.

Therefore, in the LC gauge, center vortices and monopoles appear together as local gauge defects. The LC guage ﬁxing is then followed by center projection where the

22 links are replaced by center element. Recent lattice simulations [28] have shown center dominance where the string tensions agree to a better precision than in the monopole picture with that calculated from the full theory.

1.2.5 Monopoles vs. Vortices

The strong evidence in support of a vortex condensation theory of conﬁnement pre- sented in the previous section, is reminiscent of the data that support the monopole condensation theory. However, these seem to be mutually exclusive propositions, for one thing, monopoles are of codimension 3, while vortices are codimension 2 objects.

So, this raises the natural question: Which one of these models are correct? i.e. if the QCD vacuum is dominated by vortec conﬁgurations (”spaghetti vacuum”), then how do we explain the numerical success of the abelian projection in the maximal abelian gauge?

Recently, there has been several exhaustive studies carried out on this problem

[25]. The current opinion is that a center vortex conﬁguration, transformed to maximal abelian gauge and then abelian-projected, will appear as a chain of monopoles alternating with antimonopoles. Therefore, these monopoles are essentially an arti- fact of the projection and they are condensed because the long vortices from which they emerge are condense. If this picture is accurate, then the ”spaghetti vaccum”

(of condensed vortices) appears, under abelian projection, as a ”monopole vacuum”

(see ﬁgure 1.4).

23 +

+ - + - - + + + -

- -

(a) (b)

Figure 1.4: (a) The ”Spaghetti vaccum” of condensed vortices (b) The ”monopole vacumm” appears as a result of abelian projection [25]

From the preceding discussions, clearly there is convincing numerical evidence that

monopoles and mostly vortex condensation picture of conﬁnement is valid. although

lattice simulation evidence is still very important, it is also important to use all means

possible to investigate the properties of classical objects responsible for conﬁnement.

Few important aspects of these classical solutions that can possibly be investigated

in Higgs theories include, the relationship between monopole and vortex induced conﬁnement, interaction of vortices, and the very ambitious goal of studying the condensation of vortices. However, in order to carry out such investigations one needs to know how to correctly deﬁne the energy of a single object. i.e. we need to be able to determine the energy dependence of extended classical solutions on the shape as well as the length of the object.

24 Chapter 2

The Undulating Kink

2.1 Introduction

Classical solutions of ﬁeld theories play an important role in a host of applications.

In particular vortices, abelian and nonabelian, acquired central importance, among

others, in superconductors [3], cosmic strings [4], and as objects responsible for con-

ﬁnement [5] [6]. Classical solutions of recent interest are p-branes, appearing in string

theories [32].

Vortices and some other solutions of ﬁeld equations distinguish themselves by the

fact that they are extensions of ﬁnite energy classical solutions (solitons) into one

or more additional spatial dimensions. As such, they have inﬁnite energy, but ﬁnite

energy per unit length if the additional space dimensions are inﬁnite. The energy of

these solutions becomes ﬁnite if the additional spatial dimensions are ﬁnite. In what

follows I will only consider ﬁnite additional dimensions that make analytical studies

of these objects possible.

25 To the best of my knowledge, analytical studies of these objects have not been ex-

tended to classical solutions. One important aspect of classical, ﬁnite energy solution

lifted into higher dimensions, is that their energy is dependent on the shape and length of these objects. In most applications it has been assumed [33], [34] that the energy of a 3 dimensional vortex is simply its length multiplied by the energy of unit length (which is equal to the energy of the two dimensional soliton), irrespective of its shape.

Before proceeding any further I need to clarify what I mean under the dependence of the energy on shape. Topologically stable classical solutions are characterized by the

vanishing of an order parameter (usually a scalar ﬁeld) at a ﬁnite coordinate, which

is, by deﬁnition, the location of the center or core of the object. In trivially extended

classical solutions the zeros form a straight line. These extended objects are solutions

of higher dimensional ﬁeld equations. Suppose now I impose a constraint that the

locus of zeros lies on a predetermined curve and minimize the Hamiltonian with this

constraint. Then, in general, the value of the Hamiltonian will depend on the shape

and length of the locus of zeros. This dependence is important, because before one

can attempt an analytic or numerical study of the interaction and condensation of

vortices, or of other classical extended objects, one needs to clarify the energy of a

single object. As I show in this chapter, the dependence of the energy on the length

and curvature of an undulating object is highly nontrivial.

26 One might think that the deviation of an extended object from the minimum energy straight line is a purely quantum phenomenon and should be treated in perturbation theory. Such a treatment would not, however, adequately describe large scale features of extended objects. Such, macroscopic features of vortices are clearly seen in superconductors or simulations of cosmic strings. [34] Therefore it should be possible to calculate the properties, at least in some approximation, studying the classical equations of motion. Naturally, if the energy of curved vortices can be calculated their contribution to the functional integral, i.e. their full quantum contribution, can possibly be evaluated.

One of the ultimate targets of this investigations is vortex solutions in gauge theories. As no analytic vortex solutions exist, the problem of curved vortices is fairly diﬃcult. Though my aim is to understand issues pertaining to the curvature dependence of classical solutions in more complicated quantum theories, I will ﬁrst discuss the curvature energy for a simpler problem, the kink solution of the 2+1 dimensional linear sigma model. I need to gain an understanding of this problem so

I can be apply such information to solve more diﬃcult problem of vortices and other extended classical objects as described in chapter 3.

2.2 Kink solution

One of the simplest examples of classical solutions of interacting ﬁeld theories the one dimensional kink, which is a soliton solution of the two dimensional linear sigma

27 model. Kinks are toplogically stable ﬁnite energy solutions that can be extended to 2 dimensions in a straightforward manner. I start with a brief description of these, well

known solutions, [35] with the intention of establishing notations. As the solution can

be chosen to be static, the Lagrangian coincides with the negative of the Hamiltonian.

The Hamiltonian for a time independent solution is solution is

1 ∂Φ 2 λ m2 2 H = dξ + Φ2 . (2.1) 2 ∂ξ 2 − λ Z " # The vacuum expectation value can be scaled out if I use the transformation

m Φ(ξ ξ0) = Ψ(x x0), (2.2) − √λ −

where m(ξ ξ )/√2 = x x . The rescaled Hamiltonian takes the form − 0 − 0 m3 ∂Ψ 2 H = dx + Ψ2 1 2 = dxh(x). (2.3) 2√2λ ∂x − Z " # Z with a minimizing kink (anti-kink) solution

Ψ (x x ) = tanh(x x ). (2.4) c − 0 ± − 0

The energy density of the kink solutions is

m3 1 h(x) = (2.5) √2λ cosh4(x x ) − 0 The value of the Hamiltonian at this minimizing solution is

m3 4 H = H0 = . (2.6) √2λ 3

In what follows I will set m2 = λ = 2. The correct units can be easily restored.

28 (a) (b)

Figure 2.1: (a) A plot of the static kink(anti-kink) solution (2.4) (b) The energy density

of the kink (2.5). It is localized.

2.3 Extension to 2+1 dimensions

In the three dimensional linear sigma model there is an extra term in the Hamiltonian

density, (∂Φ/∂y)2. Solution (2.4) also satisﬁes equations in 2 dimensions, but it would

have infinite energy if the space is infinite in the y direction. As indicated earlier, I choose a finite y dimension, with periodic boundary condition. In other words, I seek

a kink solution on an inﬁnite strip of width L.

The minimum energy kink on the strip is a straight line at a ﬁxed value of x. It

is interesting to study, however, the dependence of the energy on the shape of the

kink. For this purpose I distort the shape of the kink by assuming that the zero of

the solution is not at x0 = 0 (this value is chosen arbitrarily) but rather at x = (y).

The periodicity condition, with the translation symmetry along the x axis allows me

to choose (0) = (L) = 0.

29 The problem I pose is the following. Suppose a close curve, defined by x = (y), winds through the strip. This curve is the locus of the zeros of Φ(x, y). In other words, I impose the constraint on the field, Φ((y), y) = 0. We wish to find the field configuration, Φ(x, y), satisfying this constraint and minimizing the Hamiltonian. I am mostly interested in the dependence of the minimum of the Hamiltonian on the shape of the curve.

The two-dimensional Hamiltonian has the form

1 dΦ 2 dΦ 2 H = dx dy + + (Φ2 1)2 . (2.7) 2 dx dy − Z " #

It is assumed here that Φ is a conﬁguration minimizing H for a given shape of the core.

Deﬁne the ‘length-energy,’ Hl, of a two a dimensional curved kink solution as

Hl = lH0, where l is the total length of the kink,

2 l = dy 1 + 0 . Z p and H0 is the energy of the one dimensional kink (energy of unit length of straight two dimensional kink). Then the curvature energy of a two dimensional kink solution is deﬁned by subtracting the ‘length-energy,’ from the total energy, H = H H . I c − l can then prove the following theorem:

The curvature energy of the two dimensional kink is non-positive.

My detailed calculations will show that the curvature energy is negative deﬁnite. In

30 other words, discounting the length energy, the kink prefers the curled state to the

straight one.

Proof: Consider a class of solutions that depend on the coordinate x and y only C through the combination x (y). The Hamiltonian minimized on functions of this − class will have a minimum that is not smaller than the unrestricted minimum. To

prove the theorem I will show now that the Hamiltonian minimized on is exactly C the length energy.

Assume that Φ depends on x and y through x (y). Then upon substitution − into (2.7) and shift of coordinate x by (y) I obtain

2 1 2 dΦ 2 2 H = dx dy (1 + 0 ) + (Φ 1) . (2.8) 2 dx − Z " #

The substitution x x√1 + 2 leads to → 0

2 1 2 dΦ 2 2 H = dx dy 1 + 0 + (Φ 1) = lH0. (2.9) 2 " dx − # Z p where,

2 l = dy 1 + 0 . Z p Φ depends only on x. As the one dimensional Hamiltonian is minimized by the original kink solution, (2.4), we obtain H = 4l/3, as asserted above. This completes the proof of the theorem.

31 2.4 Exact solutions for small deviations from the

straight line

In the previous section illustrated that the curvature energy of an arbitrary curved kink is negative. I am able to calculate the curvature energy analytically only if it is assumed that the deviation of the kink from the straight line, (y), is small.

The general form of the two dimensional kink solution is written in the form

Φ(x, y) = Φ (x )+χ(x , y), where Φ = tanh(x). As the constraint is Φ((y), y) = c − − c

0, and having Φc(0) = 0 one gets

χ(0, y) = 0. (2.10)

Since at = 0 the solution Φc is exact, χ = O(), as well.

Now I can vary the Hamiltonian with respect to χ with the subsidiary condition

(2.10). Keeping terms of O() I obtain the following equation of motion for χ:

2 χ + χ 00(y)Φ0 2(3Φ 1)χ = δ(x)λ(y), (2.11) xx yy − c − c − where the Lagrange multiplier λ(y) = χ ( + 0, y) χ ( 0, y) and where the x − x − subscripts x and y indicate partial derivatives. The appearance of the delta function in (2.11) has no other signiﬁcance in our subsequent calculations other than removing a condition on the continuity of the derivative of χ at x = 0.

32 Notice that the inhomogeneous driving term 00(y)Φc0 in (2.11) is an even function of x. Then due to the x x symmetry of (2.11) we also have χ(x, y) = χ( x, y). It → − − is sufficient to solve (2.11) for x > 0, with boundary conditions χ(0, y) = χ( , y) = 0. ∞ Note that though χ(x, y) is even χ (0, y) = 0 because it is not continuous. x 6 To find the appropriate solution one must find the solutions of the homogeneous and inhomogeneous equations that vanish at x and find a combination of these → ∞ two solutions that also vanishes at the origin.

Let me expand χ(x, y) in a Fourier series of y as shown below.

1 χ(x, y) = e2πıny/LF (x), (2.12) √ n L n X ? where F n = Fn . Using Φc = tanh x the modes satisfy the one-dimensional inhomo- − geneous Schr¨odinger equation

2 2 2 2 6 π n 4π n n F 00 + Fn 4 1 + Fn = , (2.13) n cosh2 x − L2 − L2 cosh2 x where n is the Fourier expansion coeﬃcient of (y).

The solution of the inhomogeneous Schr¨odinger equation satisfying boundary conditions F (0) = 0 and F (x) 0 at x is n n → → ∞

1 αx 3α 3 2 Fn(x) = n e− 1 + tanh x + tanh x , (2.14) cosh2 x − α2 1 α2 1 − − where

π2n2 α = 2 1 + 2 . (2.15) r L 33 The O(2) deviation of the Hamiltonian to that of a straight kink becomes

1 2 2 2 2 ∆H = dxdy (χ ) + (χ 0Φ0 ) + 2((3Φ 1)χ . (2.16) 2 x y − c c − Z After integrating by parts and utilizing (2.11) the Hamiltonian can be written as,

1 ∆H = dxdy0Φ0 (0Φ0 χ ) (2.17) 2 c c − y Z Substituting solution (2.14) into (2.17) the integral can be easily evaluated to give

4n2π2 2 ∆H = 2 (2.18) n L2 α(α2 1) X − using (2.15) the ﬁnal result can be derived,

2 2 2 2 π n 8n π 1 + L2 ∆H = 2 (2.19) n L2 q4π2n2 L2 + 3 X Compare (2.19) with (2.9). Given that, ∆l = l L is the increase of the length − of the kink due to undulation, the increase of length energy, as obtained from the

expansion of (2.9) is

2 2 H0 2 2 8π n ∆H = H ∆l dy[0(y)] = . (2.20) 0 0 ' 2 n 3L2 n Z X

Then comparing (2.20) with (2.19) shows that ∆H0 > ∆H, or in other words, the

curvature energy is always negative. This statement is analogous to the general statement that the second order perturbation to the ground state is always negative.

Assuming that long wavelength modes dominate I can expand the expression of the excess energy in π2n2/L2. Then using

1 + 2 1 R = 0 , ' | 00| | 00| 34 where, R is the curvature radius we get the leading correction term in long wavelength

modes

8π2n2 2π4n4 1 ds ∆H 2 + . . . = H ∆l + . . . . ' n 3L2 − 3L4 0 − 32 R2 n X Z Omitted higher order terms will contain derivatives of the curvature radius. The

minimum of the total Hamiltonian is

1 2 H = H ds ds(x00) + . . . , 0 − 32 Z Z where x is the 2-dimensional coordinate vector of the undulating kink core in the path-length gauge. Clearly, the curvature term is negative, as required by theorem proven in the previous section.

To reconcile this result with another to be obtained in the next section, consider the low frequency approximation from the beginning. I will take the infrared limit of the small approximation and then take the small limit of the infrared approximation. Note that for this comparision it is better to use the view that the unperturbed solution is Φ (x ) rather than Φ (x) = tanh x. Then the method is modiﬁed and one c − c needs to solve an inhomogeneous Schrodinger equation, rather than a homogeneous one. The diﬀerential equation becomes

2 2 2 2 2 4π n 4π n F 00 2(3Φ 1)F F = Φ0 (2.21) n − c − n − L2 n − L2 n c

After integration by parts, the Hamiltonian transforms to

1 ∆H = dx dy 0Φ0 (0Φ0 χ ). (2.22) 2 c c − y Z 35 Using the approximation that at most O(δ4) terms are kept, where δ = nπ/L,

2 only O(δ ) terms in χ are needed. But then, following from (2.21), the function Fn

is proportional to δ2 anyway so correction terms of O(δ2) are not needed. Therefore

to order δ4 in the Hamiltonian one can use a simpliﬁed equation

2 2 2 2 2 4π n 4π n n F 00 2(3Φ 1)Fn = nΦ0 = . (2.23) n − c − − L2 c − L2 cosh2 x

It is feasible to check how this works. One solution of the homogeneous equation is

obviously 1 F 0 = . n cosh2 x

Then the second solution can be determined as stated in (2.37). The solution is

given by (2.39). Now, a linear combination that decreases at inﬁnity and vanishes at

the origin is

2 2 2x 4x 4π n 8e− + e− 12x 9 Fn = n − − (2.24) − L2 48 cosh2 x

This solution substituted into (2.22) results in the following contribution

2 5 ∆H = 00 . (2.25) − 36

2.5 Non-inﬁnitesimal undulation

Now consider results in the interesting domain of ﬁnite, O(1), 0(y), restricted our-

2 selves to the low frequency domain, when 00 << (0) /L. This implies that derivatives

of the ﬁeld with respect to y are small and decrease with the number of derivatives.

36 In this domain the low frequency region is the most interesting, because this leads to the most probable macroscopic deformations.

The trick for the analytic solution of this approximation is that the solution will

be dominated by the trivially transformed solution of the straight vortex,

x (y) Ψ0(x, y) = tanh − tanh u, √ 2 ≡ 1 + 0

where the variable u is deﬁned as u = (x )/√1 + 2. This function alone would − 0

give an energy of E = H0l, where l is the length of the kink, up to a correction term

2 of O(00 ). I will assume a small deviation from this solution, which, admittedly, gives

only a small deviation in the energy, as well. One would guess that increasing 00

would lead to larger deviations. The importance of this result is that it is completely

beyond the capability of calculating quantum corrections to the straight string, as the starting conﬁguration deviates already from the straight string by a ﬁnite amount.

First calculate the energy following from substituting Φ0 into the Hamiltonian.

2 1 0 u 00 2 2 2 H = dx dy + + Φ0 (u) + (Φ 1) 2 2 2 3/2 0 0 1 + 0 √1 + (1 + 0 ) − Z (" 0 # ) 2 2 2 u 00 2 2 2 = dx du √1 + 1 + Φ0 (u) + (Φ 1) . 0 (1 + 2)3 0 0 − Z 0

2 Neglecting the subleading term, of O(00 ), the Hamiltonian is clearly optimized by the function tanh(u). Then, using this form the following expression for the leading

order approximation to the Hamiltonian can be obtained

π2 6 ds H = H ds + − , (2.26) 0 9 R2 Z Z 37 2 2 where s is the arclength and also ds = dy√1 + 0 , and R = (1 + 0 )/00. Notice that

(2.26) also gives an exact upper bound to the energy of the kink, as no approximation was made, other than the assumption of the form tanh(u).

Next write the equation of motion for (2.7), assuming that Φ depends on x and y through the variables z = x (y), and y. Then, as before Φ(z, y) vanishes at z = 0 − and z . Furthermore, Φ is a periodic function of y, with period L. This is so → ±∞ because (y) is assumed be a periodic function, as well. Then the y-component of the kinetic energy term takes the form

2 (Φ 0Φ ) , y − z where as usual lower subscripts signify partial derivatives. The equation of motion becomes

2 2 Φ + Φ + 0 Φ 20Φ 00Φ 2Φ(Φ 1) = 0. (2.27) zz yy zz − yz − z − −

If 00 << 1/L then diﬀerentiations with respect to y are small. In leading order of the frequency derivatives of Φ with respect to y can be neglected. Then in the leading order (2.27) reads as

2 2 (1 + 0 )Φ 2Φ(Φ 1) = 0. (2.28) zz − −

The solution of (2.45) is exactly the form that was used previously to derive the energy (2.26).

z Φ(0) = tanh u tanh , (2.29) ≡ √ 2 1 + 0 38 where the variable u can be deﬁned as

z x u = = − . (2.30) 2 2 √1 + 0 √1 + 0

(0) Φ when substituted into the Hamiltonian gives H = H0l.

Now perform a perturbative expansion around Φ(0). Writing Φ = Φ(0) + χ. The

leading driving terms are clearly

(0) (0) 20Φ 00Φ . − zy − z

Then keeping ﬁrst order terms in χ only and omitting derivatives of χ with respect

to y one can obtain the following equation for χ:

2 (0)2 (0) (0) (1 + 0 )χ χ2(3Φ 1) = 20Φ + 00Φ . (2.31) zz − − zy z

(0) Note that the term Φyy or derivatives of χ with respect to y would contain at least

2 (00) or 000, negligible compared to 00.

The solution of (2.31) requires to ﬁnd two solutions, one of the homogeneous

equation, one of the inhomogeneous equation, both vanishing at inﬁnity and then

taking a linear combination that vanishes at z = 0, as well. First rewrite (2.31) using

variables u and y.

2 2 6 00 20 40 χuu 4 χ = 1 + u tanh u . (2.32) 2 2 2 2 2 − − cosh u u√ − 1 + 0 1 + 0 cosh 1 + 0

The solution of (2.32) satisfying the boundary conditions has the form

2 2 1 00 000 00 2000 χ = g1(0) + g2(0) + g1(u) + g2(u(2.33)), 2 2 2 3/2 2 2 3/2 −cosh u √ (1 + 0 ) √ (1 + 0 ) 1 + 0 1 + 0 39 where the ﬁrst term on the right hand side is a solution of the homogeneus equation,

while g1(u) and g2(u) satisfy the equations

6 1 g00 4 g1 = (2.34) 1 − − cosh2 u cosh2 u

and

6 2u tanh u 1 g00 4 g2 = − , (2.35) 2 − − cosh2 u cosh2 u with g (u), g (u) 0 when u . 1 2 → → ∞ One solution of the homogenous equation is

1 g0 = (2.36) cosh2 u

Then it is easy to generate another solution of the homogeneous equation as

du0 1 g˜0(u) = g0(u) = [12u + 8 sinh(2u) + sinh(4u)] . (2.37) g2(u ) 32 cosh2 u Z 0 0

Having known a solution of the homogeneous equation like

g00 + h(u)g(u) = 0

it is easy to write a solution of the inhomogeneous equation,

g00 + h(u)g(u) = f(u).

It is given by

u u0 du0 g(u) = g (u) du00g (u00)f(u00), (2.38) 0 g2(u ) 0 Z 0 0 Z 40 where f(u) is the inhomogeneity. Using the above deﬁnition of g0 inhomogeneity can be written as 1 f(u) = cosh2 u

8 cosh(2u) + cosh(4u) g¯1(u) = , (2.39) 48 cosh2 u

where a multiple of g0 has been added.

Note that the second solution of the homogeneous equation is

8 sinh(2u) + sinh(4u) + 12 u f 2(u) = (2.40) cosh2 u

Thus the solution of the inomogeneous equation with the right asymptotic behavior is

2u 4u 8 e− + e− 12 u g1(u) = − , (2.41) 48 cosh2 u

For an inhomogneity 2u tanh(u) 1 f(u) = − cosh2 u

I obtain 24u2 + 8 cosh(2u) + cosh(4u) g¯2(u) = . − 96 cosh2 u

Now the solution with the correct asymptotic behavior is

2 2u 4u 24u + 8e− + e− 12u g2(u) = − . (2.42) − 96 cosh2 u

41 Then, in view of (2.43) one can write

1 3 3 2 χ = 00 00 0 2 2 2 3/2 −cosh u 16√1 + 0 − 32(1 + 0 ) 2u 4u 2 2 2u 4u 00 8 e− + e− 12 u 2000 24u + 8e− + e− 12u + − − 2 2 2 3/2 2 √1 + 0 48 cosh u − (1 + 0 ) 96 cosh u 2 00 0 2 2u 4u = 9 24u 8e− e− + 12u 2 2 2 √ u 1 + 0 − − − 96 1 + 0 cosh 2u 4u 18 + 16e− + 2e− 24u . (2.43) − − }

Now the relevant Hamiltonian density for the ﬁeld χ, which is second order in χ

or derivatives with respect to y is

1 2 2 2 (0)2 (0) (0) (0)2 h = (1 + 0 )χ + 2χ (3Φ 1) 20(Φ χ + Φ χ ) + Φ . (2.44) χ 2 x − − y x x y y Note that it is necessary to add the contribution of the pure length term to the

Hamiltonian density. The last term of (2.44) corresponds exactly to the last term of

(2.26).

Now integrating by parts in (2.44) and using the ﬁeld equation for χ, (2.31),

1 (0) (0) (0)2 1 (0) (0) (0)2 h = χ(20Φ + 00Φ ) + Φ = 0(Φ χ Φ χ ) + Φ . (2.45) χ 2 yz z y 2 yz − z y y When one integrates this expression, one obtains a correction term of the form

1 ds . R2 Z This must agree with the expression that was derived in the previous sectionr, as the

0 limit of this must be the same. After diﬀerentiation →

(0) (0) 00 1 2 2 20Φyz + 00Φz = 2 3/2 2 1 0 + 40 u tanh u . (1 + 0 ) cosh u − 42 Furthermore, the term

2 2 2 (0)2 00 0 u Φy = 2 2 4 . (1 + 0 ) cosh u

So ﬁnally, the hamiltonian density takes the form

2 00 1 2 2 1 2 2 hχ = u 0 + 1 0 + 40 u tanh u 2(1 + 2)2 cosh4 u 96 − 0 2 0 2 2u 4u 2u 4u 9 24u 8e− e− + 12u 18 + 16e− + 2e− 24u(2.46). × 1 + 2 − − − − − 0 Substituting this expression into the expression of the integrated Hamiltonian it can

be shown that

2 2 00 5 1 2 2 5 0 ∆H = + (π 6)0 + . (2.47) (1 + 2)3/2 −36 72 − 72 1 + 2 0 0

2 where I have included the multiplier √1 + 0 coming from the change of variable

from x to u. In the limit of 0 0 this agrees with the infrared limit of the small → solution, (2.25). The interesting fact about the above expression for the energy is that

the curvature contribution is negative only if 0 < 1.32674. In eﬀect, the functional | |

integration then cuts oﬀ at 0 = 1.32674. ±

43 Chapter 3

The Circular Vortex

3.1 Introduction

In the preceding chapter, we investigated the properties of classical solutions of inter-

acting scalar ﬁeld theories in (2+1) dimensions. We now turn to more complicated systems in higher dimensions. i.e we will investigate classical solutions of gauge ﬁeld

theories in (3+1) dimensions.

Such interacting field theories give rise to non-linear equations of motion and such systems of equations often contain classical solutions [35]. The solutions of the field equations of these models can be classified by the number of dimensions of the object.

One type of classical solutions, referred to as solitons are particle like objects that

carry ﬁnite energy. The kink of the preceding chapter is a (1+1) dimensional soliton.

In this chapter I will look at another type of classical solution - the Vortex. Vortex

solutions exist in ﬁeld theories where gauge ﬁelds are coupled to a complex scalar

ﬁeld with a topologically nontrivial degenerate vacuum. The vortex solutions are

44 particle-like ﬁnite energy (2+1) dimensional solitons. If we extend the vortex into

(3+1) dimensions it becomes a two dimensional object like a string, and in this case the energy will in general be infinite, given that the extension of space is infinite. In other words, vortices carry finite energy per unit length.

One of the ﬁrst relativistic vortex models was developed by Nielsen and Olsen [38].

They investigated an example of a ﬁeld theory with a vortex solution by considering an

Abelian gauge ﬁeld coupled to a charged scalar. Nielsen and Olsen were motivated to study this system because it stimulates the dual string picture of elementary particles

[8, 39, 40]. Their system is also a relativistic analogue of what happens in a type-

II superconductor, i.e. the Landau-Ginzburg model[3]. Nielsen and Olsen found approximate analytical solutions of their abelian model. [38].

As I mentioned in the previous chapter, the energy of vortex solutions of (3+1) dimensional gauge theories is always determined by simply multiplying its length by the energy of the two dimensional vortex[33], [34]. In most of the literature on the subject of classical solutions lifted into higher dimensions, the energy dependance on the shape of the solitons has been neglected. However, as observed previously in the case of the kink, the curvature of the soliton affects the energy in a highly nontrivial fashion. One ultimate goal of this investigations is to determine the curvature dependance of vortex solutions. However, unlike in the case of the kink, no analytical vortex solutions exist, making the study of curved vortices a very difficult task. The effects of curvature can be studied easiest on circular vortices. In this chapter I will

45 investigate circular vortex solutions of large radius of the U(1) gauge theory. How- ever, before solving the equations of motion of the circular vortex, ﬁrst it is necessary to study and obtain a complete straight vortex solution. This is necessary since the straight vortex solution appears in the ﬁeld equations of the circular vortex. In the next section I shall describe the development of the straight Abelian vortex solution, and subsequently, develop the circular-vortex. Circular vortices are not stable if they are allowed to shrink. Therefore, I will investigate vortices the core of which is pinned down along a circle.

It may seem logical to consider the fluctuations of the extended vortex solutions from its straight line minimum is purely quantum mechanical. Unstable configurations are important in a quantum field theory as they contribute to partition functions when one integrates over all configurations. However, integrations over small fluctuations around classical solutions, such as the straight vortex would not include curved vortices. i.e. in order to capture the macroscopic features of extended classical solutions one needs to study the classical equations of motion.

3.2 The Straight Vortex

Consider a (2+1) dimensional U(1) gauge theory with a complex scalar Higgs (Φ).

For a static solution the Lagrangian corresponds to the negative of the Hamiltonian,

46 given by,

1 1 H = d2x (∂ A ∂ A )2 + (∂ + ıeA ) Φ 2 + λ Φ 2 η2 2 , (3.1) 2 2 ν µ − µ ν | µ µ | | | − Z where λ is the scalar coupling constant and η is the Higgs vacuum expectation value.

Aµ is the U(1) gauge ﬁeld.

µ The corresponding equations of motion in the ∂ Aµ = 0 gauge take the form,

1 2A = ıe (Φ?∂ Φ Φ∂ Φ?) e2ΦA , (3.2) µ 2 µ − µ − µ 2Φ + 2ıeA ∂µΦ e2A AµΦ = λ Φ2 η2 . (3.3) µ − µ − Finding ﬁnite energy solutions of gauge theories is a daunting task, specially if stability of the solutions are to be investigated. However, this can be facilitated by employing symmetry considerations and topological conservations laws.

It can be seen from equation (3.1) that zero energy conﬁgurations must have

Φ(x, t) = η and hence in order to ensure ﬁnite energy solutions the ﬁelds should | | statisfy,

Φ η as x , and | | → → ∞ (∂ + ıeA ) Φ 0 as x . µ µ → → ∞

The phase of the complex scalar field Φ Φ eıα is not fixed by these boundary ≡ | | conditions at infinity. Since the phase angle α takes values on a circle, S1, the boundary conditions map the surface at spatial infinity onto this circle. Now, consider the

47 following results on homotopy groups.

πn(Sn) = Z (3.4)

πn(Sm) = 0 for n < m, (3.5)

and πn(S1) = 0 for n > 1, (3.6)

where πn(Sm) refers to the homotopy group for the mapping of Sn into Sm, and Z refers to the group of integers. i.e., according to (3.4) the mapping of S S come n → n in a discrete inﬁnity of homotopy classes, each characterized by an integer. The zero on the right hand side of (3.5) and (3.6) implies that the group is trivial, i.e. in those cases all mappings can be deformed into one another. These topological relationships are important when investigating classical solutions [41], because theories that contain nontrivial homotopy classes of mappings are guaranteed to have at least one stable

ﬁnite energy solution in each of the homotopy classes.

In the Abelian model the boundary conditions at requires the symmetry group ∞ of the target space to be S1. Consider the spatial dimension to be D = 3, then the surface at inﬁnity is a sphere S2. However, (3.6) tells us that π2(S1) = 0, and therefore there are no non-trivial homotopy sectors of (3+1) dimensional solutions for this model. For the abelian model this is true for all D 3 since πD 1(S1) = 0 ≥ − ∀ D 3. However, when D = 2 the surface at inﬁnity becomes a circle, giving rise to ≥ mappings of circles into circles which leads to the non-trivial homotopy group π1(S1).

This is the group of winding numbers. Therefore, in this theory stable ﬁnite energy

48 solitons will be found only in D = 2. These solutions can be characterized by the winding number which is a conserved quantity. However, consider stable solutions in (3+1) dimensions where the solution is independent of the third direction. In this case, unless the third dimension is compact, the energy of the classical solution becomes inﬁnite, only the energy per unit length remains ﬁnite.

Given that the solutions are two dimensional, the z component of the vector potential can be chosen to be zero. Since only time independent solutions are of interest, I can choose the A0 = 0 gauge. Also note that the winding number describes the number of times the phase α of the Higgs is wound around the circle at inﬁnity,

α(2π) α(0) − n = 2π . Therefore, we may seek nontrivial classical solutions of the form,

a(ρ) A (ρ, ψ) = a(ρ)∂ ψ = ψˆ, (3.7) µ µ ρ Φ(ρ, ψ) = ηeınψχ(ρ), (3.8)

The solution only depends on the polar coordinates (ρ, ψ) and is independent of z. Now, integrating over ψ the Hamiltonian (3.1) can easily be rewritten in terms of the two ﬁelds a(ρ) and f(ρ) as,

1 da 2 η2 dχ 2 H = π dρρ + (ea + n)2 χ2 + η2 + λη4 χ2 1 2 (3.9) ρ2 dρ ρ2 dρ − Z " # The appropriate boundary conditions for ﬁnite-energy vortex solutions are

χ(0) = a(0) = 0,

lim χ(ρ) = 1, and lim a(ρ) = n/e. (3.10) ρ ρ →∞ →∞ −

49 These conditions ensure that each one of the terms of the integrand of (3.9) is ﬁnite.

For these cylindrically symmetric solutions, equations of motion (3.2, 3.3) take the

form,

d2a 1 da eη2 (ea + n) χ2 = 0 and dρ2 − ρ dρ − (3.11) d2χ 1 dχ χ + (ea + n)2 2λη4χ χ2 1 = 0. dρ2 ρ dρ − ρ2 − − Given the boundary conditions (3.10), the ﬂux Ψ of the vortex line can be derived,

a(ρ) 2πn Ψ = A (x)dxµ = dψρ = . µ ρ e I I

2π Thus, the ﬂux of the vortex line is quantized, e being the quantum.

The ﬁeld equations (3.11) can only be solved numerically, and these calculations provide monotonic solutions for both a(ρ) and χ(ρ), as shown in ﬁgure(3.1). However, a lower bound for the energy can be found by adopting the Bogomol’nyi linearization technique [43]. Assuming that n > 0, the Hamiltonian can be written as,

1 da 2 dχ (ea + n) χ 2 H = π dρρ + γeη2 χ2 1 + η2 + 2γ (3.12) ρ dρ − dρ ρ Z "

2 2 2 2 2 2 4 2 2 (ea + n) η χ (1 4γ ) 1 2 d 2 + λ γ e η χ 1 + 2 − 2γη (ea + n) χ 1 − − ρ − ρ dρ − # √λ where the parameter γ > 0. It can be observed that by choosing γ < 1/2 and γ < e

fourth and third terms in (3.12) become positive. Also, the last term can be integrated

50 and using the boundary conditions one obtain,

H = 2πγη2n + non negative terms

This shows that the energy of the soliton is greater than 2πγη2n. Using the limits imposed on γ, the lower bound of the hamiltonian will be the smaller number of either

H πη2n or ≥ H 2π√λη2n/e. (3.13) ≥

The limiting case is where the Higgs mass mH = 2η√2λ becomes identical to the

√ λ 1 gauge mass, mA = 2eη. i.e. when e2 = 4 . At this critical coupling limit the vortex- vortex forces vanish. Below this point the vortices attract each other and above it they repel each other. The Hamiltonian in the critical coupling case reduces to

1 da 1 2 dχ (ea + n) χ 2 H = πη2n + π dρρ + eη2 χ2 1 + η2 + ρ dρ 2 − dρ ρ Z " # which yields an exact value for the energy of the vortex,

H = πη2 n (3.14) | | and the ﬁelds can be shown to satisfy linearized Bogoml’ny type equations,

dχ χ = (n + ea) , and dρ − ρ da η2eρ = 1 χ2 dρ 2 −

51 3.2.1 Solutions of the straight Vortex

I am interested in determining the solutions of the coupled system of equations (3.11).

However, as mentioned previously, solving these nonlinear coupled second order dif-

ferential equations necessitate the use of numerical techniques. Begin by going over

to dimensionless variables

r ρ = , and eη (n + ea) = α. (3.15)

λ Now, by deﬁning β = e2 the hamiltonian (3.9) can be rewritten using the new vari-

ables,

2 2 1 dα dχ 1 2 H = πη2 drr + + α2χ2 + β χ2 1 . (3.16) r2 dr dr r2 − Z " # The equations of motion (3.11) of the straight vortex become,

d2α 1 dα αχ2 = 0 dr2 − r dr − d2χ 1 dχ 1 + α2χ 2βχ χ2 1 = 0, (3.17) dr2 r dr − r2 − − with the boundary conditions (3.10) transforming to,

χ(0) = 0 and α(0) = n,

lim χ(r) = 1, and lim α(r) = 0 (3.18) r r →∞ →∞

The numerical technique used to solve the ﬁeld equations (3.17) is the fourth order

Runge-Kutta method with an adaptive step size [42]. First, it is necessary to reduce

52 the second-order diﬀerential equation (3.17) to a set of ﬁrst-order ODEs,

dα = v(x), dr dv v(r) = + e2α(r)χ(r), dr r dχ = w(r), and dr dw w(x) α2(r)χ(x) = 2βχ(r) χ2(r) 1 dr − r − r2 − − where two new variables v(x) and w(x) have been introduced. The problem of solving

the field equations is thus reduced to studying a set of 4 first order ordinary differential

equations for the function yi, i = 1, ..., 4, having the general form

dy (r) i = z (r, y , y , y , y ), i=1,2,3,4 dr i 1 2 3 4 where the functions zi are known. The Runge-Kutta method uses information from previous steps as well as the values of the functions zi to propagate the solutions (yi)

over an interval from r r , where the step size is h = r r . Each stage in n → n+1 n+1 − n the fourth order Runge-Kutta method requires the following calculations,

k k k k y = y + 1 + 2 + 3 + 4 + O(h5), n+1 n 6 3 3 6 where,

k1 = hz(ρn, yn) h k k = hz ρ + , y + 1 2 n 2 n 2 h k k = hz ρ + , y + 2 3 n 2 n 2

k1 = hz(ρn + h, yn + h)

53 Now, the solutions (yi) can be propagated starting at r0, i.e. at r = 0 where the

boundary conditions gives the values of ﬁelds at the starting point. However, given

the nature of the diﬀerential equations and the boundary conditions this process has

two complications;

1. Boundary conditions (at r = 0) are only known for two of the ﬁelds, α(r) and

dα dχ χ(r). i.e. dr r=0 = v(0) and dr r=0 = w(0) are not known.

2. Two of the functions zi are singular at ρ = 0, and therefore the simulation will

diverge near this point.

The second issue can be addressed by displacing the starting point of the Runge-Kutta method by a small amount to r = , so that the singular nature of the equations at

r = 0 does not aﬀect the simulation. However, this trick will only work if the values

of the ﬁelds at the new starting point can be determined, i.e. I need to resort to a

diﬀerent technique to propagate the solution from r = 0 . This is done using a → power series method. Taylor series expansion of the ﬁelds together with the boundary

conditions at r = 0 can be used to analytically solve the ﬁeld equations (3.17). This

technique generates expressions for the ﬁelds α(r) and χ(r) in powers of r.

However, note that these solutions are only valid for very small values of r, and

therefore, has to be kept very close to zero. Also, the derivatives of the two ﬁelds at

r = 0, α0(0) and χ0(0), appear in the solutions as two unknown parameters, a point

54 previously mentioned as one of the complication that arise when using the Runge-

Kutta method. This issue can be overcome by making use of the the fact that the form of the ﬁelds are known at large r, i.e. use the boundary conditions at r , → ∞ (3.18). I use a parameter ﬁtting algorithm to pin down the two unknown boundary values at r = 0 that would produce the desired large r behavior.

3.2.2 Results

I solved the field equations numerically using the technique mentioned previously for the case where the winding number n = 1. The resulting field configurations of the

λ straight vortex for diﬀerent values of the coupling constant β = e2 are shown in ﬁgure

(3.1).

Once the fields configurations are known I can calculate the energy of the straight vortices by evaluating the hamiltonian (3.16). The numerical integration is done using the adaptive Gaussian quadrature method. The values of the Hamiltonian calculated for each case is given in the graphs. Figure (3.1a) shows the field configurations at critical coupling β = 1/4, and the Hamiltonian yields

H = πη2 (1.0000027) ×

As stated previously in (3.14) the Hamiltonian for the critical coupling can be determined analytically to be exactly H = πη2 n . Given that n = 1, the analytical value | | of the hamiltonian is H = πη2(1), and the numerical evaluation is in agreement to

6 decimal places. It can also be observed that the energy calculation of the remain-

55 HaL b=bc=1ê4

1 cHrL 0.8

0.6 H=ph2 äH1.0000027L 0.4

0.2 aHrL

r 2 4 6 8 10 12 14

HcL b=2bc HbL b=3bc

1 1 cHrL cHrL 0.8 0.8

0.6 0.6 2 H=ph2 äH1.260910L H=ph äH1.156767L 0.4 0.4 aHrL 0.2 aHrL 0.2

r r 2 4 6 8 10 12 14 2 4 6 8 10 12 14

b bc HdL b= ÅÅÅÅcÅÅÅ HeL b= ÅÅÅÅÅÅÅ 2 3

1 1 cHrL 0.8 0.8 cHrL

0.6 0.6 H=ph2 äH0.867911L H=ph2 äH0.800865L 0.4 0.4

0.2 aHrL 0.2 aHrL

r r 2 4 6 8 10 12 14 2 4 6 8 10 12 14

Figure 3.1: Plot of the straight vortex field configurations for different values of the coupling constant. (a) β = βc = 1/4, critical coupling. (b) β = 3βc. (c) β = 2βc. (d) β = βc/2. (e)

β = βc/3.

56 ing cases agrees with the lower bounds stated in (3.13). These agreements give us conﬁdence in the numerical method we employ for the solutions.

Next I turn to the circular vortex. The ﬁeld conﬁgurations of the straight vortex

developed previously will be used in the numerical treatment of the circular vortex

solutions.

3.3 The Circular Vortex

I start with the Hamiltonian of the U(1) model in 3 dimensions.

1 1 H = d3x (∂ A ∂ A )2 + (∂ + ieA ) Φ 2 + λ Φ 2 η2 2 (3.19) 2 2 ν µ − µ ν | µ µ | | | − Z Going over to dimensionless variables

x Φ ηΦ, A ηA, and x → → → eη

Since I am only interested in time independent solutions, the Hamiltonian takes the

form,

η2 (∂ A ∂ A )2 ∂Φ 2 H = d3x i j − j i + + iA Φ + β( Φ 2 1)2 , (3.20) 2 2 ∂x i | | − Z " i #

where I have introduced β = λ/e2. A toroidal coordinate system can be used to studt

this system. In other words, measure the distance ρ from a circle of radius R in the xy plane, the angle φ is the inclination out of the xy plane measured from the circle,

and the angle ψ is the angle of the projection in the xy plane on to the x axis. Then

57 the coordinate transformation from the Cartesian one is

x = cos ψ(R + ρ cos φ),

y = sin ψ(R + ρ cos φ),

z = ρ sin φ. (3.21)

The metric tensor takes the form,

cos ψ cos φ cos ψρ sin φ sin ψ(R + ρ cos φ)  − −  g = . (3.22)  sin ψ cos φ sin ψρ sin φ cos ψ(R + ρ cos φ)   −       sin φ ρ cos φ 0      This implies that the Jacobian is

det g = ρ(R + ρ cos φ). (3.23) −

Given the form of the determinant, the measure is

2π 2π ∞ d3x dψ dφ dρ ρ(R + ρ cos φ)Θ(R + ρ cos φ), (3.24) → Z Z0 Z0 Z0

Since the action is independent of ψ, the integration can be performed. The leading term of the integral will be of the form

2πR ρ dρdφ... Z which is just the two dimensional action multipled by the length of the circular vortex, 2πR. The purpose of the subsequent calculations is the extraction of the leading correction terms of total energy dependent on the curvature radius, R.

58 The Θ-function under the integral is irrelevant to any ﬁnite power correction in

R to the straight vortex contribution, as the the integrand (powers of the vortex ﬁeld

and its derivatives) decreases exponentially away from the circle where the vortex is pinned down. I use the following ansatz for the gauge field and the higgs field. The ansatz for the gauge field is

1 A (x) = φˆ a(ρ, φ), (3.25) µ µ ρ

ˆ where φµ is the unit vector in the azimuthal direction,

ˆ φµ = ρ∂µφ.

The ansatz for the Higgs ﬁeld is

Φ(x) = ηeinφf(ρ, φ), (3.26)

where the function f is real. The gauge was ﬁxed such that the phase of the higgs

ﬁeld is exactly nφ.

Now I calculate the action using our ansatz. First of all, I need to calculate ∂ ∂ µ · µ in toroidal coordinates. This becomes equivalent to performing the calculation in two

dimensional polar coordinates as there is no dependence on angle ψ.

(∂ A ∂ A )2 (∂ φ∂ ρ ∂ φ∂ ρ)2 ∂a 2 1 ∂a 2 µ ν − ν µ = µ ν − ν µ = . (3.27) 2 2 ∂ρ ρ2 ∂ρ

The kinetic term for the higgs ﬁeld is the following

∂Φ 2 1 ∂f 2 ∂f 2 + iA Φ = (a + n)2f 2 + + . (3.28) ∂x i ρ2 ∂ρ ∂φ i

59 Thus, the complete action, using this ansatz, takes the form

2π 2 2 2 ∞ ρ 1 ∂a ∂f 2 2 H = πη R dφ dρ ρ 1 + cos φ 2 + + (a n) f 0 0 R (ρ " ∂ρ ∂φ − # Z Z ∂f 2 + + β(f 2 1)2 (3.29) ∂ρ − )

This action diﬀers from the one for the straight vortex in the driving correction

term in the measure and in the term containing the derivative of the Higgs ﬁeld with

respect to the asimuthal angle. Again, it is easy to see, just like for the d-dimensional

1 1 kink, that to second order in R− only the leading correction, of O(R− ), to the Higgs

and gauge ﬁelds contributes. Thus, these ﬁelds can be written as

1 a = a0(ρ) + a1(ρ, φ), R 1 f = f 0(ρ) + f 1(ρ, φ). (3.30) R

The correction term contributes in the form

π H = H + H = H + (A + A ), (3.31) 0 1 0 R 1 2 where

2π 1 2 1 2 ∞ 1 ∂a ∂f A = dφ dρ ρ + + (a0 + n)2(f 1)2 + (a1)2(f 0)2 1 ρ2 ∂ρ ∂φ Z0 Z0 ( " ∂f 1 2 + 4(a0 + n)a1f 0f 1 + + β(f 1)2[3(f 0)2 1] , ∂ρ − ) 2π 1 0 ∞ ∂a ∂a A = 2 dφ dρ cos φ + (a0 + n)a1(f 0)2 + (a0 + n)2f 0f 1 2 ∂ρ ∂ρ Z0 Z0 ∂f 1 ∂f 0 + ρ2 + ρ2βf 1f 0[(f 0)2 1] (3.32) ∂ρ ∂ρ −

60 Clearly the variation gives inhomogeneous equations for the two unknown functions, a1 and f 1, where the inhomogeneity is proportional to cos φ. Then obviously, the

solution is such that both of these functions depend on φ such that

a1 = cos φ b(ρ),

f 1 = cos φ g(ρ). (3.33)

In terms of b and g the correction term to the hamiltonian is (using the notation

a0 a, f 0 χ to denote the straight vortex solutions) → →

2 π ∞ 1 2 2 2 2 2 2 H = ... + dρ [b0 + g + (a + n) g + b χ + 4(a + n)bχg] R ρ Z0 2 2 2 2 + ρg0 + 2βρg (3χ 1) + 2a0b0 + 2(a + n)bχ −

2 2 2 2 + 2(a + n) χg + 2ρ χ0g0 + 4ρ βχg(χ 1) . (3.34) −

The boundary conditions that ensure ﬁnite-energy solutions are

b(0) = g(0) = χ(0) = a(0) = 0,

lim b(ρ) = 0, lim g(ρ) = 0, lim χ(ρ) = 1, and lim a(ρ) = n (3.35) ρ ρ ρ ρ →∞ →∞ →∞ →∞ −

Now the equations of motion for b and g can be determined.

1 2 2 b00 b0 bχ 2(a + n)χg = ρa00 + ρ(a + n)χ , − ρ − − −

1 1 2 2 g00 + g0 [g + (a + n) g + 2(a + n)bχ] 2βg(3χ 1) ρ − ρ2 − −

1 2 1 d 2 2 2 = (a + n) χ (ρ χ0) + 2ρ βχ(χ 1). (3.36) ρ − ρ dρ −

61 Using the straight vortex ﬁeld equations (3.11) for a and χ the right hand side of

(3.36) can be simpliﬁed to get

1 2 b00 b0 bχ 2(a + n)χg = a0, − ρ − − −

1 1 2 2 g00 + g0 [g + (a + n) g + 2(a + n)bχ] 2βg(3χ 1) = χ0. (3.37) ρ − ρ2 − − −

3.4 Numerical Treatment

I begin as I did in section(3.2.1) by redeﬁning the straight vortex ﬁeld

(n + a) = α.

The correction term H1 to the hamiltonian take the form,

2 π ∞ 1 2 2 2 2 2 2 H = dρ [b0 + g + α g + b χ + 4αbχg] 1 R ρ Z0 2 2 2 2 + ρg0 + 2λρg (3χ 1) + 2a0b0 + 2αbχ −

2 2 2 2 + 2α χg + 2ρ χ0g0 + 4ρ λχg(χ 1) , (3.38) − and the ﬁeld equations change to

1 2 b00 b0 bχ 2αχg = α0, − ρ − − −

1 1 2 2 g00 + g0 [g + α g + 2αbχ] 2βg(3χ 1) = χ0, (3.39) ρ − ρ2 − − −

The boundary conditions for α can be stated as

α(0) = n and lim α(ρ) = 0. ρ →∞

62 Compared to the equations of motion of the straight vortex (3.17), the above system of equations for b and g are more complex. Accordingly, the numerical techniques that is used to solve (3.39) diﬀer from that employed previously. Next, I will describe the two techniques used to solve the ﬁeld equations of the circular vortex.

3.4.1 Variational Method

I start by using a variational method to obtain approximate solutions for the ﬁelds b and g that minimize the hamiltonian (3.38). Subsequently, I will use these approximate forms of the ﬁelds as the initial ”guess” in an iterative technique to solve the

ﬁeld equations (3.39). As indicated by the boundary conditions (3.35) both b an g tend to zero as ρ 0 and when ρ . The two trial functions given below has all → → ∞ these features.

2 γρ δρ b(ρ) = ρ b1e− + b2e−

σρ τρ g(ρ) = ρ g1e− + g2e− (3.40) Once these leading order approximations of the ﬁelds are substituted in the hamiltonian (3.34) I search for parameter values that will minimize the hamiltonian. This minimum provides an upper bound to the energy. A plot of the approximate solution of the circular vortex for the critical coupling case (β = βc = 1/4) is shown in ﬁgure

(3.2). The next step is to obtain a more exact solutions by using the approximate forms of the two ﬁelds b and g as a starting point.

63 b=bc=1ê4

0.3 p2 0.2 gHrL H= ÅÅÅÅÅÅÅ äH-0.33430L R 0.1

r 2 4 6 8 10 12 -0.1

-0.2 bHrL -0.3

Figure 3.2: Plot of the variation method approximate circular vortex ﬁeld conﬁgurations for critical coupling, β = βc = 1/4

3.4.2 Jacobian Iteration Method

In contrast to the previous technique of obtaining the circular vortex ﬁelds by minimizing the Hamiltonian, an iterative method is employed to solve the ﬁeld equations

(3.39). For the purpose of describing the technique, suppose that the task at hand is to solve for Φ given the following system of equations;

D[Φ] = 0, (3.41)

Φ1    Φ2  where D (d ), and Φ   ≡ ij ≡    .   .         ΦN      Here I have used the notation Φ Φ(ρ ), where the discretization of ρ deﬁnes the n ≡ n

ρ ρ1 grid size to be h = N − . D is a non-singular N N matrix. General iterative N × techniques proceed from some initial guess ”guess” Φ(0), and deﬁne a sequence of successive approximations Φ(1), Φ(2), .... which, in principle converge to the exact

64 solution. For any N N matrix Q - called the splitting matrix- I can rewrite (3.42) × in the equivalent form

QΦ + (D Q) Φ = 0. (3.42) −

The only condition imposed on Q is that it is non-singular. Then, equation (3.42) can be expressed as

1 Φ = (I Q− D)Φ. (3.43) −

Now, one can search for an iterative solution of the form

(k+1) k 1 k Φ = Φ Q− DΦ , (3.44) − starting from an initial conﬁguration Φ(0). There is a general theorem [44] which

1 states that given the norm I Q− D < 1, the sequence produced by equation k − k (3.44) will converge to the solution of D[Φ] = 0 for any initial vector Φ(0). I use this technique to solve the ﬁeld equations (3.39) of the circular vortex. The approximate solutions obtained previously provide the initial ”guess” of the iterative process. For the splitting matrix Q I use a diagonal matrix with the diagonal entries of form

ρi/m (qii) = α(e− ). The two parameters α and m can be varied to optimize the rate of convergence.

3.5 Results

I numerically solve the field equations using the techniques mentioned previously for the case where the winding number n = 1. The resulting field configurations of

65 λ the circular vortex for diﬀerent values of the coupling constant β = e2 are shown in

figure (3.3). Once the field configurations are determined the leading order correction

HaL b=bc=1ê4

0.3 p2 0.2 gHrL H= ÅÅÅÅÅÅÅ äH-0.33588L R 0.1

r 2 4 6 8 10 12 -0.1

-0.2 bHrL -0.3

HbL b = 2bc HbL b=3bc 0.3 0.3 p2 p2 0.2 gHrL H= ÅÅÅÅÅÅÅ äH-0.24309L 0.2 gHrL H= ÅÅÅÅÅÅÅ äH-0.20554L R R 0.1 0.1

r r 2 4 6 8 10 12 2 4 6 8 10 12 -0.1 -0.1

-0.2 -0.2 bHrL bHrL -0.3 -0.3

bc b HdL b= ÅÅÅÅÅÅÅ HeL b= ÅÅÅÅcÅÅÅ 2 3

0.4 0.4 gHrL gHrL p2 p2 0.2 H= ÅÅÅÅÅÅÅ äH-0.48179L 0.2 H= ÅÅÅÅÅÅÅ äH-0.602298L R R

r r 2 4 6 8 10 12 2 4 6 8 10 12

-0.2 bHrL -0.2 bHrL

-0.4 -0.4

Figure 3.3: Plot of the circular vortex field configurations for different values of the coupling constant. (a) β = βc = 1/4, critical coupling. (b) β = 3βc. (c) β = 2βc. (d) β = βc/2. (e)

β = βc/3. to the energy of the vortex due to the curvature can be evaluated by evaluating the hamiltonian H1 (3.38). The values of this correction term to the Hamiltonian calculated for the diﬀerent values of β are given in the graphs (3.3). These results

66 show that the curvature energy is negative. This clearly demonstrate that the energy of the (3+1) dimensional circular vortex is smaller than the naive evaluation of energy by multiplying the (2+1) dimensional vortex energy by its length. Thus, the Abelian vortex prefers the curved state to the straight one.

67 Chapter 4

Conclusions

It has been known for quite sometime that interacting field theories often have classical solutions such as, Solitons and Instantons. These topological defects in field theories are crucial in describing a wide variety of phenomena [36]. For an example, in superconducting materials magnetic-field lines (which are normally excluded from the medium by Meissner effect) can become trapped in quantized flux tubes known as vortices [3]. In grand unified theories (GUTs) of elementary-particle physics the spontaneous breaking of a symmetry can lead to the formation of line vortices, called cosmic strings [4]. Cosmic strings may have played an important role in the formation of large-scale structure in the early universe. Recently, classical solutions of string theories referred to as p-branes have attracted a lot attention. A p-brane is an extended object in space with p spatial dimensions. These objects play a crucial role in the duality scenarios that unify all the known superstring theories. Classical solutions also play and important role in dynamical models of quark confinement.

68 4.1 Classical Objects and Conﬁnement

The main motivation behind this study of classical solutions is confinement. Quark confinement is now an old and familiar idea, routinely incorporated into the standard model and all its proposed extensions. However, familiarity is not the same as understanding the phenomena. Despite efforts stretching over three decades there exists no analytical derivation of confinement. It is fair to say that no theory of confinement is generally accepted. However, recent lattice QCD studies makes a fairly convincing case in support of classical objects being responsible for confinement.

4.1.1 ’t Hooft’s dual superconductivity model

One of the most widely accepted models of conﬁnement is due to ’t Hooft [7, 8, 6]. The

BCS theory of superconductivity relies on phonon mediated electron-electron pairs

(Cooper pairs). At low temperatures, these charge 2e, spin zero bosons undergo Bose-

Einstein condensation, and the condensate has inﬁnite conductivity. Such a medium

cannot sustain long range electric ﬁelds , and so the photons become massive due

to screening (a.k.a. the Higgs mechanism in particle physics). Now, if a magnetic

ﬁeld is applied the magnetic ﬁeld is either expelled from the medium or squeezed

into thin vortices that are not superconducting. If magnetic monopoles exist then

the ﬂux between a monopole-antimonopole pair would be squeezed into a thin ﬂux

(called Abrikosov ﬂux tubes) which gives rise to a linear magnetic potential between

the monopoles. Then the monopoles are conﬁned in pairs.

69 ’t Hooft’s model of conﬁnement gives a dual picture where the roles of the chromo-

magnetic and the chromo-electric ﬁelds are reversed. It supposes the QCD vacuum to

be a condensate of monopoles, then the electric ﬁeld would become screened, gluons

would gain mass, and the electric ﬁeld would be expelled from this ’superconducting

medium. Then the chromo-electric field produced by a pair of charges QQ¯ is channeled by dual Meissner effect into Abrikosov flux tubes in the same way as magnetic field

is conﬁned in type II superconductors. The energy is proportional to the distance

E = σR, where σ is the string tension. This means conﬁnement.

This picture has two problems that have to be dealt with if it is to become a viable

theory of conﬁnement. Firstly, in order for monopoles to exist in non-abelian systems

one needs to break the gauge symmetry of the theory to the Cartan subgroup. The

symmetry is then reduced to an appropriate electro-magnetic ﬁeld needed for the

interpretation of monopoles. The second problem is that there are no monoples in pure non-abelian gauge theories. In other words, where are the monopoles in QCD?

4.1.2 Abelian Projection

’t Hooft suggested the following idea [6, 9]. Suppose QCD monopoles carry charges

N 1 that are magnetic with respect to the [U(1)] − Cartan subgroup of color SU(N).

Then SU(N) gauge symmetry obscures the magnetic charges and it is necessary to

N 1 gauge ﬁx at least the SU(N)/[U(1)] − symmetry to expose them. The abelian

gauge transformation ﬁxes the non-abelian part of SU(N) and the leftover abelian

70 gauge group remains an exact symmetry. However, since the original SU(N) gauge symmetry group is compact, the remaining abelian gauge group is alos compact, and it is know that compact gauge symmetry groups contain abelian monoples. Therefore

SU(N) gauge theory in the abelian gauge has monopoles.

The main conjecture of ’t Hooft’s dual superconductor model is that after abelian transformation the remaining abelian gauge freedom together with the appearance of monopoles is alone responsible for confinement. Therefore, after the gauge fixing one can perform a so called Abelian projection to discard of all the component in the gauge field but the abelian ones and still have a confining theory, with the same string tension as the original theory.

However, finding monopoles in QCD is not the end of the story. To solve the problem of confinement one needs to show that the monopoles condense in the ground state of the non-abelain gauge theory. The nonperturbative nature of this task requires calculations that were thought to be close to impossible until it was relaized that the relevant numerical calculations are feasible in lattice QCD [10]. Yet, there was no elementary or otherwise natural way to pick the appropriate abelian projection from the infinite number of possibilities. There is compelling evidence in support of the maximal Abelian (MA) gauge. This is the abelian gauge in which the diagonal components of the gauge fields are maximized by gauge transformations. In this gauge, physical information of the gauge configuration is concentrated in the diagonal components. Lattice studies of theories in MA gauge give strong indication of abelian

71 dominance, which means that the expectation values of physical parameters (such as string tension) in he non-abelian theory coincides with the corresponding values from the abelian projected theory [12, 13, 17]. There is also evidence that in the MA gauge the monopoles are responsible for most of the string tension [17, 16], a feature referred to as monopole dominance. The presence of these features is considered to be strong evidence in favor of the dual superconductor model of conﬁnement.

4.1.3 Center Vortex Model of Conﬁnement

The most popular model of colour conﬁnement in QCD relies on the idea of dual superconductivity, due to ’t Hooft. A realization of the idea is the abelian-projection theory of ’t Hooft [1]: he suggested to ﬁx to an abelian projection gauge, reducing

N 1 the SU(N) gauge symmetry to U(1) − , and identifying abelian gauge ﬁelds (with respect to the residual symmetry) and magnetic monopoles.

The abelian projection theory is not the only proposal for explaining the conﬁn- ing force; there have been many other suggestions over the years. One idea that was

first introduced in the late 1970s was the Vortex Condensation theory, put forward in various forms, by ’t Hooft[21], Mack[23], and by Nielsen and Olesen[20]. The idea is that the QCD vacuum is filled with closed magnetic vortices, which have the topol- ogy of tubes (in 3 spacial dimensions) of finite thickness, and which carry magnetic

ﬂux in the center of the gauge group (hence center vortices). In the center vortex model of SU(N) gauge theories, the gauge ﬁxing condition is chosen to reduce the full

72 SU(N) gauge symmetry to the discrete center group, ZN . The is referred to as center

gauge fixing. One starts by fixing to the maximal abelian gauge. As described in the previous section this gauge has the effect of making the link variables as diagonal as

N 1 possible, leaving [U(1)] − gauge symmetry. Abelian projection means the replace- ment of the full link variables by the abelian links. But while dual superconductor

N 1 idea focuses on the remnant [U(1)] − subgroup of gauge symmetry, it is the ZN center of SU(N) guage group that is most relevant in the vortex condensation picture.

This suggests making a further gauge ﬁxing, which would bring the abelian links as close as possible to the center elements. Then the Center Projection is carried out by replacing the full link variables by the center element.

Now one can use the center projected theory to do similar measurement as one did in the case of the Abelian projection case. The dependence of average Wilson loops on the area of the loop can be measured and these calculations indicate that Wilson loops follow the area law, which means conﬁnement. The numerical value of the string tension agrees even better than in the case of the abelian projected theroy[26][28].

This is referred to as Center Dominance. These results seem to support the view that

the center vortices are responsible for conﬁnement.

As illustrated in the previous two sections the recent lattice data support both

the vortex condensation theory as well as the the abelian projection theory (dual-

superconductivity theory). The question is which one of these seemingly mutually

exclusive propositions is the correct one? Recent work [25] indicate that center vortex

73 conﬁguration, transformed to maximal abelian gauge and then abelian projected, will appear as a chain of monopoles alternating with antimonopoles. These monopoles are essentially artifacts of the projection and they are condensed because the long vortices from which they emerge are condensed. Therefore, the vortex picture is currently accepted to be more fundamental than the monopole picture.

My investigation of the properties of extended classical solutions was mainly motivated by these recent successes that classical objects have had in explaining con-

ﬁnement. I was primarily interested in determining the curvature dependence of the energy, so that the energy of extended vortex solutions can be deﬁned accurately.

Such studies are highly relevant especially when investigating properties such as interactions and condensation of these vortices. Although, the ultimate goal should be to study curved extended non-abelian vortices, it is an extremely diﬃcult task. I ﬁrst started with the extended undulating kink solution of the non-linear sigma model.

This work was carried out as an exercise preceding to the investigation of curvature dependence of extended vortex solutions. However, such studies of vortices are diﬃ- cult since no analytic vortex solutions exist. This diﬃculty can be reduced by looking at circular vortices. Therefore, I next investigated a somewhat simpler problem - the circular vortex of the U(1) gauge theory coupled to a scalar Higgs. The results of these studies are described in chapter 3 of thisis.

74 4.2 The Undulating Kink

In most applications that deal with finite energy classical solutions lifted into higher dimensions, it has been naively assumed that the energy depends only on the total length of the object and the curvature effects have been neglected. In this investigations the energy dependence on the shape of the object was investigated. This dependence is of great importance, because to study the interaction and the condensation of these classical objects the energy of a single object has be correctly defined

In the trivially extended classical solutions the location of the center or the core of the object forms a straight line. Now if the Hamiltonian is minimized with the constraint that the locus of zeroes (cores) lies on a predetermined curve, then the

energy will depend on the shape as well as the length of the locus of zeros. It may seem logical that these ﬂuctuations of the extended object from its straight line minimum

energy conﬁguration is a purely quantum mechanical eﬀect. However, in order to describe the macroscopic features of extended classical objects (such as vortices) one needs to study the classical equations of motion. If the energy of curved vortices can be calculated then their full quantum contribution, can also be evaluated, at least in perturbation theory. Although, the ultimate goal is to investigate curved vortices, I began by studying the curvature energy of a much simpler problem, the kink solution of the (2+1) dimensional linear sigma model.

75 Classical solutions can be classiﬁed by the dimension of the object. The kink of the (2+1) dimensional nonlinear sigma model(see equation 2.1) is a one dimensional solition (a particle like ﬁnite energy solution), and it is one of the simplest examples of classical solutions. The kink solutions (2.4)can be easily derived analytically and have been well studied[35]. When the non-linear sigma model is raised to three

dimensions the kink is a two dimensional object having inﬁnite energy in general. I

imposed periodic boundary condition in the direction of the extra dimension so that

the energy of the kink is ﬁnite. Then the trivially extended minimum energy kink

solution is a straight line on a strip of length L. In order to investigate the dependence

of the energy on the shape of the kink, I let the locus of the zeros of the kink solution

to be a closed curve that winds through the strip. The curvature energy (Hc) of the

two dimensional kink was then deﬁned as the subtraction of the length energy, Hl,

from the total energy of the kink, H. i.e. H = H H . The length energy is obtained c − l

by multiplying the energy of the one dimensional kink, H0 by it the total length of

the one dimensional kink.

I then used analytical techniques to prove that the curvature energy of the ex-

tended kink is negative. Consequently, the energy of the two dimensional extension

of the one dimensional kink solution is always smaller than the energy of the one

dimensional kink multiplied by the length of the two dimensional kink. We found

the exact solution of the constrained two dimensional ﬁeld equations using the ap-

proximation that the deviation of the shape of the kink from the from the straight

76 line is small. The low frequency contributions to this energy can be interpreted as a positive length term and a negative curvature term. Therefore, the kink prefer the curved state to the straight one.

4.3 The Circular Vortex

In chapter 2, I investigated properties of the kink, which is a (1+1) dimensional soliton. Next I looked at a another type of classical solution -the vortex. Vortices are

(2+1) dimensional ﬁnite energy soliton solutions that exist in theories where gauge

fields are coupled complex scalar Higgs fields. As mentioned previously the energy of vortex solutions of (3+1) dimensional gauge theories is in all most all instances determined by simply multiplying its length with the energy of the two dimensional vortex. However, as illustrated in the case of the kink that the curvature dependence of energy is highly nontrivial. Unlike in the case of the kink solution, no analytical vortex solutions exist, which complicates the task of studying curved vortices. The difficulty of studying curvature effects of vortex energies are somewhat simplified in the case of the circular vortex. In chapter 3, I studied the circular vortex of the U(1) gauge theory[38].

I ﬁrst started with the (2+1) dimensional U(1) gauge theory with a complex scalar

Higgs (see equations 3.1). Finding ﬁnite energy stable classical solutions of gauge theory is a prohibitively a diﬃcult task. However, these complexities can be overcome by making use of symmetry considerations and topological conservation laws (see equa-

77 tions 3.17). These equations were solved using numerical techniques (the fourth order

Runge-Kutta method). Subsequently the energy of the (2+1) dimensional vortex was calculated. This energy multiplied by the total length of vortex in the extended dimension gives the energy of the trivially extended vortex (the straight vortex). Also the (2+1) dimensional vortex conﬁgurations were used when deriving the circular vortex solutions.

The next step was to extend the theory to (3+1) dimensions (see equation 3.19)

in order to study circular vortices. However, circular vortices are not stable if they

are allowed to shrink. Therefore, I investigated vortices where the core of the vortex

is pinned down along a circle of radius R. The use of a toroidal system of coordinates

proved to be a very convenient choice for this problem. i.e. the distance in the xy

plane was measured from a ﬁxed circle of radius R, and two angles gave the projection

out of the plane and within the plane onto the x axis. Using similar anzats for the

Higgs ﬁeld and the gauge ﬁeld as in the case of the straight vortex, the leading

order correction terms to the energy was determined. This provided (to leading order

1 O(R− )) curvature term in the hamiltonian (see equation 3.34), and consequently,

the ﬁeld equations (see equations 3.37).

The task of solving this set of equations proved to be a much more complex

problem than the straight vortex. Solving equations of motion of the circular vortex

required a two step process. First a variational method was used to ﬁnd the ﬁeld

conﬁgurations that minimizes the Hamiltonian. These approximate solutions gave an

78 upper bound to the curvature energy of the vortex. Then I used an iterative technique

using the variational method ﬁeld conﬁgurations as the initial ’guess’ to determine

the exact solutions of the circular vortex ﬁeld equations (3.37).

The results shown in ﬁgure (3.3) prove that the curvature energy of the circular

vortex is negative. This clearly demonstrate that the total energy of the (3+1) dimen-

sional circular vortex is smaller than the energy calculated by multiplying the (2+1) dimensional vortex by its length in the extended dimension. Thus, I have shown that

the U(1) vortex prefers the curved state to the straight one.

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