Yang-Mills-Higgs Theory Zhu Da

Home , Joseph Polchinski

BIFURCATION AND TRANSITION PHENOMENA OF MULTIPLE CHARGED MONOPOLE PLUS HALF-MONOPOLE OF THE SU(2) YANG-MILLS-HIGGS THEORY

ZHU DAN

UNIVERSITI SAINS MALAYSIA 2019 BIFURCATION AND TRANSITION PHENOMENA OF MULTIPLE CHARGED MONOPOLE PLUS HALF-MONOPOLE OF THE SU(2) YANG-MILLS-HIGGS THEORY

ZHU DAN

Thesis submitted in fulﬁllment of the requirements for the degree of Master of Science

February 2019 ACKNOWLEDGEMENT

First and foremost, I wish to express my sincere gratitude to my supervisor, Dr. Wong Khai Ming. For without his patient teachings and contributive advices, this dissertation would most certainly not be completed. Secondly, I am very grateful for the fact that School of Physics and Theoretical Lab provide me with top-of-the-line apparatus for computational work. For without those, the data collection process of this undertaking would have been much longer. I also thank my senior, Mr. Timothy Tie Tong Bing, for giving me useful advices and helping me study the programming language used in many aspects of this thesis. Last bust not least, I beneﬁt much from my parents, they give me a lot of support for studying aboard, both ﬁnanically and spiritually. Their support made this all possible. A special acknowledgement to my loving wife. She took care of my parents while I was away studying aboard. She shared the burden with me without any complaint. I am nothing but grateful. Once again, I would like to thank everybody who played an important role in helping me make this research possible.

ii TABLE OF CONTENTS

ACKNOWLEDGEMENT ii

TABLE OF CONTENTS iii

LIST OF FIGURES vi

LIST OF TABLES viii

ABSTRAK ix

ABSTRACT xi

CHAPTER 1 - INTRODUCTION 1

1.1 A Brief History of Theoretical Physics 1 1.2 Standard Model, Achievements and Flaws 3 1.3 Beyond the Standard Model 5 1.3.1 The Grand Uniﬁed Theory 5 1.3.2 Supersymmetry (SUSY) 7 1.4 Magnetic Monopoles 9 1.5 Objective and Research Gap 11 1.6 Dissertation Outline 11

CHAPTER 2 - GAUGE FIELD THEORY 13

2.1 Introduction to Gauge Theory 13 2.2 Abelian Gauge Theory 14 2.2.1 Global Gauge Invariance 14 2.2.2 Local Gauge Invariance 16 2.3 Non-Abelian Gauge Theory of SU(2) 19 2.3.1 Global Gauge Invariance of SU(2) Yang-Mills Theory 20

iii 2.3.2 Local Gauge Invariance of SU(2) Yang-Mills Theory 23 2.4 Spontaneous Symmetry Breaking 26 2.5 Higgs Mechanism 28 2.6 SU(2) Yang-Mills-Higgs Model 32

CHAPTER 3 - LITERATURE REVIEW ON MAGNETIC MONOPOLES 39

3.1 Maxwell’s Equations 39 3.2 Dirac Monopole 41 3.3 Wu-Yang Formalism 45 3.4 ’t Hooft-Polyakov Monopole 48 3.5 Bogomol’nyi-Prasad-Sommerﬁeld (BPS) Solutions 53 3.6 Multimonopoles 55

CHAPTER 4 - CONSTRUCTION OF ONE-PLUS-HALF MONOPOLE SOLUTIONS 58

4.1 Introduction 58 4.2 Magnetic Ansatz 58 4.3 Higgs Field 59 4.4 Magnetic Field and Magnetic Charge 61 4.5 Magnetic Dipole Moment 63 4.6 Total Energy and Energy Density 64 4.7 Exact Asymptotic Solution 65 4.8 Numerical Construction of Solutions 68

CHAPTER 5 - ONE-PLUS-HALF MONOPOLE SOLUTIONS 72

5.1 Introduction 72 5.2 Numerical Results and Discussion 73 5.2.1 Case n = 2 73 5.2.2 Case n = 3 78 5.2.3 Case n = 4 88 5.3 Comment 100

iv CHAPTER 6 - CONCLUSIONS AND FUTURE RESEARCH 101

6.1 Conclusions 101 6.2 Future Research 103

REFERENCES 104

LIST OF PUBLICATIONS

v LIST OF FIGURES

Page Figure 3.1 Pictorial representation of the line and surface integrals appeared in equation (3.20). 44

Figure 3.2 Pictorial representation of the overlapping regions, R+ and R−, in Wu-Yang formalism. 46

Figure 5.1 The 3D surface and contour plots of the Higgs ﬁeld modulus |Φ|((a) and (b)), the weighted energy density E ((c) and (d)), and the weighted magnetic charge density M ((e) and (f)), for n = 2 FB when λ = 4. 74

Figure 5.2 Contour plot of the ’t Hooft magnetic field lines and vector field plots of the ’t Hooft magnetic field unit vector for n = 2 FB when λ = 4. 76

Figure 5.3 Plots of the pole separation (a), magnetic dipole moment (b) and total energy (c) versus λ 1/2 for n = 2 FB and the magnetic charges carried by Higgs and gauge ﬁelds (d) versusx ¯ when λ = 4. 77

Figure 5.4 The 3D surface and contour plots of the Higgs ﬁeld modulus |Φ|((a) and (b)), the weighted energy density E ((c) and (d)), and the weighted magnetic charge density M ((e) and (f)), for n = 3 FB when λ = 4. 80

Figure 5.5 Contour plot of the ’t Hooft magnetic field lines and vector field plots of the ’t Hooft magnetic field unit vector for n = 3 FB when λ = 4. 81

Figure 5.6 The 3D surface and contour plots of the Higgs ﬁeld modulus |Φ|((a) and (b)), the weighted energy density E ((c) and (d)), and the weighted magnetic charge density M ((e) and (f)), for n = 3 HEB when λ = 4. 82

Figure 5.7 The 3D surface and contour plots of the Higgs ﬁeld modulus |Φ|((a) and (b)), the weighted energy density E ((c) and (d)), and the weighted magnetic charge density M ((e) and (f)), for n = 3 LEB when λ = 4. 83

Figure 5.8 Contour plot of the ’t Hooft magnetic field lines and vector field plots of the ’t Hooft magnetic field unit vector for n = 3 HEB and LEB when λ = 4. 85

Figure 5.9 The 3D surface and contour plots of the Higgs ﬁeld modulus |Φ|, for n = 3 HEB when λ = 3 and λ = 10. 86

vi Figure 5.10 Plots of the pole separation (a), magnetic dipole moment (b) and total energy (c) versus λ 1/2 for n = 3 FB, HEB and LEB and the magnetic charges carried by Higgs and gauge ﬁelds (d) versusx ¯ when λ = 4 for n = 3 FB. 87

Figure 5.11 The 3D surface and contour plots of the Higgs ﬁeld modulus |Φ|((a) and (b)), the weighted energy density E ((c) and (d)), and the weighted magnetic charge density M ((e) and (f)), for n = 4 FB when λ = 4. 89

Figure 5.12 Contour plot of the ’t Hooft magnetic field lines and vector field plots of the ’t Hooft magnetic field unit vector for n = 4 FB when λ = 4. 90

Figure 5.13 The 3D surface and contour plots of the Higgs ﬁeld modulus |Φ|((a) and (b)), the weighted energy density E ((c) and (d)), and the weighted magnetic charge density M ((e) and (f)), for n = 4 HEB when λ = 4. 92

Figure 5.14 The 3D surface and contour plots of the Higgs ﬁeld modulus |Φ|((a) and (b)), the weighted energy density E ((c) and (d)), and the weighted magnetic charge density M ((e) and (f)), for n = 4 LEB when λ = 4. 93

Figure 5.15 Contour plot of the ’t Hooft magnetic field lines and vector field plots of the ’t Hooft magnetic field unit vector for n = 4 HEB and LEB when λ = 4. 94

Figure 5.16 The 3D surface and contour plots of the Higgs ﬁeld modulus |Φ|((a) and (b)), the weighted energy density E ((c) and (d)), and the weighted magnetic charge density M ((e) and (f)), for n = 4 NB when λ = 4. 96

Figure 5.17 Contour plot of the ’t Hooft magnetic field lines and vector field plots of the ’t Hooft magnetic field unit vector for n = 4 NB when λ = 4. 97

Figure 5.18 Plots of the pole separation (a), magnetic dipole moment (b) and total energy (c) versus λ 1/2 for n = 4 FB, HEB, LEB and NB and the magnetic charges carried by Higgs and gauge ﬁelds (d) versusx ¯ when λ = 4 for n = 4 FB. 98

vii LIST OF TABLES

Page Table 1.1 A comparison between formulations of Maxwell’s equations with or without magnetic monopoles in SI units 10

Table 5.1 Peak value and position of weighted energy density and weighted magnetic charge density for n = 2 FB when λ = 4. 75

Table 5.2 Peak value and position of weighted energy density and weighted magnetic charge density for n = 3 FB, LEB and HEB when λ = 4. 85

Table 5.3 Lower bounds, λb, critical point for bifurcation, λc, and transition point, λt, for all cases and all branches. 95

Table 5.4 Peak value and position of weighted energy density and weighted magnetic charge density for n = 4 FB, LEB, HEB and NB when λ = 4. 97

viii FENOMENA PENCABANGAN PERALIHAN MONOKUTUB DAN SESETENGAH MONOKUTUB BERCAJ BERGANDA UNTUK TEORI SU(2) YANG-MILLS-HIGGS

ABSTRAK

Monokutub magnet dan multi-monokutub adalah penyelesaian soliton bertopologi dalam tiga dimensi yang timbul apabila simetri SU(2) tak-Abelian dipecah secara spontan oleh medan Higgs. Teori tolok yang boleh menerangkan kewujudan meraka adalah teori SU(2) Yang-Mills-Higgs, yang juga dikenali sebagai model SU(2) Georgi- Glashow. Baru-baru ini, kewujudan penyelesaian monokutub separuh telah dicadan- gkan dan konﬁgurasi yang melibatkan satu monokutub separuh dan satu monokutub bi- asa ‘t Hooft-Polyakov dalam model SU(2) Georgi-Glashow juga dilaporkan walaupun demikian, disebabkan monokutub separuh merupakan bidang penyelidikan yang baru, topik yang berkaitan dengan interaksi antara monokutub dan monokutub separuh adalah sedikit. Dalam tesis ini, kami mengkaji tentang penyelesaian monokutub dengan monokutub separuh dalam teori SU(2) Yang-Mills-Higgs yang mempunyai nilai nombor penggulungan φ yang lebih besar, n (2 ≤ n ≤ 4), antara sesuatu jarak terhad malar gandingan Higgs, λ (0 < λ ≤ 40). Penggunaan grid beresolusi (110 × 100) dalam kaedah be- rangka kami untuk mendapat penyelesaian juga adalah lebih besar berbanding dengan kajian yang lepas. Matlamat disertasi ini adalah untuk mendapatkan maklumat tentang ciri-ciri konﬁgurasi monokutub magnet dengan monokutub separuh, untuk mengkaji interaksi antara konstituen melalui fenomena pencabangan dan peralihan penyelesaian, danjuga untuk memperdehi pemahaman lebih mendalam tentang struktur teori tolok dan pada masa yang sama, memperoleh perfahaman yang lebih mendalam tentang struktur teori tolok terlibat. Apabila n ≥ 2, kami mendapati monokutub menjadi n-monokutub yang bertindih di tempat yang sama. Pada masa yang sama, monokutub separuh yang bertempat di titik asalan menjadi satu n-monokutub separuh yang bertindih. Apabila n = 2, penyelesaian berperangai ganjil, dan mencapah selepas

λ = 8.00 dan apabila n ≥ 3, bertentangan dengan pemerhatian yang didapati pada

ix konﬁgurasi pasangan monokutub-anti monokutub (MAP) atau rantai monokutub-anti monokutub (MAC), monokutub tidak bergabung dengan monokutub separuh untuk membentuk gegelung vorteks. Sebaliknya, apabila pemalr gandingan Higgs mencapai nilai peralihan fasa kritikal, λt, n-monokutub separuh kekal tidak berubah di titik asalan tetapi gegelung vorteks dibentuk pada n-monokutub yang bertindih. Ini dikenali sebagai fenomena peralihan. Pada masa yang sama, pencabangan dapat diperhatikan apabila n ≥ 3, yang masa selain penyelesaian asas, satu lagi cabangan penyelesaian baru yang mempunyai tenaga yang lebih tinggi muncul apabila λ mencapai sesuatu nilai kritikal λc, dan dalam kes istimewa di mana n = 4, satu cabangan penyelesaian baru muncul. Selain itu, untuk n ≥ 2, wujud suatu nilai batasan bawah kritikal λb untuk penyelesaian asas, yang mana tiada penyelesaian boleh didapati apabila λ < λb. Perangai ganjil untuk penyelesaian apabila n = 2 dan penyelesaian cabang baru yang dijumpai apabila n = 4 boleh diatributkan kepada kewujudan monokutub setengah dalam model ini. Dengan mengambilkira penyelidikan sebelum dan membandingkan penyelidikan ini dengan penyelesaian MAP piawai, satu spekulasi yang menarik boleh dibuat, yang mana kewujudan monokutub setengah hanya mempengaruhi penyelesaian dengan nombor penggulungan φ, n yang genap. Selain itu, di samping impak besar keatas penyelesaian, monokutub setengah adalah dorman dan tak-aktif, perubahan dalam kuantiti ﬁzik nampaknya disumbangkan hanya oleh monokutub.

x BIFURCATION AND TRANSITION PHENOMENA OF MULTIPLE CHARGED MONOPOLE PLUS HALF-MONOPOLE OF THE SU(2) YANG-MILLS-HIGGS THEORY

ABSTRACT

Magnetic monopoles and multimonopoles are three-dimensional topological soliton solutions, which arise when the non-Abelian SU(2) symmetry is spontaneously broken by the Higgs field. The gauge theory describing their existence is the SU(2) Yang-Mills-Higgs theory, which is also known as the SU(2) Georgi-Glashow model. Recently, the existence of half-monopole solutions had been proposed, and a configuration involving a half-monopole and an ordinary ’t Hooft-Polyakov monopole within the SU(2) Georgi-Glashow model was also reported. However, since half-monopole is a relatively new field of research, topics regarding the interactions between one- monopoles and half-monopoles are rather scarce. In this thesis, the one-monopole plus half-monopole solution of the SU(2) Yang-Mills-Higgs theory with higher value of φ-winding number, n (2 ≤ n ≤ 4) is studied for a range of the Higgs coupling constants, λ (0 < λ ≤ 40), and the resolution of the grids used (110 × 100) in the numerical method for calculating the solutions is also greater than previous research. The goal of this dissertation is to gain information about the general behaviors and properties of the one-plus-half monopole configuration, to probe the interactions between constituents through phenomena manifested as bifurcations and transitions of solutions, as well as to obtain a deeper understanding of the structure of gauge theories. We noticed that for n ≥ 2, the one-monopoles become an n-monopole superimposed at the same location. At the same time, the half-monopoles at the origin, in the same manner, becomes a superimposed n-half-monopole. When n = 2, the solutions behave strangely and diverge after λ = 8.00 and when n ≥ 3, in contrary to the observation in monopole-antimonopole pair (MAP) or monopole-antimonopole chain (MAC) configurations, the one-monopoles do not merge with the half-monopoles to form vortex-rings. Instead, when the Higgs coupling constant reaches a certain critical

xi phase transition value, λt, the n-half-monopole remains unchanged at the origin while vortex-rings were formed among the superimposed n-monopole. This is known as the transition phenomenon. At the same time, bifurcation phenomenon is also observed when n ≥ 3, where besides the fundamental branch, new branches of solution with higher energies emerge at some critical value of λ, λc and in the special case where n = 4, a completely new branch of solutions appeared. It is also noticed that for n ≥

2, there exists a critical lower bound λb for the fundamental branch, for which when λ

< λb, no solution can be found. The peculiar behavior of the solution when n = 2 and the new branch of solution found when n = 4 can all be attributed to the existence of half-monopoles within this model. Taking previous researches into account and comparing this research with the standard MAP results, one interesting speculation can be drawn, which is that the existence of half-monopoles seems only affect solutions with even number of the φ-winding number, n. Furthermore, despite the huge impact on solutions, half-monopoles appear rather dormant and inactive, the change in physical quantities seems only contributed by one-monopoles.

xii CHAPTER 1 - INTRODUCTION

1.1 A Brief History of Theoretical Physics

The foundation of modern physics is built upon two major achievements of the 20th century, relativity and quantum mechanics. While both theories of relativity are the masterpiece of Albert Einstein, quantum mechanics is the wisdom from hundreds of scientists and through decades of discoveries, Max Planck, Erwin Schrodinger,¨ Paul Dirac, Enrico Fermi, just to name a few. Relativity deals with physics on a grand scale whereas quantum mechanics focuses on the minuscule structure of a nucleus. The undertaking to combine these two extremely different theories into a single, all- inclusive theory is still one of the most formidable task even to this day. The first and most successful attempt to combine quantum mechanics and special relativity is quantum electrodynamics (QED) in the 1940s. Though the very first formulation of a quantum theory describing radiation and matter interaction was done by Paul Dirac (Dirac, 1927), not until 1947 when the idea of renormalization was proposed by Hans Bethe (Bethe, 1948) did quantum electrodynamics attain its present, fully-accepted form. QED is one of the most strictly tested theories in physics, the agreement of theoretical predictions of QED and experimental results is within ten parts in a billion (10-8) (Peskin and Schroeder, 1995, p.198). Although combining electrodynamics with special relativity was done perfectly as early as in the 1940s, combining quantum theories with general relativity remains an active field of research nowadays and no decisive conclusion has been drawn or experimental evidences to show a clearer path. The mathematical formulation of QED shows a kind of group symmetry. A mathematical group is defined as a set with a symmetry operation. In mathematics, any operation uses two elements of the set as input and if the output satisfies 4 conditions (namely, closure, associativity, identity element and inverse element), the set and the operation together form a group, G, as denoted in mathematics. The group symmetry exhibited in QED is called U(1) or unitary group of order 1. The set consists of all first

1 order complex unitary matrix (numbers) as elements. As the elements are numbers, the group possesses one additional feature, that is the operations commute. Hence the U(1) group is abelian. The manifestation of a group symmetry in QED is actually the result of selecting a particular gauge (Gauge theory wil be discussed in details in Chapter 2 of this dissertation.). Inspired by the massive success of QED, the mathematical tool, group theory, was soon applied to other theories. In 1954, Chen-Ning Yang and Robert Mills together published a paper in which they imposed a similar group symmetry upon the isospin doublet in the hope of providing an explanation for the nuclear strong interactions (Yang and Mills, 1954). Unlike G = U(1) in QED, the type of group symmetry exhibited in their formulation was actually G = SU(2) or special unitary group of order 2. The set consists of all second order complex unitary matrix with determinant 1 as elements. As the elements in the set are no longer numbers, the commutation feature is not preserved. Thus, the SU(2) group symmetry is non-abelian. The work of Yang and Mills was shortly abandoned as it was later found to be incomplete on its own. Wolfgang Pauli pointed out that Yang and Mills’ theory alone describes a long-distance interaction with massless mediator, which was something he encountered back in 1953 and caused him refraining from publishing his results formally (Straumann, 2000). A massless mediator contradicts the features of the extremely short ranged nuclear strong force which requires massive mediators. This is- sue was resolved when the concept of spontaneous symmetry breaking and the Higgs mechanism were put forward. In Peter Higgs’ formulation, mediators acquire mass through the spontaneous symmetry breaking caused by the Higgs field, a scalar field permeating all space (Higgs, 1964). This theory was later known by the name of Yang- Mills-Higgs (YMH) field theory. The spontaneous symmetry breaking and Higgs mechanism not only saved Yang- Mills theory, they also reassured theoretical physicists that imposing group symmetry was the right approach to a deeper understanding of the fundamental interactions of our universe. Notable achievements after this were quantum chromodynamics (QCD) and

2 the electroweak theory (EWT). They both went through a series of developments and it’s hard to put a date on them. QCD is the correct interpretation of strong interactions and belongs to the group symmetry SU(3). Although Yang-Mills theory is monumental as it provides the mathematical basis for all later theories, it was also an early attempt to describe the strong interactions and it was on the wrong track in that respect. QCD was only possible when quark model was brought forth and when the colour scheme was proposed in order to solve the violation of Pauli exclusion principle appeared in resonance particles (Griffiths, 2008, p.43). The latter, EWT, is a marvelous masterpiece of Sheldon Glashow, Steven Weinberg and Abdus Salam. And for this reason, EWT sometimes is called the Glashow-Weinberg-Salam (GWS) model. In this theory, the coupling constants of both electromagnetic interactions and weak interactions become identical when a critical energy limit is reached (Glashow, 1959; Weinberg, 1967; Salam, 1959), which indicates that two of the fundamental interactions of our universe are just different manifestations of the same interaction called electroweak interaction. In this theory, G = SU(2)×U(1). The above is the brief history of theoretical physics up until the 1970s and to final- ize this section, I would like to quote David J. Griffiths from his book, Introduction to Elementary Particles (Griffiths, 2008, p.3):

This theory - or, more accurately, this collection of related theories, based on two families of elementary particles (quarks and leptons), and incorporating quantum electrodynamics, the Glashow-Weinberg-Salam theory of electroweak processes, and quantum chromodynamics - has come to be called the Standard Model.

1.2 Standard Model, Achievements and Flaws

The success of a theory is determined by the experimental veriﬁcations of its predictions. In the 1970s, when the majority of the theories, which belongs to the Standard Model, had been proposed, only 3 types of quarks (up, down and strange) and 4 types of leptons (electron, muon and their corresponding neutrinos) were observed experi-

3 mentally and the idea of generations of matter had not yet been fully appreciated. The first sign of success of the Standard Model could be attributed to the confirmation of neutral current in 1973, shortly after being predicted by the GWS model. This so- lidified the foundation of electroweak theory, a pillar of the Standard Model that we now know today. And later, the detection of charm quark was made in 1974, which completed the first two generations of matter. Then the Standard Model set out on a road of success and triumph. Tau lepton was detected using the Stanford Positron Electron Asymmetric Rings (SPEAR) with a series of experiments conducted between 1974 and 1977 (Okun, 1980, p.103; Perl et al., 1975). In 1977, bottom quark was detected at Fermilab (Herb et al., 1977). The mediators of the weak force (W+,W- and Z) were all discovered in 1983 at CERN. After that, things went quite for a decade as the precision and operating energy of the then-top-of-the-line labs were not enough to detect or produce the remaining particles predicted by the Standard Model. With the construction of new detectors like those used in the Collider Detector at Fermilab (CDF), D0 experiment (D0 or D∅) and Direct Observatrion of the Nu Tau (DONUT) at Fermilab, top quark was detected in 1995 (Abe et al., 1995; Abachi et al., 1995) and tau neutrino was detected in 2000, which completed the 3 generations of quarks and leptons. All of these breathtaking achievements culminated in 2012 when the last piece of the puzzle, the Higgs boson, was finally detected. These discoveries of the past 40 years clearly indicate that the Standard Model is indeed the correct interpretation of our universe. But no one would say that it is the final picture as there are still lots of questions left unexplained and unanswered by the Standard Model. The most obvious problem is that only three of the four fundamental forces are being described by the Standard Model, the fourth one, gravity, is not accounted for. Another problem into which the Standard Model offers no insight is the matter-antimatter asymmetry or the baryon asymmetry problem. The world we live in is made of baryonic matter, if the Big Bang created the same amount of matter and antimatter, where did all the antimatter go? The Standard Model also provides no

4 explanation for dark matter and dark energy, something we now know that made up roughly 95 percent of our universe’s total mass energy. On the other hand, the Standard Model itself is also not as elegant as physicists hope it could be. It contains around 20 unrelated, arbitrary constants (Blumhofer and Hutter, 1997) whose value can only be determined through experiments and the theoretical explanations on such constants are completely inadequate (Cahn, 1996). The Standard Model thus received a lot of criticism from an aesthetic point of view. All of these flaws and insufficiencies hint at the possiblity of an even more all- inclusive theory. A few tries have already been made in the past several decades and some of them will be briefly discussed in the next section.

1.3 Beyond the Standard Model

Looking back at history, physics is all about unification. Back in the 19th century, James Clerk Maxwell unified electric and magnetic forces. Then following the huge success of combining nuclear weak force with electromagnetic force in the 1960s, it is natural to try including nuclear strong force into the picture as well. Though in the Standard Model, the theory responsible for describing strong force, QCD, is already an important constituent, it is presented only as a parallel to the EWT and no unification has been made.

1.3.1 The Grand Uniﬁed Theory

Mathematically, in EWT, G = SU(2)×U(1) and in QCD, G = SU(3). Thus, a theory incorporating both of them must show G = U(1)×SU(2)×SU(3) at least or possesses an even higher order symmetry which includes U(1)×SU(2)×SU(3) as a subgroup. Historically, such a theory has come to be called a Grand Uniﬁed Theory or GUT for short. In 1974, Howard Georgi and Sheldon Glashow attempted to construct a GUT (Georgi and Glashow, 1974) and their theory is sometimes called the Georgi-Glashow (GG) model. The GG model is based on the smallest, simplest simple Lie group (The

5 mathematical details of a simple Lie group will not be discussed here as it is outside of the scope of this dissertation. Interested readers are encouraged to refer to advanced textbooks on abstract algebra.) that contains the Standard Model, G = SU(5). In their model, under some extremely high energy, the SU(5) symmetry is spontaneously broken into smaller symmetry groups, those possessed by the Standard Model. This is very reminiscent of, as it should be, what we have seen in EWT in which under certain energy limit the SU(2)×U(1) symmetry is broken into smaller symmetry group which is U(1). Apart from incorporating strong force into the picture, expressing the strong force as a different manifestation of a single, unified interaction, the GG model predicted one other phenomenon, the proton decay (Griffiths, 2008, p.33). When the SU(2)×U(1) symmetry is broken, 4 force mediating particles are produced, W+,W-, Z and γ. By the same token, when SU(5) symmetry is broken, several other force carrying particles, the X bosons (Cheng and Li, 1983, p.437), are produced and they provide ways for protons to decay. In addition, allowing protons to decay breaks the conservation law of baryon number and theoretically solves the baryon asymmetry problem mentioned in the previous section. In the GG model, the proton half-life is predicted to be at least 1030 years (Griffiths, 2008, p.406). In 1998, a research team conducted a 414-days- long experiment to detect proton decay. Though not a single sign of the decay was detected, they managed to push the lower bound of proton half-life and the result they obtained back then was at least 1.6 × 1033 years (Abe et al., 1998). This obviously vetoes the GG model, but it’s not the only GUT we have at hand. One of the major difficulties when constructing a GUT is the choice of symmetry group. There are simply too many of them and we don’t know which one truely describes our universe. Except the SU(5) used by the GG model, there are other symmetry groups that have been tested, some of them are: SO(10) (special orthogonal group of order 10, consists of all orthogonal matrices of order 10 with determinant 1), SO(16), SU(8), Sp(8) (symplectic group of order 8, consists of all symplectic matrices of order 8), etc. The proton half-lifes predicted by these models are slight different

6 from each other, but they all lie within the range of 1030 to 1036 years (Nath and Perez, 2007). Since the experiment results indicate the lower bound of proton half-life should not be lower than 1033 years, simpler GUTs, like the GG model, have been ruled out. Although there are still possible GUT candidates, the lack of hard experimental evidence and aesthetic elegance put the whole thing into question for some physicists. They resort to a different approach and hence comes the Supersymmetry (SUSY).

1.3.2 Supersymmetry (SUSY)

GUTs, being an enlarged version of the Standard Model, share both its features and defects. The fourth fundamental force, gravity, is still being excluded and even more arbitrary parameters have been added to them. Also, problems regarding dark matter and dark energy are still not addressed in almost all versions of GUT. Interestingly enough, nearly all these conundrums are marvelously solved by the introduction of SUSY, if it’s eventually proven to be correct. Physics has always been dealing with internal symmetries that link closely related subatomic particles, like the color symmetry that links red, blue and green charges and flavor symmetry that links 6 different types of quark. Physical systems are invariant under rotations within the corresponding space (color space, flavor space, etc.). In the 1960s, Japanese physicist Hironari Miyazawa put forward an idea that could be counted as the prototype of SUSY (Miyazawa, 1966, 1968). He proposed a kind of symmetry that transcends normal internal symmetry and links mesons and baryons together. Since obviously the symmetry, if there is any, is extremely broken, his work was mostly ignored at the time. A few year later, Miyazawa’s original idea was generalized to all particles linking fermions with bosons, most notably in Wess and Zumino’s work (Wess and Zumino, 1974). In that particular symmetry space, a physical state representating a fermion turns into a boson after rotation (and vice versa). The system remains invariant, the fermion and boson are just different manifestations of a single state. Invariance of this kind is called SUSY (Griffiths, 2008, p.412). Since SUSY links fermions with bosons, it assigns a partner to each and every par-

7 ticle known to the Standard Model. Each particle and its partner (called superpartners and collectively known as sparticles) share exactly the same quantum numbers, except their spins. Particles and sparticles should have the same mass and some of the superpartners of relatively light particles like electrons should easily be detected. The reason behind why selectrons (superpartner of electrons) are never detected could be due to the SUSY of our universe is severely broken by the spontaneous symmetry breaking mechanism and superpartners like selectrons gain an enormous amount of mass energy through interacting with the Higgs field, thus rendering them impossible to be produced by our current equipments. There are other possibilites, however, especially if gravity is brought into the picture (Griffiths, 2008, p.412). All of these seems like a sleight of hand. They are, if not for their deeper theoretical implications. As SUSY does not belong to the scale of this dissertation, its theoretical achievements will be just briefly mentioned here in this section. Among them, there are four major accomplishments of SUSY that worth mentioning. First of all, SUSY could naturally incorporate the fourth natural force, gravity, into its scheme, the resulting theory is called Supergravity (SUGRA), details of this theory can be found in Van Nieuwenhuizen’s publication (Van Nieuwenhuizen, 1981). Theories like this, incorporating all four natural forces, are referred to as Theory of Everything, or ToE for short. The second major achievement of SUSY could be attributed to the theoretical fact that all three running coupling constants of electromagnetic force, weak force and strong force can perfectly converge at a single point when energy reaches some critical value (Griffiths, 2008, p.406). Something all GUTs are trying to do but failed. This marvelous convergence is so elegant that most scientists believe that there is no coincidence and it’s a clear indication that SUSY must be correct. There is another problem left unanswered and that has haunted the Standard Model since the very beginning, the so-called, Hierarchy Problem. The values of coupling constants we measured in experiments are their effective values, after the renormalization process (Bethe, 1948). However, the real values of some coupling constants are

8 vastly different from their effective values and require a huge amount of fine-tunings. This is the dilemma the Hierarchy Problem addresses. Even though lots of scientists devoted a large portion of their careers trying to solve this conundrum, no one could offer an answer to the fact that while the coupling constants of the other three forces are relatively close, gravity is 1024 times weaker than weak force (Hughes, 2005). In SUSY, on the other hand, divergent parts occurred in the renormalization process cancelled out naturally between superpartners and do not require fine-tunings at all (Haber, 2013). Last but not least, among the huge amount of new particles predicted by different versions of SUSYs, some of them, known collectively as the neutralinos (Griffiths, 2008, p.416), could be considered as candidates of the WIMPs (weakly interacting massive particles) we need to solve the dark matter problem. Indeed, SUSY looks promising, but despite all these theoretical achievements, there is no exprimental evidence whatsoever to prove whether SUSY is right or wrong, slowly but surely time will tell. Except SUSY, there are other approaches as well, su- perstring theory, multiverse, loop quantum gravity, just to name a few. Interestingly enough, nearly all of them share one thing in common, that is they require the existence of one particular thing, magnetic monopoles.

1.4 Magnetic Monopoles

The word “magnetic monopole” was coined by early physicists even before James Clerk Maxwell’s uniﬁcation of electricity and magnetism. The concept was put forward in order to give an explanation to the naturally magnetized nature of lodestones. It was believed that magnetic monopoles carrying different charges (corresponding to north and south pole) accumulate at opposite sides of a lodestone, forming the so-called magnetic ﬂuids and thus give lodestones a magnetized nature. Of course, this idea was quickly vetoed by a better understanding of electromagnetism in the nineteenth century when French physicist Andre-Marie´ Ampere` discovered the circuital law. Then the word “magnetic monopole” was rarely seen in the

9 physics community until Paul A. M. Dirac brought it back under the spotlight. In 1931, Dirac published a paper (Dirac, 1931) in which he demonstrated that the existence of even a single magnetic monopole, given that the form of Maxwell’s equations is intact, would force all the electric charges in the entire universe to be quantized. This is called the Dirac quantization condition. Even though the quantization of electric charges is a well-observed phenomenon, it is only a necessary condition and thus logically does not prove the existence of magnetic monopoles, but a lack of proper explanations as to why all electric charges in our universe are quantized have led lots of physicists to believe that magnetic monopoles must exist. Another compelling theoretical evidence is the aesthetically pleasing form which the Maxwell’s equations exhibit when magnetic monopoles are incorporated (Moulin, 2002), as tabulated in Table 1.1, this symmetric form exhibited by the new formulation rendered the classical Maxwell’s equations artiﬁcial and unnatural. Table 1.1: A comparison between formulations of Maxwell’s equations with or without magnetic monopoles in SI units

Name of Laws without magnetic monopoles with magnetic monopoles ∇ · E = ρe Gauss’s Law ε0 ∇ · B = 0 ∇ · B = µ0ρm ∂B ∂B Faraday’s Law −∇ × E = ∂t −∇ × E = ∂t + µ0Jm ∂E Ampere’s` Law ∇ × B = µ0ε0 ∂t + µ0Je

Finally in 1974, the first topologically smooth monopole solution was proposed independently by Gerard ’t Hooft and Alexander Polyakov (’t Hooft, 1974; Polyakov, 1974). The ’t Hooft-Polyakov monopole is similar to Dirac monopole, but possesses a finite total energy and no singularities (Details of ’t Hooft-Polyakov monopole will be discussed in Chapter 3). The mathematical formulation of ’t Hooft-Polyakov monopoles and the methodologies involved later became the cornerstone of a vast majority of the researches done in this field, including the model described in this dissertation. On top of all the above, almost all versions of GUTs and candidates of ToEs pre- dict the existence of magnetic monopoles and the masses predicted are seem to be very model-dependent. Hence, the experimental confirmation of magnetic monopoles become vital as it could show us as to which GUTs or ToEs are on the right track and

10 which should be discarded. Sadly, ever since Dirac’s paper brought this mysteriously rare particle back to the spotlight, tons of systematic and thorough searches have been performed, but all attemps returned null results. However, just as the string-theorist, Joseph Polchinski, once said, “The existence of magnetic monopoles seems like one of the safest bets that one can make about physics not yet seen.” (Polchinski, 2003), magnetic monopoles, with its theoretical feasibilities and the state of being the logically natural next step of modern physics, remain the most long-waited particles in the wake of the discovery of Higgs bosons. Here, waits the future of modern physics.

1.5 Objective and Research Gap

The objective of this research is to study the interactions between multi-monopoles and multiple half-monopoles over a large range of the Higgs coupling constant by investigating the bifurcation and transition phenomena, which are the results or manifestations of the interactions between constituents of the one-plus-half monopole configuration. This particular goal is chosen out of the consideration that half-monopoles are themselves still a relatively new concept and a study regarding their interactions with the more commonly known one-monopoles, ’t Hooft-Polyakov monopoles in particular, will surely shine some light into the nature of this type of exotic particles. Cur- rently, the newly found one-plus-half monopole configuration is the only platform in the field which made this effort possible. Finally, the scope of this research concerns only the numerical aspect of the solutions found, specifically, the trending behaviours. Physical quantities such as total energy, magnetic dipole moment, pole separation, Higgs modulus, magnetic charge density, energy density, magnetic charge of the system are plotted, analyzed and discussed in this research.

1.6 Dissertation Outline

This dissertation is divided into 6 chapters. The mathematical framework upon which all modern particle physics are based on, gauge ﬁeld theory, is discussed in detail in Chapter 2. A review of monopole solutions is given in Chapter 3. Theo-

11 retical details about constructing the one-plus-half monopole solutions, the numerical method employed in this research and the physical quantities investigated are presented in Chapter 4. Results and discussions are in Chapter 5 and some comments and future research suggestions are saved for the last chapter.

12 CHAPTER 2 - GAUGE FIELD THEORY

2.1 Introduction to Gauge Theory

Gauge theories are the theories that describe literally all elementary particle interactions in modern physics. The word “gauge” (German “eich”) was coined by German physicist Hermann Weyl and first appeared in his paper in 1929 (Weyl, 1929). Its meaning can be taken as “scale” or “measure”. Technically, it refers to the mathematical formalism used to regulate the redundant degrees of freedom in the Lagrangian. Even though gauge theories are notoriously mathematically heavy, the method used to construct a gauge theory is rather simple. We are interested in transformations made to constituents of the Lagrangian that leave it unchanged, in other words, gauge transformations. The Lagrangian is said to be invariant under these transformations and their specific mathematical form is the gauge. If the transformations do not depend on spatial coordinates, then the invariance involved is referred to as the global gauge invariance (German “eichinvarianz”). A gauge theory is then constructed by demanding the global gauge invariance to hold locally, that is, to require the transformations to depend on spatial-temporal coordinates and at the same time, leaves the Lagrangian untouched. This is called the local gauge invariance. Furthermore, gauges of a particular Lagrangian form a Lie group which is referred to as the gauge group. Generators of this group generate fields which are called gauge fields and the field quanta associated to these fields are the gauge bosons. It is now clear that Maxwell’s unifying theory of electricity and magnetism also exhibits local gauge invariance with electromagnetic four-potential as the gauge field and photon being the only gauge boson. It manifests a U(1) symmetry just as QED does and can be taken as the very first and simplest gauge theory in the history of physics. Mathematically, gauge theories can be classified into two categories, Abelian gauge theory and Non-Abelian gauge theory according to the commutative property of their underlying operation (as discussed in section 1.1).

13 2.2 Abelian Gauge Theory

In electrostasis, if we have an electric potential, V, the electric field, E, can be obtained through E = −∇V. The gradient indicates that the electric field is directly related to the change in electric potential. That is to say, if the electric potential, V, transforms according to V → V0 = V + C, where C is some constant, then, the electric field, E, stays the same. The transformation made to V is precisely a type of gauge transformation. In the following subsections, we will discuss the global and local gauge invarianc to show mathematically that the gauge transformation made to electric potential is what we called, an Abelian gauge transformation, and at the same time, gain some insights into gauge theories.

2.2.1 Global Gauge Invariance

In quantum field theory, there are three Lagrangian densities that are of the utmost importance. The first one being the Klein-Gordon Lagrangian density (Griffiths, 2008, p.355), it describes a scalar field, φ(x,y,z,t), with its quanta having spin-0:

1 µ 1 2 2 L φ,∂ φ = ∂ φ (∂ φ) − m φ . (2.1) µ 2 µ 2

The second one is the Dirac Lagrangian density (Grifﬁths, 2008, p.355). It describes a

1 fermionic ﬁeld with its quanta having spin- 2 and are represented by the Dirac bispinor, ψ:

µ L ψ,∂µ ψ = iψγ ∂µ ψ − mψψ, (2.2) where m is the mass of the ﬁeld quanta, γ µ is the Dirac matrices, ψ stands for the adjoint spinor and is deﬁned as:

ψ ≡ ψ†γ0, (2.3) here, the dagger stands for the Hermitian conjugate and γ0 is the zeroth Dirac matrix. Lastly, the third one is the Proca Lagrangian density (Grifﬁths, 2008, p.356) describing a vector ﬁeld and is represented by a potential four-vector, Aµ (V,A), with its quanta

14 having spin-1:

µ ν 1 µν 1 2 ν L A ,∂ A = − F F + m A A , (2.4) µ 4 µν 2 ν where Fµν is the ﬁeld strength tensor and is deﬁned as:

F µν ≡ ∂ µ Aν − ∂ ν Aµ , (2.5) or, it can be equally represented by matrix:

  0 −Ex −Ey −Ez     Ex 0 −Bz By  µν   F =  . (2.6) E B −B   y z 0 x   Ez −By Bx 0

All the above equations are given in natural units. Once all the cards are on the table, we can start constructing a gauge theory using these building blocks. Now, consider the Dirac Lagrangian density, if we make a transformation, G, to ψ, ψ → ψ0 = Gψ, then the corresponding adjoint spinor ﬁeld would transform according to:

ψ → ψ0 = (Gψ)† γ0 = ψ†G†γ0. (2.7)

In gauge theory, we are interested with the invariance of Lagrangian. In this case, the product ψψ must satisfy the following criterion when being transformed according to G:

ψψ → (ψψ)0 = ψ†G†γ0Gψ = ψψ. (2.8)

Currently, we take G as a number, thus G†G can be brought together and if G†G = 1 is satisﬁed, ψψ is invariant under the transformation. At this point, it’s obvious that G is precisely the group U(1) and all elements in G can be expressed as G = eiθ and this is called the phase factor. Thus, transformations of the form, G = eiθ , made to the Dirac Lagrangian density

15 manifest what we called the global gauge invariance as the parameter θ in this case is independent of position and time. U(1) is clearly Abelian, but only global gauge invariance alone does not construct a gauge theory, that’s where local gauge invariance comes in.

2.2.2 Local Gauge Invariance

Now, if the phase factor depends on the position four-vector, xµ . The second term in equation (2.2) stays the same under the transformation G(x, t) = eiθ (x, t) (For simplicity, (x, t) will be omitted when the context is clear). The ﬁrst term, however, splits into two parts:

µ iθ µ iθ µ iθ iψγ ∂µ e ψ = −ψγ ∂µ θ e ψ + iψγ e ∂µ ψ. (2.9)

For reasons shall become clear later, we introduce a new variable, λ(x, t), by pulling a factor of −q out of θ, that is: 1 λ ≡ − θ, (2.10) q now the Lagrangian density changes in the following way when it is being transformed by G:

0 µ L → L = L + (qψγ ψ)∂µ λ. (2.11)

This extra term is clearly not zero under normal circumstances and to maintain the

Lagrangian density’s invariance, we are obliged to add an additional term to L in order to soak up the extra term in equation (2.11):

µ µ L = iψγ ∂µ ψ − mψψ − (qψγ ψ)Aµ , (2.12)

here, Aµ transforms, under the inﬂuence of G, according to:

0 Aµ → Aµ = Aµ + ∂µ λ. (2.13)

This way, when a local gauge transformation is applied to the newly modiﬁed L , both the original L (the two terms in the ﬁrst parentheses in equation (2.12)) and Aµ pick

16 up an extra term and they cancel each other. The invariance of L is thus restored and judging from the way Aµ transforms, it is yet another four-vector and as we shall see later, Aµ is precisely the electromagnetic potential four-vector. The concept of covariant derivative needs to be introduced here before we go any further. Note that the steps we’ve shown thus far to restore the invariance of L is equivalent to replacing all ∂µ with:

Dµ ≡ ∂µ + iqAµ . (2.14)

Dµ is called the covariant derivative and the technique of replacing all ∂µ with Dµ in order to convert a globally invariant Lagrangian density into a locally invariant one is called the minimal coupling rule (Grifﬁths, 2008, p.360). Thus, the locally invariant Lagrangian density can be also be written as:

µ L = iψγ Dµ ψ − mψψ. (2.15)

While trying to restore the gauge invariance of L , the new four-vector, Aµ , is inevitably introduced into L , either by following the steps shown from the beginning of this section or by invoking the minimal coupling rule mentioned just now. However, a new term cannot be simply added to the Lagrangian density without considering its effects. Otherwise, it is just a mathematical construct. In our case, an additional term signifying the physics of Aµ must be added to L as well and at the same time, it must not spoil the overall invariance we are trying to maintain. As Aµ is a four-vector. Naturally, we look to the Proca Lagrangian density, equation (2.4). In this case, it can be shown that the first term in Proca Lagrangian density is invariant under the transformation G, but the second term is not. So, in order to maintain the locally invariant properity of L , we must set the mass of the field, m, to zero and it is fairly clear now that this particular particle of spin-1 and possesses zero mass is exactly the photon, the only gauge boson of this particular gauge theory and finally,

17 the complete Lagrangian density reads:

µ 1 µν µ L = iψγ ∂ ψ − mψψ − F F − (qψγ ψ)A . (2.16) µ 4 µν µ

Now we can see that the last two terms in the above equation reproduce the Maxwell Lagrangian density and we can identify the current density as:

Jµ ≡ qψγ µ ψ. (2.17)

Equation (2.16) is invariant under the local gauge transformation G, which is also Abelian. The transformation condition described in equation (2.13) is the general form of the change in electric potential, V → V0 = V + C, mentioned in the beginning. Furthermore, equation (2.16) is also clearly the Lagrangian density for QED, which describes the interaction of two fields, a Dirac field and a Maxwell field. The first two

1 terms belong to the original Dirac Lagrangian density, describing particles of spin- 2 . The third term describes photons and the last term depicts an all-permeating massless vector ﬁeld, which is exactly the electromagnetic ﬁeld. All of QED can be obtained from this equation. As seen from the above, demanding the global U(1) gauge invariance of the Dirac Lagrangian density to hold locally (This will be referred to as “the principle of local gauge invariance” for the rest of this dissertation.) generates QED. This is a breathtaking achievement and is done by the simplest of the simplest gauge theories, the Abelian ones. Gauge theories are not some particular physics theory, they are a powerful mathematical tool at our disposal. In the example above, we used a Dirac Lagrangian density to demonstrate, but we can equally well use a Klein-Gordon Lagrangian density, apply the same procedure and another Abelian gauge theory will be produced. In the next section, we will discuss the more general non-Abelian gauge theories.

18 2.3 Non-Abelian Gauge Theory of SU(2)

In previous sections, our starting point was only one Dirac Lagrangian density with

1 no other interactions presented. Now, if there are two spin- 2 ﬁelds interacting with each other, the new Lagrangian density is nothing but the sum of two Dirac Lagrangian densities:

µ µ L = iψ1γ ∂µ ψ1 − m1ψ1ψ1 + iψ2γ ∂µ ψ2 − m2ψ2ψ2 . (2.18)

The above equation can be compactiﬁed if we introduce a two-component column vector:   ψ  1 ψ ≡  , (2.19) ψ2 and the corresponding adjoint spinor matrix is: ψ = (ψ1 ψ2). Then equation (2.18) can be compactly written as:

µ L = iψγ ∂µ ψ − Mψψ, (2.20) where M is the mass matrix:   m 0  1  M =  . (2.21) 0 m2

In particular, if the mass difference between m1 and m2 are negligable, then:

    m 0 1 0     M =   = m  = mI, (2.22) 0 m 0 1 and thus equation (2.20) can be expressed as:

µ L = iψγ ∂µ ψ − mψψ, (2.23) which is exactly the same as equation (2.2) except the spinors, adjoint spinors and masses become matrices.

19 The structural similarities between equation (2.2) and equation (2.23) are only made possible if the particles presented in the theory have negligable mass difference, just like protons and neutrons, which is precisely from where Chen-Ning Yang and Robert Mills got their inspirations. In the previous sections, we see that demanding the principle of local gauge invariance to hold true on one Dirac Lagrangian density generates QED. Similarly, the combination of two Dirac Lagrangian densities with the principle of local gauge invariance constructs the entire SU(2) Yang-Mills theory, the theory which shaped modern physics. Although the SU(2) Yang-Mills theory alone is physically impossible and does not describe any real physical process, its importance and position in the history of physics is widely acknowledged and appreciated. In the following subsections, we are going to discuss the global and local gauge invariance of this theory.

2.3.1 Global Gauge Invariance of SU(2) Yang-Mills Theory

Just as equation (2.2) is globally invariant under a transformation of the form, G = eiθ , a similar global gauge transformation can be applied to equation (2.23). This time, the new transformation G takes the form, G = eiH, where H is a 2 × 2 matrix. The column matrix ψ transforms like ψ → ψ0 = Gψ. For the row matrix ψ, treat γ0 as a number as it goes into each element of the row matrix. It transforms according to:

0 0 0 ψ → ψ0 = = † 0 † 0 = † † γ0 ψ1 ψ2 ψ1 γ ψ2 γ ψ1 ψ2 0 = ψ†γ0 = (Gψ)† γ0 = ψ†G†γ0 = ψ†γ0G† = ψG†. (2.24)

Obviously, ψψ is invariant if G†G = I. Thus, G belongs to U(2). Now suppose G is also Hermitian and any 2 × 2 Hermitian matrices can be expressed as (Grifﬁths, 2008, p.362):

H = θI + τ · a, (2.25) where θ is any real number, τ is a vector-like construct made of Pauli matrices and a is

20 any real vector. Then, G can be written as G = eiθ eiτ·a. We’ve already seen eiθ belongs to U(1). So now, we are more interested in the second factor. We want to calculate the determinant of matrix eiτ·a and in order to do so, we ﬁrst

1 1 pull out a factor of − 2 out of a for reasons shall become clear later. So, a = - 2 b and i ·a - i ·b matrix e τ becomes e 2 τ , then expand the matrix:

2 3 − i ·b τ · b 1 τ · b 1 τ · b e 2 τ = 1 + −i + −i + −i + ... (2.26) 2 2 2 3! 2

Now, multiplication of the form (τ·a)(τ·b) can be easily calculated using summation notations:

(τ · a)(τ · b) = ∑τiaiτ jb j = ∑aib j τiτ j = ∑aib j δi j + iεi jkτk i, j i, j i, j

= ∑aib jδi j + i∑εi jkaib jτk = a · b + iτ · (a × b). (2.27) i, j i, j

Then, in our case, the second term in the expansion becomes:

1 τ · b2 1 b b 1 b b b b 1 b2 −i = − τ · τ · = − · + iτ · × = − , 2 2 2 2 2 2 2 2 2 2 2 2 (2.28) and thus the expansion can be simpliﬁed to:

2 2 − i ·b iτ · b 1 b 1 b e 2 τ = 1 − − + i(τ · b) + ... 2 2 2 3! 23 " # " # 1 b2 1 b4 iτ · b b 1 b3 = 1 − + − ... − − + ... 2 2 4! 2 b 2 3! 2 b b = cos − ibˆ · τsin , (2.29) 2 2 here bˆ is a unit vector. Then we express the matrix in its traditional block form, ﬁrst

21 calculate bˆ · τ:

      0 1 0 −i 1 0 ˆ ˆ   ˆ   ˆ   b · τ = bx   + by   + bz   1 0 i 0 0 −1   bˆz bˆx − ibˆy =  , (2.30)   bˆx + ibˆy −bˆz

- i ·b then the matrix e 2 τ can be expressed as (Note that the ﬁrst term in equation (2.29) is actually a matrix):

    1 0 bˆ bˆ − ibˆ − i ·b b b z x y e 2 τ = cos   − isin   2   2   0 1 bˆx + ibˆy −bˆz   b b b cos − ibˆz sin −i bˆx − ibˆy sin =  2 2 2  (2.31)  ˆ ˆ b b ˆ b  −i bx + iby sin 2 cos 2 + ibz sin 2 .

And now we are in a position to calculate the determinant:

− i τ·b b b b b 2 b det e 2 = cos − ibˆ sin cos + ibˆ sin + sin bˆ − ibˆ bˆ + ibˆ 2 z 2 2 z 2 2 x y x y b b b b b = cos2 + bˆ2 sin2 + sin2 bˆ2 + bˆ2 = cos2 + sin2 bˆ2 + bˆ2 + bˆ2 2 z 2 2 x y 2 2 x y z b b = cos2 + sin2 = 1. (2.32) 2 2

Up until now, we’ve shown equation (2.23) exhibits a U(2) global gauge invariance and it can be factored out into a U(1) factor plus another one expressed by eiτ·a. In equation (2.32) we proved the determinant of eiτ·a is 1 and G†G = 1 shows that eiτ·a is unitary. These indicate that eiτ·a belongs to SU(2). Thus, equation (2.23) is not only invariant under the larger U(2) global gauge transformations, but also invariant under the smaller SU(2) global guage transformations. Next, we are going to show how SU(2) local gauge invariance is achieved.

22 2.3.2 Local Gauge Invariance of SU(2) Yang-Mills Theory

Suppose a in eiτ·a now depends on the position four-vector, xµ . Once again, we redeﬁne a new variable, λ, by pulling out a factor of −q out of a:

1 λ ≡ − a, (2.33) q here, q is a coupling constant analogous to the electric charge. Then, the local SU(2) gauge transformation, G, now takes the form:

G = e−iqτ·λ . (2.34)

We’ll focus on ∂µ ψ only as it is the only factor that will affect the invariance. Now, apply the local gauge transformation G to ∂µ ψ:

∂µ (Gψ) = G∂µ ψ + ∂µ G ψ, (2.35)

then, invoke the minimal coupling rule mentioned in section 2.2.2 to replace all ∂µ with Dµ , in this case, the covariant derivative takes the form:

Dµ ≡ ∂µ + iqτ · Aµ , (2.36)

here, Aµ is a vector-like construct made of three four-vectors, that is, similar to the case in section 2.2.2, but rather than one, there are three new ﬁelds introduced into the Lagrangian dentisy. Alternatively, they are called the Yang-Mills ﬁelds. The minimal coupling rule obliterates the second term in equation (2.35). So,

Dµ (Gψ) = GDµ ψ and in order for this to hold true, Aµ must satisfy a certain rule and to ﬁnd it, we go from Dµ (Gψ) = GDµ ψ, write out the covariant derivates long- hand: 0 0 ∂µ + iqτ · Aµ ψ = G ∂µ + qτ · Aµ ψ, (2.37) where primed terms indicates they were already transformed by G, like ψ0 = Gψ.

23 Together with equation (2.35), equation (2.37) now becomes:

0 ∂µ G ψ + G ∂µ ψ + iq τ · Aµ Gψ = G ∂µ ψ + iqG τ · Aµ ψ. (2.38)

−1 Cancel the G(∂µ ψ) on each side of the equation and multiply G on the right to obtain 0 a condition for τ · Aµ :

1 τ · A0 = G(τ · A )G−1 + i (∂ G)G−1. (2.39) µ µ q µ

From this point on, to find the exact solution is extremely formidable. The approximate transformation rule in the limiting case of very small |λ|, however, is rather straightforward and as a finite gauge transformation is bulit upon infinitesimal ones, finding the approximate transformation is equivalent to finding the exact one. Now, expand the relative matrices and keep only the first-order terms:

−1 G ≈ 1 − iqτ · λ,G ≈ 1 + iqτ · λ,∂µ G ≈ −iqτ · ∂µ λ. (2.40)

In this approximation, equation (2.39) becomes:

0 τ · Aµ ≈ τ · Aµ + iq τ · Aµ ,τ · λ + τ · ∂µ λ, (2.41) the square bracket stands for commutator and we’ve already shown in the previous section that:

(τ · a)(τ · b) = a · b + iτ · (a × b). (2.42)

So the commutator [τ · Aµ ,τ · λ] becomes:

τ · Aµ ,τ · λ = τ · Aµ (τ · λ) − (τ · λ) τ · Aµ = iτ · Aµ × λ − iτ · λ × Aµ = 2iτ · Aµ × λ . (2.43)

24 Thus, the approximate transformation rule for Aµ is:

0 Aµ → Aµ ≈ Aµ + ∂µ λ − 2q Aµ × λ . (2.44)

This rule ensures the minimal coupling rule holds and the Lagrangian density ob- tains its local guage invariance status. However, this is not the ﬁnal step as those three newly introduced ﬁelds haven’t been interpreted. As in section 2.2.2, each one of them requires an additional term, thus, three new terms need to be added to the Lagrangian density. Once again, we look to the Proca Lagrangian density, equation (2.4). This time, it takes three of them:

1 µν 1 µν 1 µν 1 L = − F F − F F − F F = − Fµν · F . (2.45) A 4 1 µν1 4 2 µν2 4 3 µν3 4 µν

In order not to spoil the local gauge invariance, the second term in Proca Lagrangian density is again ignored by letting the mass of the field, m, equals to zero. However, for the case at hand, the field strength tensor, F µν , itself also need to be modified as the original definition, equation (2.5) is no longer invariant under local SU(2) transformation. The modified field strength tensor takes the form (Griffiths, 2008, p.364):

Fµν ≡ ∂ µ Aν − ∂ ν Aµ − 2q(Aµ × Aν ), (2.46)

µν this new form restores the invariance of F · Fµν under local SU(2) gauge transformations and the ﬁnal Lagrangian density becomes:

µ 1 µν µ L = iψγ ∂ ψ − mψψ − F · F − (qψγ τψ) · A . (2.47) µ 4 µν µ

By the same token, we can identify three current densities associate with each newly introduced ﬁeld from equation (2.47):

Jµ ≡ qψγ µ τψ. (2.48)

25 The above presents the mathematical formulation of Yang-Mills SU(2) gauge theory. By the same token, gauge theories of higher order symmetry can be constructed. For instance, imposing the principle of local gauge invariance upon a Dirac Lagrangian density, which possesses SU(3) symmetry, will generate quantum chromodynamics (QCD), the covariant derivative of that particular gauge theory introduces eight new massless gauge fields into the Lagrangian density and the gauge bosons produced by these fields are precisely the eight massless gluons we are so familiar with. In short, it is suffice to say that mathematical implementations of this kind formulated the entire Standard Model. However, there is one, yet fatal problem with this powerful mathematical tool. The gauge theory of SU(2) is strikingly similar to that of U(1). Yet, it is far less successful than QED. The reason behind this is clear, except photons for U(1), gluons for SU(3) and currently-still-hypothetical gravitons, there are no more massless bosons in nature. The gauge bosons described by Yang-Mills theory are nowhere to be found. The three Yang-Mills fields introduced in the derivation above are somewhat familiar, reminiscent of the electroweak interactions. The corresponding gauge bosons are strikingly similar to W+ and W- bosons, however, these particles are hardly massless. Of course there is no coincidence and in order to solve this problem, we need to first address the concept of spontaneous symmetry breaking.

2.4 Spontaneous Symmetry Breaking

In a Lagrangian density, terms can be divided into a kinetic part and several potential parts, like the traditional definition of the Lagrangian, L = T −U. The term indicating mass is usually of the second order in the field and with a sign opposite that of the kinetic term, like the second term in the Proca Lagrangian density. However, sometimes the mass term is concealed, consider the following scalar field, φ:

1 µ −αφ 2 L = ∂ φ (∂ φ) + e , (2.49) 2 µ

26 where α is some constant. The ﬁeld represented by this Lagrangian density seems to be massless since there is no term of the second order in the ﬁeld, but if we expand the exponential into a power series:

1 µ 2 2 1 4 4 L = ∂ φ (∂ φ) + 1 − α φ + α φ + ..., (2.50) 2 µ 2 then the third term satisfies our profile, it is of the second order in the field and has a sign opposite that of the first (kinetic) term. In the Lagrangian density, the constant one has no effect on the field equation and the rest of the terms that come from the expansion represent higher order couplings. In other cases, there might be a term of the second order in the field, but the sign is not correct, like:

1 µ 1 2 2 1 2 4 L = ∂ φ (∂ φ) + β φ − λ φ , (2.51) 2 µ 2 4 here β and λ are again constants. A plus sign indicate the mass is imaginary which is obviously impossible. To get around this, we identify the kinetic part and potential part. In this case, the first term signifies interactions which belongs to the kinetic part, the second and the third term are related to the field which belong to the potential part. We define: 1 1 U (φ) ≡ − β 2φ 2 + λ 2φ 4. (2.52) 2 4

Then, take the ﬁrst order derivative with respect to φ and set the result equal to zero. The solutions are: φ = 0,±β/λ and the minima occur at φ = ±β/λ. We, then, rede- ﬁne a new variable as deviations from these minima:

β η ≡ φ ± . (2.53) λ

Rewrite the Lagrangian density in terms of η:

4 1 µ 2 2 3 1 2 4 1 β L = ∂ η (∂ η) − β η ± βλη − λ η + . (2.54) 2 µ 4 4 λ 2

27 The second order term now has the correct sign. The third, fourth terms in the La- grangian density represent higher order couplings and the last term is a constant which signiﬁes nothing.

Note here, the original Lagrangian dentisy, equation (2.51), is even in φ. The modified Lagrangian density, however, no longer possesses this symmetry. Yet, those two expressions represent the same field. Its symmetrical property has been broken. Phe- nomena like this are called spontaneous symmetry breaking, the word “spontaneous” indicates there is no external agents at work. These kind of phenomena occur is because when working with a specific Lagrangian density, in order to obtain its mass term or to make the physics more transparent, we are obliged to choose only one of the possible ground states. The set of all ground states indeed shares the same symmetry as the original Lagrangian density, but only one does not and the asymmetry is inevitably introduced into the picture once we start working. More interesting effects appear when the symmetry of the original Lagrangian density is continuous. The Goldstone’s theorem states that spontaneous breaking of a continuous global symmetry is always accompanied by the appearance of one or more massless scalar particles (Griffiths, 2008, p.377). These particles are referred to as the Goldstone bosons, but as they are still massless, the masses of W+,W- and Z are unaccounted for as before. This will be ultimately solved by combining the idea of spontaneous symmetry breaking with the principle of local gauge invariance or it is more commonly known as the Higgs mechanism.

2.5 Higgs Mechanism

The combination of local gauge invariance and spontaneous symmetry breaking brought forth one of the greatest achievements in theoretical physics, the Higgs Mech- anism, which is responsible for all elementary particles acquiring mass. To understand

28 this, consider the following Lagrangian density:

1 µ 1 µ 1 2 2 2 1 2 2 22 L = ∂ φ (∂ φ ) + ∂ φ (∂ φ ) + β φ + φ − λ φ + φ , 2 µ 1 1 2 µ 2 2 2 1 2 4 1 2 (2.55) with β and λ being some random constants. This equation possesses continuous global rotational symmetry, that is L is invariant under transformations of the form:

0 φ1 → φ1 = φ1 cosθ + φ2 sinθ

0 φ2 → φ2 = φ1 sinθ + φ2 cosθ, (2.56) for any angle θ. Next, divide the Lagrangian density into kinetic and potential parts, it is obvious the minima lie on a circle of radius β/λ. Then redeﬁne two new variables:

β η ≡ φ − ;ξ ≡ φ . (2.57) 1 λ 2

Up until now, the procedure is pretty much the same as what we have seen in the previous section. But before we rewrite L in terms of the new variable, η and ξ, let us ﬁrst rearrange equation (2.55) into a more compactiﬁed version in order to apply the principle of local gauge invariance. Let:

φ ≡ φ1 + iφ2, (2.58)

2 2 then φ1 + φ2 can be written as: φφ, where φ is the complex conjugate of φ. The third and fourth term in equation (2.55) can be rewritten directly. For the ﬁrst term,

µ µ replace (∂µ φ1)(∂ φ1) with [∂µ (φ − iφ2)][∂ (φ − iφ2)] and expand it, the second term in equation (2.55) will be cancelled and the ﬁnal expression becomes:

1 µ 1 2 1 2 2 L = ∂ φ (∂ φ) + β φφ − λ φφ . (2.59) 2 µ 2 4

Note here, combining the variables φ1, φ2 into φ also caused spontaneous symmetry breaking, but instead of breaking the symmetry completely, the new formulation ob-

29 tained a global U(1) symmetry as equation (2.59) is invariant under transformations of the form G = eiθ . Now, in order to convert the global U(1) symmetry into a local one, invoking the minimal coupling rule and this time the covariant derivative takes the form:

Dµ ≡ ∂µ + iqAµ , (2.60)

where Aµ is the newly introduced gauge ﬁeld. We are not going to explore the corresponding transformation rule for Aµ as it is irrelevant to current topic. Replacing all

∂µ with Dµ and incorporating the term signifying the physics of Aµ , we now have the locally invariant Lagrangian density:

1 µ µ 1 2 1 2 2 1 µν L = ∂ − iqA φ [(∂ + iqA )φ] + β φφ − λ φφ − F F . 2 µ µ 2 4 4 µν (2.61) We can now start combining the power of both local gauge invariance and spontaneous symmetry breaking. First, switch the variables of equation (2.61) back to φ1,

φ2 and then use η, ξ in equation (2.57) as the ﬁnal choice of variables. The math is extremely lengthy but just basic algebra, the result is:

1 µ 2 2 1 µ L = ∂ η (∂ η) − β η + ∂ ξ (∂ ξ) 2 µ 2 µ " # 1 1 qβ 2 + − F µν F + A Aµ + { qη ∂ ξ − ξ ∂ ηAµ 4 µν 2 λ µ µ µ β 1 + q2η A Aµ + q2 η2 + ξ 2A Aµ − λβ η3 + ηξ 2 λ µ 2 µ 2 1 qβ β 2 − λ 2 η4 + 2η2ξ 2 + ξ 4 } + ∂ ξAµ + . (2.62) 4 λ µ 2λ

1 qβ 2 µ We are interested in one term only, 2 ( λ ) Aµ A . This is a mass term and it is clearly not zero. The gauge ﬁeld has acquired a mass and hence it produces massive gauge bosons. However, this is not the whole story. Further examining the equation, we

1 µ found the massless Goldstone boson still exists as indicated by the term 2 (∂µ ξ)(∂ ξ). qβ µ The rest of the terms can be attributed to various couplings, all but one, λ (∂µ ξ)A ,

30 which cannot be interpreted as either a kinetic term or a coupling.

Both problem can be easily solved by noting both terms have ξ in it. We only need to exploit the local gauge invariance of L and choose a particular gauge which will set ξ equal to zero. One such gauge is:

0 −1 φ2 −1 φ2 φ → φ = φ1 cos −tan − φ2 sin −tan φ1 φ1 −1 φ2 −1 φ2 + i φ1 sin −tan + φ2 cos −tan , (2.63) φ1 φ1 which looks formidable, but it is rather straightforward. Noting:

1 x −tan−1 x = tan−1 (−x),costan−1 x = √ ,sintan−1 x = √ , (2.64) 1 + x2 1 + x2

0 0 φ is always real, which means φ2 is zero and since previously we chose ξ = φ2 from equation (2.57), ξ will be zero once the new gauge is implemented. To apply the new gauge, return to equation (2.61). The implementation is even lengthier, but still just basic algebra and the ﬁnal result is:

" 2 # 1 µ 2 2 1 µν 1 qβ µ L = ∂ η (∂ η) − β η + − F F + A A 2 µ 4 µν 2 λ µ 2 β 1 1 β 2 + q2η A Aµ + q2η2 A Aµ − λβη3 − λ 2η4 + , (2.65) λ µ 2 µ 4 2λ which is just equation (2.62) without any ξ term, as expected. The terms in the first square bracket describe a massive interacting field, the Higgs field. Terms in the second square bracket describe the newly introduced gauge field and it acquired a mass, which is precisely the result we eager to get. Terms in the curly bracket represent various couplings and the last term is just a constant. We’ve demonstrated that combining local gauge invariance and spontaneous symmetry breaking, then carefully selecting a particular gauge would solve the massless gauge boson problem. This marvelous technique is now known as the Higgs mechanism and with its help, Yang-Mills theory finally bore fruit and the resulting theory

31 was called Yang-Mills-Higgs (YMH) theory or synonymously known as the SU(2) Georgi-Glashow model (Georgi and Glashow, 1972), the theoretical framework of this dissertation, which will be discussed in the next section.

2.6 SU(2) Yang-Mills-Higgs Model

The derivations we have shown thus far are mostly based on Dirac Lagrangian density, equation (2.2). However, Higgs bosons do not ﬁt this proﬁle as they are scalar bosons with spin-0. Instead, we turn to the Klein-Gordon Lagrangian density, equation (2.1). Now, consider its kinetic part:

1 µ LK = ∂ Φ (∂ Φ), (2.66) KG 2 µ with Φ being the Higgs field. Our next step is to invoke the minimal coupling rule to replace all ∂µ with Dµ as usual. However, the case at hand is more subtle as the Higgs field is presented in the YMH theory in a unique way called the adjoint representation, a concept deserves a bit more explanation. In group theory, groups like U(N), SU(N) are also known as Lie groups as they can be represented by a smooth curve in a 2n2-dimensional real Euclidean space. For example, the group U(1) is a circle with unit radius located at the origin of a two- dimensional plane. 2n2 comes from the fact that the real and imaginary parts of each element of an n × n complex matrix can be taken as coordinate axes. A representation of a group is a one-to-one mapping performed onto this space, like the circle group U(1) can also be represented by a group which can be depicted as an ellipse (But not a sphere as then it is no longer one-to-one.). Among those representations, there is a particular type called the adjoint representation which arises naturally for any Lie group as it is defined on the tangent space near the identity element, the Lie algebra.

Suppose we have an element, φ, that belongs to the adjoint representation of SU(2). By deﬁnition, a transformation of the form, φ → φ 0 = GφG−1 for any G in SU(2), made to φ, the resulting element, φ 0, also belongs to the adjoint representation. Recall equation (2.39), absorb the constant vector, τ, into the gauge ﬁeld and set G to be

32 independent of xµ , we obtain:

0 −1 Aµ = GAµ G , (2.67)

0 0 where Aµ = τ · Aµ . This indicates Aµ takes values in the SU(2) Lie algebra and thus belongs to the adjoint representation of SU(2). Moreover, similar to a vector space, one could choose a base to represent all elements in a group. Elements of this base are called the generators of the group. In SU(2), there are three generators which are the familiar Pauli matrices. The generators can be used to span all elements inside the Lie algebra and exponentiated to span the entire Lie group, that is, for an element, say a, which takes values in the Lie algebra, one could write:

a = τ1a1 + τ2a2 + τ3a3 = τ · a. (2.68)

This represents all elements in the SU(2) Lie algebra and all elements in the Lie group SU(2) could be expressed as eiτ·a, which is exactly what we have shown in section 2.3.1. i is introduced into the exponential because the generators of SU(2) are actually a set of traceless skew-Hermitian matrices. Physicists tend to use Pauli matrices because we mostly work with Hermitian matrices and any skew-Hermitian matrix can be expressed as a product of a Hermitian matrix and i. Furthermore, it can be shown that adjoint representations are always real and since SU(2) has three generators, we can express the gauge ﬁeld by a triplet of real components which take values in the adjoint representation (Rubakov, 2002, p.67):

τa A = −ig Aa , (2.69) µ 2 µ where a = 1,2,3 is the SU(2) internal group indices and g is the gauge coupling con-

1 stant. The factor, 2 , is called structural constant and comes from the SU(2) Lie algebra.

Similarly, the ﬁeld strength tensor, Fµν can be expressed in terms of real compo- 0 nents as well. Starting from equation (2.39), we notice that Aµ belongs to the adjoint

33 representation even if G depends on xµ :

1 A0 = GA G−1 + i ∂ GG−1 µ µ q µ i = GA G−1 + G∂ G−1, (2.70) µ q µ

0 0 again, here Aµ = τ · Aµ and we exploit the relation:

−1 −1 −1 ∂µ GG = ∂µ G G + G ∂µ G = 0, (2.71) since GG−1 = 1 and thus, the form of the second term indicates it belongs to the adjoint

0 representation as well and consequently, so does Aµ . Speciﬁcally, in YMH theory, the gauge ﬁeld Aµ in the adjoint representation transforms according to:

0 −1 −1 Aµ → Aµ = GAµ G + G ∂µ G . (2.72)

0 −1 We now demand Fµν transform according to Fµν → Fµν = GFµν G . This requires equation: 0 0 −1 ∂µ Aν − ∂ν Aµ = G ∂µ Aν − ∂ν Aµ G , (2.73) to hold true. However, it obviously doesn’t. We then write out the differentiations and it shows that incorporating a commutator of Aµ and Aν would cancel any extra terms and thus satisfy the condition (Rubakov, 2002, p.65). So, the ﬁeld strength tensor in the adjoint representation takes the form:

Fµν = ∂µ Aν − ∂ν Aµ + Aµ ,Aν . (2.74)

Now, express Fµν in terms of a triplet of real components similar to equation (2.69):

τa F = −ig Fa , (2.75) µν 2 µν

34 then it follows:

τa τb τc F = −ig ∂ Aa − ∂ Aa + (ig)2 Ab Ac , µν 2 µ ν ν µ µ ν 2 2 τa τa = −ig ∂ Aa − ∂ Aa − g2Ab Ac iεabc 2 µ ν ν µ µ ν 2 τa = −ig ∂ Aa − ∂ Aa + gεabcAb Ac , (2.76) 2 µ ν ν µ µ ν

Here, we exploit the commutation relations of the Pauli matrices:

[τi,τ j] = 2iεi jkτk, (2.77) where εi jk is the Levi-Civita symbol. Hence, the real components of the ﬁeld strength

a tensor, Fµν , can be expressed as:

a a a abc b c Fµν = ∂µ Aν − ∂ν Aµ + gε Aµ Aν . (2.78)

Now go back to the covariant derivative. Similar to other gauge theories, in YMH model, the covariant derivative is constructed from:

Dµ = ∂µ + Aµ . (2.79)

0 −1 Again, we demand the covariant derivative transform according to Dµ → Dµ = GDµ G while acting on the Higgs ﬁeld, Φ. The procedures are the same as before and the result (Kleihaus and Kunz, 1999) is:

Dµ Φ = ∂µ Φ + Aµ ,Φ , (2.80)

35 then it follows:

τa τa τb τc τc τb D Φa = ∂ Φa + −ig Ab Φc − Φc −ig Ab µ 2 µ 2 2 µ 2 2 2 µ τa τb τc τc τb = ∂ Φa − ig Ab Φc + Φc ig Ab µ 2 2 µ 2 2 2 µ τa τa τa = ∂ Φa − i2gεabc Ab Φc + i(−i)gεabc ΦcAb µ 2 4 µ 4 µ τa τa = ∂ Φa + gεabc Ab Φc. (2.81) µ 2 2 µ

Hence, the covariant derivative in YMH theory takes the form:

a a abc b c Dµ Φ = ∂µ Φ + gε Aµ Φ . (2.82)

Next, the kinetic part of the gauge ﬁeld Lagrangian density involves a factor of

µν F Fµν , which can be more conveniently expressed as a trace of a matrix product, more generally we choose:

1 µν L = Tr F F , (2.83) Aµ 2g2 µν

a then, LAµ can be expressed in terms of real components, Fµν :

a a 1 a aµν 2 τ τ 1 a aµν LK = F F (−ig) Tr = − F F . (2.84) Aµ 2g2 µν 2 2 4 µν

Henceforth, the kinetic part of the Yang-Mills-Higgs Lagrangian density can be expressed as:

1 a aµν 1 µ a a LK = LK + LK = − F F − D Φ D Φ , (2.85) YMH Aµ KG 4 µν 2 µ aftering invoking the minimal coupling rule.

Now, the potential part of the YMH Lagrangian density, LPYMH , similar to the potential part of equation (2.49), can be expressed as a power series in the most generic form. However, in order for the theory to be renormalizable, terms of higher order in

36 the ﬁeld must be discarded, otherwise, physical quantities would blow up to inﬁnity.

As a result, LPYMH can only have two terms, a term of the second order representing ﬁeld mass and a term of the fourth order signifying interactions. Thus, in YMH model,

LPYMH takes the form: 1 a a 22 LP = λ Φ Φ − ξ , (2.86) YMH 4 where λ is the strength of the Higgs field, µ refers to the Higgs field mass and ξ is √ the vacuum expectation value of the Higgs field, which is defined as ξ = µ/ λ. The

1 factor 4 is chosen simply for mathematical convenience and there is an extra term if we work out the square, but it has no inﬂuence on the Lagrangian density being a constant. Finally, we have the complete Yang-Mills-Higgs Lagrangian density:

LYMH = LKYMH − LPYMH 1 1 1 2 = − Fa Faµν − Dµ ΦaD Φa − λ ΦaΦa − ξ 2 . (2.87) 4 µν 2 µ 4

Now, in YMH theory, the SU(2) symmetry is broken by the vacuum expectation value of the Higgs field. The procedure is quite similar to what we have shown in section 2.5. The Lagrangian density is reformulated by identifying the ground states of the Higgs potential, which correspond to the lowest energy configurations, and rein- terpreting the theory as a perturbation about one particular chosen ground state. This action breaks the symmetry of the YMH model, however, it is not complete as the final result still exhibit a U(1) symmetry. After the spontaneous symmetry breaking, three massive gauge bosons can be iden- tified. Two correspond to the W- and W+ bosons which are responsible for the charged weak interactions, the other one is the famous Higgs boson. The only massless gauge boson in the theory is the photon which corresponds to the unbroken U(1) symmetry. This model was once considered the correct interpretation of the electroweak interactions, however, its lack of explanations of the Z bosons and the neutral current clearly indicates this is not the whole story. It is now known that the electroweak interactions belong to an even larger symmetry group, the SU(2) × U(1) gauge group, which is

37 described by the Glashow-Weinberg-Salam model. The YMH model is also well-known for the wide spectrum of magnetic monopole, antimonople and vortex-ring configurations found within this thoery. This dissertation is dedicated to finding and understanding new magnetic monopole configurations in the YMH theory and in the next chapter, we first present a literature review on magnetic monopole solutions.

38 CHAPTER 3 - LITERATURE REVIEW ON MAGNETIC MONOPOLES

3.1 Maxwell’s Equations

Although the word “magnetic monopole” appeared rather early in the history of modern physics, the necessity of a magnetic charge carrier was not obvious until the mid-19th century. In 1865, Scottish scientist James Clerk Maxwell published a paper, “A dynamical theory of the electromagnetic ﬁeld” (Maxwell, 1865) and later became a milestone in human history. A set of simple ﬁrst order partial differential equations was elegantly formulated which governs all classical electromagnetic phenomena. From where the existence of a magnetic charge carrier, analogous to electrons or protons, became necessary out of consideration for the electromagnetic duality (Song, 1996). To understand this, let us take a look at Maxwell’s equations without any external source:

∇ · E = 0, ∂E ∇ × B − µ ε = 0, 0 0 ∂t ∇ · B = 0, ∂B ∇ × E + = 0, (3.1) ∂t

where E, B are the electric and magnetic ﬁeld strength and µ0, ε0 are the permeability and permittivity of free space respectively. The above equations are clearly invariant under the transformation:

E → B;B → −µ0ε0E. (3.2)

This property of the sourceless Maxwell’s equations is called the electromagnetic duality, a beautiful symmetry between electric ﬁeld and magnetic ﬁeld. However, this symmetry is easily broken once an external electric source is introduced into the pic-

39 ture, the Maxwell’s equations then become:

ρ ∇ · E = e , ε0 ∂E ∇ × B − µ ε = Je, 0 0 ∂t ∇ · B = 0, ∂B ∇ × E + = 0, (3.3) ∂t

where ρe refers to the electric charge density and Je stands for the electric current density. The first two entries in equations (3.3) are known as the inhomogeneous Maxwell’s equations, whereas the last two entries are the homogeneous ones. They remain unchanged as unlike electric charge carriers, no magnetic counterpart has been observed. Yet, there is no scientific evidence whatsoever as to why it is so. Now suppose there exists a magnetic source, such as a magnetic monopole, theoretically we could introduce the magnetic counterparts of electric charge density and electric current density, ρm and Jm into the homogeneous Maxwell’s equations and thus the Maxwell’s equations with both electric and magnetic sources are then given by (Griffiths, 1999, p.327):

ρ ∇ · E = e , ε0 ∂E ∇ × B − µ ε = µ Je, 0 0 ∂t 0

∇ · B = µ0ρm, ∂B ∇ × E + = −µ Jm. (3.4) ∂t 0

This way, the symmetry of Maxwell’s equations is restored and so does the electro-

40 magnetic duality. Equations (3.4) are invariant under the transformation:

E → BB → −µ0ε0E

ρe → µ0ε0ρm ρm → −ρe

Je → µ0ε0Jm Jm → −Je (3.5)

This aesthetically pleasing form of the Maxwell’s equations is simply charming. Though it does not supply us with any ironclad evidence of the existence of magnetic monopoles, it sure gives us hope and it is this very hope that motivates tens of thousands of physicists to hunt this mysterious particle to this day. Of course, the symmetry of the modiﬁed Maxwell’s equations only suggests the existence of magnetic monopoles from a purely mathematical point of view, but then comes Paul Dirac, whose work has a more physics touch.

3.2 Dirac Monopole

One of the things that physicists take as nature’s axiom without further explanations is the quantization of electric charge. Over the decades, we know it does not do us any harm, but it is there, like a benign tumor. In 1931, English physicist Paul Dirac published a paper, “Quantised Singularities in the Electromagnetic Field” (Dirac, 1931), in which he showed us a potential way to solve the electric charge quantization conundrum. Surprisingly enough, the protagonist is precisely a type of magnetic monopole known as the Dirac monopole.

In electrostasis, the physics between an electric source charge, qe2 , and a test charge, qe1 , is governed by the Coulomb’s law (in natural unit):

qe1 qe2 F = 2 rˆ21 = qe1 E. (3.6) |r21|

The electric ﬁeld of the source charge can be expressed as:

q 1 E = e rˆ = −q ∇ , (3.7) r2 e r

41 and the electric ﬂux through a sphere enclosing the source charge is:

Z Z π Z 2π qe 2 Φe = E · dS = 2 rˆ · r sinθdθdφ rˆ = 4πqe. (3.8) s 0 0 r

By the same token, the physics around a theoretical magnetic monopole can be similarly constructed. The magnetic ﬁeld, B, around a monopole with magnetic charge, qm, and the magnetic ﬂux, Φm, through a sphere enclosing it can be written as:

q 1 B = m rˆ = −q ∇ , (3.9) r2 m r Z Z π Z 2π qm 2 Φm = B · dS = 2 rˆ · r sinθdθdφ rˆ = 4πqm. (3.10) s 0 0 r

However, in vector calculus, according to Gauss theorem, the ﬂux of a vector ﬁeld through a closed surface is equal to the volume integral of the divergence over the space inside the surface. This means:

Z Z Φm = B · dS = ∇ · BdV. (3.11) s v

Now, the magnetic ﬁeld, B, and the magnetic vector potential, A, is related through:

B = ∇ × A, (3.12)

and since the divergence of the curl of any vector field is zero, this leads to Φm = 0, which is in direct contradiction to equation (3.10). To fix this, first notice that from equation (3.9), we can express the divergence of the magnetic field as:

1 ∇ · B = −q ∇2 . (3.13) m r

Note that the above expression has a singularity at the origin. This is what is causing the contradiction we encountered earlier. Fortunately, this can be avoided by introduc-

42 ing the Dirac delta function:

  0 if r 6= 0 δ 3(r) = . (3.14) ∞ if r = 0

Then, equation (3.13) can be recast into the form:

3 ∇ · B = 4πqmδ (r). (3.15)

Finally, performing the volume integral, we have:

Z Z 3 Φm = ∇ · BdV = 4πqmδ (r)dV = 4πqm, (3.16) v v which is consistent with equation (3.10). Let us now consider a magnetic monopole situated at the origin. The wave function of an electrically charged particle with charge, qe, has the form (Ryder, 1996, p.402):

Ψ = |Ψ|ei(p·r−Et), (3.17) with p,r being the momentum and position vector of the particle, E denotes the energy of said particle and the instant, t, at which these data is taken. Under the inﬂuence of the magnetic ﬁeld of the monopole, the momentum, p, in the above equation becomes:

0 p → p = p − qeA. (3.18)

This means the phase, α, is changed to:

0 i(p · r − Et) = α → α = α − qeA · r. (3.19)

For a closed path at ﬁxed r, θ, with φ ranging from 0 to 2π, the total change in phase,

43 Figure 3.1: Pictorial representation of the line and surface integrals appeared in equation (3.20).

∆α, can be calculated:

I Z ∆α = qe A · dl = qe (∇ × A) · dS Z = qe B · dS = qeΦm, (3.20) here, in the first line, Stokes’ theorem is invoked to change a line integral on a vector field into a surface integral of its curl. In Figure 3.1, the path and area used by the line and surface integrals are represented by green line, l, and red shaded area, S, respectively (Note here, S is a closed surface as the area formed by l is also a part of S.). It can be seen clearly from the figure that for any sphere of fixed radius, the surface area of S is θ dependent. As θ becomes larger, so does S and it reaches a maximum when θ equals π. In particular, when θ > π/2, the shaded area, S, then encloses the magnetic monopole at the origin and thus represents the magnetic flux,

Φm. Now, according to equations (3.10) and (3.16), we would expect the value of Φm to be 4πqm when θ reaches π. However, the path taken for the original line integral in equation (3.20) now shrinks to a point, this means dl is zero and so does Φm. Once again, we are facing a similar problem. The magnetic flux is zero. To fix this, Dirac pointed out that on a sphere of fixed radius, the magnetic vector potential, A, is well-defined everywhere expect the point located at the negative z-axis. This argument

44 is, in fact, applicable to spheres of any radius and thus forming the so-called “Dirac String”, a semi-inﬁnitely long singularity starting from the origin, on the negative z- axis, extending towards inﬁnity. Furthermore, in quantum mechanics, in order to guarantee that there is only a single value for the probability of a system being in a given state, the wave function, Ψ, must be single-valued. This leads to:

n ∆α = 2πn = q × 4πq =⇒ q q = , (3.21) e m e m 2 for any integer n. This is the famous Dirac quantization condition. It states that if there exists even if only one magnetic monopole in the entire universe, all electric charges will be quantized. However, as pointed out in Chapter 1, the Dirac quantization condition is only a necessary condition and thus logically does not prove the existence of magnetic monopoles, but this statement is too powerful and a lack of proper explanations as to why all electric charges are quantized have led lots of physicists to believe that magnetic monopoles must exist. Nevertheless, despite all those accomplishments of Dirac’s work, there is still a major ﬂaw in Dirac’s interpretation of monopoles. The string singularity is unphysical and must be dealt with. In the next section, we are going to discuss Wu and Yang’s formalism and how the string singularity is evaded.

3.3 Wu-Yang Formalism

The string singularity appeared in Dirac’s interpretation of magnetic monopoles is essentially a series of points with infinite magnetic potential. The occurrence of singularities in a theory is undesirable and harder to work with, but fortunately for us, it was demonstrated that the Dirac string can be eliminated by dividing the vicinity of the magnetic monopole into two overlapping regions and defining them separately. This method was first reported by Tai-Tsun Wu and Chen-Ning Yang in 1975 (Wu and Yang, 1975). According to Wu-Yang formalism, the space around the magnetic monopole at the origin is divided into two overlapping regions, R+ and R−, with R+ excludes the

45 Figure 3.2: Pictorial representation of the overlapping regions, R+ and R−, in Wu-Yang formalism.

negative z-axis and R− excludes the positive z-axis. This is depicted in Figure 3.2, R+ is made by rotating sector AOCB about the z-axis and R− is similarly made by rotating sector DOFE.

In R+, the magnetic vector potential, A+, is deﬁned as (Ryder, 1996, p.404):

qm 1 − cosθ A+r = A = 0,A = . (3.22) +θ +φ r sinθ

Note that A+φ is well-defined everywhere except when θ = 0 and π. However, although A+φ is not defined when θ = 0, it is indeed finite as:

1 − cosθ lim = 0, (3.23) θ→0 sinθ

and thus the magnetic vector potential, A+, vanishes along the positive z-axis. On the other hand, (1 − cosθ)/sinθ does blow up when θ approaches π. So A+ is infinite along the negative z-axis and it is exactly where the Dirac string is located. Hence, by excluding the negative z-axis, A+ is finite everywhere except at the origin. Now, if we choose the Dirac string to be along the positive z-axis, then the magnetic vector potential must be modified accordingly:

qm 1 + cosθ A−r = A = 0,A = − , (3.24) −θ −φ r sinθ

46 and readers could easily tell from the labels that the above equations are exactly how

A− is deﬁned in region R−. The singularities along the positive z-axis are similarly excluded to ensure A− is ﬁnite everywhere in R− except at the origin.

The imminent problem is that in the overlapping region, A+ and A− are deﬁned differently. However, they are related by a U(1) gauge transformation, which has the form:

g = e2iqeqmφ , (3.25)

and thus A+φ , A−φ are related by:

2qm i −1 A−φ = A+φ − = A+φ − g∇φ g . (3.26) r sinθ qe

This indicates that A+ and A− describe exactly the same physics and the awkward Dirac string is eliminated completely, leaving only one singularity at the origin where the monopole is located. Moreover, for the gauge transformation, g, to be well-deﬁned, the following relation must be satisﬁed:

g(φ) = g(φ + 2π), (3.27) otherwise, the same point in space would correspond to multiple values, which renders A ill-deﬁned. That is, g must be single-valued. Then, it follows:

n g = e2iqeqmφ = e2iqeqm(φ+2π) =⇒ q q = , (3.28) e m 2 for any integer, n, and once again, we arrived at the Dirac quantization condition without any string singularities. This is monumental as it means that if magnetic monopoles do exist, their magnetic charge must be quantized under any circumstances (Preskill, 1984, p.478) and in unit of:

1 hc¯ qm = in natural units or qm = in Gaussian units. (3.29) 2qe 2qe

47 This is called the unit Dirac charge.

The integer, n, is also called the φ-winding number. Topologically, it represents the number of times a curve travels counterclockwise around a speciﬁc point. Here, it means the number of times g revolves and coveres the entire U(1) group. It also represents the number of magnetic charge in Dirac units, which plays an important role in constructing multimonopole solutions (see section 3.6). Wu and Yang took Dirac’s work one step further. Their interpretation of magnetic monopoles is more generalized and enlightening. However, the golden age of magnetic monopole research has only begun when non-Abelian gauge groups were brought into the picture. Dutch physicist, Gerard ’t Hooft, and Russian physicist, Alexander Markovich Polyakov, are amongst the pioneers in this ﬁeld. Their work will be discussed in the next section.

3.4 ’t Hooft-Polyakov Monopole

In the previous three sections, the discussions made all lie within the context of Maxwell’s electrodynamics or the Abelian group, U(1). Magnetic monopoles were added into the theoretical framework axiomatically. It is important to note that either a theory with magnetic monopoles is more symmetric (as discussed in section 3.1) or the inclusion of magnetic monopoles might provide a potential explanation to the electric charge quantization problem (in section 3.2 and 3.3) does not guarantee the existence of magnetic monopoles. However, solutions with magnetic charge, which can be identiﬁed as magnetic monopoles, are found in a number of non-Abelian gauge groups with spontaneous symmetry breaking introduced. These solutions can be found even when the matter and gauge ﬁelds in the theory carry electric charge only (Ryder, 1996, p.406). This means the magnetic charge is topological, that is, it originated from the complex structure of the gauge group itself. Moreover, if the underlying gauge group truly describes our universe, then the corresponding solutions found within are the ironclad evidence of the existence of magnetic monopoles. The theoretical possibility of this type of

48 magnetic monopoles were ﬁrst reported by ’t Hooft and Polyakov (’t Hooft, 1974; Polyakov, 1974) independently.

a a In monopole researches, the particular form the real triplets, Aµ and Φ , take in a certain model is called the magnetic ansatz. Now, let us consider the time-independent magnetic ansatz ’t Hooft put forward. It has the asymptotic form when r approaches inﬁnity:

a A0 = 0, rb Aa = −εiab , i gr2 ra Φa = ξ , (3.30) r here, i = 1,2,3. This ansatz, when applied to the YMH model discussed in section 2.6, gives rise to a radial magnetic ﬁeld which could be ascribed to the so-called ’t Hooft-

Polyakov monopole. To see this, a gauge invariant Fµν was constructed (’t Hooft, 1974) which has the form:

1 a a 1 a b c Fµν = Φ Fµν − εabcΦ Dµ Φ (Dν Φ ). (3.31) |Φ| g|Φ|3

a a Recall the field strength tensor, Fµν , and covariant derivative of the Higgs field, Dµ Φ , defined in equations (2.78) and (2.82). Inserting the spherical symmetic magnetic ansatz (3.30) leads to:

ξ ΦaFa = − ε µνara, µν gr3 ra rc D Φa = ξ∂ + gεabcAb ξ µ µ r µ r δ µa rµ ra ξrmrc = ξ − − εabcε = 0. (3.32) r r3 ibm r3

Substitute the above into equation (3.31) and we have an expression for the newly deﬁned Fµν : 1 F = − ε µνara. (3.33) µν gr3

49 Now recall the matrix definition of the field strength tensor defined in section 2.2.1. It is easy to see that there exists a radial magnetic field, Ba, which can be expressed as:

ra B = , (3.34) a gr3

Also note that the electric ﬁeld is zero as Ei = F0i = 0. Then, the total magnetic ﬂux,

Φm, through a sphere of radius, r, can be easily calculated:

|ra| 1 4π Φ = 4πr2|B | = 4πr2 = 4πr2 = . (3.35) m a gr3 gr2 g

Compare the above result with equation (3.10) and we conclude that the magnetic charge, qm, is such that:

gqm = 1. (3.36)

So, if the Higgs coupling constant, g, takes on the value of unit electric charge, qe, the above expression is twice the Dirac unit (see equation (3.29)).

Furthermore, the potential four-vector, Aµ , can be expressed as:

1 A = ΦaAa , (3.37) µ |Φ| µ

and thus, under the ’t Hooft ansatz (3.30), Aµ = 0, which means the magnetic field, Ba, we arrived at earlier, is solely contributed by the Higgs field. Moreover, with equation (3.37), the gauge invariant field strength tensor constructed by ’t Hooft can be recast into the following form (Arafune et al., 1975):

1 F = ∂ A − ∂ A − ε Φˆ a ∂ Φˆ b ∂ Φˆ c, (3.38) µν µ ν ν µ g abc µ ν where Φˆ i = Φi/|Φ|. Additionally, this form can be divided into a gauge and a Higgs

50 part:

Fµν = Gµν + Hµν , 1 G = ∂ A − ∂ A ,H = − ε Φˆ a ∂ Φˆ b ∂ Φˆ c. (3.39) µν µ ν ν µ µν g abc µ ν

It can be seen that the field strength tensor in this form is regular everywhere. This is another outstanding feature of the ’t Hooft-Polyakov monopole indicating that it has a finite energy and its mass can be calculated. ’t Hooft estimates the monopole mass to be of the order of 137Mw, where Mw is a typical vector boson mass, so the monopoles are extremely heavy and the mass is inversely proportional to g2 (Ryder, 1996, p.409). ’t Hooft-Polyakov monopoles are, in fact, solutions to the equations of motion of the SU(2) Yang-Mills-Higgs model. Recall the Lagrangian density we arrived at in equation (2.87) and then apply the relativistic Euler-Lagrange equation, taking the gauge field and Higgs field as independent field variables, we obtain the equations of motion:

µ a µ a abc bµ c abc b c D Fµν = ∂ Fµν + gε A Fµν = gε Φ Dν Φ

µ a a b b 2 D Dµ Φ = λΦ Φ Φ − ξ (3.40)

a a with field strength tensor, Fµν , and covariant derivative of the Higgs field, Dµ Φ , defined in equations (2.78) and (2.82) as usual, g is the Higgs coupling constant, λ stands √ for the Higgs field strength, µ refers to the Higgs field mass and ξ = µ/ λ is the vacuum expectation value of the Higgs field. Equations (3.40) are coupled nonlinear second order partial differential equations. Solving equations of motion of this kind is indeed a formidable task, however, not impossible. It is approached by employing certain magnetic ansatz to lower the numerical difficulty. In this case, a more generic ansatz is given by (Prasad and Sommerfield,

51 1975):

Jra Aa = , 0 gr2 r j (1 − K) Aa = εai j , i gr2 Hra Φa = , (3.41) gr2 here, J,K,H are all functions of r and under this ansatz, the equations of motion given above are reduced to:

r2K00 = K K2 − 1 + K H2 − J2,

r2J00 = 2JK2, λ r2H00 = 2HK2 + H3 − ξ 2g2r2H, (3.42) g2 and the set of solution which corresponds to the ’t Hooft-Polyakov monopole is:

ξgr K = , sinh(ξgr) J = 0,

H = ξgr coth(ξgr) − 1. (3.43)

Clearly, when r approaches infinity, we have K → 1,H → ξgr, which reproduces the ’t Hooft ansatz (3.30). So far, exact solutions (Prasad and Rossi, 1981; Prasad, 1981; Forgacs,´ et al. 1981a, b; Ward, 1981; Rebbi and Rossi, 1980) can only be found in the so-called Bogomol’nyi-Prasad-Sommerfield (BPS) limit when the Higgs field is vanishing (Bo- gomol’nyi, 1976; Prasad and Sommerfield, 1975). This will be discussed in the next section.

52 3.5 Bogomol’nyi-Prasad-Sommerﬁeld (BPS) Solutions

The solution to the equations of motion of the YMH model shown in the previous section, given by Prasad and Sommerfield, was actually the result of fitting numerical solutions with simple analytic functions. Indeed, solving coupled nonlinear second order partial differential equations analytically was proven to be a tough task even for accomplished scientists like them. However, this difficulty can be circumvented, under certain scenarios, by reducing the equations of motion to of first order only. As we shall see later, solutions of this type correspond to energy global minima.

In the YMH model, the symmetric energy-momentum tensor, Tµν , which follows directly from the Lagrangian density given by equation (2.87) is (Prasad and Sommer- ﬁeld, 1975):

a aλ a a Tµν = Fµλ Fν + Dµ Φ Dν Φ + gµν L

a aλ a a = Fµλ Fν + Dµ Φ Dν Φ 1 1 1 2 + g − Fa Faµν − Dµ ΦaD Φa − λ ΦaΦa − ξ 2 , (3.44) µν 4 µν 2 µ 4 and the static energy of a system is thusly given by:

Z Z 3 a aα a a E = T00d x = {F0α F0 + D0Φ D0Φ

1 1 1 2 +g − Fa Faαβ − Dα ΦaD Φa − λ ΦaΦa − ξ 2 d3x 00 4 αβ 2 α 4 Z 1 1 = EaEa + D ΦaD Φa − (EaEa − BaBa) + (D ΦaD Φa − D ΦaD Φa) i i 0 0 2 i i i i 2 i i 0 0 1 2 + ΦaΦa − ξ 2 d3x 4 Z 1 = (EaEa + BaBa + D ΦaD Φa + D ΦaD Φa) 2 i i i i 0 0 i i 1 2 + ΦaΦa − ξ 2 d3x, (3.45) 4 with the electric and magnetic ﬁelds given by:

1 Ea = Fa,Ba = − ε Fa . (3.46) i 0i i 2 i jk jk

53 a Now, according to Bianchi identity for tensors of non-Abelian magnetic ﬁeld, DiBi = 0. We have the following identity:

a a a a a a a a ∂i (Bi Φ ) = (DiBi )Φ + Bi DiΦi = Bi DiΦi . (3.47)

And using the ﬁrst equation of motion in equations (3.40), noting that we assume

a a D0Φ = 0, we have DiEi = 0. Then the static energy, E, can be recast into a more general form:

Z 1 1 2 E = (EaEa + BaBa + D ΦaD Φa + D ΦaD Φa) + ΦaΦa − ξ 2 d3x 2 i i i i 0 0 i i 4 Z 1 Z 1 = (Ea ∓ D Φa sinα)2 d3x + (Ba ∓ D Φa cosα)2 d3x 2 i i 2 i i Z 1 2 Z Z + ΦaΦa − ξ 2 d3x ± cosα EaD Φad3x ± sinα BaD Φad3x, (3.48) 4 i i i i where α is an arbitrary constant. The last two terms in the above equation represent the electric and magnetic charges of the system respectively:

Z Z Z 1 a a 1 a a 3 qe = EidSi = E Φ dSi = E DiΦ d x. ξ i ξ i Z Z Z 1 a a 1 a a 3 qm = BidSi = B Φ dSi = B DiΦ d x. (3.49) ξ i ξ i

Note that when λ and µ2 approach 0, ξ is finite and the potential energy of the Higgs field vanishes. This is called the Bogomol’nyi-Prasad-Sommerfield (BPS) limit. In this limit, the total energy is minimized by functions of the form (Bogomol’nyi, 1976; Prasad and Somerfield, 1975):

a a a a Ei = ±DiΦ sinα,Bi = ±DiΦ cosα, (3.50) this is called the BPS equations and the energy is simply given by the last two terms of equation (3.48). That is, the energy of the system is solely determined by the electric and magnetic charges of the solution. So, the lower bound of the energy of the system

54 is then given by:

2 2 1/2 E ≥ ξ (|qe|sinα + |qm|cosα) = ξ qe + qm , (3.51)

2 21/2 2 21/2 where sinα = qe/ qm + qe ,cosα = qm/ qm + qe and it is also known as the BPS bound. In this dissertation, we are only concerned with electrically neutral system with qe = 0, consequently we have:

a a Bi = ±DiΦ ,E ≥ ξ|qm|. (3.52)

Now, plugging the ’t Hooft magnetic ansatz (3.30) into the purely magnetic BPS equation. The original coupled second order partial differential equations become:

dK dH r = −KH,r = H + 1 − K2, (3.53) dr dr which is clearly more tractable and the solution is precisely what we have shown in equation (3.43). The BPS equation provides us a way to find the analytic solutions, but only when the Higgs field is massless, µ = 0. In this special circumstances, the force mediated by Higgs field becomes long range and according to Manton’s work (1977), the mag- nitude of this force is found to have an inverse square law behavior, like the Coulomb force, but the net force between equally charged monopoles is zero. Moreover, any solution to the BPS equation is found to be in static equilibrium and noninteracting (O’Raifeartaigh et al., 1979). These features of the BPS solution make the coexistence of more than one monopole becomes possible. Configurations of this type is referred to as multimonopoles which will be briefly discussed in the next section.

3.6 Multimonopoles

As mentioned earlier, the φ-winding number, n, indicates the number of magnetic charge in Dirac unit that a solution carries. A system of two or more magnetic

55 monopoles with n ≥ 2 is a natural generalization. Comparing to solutions we have shown thus far, configurations with higher φ-winding number offers more insight into the interactions between monopoles instead of the monopoles themselves. However, finding solutions like that is no mean feat. The same method ’t Hooft and Polyakov find their monopole does not apply here. It is demonstrated by Weinberg and Guth (1976) that it is not possible for a spherically symmetric multimonopole configuration to possess finite energy. Solutions with n ≥ 2 can have axial symmetry at most. Generally, multimonopole solutions are classified into two categories depending on where the magnetic charges are located. In one type, all magnetic charges are located at the same point in space, whereas in the other, the monopoles are isolated and their total charge add up to n. The first ever exact multimonopole solutions was found in 1981 by Ward (1981). It is a BPS solution with n = 2 and possesses axial symmetry. In this solution, all magnetic charges are located at a single point. Rebbi and Rossi (1980) further studied this configuration and their numerical result showed that the interaction energy between monopoles is less than 1% of the total energy which confirms the work by O’Raifeartaigh et al. (1979). Multimonopole configuration of this type was soon generalized to have even higher φ-winding number by Prasad and Rossi (1981a, b). On the other hand, multimonopole configurations with magnetic charges located at different points in space were found when n ≥ 3 in the BPS limit (Hitchin et al., 1995; Houghton and Sutcliffe, 1996; Houghton etal., 1998). These solutions were inspired by skyrmion solutions (Skyrme, 1961; Kopeliovich and Shtern, 1987) as they possess similar symmetry shown by ’t Hooft-Polyakov monopole and Ward monopole solutions. Multimonopole solutions were not limited for BPS solutions. Configurations which only satisfy the second order SU(2) YMH equations of motion but not the first order BPS equation were investigated over the years. These non-BPS solutions possess energy higher than the BPS bound and the first such non-BPS solution was found by Ruber¨ (1985). It was extended by Kleihaus and Kunz (1999) to accommodate finite

56 Higgs potential. These solutions possess axial symmetry and represent a monopole- antimonopole pair (MAP) with non-zero magnetic dipole moment and zero net magnetic charge. Later, solutions of this kind was generalized to a monopole-antimonopole chain (MAC) configuration (Kleihaus et al., 2003a, b, 2004) in which monopoles and antimonopoles are arranged in alternating order along the z-axis. Unlike MAP solutions, they possess unit magnetic charge but no net magnetic dipole moment. In addition to MAP and MAC solutions, vortex ring solutions were found, a new configuration where the Higgs field vanishes on closed rings around the symmetry axis, when n ≥ 3.

57 CHAPTER 4 - CONSTRUCTION OF ONE-PLUS-HALF MONOPOLE SOLUTIONS

4.1 Introduction

In this chapter, we are going to discuss the construction of one-plus-half monopole solutions (Teh, Ng and Wong, 2014b) in YMH model. We start off by identifying the magnetic ansatz used in this particular model and explaining some of its properties, then followed by a discussion of the physical quantities investigated in this research which include the Higgs field, magnetic field and charge, magnetic dipole moments and the energy of the system. We will show the asymptotic behaviours of the solutions when r approaches infinity and finally, the numerical construction of the solutions is presented at the end of this chapter.

4.2 Magnetic Ansatz

a The real triplets, Aµ and Φa, take the axially symmetric form in constructing the one-plus-half monopole model (Teh, Ng and Wong, 2014b):

a 1 a 1 a 1 a 1 a gA = − ψ (r,θ)nˆ θî + P (r,θ)nˆ φî + R (r,θ)nˆ rî − P (r,θ)nˆ φî, i r 1 φ r sinθ 1 θ r 1 φ r sinθ 2 r a gA0 = 0,

a a a gΦ = Φ1 (r,θ)nˆr + Φ2 (r,θ)nˆθ , (4.1)

where P1 (r,θ) = sinθ ψ2 (r,θ) and P2 (r,θ) = sinθ R2 (r,θ). The above constitutes the magnetic ansatz used in this work. In particular,r î,θî,φî are the spatial spherical coordinate orthonormal unit vectors:

rˆi = sinθ cosφ δi1 + sinθ sinφ δi2 + cosθ δi3,

θˆi = cosθ cosφ δi1 + cosθ sinφ δi2 − sinθ δi3,

φˆi = −sinφ δi1 + cosφ δi2, (4.2)

58 a a a andn ˆr ,nˆθ ,nˆφ are the isospin coordinate orthonormal unit vectors:

a a a a nˆr = sinθ cosnφ δ1 + sinθ sinnφ δ2 + cosθ δ3 ,

a a a a nˆθ = cosθ cosnφ δ1 + cosθ sinnφ δ2 − sinθ δ3 ,

a a a nˆφ = −sinnφ δ1 + cosnφ δ2 . (4.3)

The φ-winding number, n, is an integer and in this work, it takes the value of n = 2,3,4. The magnetic ansatz (4.1) is invariant under gauge transformations of the form:

i anˆa f (r, ) ω = e 2 σ φ θ , (4.4) where σ a is the Pauli matrices. The transformed gauge potential and Higgs ﬁeld take the form:

0 1 gA a = − (ψ − ∂ f )nâ θˆ i r 1 θ φ i 1 a + {P cos f + P sin f + n[sinθ − sin( f + θ)}nˆ φî r sinθ 1 2 θ 1 + (R + r∂ f )nâ rˆ r 1 r φ i 1 a − {P cos f − P sin f − n[cosθ − cos( f + θ)]}nˆ φî, r sinθ 2 1 r 0a gA0 = 0,

0a a a gΦ = (Φ1 cos f + Φ2 sin f )nˆr + (Φ2 cos f − Φ1 sin f )nˆθ . (4.5)

4.3 Higgs Field

a In order to calculate the magnetic ﬁeld, Bi, the Higgs ﬁeld, Φ , needs to be presented in the three dimensional Cartesian coordinate system. Consider the spherical expression:

a a a a gΦ = Φ1 (x)nˆr + Φ2 (x)nˆθ + Φ3 (x)nˆφ . (4.6)

Recall the isospin coordinate unit vectors deﬁned in equations (4.3). Substitude them into the above expression and combine terms with the same δ factor, we would arrive

59 at:

a a1 a2 a3 gΦ = Φ˜ 1 (x)δ + Φ˜ 2 (x)δ + Φ˜ 3 (x)δ , (4.7) and it is obvious that the tilded symbols are:

Φ˜ 1 = sinθ cosnφ Φ1 + cosθ cosnφ Φ2 − sinnφ Φ3,

Φ˜ 2 = sinθ sinnφ Φ1 + cosθ sinnφ Φ2 + cosnφ Φ3,

Φ˜ 3 = cosθ Φ1 − sinθ Φ2, (4.8) respectively.

Now, if we express the projections of Φa unto the Cartesian coordinate axes (in this case, Φ˜ 1, Φ˜ 2 and Φ˜ 3) using modulus of Φ (|Φ|), inclination (α) and azimuth (β), we would have:

Φ˜ 1 = |Φ|sinα cosβ,

Φ˜ 2 = |Φ|sinα sinβ,

Φ˜ 3 = |Φ|cosα. (4.9)

Consequently, the Higgs unit vector in three dimensional Cartesian coordinate system can be written as:

Φˆ a = sinα cosβδ a1 + sinα sinβδ a2 + cosαδ a3. (4.10)

In our case, the magnetic ansatz (4.1) takes an axially symmetrical form and Φ3 = 0. It is easy to see that:

cosα = h1 (r,θ)cosθ − h2 (r,θ)sinθ,β = nφ, Φ Φ h (r,θ) = 1 ,h (r,θ) = 2 . (4.11) 1 |Φ| 2 |Φ|

2 2 2 Additionally, the analytic form of sinα can be derived using h1 + h2 = 1 and cosα +

60 sinα2 = 1. The calculation is straightforward:

sinα = h1 (r,θ)sinθ + h2 (r,θ)cosθ. (4.12)

Finally, using the inclination angle, α, the Higgs ﬁeld and the gauge transformed Higgs ﬁeld from equations (4.1) and (4.5) can be expressed in a more symmetric and transparent form:

a a a Φ = |Φ|[cos(α − θ)nˆr + sin(α − θ)nˆθ ], 0a 0 a 0 a Φ = |Φ| cos α − θ nˆr + sin α − θ nˆθ , (4.13) where α0 = α − f .

4.4 Magnetic Field and Magnetic Charge

Recall the electromagnetic ﬁeld strength tensor proposed by ’t Hooft given by equation (3.31) and it can be decomposed into a gauge part and a Higgs part as shown in equations (3.39). The magentic ﬁeld, Bi, can be extracted from Fµν and similarly be divided into two parts:

1 B = − ε F = BG + BH, where i 2 i jk i j i i 1 BG = −ε ∂ A and BH = ε εabcΦˆ a∂ Φˆ b∂ Φˆ c. (4.14) i i jk j k i 2g i jk j k

H Substitude equation (4.10) into the expression for Bi above and with some calculation, it can be reduced to:

H gBi = −nεi jk∂ j cosα∂kφ n ∂ cosα n ∂ cosα = − rˆi + θˆi. (4.15) r2 sinθ ∂θ r sinθ ∂r

61 G Similarly, Bi can be simpliﬁed as well using equation (3.37) and the magnetic ansatz (4.1):

G gBi = −nεi jk∂ j cosκ∂kφ n ∂ cosκ n ∂ cosκ = − rˆi + θˆi, r2 sinθ ∂θ r sinθ ∂r 1 where cosκ = (h P − h P ). (4.16) n 2 1 1 2

The net magnetic charge of the system can then be easily calculated from these quantities:

Z I 1 i 3 1 2 M = ∂ Bid x = d σiBi = M + MH, where 4π 4π G I I 1 2 G 1 2 H M = d σiB and MH = d σiB , (4.17) G 4π i 4π i are the magnetic charge carried by the gauge ﬁeld and the Higgs ﬁeld respectively.

G H Moreover, the ’t Hooft magnetic ﬁeld, which is the sum of Bi and Bi , can be expressed as:

G H gBi = g Bi + Bi = −nεi jk∂ j (cosα + cosκ)∂kφ = −εi jk∂ jAk, (4.18)

where Ak is the ’t Hooft gauge potential. In particular, the contour of (cosα +cosκ) = C for any constant C on the plane described by φ = 0 is used to draw the magnetic field lines of the configuration and the orientation of the magnetic field is plotted using the magnetic field unit vector:

−∂θ (cosα + cosκ)rî + r∂r (cosα + cosκ)θî Bî = . (4.19) q 2 2 [∂θ (cosα + cosκ)] + [r∂r (cosα + cosκ)]

However, there is one downside of using the ’t Hooft electromagnetic ﬁeld strength tensor. It is singular when |Φ| = 0 and the magnetic charge density vanishes whenever |Φ| 6= 0. This indicates that the magnetic charges are discrete and reside on zeros of the Higgs ﬁeld. Unfortunately, there is no magnetic charge distribution under this scenario

62 and thus the sign of the charges can not be determined. Yet, it has been shown that the form of the electromagnetic field strength tensor is not unique outside the Higgs vacuum at finite values of r (Coleman, 1975). So, there exist other possible definitions that are less singular. One such definition was proposed in 1976 by Bogomol’nyi (1976) and Faddeev (1976a, b). The magnetic field takes the following form if the new definition is employed: a a Φ Bi = B , (4.20) i ξ where ξ is the vacuum expectation value of the Higgs field. Hence, instead of becoming singular when the Higgs field vanishes, the magnetic field becomes zero. In this case, the net magnetic charge can be calculated as:

Z Z 1 i 3 M = ∂ Bid x = M dθdr, (4.21) 4π where, 1 2 i M = r sinθ ∂ Bi (4.22) 2 is the magnetic charge density. We can also numerically evaluate the different magnetic charges at different dis- tances, r, from the origin by the following equations:

n n M = − {cosκ}|θ=π ,M = − {cosα}|θ=π , G 2g θ=0,r H 2g θ=0,r n M = − {cosα + cosκ}|θ=π . (4.23) 2g θ=0,r

4.5 Magnetic Dipole Moment

According to Maxwell’s theory of electromagnetism, the ’t Hooft gauge potential,

Ai, from equation (4.18), at large r tends to:

φˆi FG (θ) Ai = n(cosα + cosκ)∂iφ|r→ = n(acosθ + b) + , (4.24) ∞ r sinθ r

63 where, a and b are constants, the function, FG (θ), has the following form as r approaches inﬁnity:

FG (θ) = r [h2 (P1 − nsinθ) − h1 (P2 − ncosθ) − n(acosθ + b)]|r→∞. (4.25)

By plotting the graph of FG (θ) versus θ, we can see that the graph satisﬁes the relation:

2 FG (θ) = µm sin θ, (4.26)

here, µm is the dimensionless magnetic dipole moment and in this research, the value π of µm is taken from the graph of FG (θ) versus θ at θ = 2 .

4.6 Total Energy and Energy Density

In the BPS limit, the total energy of the system is minimized as demonstrated in section 3.5. For a model which is electrically charged, this energy is given by (Hartmann etal., 2000): 4πξ q E = M2 + Q2, (4.27) min g H where Q is the total electric charge of the system and the corresponding total energy for a non-BPS solution is:

Z g a a a a a a a a λ a a 22 3 E = B B + E E + DiΦ DiΦ + D Φ D Φ + Φ Φ − ξ dx 8πξ i i i i 0 0 2 Z = (ED)dx3, (4.28) where ED is the dimensionless energy density. However in this research, the model is electrically neutral and the corresponding formulae for energy are obtained by setting the total electric charge, Q, the electric

a a ﬁeld, Ei , and time component of the covariant derivative of Higgs ﬁeld, D0Φ , to zero (Manton, 1977; Actor, 1979). Additionally, this model has singularities but possess an integrable singular energy density (Teh et al., 2009, 2014a, b) and thus, it would be

64 useful to deﬁne a weighted energy density:

E = ED × 2πr2 sinθ, (4.29) and the total energy can be calculated as E = R E drdθ.

4.7 Exact Asymptotic Solution

So far, all of the physical quantities investigated in this research have been introduced in the previous several sections. Now, let us refocus on the construction of the one-plus-half monopole model by first discussing the exact asymptotic solutions when r approaches infinity. To obtain these solutions, we first apply the gauge transformation, descirbed in equation (4.4), to the following solution (Teh et al., 2014b):

a a a a a acosθ + b − 1 gA = 0,gA = A(r,θ)δ φˆi,gΦ = ξδ ,A(r,θ) = , (4.30) 0 i 3 3 r sinθ where a and b are constants. This solution is the least singular and align along one of the SU(2) isospin direction (Wu and Yang, 1975). Then, after the transformation, the gauge potential and Higgs ﬁeld would have the form:

a0 gA0 = 0, 0 1 1 gAa = ∂ f (r,θ)nâ θˆ + ∂ f (r,θ)nâ rˆ i r θ φ i r r φ i n a + {sinθ − sin( f (r,θ) + θ)[1 + rA(r,θ)sinθ]}nˆ φî r sinθ θ n a − {cosθ − cos( f (r,θ) + θ)[1 + rA(r,θ)sinθ]}nˆ φî, r sinθ r 0a a a gΦ = ξ [cos( f (r,θ) + θ)nˆr − sin( f (r,θ) + θ)nˆθ ]. (4.31)

Compare the above set of equations with the magnetic ansatz (4.1), we can see that the

65 proﬁle functions, ψ1, P1, R1, P2, Φ1 and Φ2, are:

ψ1 = −∂θ f ,P1 = nsinθ − nsin( f + θ)[acosθ + b],

R1 = ∂r f ,P2 = ncosθ − ncos( f + θ)[acosθ + b],

Φ1 = ξ cos( f + θ),Φ2 = −ξ sin( f + θ), (4.32) respectively and from the last two entries in equations (4.32), we would have |Φ| = q 2 2 Φ1 + Φ2 = ξ, as expected. Now, recall the expression we have arrived at in section 4.3 for the Higgs ﬁeld using the inclination angle, α, in equations (4.13). It is clear that α = − f and consequently:

cosα = cos(− f ) = cos f ,

cosκ = −cos f + [1 + r sinθA(r,θ)] = −cos f + acosθ + b. (4.33)

Moreover, the ’t Hooft gauge potential and net magnetic ﬁeld of the conﬁguration can be written as:

acosθ + b Ai = [1 + r sinθA(r,θ)]∂iφ = φˆi, r sinθ na gB = −ε ∂ [1 + r sinθA(r,θ)]∂ φ = rˆ . (4.34) i i jk j k r2 i

In this model, the function, f (r,θ) = pθ, in which p is a constant that controls the number of magnetic poles in the conﬁguration, whereas a and b decide the magnetic charge and magnetic dipole moment. The one-plus-half monopole solution is obtained

3 1 by choosing p = − 2 and a = b = 2 . These parameters are chosen in such a way that the resulting numerical results have ﬁnite energies and possess the least singularities.

66 After substituting those values, the proﬁle functions now have the form:

3 n θ ψ = ,P = nsinθ + sin [cosθ + 1], 1 2 1 2 2 n θ R = 0,P = ncosθ − cos [cosθ + 1], 1 2 2 2 θ θ Φ = ξ cos ,Φ = ξ sin , (4.35) 1 2 2 2 the above constitutes the exact asymptotic solution and moreover, the magnetic ﬁelds can be calculated accordingly:

( 3n 3 ) G 2 sin 2 θ n gB = − + rˆi i r2 sinθ 2r2 3n 3 H 2 sin 2 θ gB = rˆi. (4.36) i r2 sinθ

G H Note that the polar angle, θ, takes value from the interval [0,π] and both gBi and gBi are undeﬁned when θ = π. This indicates the magnetic ﬁelds are singular along the negative z-axis, where the Dirac string is located. For this reason, the net magnetic

G H ﬁeld, gBi, which can be expressed as gBi = gBi + gBi , needs to be modiﬁed due to the presence of a Dirac string. In this case, it is given by:

n gB = rˆ − 2nπδ (x )δ (x )θ (x )δ 3. (4.37) i 2r2 i 1 2 3 i

Please refer to Shnir’s book (Shnir, 2005, p.14) for detailed calculation about obtaining the above expression. This particular form contains several pieces of important information about the Dirac string. First, it possesses a magnetic flux of −2nπ/g as indicated by the coefficient of the second term, which is equivalent to a magnetic charge of −n/2g. The two delta functions, the second and third factor in the second term, show that the singularities are located along the third axis and lastely, the step function, θ (x3), tells us that the line singularity is half-infinite and only exists along 3 the negative x3-axis. δi is just a unit vector indicating direction. Equations (4.23) allows us to evaluate different magnetic charges at different dis-

67 tances from the origin. As seen from inﬁnity, the topological charges or the magnetic charges carried by the Higgs ﬁeld can be calculated as follows:

2 n θ=π n 3 θ= 3 π 3 θ=π MH = − cosα|θ=0,r→∞ = − cos θ|θ=0 + cos θ| 2 2g 2g 2 2 θ= 3 π n n = (+n) + − = + , (4.38) 2 2 when the gauge coupling constant, g, is set to 1. The splitting point of θ, which is equal to 2π/3 in this situation, is obtained by observing the vector plot of magnetic field. Along θ = 2π/3, magnetic fields of one monopoles and anti half-monopoles (as indicated by the negative sign) completely cancel each other and thus the Gaussian surface must be form along this angle to fully accommodate each magnetic charge carrier. Also, recall that there is also a magnetic charge contritubed by the Dirac string. Its value is −n/2g or −n/2 when g is set to 1. Hence, when the Dirac string is taken into consideration, the net magnetic charge of the configuration is zero. Thus, this particular configuration describes n ’t Hooft-Polyakov monopoles, each with a unit charge of 1/g superimposed at one point somewhere along the positive z-axis and n anti magnetic half-monopoles, each possesses a unit charge of −1/2g superimposed at the origin accompanied by a Dirac string, which itself contributes to the magnetic charge of the system by an amount of −n/2g. The resultant net magnetic charge of the system is consequently zero.

4.8 Numerical Construction of Solutions

In this section, we will focus on the actual construction process of the numerical solutions of the one-plus-half monopole conﬁguration. First of all, the set of magnetic ansatz (4.1) is substituted into the equations of motion (3.40). After some calculation, we noticed that each term in the equation contains a tensor made of three unit vectors and naturally, terms with the same combination of unit vectors are grouped together. Now, in order for the equations of motion to work out in this scenario, we need to set all tensor coefﬁcients to zero and subsequently, the

68 equations of motion are reduced to six coupled nonlinear second order partial differential equations.

r Next, the radial coordinate is compactified by changing the argument, r, tox ¯ = r+1 2 in order to avoid infinity. This is done by substitutions of the form: ∂r → (1 − x¯) ∂x¯ 2 2 and ∂ → (1 − x¯)4 ∂ − 2(1 − x¯)3 ∂ . This way, instead of dealing with the interval ∂r2 ∂x¯2 ∂x¯ 0 ≤ r < ∞, we are left with 0 ≤ x¯ ≤ 1. Then, the finite difference method is employed to approximate partial derivatives using algebraic fractions, turning the six coupled partial differential equations into a system of nonlinear algebraic equations. For a detailed explanation on finite difference method, readers are encouraged to refer to the textbook written by LeVeque (2007). The truncation error associated with the finite difference method is estimated at the order of 10−4. This system of algebraic equations are then discretized onto a non-equidistant grid.

The region of integration is covered by 0 ≤ x¯ ≤ 1 and 0 ≤ θ ≤ π. The grid size used in this research is 110×100 and the Jacobian sparsity pattern matrix corresponding to this grid is subsequently constructed using Maple. The Jacobian is a pattern matrix with each entry has only two values to take, 0 or 1, indicating the structure of each algebraic equation in the system. This information is then supplied to MATLAB for executing the numerical computations. In the process, entries with null value will automatically be skipped and thus greatly simplifies the calculation and reduces computing time. Nonlinear system solver “fsolve” from the MATLAB Optimization Toolbox is used to solve the system of algebraic equations numerically. The solver uses trust-region- reflective algorithm and a detailed description of this solver and its algorithm can be found in the Optimization Toolbox User’s Guide (The Mathworks, 2018, p. 2-10). An initial guess is required, which is defined as a constant coefficient multiplied by a matrix of ones, in a sense of a plane approximation. The size of the matrix corresponds to the size of the grid. When this initial guess is supplied to the MATLAB program, the solver will iterate to find the numerical solution. However, the convergence of numerical solutions is largely dependent on the closeness of said initial guess to the actual solution. If the initial guess fails to converge, we simply tweak the constant

69 coefficient until a convergence is reached. The result is then saved and used as a new initial guess of subsequent calculations. In our work here on the monopole plus half-monopole configuration, the equations of motion (3.40) are solved when the φ-winding number n = 2,3 and 4, with non-zero expectition value ξ = 1, gauge coupling constant g = 1, and when the Higgs self- coupling constant λb < λ ≤ 40 where λb stands for the critical lower bound. However, there is a special case when n = 2 in which λ only reaches 8 as the solution blows up afterwards. Bifurcation and transition phenomena are observed for n > 2. The exact point at which these phenomena happened are labelled as λc and λt respectively. More specifically, the boundary conditions used in obtaining the numerical solutions consist of four set of equations applied, respectively, to r → 0, r → ∞, θ = 0 and θ = π. The exact asymptotic solution discussed in previous section serves as the set of boundary condition applied when r → ∞. We dedicated a section to it as it is the more difficult one to obtain among all four. The method used to obtain the other three sets is rather straightforward and they are listed down below. Near the origin, when r → 0, we have the common trivial vacuum solution:

ψ1 = P1 = R1 = P2 = 0,Φ1 = ξ0 cosθ,Φ2 = −ξ0 sinθ,

sinθΦ1 (0,θ) + cosθΦ2 (0,θ) = 0,

∂r [cosθΦ1 (0,θ) − sinθΦ2 (0,θ)]|r=0 = 0. (4.39)

When θ = 0, we have:

∂θ ψ1 (r,θ)|θ=0 = 0,P1 (r,0) = 0,R1 (r,0) = 0,P2 (r,0) = 0,

∂θ Φ1 (r,θ)|θ=0,Φ2 (r,0) = 0. (4.40)

70 And lastly, when θ = π, we have:

∂θ ψ1 (r,θ)|θ=π = 0,P1 (r,π) = 0,R1 (r,π) = 0,∂θ P2 (r,θ)|θ=π = 0,

Φ1 (r,π) = 0,∂θ Φ2 (r,θ)|θ=π = 0. (4.41)

Together with the gauge ﬁxing condition (Kleihaus et al., 1998, 2003a):

r∂rR1 − ∂θ ψ1 = 0. (4.42)

This eliminates gauge related solutions, which are physically equivalent to each other, and thus simpliﬁes the calculation. The value of the physical quantities introduced in the previous sections will be evaluated and plotted according to the formulae given. To summarize, total energy E

(4.28), magnetic charge M (4.21), magnetic dipole moment µm by equations (4.25) and

(4.26), ’t Hooft magnetic field lines Bi (4.18) and its orientation Bî (4.19), weighted energy density E (4.29) and magnetic charge density M (4.22) and lastly, the Higgs mod- q 2 2 ulus, which is given by |Φ| = Φ1 + Φ2, will be presented and discussed in Chapter 5. Moreover, additional graphs are plotted as well to aid our study of the relationships between the corresponding physical quantities. Finally, we would like to point out that in addition to the fact that using dimensionless unit for quantities like the Higgs coupling constant, the strength of Higgs potential and the expectation value of the Higgs field greatly simplifies calculation, more impor- tantly, a model like the SU(2) Yang-Mills-Higgs theory describes an overly simplified reality. In other words, it is not physical. Exact and dimensional values are only of interest in a physical model like the SU(2)×U(1) Weinberg-Salam Model and thus, we mainly focus on the general trending behaviour of the data collected. In our work, a numerical solution is accepted as long as the plots of its physical quantites show a localized and smooth distribution.

71 CHAPTER 5 - ONE-PLUS-HALF MONOPOLE SOLUTIONS

5.1 Introduction

Back in Chapter 3, we briefly described the development of magnetic monopole research, starting from the Dirac monopoles all the way to the multimonopole configurations. Despite their theoretical bases are vastly different from each other, they all have one thing in common, that is, the individual magnetic charge carrier only possesses integer topological charges. In fact, there are relatively few papers discussing the possibility of magnetic charge carrier carrying half topological charges in the field. Hence, it would be interesting to delve into this particular aspect of the field. The existence of generic smooth Yang-Mills potentials of one-half monopole was demonstrated in the work of Harikumar et al. (2003), though no exact or numerical solutions had been given. Shortly after, exact one-half monopole mirror symmetric solutions and one-half monopole axially symmetric solutions with Dirac-like string formations were presented by Teh and Wong (2005a), yet these configurations do not possess finite energies. Later, more exact solutions were constructed, four gauge related solutions labelled as type A1, A2, B1, B2 were reported in the paper published by Teh et al. (2009). With these solutions used as boundary conditions as r → ∞, finite energy solutions of one-half monopole were finally found (Teh et al., 2012a, b). Lately, a new configuration where a ’t Hooft-Polyakov monopole coexists with a one-half monopole in the SU(2) Yang-Mills-Higgs theory have been found (Teh et al., 2014a, b), which is also known as the one-plus-half monopole configuration. Our work here is based on the one-plus-half monopole configuration mentioned above. In the original paper (Teh et al., 2014b), the φ-winding number, n, is 1, the resolutions used is 70×60 and the range of Higgs coupling constant considered is

0 ≤ λ ≤ 12. In this dissertation, the φ-winding number runs from 2 to 4, that is, we consider the multiple charged monopole plus half-monopole situation to further investigate the interactions between one-monopoles and half-monopoles. The grid size or resolution chosen in this research is 110×100 which is signiﬁcantly higher than the

72 one used in the original paper and thus provides more numerical accuracy. The range of Higgs coupling constant considered here is λb < λ ≤ 40 which makes the trending behaviour when λ gets larger more obvious. The main focus of our research is the bifurcation and transition phenomena that appeared in cases such as MAP and MAC (Kleihaus et al., 2003b, 2004; Teh and Wong, 2005b). They are to be expected in the multiple charged monopole plus half-monopole conﬁguration as well and indeed, these are observed in our numerical solutions for cases n = 3 and n = 4. These ﬁndings will be discussed in later sections of this chapter.

5.2 Numerical Results and Discussion

5.2.1 Case n = 2

There is no bifurcation or transition phenomenon observed for our one-plus-half monopole configuration when n = 2. Only one branch of numerical solution exists which is the fundamental branch and this will be abbreviated as FB from now on. The critical lower bound, λb, for n = 2 FB is found to be λb = 1.96. The critical lower bounds for different cases will be tabulated at the end of this chapter. The 3D surface and contour plots of the Higgs field modulus, |Φ|, the weighted energy density, E , and the weighted magnetic charge density, M , of n = 2 FB for λ = 4 are shown in Figure p 5.1 and in the plots, the ρ-axis is defined as ρ = x2 + y2. This particular value of λ is chosen for the ease and consistency of comparison as for n ≥ 3, numerical solutions might not exist for λ < 4 for all branches. In the Higgs Modulus 3D surface plots, Figure 5.1(a), the characteristic inverted circular and flatten cone structures of the ’t Hooft-Polyakov monopoles and the half- monopoles are easy to identify. The ’t Hooft-Polyakov one-monopoles and the half- monopoles are located at the vertices of the cones respectively. As shown in Fig- ure 5.1(b), the half-monopoles are located at the origin extending towards the negative z-axis forming a string-like structure, on the other hand, the ’t Hooft-Polyakov monopoles are located at z = 7.304 on the positive z-axis. This value is denoted as dz or pole separation.

73 (a) (b)

(e) (f)

Figure 5.1: The 3D surface and contour plots of the Higgs ﬁeld modulus |Φ|((a) and (b)), the weighted energy density E ((c) and (d)), and the weighted magnetic charge density M ((e) and (f)), for n = 2 FB when λ = 4.

74 As mentioned earlier, inspite of the fact that our numerical solutions are singular along the negative z-axis, they still possess an integrable energy density and because of this, our solutions have ﬁnite energies. The 3D surface plot of the weighted energy density, E , is presented in Figure 5.1(c). The peak of the weighted energy density of the ’t Hooft-Polyakov one-monopoles deviates from the z-axis. The double peak forms a torus which can be seen clearly from the contour plot as shown in Figure 5.1(d). The peaks are located at z = 7.295, ρ = ±0.989 and the peak value is 3.465. The weighted energy density of the half-monopoles, however, shaped like a long, narrow rugby ball. The peak is located at z = −4.822 along the negative z-axis and rapidly goes to zero along either side of ρ-axis. The peak value is found to be 1.164.

The 3D surface and contour plots of the weighted magnetic charge density, M , are shown in Figure 5.1(e) and (f). The plots are similar to that of the weighted energy density except the peak of the half-monopoles is now inverted. This indicates that the half-monopoles carry negative magnetic charges and could be interpreted as anti- half-monopoles. The peak of the weighted magnetic charge of the half-monopoles are located along the negative z-axis at z = −4.765 with a value of −2.136. The ’t Hooft- Polyakov one-monopoles carry positive magnetic charges and the peaks are located at z = 7.341, ρ = ±1.030. The peak value is evaluated to be 5.705. The relevant data are tabulated in Table 5.1. The contour plot of the ’t Hooft magnetic ﬁeld lines and vector ﬁeld plot of the ’t

Hooft magnetic field unit vector, Bî, for n = 2 FB when λ = 4 are shown in Figure 5.2(a) and (b). The magnetic field around the half-monopoles shaped like a string which stretches from z ≈ 0.212 to z ≈ −5.303. The magnetic field lines then diverge, but eventually converge on to the ’t Hooft-Polyakov one-monopoles. The direction of the magnetic field as shown in the vector field plot clearly indicates that

Table 5.1: Peak value and position of weighted energy density and weighted magnetic charge density for n = 2 FB when λ = 4.

Type of Monopole (ρ,z) Emax (ρ,z) Mmax Half-monopole (0,-4.822) 1.164 (0,-4.765) -2.136 FB One-monopole (±0.989,7.295) 3.465 (±1.030,5.705) 5.705

75 (a) (b)

Figure 5.2: Contour plot of the ’t Hooft magnetic field lines and vector field plots of the ’t Hooft magnetic field unit vector for n = 2 FB when λ = 4. the ’t Hooft-Polyakov one-monopoles carry positive magnetic charges and the half- monopoles carry negative magnetic charges which is in accordance with our discussions regarding the weighted magnetic charge density plots. The location of the one- monopoles and half-monopoles are indicated by red and black dots in the figure.

The plots of the pole separation, dz, magnetic dipole moment, µm, and the total energy, E, versus λ 1/2 for n = 2 FB and the magnetic charges carried by Higgs and gauge ﬁelds, MH and MG, versusx ¯ when λ = 4, are shown in Figure 5.3. The range of λ is 1.96 = λb ≤ λ ≤ 8 for case n = 2. The pole separation, dz, starts at

7.504 when λ = λb = 1.96, then goes down as we increase the value of λ. How- ever, when λ reaches roughly 3.31, the pole separation between the one-monopoles and half-monopoles reaches a plateau region with dz slightly rising upwards. When

λ reaches around 5.29, dz starts decreasing once again with the smallest value being 7.151 when λ = 8.00.

1/2 In Figure 5.3(b), we have the plot of magnetic dipole moment, µm versus λ .

Starting from µm = 5.640, it then goes down to 5.571 where µm reaches its minimum when λ = 2.10. Afterwards, the value increases all the way until µm = 10.56 when

λ = 6.60. Then decreases once again. The last data recorded is µm = 8.957 when λ = 8.00. As for the total energy, the shape of the graph looks like disproportioned sin wave

76 (a) (b)

Figure 5.3: Plots of the pole separation (a), magnetic dipole moment (b) and total energy (c) versus λ 1/2 for n = 2 FB and the magnetic charges carried by Higgs and gauge ﬁelds (d) versusx ¯ when λ = 4. as shown in Figure 5.3(c). E = 4.012 at the critical lower bound, the local maximum and minimum are located at λ = 4.12 and 6.70 where E = 4.155 and 3.774. From the last several data points we can clearly see, the increment is getting larger. After

λ = 8.00, we can only obtain extremely poorly converged data. One of the tested data is located at λ = 8.0001, its value is so high that the point is not even located in the current region. The convergence is so poor and the accuracy is over 109 times less than all the rest of the data collected, so we decided not to include these points in the plot.

We conclude that the numerical solution for this case blows up rapidly after λ = 8.00. Normally, the curves are expected to be monotonic in most cases, such as in a standard MAP result. Total energy of the system is expected to be increasing with

77 increasing λ. However, the sudden, drastic energy drop appeared roughly after when λ = 4 strongly suggest that there are some unknown interactions taking place within the model, even if no bifurcation or transition phenomenon is observed for this case. Lastly, in Figure 5.3(d), we can see clearly the change in magnetic charges carried by the Higgs and gauge fields as we move further away from the configuration. As r → ∞,x ¯ → 1, the magnetic charge carried by the Higgs field, MH, approaches 1, which is its predicted value. However, the magnetic charge carried by the gauge field, MG, has a limiting value of −0.103. Theoretically it should have a value of −1 so that the total magnetic charge of the configuration, M = MH + MG, is zero as discussed in the previous chapters. Unfortunately, our numerical data cannot account for the magnetic charge carried by the Dirac string and this non-vanishing value is actually due to the existence of a line singularity in the gauge field profile function, R2.

5.2.2 Case n = 3

Bifurcation phenomenon is observed in this case. The critical point is found to be λc = 2.28 after which two more branches of numerical solutions emerge. For the n = 3 FB, the critical lower bound is found to be λb = 0.48. The 3D surface and contour plots of the Higgs field modulus, |Φ|, the weighted energy density, E , and the weighted magnetic charge density, M , of n = 3 FB for λ = 4 are shown in Figure 5.4. In the Higgs Modulus 3D surface plot, Figure 5.4(a), once again we have the characteristic inverted circular and flatten cone structures of the ’t Hooft-Polyakov monopoles and the half-monopoles, yet in this case, the one-monopole cone is rather shallow. Conventionally, the location of the monopoles are indicated by the zeros in the Higgs field. In other words, the inverted cone structure should touch all the way down to zero. However, as the φ-winding number, the Higgs self-coupling constant and the grid size of the numerical solutions increase and with it, the complexity of the model, the convergence of the solutions are getting poorer and such ideal and desirable results become extremely hard to come by. So instead, without losing generality, we take the position of the vetices as the location of the monopoles as in Figure 5.4(b), it can be

78 seen that the ’t Hooft-Polyakov monopoles are located at z = 10.595 on the positive z-axis. The pole separation, dz, between the one-monopoles and the half-monopoles are higher as compared to case for n = 2 FB. The increase is about 45.1%.

The 3D surface and contour plots of the weighted energy density, E , for this case are shown in Figure 5.4(c) and (d). The double peaks of the ’t Hooft-Polyakov one- monopoles are located at z = 10.020, ρ = ±1.607 with a value of 3.283. As compared with n = 2 FB, the peak deviation from the z-axis is getting larger significantly with roughly 62.5% increase. However, the peak vaue is, in fact, lower than that of the n = 2 FB by about 5.3%. On the other hand, the peak of the half-monopoles is located at z = −6.801 along the negative z-axis. The peak value is found to be 1.456. The position of the peak is shifted further along the negative z-axis for about 41.0% and its value is larger than that of n = 2 FB by roughly 25.1%. In Figure 5.4(e) and (f), we have the 3D surface and contour plots of the weighted magnetic charge density, M . The ’t Hooft-Polyakov one-monopoles carry positive magnetic charges and the peaks are located at z = 9.917, ρ = ±1.545 with a value of 5.548 whereas the half-monopoles carry negative magnetic charges and the peak is located along the negative z-axis at z = −6.826 with a value of −2.644. As compare to n = 2 FB, the peak value of the one-monopoles is lowered by roughly 2.8% and that of the half-monopoles is increased by about 23.8%. The contour plot of the ’t Hooft magnetic field lines and vector field plot of the

’t Hooft magnetic field unit vector, Bî, for n = 3 FB when λ = 4 are shown in Figure 5.5(a) and (b). The string of the half-monopoles stretches from z ≈ 0.212 to z ≈ −7.848 now. And again, the magnetic field lines then diverge and subsequently converge on to the ’t Hooft-Polyakov one-monopoles. The point of convergence is roughly z ≈ 10.390 which is consistent with the location of the one-monopoles we obtained from the above discussion. The directions of the magnetic field are shown in the vector field plot and the one-monopoles and half-monopoles are indicated by the red and black dots in the graph. Bifurcation and transition phenonema are observed when we gradually increase

79 (a) (b)

(e) (f)

Figure 5.4: The 3D surface and contour plots of the Higgs ﬁeld modulus |Φ|((a) and (b)), the weighted energy density E ((c) and (d)), and the weighted magnetic charge density M ((e) and (f)), for n = 3 FB when λ = 4.

80 (a) (b)

Figure 5.5: Contour plot of the ’t Hooft magnetic field lines and vector field plots of the ’t Hooft magnetic field unit vector for n = 3 FB when λ = 4.

λ until some critical value. Two new branches of numerical solutions with higher energies than FB emerge. This particular value of λ for case n = 3 is 2.28 and is labelled as λc. The critical values for bifurcating branches to emerge for different cases will be tabulated at the end of this chapter. These solutions start from the same point, then diverge into two branches as we keep increasing λ. The one with the highest energy among all three branches (including FB) is labelled as Higher Energy Branch (HEB) and the other one with relatively lower energy (yet still higher than FB) is labelled as Lower Energy Branch (LEB). The 3D surface and contour plots of the

Higgs ﬁeld modulus, |Φ|, the weighted energy density, E , and the weighted magnetic charge density, M , of n = 3 HEB and LEB for λ = 4 are shown in Figure 5.6 and Figure 5.7. For n = 3 HEB and LEB, the ’t Hooft-Polyakov one-monopoles are located at z = 7.364 and z = 7.717 along the positive z-axis respectively as can be seen from the Higgs Modulus 3D surface and contour plots in 5.6(a), (b) and 5.7(a), (b). The pole separation is smaller than that of n = 3 FB, 20.5% decrease for HEB and 27.2% for LEB. With regards to the weighted energy densities, for n = 3 HEB, the one-monopole peaks are located at z = 7.295, ρ = ±1.360 with a value of 5.416. The half-monopole peak is located at z = −6.801 and its value is calculated to be 1.454. For n = 3 LEB,

81 (a) (b)

(e) (f)

Figure 5.6: The 3D surface and contour plots of the Higgs ﬁeld modulus |Φ|((a) and (b)), the weighted energy density E ((c) and (d)), and the weighted magnetic charge density M ((e) and (f)), for n = 3 HEB when λ = 4.

82 (a) (b)

(e) (f)

Figure 5.7: The 3D surface and contour plots of the Higgs ﬁeld modulus |Φ|((a) and (b)), the weighted energy density E ((c) and (d)), and the weighted magnetic charge density M ((e) and (f)), for n = 3 LEB when λ = 4.

83 the corresponding one-monopole peaks are found at z = 7.543, ρ = ±1.360 with its value being 4.461 and finally, the half-monopole peak for n = 3 LEB is situated, once again, at z = −6.801 with a value of 1.455. Pleas refer to 5.6(c), (d) and 5.7(c), (d) for pictorial representations if needed. The data is clear, the value and position of the peak of half-monopoles is not affected by different solutions, the deviation between solutions is less than 1%. The only change is happened to the ’t Hooft-Polyakov one- monopoles. For HEB and LEB, the one-monopoles and half-monopoles are closer to each other as compared to FB and the peak value is increased by a substantial amount, 65.0% increase for HEB and 35.9% for LEB. The peak deviation from the z-axis is smaller as compared to n = 3 FB with both HEB and LEB having roughly 15.4% decrease. Concerning the weighted magnetic charge densities, for HEB, one-monopole peaks are located at z = 7.341, ρ = ±1.417 with a value of 8.336. The half-monopole peak is located at z = −6.826 and its value is calculated to be −2.645. For LEB, the corresponding one-monopole peaks are found at z = 7.341, ρ = ±1.417 with its value being 7.162 and finally, the half-monopole peak for LEB is unchanged, the same as that of HEB. The corresponding 3D surface and contour plots are presented in 5.6(e), (f) and 5.7(e), (f). Again, the weighted magnetic charge density of the half-monopoles is unaffected by different solutions. The increases in peak value in the ’t Hooft-Polyakov one-monopoles are 50.3% for HEB and 29.1% for LEB respectively. The peak deviation from the z-axis is smaller as compared to n = 3 FB with both HEB and LEB have roughly 8.3% decrease. The contour plot of the ’t Hooft magnetic field lines and vector field plot of the ’t

Hooft magnetic field unit vector, Bî, for n = 3 HEB and LEB when λ = 4 are shown in Figure 5.8. The string of the half-monopoles is still stretching from z ≈ 0.212 to z ≈ −7.848 for both HEB and LEB. Again, half-monopoles are not influenced by different solutions. The one-monopoles and half-monopoles are indicated by the red and black dots in the graph. The relevant data are tabulated in Table 5.2. Transition phenonemon from one-monopole to vortex ring solution occurs for both

84 (a) (b)

Figure 5.8: Contour plot of the ’t Hooft magnetic field lines and vector field plots of the ’t Hooft magnetic field unit vector for n = 3 HEB and LEB when λ = 4. n = 3 HEB and LEB. In our model, there are n ’t Hooft-Polyakov one-monopoles superimposed at one point somewhere along the positive z-axis and n half-monopoles superimposed at the origin. This is true for all cases. As we increase the Higgs self-coupling constant, λ, certain phase transition might occur. If this happens, the n one-monopoles will merge together forming a different structure other than the usual

Table 5.2: Peak value and position of weighted energy density and weighted magnetic charge density for n = 3 FB, LEB and HEB when λ = 4.

Type of Monopole (ρ,z) Emax (ρ,z) Mmax Half-monopole (0,-6.801) 1.456 (0,-6.826) -2.644 FB One-monopole (±1.607,10.020) 3.283 (±1.545,9.917) 5.548 Half-monopole (0,-6.801) 1.455 (0,-6.826) -2.645 LEB One-monopole (±1.360,7.543) 4.461 (±1.417,7.341) 7.162 Half-monopole (0,-6.801) 1.454 (0,-6.826) -2.645 HEB One-monopole (±1.360,7.295) 5.416 (±1.417,7.341) 8.336

85 inverted circular cone. Similar to the double peaks and torus formation as seen earlier in the weighted energy density and magnetic charge density 3D surface and contour plots, the vortex ring solutions are just as easy to identify. The exact point of transition is labelled as λt and the values for case n = 3 are found to be λt = 3.32 for HEB and

λt = 8.83 for LEB. The phase transition points for different branches of solutions of different cases will be tabulated at the end of this chapter.

The Higgs modulus 3D surface plots for n = 3 HEB when λ = 3 and λ = 10 are shown in Figure 5.9 for direct pictorial comparison. The vertices of the inverted circular cone are circled in red. It is obvious that when λ = 3, the conﬁguration is still a one-monopole solution as the vertex is still located along the positive z-axis whereas when λ = 10, the vertex shifted along the ρ-axis and thus a vortex ring is formed.

The composed plots of the pole separation, dz, magnetic dipole moment, µm, and the total energy, E, versus λ 1/2 for n = 3 FB, HEB and LEB and the magnetic charges carried by Higgs and gauge ﬁelds, MH and MG, versusx ¯ when λ = 4 for n = 3 FB, are shown in Figure 5.10. The range of λ is 0.48 = λb ≤ λ ≤ 40 for case n = 3. In Figure 5.10(a), the pole separation for n = 3 FB increases with λ 1/2 concaving downwards, it starts from 9.946 at the critical lower bound and reaches saturation around λ = 7.03 with a value of dz = 10.620. For LEB and HEB, dz initially have

(a) (b)

Figure 5.9: The 3D surface and contour plots of the Higgs ﬁeld modulus |Φ|, for n = 3 HEB when λ = 3 and λ = 10.

86 (a) (b)

Figure 5.10: Plots of the pole separation (a), magnetic dipole moment (b) and total energy (c) versus λ 1/2 for n = 3 FB, HEB and LEB and the magnetic charges carried by Higgs and gauge ﬁelds (d) versusx ¯ when λ = 4 for n = 3 FB.

the same value dz = 7.517 when λ = 2.28 at the critical point. Afterwards, the pole separation for LEB goes upwards while it is quite the opposite for HEB. The saturation value for LEB is dz = 7.817 and for HEB, it seems the pole saturation has not yet reached its saturation value even when λ = 40. The minimum value is dz = 7.181. The composed magnetic dipole moment plot is shown in Figure 5.10(b). The be-

1/2 haviour of µm for n = 3 FB is similar to that of its curve for dz. It increases with λ while concaving downwards. The initial value is µm = 5.774 and the saturation value is

µm = 6.306. As for HEB, the local maximum appears near λ = 5.00 with µm = 4.591.

Then it keeps decreasing until it reaches saturation at λ = 17.93, µm = 4.548. The magnetic dipole moment for LEB is like a mirror reverse of that of the HEB. Its satu-

87 ration value is µm = 4.572 and it reaches saturation at the same time with HEB. The local minimum of LEB is µm = 4.453 at λ = 4.02. Once again, for both HEB and LEB, the magnetic dipole moment starts off at the same point with a value of µm = 4.517. Generally speaking, LEB has slightly larger dipole moment at large λ. The total energy plot is typical. The curves of all three branches of solutions are monotonic increasing functions of λ 1/2 concaving downwards with HEB having the highest energy, FB having the lowest. For FB, total energy, E, runs from 5.863 to

6.615 in the interval of 0.48 ≤ λ ≤ 40. As for LEB, E ranges from 7.013 to 7.511. And ﬁnally, E goes from 7.013 to 8.641 for HEB in the interval. Lastly, the magnetic charges carried by Higgs and gauge ﬁelds are presented in

Figure 5.10(d). Once again, MH approaches its predicted value of 1.5 as n is now 3 and MG has a limiting value of −0.055 which can be attributed to the singularity in R2 as before.

5.2.3 Case n = 4

Bifurcation phenomenon is also observed in n = 4. The critical point is found to be λc = 2.87 after which two more branches of numerical solutions emerge. For the n = 4 FB, the critical lower bound is found to be λb = 0.57. The 3D surface and contour plots of the Higgs ﬁeld modulus, |Φ|, the weighted energy density, E , and the weighted magnetic charge density, M , of n = 4 FB for λ = 4 are shown in Figure 5.11. In Figure 5.11(a) and (b), it can be seen from the Higgs Modulus 3D surface and contour plots that the ’t Hooft-Polyakov monopoles are located at z = 10.524 on the positive z-axis. The pole separation, dz, between the one-monopoles and the half- monopoles is almost the same as n = 3 FB. The deviation is less than 1%.

The 3D surface and contour plots of the weighted energy density, E , for this case are shown in Figure 5.11(c) and (d). The double peaks of the ’t Hooft-Polyakov one- monopoles are located at z = 10.020, ρ = ±1.978 with a value of 3.974. As compared with n = 3 FB, the peak deviation from the z-axis is getting larger with roughly 23.1%

88 (a) (b)

(e) (f)

Figure 5.11: The 3D surface and contour plots of the Higgs ﬁeld modulus |Φ|((a) and (b)), the weighted energy density E ((c) and (d)), and the weighted magnetic charge density M ((e) and (f)), for n = 4 FB when λ = 4.

89 increase. The peak vaue is also larger than that of n = 3 FB by about 21.0%. On the other hand, the peak of the half-monopoles is located at z = −8.037 along the negative z-axis. The peak value is found to be 1.641. The position of the peak is shifted further along the negative z-axis for about 18.2% and its value is larger than that of n = 3 FB by roughly 12.7%. In Figure 5.11(e) and (f), we have the 3D surface and contour plots of the weighted magnetic charge density, M . The ’t Hooft-Polyakov one-monopoles carry positive magnetic charges and the peaks are located at z = 9.917, ρ = ±1.932 with a value of 6.578 whereas the half-monopoles carry negative magnetic charges and the peak is located along the negative z-axis at z = −8.114 with a value of −2.979. As compare to n = 3 FB, the peak value of the one-monopoles is increased by roughly 18.6% and that of the half-monopoles is increased by about 12.7%. The contour plot of the ’t Hooft magnetic ﬁeld lines and vector ﬁeld plot of the ’t

Hooft magnetic field unit vector, Bî, for n = 4 FB when λ = 4 are shown in Figure 5.12(a) and (b). The string stretch starts from z ≈ 0.212 to z ≈ −9.545. Two new branches of numerical solutions (HEB and LEB) with higher energies than FB emerge. The critical value λc is found to be 2.87. Similar to case n = 3, these solutions start from the same point, then diverge into two branches as we keep increasing λ. The 3D surface and contour plots of the Higgs field modulus, |Φ|, the

(a) (b)

Figure 5.12: Contour plot of the ’t Hooft magnetic field lines and vector field plots of the ’t Hooft magnetic field unit vector for n = 4 FB when λ = 4.

90 weighted energy density, E , and the weighted magnetic charge density, M , of n = 4 HEB and LEB for λ = 4 are shown in Figure 5.13 and Figure 5.14. For n = 4 HEB and LEB, the ’t Hooft-Polyakov one-monopoles are located at z = 7.325 and z = 7.717 along the positive z-axis respectively as can be seen from the Higgs Modulus 3D surface and contour plots in 5.13(a), (b) and 5.14(a), (b). The pole separation, dz, is smaller than that of n = 4 FB, 30.4% decrease for HEB and 26.7% for LEB. With regards to the weighted energy densities, for n = 4 HEB, the one-monopole peaks are located at z = 7.295, ρ = ±1.607 with a value of 6.844. The half-monopole peak is located at z = −8.307 and its value is calculated to be 1.652. For n = 4 LEB, the corresponding one-monopole peaks are found at z = 7.790, ρ = ±1.978 with its value being 5.559 and ﬁnally, the half-monopole peak for n = 4 LEB is situated, once again, at z = −8.307 with a value of 1.651. Pleas refer to 5.13(c), (d) and 5.14(c), (d) for pictorial representations if needed. Again, the data is clear, the value and position of the peak of half-monopoles is not affected by different solutions, the deviation between solutions is less than 1%. The only change is happened to the ’t Hooft-Polyakov one- monopoles. For HEB and LEB, the one-monopoles and half-monopoles are closer to each other as compared to FB and the peak value is increased by a substantial amount, 72.2% increase for HEB and 39.9% for LEB. The peak deviation from the z-axis is smaller as compared to n = 4 FB for HEB with roughly 18.2% decrease. Yet it is not the case for LEB. It seems the peak deviation for LEB stays the same as n = 4 FB. Concerning the weighted magnetic charge densities, for HEB, one-monopole peaks are located at z = 7.341, ρ = ±1.803 with a value of 9.408. The half-monopole peak is located at z = −8.114 and its value is calculated to be −3.000. For LEB, the corresponding one-monopole peaks are found at z = 7.856, ρ = ±2.061 with its value being 8.830 and ﬁnally, the half-monopole peak for LEB is unchanged, the same as that of HEB. The corresponding 3D surface and contour plots are presented in 5.13(e), (f) and 5.14(e), (f). Again, the weighted magnetic charge density of the half-monopoles is unaffected by different solutions. The increases in peak value in the ’t Hooft-Polyakov

91 (a) (b)

(e) (f)

Figure 5.13: The 3D surface and contour plots of the Higgs ﬁeld modulus |Φ|((a) and (b)), the weighted energy density E ((c) and (d)), and the weighted magnetic charge density M ((e) and (f)), for n = 4 HEB when λ = 4.

92 (a) (b)

(e) (f)

Figure 5.14: The 3D surface and contour plots of the Higgs ﬁeld modulus |Φ|((a) and (b)), the weighted energy density E ((c) and (d)), and the weighted magnetic charge density M ((e) and (f)), for n = 4 LEB when λ = 4.

93 (a) (b)

Figure 5.15: Contour plot of the ’t Hooft magnetic field lines and vector field plots of the ’t Hooft magnetic field unit vector for n = 4 HEB and LEB when λ = 4. one-monopoles are 43.0% for HEB and 34.2% for LEB respectively. The contour plot of the ’t Hooft magnetic field lines and vector field plot of the ’t

Hooft magnetic ﬁeld unit vector, Bˆi, for n = 4 HEB and LEB when λ = 4 are shown in Figure 5.15. The string stretch stays the same for both HEB and LEB. Similar to case n = 3 HEB and LEB. Transition phenomenon is observed for both branches. The critical phase transition value, λt is 3.62 for HEB and 9.12 for LEB.

In both cases, HEB has smaller λt. We also notice that n = 3 has smaller λt for either HEB or LEB as compared to case n = 4. Interestingly enough, there is a completely new branch of solution emerged for case n = 4 with energy even higher than that of n = 4 HEB. We have never encountered such a situation in the ﬁeld before and this new branch of numerical solution is simply

94 Table 5.3: Lower bounds, λb, critical point for bifurcation, λc, and transition point, λt, for all cases and all branches.

n = 2 n = 3 n = 4 FB FB HEB LEB FB HEB LEB NB λb 1.96 0.48 - - 0.57 - - 0.20 λc - - 2.28 2.28 - 2.87 2.87 - λt - - 3.32 8.83 - 3.62 9.12 - labelled as New Branch (NB) in this dissertation. The critical lower bound of this branch of solution is λb = 0.20. So far, we have already introduced all the critical values of λ for all cases and all branches, these data are tabulated in Table 5.3. Now, the 3D surface and contour plots of the Higgs ﬁeld modulus, |Φ|, the weighted energy density, E , and the weighted magnetic charge density, M , of n = 4 NB for λ = 4 are shown in Figure 5.16. It is obvious that we can see this is a vortex ring solution from the Higgs Modulus 3D surface plot shown in Figure 5.16(a). The inverted torus structure is easy to identify. The vortex ring formed by three ’t Hooft-Polyakov one-monopoles is located at z = 10.748 along the positive z-axis as in the contour plot in Figure 5.16(b). No transition is observed, this NB is a vortex ring solution for the whole branch.

The 3D surface and contour plots of the weighted energy density, E , for NB λ = 4 are shown in Figure 5.16(c) and (d). The double peaks of the ’t Hooft-Polyakov one-monopoles are located at z = 11.000, ρ = ±1.731 with a value of 7.956. As compared with n = 3 FB, the peak deviation from the z-axis is smaller with roughly 12.5% decrease. The peak vaue is signiﬁcantly larger than that of n = 3 FB, almost doubling its value with 100.2% increase. On the other hand, the peak of the half- monopoles is located at z = −8.037 along the negative z-axis. The peak value is found to be 1.640. The half-monopoles is unaffected. In Figure 5.16(e) and (f), we have the 3D surface and contour plots of the weighted magnetic charge density, M . The ’t Hooft-Polyakov one-monopoles carry positive magnetic charges and the peaks are located at z = 11.200, ρ = ±1.803 with a value of 9.460 whereas the half-monopoles carry negative magnetic charges and the peak is located along the negative z-axis at z = −8.114 with a value of −2.974. As compare

95 (a) (b)

(e) (f)

Figure 5.16: The 3D surface and contour plots of the Higgs ﬁeld modulus |Φ|((a) and (b)), the weighted energy density E ((c) and (d)), and the weighted magnetic charge density M ((e) and (f)), for n = 4 NB when λ = 4.

96 Table 5.4: Peak value and position of weighted energy density and weighted magnetic charge density for n = 4 FB, LEB, HEB and NB when λ = 4.

Type of Monopole (ρ,z) Emax (ρ,z) Mmax Half-monopole (0,-8.037) 1.641 (0,-8.114) -2.979 FB One-monopole (±1.978,10.020) 3.974 (±1.932,9.917) 6.578 Half-monopole (0,-8.307) 1.651 (0,-8.114) -3.000 LEB One-monopole (±1.978,7.790) 5.559 (±2.061,7.856) 8.830 Half-monopole (0,-8.037) 1.652 (0,-8.114) -3.000 HEB One-monopole (±1.607,7.295) 6.844 (±1.803,7.341) 9.408 Half-monopole (0,-8.037) 1.640 (0,-8.114) -2.974 NB One-monopole (±1.731,11.000) 7.956 (±1.803,11.20) 9.460 to n = 3 FB, the peak value of the one-monopoles is increased by roughly 43.8% and that of the half-monopoles is unaffected. The relevant data are tabulated in Table 5.4. The contour plot of the ’t Hooft magnetic ﬁeld lines and vector ﬁeld plot of the ’t

Hooft magnetic ﬁeld unit vector, Bˆi, for n = 4 NB when λ = 4 are shown in Figure 5.17. The string stretch stays the same as it is in FB. It appears the physical quantities related to half-monopoles only reacts to changes in the φ-winding numer, n, and unaffected by anything else.

The composed plots of the pole separation, dz, magnetic dipole moment, µm, and the total energy, E, versus λ 1/2 for n = 4 FB, HEB, LEB and NB and the magnetic charges carried by Higgs and gauge ﬁelds, MH and MG, versusx ¯ when λ = 4 for n = 4

FB, are shown in Figure 5.18. The range of λ is 0.20 = λb ≤ λ ≤ 40 for case n = 4.

(a) (b)

Figure 5.17: Contour plot of the ’t Hooft magnetic field lines and vector field plots of the ’t Hooft magnetic field unit vector for n = 4 NB when λ = 4.

97 (a) (b)

Figure 5.18: Plots of the pole separation (a), magnetic dipole moment (b) and total energy (c) versus λ 1/2 for n = 4 FB, HEB, LEB and NB and the magnetic charges carried by Higgs and gauge ﬁelds (d) versusx ¯ when λ = 4 for n = 4 FB.

In Figure 5.18(a), the pole separation for n = 4 FB, similar to case n = 3, increases with λ 1/2 concaving downwards, it starts from 9.858 at the critical lower bound and reaches saturation around λ = 7.98 with a value of dz = 10.620. The pole separation for both n = 3 and n = 4 FB have exactly the same saturation value. For LEB and HEB, the graph of dz shaped like a sound fork, initially they have the same value dz = 7.483 when λ = 2.87 at the critical point. Afterwards, the pole separation for LEB goes upwards while that of HEB decreases monotonically. The saturation value for LEB is dz = 7.820 and dz = 7.167 for HEB. As for NB, the pole separation is decreasing, generally speaking. It starts from dz = 10.970 at the critical lower bound, then its curve merges with that of FB. They have the same saturation value.

98 The composed magnetic dipole moment plot is shown in Figure 5.18(b). µm for FB increases slightly with λ 1/2 then reaches a state of constant decreasing. The initial value is µm = 5.915, the local maximum is 6.010 and the saturation value is 5.795 as λ keeps increasing. As for HEB and LEB, the magnetic dipole moments are decreasing, generally speaking, with HEB decreases faster than LEB. However, they have the same saturation value of µm = 4.584. The curve for µm of NB lies way above the other three branches of solution. Its value decreases initially for just a bit. The local minimum is

µ = 6.402 at λ = 1.02. It goes upwards from this point onwards and the saturation value is 6.838. Apart from NB, the total energy plot is typical. The curves of all three branches of solutions are monotonic increasing functions of λ 1/2 concaving downwards with HEB having the highest energy, FB having the lowest. For FB, total energy, E, runs from

8.379 to 9.566 in the interval of 0.57 ≤ λ ≤ 40. As for LEB, E ranges from 10.020 to 10.710. And ﬁnally, E goes from 10.020 to 12.620 for HEB in the interval. The total energy for NB lies above the other three. However, its starting point has rather low energy, even lower than FB, with E = 7.875 at λ = 0.20. This has not been observed. Typically, the range of total energies of one branch of solution does not go across that of the other. The total energy for NB has a range of 7.875 ≤ E ≤ 13.110 for the interval

0.20 ≤ λ ≤ 40. Taking previous study into account (Teh, et al., 2014a), in which case n = 1 was investigated, the results show extreme similarities with standard MAP solutions when n = 1 and n = 3. Combining with the fact that half-monopoles appear inactive on the weighted energy density, weighted magnetic charge density as well as Higgs modulus 3D plots, one speculation could be drawn that even if the half-monopoles appear “dormant”, they are actually rather interactive among themselves and the interaction seems tied with the fractal nature of the charge they carry as this type of interactions seems only affect solutions with even number of the φ-winding number, n. The numerical method implemented in this research does not offer much to unveil the nature of this interaction, further theoretical studies are required.

99 Lastly, the magnetic charges carried by Higgs and gauge ﬁelds are presented in

Figure 5.18(d). Once again, MH approaches its predicted value of 2 as n is now 4 and

MG has a limiting value of −0.026 which can be attributed to the singularity in R2 as before.

5.3 Comment

We have thoroughly discussed the data we gathered for the one-plus-half monopole configuration with φ-winding number, n, ranging from 2 to 4. The idea is to superim- pose n ’t Hooft-Polyakov monopoles and n half-monopoles, yet the result is not a simple addition. Each case we have investigated has its own feature and behaves differently. For n = 2, the solution blows up for some reason which is totally unexpected. This indicates that there exists some unknown physical interactions between the ’t Hooft- Polyakov monopoles and half-monopoles which begs for us to discover. The case n = 3 looks like something right out of the monopole research textbook. Extremely similar solutions that behaves in almost exactly the same way, like the MAP configurations, can be found regarding either the bifurcation and transition phenomena occured in this case. The same can be said for n = 4. However, the emergence of NB tells a completely different story. While investigating the peak value and location of the weighted energy density and weighted magnetic charge density, it is clear to see that the half-monopoles are rather “dormant”. They are less reactive and sensitive to changes in most physical quantities. Once n is fixed, the half-monopoles will behave in the exactly same way in different branches of solutions. So, the change in physical quantities in the configuration are solely contributed by the ’t Hooft-Polyakov one-monopoles. On the other hand, while we look at the trending behaviour of our model across a range of the Higgs self-coupling constant, λ, we can conclude that, apart from a special case presented in n = 2 when λ > 8.00, all of our solutions possess finite energy and non-vanishing magnetic dipole moment.

100 CHAPTER 6 - CONCLUSIONS AND FUTURE RESEARCH

6.1 Conclusions

A large amount of research have been conducted in the field regarding integer topological charged monopoles of the non-Abelian SU(2) Georgi-Glashow model, These include the MAP and vortex-rings configurations. Reports of monopoles with half topological charge is rather limited, yet this particular field has gradually becoming one of the active research topic. The possibility of a one-monopole coexisting with a half-monopole (Teh et al., 2014a, b) provides us a way to investigate the interaction between these two types of mysterious particles. In this dissertation, we present a numerical approach to the one-plus-half monopole model by using an appropriate magnetic ansatz (4.1) and boundary conditions (in Chapter 4, section 7 and 8) as well as providing good initial guesses. Through varying the φ-winding number, n, and using greater resolution, we managed to obtain different sets of solutions with higher numerical accuracy, offering more insight into the interaction between ’t Hooft-Polyakov monopoles and half-monopoles. In Chapter 4, we introduced the axially symmetric magnetic ansatz (4.1) and some important formulae that are used in our research to calculate different physical quantities. The parameter a, b and f (r,θ) in the exact asymptotic solutions at large r de- termine the types of monopole configurations we are going to obtain. By choosing

1 3 a = b = 2 and p = − 2 , the one-plus-half monopole configuration is obtained. Subse- quently, we solved the system for a range of the Higgs self-coupling constant, λ, for different φ-winding number, n. In Chapter 5, we analyzed the one-plus-half monopole configuration according to different cases, compared different branches of solutions for a particular value of λ and observed over a large range of λ for each single branch of solution. We looked into this matter with depth as well as length and width. The n ’t Hooft-Polyakov monopoles carry n positive magnetic charges and, n half monopoles carry n/2 negative magnetic charges and with the Dirac string contributing a magnetic flux of −2nπ/g, we conclude

101 that our model has a zero net magnetic charge for all cases. Apart from case n = 2, the total energy of all solutions increases with increasing λ. The pole separation and magnetic dipole moment behave differently for different branches of solutions, but generally speaking, HEB and LEB will have smaller dz and µm as compared to their respective FB. The half-monopoles seem to have a “dormant” character which further theoretical studies might provide some explanations to the odd behaviours of case n = 2 and the emergence of NB in n = 4. To sum up, in this dissertation, we investigated the one-plus-half monopole configuration of the SU(2) Yang-Mills-Higgs theory with higher value of φ-winding number, n (2 ≤ n ≤ 4) and for a range of the Higgs coupling constants, λ (0 < λ ≤ 40). The resolution of the grids used in the numerical method for calculating the solutions, which is 110 × 100, was greater than previous research. The aim of our work here is to gain information of the general behaviors and properties of the one-plus-half monopole configuration as well as obtaining a deeper understanding of the structure of gauge field theory itself. We noticed that for n ≥ 2, the one-monopoles become a n-monopole superimposed at the same location. At the same time, the half-monopoles at the origin, in the same manner, become a superimposed n-half-monopole. As we increase the Higgs self-coupling constant, λ, in contrary to the observation in monopole-antimonopole pair (MAP) or monopole-antimonopole chain (MAC) configurations, the n-monopole do not merge with the n-half-monopole to form vortex-ring structures. Instead, when the Higgs coupling constant reaches a certain critical phase transition value, λt, the n- half-monopole remain unchanged at the origin while vortex-rings were formed among the superimposed n-monopole. Bifurcation phenomena are observed when n ≥ 3, where besides FB, new branches of solution with higher energies emerged (labelled as

HEB and LEB and in case n = 4, NB) at some critical value of λ, λc. It is also noticed that for n ≥ 2, there exists a critical lower bound λb for FB, below which no solution can be found.

102 6.2 Future Research

Follow from our work here, one possible direction of research is to move to the SU(2)×U(1) Weinberg-Salam model. As mentioned earlier, the SU(2) Yang-Mills theory is but a toy model. It is not physical and that being the reason we chose to use the dimensionless units in our research. However, the numerical complexity of moving from SU(2) to SU(2)×U(1) is formidable and so does the difficulty of finding a suitable initial guess. Another possible direction of research is to focus on half- monopole configuration only, but with higher φ-winding number. For instance, the numerical result of half-monopole model with n = 2 would be useful in understanding the peculiar dormant behaviours encounterd in our research. Other than these, gravity itself can also be introduced into the system by incoroprating the theory of general relativity into the theory, this opens up more possibilities and enables us to investigate more interactions. In fact, there are many fields one could explore, the most difficult problem is the numerical complexity presented in the process.

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108 LIST OF PUBLICATIONS

Zhu, D., Wong, K. M. and Tie, T. (2018). “Bifurcation and Transition of Multiple Charged One-Plus-Half Monopole Solutions of the SU(2) Yang-Mills-Higgs Theory”. Jurnal Fizik Malaysia 39: 30027.

Zhu, D., Wong, K. M. and Tie, T. (2016). “Coexistence of Multimonopole and Multiplu- charged Half-monopole”, an oral presentation in the 2016 National Conference of Physics, PERFIK 2016, Pullman Hotel, Kuala Lumpur, Malaysia, 21 - 22 December 2016, submitted for publication.