nontopological solitons in a spontaneously broken...

Nontopological Solitons

in a Spontaneously Broken U(1) Gauge Theory

(対称性が自発的に破れたU(1)ゲージ理論に於ける

ノントポロジカルソリトン)

Tatsuya Ogawa(小川達也)

Nontopological Solitonsin a Spontaneously Broken U(1) Gauge Theory

(対称性が自発的に破れたU(1)ゲージ理論に於ける

ノントポロジカルソリトン)

Department of Mathematics and Physics,

Graduate School of Science

令和元年度

Tatsuya Ogawa

(小川達也)

Abstract

We construct spherically symmetric nontopological solitons, Q-balls, in

a system consists of a complex matter scalar field, a U(1) gauge field, and

a complex Higgs scalar field with a potential that causes the spontaneous

symmetry breaking. This is a generalized system based on the model by

Friedberg, Lee, and Sirlin.

In the Q-balls of our system, the U(1) charge densities are induced by

both the complex scalar matter field and the complex Higgs scalar fields.

Owing to the Higgs mechanism the gauge field aquires a mass, that it me-

diates short-range force in the vacuum state. As a result, influence of the

charge induced by the complex scalar matter field is limited in a finite range

of distance. In other words, the charge induced by the complex scalar matter

field should be screened by some appropriate configuration of the complex

Higgs scalar field.

If we restrict the system stationary and spherically symmetric, the sys-

tem considered in this paper is analogous to a dynamical system of a particle

in three dimensions, and then a configuration of the fields can be interpreted

as a motion of the particle. There are some stationary points in the dynam-

ical system, whose one of them corresponds to the vacuum. We find bounce

solutions that describe large Q-balls: a particle starts from a stationary point

traverses toward the vacuum stationary point. By numerical calculations,

we show that Q-balls which can be interpreted as bounce solutions have the

following properties: (i) the size can be arbitrarly large, (ii) energetically

stable, and (iii) charge is screened everywhere.

2

Acknowledgements

I am very grateful to Prof. Hideki Ishihara for his continuous encouragement,

meaningful discussions, and genuine support throughout my post graduate

study. Also, I would like to thank to Prof. Ken-ichi Nakao, Prof. Hiroshi

Itoyama, Prof. Nobuhito Maru, and Prof. Sanefumi Moriyama for valu-

able suggestions. It is also my pleasure to thank all of the members of

research group for theoretical astrophysics, particle physics, and mathemat-

ical physics in Osaka City University, especially, Dr. Ryusuke Nishikawa,

Dr. Hirotaka Yoshino, Dr. Yoshiyuki Morisawa, Dr. Ryotaku Suzuki, Dr.

Hiroyuki Negishi, Dr. Atsuki Masuda. I am thankful to Prof. Masaki

Arima, Prof. Nobuyuki Sakai, and Dr. Masato Minamitsuji for helpful ad-

vice. Finally, I would like to thank my family for trusting and supporting

me.

3

Contents

1 Introduction 6

2 U(1) Gauge theory and Higgs mechanism 9

2.1 Global symmetry . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Local symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Higgs Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Charge Screening of Extended Sources in a Spontaneously

Symmetry Broken System 12

3.1 Charge screening with a Proca field . . . . . . . . . . . . . . . 12

3.2 Basic System with the spontaneous symmetry breaking . . . 14

3.3 Spherically symmetric model . . . . . . . . . . . . . . . . . . 16

3.4 Numerical Calculations . . . . . . . . . . . . . . . . . . . . . 18

3.4.1 Gaussian distribution source . . . . . . . . . . . . . . 18

3.4.2 Homogenious ball source . . . . . . . . . . . . . . . . . 23

4 Q-balls in a Spontaneously Broken U(1) Gauge Theory 27

4.1 Q-balls and gauged Q-balls in a coupled two scalar fields model 27

4.2 Basic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.3 Numerical Calculations . . . . . . . . . . . . . . . . . . . . . 31

4.4 Homogeneous ball solutions . . . . . . . . . . . . . . . . . . . 34

4.5 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

5 Summary 41

Appendicies 45

Appendix A Charge Screning of a Point source . . . . . . . . . . 45

A.1 Asymptotic behaviors for the point source . . . . . . . 45

A.2 Distant region . . . . . . . . . . . . . . . . . . . . . . 46

4

A.3 Numerical calculations . . . . . . . . . . . . . . . . . . 47

Appendix B Approximate solutions for the Gaussian distribu-

tion sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

Appendix C Approximate solutions for the homogeneous ball

sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

Appendix D Energy-Momentum Tensor of the System . . . . . . 55

Appendix E Other bounce solutions . . . . . . . . . . . . . . . . 56

E.1 Stationary Points of the System and Bounce Solutions 56

E.2 Numerical calculations . . . . . . . . . . . . . . . . . . 56

5

Chapter 1

Introduction

In 1834, John Scott Russell observed that a solitary wave were created and

propagated on a canal. The wave has the following characteristic properties:

velocity and shape of the wave do not change while it propergates over a

long distance. Since the wave behaves like a particle, such solitary waves

are called solitons.

The soliton solutions appear in various research fields of science: op-

tics, biology, and field theories in physics. The solitons in the field theories

are classical solutions to nonlinear field equations, where the each energy

is localized in a finite spatial region. The solitons are divided into two

types: topological solitons, and nontopological solitons. The former are ho-

motopically distinct solutions from the vacuum solution, namely, field con-

figurations with topological charge, which are invariant under continuous

deformations of the fields with fixed boundary conditions. The topologi-

cal solitons cannot relax to the zero energy configuration due to conserved

topological quantities. For example, domain walls, cosmic strings [1], and

’t Hooft-Polyakov-monopoles [2] are topological solitons [3]. The latter,

nontopological soliton, represent field configurations which have the lowest

energy for fixed conserved charge in global U(1)-invariant theories, and can

be interpreted as bound states of bosonic particles.

The first nontopological soliton solutions to a nonlinear scalar field the-

ory was first found by Rosen in 1968 [4, 5]. In 1976, Friedberg, Lee and

Sirlin [6] introduced the nontopological solitons in a coupled system of a

complex scalar field and a real scalar field with a potential that causes the

spontaneous Z2 symmetry breaking. Some years later, the simplest example

of the nontopological solitons was proposed in a system of a self-interacting

6

single complex scalar field by Coleman in 1985 [7]. He called the spherically

symmetric nontopological solitons Q-balls1. The models with a single scalar

field are assumed to have complicated self-interactions, e.g., third or sixth

order potentials, or non-polynomial potentials, for the existence of Q-balls.

On the other hand, in the models with two scaler fields, more natural poten-

tials, e.g., forth order potentials, and interaction between two scalar fields

are assumed.

The Q-balls have been widely investigated in a context of cosmologies

and astrophysics. A. Kusenko constructed Q-balls in a minimal supersym-

metric standard model that include global U(1) symmetries in 1997 [8, 9].

It is known that a Q-ball can be a candidate of dark matter of the Universe

[11, 12, 13, 14, 15, 16] and a source for a baryogenesis [17, 18, 10, 19].

If taking a gravity into account, the stable Q-balls can be boson stars.

[36, 37, 38, 39, 40, 41]. The boson stars have been investigated as a can-

didate of central core of galaxies [40], and a source of gravitational wave

[42].

Genelalization of the Q-balls by requiring a local U(1)-invariance, achieved

by a U(1) gauge field, so-called gauged Q-balls, was first studied by K.M.

Lee, J.A. Stein-Schabes, R. Watkins and L.M. Widrow in 1988 [20] based

on the system by Coleman, and many works follow it [21, 22]. Their system

consisis of a complex scalar field coupled to a U(1) gauge field. On the other

hand, gauged Q-balls based on the system by Friedberg, Lee, and Sirlin was

proposed in 1991 [21]. This system consists of a real scalar fields with a po-

tential that causes spontaneously the Z2 symmetry breaking and a complex

scalar field coupled to a U(1) gauge field. The both types of the gauged Q-

balls have significant properties compared with the nongauged Q-balls. For

example, nongauged Q-balls with arbitrary large charge are allowed, while

gauged Q-balls have upper bound of charge because of the Coulomb repul-

sive force mediated by the gauge field. Another genalirized Q-balls with

a fermionic field called fermionic Q-balls are proposed [49, 50, 51, 52]. In

this case, a large fermionic soliton is hardly produced because of the Pauli

exclusion principle.

In this paper, we generalize the gauged Q-balls based on the Friedberg,

Lee, and Sirlin system. We consider a system consist of a complex matter

scalar field, a U(1) gauge field, and a complex Higgs scalar field with a

1Hereafter, we call a spherically symmetric nontopological soliton a Q-ball, in short.

7

potential that causes the spontaneous symmetry breaking. The local U(1) ×global U(1) symmetry breaks to the global U(1) symmetry through the Higgs

mechanism. We show the existence of Q-balls as stationary and spherically

symmetric solutions in the model that has simple natural interaction terms.

Then, this work would suggest Q-balls can appear in a wide class of gauge

theories.

In our system of the gauged Q-balls, the U(1) charge densities are in-

duced by both the complex scalar matter field, and the complex Higgs scalar

fields. Owing to the Higgs mechanism the gauge field which aquires a mass

mediates a short-range force in the vacuum state. As a result, influence of

the charge induced by the complex scalar matter field is limited in a finite

range of distance. In other words, the charge induced by the complex scalar

matter field should be screened by some appropriate configuration of the

complex Higgs scalar field.

If we restrict the system stationary and spherically symmetric, the sys-

tem considered in this paper is analogous to a dynamical system of a particle

in three dimensions, and then a configuration of the fields can be interpreted

as a motion of the particle.

The dynamical system of the particle has some unstable stationary points.

One of the stationary point corresponds to the vacuum. The particle that

travels toward the vacuum stationary point describes a Q-ball solution. We

construct Q-balls by solving the dynamical system numerically.

If the particle stays an initial stationary point long time, and travels to

the vacuum stationary point quickly, the size of the corresponding Q-ball is

large. We show that a Q-ball that has the following properties is allowed:

(i) the size can be arbitrarly large, (ii) energetically stable, (iii) charge is

screened everywhere. It would be preferable properties for a candidate of

dark matter in the universe.

The paper is organized as follows. In Chap.2, we review a U(1) gauge

theory and a Higgs mechanism, and define a total charge screening used in

this paper. In Chap.3, we investigate charge screening in a system consists

of a complex Higgs scalar field and a U(1) gauge field with a external source

charge. In Chap.4, we construct numerical solutions of Q-ball in our system

and investigate their properties. In Chap.5, we summarize the paper.

8

Chapter 2

U(1) Gauge theory and

Higgs mechanism

2.1 Global symmetry

First, we consider the Lagrangian consisits of a complex scalar filed with

mass m given by

L = −(∂µϕ)∗(∂µϕ)−m2ϕ∗ϕ. (2.1)

This Largangian is invariant under a global U(1) transformation

ϕ(t, x) → ϕ′(t, x) = eiγϕ(t, x), (2.2)

where γ is an arbitrary constant. This corresponds to a phase rotation of

the complex scalar field in an inner space. “Global” means that the trans-

formation does not depend on spacetime coordinate (x, t). If Lagrangian

has this symmetry, the four-current density defined as

jµ := i (ϕ∗(∂µϕ)− ϕ(∂µϕ)∗) , (2.3)

is conserved, namely, ∂µjµ = 0 is satisfied. Consequently, the charge defined

by

Q :=

∫jtd3x, (2.4)

is conserved.

9

2.2 Local symmetry

Next, we consider that a transformation that depends on spacetime coordi-

nate (t, x) as

ϕ(t, x) → ϕ′(t, x) = eiχ(t,x)ϕ(t, x). (2.5)

This is called local U(1) transformation. In this case, (2.1) is not invariant

under (2.5). We require that the Lagrangian is invariant under local U(1)

transformation. A vector field Aµ(t, x) should be introduced and trans-

formed by

Aµ(t, x) → Aµ(t, x) = Aµ(t, x) + e−1∂µχ(t, x), (2.6)

where e is a coupling constatnt between the vector and the complex scalar

fields. This vector field is called U(1) gauge field. By introducing the gauge

field, the Lagrangian (2.1) is changed to

L = −(Dµϕ)∗(Dµϕ)− 1

2m2ϕ∗ϕ− 1

4FµνF

µν , (2.7)

where

Dµϕ := ∂µϕ− ieAµϕ, (2.8)

Fµν := ∂µAν − ∂νAµ (2.9)

are the covariant derivative and the field strength of a U(1) gauge field

Aµ respectively. Owing to the local U(1) symmetry, a four-current density

defined by

jµ := ie ϕ∗(Dνϕ)− ϕ(Dνϕ)∗ , (2.10)

and therefore the charge (2.4) are conserved.

2.3 Higgs Mechanism

In the previous sections, we introduced U(1) gauge symmetry. However,

assuming the gauge symmetry to the Lagrangian, we can not introduce a

mass term of the gauge field explicitly in the Lagrangian. In fact, the mass

term of the gauge field breaks the gauge symmetry as

m2AAµA

µ = m2AA

′µA

′µ. (2.11)

10

However, it is well known that the weak gauge boson have finite mass exper-

imentally. This problem was solved by using idea of spontaneous symmetry

breaking. The gauge field acquire the mass in symmetry breaking vacuum

naturally.

We consider a Lagrangian of the complex scalar field and the gauge field

given by

L = −(Dµϕ)∗(Dµϕ)− λ

4(ϕ∗ϕ− η2)2 − 1

4FµνF

µν , (2.12)

where λ and η are positive constants. Minima of the potential, say ϕ0(x),

are degenerate infinitely as given by

ϕ0(x) = ηeiθ(x), (2.13)

where θ is an arbitrary function. Without loss of generality, we can choose

θ = 0. Expanding small perturbation of the complex scalar field around

ϕ0 = η by

ϕ = η +1√2φ1 + iφ2

∼(η +

1√2φ1

)ei 1√

2ηφ2 , (2.14)

and choosing the gauge in (2.5), which is called unitary gauge,

χ(x) = − 1√2ηφ2, (2.15)

we can transform the complex scalar field by

ϕ′(x) = η +1√2φ1, (2.16)

where φ1 is a real scalar field. Substituting (2.16) into (2.12), we can rewrite

the Lagrangian as

L = −1

2(∂µφ1)(∂

µφ1)−1

2m2ϕφ

21 −

1

4FµνF

µν − 1

2m2AAµA

µ +O(φ31, A

3µ),

(2.17)

where mϕ :=√λη is a mass of the real scalar field. This Lagrangian is not

invariant under the local U(1) transformations, namely, symmetry is broken

spontaneously. As a result, the gauge field acquire the mass mA :=√2eη.

The scheme is called the Higgs mechanism.

11

Chapter 3

Charge Screening of

Extended Sources in a

Spontaneously Symmetry

Broken System

In this chapter, we introduce a total charge screening by classical solutions

in field theories. The total charge screening by the fields are proposed in

1970s to 1980s which is motivated to explain color confinment in quantum

chromodynamics [53, 54, 55, 56].

3.1 Charge screening with a Proca field

In this section we show that if massive vector field, Proca field, mediate the

interaction between charge, the charge screening should occur.

We consider a Proca field theory with an external source described by

the Lagrangian density

L = −1

4FµνF

µν − 1

2m2AAµA

µ − eAµJµ , Fµν := ∂µAν − ∂νAµ (3.1)

where Jµ is the external source.

The energy of the system is given by

E =

∫d3x

(1

2

(EiE

i +BiBi))

, (3.2)

12

where Ei := Fi0, Bi := 1/2ϵijkFjk, and i denotes spatial index. In the

vacuum state, which minimizes the energy (3.2), Aµ should take the form

Aµ = ∂µθ(x), (3.3)

where θ(x) is an arbitrary function.

By varying (3.1) with respect to Aµ, we obtain the equations of motion

∂µFµν = m2

AAν + eJν . (3.4)

We can interpret the right-hand side of (3.4) as the source of the Proca field.

We consider a static point source in the form

J t = δ3(r), and J i = 0, (3.5)

where t and r are the time and the radial coordinates. We also assume that

the fields are spherically symmetric and static in the form,

At = At(r), and Ai = 0. (3.6)

We impose that the Proca field should be in the vacuum state at infinity

At → 0. (3.7)

Substituting (3.6) into (3.4), we obtain

d2Atdr2

+2

r

dAtdr

= m2AAt − eδ3(r), (3.8)

and solving (3.8), we get

At(r) =e

4πrexp(−mA r). (3.9)

Therefore, in the vacuum state, the Proca field decays exponentially with

the mass scale. Then the influence of the source charge by the massive vector

field is limited in a finite range of distance.

From another point of view, we reinterpret the solution (3.9). Integrating

(3.8), we get

limr→∞

4πr2A′t = 4π

∫ ∞

0r2 ρind(r)− ρext(r) dr

: = Qind +Qext, (3.10)

13

where ρind := m2AAt is a charge density induced by the vector field and

ρext := eδ3(r), and Qind and Qext are total charge by the vector field and

the external source respectively. Substituting (3.9) into (3.10), we get

Qind +Qext = 0. (3.11)

This means that the charge of the external source is totally screened by the

counter charge of the vector field. From (3.10), At should decay faster than

r−1 for the total charge screening.

3.2 Basic System with the spontaneous symmetry

breaking

In the previous sections, it is shown that if a vector field have a mass,

the total screening of an external source occurs. On the other hand we

showed that a gauge field, vector field, can acquire a mass through the

Higgs mechanism in Chap.2. Therefore, we expect that if an external source

exist in abelian Higgs model, the source charge should be screened through

the Higgs mechanism.

We consider an abelian Higgs system described by the Lagrangian den-

sity

L = −(Dµϕ)∗(Dµϕ)− V (ϕ)− 1

4FµνF

µν , (3.12)

where Fµν := ∂µAν − ∂νAµ is the field strength of a U(1) gauge field Aµ,

and ϕ is a complex Higgs scalar field with the potential

V (ϕ) =λ

4(ϕ∗ϕ− η2)2, (3.13)

where λ and η are positive constants soliton. The Higgs field ϕ couples to

the gauge field by the covariant derivative given by

Dµϕ := ∂µϕ− ieAµϕ, (3.14)

where e is a coupling constant. The Lagrangian density (3.12) is invariant

under local U(1) gauge transformations,

ϕ(x) → ϕ′(x) = eiχ(x)ϕ(x), (3.15)

Aµ(x) → A′µ(x) = Aµ(x) + e−1∂µχ(x), (3.16)

14

where χ(x) is an arbitrary function.

The energy of the system is given by

E =

∫d3x

(|Dtϕ|2 + (Diϕ)

∗(Diϕ) + V (ϕ) +1

2

(EiE

i +BiBi))

, (3.17)

where Ei := Fi0, Bi := 1/2ϵijkFjk, and i denotes spatial index. In the

vacuum state, which minimizes the energy (3.17), ϕ and Aµ should take the

form

ϕ = ηeiθ(x) and Aµ = e−1∂µθ, (3.18)

where θ is an arbitrary function. Equivalently, eliminating θ we have

ϕ∗ϕ = η2 and Dµϕ = 0. (3.19)

After the Higgs scalar field takes the vacuum expectation value η, the gauge

field Aµ absorbing the Nambu-Goldstone mode, the phase of ϕ, forms a

massive vector field with the mass mA =√2eη, and the real scalar field

that denotes a fluctuation of the amplitude of ϕ around η acquires the mass

mϕ =√λη.

In order to study the charge screening, adding an extremal source cur-

rent, Jµ, coupled with Aµ to the original Lagrangian (3.12), we consider the

action1

S =

∫d4x

(−(Dµϕ)

∗(Dµϕ)− λ

4(ϕ∗ϕ− η2)2 − 1

4FµνF

µν − eAµJµ

).

(3.20)

By varying (3.20) with respect to ϕ∗ and Aµ, we obtain the equations of

motion

DµDµϕ− λ

2ϕ(ϕ∗ϕ− η2) = 0, (3.21)

∂µFµν = ejνind + eJν , (3.22)

where jνind is the gauge invariant current density that consists of ϕ and Aµ

defined by

jνind := i (ϕ∗(∂ν − ieAν)ϕ− ϕ(∂ν + ieAν)ϕ∗) . (3.23)1The case of vanishing potential, V (ϕ) = 0, in which the symmetry does not break,

is studied in ref.[53], and the case V (ϕ) = 12m2|ϕ|2, in which partial screening occurs, is

studied in ref.[54].

15

3.3 Spherically symmetric model

We consider a spherically symmetric and static external source in the form

eJ t = ρext(r), and eJ i = 0, (3.24)

where t and r are the time and the radial coordinates. We also assume that

the fields are spherically symmetric and stationary in the form,

ϕ = eiωtf(r), (3.25)

At = At(r), and Ai = 0, (3.26)

where ω is a constant, and f(r) is a real function of r. By using the gauge

transformation (3.15) and (3.16) to incorporate the phase rotation of ϕ, i.e.,

Nambu-Goldstone mode, with At, we introduce a new variable α(r) as

α(r) := At(r)− e−1ω. (3.27)

The charge density induced by the fields ϕ and Aµ defined by (3.23) is

written as

ρind := ejtind = −2e2f2α. (3.28)

Substituting (3.24) - (3.27) into (3.21) and (3.22), we obtain

d2f

dr2+

2

r

df

dr+ e2fα2 − λ

2f(f2 − η2) = 0, (3.29)

d2α

dr2+

2

r

dα

dr+ ρind(r) + ρext(r) = 0. (3.30)

Using the ansatz (3.25) and (3.26), we rewrite the energy (3.17) for the

symmetric system as

E = 4π

∫ ∞

0r2ϵ(r)dr, (3.31)

ϵ := ϵKin + ϵElast + ϵPot + ϵES, (3.32)

where

ϵKin := |Dtϕ|2 = e2f2α2, ϵElast := (Diϕ)∗(Diϕ) =

(df

dr

)2

, (3.33)

ϵPot := V (ϕ) =λ

4(f2 − η)2, ϵES :=

1

2EiE

i =1

2

(dα

dr

)2

, (3.34)

16

are density of kinetic energy, elastic energy, potential energy, and electro-

static energy, respectively.

We consider two extended source cases separately. As the extended

source cases, we discuss Gaussian distribution sources and homogeneous

ball sources. The both are smoothly distributed and have finite supports.

The charge density of the Gaussian distribution source is given by

ρext(r) = ρ0 exp

[−(r

rs

)2], (3.35)

where rs is the width of the extended source. The total external charge is

assumed to be normalized as

4π

∫ ∞

0r2ρext(r)dr = q, (3.36)

then the central density ρ0 is given by

ρ0 =q

π3/2 r3s. (3.37)

In the limit rs → 0, the charge density (3.35) with (3.37) reduces to the

point source case2.

The charge density of the homogeneous ball considered in this paper is

given by

ρext(r) =ρ02

[tanh

(rs − r

ζs

)+ 1

], (3.38)

where rs is the radius of the external source, and ζs is the thickness of surface

of the ball. We assume rs ≫ ζs so that the charge density within the radius

rs is almost constant value ρ0. Then, the total external charge is (4π/3)r3sρ0

in this case.

We impose boundary conditions so that the fields are regular at the

origin. The regularity conditions for the spherically symmetric fields at the

origin are

df

dr→ 0 ,

dα

dr→ 0 as r → 0. (3.39)

The energy density at the origin is finite for finite central values of f and α

. At infinity, the fields should be in the vacuum state, which minimizes ϵ in

(3.32). Therefore, we impose the conditions

f → η , α→ 0 as r → ∞. (3.40)2In the cases of a point source is discussed in Appendix A.

17

3.4 Numerical Calculations

We use the relaxation method to obtain numerical solutions to the coupled

ordinary differential equations (3.29) and (3.30). In numerics, hereafter,

we set η = 1, and scale the radial coordinate r as r → ηr, and scale the

functions f , α as f → η−1f , α→ η−1α, respectively.

3.4.1 Gaussian distribution source

For the first example of smoothly extended source, we consider the external

charge density given by the Gaussian distribution (3.35). As the boundary

conditions, we impose the regularity conditions (3.39) at the origin, and the

vacuum condition (3.40) at infinity.

For the extended external sources, the behaviors of f and α do not

depend critically on the value of κ := eq/(4π) unlike the point source case.

So, we concentrate on the case q = 1. We set e = 1/√2 and λ = 1 so that

rϕ := m−1ϕ = 1 and rA := m−1

A = 1, and perform numerical calculation with

several values of rs that denotes the thickness of the external source.

Field configurations

By numerical calculations, we show typical behaviors of the function f and

α with the external charge density ρext in the cases of rs = 0.1, 1, 10, and 100

in Fig.3.1. We see that the function f and α change in their shapes with rs.

Especially, for the thin source case, rs ≪ rA, numerical solutions are shown

in Fig.3.2. As rs approaches to zero, since the normalized Gaussian function

(3.35) with (3.37) reduces to the δ-function (A.1), then as we expected, the

function f and α approach to the solutions for the point source case discussed

in the previous subsection. In the thick source case, rs > rA, the widths of

f and α are order of rs. Typical behaviors can be understood by analytical

method given in Appendix B.

18

102*( f-η)/(ρ0/η2)α/(ρ0/η

2)

ρext/ρ0

0.0 0.5 1.0 1.5 2.0 2.5 3.00.000

0.002

0.004

0.006

0.008

0.010

η r

rs=0.1

102*( f-η)/(ρ0/η2)α/(ρ0/η

2)

ρext/ρ0

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0

η r

rs=1.0

102*( f-η)/(ρ0/η2)α/(ρ0/η

2)

ρext/ρ0

0 10 20 30 40 500.0

0.2

0.4

0.6

0.8

1.0

η r

rs=10

102*( f-η)/(ρ0/η2)α/(ρ0/η

2)

ρext/ρ0

0 100 200 300 400 5000.0

0.2

0.4

0.6

0.8

1.0

η r

rs=100

Figure 3.1: Numerical solutions in the case of Gaussian distribution

sources. Behaviors of f(r) and α(r) for rs = 0.1, 1, 10, 100 are drawn

together with ρext as functions of r. In the case of rs = 100, α(r)

coincides with ρext (see the lower right panel).

ρext=qδ3(r)rs=0.01

rs=0.1

rs=1.0

-7 -6 -5 -4 -3 -2 -1 00.000

0.005

0.010

0.015

0.020

loge[ηr]

loge[f/η]

ρext=qδ3(r)

rs=0.01

rs=0.1 rs=1.0

-7 -6 -5 -4 -3 -2 -1 0-4

-2

0

2

4

loge[ηr]

loge[α/η]

Figure 3.2: Behaviors of f(r) and α(r) for various rs. As rs decreases

to zero, the configurations of f and α approach to the ones in the

point source case.

19

Charge screening

We depict the induced charge density ρind(r) with the external charge den-

sity ρext(r) in Fig.3.3. The sign of ρind is opposite to ρext. In the central

region of rs = 0.1 and rs = 1 cases, we find that |ρext| is larger than |ρind|,i.e., total charge density ρtotal(r) has the same sign with ρext(r). As r in-

creases, |ρind| exceeds |ρext|. In the region r ≫ rA, the both ρext and ρind

decrease quickly to zero. As shown in Fig.3.4, Q(r), the total charge within

radius r, decreases to zero in the region r ≫ max(rA, rs), it means the ex-

ternal charge is totally screened by the induced charge cloud for a distant

observer.

ρext/ρ0ρind/ρ0ρtotal/ρ0

0.0 0.5 1.0 1.5 2.0 2.5 3.0-0.004-0.0020.0000.0020.0040.0060.0080.010

η r

rs=0.1 ρext/ρ0ρind/ρ0ρtotal/ρ0

0 2 4 6 8 10-0.2

0.0

0.2

0.4

0.6

0.8

1.0

η r

rs=1.0

ρext/ρ0ρind/ρ0ρtotal/ρ0

0 10 20 30 40 50-1.0

-0.5

0.0

0.5

1.0

η r

rs=10 ρext/ρ0ρind/ρ0ρtotal/ρ0

0 100 200 300 400 500-1.0

-0.5

0.0

0.5

1.0

η r

rs=100

Figure 3.3: The external charge density, ρext(r), the induced charge

density, ρind(r), and sum of them, ρtotal(r), are plotted as functions

of r for rs = 0.1, 1, 10, 100.

In the case of rs ≪ rA, the width of the induced charge cloud is the

order of rA, while in the case of rs ≥ rA, the width is almost same as rs. In

the case of rs = 100, we have

ρind(r) = −ρext(r), (3.41)

as is justified by (B.7). Then, ρtotal vanishes everywhere, equivalently Q(r)

vanishes everywhere. We call this ‘perfect screening’.

20

rs=0.1

rs=1.0rs=5.0 rs=10

0 5 10 15 20 25 300.00.10.20.30.40.50.60.7

η r

Q/q

Figure 3.4: The total charges within radius r, Q(r), are plotted for rs =

0.1, 1, 5, 10.

Energy of the cloud

We inspect the energy density of the numerical solutions. The components

soliton of energy density given by (3.33) and (3.34) are shown in Fig.3.5.

The dominant components soliton of energy density ϵ are ϵKin and ϵES, while

ϵElast and ϵPot are negligibly small in the present cases.

In the thin source case, rs ≪ rA, the electrostatic energy density domi-

nates the total energy density (see rs = 0.1 case in the first panel of Fig.3.5

for example), i.e.,

ϵ(r) ≃ ϵES(r) =1

2

(dα

dr

)2

. (3.42)

In the near region 0 ≤ r ≤ rs, as shown in Appendix B, the asymptotic

behavior of the function α(r) near the origin is given by (B.4), i.e.,

α(r) ∼ α0 −ρ0r

2s

6

(r

rs

)2

, (3.43)

where α0 is the central value of α. Substituting (3.43) into (3.42), the energy

within rs is given by

E|r≤rs ≃4πρ209

∫ rs

0r4dr =

4q2

45π2rs. (3.44)

In the region r > rs, since α is given by (A.16), then the energy of the

system (3.31) in this range can be written by

E|r>rs ≃ 2π

∫ ∞

rs

r2(

1

r2+

1

rrA

)2

exp

(− 2r

rA

)dr

= 2π exp

(−2rsrA

)(1

2rA+

1

rs

)≃ 2π

rs. (3.45)

21

Therefore, the total energy E = E|r≤rs + E|r>rs is proportional to r−1s .

ϵKin/ϵMaxϵES/ϵMax

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

η r

rs=0.1


0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0

η r

rs=1.0


0 10 20 30 40 500.0

0.2

0.4

0.6

0.8

1.0

η r

rs=10


0 100 200 300 400 5000.0

0.2

0.4

0.6

0.8

1.0

η r

rs=100

Figure 3.5: The kinetic energy density, ϵKin, and the electrostatic

energy density, ϵES, normalized by the maximum values of ϵ are

plotted for rs = 0.1, 1, 10, 100.

In contrast, in a thick source case, rs ≫ rA, as shown in (B.9) of Ap-

pendix B, we see that f(r) ≃ η and α(r) = ρext(r)/m2A, then the energy

density becomes

ϵ(r) ≃ ϵKin(r) = e2f2α2 =1

2m2A

ρ20 exp

[−2

(r

rs

)2]. (3.46)

Therefore, the energy E given by

E = 4π

∫ ∞

0r2ϵ(r)dr =

q2

4√2π3/2m2

A r3s. (3.47)

is proportional to r−3s .

22

-2 -1 0 1 2

-6

-4

-2

0

log10[η rs]

log 10[E/η3 ]

rϕ=1.0 , rA=1.0

-2 -1 0 1 2

-6

-4

-2

0

log10[η rs]

log 10[E/η3 ]

rϕ=10 , rA=1.0

-2 -1 0 1 2 3

-8

-6

-4

-2

0

log10[η rs]

log 10[E/η3 ]

rϕ=1.0 , rA=10

-2 -1 0 1 2 3

-8

-6

-4

-2

0

log10[η rs]

log 10[E/η3 ]

rϕ=10 , rA=10

Figure 3.6: Log-log plot of the total energy E versus width of the

external charge rs. The broken line means r−3s while the dot-dashed

line means r−1s .

By numerical calculations for some values of the parameter sets (rϕ, rA),

the energy E is plotted as a function of rs in Fig.3.6. In all cases, we see that

E ∝ r−1s for small rs, and E ∝ r−3

s for large rs. The power index changes

around rs = rA.

3.4.2 Homogenious ball source

As the second example of smoothly extended source, we consider the ball

of constant charge density expressed by (3.38). As same as the Gaussian

distribution case discussed above, we set e = 1/√2, and λ = 1. We fix the

central charge density ρ0, and find numerical solutions for several values of

rs, radius of the ball, and ζs, surface thickness parameter. Note that the

total external charge is in proportion to r3s .

By numerical calculations, f(r) and α(r) with ρext(r) are shown in the

cases of rs = 1, 10, and 100 for fixed surface thickness as ζs = 0.01 in Fig.3.7.

As is shown in AppendixC, in the region r < rs − rA, where ρext ≃ ρ0 =

23

const., we see

f ≃ f0 and α ≃ α0, (3.48)

where f0 and α0 are constants soliton given in Appendix C. In the region

r ≥ rs+rA, where ρext ≃ 0, we see simply f ≃ η and α ≃ 0. The functions f

and α change the values quickly in the vicinity of the ball surface rs− rA ≤r ≤ rs + rA. The profiles of f and α near the ball surface rs are almost

identical if rs ≫ rA (see Fig.3.8 ).

( f-η)/(ρ0/η2)

α/(ρ0/η2)

ρext/ρ0

0 1 2 3 4 50.0

0.2

0.4

0.6

0.8

1.0

η r

rs=1.0

( f-η)/(ρ0/η2)

α/(ρ0/η2)

ρext/ρ0

0 5 10 15 200.0

0.2

0.4

0.6

0.8

1.0

η r

rs=10

( f-η)/(ρ0/η2)

α/(ρ0/η2)

ρext/ρ0

0 50 100 150 2000.0

0.2

0.4

0.6

0.8

1.0

η r

rs=100

Figure 3.7: Behaviors of f , α and ρext for rs = 1, 10, 100 with fixed

ζs = 0.01.

rs=1.0rs=10rs=100

-10 -5 0 5 100.00

0.05

0.10

0.15

0.20

η (r - rs)

(f-

η)/(ρ0/η2 )

rs=1.0rs=10rs=100

-10 -5 0 5 100.0

0.1

0.2

0.3

0.4

0.5

η (r - rs)

α/(ρ0/η2 )

Figure 3.8: Behaviors of f (left panel) and α (right panel) in the

vicinity of the ball surface rs. Three cases rs = 1, 10, 100 are super-

posed.

Next, we consider variation of surface thickness ζs for fixed ball radius rs.

The profile of the functions f and α, the charge density, and energy density

are shown in the cases of ζs = 0.1, 1, and 10 for fixed ball radius as rs = 100

in Fig.3.9. Inside the homogeneous ball source, the induced charge density

cancels the external charge density except the vicinity of the ball surface. In

the thin ball surface case, ζs ≪ rA, at the surface, where α changes its value

24

quickly, the induced charge exceeds the external charge inside the surface,

and vice versa outside. Therefore, an electric double layer emerges at the

surface of the ball. For the thick surface case, ζs ≫ rA, charge cancellation

occur everywhere even at the surface. Namely, the perfect screening occurs

in this case.

The components soliton of energy density given in (3.33) and (3.34) are

shown in Fig.3.9. Inside the homogeneous ball, the kinetic energy dominate

the energy density and the electrostatic energy density caused by the electric

double layer appears at the neighborhood of surface for the thin surface case.

25

( f-η)/(ρ0/η2)

α/(ρ0/η2)

ρext/ρ0

70 80 90 100 110 120 130 1400.0

0.2

0.4

0.6

0.8

1.0

η r

ζs=0.1 ρext/ρ0ρind/ρ0ρtotal/ρ0

70 80 90 100 110 120 130 140-1.0

-0.5

0.0

0.5

1.0

η r

ζs=0.1ϵKin/ϵMaxϵPot/ϵMaxϵES/ϵMax

70 80 90 100 110 120 130 1400.0

0.2

0.4

0.6

0.8

η r

ζs=0.1

( f-η)/(ρ0/η2)

α/(ρ0/η2)

ρext/ρ0

70 80 90 100 110 120 130 1400.0

0.2

0.4

0.6

0.8

1.0

η r

ζs=1.0 ρext/ρ0ρind/ρ0ρtotal/ρ0

70 80 90 100 110 120 130 140-1.0

-0.5

0.0

0.5

1.0

η r

ζs=1.0ϵKin/ϵMaxϵPot/ϵMaxϵES/ϵMax

70 80 90 100 110 120 130 1400.0

0.2

0.4

0.6

0.8

η r

ζs=1.0

( f-η)/(ρ0/η2)

α/(ρ0/η2)

ρext/ρ0

70 80 90 100 110 120 130 1400.0

0.2

0.4

0.6

0.8

1.0

η r

ζs=10 ρext/ρ0ρind/ρ0ρtotal/ρ0

70 80 90 100 110 120 130 140-1.0

-0.5

0.0

0.5

1.0

η r

ζs=10ϵKin/ϵMaxϵPot/ϵMaxϵES/ϵMax

70 80 90 100 110 120 130 1400.0

0.2

0.4

0.6

0.8

η r

ζs=10

Figure 3.9: Behaviors of f and α, the charge densities, the energy densities

for various surface thickness parameters ζs. The functions f and α are shown

together with ρext in the first column, ρext, ρind and ρtotal are shown in the

middle column, and ϵKin, ϵPot and ϵES are shown in the right column. The

first row is for ζs = 0.1, the second for ζs = 1, and the last for ζs = 10.

26

Chapter 4

Q-balls in a Spontaneously

Broken U(1) Gauge Theory

In the previous charpter, we investigated the total charge screening of an

external souorce in a spontaneously broken U(1) gauge theory. As a result,

we found that an arbitrary localized source is always totally screened by the

charge density of the complex Higgs scalar field coupled with a U(1) gauge

field.

In this chapter, we construct a Q-ball solution by introducing another

complex matter scalar field instead of the external source. We show that

the total charge screening occurs for the Q-balls.

4.1 Q-balls and gauged Q-balls in a coupled two

scalar fields model

Spherically symmetric nontopological solitons, i.e., Q-balls are first proposed

by R. Friedberg, T.D. Lee, and A. Sirlin [6]. They assumed a system consisits

of a real scalar field with a potential and a complex matter scalar field given

by

S =

∫d4x

(−(∂µψ)

∗(∂µψ)− 1

2(∂µϕ)(∂

µϕ)− λ

4(ϕ2 − η2)2 − µψ∗ψϕ2

),

(4.1)

where the firld ϕ breaks Z2 symmetry spontaneously. In 1991 X. Shi and

X.Z. Li generalized the Q-balls by assuming a local U(1) gauge invariance

27

[21]. The action is given by

S =

∫d4x

(−(Dµψ)

∗(Dµψ)− 1

2(∂µϕ)(∂

µϕ)− λ

4(ϕ2 − η2)2 − µψ∗ψϕ2 − 1

4FµνF

µν

).

(4.2)

In the sysytem, since the U(1) gauge field is coupled only with the matter

scalar field, charge screening can not occur. We replace a real scalar field to

a complex Higgs scalar field.

4.2 Basic Model

We consider the action given by

S =

∫d4x

(−(Dµψ)

∗(Dµψ)− (Dµϕ)∗(Dµϕ)− V (ϕ)− µψ∗ψϕ∗ϕ− 1

4FµνF

µν

),

(4.3)

where ψ is a complex matter scalar field, ϕ is a complex Higgs scalar field

with the potential

V (ϕ) :=λ

4(ϕ∗ϕ− η2)2, (4.4)

where λ and η are positive constants, and Fµν := ∂µAν − ∂νAµ is the field

strength of a U(1) gauge field Aµ. The covariant derivative Dµ in (4.3) is

defined by

Dµψ := ∂µψ − ieAµψ, Dµϕ := ∂µϕ− ieAµϕ, (4.5)

where e is a gauge coupling constant. This model is a generalization of the

Friedberg-Lee-Sirlin model by introducing a complex Higgs scalar field and

a U(1) gauge field.

The action (4.3) is invariant under the local U(1) times the global U(1)

gauge transformations,

ψ(x) → ψ′(x) = ei(χ(x)−γ)ψ(x), (4.6)

ϕ(x) → ϕ′(x) = ei(χ(x)+γ)ϕ(x), (4.7)

Aµ(x) → A′µ(x) = Aµ(x) + e−1∂µχ(x), (4.8)

where χ(x) and γ are an arbitrary function and a constant, respectively.

Owing to the gauge invariance, there are the conserved current

jνψ := ie ψ∗(Dνψ)− ψ(Dνψ)∗ , (4.9)

jνϕ := ie ϕ∗(Dνϕ)− ϕ(Dνϕ)∗ (4.10)

28

satisfying ∂µjµψ=0 and ∂µj

µϕ=0. Consequently, the total charge of ψ and ϕ

defined by

Qψ :=

∫ρψd

3x, (4.11)

Qϕ :=

∫ρϕd

3x, (4.12)

are conserved, where ρψ := jtψ and ρϕ := jtϕ.

The energy of the system is given by1

E =

∫d3x

(|Dtψ|2 + (Diψ)

∗(Diψ) + |Dtϕ|2 + (Diϕ)∗(Diϕ)

+ V (ϕ) + µ|ψ|2|ϕ|2 + 1

2

(EiE

i +BiBi))

, (4.13)

where Ei := Fi0, Bi := 1/2ϵijkFjk, and i denotes a spatial index. In the

vacuum state, which minimizes the energy (4.13), the fields ψ, ϕ, and Aµ

should satisfy

ψ = 0, ϕ∗ϕ = η2, Dµϕ = 0, and Fµν = 0, (4.14)

equivalently,

ψ = 0, ϕ = ηeiθ(x), and Aµ = e−1∂µθ, (4.15)

where θ is an arbitrary continuous regular function. We exclude topologi-

cally non-trivial case in this paper. The Higgs scalar field ϕ has the vacuum

expectation value η, then the Ulocal(1)×Uglobal(1) symmetry is broken into

a global U(1) symmetry, so that the gauge field Aµ and the complex scalar

field ψ acquire the mass mA =√2eη and mψ =

√µη, respectively. The

real scalar field that denotes a fluctuation of the amplitude of ϕ around η

acquires the mass mϕ =√λη.

By varying (4.3) with respect to ψ∗, ϕ∗, and Aµ, we obtain the equations

of motion

DµDµψ − µϕ∗ϕψ = 0, (4.16)

DµDµϕ− λ

2ϕ(ϕ∗ϕ− η2)− µϕψ∗ψ = 0, (4.17)

∂µFµν = jνϕ + jνψ. (4.18)

1See Appendix D.

29

We assume that the fields are stationary and spherically symmetric in

the form,

ψ = eiωtu(r), (4.19)

ϕ = eiω′tf(r), (4.20)

At = At(r), and Ai = 0, (4.21)

where ω and ω′ are constants, u(r) and f(r) are real functions of r. Using

the gauge transformation (4.6), (4.7) and (4.8), we fix the variables as

ϕ(r) → f(r), (4.22)

ψ(t, r) → eiΩtu(r) := ei(ω−ω′)tu(r), (4.23)

At(r) → α(r) := At(r)− e−1ω′, (4.24)

where we assume Ω := ω − ω′ > 0 without loss of generality.

Substituting (4.22), (4.23), and (4.24) into (4.16), (4.17), and (4.18), we

obtain a set of the ordinary differential equations:

d2u

dr2+

2

r

du

dr+ (eα− Ω)2u− µf2u = 0, (4.25)

d2f

dr2+

2

r

df

dr+ e2fα2 − λ

2f(f2 − η2)− µfu2 = 0, (4.26)

d2α

dr2+

2

r

dα

dr+ ρtotal = 0, (4.27)

where ρtotal is defined by

ρtotal(r) := ρψ(r) + ρϕ(r). (4.28)

The charge densities ρψ and ρϕ are given by the variables u, f , and α as

ρψ = −2e(eα− Ω)u2, (4.29)

ρϕ = −2e2αf2. (4.30)

We seek configurations of the fields with a non-vanishing value of Ω that

characterizes the solutions.

We require boundary conditions for the fields so that the fields should

be regular at the origin. Then, we impose the conditions for the spherically

symmetric fields as

du

dr→ 0 ,

df

dr→ 0 ,

dα

dr→ 0 as r → 0. (4.31)

30

On the other hand, fields should be in the vacuum state at the spatial

infinity. Therefore, from (3.18) we impose the conditions

u→ 0 , f → η , α→ 0 as r → ∞. (4.32)

4.3 Numerical Calculations

In this section, we present numerical solutions of the Q-ball by using the

relaxation method. In numerics, hereafter, we set η as the unit, and scale the

radial coordinate r as r → ηr, and scale the functions f , u, α as f → η−1f ,

u→ η−1u, α→ η−1α, respectively, and scale the parameter Ω as Ω → η−1Ω.

We set λ = 1, µ = 1.4 and e = 1, as an example, in this paper.

In Fig.4.1, we plot u(r), f(r), and α(r) as functions of r with four values

of Ω. In the all cases of Ω, the functions, whose shapes depend on Ω, have

finite support, namely, solitary solutions are obtained.

( f-η)/η

α/η

u/η

0 50 100 150 200

0.00

0.02

0.04

0.06

0.08

η r

Ω=1.183

( f-η)/η

α/η

u/η

0 50 100 150 200

-0.1

0.0

0.1

0.2

0.3

η r

Ω=1.178

( f-η)/η

α/η

u/η

0 50 100 150 200

-0.1

0.0

0.1

0.2

0.3

η r

Ω=1.171

( f-η)/η

α/η

u/η

0 50 100 150 200

-0.1

0.0

0.1

0.2

0.3

η r

Ω=1.170

Figure 4.1: Numerical solutions f(r), u(r), and α(r) are drawn for Ω =

1.183, 1.178, 1.171 , and 1.170.

31

In the case of Ω = 1.183 and Ω = 1.178, the field profiles are Gaussian

function like. On the other hand, for Ω = 1.171, Ω = 1.170, the field

profiles are step function like. The solutions in the latter cases represent

homogeneous balls, namely, the functions u, f and α take constant values

inside the ball, and they change the values quickly in a thin region of the

ball surface, r = rs, and u, α vanish and f takes the vacuum expectation

value η outside the ball.

ρψ/ρψ(0)ρϕ/ρψ(0)ρtotal/ρψ(0)

0 50 100 150 200-1.0

-0.5

0.0

0.5

1.0

η r

Ω=1.183


0 50 100 150 200-1.0

-0.5

0.0

0.5

1.0

η r

Ω=1.178


0 50 100 150 200-1.0

-0.5

0.0

0.5

1.0

η r

Ω=1.171


0 50 100 150 200-1.0

-0.5

0.0

0.5

1.0

η r

Ω=1.170

Figure 4.2: The charge densities ρψ, ρϕ and ρtotal := ρψ + ρϕ normalized by

the central value of ρψ are shown for Ω = 1.183, 1.178, 1.171, and 1.170.

By numerical calculations, we depict the charge densities ρψ(r) and ρϕ(r)

in Fig.4.2 as functions of r. We find that the charge density ρψ is com-

pensated by the counter charge density ρϕ. Then, ρtotal almost vanishes

everywhere, namely, perfect screening occurs [26].

32

As the parameter Ω varies, the total charge of ψ, Qψ, defined by (4.11)

varies as shown in Fig.4.3. The solution exists for Ω in the range

Ωmin < Ω < Ωmax, (4.33)

where the values of Ωmin and Ωmax are discussed later. As seen in Fig.4.3,

Qψ diverges at Ω = Ωmin and Ω = Ωmax. For Ω near Ωmin in the range

(4.33), the solutions represent homogeneous balls, where the radius of the

ball increases as Ω approaches to Ωmin, while the constant values of u, f and

α are independent of Ω.

Ωmin 1.171 1.174 1.178 Ωmax

103

104

105

106

107

108

Ω/η

Qψ

Figure 4.3: The total charge of ψ, Qψ, is plotted as a function of Ω. Qψ

diverges at Ω = Ωmin, dot-dashed line (left), and Ω = Ωmax, dot-dashed line

(right). The value of Ωmin and Ωmax are defined by Eqs.(4.35) and (4.43),

respectively. The circle, square, triangle, and diamond marks in the figure

correspond to the cases of Ω = 1.183, 1.178, 1.171, and 1.170 that are shown

in Fig.4.1 and Fig.4.2, respectively.

Here, we estimate the value of Ωmax. Since u is small at a large distance,

and f − η and α are smaller than u there (see Fig.4.1), then solving the

linearized equations of (4.25), we have

u(r) ∝ 1

rexp

(−√m2ψ − Ω2 r

). (4.34)

If we require the solutions are localized in a finite region, the parameter Ω

should satisfies

Ω2 < Ω2max := m2

ψ = µη2. (4.35)

33

4.4 Homogeneous ball solutions

For the parameter Ω very closed to Ωmin, the homogeneous ball solutions

with large radius appear. We inspect the homogeneous ball solutions in

detail.

The set of equations (4.25), (4.26), and (4.27) can be derived from the

effective action in the form

Seff =

∫r2dr

((du

dr

)2

+

(df

dr

)2

− 1

2

(dα

dr

)2

− Ueff(u, f, α)

), (4.36)

Ueff(u, f, α) := −λ4(f2 − η2)2 − µf2u2 + e2f2α2 + (eα− Ω)2u2. (4.37)

If we regard the coordinate r as a ‘time’, the effective action (4.36) describes

a mechanical system of three degrees of freedom, u, f and α, where the

‘kinetic’ term of α has the wrong sign. In the case of the homogeneous ball

solution with a large radius, the damping terms that proportional to 1/r in

(4.25), (4.26), and (4.27) are negligible. In this case,

Eeff :=

(du

dr

)2

+

(df

dr

)2

− 1

2

(dα

dr

)2

+ Ueff(u, f, α) (4.38)

is conserved during the motion in the fictitious time r.

There are stationary points of the dynamical system on which

∂Ueff

∂u= 0,

∂Ueff

∂f= 0, and

∂Ueff

∂α= 0 (4.39)

are satisfied. Two stationary points exist in the region u ≥ 0, f ≥ 0, and

α ≥ 0. One stationary point, say Pv, exists at (u, f, α) = (0, η, 0), that is

the true vacuum. The other stationary point, say P0, exists at (u, f, α) =

(u0, f0, α0), where u0, f0, and α0 are given by solving (4.39) as

α0 =1

e(4µ− λ)

((µ− λ)Ω +

√µ(2λ+ µ)Ω2 − µλ(4µ− λ)η2

),

f0 =1√µ(Ω− eα0),

u0 =1√µ

√eα0(Ω− eα0).

(4.40)

We note that 0 < eα0 < Ω should hold for real value of u0. This condition

with (4.35) requires

λ < µ. (4.41)

34

A homogeneous ball solution with a large radius is described by a bounce

solution from P0 to Pv. Namely, the point that starts in the vicinity of the

stationary point P0 spends much ‘time’, r, near P0, and traverses to the

stationary point Pv in a short period, and finally stays on Pv. In Fig.4.4,

the homogeneous ball solution for Ω = 1.170 is shown as a trajectory in the

(u, f, α) space.

Figure 4.4: Trajectory of the numerical solution for Ω = 1.170 in the (u, f, α)

space. It starts from a point in a vicinity of P0 and ends at Pv. Dots on the

trajectory denote laps of the fictitious time r.

If Ω approaches to Ωmin, the radius of the homogeneous ball diverges.

It means that the solution with infinitely large radius starts from P0. Since

Eeff is conserved for the homogeneous ball solution with a large radius,

the bounce solution that describes the homogeneous ball connects the two

stationary points with equal potential heights, i.e.,

Ueff(Pv) = Ueff(P0). (4.42)

35

We see that this occurs for

Ω = Ωmin :=

√2√λµ− λ η =

√mϕ(2mψ −mϕ). (4.43)

Then, for the parameters satisfying (4.41), we see

Ωmin < Ωmax. (4.44)

Then, the non-topological soliton solutions exist for the model parameters

with (4.41).

We can estimate the value of Ωmax and Ωmin given by (4.35) and (4.43)

using the parameters λ, µ, and η in the numerical calculations as Ωmax ∼1.1832 and Ωmin ∼ 1.1689. We see in Fig.4.3 that numerical calculation

reproduce these values.

Using the ansatz (4.22), (4.23), and (4.24), we rewrite the energy (3.17)

for the symmetric system as

ENTS = 4π

∫ ∞

0r2ϵ(r)dr, (4.45)

ϵ := ϵψKin + ϵϕKin + ϵψElast + ϵϕElast + ϵInt + ϵPot + ϵES, (4.46)

where

ϵψKin := |Dtψ|2 = (eα− Ω)2u2, ϵϕKin := |Dtϕ|2 = e2f2α2,

ϵψElast := (Diψ)∗(Diψ) =

(du

dr

)2

, ϵϕElast := (Diϕ)∗(Diϕ) =

(df

dr

)2

,

ϵPot := V (ϕ) =λ

4(f2 − η)2, ϵInt := µ|ϕ|2|ψ|2 = µf2u2, ϵES :=

1

2EiE

i =1

2

(dα

dr

)2

,

(4.47)

are densities of kinetic energy of ψ and ϕ, elastic energy of ψ and ϕ, po-

tential energy of ϕ, interaction energy between ψ and ϕ, and electrostatic

energy, respectively. For the homogeneous ball solutions, these components

of energy density are shown in Fig.4.5. The dominant components of the

energy density ϵ are ϵψKin and ϵInt, and subdominant components are ϵϕKin

and ϵPot for the present cases. The densities of the elastic energy and the

electrostatic energy, which appear near the surface of the ball, are negligibly

small, then, they are not plotted.

36

We see, from (4.40), that the dominant and subdominant components of

energy density inside the balls are constants with the values

ϵψKin = ϵInt =1

µeα0(Ω− eα0)

3,

ϵϕKin =1

µ(eα0)

2(Ω− eα0)2, ϵPot =

λ

µ2((Ω− eα0)

2 − η2)2. (4.48)

Then the energy density and pressure2 for the homogeneous ball are con-

stants given by

ϵ ≃ ϵψKin + ϵϕKin + ϵInt + ϵPot

=2

µeα0(Ω− eα0)

3 +1

µ(eα0)

2(Ω− eα0)2 +

λ

µ2((Ω− eα0)

2 − η2)2,

p = pr ≃ pθ = pφ ≃ ϵψKin + ϵϕKin − ϵInt − ϵPot

=1

µ(eα0)

2(Ω− eα0)2 − λ

µ2((Ω− eα0)

2 − η2)2. (4.49)

We see that the pressure is almost isotropic, and p ∼ 0.05ϵ for the homoge-

neous ball of Ω = 1.170. The equation of state of the homogeneous balls is

like non-relativistic gas.

ϵψKin/ϵMax

ϵϕKin/ϵMax

ϵPot/ϵMax ϵInt/ϵMax

0 50 100 150 2000.0

0.1

0.2

0.3

0.4

η r

Ω=1.171

ϵψKin/ϵMax

ϵϕKin/ϵMax

ϵPot/ϵMax ϵInt/ϵMax

0 50 100 150 2000.0

0.1

0.2

0.3

0.4

η r

Ω=1.170

Figure 4.5: Components of energy densities of the homogeneous balls nor-

malized by the central value of total energy density are drawn for Ω = 1.171

(left panel) and for Ω = 1.170 (right panel). The profiles of ϵψKin and ϵInt

overlap each other.

2See Appendix D.

37

In the limit Ω → Ωmin, so that Qψ → ∞, we see

ϵψKin = ϵInt →λ(√µ−

√λ)√µ

(2√µ−

√λ)2

η4,

ϵϕKin → λ

( √µ−

√λ

2√µ−

√λ

)2

η4, ϵPot → λ

( √µ−

√λ

2√µ−

√λ

)2

η4, (4.50)

then we have

ϵ→2λ(

√µ−

√λ)

2√µ−

√λ

η4,

p→ 0. (4.51)

Therefore, in the large homogeneous ball limit, the ball becomes dust ball

with constant energy density given by (4.51).

4.5 Stability

The nontopological soliton, called Q-ball, can be interpreted as a condensate

of particles of the scalar field ψ, where the Higgs field plays the role of glue

against repulsive force by the U(1) gauge field. We compare energy of the

soliton, ENTS, given by (4.45) with mass energy of the free particles of ψ

that have the same amount of charge of the soliton as a whole. Then, the

numbers of the particles is defined by

Nψ :=Qψe, (4.52)

and the mass energy of the free particles of ψ is given by Efree = mψNψ.

Fig.4.6 shows the energy ratio ENTS/Efree as a function of Ω and as a

function of Nψ, respectively. We find a critical value of Ω, Ωcr, such that if

Ω < Ωcr, ENTS < Efree holds. Therefore, a Q-ball for Ω in the range

Ωmin < Ω < Ωcr (4.53)

is energetically preferable than the free ψ particles with the same charge of

the Q-ball as a whole. From the Fig.4.6, there exist stable Q-balls that are

condensates of large numbers of ψ particles.

Since the energy density and charge density are constant inside the ball,

the total energy and the total charge of matter field of the homogeneous ball

are written by

ENTS = ϵV, and Qψ = ρψV, (4.54)

38

where V is the volume of the ball. Then, the energy ratio ENTS/Efree for

the homogeneous ball is calculated as

ENTS

Efree=

ϵV

mψNψ=

ϵQψ/ρψmψQψ/e

=eϵ

mψρψ. (4.55)

In the limit Ω → Ωmin, so that Qψ → ∞, we obtain ENTS/Efree as

ENTS

Efree→((

2−√λ/µ

)√λ/µ

)1/2. (4.56)

It is clear that ENTS/Efree < 1 for λ < µ in the limit Ω → Ωmin. Therefore,

in the large limit of the homogeneous ball solution is stable.

Ωcr

1.170 1.174 1.178 1.182

0.990

0.992

0.994

0.996

0.998

1.000

Ω/η

ENTS/Efree

Ncr

103 5*103 5*104 5*105

0.990

0.995

1.000

Nψ

ENTS/Efree

Figure 4.6: The energy ratio ENTS/Efree is plotted as a function of Ω

(left panel), and as a function of Nψ (right panel). The circle, square,

triangle, and diamond marks in the figure correspond to the cases of

Ω = 1.183, 1.178, 1.171, and 1.170 that are shown in Figs.4.1 and 4.2, re-

spectively.

We show ENTS/Efree for various Qψ in Table.4.1. We see the inequality

ENTS(Qψ1) + ENTS(Qψ2) > ENTS(Qψ1 +Qψ2) (4.57)

holds for any Qψ1 and Qψ2 in the table. It means that one large Q-ball

is energetically preferable to two small Q-balls. Therefore, two Q-balls can

merge into a Q-ball, but a Q-ball does not decay into two Q-balls.

39

Ω Qψ ENTS

1.17771 2000 2363.4

1.17559 4000 4716.3

1.17465 6000 7066.4

1.17407 8000 9415.1

1.17368 10000 11762.8

1.17262 20000 23493.5

1.17213 30000 35217.0

1.17182 40000 46936.7

1.17161 50000 58653.7

1.17103 100000 117217

1.17059 200000 234295

1.17037 300000 351342

1.17024 400000 468373

1.17015 500000 585392

Table 4.1: The total charge of ψ, Qψ, and total energy, ENTS, of Q-balls for

various values of parameters Ω.

40

Chapter 5

Summary

In this paper, we studied the Q-ball solutions whose charge is totally screened

in a spontaneouly broken U(1) gauge theory. We first investigated the total

charge screening of an external source before constructing Q-balls.

In Chap.2, we introduced a scalar field theories with a U(1) symmetry.

We reviewed the standard idea, that is, by promoting the global U(1) sym-

metry to the local symmetry by introducing a gauge field, and it acquires a

mass through the Higgs mechanism.

In Chap.3, first, we showed the total charge screening by using a massive

vector field, i.e., Proca field always occurs. Based on the result of the Proca

field, we studied the classical system that consists of a U(1) gauge field and

a complex Higgs scalar field with a potential that causes the spontaneous

symmetry breaking. We presented numerical solutions in the presence of a

smoothly extended external source with a finite size. We have investigated

two kinds of extended external sources: Gaussian distribution sources and

homogeneous ball sources. In the case of Gaussian distribution source, the

profile of the total charge within radius r, Q(r), depends on the width of the

external source, rs. In the thin source case, where rs is much smaller than the

mass scale of the vector field, rA = m−1A , non-vanishing peak of Q(r) appears

at a radius in the range r < rs. Then, the charge density is detectable in the

region r < rs. The maximum value of Q(r) is less than the total external

charge, then the partial screening occurs in a finite distance. As r increases,

Q(r) damps quickly, then the total charge screening occurs by the induced

charge cloud for a distant observer. In the thick source case, where rs is much

larger than rA, Q(r) is almost zero everywhere, equivalently, ρtotal almost

vanishes everywhere. In this case, the charge is perfectly screened so that

41

the charge is not detectable anywhere. For the homogeneous ball source, we

have considered that the charge density is constant within the ball radius, rs,

which is assumed to be much larger than rA, and the charge density varies

with the surface thickness scale, ζs, at the ball surface. We found that inside

the ball, r < rs−rA, the amplitude of the scalar field and the gauge field take

constant values, respectively, and outside the ball, r > rs + rA, the scalar

field takes the vacuum expectation value and the gauge field vanishes. At

the ball surface, the both fields change their values quickly. The external

charge is canceled out by the induced charge cloud except the vicinity of ball

surface. In the thin surface case, ζs ≪ rA, electric double layer appears at

the ball surface. In the thick surface case, ζs ≫ rA, the charge cancellation

occurs even at the ball surface, namely, the perfect screening occurs. The

kinetic energy and the potential are main components of the energy density

inside the ball. For the thin surface case, the electrostatic component of the

energy density by the electric double layer appears at the ball surface.

In Chap.4, we studied the coupled system of a complex matter scalar

field, a U(1) gauge field, and a complex Higgs scalar field with a potential

that causes the spontaneous symmetry breaking. This is a generalization

of the Friedberg-Lee-Sirlin model [6]. In this model a global U(1) × Z2

symmetry is assumed. On the other hand, our system has a local U(1) ×global U(1) symmetry. Promotion of a global symmetry to a local one is a

natural extention in field theories. The local U(1)× global U(1) symmetry

in this system is broken spontaneously into a global U(1) symmetry by the

Higgs field. We have shown numerically that there are spherically symmetric

nontopological soliton solutions, Q-balls, that are characterized by phase

rotation of the complex matter scalar field, Ω. The solutions exist for Ω

in the range Ωmin < Ω < Ωmax. The lower bound, Ωmin, appears in order

to localiz the fields, and it is given by mass of the complex scalar matter

field as Ωmin = mψ. In the nontopological soliton solutions, charge densities

are induced by the matter scalar field and the Higgs field, respectively, and

they are totally screened each other. As a result, the Q-balls behave as

electrically neutral objects for distant observers.

If we assume the stationary and spharically symmetry, and if we regard

the radial coordinate as a “time” and the value of the fields as a position of

a particle, the system is analogous to a dynamical system of the particle in

three dimensions. The equations of motion for the fields are derived from an

42

effective action Seff which includes an effective potential Ueff. The system

has a friction that is in proportion to the inverse of “time”, i.e., r−1. If the

motion of the particle occurs in a range of large r, the frictional term can

be negligible, and the effective “energy” Eeff is conserved during the motion

in the fictitious “time”. The present dynamical system has some stationary

points. One of them corresponds to the vacuum. We found typical solutions

that the particle starts from a stationary point traverse toward the vac-

uum stationary point, we call them the bounce solutions. The trajectories

for the bounce solutions in three dimensional space connect two station-

ary points with the same potential height. We found stationary points for

the bounce solutions exist in the effective potential: one corresponds to the

true vacuum, say Pv, and the other is a nontrivial point, say P0. In the

limit Ω → Ωmin, values of the effective potential take same value at the two

stationary points respectively, U(Pv) = U(P0). Then radius of the Q-ball

diverges, and therefore, the charge and the mass diverge. We called the

bounce solutions homogeneous ball, namely, the fields takes constant val-

ues inside the ball. For the homogeneous balls, perfect screening occurs,

namely, the charge density of matter scalar field is canceled out everywhere

by the counter charge cloud of the Higgs and the gauge fields. Inspecting the

energy-momentum tensor of the fields, we showed that the pressure inside

the ball is almost isotropic, and the value is much less than the energy den-

sity. Then, a homogeneous ball is like a ball of homogeneous nonrelativistic

gas. Especially, in the limit Ω → Ωmin, the pressure vanishses and then the

ball behaves as dust with constant energy density.

The Q-balls obtained in this paper would have applications in cosmology

and astrophysics. The total screening of the charge is a preferable property

for the gauged Q-balls to be dark matter [12, 13, 14, 15, 16]. It is an impor-

tant issue how much amount of the Q-balls are produced in the evolution

of the universe [43, 44, 45, 46, 47, 48]. It would be an interesting problem

to clarify the mass distribution spectrum of the Q-balls, which would evolve

by merging process of Q-balls, in the present stage of the universe.

Furthermore, our planned future works are divided into three issues:

searching other types of the large Q-balls; investigating stability for the Q-

balls in our system; and taking the Einstein gravity into account. Large

Q-balls studied in this paper could be interpreted as bounce solutions from

one stationary point P0 to the vacuum stationary point Pv. In addition,

43

other types of the bounce solutions may be constructed since the effective

potential of the system admits other stationary points1. If other types of

the large Q-balls exist, they would behaves as other components compared

with them in this paper.

We investigated stability for the Q-balls energetically in this paper. This

is the simplest discussion for stability . However, it is not enough for proov-

ing stability of the Q-balls. An alternative analysis is the dynamical sta-

bility. This is achived by solving eigenvalue problems for perturbed field

equations. This is a very important issue in order to apply our Q-balls

to the cosmology and the astrophysics. There are various view points for

stability [30, 31, 32, 33, 34, 35].

We proved that the dust ball is energetically stable. Although the dust

balls have no limit of mass, if we take the gravity into account, arbitrary

large mass can not be allowed. If the mass of the dust ball becomes too

large so that the pressure fails to sustain the gravity, the ball would collapse

to a black hole. It is an interesting issue to study the gravitational effects

on the Q-balls [36, 37, 38, 39, 41]. In the case of the boson stars, upper

bound of mass exist for stability. It would be also an interesting problem

to clarify that these boson stars could be origins of primordial black holes

and/or super massive black holes in the central region of galaxies.

1We obtain other types of Q-ball solutions as shown in Appendix E.

44

Appendicies

Appendix A Charge Screning of a Point source

We analyze asymptotic behaviors of the scalar and gauge fields governed by

(3.29) and (3.30) for a point source, where the charge density is given by

the δ-function as

ρext(r) = qδ3(r), (A.1)

where q denotes the total external charge [57, 58]. The equations admit

an exact solution α(r) = q/(4πr) and f(r) = 0, the Coulomb solution.

However, this configuration does not minimize the energy (3.17), i.e., not

the vacuum. To seek other solutions with non-vanishing f(r), we discuss

asymptotic behavior of the fields near the point source and at infinity. We

set e = 1/√2 and λ = 1 so that rϕ := m−1

ϕ = 1 and rA := m−1A = 1.

A.1 Asymptotic behaviors for the point source

We assume that the asymptotic behavior of the fields in the vicinity of the

point source are given by

α(r) ∼ a1rγ , (A.2)

f(r) ∼ b1rβ. (A.3)

where a1 and b1 are non-vanishing constants. Substituting these expression

in (3.29) and (3.30), we obtain

β(β − 1)rβ−2 + 2βrβ−2 + e2a21rβ+2γ − λ

2b21r

3β +λ

2rβ = 0, (A.4)

γ(γ − 1)rγ−2 + 2γrγ−2 − 2e2b21r2β+γ = 0. (A.5)

45

First, we consider the case of β > −1. In this case, we can ignore the

third term in (A.5), and obtain γ = −1. By Gauss’ integral theorem applied

in a small volume including the point source, we have

α =a1r

=q

4πr. (A.6)

Since β > −1 and γ = −1, the first three terms in (A.4) should compensate

each other. Then, we obtain

β =1

2

(−1±

√1− 4κ2

)(A.7)

where κ := eq/4π.

If κ ≤ 1/2, β is real number. For the upper sign in (A.7), the elastic

energy density ϵEla defined in (3.33) is finite in the limit r → 0, however it

diverges for the lower sign. Then, we take the positive sign in (A.7) for the

power index of f .

If κ > 1/2, β becomes complex numbers

β =1

2

(−1± i

√4κ2 − 1

), (A.8)

then we have the real function f(r) in the form

f(r) =b1√rcos (σ log r + c1) , (A.9)

σ : =1

2

√4κ2 − 1, (A.10)

where b1 and c1 are constants soliton.

In the case of β ≤ −1, after some consideration, we see b1 should vanish.

Then, it is not the case in which the expected solution exists.

A.2 Distant region

At spatial infinity, α approaches to zero, and f does to η asymptotically.

Then, in the distant region, we rewrite f(r) as

f(r) ∼ η + δf(r), (A.11)

where δf → 0 as r → ∞. Substituting (A.11) to (3.29) and (3.30), we obtain

a set of linear differential equations

d2

dr2δf +

2

r

d

drδf − 1

r2ϕδf = 0, (A.12)

d2

dr2α+

2

r

d

drα− 1

r2Aα = 0, (A.13)

46

where higher order terms in δf and α are neglected. Solving these equations,

we obtain asymptotic behaviors of the functions as

δf(r) ∼ b2rexp

(− r

rϕ

), (A.14)

α(r) ∼ a2rexp

(− r

rA

), (A.15)

where b2 and a2 are constants soliton. These behaviors at the large distance

are general if the external source has a compact support around the origin.

A.3 Numerical calculations

As is shown in previous subsection, inspecting the equations (3.29) and

(3.30) near the origin, we obtain the asymptotic behavior of α in the form

of (A.6), and f is given by (A.3) for κ ≤ 1/2 or (A.9) for κ ≥ 1/2.

102*( f-η)/ηα/η

0 1 2 3 4 50

2

4

6

8

10

η r

κ=0.1

( f-η)/η

α/η

0 1 2 3 4 50

2

4

6

8

10

η r

κ=1.0

Figure 5.1: Numerical solutions of f(r) and α(r) for a point source

in the case κ = 0.1 (left panel), and in the case κ = 1.0 (right panel).

47

κ=0.1

κ=0.5

κ=0.2κ=0.3

κ=0.4

-7 -6 -5 -4 -3 -2 -1 0-0.50.00.51.01.52.02.53.0

log e[ηr]

loge[f/η]

κ=0.6κ=0.7

κ=0.8

κ=0.9

κ=1.0

-6 -5 -4 -3 -2 -1 0-1.5-1.0-0.50.00.51.01.5

loge[ηr]

log e[rf]

-6.5 -6.0 -5.5 -5.0

5.0

5.5

6.0

6.5

7.0

loge[ηr]

loge[α/η]

Figure 5.2: Asymptotic behaviors of

the function f(r) and α(r) near the

origin. Behaviors of f(r) for κ =

0.1 − 0.5 (left upper panel) and for

κ = 0.6 − 1.0 (right upper panel)

are shown. Behaviors of α(r) (lower

panel) are the same for κ = 0.1− 1.0.

On the other hand, the asymptotic behaviors at infinity are given by

α(r) ∼ a2rexp

(− r

rA

), (A.16)

f(r) ∼ η +b2rexp

(− r

rϕ

), (A.17)

where a2 and b2 are constants soliton.

Here, we solve equations (3.29) and (3.30) numerically, and study basic

properties of the system. Typical behaviors of the functions f(r) and α(r)

are shown in Fig.5.1. Especially, the behaviors of f and α near the origin are

shown in Fig.5.2. In the case of κ ≤ 1/2, f is given by the power function of

r, while in the case of κ > 1/2, oscillatory behaviors appear. The function α

is in proportion to r−1 independent with κ. These behaviors coincide with

(A.6) and (A.3) or (A.9). The asymptotic behaviors of the functions f and

α in a distant region coincide with (A.16) and (A.17) as shown in Fig.5.3.

48

0 2 4 6 8 10

-10

-8

-6

-4

-2

η r

loge[r(f-

η)]

0 2 4 6 8 10

-10

-8

-6

-4

-2

0

η r

loge[rα]

Figure 5.3: Asymptoric behaviors of the function f(r) and α(r) at

infinity. The functions r(f(r)−η) (left panel) and rα(r) (right panel)are shown in logarithmic scale.

The induced charge density ρind in (3.28) is plotted in the left panel of

Fig.5.4 as a function of r. The induced charge, whose sign is opposite to

the external source charge, distributes as a cloud around the point charge

source. We define the total charge within the radius r, say Q(r), by

Q(r) : = 4π

∫ r

0r2ρtotal(r)dr, (A.18)

where the total charge density ρtotal(r) is defined by

ρtotal(r) := ρext(r) + ρind(r). (A.19)

As shown in Fig.5.4, Q(r) is monotonically decreasing function of r. It means

that the positive charge of the external source is screened by the induced

negative charge cloud. In the region near the point source the charge is

partly screened, i.e., 1 > Q(r)/q > 0 and at a large distance the charge is

totally screened, i.e., Q(r)/q = 0.

0 2 4 6 8 10-0.5

-0.4

-0.3

-0.2

-0.1

0.0

η r

ρind/η3

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0

η r

Q/q

Figure 5.4: The induced charge density, ρind, (left pannel), and the

total charge within radius r, Q(r), (right pannel) are plotted for the

case of point charge source.

49

For some sets of two characteristic length scales (rϕ, rA), the function

Q(r) is plotted in Fig.5.5. We see that the shape of Q(r) does not depend

on rϕ, while the width of Q(r) is given by rA. In any case, in a distant

region where r ≫ rA, charge is totally screened. Except the neighborhood

of the origin, as shown in Fig.5.1, f takes the vacuum expectation value η.

The massive gauge mode with mass mA causes the charge screening, with

the size of rA = m−1A .

rϕ=1.0 , rA=1.0 rϕ=0.1 , rA=1.0 rϕ=1.0 , rA=0.1 rϕ=0.1 , rA=0.1

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0

η r

Q/q

Figure 5.5: The total charge Q(r) for some sets of parameters (rϕ, rA) and

fixed q = 0.1.

50

Appendix B Approximate solutions for the Gaus-

sian distribution sources

First, we consider the thin source case, rs ≪ rA. As shown in the first panel

of Fig.3.1 and Fig.3.3 for the case rs = 0.1 as an example, we see

|ρind| = 2e2f2α≪ ρext and η2 < f2 ≪ α2 (B.1)

in the near region, 0 ≤ r ≤ rs. Then, (3.29) and (3.30) reduces to

d2f

dr2+

2

r

df

dr+ α2f = 0, (B.2)

d2α

dr2+

2

r

dα

dr+ ρ0 exp

[−(r

rs

)2]= 0. (B.3)

in this region. We easily find a set of approximate solutions that satisfies

the boundary condition (3.39) in the expansion form

α(r) = α0 −ρ0r

2s

6

(r

rs

)2

+O(r

rs

)4

, (B.4)

f(r) = f0

(1− α2

0r2s

6

(r

rs

)2

+O(r

rs

)4), (B.5)

where α0 := α(0) and f0 := f(0).

In the far region, r ≫ rs, the functions f and α take the same forms of

the point source case. The constants soliton α0 and f0 should be adjusted

so that the solutions are smoothly connected from the near region to the far

region.

Next, we consider the thick source case, rs ≫ rA. Since the source is

spread widely, the variation of the external charge density is very small.

Accordingly, the variation of the functions f and α are also small as is seen

in the last panel of Fig.3.1 as an example. Then the derivative terms in

(3.29) and (3.30) can be negligible, and we have

− 2e2α2 + λ(f2 − η2) = 0, (B.6)

ρind = −ρext. (B.7)

If the external charge density ρext is small such that

ρext ≪η

rA rϕ, (B.8)

51

we have

f ≃ η, and α ≃ ρextm2A

. (B.9)

This behavior is seen in the last panel of Fig.3.1.

52

Appendix C Approximate solutions for the homo-

geneous ball sources

In the homogeneous ball sources with rs ≫ rA, except the vicinity of the

ball surface, rs − rA < r < rs + rA, f and α are almost constants soliton.

We can approach approximately to this simple behaviors.

In the region r < rs − rA, where ρext ≃ ρ0 = const., since the derivative

terms in (3.30) and (3.29) can be ignored for the solutions that satisfy the

boundary condition (3.39), then f and α take constant values. The equations

of motion reduce to

e2fα2 − λ

2f(f2 − η2) = 0, (C.1)

− 2e2f2α+ ρ0 = 0. (C.2)

By solving these coupled algebraic equations, we obtain

f2 ≃ f20 =1

3η2[1 +

(1 +X +

√X(2 +X)

)1/3+(1 +X +

√X(2 +X)

)−1/3],

(C.3)

α ≃ α0 =ρ0

2e2f20, (C.4)

where X is the constant defined by

X :=27rArϕ

2ηρ0. (C.5)

In the region r ≥ rs + rA, where ρext ≃ 0, we have simply f ≃ η and α ≃ 0.

The fields f and α change their values quickly in the vicinity of the ball

surface.

If ρ0 ≪ η/(rϕrA), a global solution can be obtained approximately. In

this case, as same as the Gaussian source case, f(r) ∼ η. Moreover, if

ζs ≪ rA, the equation of the gauge field can be reduced to

d2α

dr2+

2

r

dα

dr−m2

Aα+ ρ0θ(rs − r) = 0, (C.6)

where mA = r−1A . This is the Proca equation for a homogenious ball source.

In the region r < rs, since ρext(r) = ρ0, we have a solution

α(r) =C1

rsinh (r/rA) + ρ0r

2A, (C.7)

53

while in the region r > rs, we have

α(r) =C2

rexp (−r/rA) , (C.8)

where C1 and C2 are constants soliton that should be determined by con-

tinuity. This is achieved by junction conditions for (C.7) and (C.8) at the

surface r = rs as

C2 exp(−rs/rA) = C1 sinh(rs/rA) + ρ0rsr2A, (C.9)

−C2 exp(−rs/rA)(1 +

rsrA

)= C1

[rsrA

cosh(rs/rA)− sinh(rs/rA)

].

(C.10)

By solving (C.9) and (C.10), we obtain

C1 = −ρ0r

3A

(1 + rs

rA

)cosh(rs/rA) + sinh(rs/rA)

, (C.11)

C2 =ρ0r

3A exp(rs/rA)

[rsrA

cosh(rs/rA)− sinh(rs/rA)]

cosh(rs/rA) + sinh(rs/rA). (C.12)

Another simple case is that of ζs ≫ rA. The derivative terms in (3.30)

and (3.29) can be ignored everywhere for the solutions that satisfy the

boundary conditions (3.39) and (3.40), then the equations of motion reduce

to

e2fα2 − λ

2f(f2 − η2) = 0, (C.13)

− 2e2f2α+ ρext = 0. (C.14)

Therefore,

f2(r) ≃ 1

3η2[1 +

(1 + Y (r) +

√Y (r)(2 + Y (r))

)1/3+(1 + Y (r) +

√Y (r)(2 + Y (r))

)−1/3],

(C.15)

α(r) ≃ ρext(r)

2e2f(r)2, (C.16)

where Y (r) is the function defined by

Y (r) :=27rArϕ

2ηρext(r). (C.17)

54

Appendix D Energy-Momentum Tensor of the Sys-

tem

The energy-momentum tensor Tµν of the present system is given by

Tµν =2(Dµψ)∗(Dνψ)− gµν(Dαψ)

∗(Dαψ)

+ 2(Dµϕ)∗(Dνϕ)− gµν(Dαϕ)

∗(Dαϕ)

− gµν (V (ϕ) + µψ∗ψϕ∗ϕ)

+

(FµαF

αν − 1

4gµνFαβF

αβ

). (D.1)

Energy density and pressure components soliton are given by

ϵ =− T tt

= |Dtψ|2 + (Diψ)∗(Diψ) + |Dtϕ|2 + (Diϕ)

∗(Diϕ)

+ V (ϕ) + µ|ψ|2|ϕ|2 + 1

2

(EiE

i +BiBi), (D.2)

pr =Trr

=(Drψ)∗(Drψ) + |Dtψ|2 − (Dθψ)

∗(Dθψ)− (Dφψ)∗(Dφψ)

+ (Drϕ)∗(Drϕ) + |Dtϕ|2 − (Dθϕ)

∗(Dθϕ)− (Dφϕ)∗(Dφϕ)

− V (ϕ)− µ|ψ|2|ϕ|2

+1

2(−ErEr + EθE

θ + EφEφ −BrB

r +BθBθ +BφB

φ), (D.3)

pθ =Tθθ

=(Dθψ)∗(Dθψ) + |Dtψ|2 − (Drψ)

∗(Drψ)− (Dφψ)∗(Dφψ)

+ (Dθϕ)∗(Dθϕ) + |Dtϕ|2 − (Drϕ)

∗(Drϕ)− (Dφϕ)∗(Dφϕ)

− V (ϕ)− µ|ψ|2|ϕ|2

+1

2(−EθEθ + ErE

r + EφEφ −BθB

θ +BrBr +BφB

φ), (D.4)

pφ =Tφφ

=(Dφψ)∗(Dφψ) + |Dtψ|2 − (Drψ)

∗(Drψ)− (Dθψ)∗(Dθψ)

+ (Dφϕ)∗(Dφϕ) + |Dtϕ|2 − (Drϕ)

∗(Drϕ)− (Dθϕ)∗(Dθϕ)

− V (ϕ)− µ|ψ|2|ϕ|2

+1

2(−EφEφ + ErE

r + EθEθ −BφB

φ +BrBr +BθB

θ). (D.5)

55

Appendix E Other bounce solutions

E.1 Stationary Points of the System and Bounce Solutions

By solving (4.39), we can find stationary points. In addition to the isolated

stationary points Pv and P0, there exist ridges, R1 and R2, each ridge con-

sists of infinite stationary points aligned on a line, in the region u ≥ 0, f ≥ 0,

and α ≥ 0. The ridges are sets of points

R1 : α1 =Ω

e, f1 = 0, u1 = arbitrarly constants, (E.1)

R2 : α2 = arbitrarly constants, f2 = 0, u2 = 0. (E.2)

In addition to the bounce solution of the type P0 → Pv, we find numerical

bounce solutions of the type

P1 → Pv and Pv → Pv, (E.3)

where P1 is a point in the ridge R1. For any stationary point P2 in the ridge

R2, there is no solution of the type P2 → Pv.

E.2 Numerical calculations

Trajectories of the three solutions, P0 → Pv, P1 → Pv, and Pv → Pv, in

(u, f, α) space are shown in Fig.5.6.

56

Pv

P0

P1

R1

R2

Figure 5.6: Trajectories of the moving particle that connects the stationary

points: P0 → Pv, P1 → Pv, and Pv → Pv. The lines R1 and R2 are the

ridges that consist of infinite stationary points.

In Fig.5.7, we plot u(r), f(r), and α(r) as functions of r for the solutions:

P0 → Pv, P1 → Pv, and Pv → Pv.

( f-η)/η

α/η

u/η

0 50 100 150 200

-0.1

0.0

0.1

0.2

0.3

η r

e=1.0 , Ω=1.170

( f-η)/η

α/η

u/η

0 50 100 150 200

0

5

10

15

η r

e=0.09 , Ω=1.500

( f-η)/η

α/η

u/η

0 50 100 150 200-10123456

η r

e=0.09 , Ω=0.7794

Figure 5.7: Numerical solutions f(r), u(r), and α(r) for the solution P0 →Pv (left panel), P1 → Pv (central panel), and Pv → Pv (right panel).

57

Bibliography

[1] H. B. Nielsen and P. Olesen, “Vortex Line Models for Dual Strings,”

Nucl. Phys. B 61, 45 (1973).

[2] G. ’t Hooft, “Magnetic Monopoles in Unified Gauge Theories,” Nucl.

Phys. B 79, 276 (1974).

[3] A. Vilenkin, “Cosmic Strings And Other Topological Defects,”

[4] G. Rosen, “Particlelike Solutions to Nonlinear Complex Scalar Field

Theories with Positive‐ Definite Energy Densities,” J. Math. Phys. 9,

996 (1968).

[5] G. Rosen, “Charged Particlelike Solutions to Nonlinear Complex

Scalar Field Theories,” J. Math. Phys. 9, 999 (1968).

[6] R. Friedberg, T. D. Lee and A. Sirlin, “A Class of Scalar-Field Soliton

Solutions in Three Space Dimensions,” Phys. Rev. D 13, 2739 (1976).

[7] S. R. Coleman, “Q Balls,” Nucl. Phys. B 262, 263 (1985) Erratum:

[Nucl. Phys. B 269, 744 (1986)].

[8] A. Kusenko, “Solitons in the supersymmetric extensions of the stan-

dard model,” Phys. Lett. B 405, 108 (1997).

[9] A. Kusenko, “Q balls in the MSSM,” Nucl. Phys. Proc. Suppl. 62A/C,

248 (1998) doi:10.1016/S0920-5632(97)00664-6 [hep-ph/9707306].

[10] S. Kasuya and M. Kawasaki, “A New type of stable Q balls in the

gauge mediated SUSY breaking,” Phys. Rev. Lett. 85, 2677 (2000).

[11] G. R. Dvali, A. Kusenko and M. E. Shaposhnikov, “New physics in a

nutshell, or Q ball as a power plant,” Phys. Lett. B 417, 99 (1998).

58

[12] A. Kusenko and M. E. Shaposhnikov, “Supersymmetric Q balls as

dark matter,” Phys. Lett. B 418, 46 (1998).

[13] A. Kusenko and P. J. Steinhardt, “Q ball candidates for selfinteracting

dark matter,” Phys. Rev. Lett. 87, 141301 (2001).

[14] M. Fujii and K. Hamaguchi, “Higgsino and wino dark matter from Q

ball decay,” Phys. Lett. B 525, 143 (2002).

[15] K. Enqvist, A. Jokinen, T. Multamaki and I. Vilja, “Constraints on

selfinteracting Q ball dark matter,” Phys. Lett. B 526, 9 (2002).

[16] A. Kusenko, L. Loveridge and M. Shaposhnikov, “Supersymmetric

dark matter Q-balls and their interactions in matter,” Phys. Rev. D

72, 025015 (2005).

[17] K. Enqvist and J. McDonald, “Q balls and baryogenesis in the

MSSM,” Phys. Lett. B 425, 309 (1998).

[18] S. Kasuya and M. Kawasaki, “Q ball formation through Affleck-Dine

mechanism,” Phys. Rev. D 61, 041301 (2000).

[19] M. Kawasaki, F. Takahashi and M. Yamaguchi, “Large lepton asym-

metry from Q balls,” Phys. Rev. D 66, 043516 (2002).

[20] K. M. Lee, J. A. Stein-Schabes, R. Watkins and L. M. Widrow,

“Gauged q Balls,” Phys. Rev. D 39, 1665 (1989).

[21] X. Shi and X. Z. Li, ”Nontopological solitons in a U(1) gauge theory,”

J. Phys. A 24, 4075 (1991).

[22] I. E. Gulamov, E. Y. Nugaev, A. G. Panin and M. N. Smolyakov,

“Some properties of U(1) gauged Q-balls,” Phys. Rev. D 92, no. 4,

045011 (2015).

[23] H. Arodz and J. Lis, “Compact Q-balls and Q-shells in a scalar elec-

trodynamics,” Phys. Rev. D 79, 045002 (2009).

[24] T. Tamaki and N. Sakai, “Large gauged Q-balls with regular poten-

tial,” Phys. Rev. D 90, 085022 (2014).

59

[25] H. Ishihara and T. Ogawa, “Charge Screened Nontopological Soli-

tons in a Spontaneously Broken U(1) Gauge Theory“ arXiv:1811.10894

[hep-th], Prog. Theor. Exp. Phys. 2019, 021B01 (2019).

[26] H. Ishihara and T. Ogawa, “Perfect Charge Screening of Extended

Sources in an Abelian-Higgs Model,” arXiv:1811.10848 [hep-th].

[27] H. Ishihara and T. Ogawa, “Homogeneous Balls in a Spontaneously

Broken U(1) Gauge Theory,“ arXiv:1901.08799 [hep-th]., Phys. Rev.

D 99, 056019 (2019).

[28] M. Deshaies-Jacques and R. MacKenzie, “Maxwell-Chern-Simons

Q-balls,” Can. J. Phys. 85, 693 (2007) doi:10.1139/p07-063 [hep-

th/0606258].

[29] K. N. Anagnostopoulos, M. Axenides, E. G. Floratos and N. Tetradis,

“Large gauged Q balls,” Phys. Rev. D 64, 125006 (2001).

[30] A. G. Cohen, S. R. Coleman, H. Georgi and A. Manohar, “The Evap-

oration of Q Balls,” Nucl. Phys. B 272, 301 (1986).

[31] A. Kusenko, “Small Q balls,” Phys. Lett. B 404, 285 (1997).

[32] T. Multamaki and I. Vilja, “Analytical and numerical properties of Q

balls,” Nucl. Phys. B 574, 130 (2000).

[33] F. Paccetti Correia and M. G. Schmidt, “Q balls: Some analytical

results,” Eur. Phys. J. C 21, 181 (2001).

[34] M. Kawasaki, K. Konya and F. Takahashi, “Q-ball instability due to

U(1) breaking,” Phys. Lett. B 619, 233 (2005).

[35] N. Sakai and M. Sasaki, ”Stability of Q-balls and Catastrophe,” Prog.

Theor. Phys. 119, 929 (2008).

[36] R. Friedberg, T. D. Lee and Y. Pang, “Mini - Soliton Stars,” Phys.

Rev. D 35, 3640 (1987).

[37] R. Friedberg, T. D. Lee and Y. Pang, ”Scalar Soliton Stars and Black

Holes,” Phys. Rev. D 35, 3658 (1987).

[38] T. D. Lee and Y. Pang, “Fermion Soliton Stars and Black Holes,”

Phys. Rev. D 35, 3678 (1987).

60

[39] B. W. Lynn, “Q Stars,” Nucl. Phys. B 321, 465 (1989).

[40] D. F. Torres, S. Capozziello and G. Lambiase, “A Supermassive scalar

star at the galactic center?,” Phys. Rev. D 62, 104012 (2000)

[41] E. W. Mielke and F. E. Schunck, “Nontopological scalar soliton as

dark matter halo,” Phys. Rev. D 66, 023503 (2002).

[42] F. E. Schunck and E. W. Mielke, “General relativistic boson stars,”

Class. Quant. Grav. 20, R301 (2003)

[43] J. A. Frieman, G. B. Gelmini, M. Gleiser and E. W. Kolb, “Solitogen-

esis: Primordial Origin of Nontopological Solitons,” Phys. Rev. Lett.

60, 2101 (1988).

[44] K. Griest and E. W. Kolb, “Solitosynthesis: Cosmological Evolution

of Nontopological Solitons,” Phys. Rev. D 40, 3231 (1989).

[45] S. Kasuya and M. Kawasaki, “Q Ball formation in the gravity medi-

ated SUSY breaking scenario,” Phys. Rev. D 62, 023512 (2000).

[46] M. Postma, “Solitosynthesis of Q balls,” Phys. Rev. D 65, 085035

(2002).

[47] T. Multamaki and I. Vilja, “Simulations of Q ball formation,” Phys.

Lett. B 535, 170 (2002).

[48] T. Hiramatsu, M. Kawasaki and F. Takahashi, “Numerical study of

Q-ball formation in gravity mediation,” JCAP 1006, 008 (2010).

[49] R. Friedberg and T. D. Lee, ”Fermion Field Nontopological Solitons.

1.,” Phys. Rev. D 15, 1694 (1977).

[50] R. Friedberg and T. D. Lee, “Fermion Field Nontopological Solitons.

2. Models for Hadrons,” Phys. Rev. D 16, 1096 (1977).

[51] K. Shima, ”Charge Screening and an Extended Model of Hadrons,”

Nuovo Cim. A 44, 163 (1978).

[52] T. S. Levi and M. Gleiser, “Gauged fermionic Q balls,” Phys. Rev. D

66, 087701 (2002).

[53] J. E. Mandula, “Total Charge Screening,” Phys. Lett. 69B, 495

(1977).

61

[54] M. Bawin and J. Cugnon, “Charge Screening In Classical Scalar Elec-

trodynamics,” Phys. Rev. D 37, 2344 (1988).

[55] P. Sikivie and N. Weiss, “Screening Solutions To Classical Yang-mills

Theory,” Phys. Rev. Lett. 40, 1411 (1978).

[56] W. B. Campbell and R. E. Norton, “Total source charge and charge

screening in Yang-Mills theories,” Phys. Rev. D 44, 3931 (1991).

[57] L.D. Landau and E.M. Lifschitz, “Quantum mechanics”, Pergamon

Press (1965)

[58] S. L. Adler and T. Piran, “Relaxation Methods for Gauge field Equi-

librium Equations,” Rev. Mod. Phys. 56, 1 (1984).

62

nontopological solitons in a spontaneously broken...

Documents