comps final

Matter-Antimatter Asymmetry:

Why is there something and not nothing?∗

Hayden Tornabene†

(Dated: 25 January 2016)

In this paper, I examine the apparent discrepancy between matter and antimatter within the universe.

Extensive experimental work has been done with particle accelerators around the globe that have the

ability to recreate the conditions just after the big bang. The extreme temperature and energy density

of the early universe created an abundance of particles and antiparticles that annihilated in pairs as the

system cooled. Our universe is teeming with matter that comprises stars, planets, mountains, plants,

and animals whereas the presence of antimatter has generally been restricted to the domain of bubble

chambers and laboratories. This paper will explore the nature of the fundamental forces, symmetries,

and particles that make up the cosmos with a specific focus towards addressing possible explanations as

to way there is something rather than nothing.

∗Word Count: 7500†Carleton College, Physics Department - Northfield Minnesota

CONTENTS

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 BARYON DENSITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1 Sakharov’s Ingredients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3 THE STANDARD MODEL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3.1 Gauge Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3.2 Spontaneous Symmetry Breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.3 Electroweak Unification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4 SYMMETRY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.1 Charge Conjugation (C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.2 Space Inversion (P) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.3 Time Reversal (T) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5 NEUTRAL K MESON DECAY AND CP VIOLATION . . . . . . . . . . . . . . . . . . . . . 16

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

5.2 Christenson et al. (1964) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

5.3 Oscillations and Interference Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

5.4 CPT Conservation in K0 Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

5.5 Why is CP Violated? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

6 ANTIMATTER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

7 Baryon Asymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

7.1 Electroweak Baryogenesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

8 CONCLUDING THOUGHTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2

Matter-Antimatter Asymmetry H. Tornabene

1 INTRODUCTION

The question as to why is there something and rather than nothing is a far-reaching question that is

generally left to the preview of philosophers. Even if all the matter and antimatter in the universe was

created in a 1:1 ratio, and then pair annihilated and decayed in such a way as to leave no physical matter

behind, the quantum vacuum and photons would remain. The vacuum is still something. Photons are

still something. The rules of the vacuum, the rules that govern the sea of virtual particles, are still in fact

something. I take this brief philosophical prelude to establish the idea that the asymmetry between matter

and antimatter is merely the beginning of a larger and more complex story that I will not address in this

paper.

This paper will however address a shorter, yet still thrilling story of particles, invariance, unification,

symmetry violation, and the multiple flavors of baryogenesis. It will begin with a discussion of the Sakharov

conditions, a set of three empirical conditions theorized by the Russian physicist Andrei Sakharov as the

potential means by which a matter-antimatter asymmetry might be produced. I will then proceed to an

outline of the nature of the standard model, followed by a discussion of CPT symmetry. Once a sufficient

understanding has been established, this knowledge will help detail the surprising results observed in K

meson decay and its relation to symmetry breaking as well as our ultimate question of matter-antimatter

asymmetry. A detailed discussion of antimatter in terms of CPT symmetry will proceed a comprehensive

examination of baryon asymmetry, and electroweak baryogenesis as a possible scenario responsible for the

origin cosmological matter-antimatter asymmetry. The first portion of this paper will establish the important

concepts, vocabulary, and intellectual machinery necessary to understand and interpret the implications of

baryon asymmetry given the greater context of the current state and future of particle and fundamental

physics.

2 BARYON DENSITY

It is believed that our universe began with fiery explosion of immense force and heat where all of the

matter and antimatter were born out of an infinite density known as a singularity. As the universe cooled,

particles and antiparticles created in the Big Bang annihilated in pairs. Yet it is clear, just by looking

around, that all the matter and antimatter created did not annihilate in a 1:1 ratio, leaving the quantum

vacuum behind1. Why? Is it a question of the initial composition, that for ever billion pair annihilations,

one particle remained? Or is it some result of the subsequent 13.8 billion year history following the initial

explosion. This excess of matter is known as baryon asymmetry, where the production of that matter excess

is known as baryogenesis.

2.1 Sakharov’s Ingredients

Baryons of course do not make up the total composition of the universe, comprising only 5% of the

total energy density2. Dark matter, nonbaryonic matter, comprises about 1/3 of the energy density whereas

some negative pressure comprises the remaining 2/3. According to Cohen, De Rujula, and Glashow, 19983,

it seems that there is good evidence that large regions of antimatter do not exist in any local neighborhood

but only at large cosmic distance scales (Mpc), if at all. These islands are unlikely as we should see copious

511 keV gamma ray emission, characteristic of annihilation between matter and antimatter island overlap.

This emission is not observed to the requisite magnitude anywhere.

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The Russian nuclear physicist Andrei Sakharov posited that baryon density in the universe may not in

fact be a consequence of some initial configuration of the singularity, but as a result of physical laws that

have played out throughout the history of the universe. He is known for his groundbreaking paper in 19674

where he lists three ingredients that lead to the current state of baryon density (baryon asymmetry). His

three ingredients, which will motivate the next 4 sections (so if you get confused, fear not and read on) are

as follows:

(1). B violation, where baryon-number conservation is violated in fundamental laws. In the initial stages of

the universe, billionths of seconds following its birth, it is possible that baryon-number violating interactions

might have been in an equilibrium such that the initial baryon number of the universe was zero. Of course,

baryon-number violating processes would have taken the essentially zero baryon-number of the early universe

and forced a nonzero asymmetry, the one we see today.

(2). Charge-Parity (CP) violation would be necessary as if CP where conserved for all history of the universe,

every reaction that produced a particle would produce its antiparticle at precisely the same rate (as was

explored in neutral kaon decay). If CP were conserved for all time, the baryon number would remain zero

or unchanged.

(3). The universe, for most of its existence, resided in a thermal equilibrium. Cosmic microwave background

radiation, or CMBR shows a perfect blackbody spectrum, implying a thermal equilibrium for the majority

of the history of the universe. Thermal equilibrium however, even with Baryon and Charge-Parity violating

interactions and processes do not allow for a net baryon asymmetry. CPT symmetry implies that particles and

antiparticles have the same mass, thus at thermal equilibrium, particle and antiparticle density populations

are equal. Thus the third condition requires the presence of the first two ingredients, implying ultimately

that the universe must have resided in some thermal disequilibrium during its history. It is believed that

such a disequilibrium could have existed in the very earliest moments of the universe’s history. One theory,

developed by M.A. Markov5, suggested that such a violation of thermal equilibrium occurred because of

the existence of particles of maximum mass called ’maximoms.’ These neutral spinless particles supposedly

affect the state of quarks as they interact, causing the density population to favor the particle or antiparticle

population. What processes actually might cause a violation of thermal equilibrium is still widely by explored.

3 THE STANDARD MODEL

As of today, the Standard Model of physics successfully accurately details every known interaction of

particles and forces. A menagerie of elementary particles are governed by the four fundamental forces known

as the gravitational force, the electromagnetic force, the strong nuclear interaction, and the weak nuclear

interaction. The groundbreaking work of primarily three theoretical physicists, S. Glashow, S. Weinberg, and

A. Salam, unified the electromagnetic and weak interaction into what is dubbed the electroweak interaction.

The 1960s saw a profound leap in the understanding of the weak nuclear interaction, and its relationship to

electromagnetism, given a greater understanding of a concept known as local gauge invariance, a complex

issue which will be discussed in detail later in this paper. It is within the realm of of the electroweak

interaction where the beginning to the answer of our question lurks: why is there something rather than

nothing?

Before we embark on such a trepidatious journey, let us first discuss the observed and hypothetical

particles and quanta that populate the theory of the electroweak interaction.6 Figure 1 illustrates the el-

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ementary particles that comprise the standard model: leptons, quarks, the assorted force carrying bosons

(the photon, the weak intermediate W and Z bosons, and the gluon), and the Higgs boson. The leptons

are comprised of the electron, the muon µ−, and the tau lepton τ− and the associated neutrinos νe, νµ, ντ .

Leptons have spin one half are are divided, as seen in Figure 1 into pairs. The upper members have charge

e= 1.602 x 10−19 coulombs and are subject to both the electromagnetic and weak interactions whereas the

neutrino group have no charge and are subject to only the weak nuclear force. An important experimen-

tal fact regarding leptons shows that a quantity known as the lepton number is conserved. In all known

processes, the aforementioned leptons have number +1, where the antileptons, such as the positron and the

different flavors of antineutrinos (where helicity distinguishes the two as opposed to charge), have lepton

number -1. All other particles have lepton number 0.

Figure 1: The current Standard Model of physics describes the fundamental particles and forces that comprisenature. The fermions are broken into corresponding doublets based on particle characteristic where the gaugebosons are divided by the field with which they correspond. The graviton, the hypothesized gauge bosonthought mediate the gravitational force, is omitted here. Ongoing research into the field of gravitationalwaves hopes to find empirical evidence of the graviton’s existence7.

Particles that are subject to the strong nuclear force are known as hadrons and are further divided

into two distinct types. The first type of hadron is the half-integer-spin (fermion) baryon, which consists

of particles like the proton, the neutron, the lambda particle (Λ0), and other particles comprised of three

valence quarks8. The second type of hadron is the integer-spin (boson) meson, which consists of particles

like the pion (π), the kaon K, and other particles comprised of a pair of valence quarks, one quark and one

antiquark9. Quarks are known to come in six different flavors, up, down, charm, strange, bottom, and top.

As seen in Figure 1, the upper members of the quark doublets have electric charge Q = 23 |e|, and the lower

members have charge Q = - 13 |e|. The antiquarks (ex. u, d, etc.) simple vary by the sign of the charge. Both

quarks and antiquarks are subject to all interactions barring the gravitational force.

As was the case with lepton number, a quantity known as baryon number is similarly conserved in all

known processes. Quarks have baryon number B = 13 whereas the antiquarks have baryon number B = - 1

3 .

5


Understanding the conservation of baryon number will be an important tool in understanding different decay

processes and certain symmetry violations, particularly in the case of Kaon decay, which will be discussed

at length in section 4.

Figure 2: A Feynman Diagram describing the annihilation of an electron and positron pair into a photon.The photon then pair creates an electron an positron. A Feynman diagram is a pictorial representationof the mathematic formalism used to describe the interaction between subatomic particles. Introduced byRichard Feynman in 1948, these diagrams are meant to represent equations of interaction. Straight lines areconventionally used to show fermions whereas squiggly lines represent bosons. Feynman diagrams should beread as spacetime diagrams, read left to right11.

We now turn to a brief discussion of the intermediate bosons which, as mentioned before, transmit forces

between fermions. There exists the massless photon (γ) of quantum electrodynamics and three massive

bosons known as the W+ and the W− bosons, with respective ± |e| charge, and the neutral Z 0 boson.

While the photon mediates electromagnetic processes, like electron scattering, the massive bosons mediate

the charged and neutral weak interactions. Gluons, which mediate the strong nuclear force, are described

by a field known as quantum chromodynamics (QCD). QCD describes the strong nuclear interaction in a

complimentary way to the electroweak theory. While quarks can come in the distinct u,d,c,s,t,b flavors,they

also come with a distinct, conserved, ”color” charge, which of course has nothing to do with the actual

color of the particle10. Analogous to electric charge, the ’redness’, ’greenness’, and ’blueness’ of a quark

and a gluon describes how the particle responds to the presence or motion of color charge. Quantum

chromodynamics, in essence, is a crucial element of describing possible physical processes in spacetime, in

terms of a probability amplitude and its associated Feynman diagram. An example shown in Figure 2, a

Feynman diagram elegantly depicts events in terms of interaction vertices where charged particles (leptons

or quarks) are linked by a force carrying particle (photon, W boson, gluon, etc.). QCD, as opposed to

quantum electrodynamics (QED), introduced a wider set of ingredients where quarks (antiquarks) can carry

one positive (negative) unit of color charge, and linear superpositions of the 9 possible combinations of gluon

colors form 8 physical gluon types, where each gluon can also carry a unique color charge. There are a

variety of other details regarding QCD that I have not discussed here for the sake of brevity, but I suggest

to the reader Wilczek 2000 (Physics Today) for a concise introduction of the topic.

3.1 Gauge Invariance

I have been quite vague, thus far, concerning the interaction between particles and forces, speaking of

fermions and bosons in a way that I shall now clarify. Figure 2 shows a very simple, yet clear picture of how

6


fermions and bosons interact, specifically electrons, positrons, and photons in the given example. It should

be noted that the Feynman diagram in Figure 2 is correct to first order in the fine structure constant, α.

The full interaction is expressed with a Taylor series that includes higher order terms of the fine structure

constant. More complex diagrams emerge with greater attention to the expansion11. It is fruitful however to

discuss the first-order picture and how symmetry can govern the interaction and coupling between fermions

(matter, like mesons and baryons) to bosons (force fields, like the photon or gluon). An understanding of

how symmetry relates to the particles of the standard model reveals a great depth of information concerning

the properties and nature of bosons and fermions. Let us consider the following thought experiment.

Consider a smooth, uniformly indistinguishable table cloth upon a table12. Take the tablecloth as an

unbounded model of the universe. If we rotate the whole table cloth, nothing appears to have changed,

implying that the cloth is invariant under global rotations. We can consider some Lagrangian that describes

a state ψ,

L = δµψ†(x)δµψ(x)−m2ψ†(x)ψ(x), (1)

where we consider the state (and the Hermitian conjugate) has rotated by some complex phase,

ψ → eiθψ, ψ† → e−iθψ† (2)

The Lagrangian is identical for both the normal state and the rotated state, demonstrating the notion of

global gauge invariance. However, relativity forces us to consider only local symmetry rotations as the speed

of which information can travel is restricted by the speed of light. We enforce this local condition onto the

table cloth by placing a finger somewhere and twisting the finger. The cloth is invariant under rotation

everywhere on the cloth except for the place near the finger where wrinkles have formed. One may think

of the wrinkles as sort of fields lines where the local symmetry has created a field. As the tablecloth is not

symmetric under local rotations, no smooth connection links the local twist and the rest of the cloth. In

a Feynman diagram, the wrinkle is the vertex at which a boson is created, linking particle and field. We

consider the same state in Equation 2 but where the rotation becomes spatially (x) dependent where,

ψ → eiθ(x)ψ, ψ† → e−iθ(x)ψ†. (3)

New terms emerge in Lagrangian once we consider a spatially dependent (local) rotation, where

L = (Dµψ)†(Dµψ)−m2ψ†ψ − 1

4FµνF

µν −M2Aµ, Aµ (4)

where the first term in the sum is the kinetic term, followed by a mass term, dynamic term, and finally

the A-field in the field term. The final term in the Lagrangian describes the bosons that emerge from local

gauge rotations. In the electromagnetic formalism, the field term of the Lagrangian describes the photon

(hence massless), the W and Z bosons for the weak nuclear force, the gluon for the strong nuclear force,

and the graviton in the gravitational case. This idea will be discussed at length in the larger discussion of

spontaneous symmetry breakage, however it is important to note that local gauge rotations, as opposed to

global rotations, produce gauge bosons as a result of both the mathematic formalism and physical laws.

As this concept is of crucial importance, let us consider a secondary approach. We consider a particle

moving with constant velocity v. S is some rest frame where an observer resides and the boosted S’ frame is

where the particle resides. These two frames are related by a simple ’global’ Lorentz transformation, that

7


is applied regardless of the relative position between the frames. Now let us consider13 an elevator in free

fall, affected by the gravitational field. If the physicist within the elevator is boosted to the frame S’ of the

particle, and they attempt to measure the motion of the particle with respect to the elevator, the particle

may be described as if there were no gravitational field affecting it.

As a gravitational field is not uniform in space, the elevator is only a sufficient reference frame within

a specific arbitrarily small region of spacetime. That is to say, over a finite region, we can only consider

local symmetries of the gravitational field. The problem that arises from the distinction between local and

global symmetries is accentuated if we consider multiple physicists in different free falling elevators. Consider

a group of physicists attempting to map the strength of a gravitational by performing a series of free-fall

experiments at different places around the field. Each elevator is free-falling at a specific rate as determined

by the strength of the gravitational field at their discrete point in spacetime. Each physicist makes a series

of measurements, recording the path of the particle, also in the gravitational field in it’s frame S’, as it

appears to them. The relative position of the particle allows one to measure the strength of the gravitational

field within a local region. But how are the paths of the particles to be related to each other given that

measurements taken in different places will yield different results? A simple Lorentz transformation is not

sufficient.

Einstein developed a solution to this problem with the development of mathematical relation known as a

’connection.’ Consider some 4-vector Aµ which simply represents a measured quantity in the four dimensions

of spacetime. The components of a 4-vector are given as:

Aµ = (A0,A) where, A = (A0, A1, A2, A3), (5)

where the upper indicies indicate contravariant real vector quantities (generally displacement, velocity, etc.),

and the lower indicies represent mixed metric covariant vectors (i.e. gradient vectors). Upper and lower

indicies are related by the metric tensor where xµ = gµν xν where gµν is diagonalized 4x4 matrix with

g00=-1 for undisturbed flat spacetime. Physicist A measures quantity Aµ that changes by dAµ in at x where

Physicist B measures at x’ measures a change of dA′ν . As the differential is also a vector in special relativity,

the differential is simply related by:

dA′ν =δxµ

δx′νdAµ, (6)

where the index ν is summed over the four dimensions (0-3). In general relativity however, where x to x′ is

not linear (via a Lorentz transformation), the general expression gains a second term known as curvilinear

coefficient where,

dA′ν =δxµ

δx′νdAµ +AµΓµνλdx

′λ where, Γµνλ =δ2xµ

δx′νδx′λ. (7)

The gamma-function Γ is known as a Christoffel symbol, which concretely represents distinct geometric

features in the coordinate system of some manifold. The general Christoffel symbol is represented as a

function of the tensor over some arbitrary summed indicies k,l,m, where Γikl = 12gim( δgmk

δxl+ δgml

δxk− δgkl

δxm ).

We see that the Christoffel symbol in Equation 3 is a reduced form for indicies µ, in terms of ν and λ. The

physical meaning of this extra geometric term is clearly shown in a simplified picture given in Figure 3.

Given two separate vector measurements, one at x,y, one at x’,y’ on a circular path, the vectors both appear

to be perpendicular to the center yet point in different directions. The vector measurement is rotated by

8


some angle dφ and, thus no simple Lorentz transformation exists between the two measurements.

Figure 3: An example where a spatial transformation does not preserve physical symmetry as a result ofa curvilinear coordinate system. Values A and A’, similar to the result in Equation 3, are non identical,where in this specific case the vectors are of the same magnitude, however the unit vectors point in differentdirections14.

A German mathematician and theoretical physicist named Hermann Weyl would take the essential fea-

tures of the relativistic picture and create a new gauge theory that described the effects of a electromagnetism

in terms of the relative orientation between local frames and space time 15. Weyl suggested that the magni-

tude of a physical vector measurement should not be absolute but dependent on its spacetime coordinates.

A ’connection’, much like in the simpler case described above was created to relate the lengths of vectors

with different spacetime coordinates. This principle, as roughly outlined in Figure 3, is known as ’gauge’

invariance. While Weyl’s more specific theory was shown to be incorrect, the profound importance of locality

that determines the structure and consequence of many of the detailed features of gauge theory, which we

will not discuss here.

In brief summary16, rotation-like symmetries are called gauge groups where the associated wrinkles,

or fields, are known as a gauge fields. As we have begun to see, and will see more momentarily, the

process of spontaneous symmetry breaking (through local gauge rotations) can produce massive gauge bosons

that act as the mediator that links forces and particles. As fundamental fields cannot be measured in a

straightforward way, different configurations of the unobservable fields result in specific observable quantities.

Transformations between field configurations are known as gauge transformations where the uniformity of

the measurable quantities, in spite of the transformation, is known as gauge invariance.

The symmetry of gauge fields is described by a series of mathematic objects known as gauge groups.

There are three specific gauge groups that we will consider in this paper. The first group is known as

the U(1) one-dimensional unitary group. This group mathematically describes the symmetry of quantum

electrodynamics (QED), consisting of all complex numbers with absolute value 1 under multiplication, in

essence producing complex phase change. The SU(2) or special unitary group of dimension 2, associated with

weak charge, represents the three 2x2 complex unitary matrices with determinant one. The SU(3), group

which is associated with color charge and quantum chromodynamics (QCD), is the set of 8 3x3 complex

9


unitary matrices with determinant one. SU(3) can be used to describe rotations in color space, showing the

transformation of quark and gluon states, generally invariant under gauge rotation. These different unitary

groups are important as it is believed that during an inflationary transition period of the early universe, the

originally symmetric universe transitioned into one that had separate and partly broken symmetries, such

as the U(1), SU(2), and SU(3). Lagrangians used to describe QED or QCD produce massive particles when

certain symmetry groups are broken, and it is believed that in the early stage of the universe, it is possible

that symmetry breakage could have produced massive particles17.

3.2 Spontaneous Symmetry Breaking

In the previous section, I alluded to the fact that spontaneous symmetry breakage was responsible for

massive gauge bosons emerging from gauge fields. Now that a general understanding of gauge invariance

has been established, let us consider symmetry breakage, but first through the canonical mechanics problem

studied by Leonhard Euler involving elastic instability. With this more accessible understanding, we may

make some generalizations regarding field theory without being weighed down by complex notation and a

more abstract context.

We consider18 a thin rod of length L and of radius R. The rod is clamped on both ends and subjected to

a force along the z-axis of the rod, as seen in Figure 4. We model deflections of the rod in x and y directions

as functions of z such that X(z) and Y(z) can be written in terms of two fourth order homogeneous ordinary

differential equations where the conditions for equilibrium are,

IEd4X

dz4+ F

d2X

dz2= 0 (8)

and,

IEd4Y

dz4+ F

d2Y

dz2= 0 (9)

where I = 14 π R

4 and E is given as Young’s modulus for a thin metal rod, found empirically by,

E =FL0

A0∆L(10)

for F the force applied along the z-axis, L0 the original length, A0=π r2, and ∆L the change in the length

of the object19. The boundary conditions on the rod are given as X(0) = X(L) = X’(0) = X’(L) = 0 and

X’ = dXdz with like conditions on Y.

The equilibrium condition will hold for arbitrarily small force F, but if F is increased to some critical force

such that solutions to Equations 8 and 9 become nonzero, the rod becomes unstable to small perturbations

and the solutions to the differential equations, given the boundary conditions, become:

X(z) = C sin2(πz

L), (11)

with a similar solution for Y(z) given symmetry, which corresponds to a critical force necessary to induce

perturbations of Fc = 4 π IEL2. Small transverse oscillations in the straight rod may be shown to have

eigenfrequencies,

cos(kL) cosh(kL) = 1 (12)

10


where k = ω1/2(µ0 /EI) 1/4, µ0 is the mass per unit length, and specifically for small oscillations ω. For

the bent case where Fc has been applied, the characteristic oscillation ω will be different than Equation 12,

and the frequency in the plane of bending will also vary from the perpendicular direction. The Lagrangian

is symmetric around the z-axis and utilizes a F as its continuous parameter. We can discuss the eigenstates

of the Lagrangian where its ground state is F < Fc. For F ≥ Fc, random perturbations break the axial

symmetry of the Lagrangian and the system jumps to a new ground state. The new ground state of the

Lagrangian is degenerate as there are an infinite number of possible bent states around the z-axis.

Figure 4: The problem of elastic instability where a rod is subjected to a force at both ends. Small pertur-bations create a spontaneous symmetry breakage around the axially symmetric z-plane21.

For some quantum field φ = φ (xµ), in place of the rod, with a potential function V (φ), it is conceivable

to build a Lagrangian for that quantum field with some continuous parameter (not necessarily the same

force seen in our example) and an axial symmetry along a preferred axis. The potential function V (φ) is

seen in Figure 5 where for certain values of the continuous parameter, the ground state (or vacuum state) is

preferred and the symmetry is preserved. This ground state, at the top of the potential (consider a ball at

the top of the ’sombrero’), is unstable and can be perturbed yielding a lower energy vacuum state around

the ring of lowest potential states.

The infinite number of possible broken symmetry vacuum states is analogous to the bent rod in the

classical example. The symmetry can be broken in the same way as above if the continuous parameter

exceeds a critical value, given an new degenerate state of the Lagrangian. Small perturbative oscillations,

like those seen in the rod example and in Equation 12 are analogous in quantum field theory, and it is this

symmetry breakage that ultimately produces gauge bosons.

The previous discussion of gauge invariance and symmetry breakage are linked in profound way as there

is a way in which symmetry breaking may be built into local gauge theory that creates particles (oscillations)

with mass. The key principle resides in the fact that there exists an energy difference between the ground

11


Figure 5: A potential function V (φ) describing possible states of some system. The point at the top ofthe ’sombrero’ represents an unstable vacuum state that may be perturbed. The perturbation causes adegenerate symmetry breakage, yielding a lower energy, stable vacuum state. This symmetry breakagecauses massive gauge bosons to emerge in the Lagrangian.

state of the vacuum and the excited broken state20. As was mentioned before, we can write a Lagrangian

that describes particles and fields, and by imposing certain symmetry restrictions, massive boson terms

emerge. Symmetry breaking can lead to the appearance of massive particles as the ability to introduce some

sort of deviation, or oscillation in a field necessarily requires the vacuum not to be symmetric under a gauge

transformation. It is possible then for some state of higher energy to be broken into a lower state that has

more particles.

3.3 Electroweak Unification

In 1979, Sheldon Lee Glashow, Abdus Salam and Steven Weinberg were awarded the Nobel Prize in

physics ”for their contributions to the theory of the unified weak and electromagnetic interaction between

elementary particles, including, inter alia, the prediction of the weak neutral current”22. It turned out that

the field theory described by gauge theory could generate massive bosons which mediated a force that in turn

could describe the weak interaction and electromagnetism under the same formalism, among other things23.

A melding of these two theories seems plausible if we consider Figure 1. We see the basic weak multiplet,

otherwise known as the doublet containing the electron and the electron neutrino (νe), which falls under

SU(2) symmetry. The doublet falls into this specific symmetry given its weak isospin gauge symmetry, a value

that quantifies a particle’s charge state with respect to the weak force. We also see that this doublet contains

the singlet of electromagnetism (the electron) which is symmetric under U(1), (the group of one-dimensional

rotations). As the neutrino is neutral however, it seems that a SU(2)⊗U(1) scheme is not possible to see the

doublet as product of multiplets. According to Glashow, Salam, and Weinberg, however, there is a way out.

Let us begin by considering SU(2) symmetry producing a vector boson triplet (W+,W 0,W−) and the

U(1) group can produce some V-photon (a neutral, massless ’particle’ that will eventually become our photon

γ) that can couple (interact) to electrons and neutrinos. If this picture were complete, no differentiation

could be made between the weak and electromagnetic force as there is no distinguishing the electron and

12


the neutrino and the gauge bosons. As we know there is a distinction between the two, let us consider a

superposition of the neutral W 0 and the neutral V-photon where two new spin-1 bosons emerge:

γ = V cos(θW ) +W 0 sin(θW ) and Z = W 0 cos(θW ) + V sin(θW ), (13)

where γ is the photon, Z is known as the neutral vector boson, and θW is our superposition ’mixing’ (phase)

angle. This superposition is chosen such that the photon exclusively couples to the electron, and other

electrically charged particles that have appropriate SU(2) doublets. This superposition can also be selected

over certain other combinations. Through a spontaneous breakage of SU(2)⊗U(1) symmetry, the W+− and

Z bosons acquire mass as ’hidden’ mass terms appear in the associated descriptive Lagrangians. This solves

two problems simultaneously: first, the broken symmetry establishes the preferential direction in the vacuum,

precisely selecting the photon as a massless gauge boson for the electromagnetic force and second, the broken

symmetry provides mass to the W+− and Z, which were confirmed to have physical masses in 1983 by the

UA1 and UA2 collaborations24. As only one specific combination of the W 0 and V-boson (a degenerate

boson produced by the U(1) symmetry group that represents the photon) treats the vacuum as symmetric,

the case that gives the massless photon, the other combination must see the vacuum as asymmetric, thus

giving the bosons mass. Thus we are given four gauge bosons in total, the vector triplet (W+, Z0, W−) and

the singlet containing the photon (γ), unifying the two forces under the same formalism given the way in

which gauge bosons are formed through spontaneous symmetry breakage.

Figure 6: In this series of Feynman diagrams, vertices with W 0 and V combined into γ an Z0 verities. Thefirst case produces a massless particle, the photon, given the specific superposition of the SU(2) boson (W 0

and the U(1) boson (V). A different superposition of the states produces the Z0 with finite mass25.

13


4 SYMMETRY

Three important symmetry principles that pertain to the nuclear interaction require a detailed explanation

before a further investigation of matter-antimatter asymmetry. Charge conjugation (C), parity (P), and

time reversal invariance (T), consider the questions to whether a anything behaves in a different way if the

particle’s charge changes (C), if its spacial orientation changes (P), or if the direction of time affects the

overall nature or behavior.

Figure 7: A visual description of a physical example of parity violation where mirrored universe is dis-tinguishable from our own. It is important to note that the emission direction of the beta rays remainsunchanged while the direction of spin (the red arrow) is mirrored (and thus violates parity)26.

4.1 Charge Conjugation (C)

Charge conjugation is one of the three fundamental symmetries that constitutes the large CPT invariance.

In a classical system, charge conjugation simply implies that if a positive charge is replaced with a negative

one, the electric and magnetic fields will reverse27. In a quantum mechanical picture, charge conjugation

involves reversing the conserved quantities discussed earlier, such as lepton number, baryon number and

strangeness. The mass, energy, momentum, and spin or left unaffected.

We consider charge conjugation as an operator C. Given its nature, Maxwell’s equations, and thus the

electromagnetic processes, are invariant under C. That is to say, the equations are equally as viable whether

the charge is positive or negative. As a result, particle processes, such as decay, are specifically restricted.

Let us consider the decay of the neutral pion π0, which decays into two photons (γ). The charge parity of a

photon is defined as νC(γ) = -1. The negative eigenvalue for the photon stems from the fact that a photon is

its own antiparticle. Furthermore, the helicity of the photon forces a 180 degree rotation when the hermitian

conjugate is applied between the photon and antiphoton state, thus introducing the minus sign. Thus, the

14


behavior of a pion decay under charge conjugation is,

νC(π0) = νC(γ)νC(γ) = (−1)2 = 1, (14)

which demonstrates that neutral pion decay cannot yield one or three photons as these decay processes would

violate charge conjugation symmetry (+1 vs. -1). The weak interaction does not obey charge conjugation

symmetry as neutrinos and antineutrinos, as mentioned earlier, are determined by the helicity (right vs. left

handedness). This implies that in order to produce a right-handed antineutrino, one must change the physical

helicity of the particle with the C operator in conjunction with another operator (the parity P operator),

resulting in beta decay that is invariant under CP symmetry. Neutrinos and photons are different, with

respect to the C operator, in the way that photons may exist with both parities while all neutrinos are

left-handed and all antineutrinos are right-handed.

4.2 Space Inversion (P)

The parity operator simply produces a mirror image by changing the algebraic sign of the coordinate system.

This simple idea has profound ramifications in quantum mechanics as wave functions that describe particles

behave differently given coordinate transformations. The parity transformation is given as:

Figure 8: A parity transformation, as produced by the P operator, where a right-handed coordinate systemis changed to a left-handed one29.

Until28 1957, physicists believed that the laws of physics were invariant under parity transformations,

meaning that a system could not favor a left-handed orientation over a right-handed orientation. This is to

say that it was expected that nature would not prefer some inversion asymmetry, resulting in phenomena

that varied with which side of the ’mirror’ the particle resided on. P-symmetry however was observed to be

broken by a specific radioactive beta decay process, documented by C.S. Wu. Wu observed that a cobalt-

60 nucleus placed in a magnetic field preferentially emitted electrons from the beta decay in the direction

opposite of aligned angular momentum. As it is possible to distinguish a handedness, by using a mirror as

seen in Figure 7, the symmetry is not conserved. This is only one example of symmetry violation which will

prove crucial to the later discussion of K meson decay and CP violation.

4.3 Time Reversal (T)

Classically speaking, time reversal implies changing the sign of t with −t, inverting the direction of

time, and thus reversing the time derivative of spacial quantities like momentum and angular momentum.

Generally speaking, time reversal symmetry is not observed on the classical level given the second law

of thermodynamics, which states that the entropy of a complete system must increase, producing a t-

asymmetry. With that said, some quantum measurements are predicted to have time-reversal symmetry,

15


with little experimental success thus far. One specific example pertains to the experimental probing of the

dipole moment of the neutron, specifically the quarks that comprise it. The electric dipole moment is defined

as the shift in the energy of a state in the presence of an external electric field. As the electric dipole moment

is a vector, the expectation value in some state must be proportional to 〈ψ| J |ψ〉. Under time reversal, the

invariant state must have a vanishing EDM vector. This indicates that a non-vanishing EDM represents both

parity and time symmetry-breakage. Experimental evidence suggests that there is in fact no electric dipole

moment (and thus no symmetry violation), even though the quarks comprising the neutron are charged.

5 NEUTRAL K MESON DECAY AND CP VIOLATION

5.1 Introduction

As indicated in section 3, it was believed that if charge conjugation (C) were violated, then the combi-

nation of the CP operators (charge with parity) would conserve the overall symmetry. The decay of neutral

K0 mesons was the first experimental evidence of CP violation. I will outline the experiment fully, as well

as the theoretical formalism, to underline the importance of CPT conservation as a whole and the possible

ways in which symmetry breakage can lead to unexpected results.

K mesons30, or the Kaon, are produced in the following interaction:

π− + p −→ Λ0 +K0, (15)

where we see that both charge and strangeness are conserved given that the proton and pion have strangeness

0 while for Λ0, S=-1 and for K0 S=+1. The strangeness of a particle comes from its composition, and

the number of strange quarks present. Proton quark composition reads up-up-down (uud, S=0), pion

composition is antiup-down (ud, S=0), Λ composition is up-down-strange (uds, S=1), and K0 composition

is down-antistrange (ds, S=+1). The corresponding antiparticle of the K0 is destroyed in the following

absorption process:

K0 + p −→ Λ0 + π+ (16)

An important detail, that will prove crucial later, is that the two different flavors of neutral kaon, the short

and long lived varieties, may be described as a linear superposition of the particle and antiparticle states

(|K0〉 and |K0〉 respectively), where these two states from a two-dimensional orthonormal basis. We can

define the CP operator in terms of these states as follows,

CP |K0〉 = |K0〉 , CP |K0〉 = |K0〉 (17)

which is described in matrix form in the K0 K0 basis as

CP =

(0 1

1 0

).

As a result of detailed experimental observation, which I will explain shortly, it turns out that there are

two types of kaons, the short-lived and long-lived flavors that decay into different products. From Equation

17, the normalized eigenstates of CP can be written in terms of the short (S) and long (L) lived flavors as

follows31:

16


|KS〉 =1√2

(|K0〉+ |K0〉), |KL〉 =1√2

(|K0〉 − |K0〉). (18)

CP |KS〉 = |KS〉 , CP |KL〉 = − |KL〉 (19)

for eigenvalues of CP +1 and -1. We can similarly express the kaon particle and antiparticle as a superposition

of the long and short lived two-dimensional orthonormal basis:

|K0〉 =1√2

(|KS〉+ |KL〉), |K0〉 =1√2

(|KS〉 − |KL〉), (20)

where we are unable to assign a definite value of CP to the individual particle antiparticle states as they are

linear combinations of +1 and -1 states. As the formalism just established might suggest, K mesons decay

by the weak interaction in two distinct ways where the short lived process decays into a state that has CP

= +1 where,

KS −→ π+ + π−, KS −→ 2π0 (21)

with a lifetime of τS ≈ 0.9 x 10−10 seconds. The long lived kaon, with a lifetime of τL ≈ 560 x 10−10 seconds,

has a CP of -1 where the decay possibilities are given by:

KL −→ πππ, KL −→ πeν, KL −→ πµν. (22)

Why would nature preferentially ’choose’ a long or short lived state where the CP value for a given

process is +1 or -1? This is a profound question with profound implications, and resides at the heart of

CP violation, the very question Cronin, Fitch and their collaborates explored when they discovered the two

different flavors of kaon decay that violated CP symmetry.

5.2 Christenson et al. (1964)

As described in Christenson et al. 1964, the observation of a neutral two body kaon decay (into two

pions) challenged physicists understanding of CP symmetry in a fundamental way. I will sketch the basic

experimental apparatus and results in order to understand the consequences of such a unexpected result. As

seen in Figure 9, Kaon particles32 were beamed through a collimator into a helium bag to reduce regeneration

effects (change in relative amplitude phase shifts between the particle and antiparticle states, causing shifts

between the short and long-lived states). The charged decay products produced in the helium chamber

where analyzed in two symmetrically placed magnetic spectrometers that fed into a spark chamber (a device

used to detect charged particles). The spark chambers were triggered by scintillators, which ultimately

photographed the decay for later reconstruction.

The collaboration was able to distinguish the common three-body decay from the rarer short-lived two

body decay by considering two factors. The first was the consideration of a quantity known as the momentum

four-vector which describes the momentum of a particle in the three spatial dimensions and singular time

dimension. Given the experimental design, the two-body decay was expected to produce an invariant 4-

momentum, that is it would be the same for the pair of detected particles and the initial KS state. The

secondary marker considered the orientation of the momentum of just the three spacial dimensions. A two-

body decay would imply a vector sum that pointed along the incident beam created by the collimator, where

17


θ = 0.

Figure 9: The experimental design used in the Kaon decay experiment where decay time is measured usingsymmetric spark chambers, scintillators, and magnets33.

5.3 Oscillations and Interference Effects

Let us first consider kaon mixing before a larger consideration of CP violation. Consider a kaon produced

by Equation 11, and the state of the K(0) particle at time t is given as:

|ψ(t)〉 =1√2

(|K0S〉 (t) + |K0

L〉 (t)) =1√2

(|K0S〉 (0)e−λst + |K0

L〉 (0)e−λst), (23)

where the t = 0 state evolves to some later state t after the production of the kaon34. We can think of the

time in terms of distance where t = d / β γ for d the distance (m), β the relativistic speed of the particle

(v/c), and gamma the dimensionless Lorentz factor. The eigenvalues of the state equation given in Equation

23 are given as:

λS =1

2γS + imS , λL =

1

2γL + imL, (24)

where L and S again represent the long and short-lived kaons respectively. The change35 of the state described

in Equation 23 is affected by the KS and KL components that have well defined frequencies. The probability

amplitude that a meson is in the antiparticle state is some time t is 〈K0|Ψ(t)〉, which gives us a probability

of:

PK0(t) = | 〈K0|Ψ(t)〉 |2 =1

4[e

−tτS + e

−tτL − 2e−t(τ

−1S +τ−1

L )/2 cos(ωS − ωL)], (25)

where the ω terms are related to the energy states of the meson, ωi = Ei/~.

Figure 10 shows the plot of PK0(t) for t << τL, where the abscissa is at distance l = vt from when the

kaon is created. When the kaon is born, it has no amplitude for being in the antiparticle state, however as

18


Figure 10: The black line represents the probability of seeing a kaon at time t in the antiparticle or particlestate where the time is measured in terms of the life time τS

36.

the particle travels along, the short lived eigenstate decays away, leaving just the long lived KL eigenstate,

containing a large antiparticle component in the superposition. It is important to keep in mind that the

long lived eigenstate will also decay away, but this happens at a much later time t, at later distance l than

is shown in Figure 10. This example highlights the subtle relation between particles and antiparticles in a

very specific case where this given meson (the kaon) is a superposition between two different flavors of kaon,

KS and KL.

The Cronin and Fitch experiment, as well as the formalism we have just established, highlighted a

preference for one decay process over the over. If we use the CP operator on the KL decay process, we see:

K0L −→ π+ + e− + νe, K0

L −→ π− + e+ + νe (26)

CP [π+ + e− + νe] = π− + e+ + νe. (27)

CP invariance would suggest an identical probability of decay of these KL states, but as we have seen, the

linear superposition of the KS-KL (or in terms of the K0 and K0 basis) causes behavior that we might

not a priori expect, specifically that there is not an identical probability for the two decay processes. The

fractional excess is a mere 3.3×103, however, even a tiny discrepancy suggests a profound difference between

the way nature orients matter and antimatter, one that we will hope to begin to reconcile. This tiny violation

of CP symmetry might signal a solution to our question, why is there something (specifically matter and not

antimatter) and not nothing in our universe37.

5.4 CPT Conservation in K0 Decay

In the K0 and K0 basis38, the time evolution of a neutral kaon is described by,

d

dtΨ = −iΛΨ, Λ ≡M − i

2Γ (28)

with H represented by a two-dimensional matrix, H = M i Γ, where M and Γ are Hermitian, two-dimensional

matrices called the mass and decay matrices respectively. In order for CPT invariance to hold, the diagonal

elements of the matrix Λ must be equal. A complex violation parameter δ is introduced that combines

matrices Γ for both the particle and antiparticle state with the eigenvalues of of the mass and decay matrices

for both the long and short-lived component. The parameter δ is defined:

19


δ =ΛK0K0 − ΛK0K0

2(λL − λS(29)

The real part of γ, and thus the CPT conservation, may be experimentally measured by considering the time

evolution of the strangeness content as the Kaon state oscillates between the particle and antiparticle state, as

shown in Figure 10 (strangeness oscillates between +1 and -1). The details of this proof, which are far too long

and complex to reproduce here, may be found in P. Bloch’s, CPT Invariance Test in Neutral Kaon Decay39.

The essence of this CPT conservation argument is that while certain reactions may show slight violations in

CP symemtry, CPT symmetry is consistent with all known experimental observations, which has dramatic

implications for the nature of particles and their antiparticle pairs.

5.5 Why is CP Violated?

While CPT invariance preserves the integrity of C, P, and T symmetry, we still might ask, why is it

the case that CP is violated in K0L decay40? There are a variety of possible causes of CP violation, while

a complete picture still remains elusive. The primary difficulty in testing the limits of CP violation is that

the violation has been observed in very small select group of experiments. Probing the D meson particle

and antiparticle system as well has the bottom quark meson pair, B, seem promising candidates for further

CP violation, however increased sensitivity is required to make more decisive assertions on the nature of CP

violation. Commins and Bucksbaum list a series of possible causes for CP violation. The specifics of the

following causes are too complex for the scope of this paper, but for an elucidated discussion on points i-vi,

consult chapter 7.13 of their Weak Interactions of Leptons and Quarks. One or more of the following are

possible sources of CP violation, specifically in the long lived KL decay system:

i. T or C violation in the strong interaction of order of magnitude 10−3

ii. A larger (0.1, 0.01) T or C violation in the hadronic electromagnetic interaction.

iii. A T violation of the weak interaction of order of magnitude 10−3

iv. A ’superweak’ ∆ S = 2 (difference in strangeness) interaction

v. A mixing between the s with the c charm, b bottom, and t top quarks in a manner similar to

Cabibbo mixing (a unitary matrix that quantifies the strength of flavor-changing weak decay of

quarks as they propagate freely and interact weakly). This mixing is CP violating.

vi. The existence of extra CP violating scalar (Higgs) bosons in nature.

This last possibility seems of particular plausibility as tantalizing new evidence from CERN (the Center for

European Nuclear Research) suggests the existence of a second Higgs boson at 750 GeV. As to whether this

new scalar boson is CP violating has yet to be seen as the findings are still being reviewed and validated41.

The existence of a new Higgs boson would have profound effects on the validity of the Standard model and

might require a fundamental redrawing of our understanding of the forces of and particles of nature.

6 ANTIMATTER

We have worked long and hard to develop the necessary knowledge to understand symmetries, forces,

fields, and the particles that make up the standard model. We have alluded to the existence of antiparticles

(positrons, K0, etc.), yet we have thus far not really described the nature of an antiparticle. An antiparticle,

in short, is a particle with the same properties as its antiparticle pair, but with opposite charge. However,

20


we can still ask where such a particle comes from. Richard Feynman developed a theory of antimatter in

terms of pair creation and annihilation: consider an electron at some initial point A42 43. The electron moves

through an electric potential at a point B. As a result of the potential, the particle speeds up, and its path

is changed, until it reaches C at the other end of the potential. The particle regains its initial trajectory

(A-B) and arrives at point D. If we consider this journey, as seen in Figure 11a, as on a spacetime diagram,

the event B proceeds C. As we know from special relativity, the ordering of events depends on the inertial

reference frame from which the events are observed. We know however, given the finite speed of the electron

as < c, events B and C lie within the light cone, thus for any observer, classically speaking, B will proceed C.

If however we consider quantum effects and the Heisenberg uncertainty principle, the wave packet describing

the electron is no long restricted along the boundary of the light-cone, as seen in Figure 12, and the particle

may exist in the typically forbidden space-like region. If some event exists in the space-like region, temporal

ordering is lost, and thus some observers must see event C proceeding B (as seen in Figure 11b).

Figure 11: One possible description of pair creation and annihilation. In a., the particle moves through aplane-parallel field where it is respectively accelerated and decelerated by the potential field. In b., the sameexperiment as a. is produced, however it is observed at a different velocity (after Lorentz transformation).Heisenberg’s Uncertainty principle allows for event V to occur after F. This picture describes Feynman’spreferred description of pair creation and annihilation where instead of having particles moving backwardsin time, a particle and antiparticle pair is created at C. The particle moves to D where the antiparticle movesto B and meets the original particle from A44.

The ramifications of this theory are difficult to swallow. Given Feynman’s description of the particles

wordline proper time in terms of a path-integral, the individual trajectories, while consistent with relativity,

show that the particles are in fact traveling backwards in time. If this is the case, the particle would have

a negative frequency, and negative frequency of course implies the particle has negative energy (E = hf).

Let us understand than the events in Figure 11b through a different interpretation. An electron leaves A,

two particles are created spontaneously at C, one proceeds toward the original electron, intercepting it at B,

causing both particles to annihilate, while the other particle created at C travels to D. This picture seems

ad hoc and problematic as pair creation and annihilation seems to violate conservation of mass and energy.

If, however, we combine the energy uncertainty relation with relativity, we see that:

E = mc2, ∆E∆t > ~, ∆m∆t > ~/c2. (30)

Quantum relativity allows, through pair creation and annihilation, both commonly observed occurrences,

21


Figure 12: The purple line indicates the light cone and the different corresponding time and space regions.E3 represents the electron existing in the forbidden space-like region, causing event C to occur before eventB (Feynman picture of pair-creation and annihilation) 45.

the existence of massive antiparticles.

7 Baryon Asymmetry

Knowledge of the big bang rests principally on the understanding of two phenomena: the Hubble expan-

sion the universe and CMBR46. The first, Hubble expansion, indicates that the expansion of the universe

is accelerating. This fact was established by the collective efforts of the European Southern Observatory,

Cerro/Tololo, Keck, and Hubble telescopes through the observation of supernova remnants. The second, as

already hinted at, shows that about 300,000 years after the big bang, the temperature of the early universe

dropped to a fraction of an electron volt, allowing for the creation of neutral atoms. Theory and observation

agree yielding a baryon-to-photon ratio of,

nBnγ

= (6.1+0.3−0.2) x 10−10. (31)

The inflation of the universe, the exponential expansion of spacetime that occurred after the big bang, lasting

between 10−36 to 10−32 seconds1, tells us a few important facts. First, the observed universe grew from a

singularity, which explains homogeneity and isotropy. Second, it tells us that the universe is spatially flat.

Third, small fluctuations in the Robertson-Walker metric, a spacetime metric that describes the relative

orientation of the universe in terms of its curvature k, during inflation explain the small observed variation

in the temperature of CMBR. It seems as though these small fluctuations are the root to the formation

of observed structure in the universe. The final conclusion from the knowledge of inflation explains the

physical absence of theoretical objects, like the magnetic monopole. In essence inflation describes these sorts

of psychical phenomena as occurring during or before inflation whereas the rapidity of inflation flattens these

effects (like a magnetic monopole) to unobservable levels.

Before we address electroweak baryogenesis, one of the more compelling mechanisms believed responsible

for baryon asymmetry, let as consider the generally agreed open narrative of the possible history of the birth

22


and early life of the universe as outlined by Dine et al. 2003:

(1). The universe is very homogeneous with large density before t≈ 10−25 sec. Small patches (or bubbles)

begin to undergo inflation.

(2). The potential energy, or the binding energy of a particular field value, which describes the scale of the

inflation potential, is incredibly high at 1060 GeV4.

(3). Inflation causes scale factors to increase where approximately conserved quantities, such as monopole

number or baryon number, are reduced by an approximate factor of 1060.

(4). Inflation ends as the inflaton (hypothetical scalar field responsible for inflation) reaches minimum

potential. Inflaton decay reheats the universe.

(5). Baryon asymmetry is generated after inflation ends where the reheating caused by the scalar field decay

constrains the baryogenesis.

7.1 Electroweak Baryogenesis

While the Standard Model allows for baryogenesis, the amount baryogenesis required to produce the

observed baryon density is impossible. Thus, we must consider some sort of new physics that favors matter

creation of antimatter to sufficient order of magnitude. Electroweak baryogenesis is a promising theory that

explicitly satisfies all the Sakharov conditions, suggesting that the asymmetry we see today occurred during

the electroweak phase transition. A phase transition is the transition of a thermodynamic system phase

or state of matter to another via heat transfer. The electroweak phase transition specifically refers to the

point in the history of the universe where SU(2)×U(1) symmetry was broken and the electroweak force was

’broken’ into the separate electromagnetic and weak nuclear forces.

Baryon asymmetry is produced through a process known as the sphaleron process. A sphaleron is time-

independent solution to the electroweak field equation as constrained by the current Standard Model. The

sphaleron process is non-perturbative, meaning it cannot be described in a Feynman diagram. This simply

means that the sphaleron process does not represent the probability amplitude of some perturbation in

a quantum field. Rather, one may think of a sphaleron as a saddle point of the electroweak potential

energy, some infinite-dimensional field space, that can convert baryons to antileptons and antibaryons to

leptons (violating baryon and lepton number). It is believed that such points (sphalerons) in the electroweak

potential that permeates all of space were prominent in the high temperature of the early universe, but are

now unobservably rare47 We can think about the sphaleron process as potential difference, as in Figure 13

where we see two vacuum states separated by a potential difference Vo. If we consider the baryon state and

the antilepton state as two vacuum states separated by a potential difference, the sphaleron process acts as

the mediator between the two states. In the early universe, the ’sphaleron potential’ was low, allowing back

and forth baryon-antilepton conversion. The same is theoretically true for back and forth antibaryon-lepton

conversion. As the universe expanded and cooled, the potential increased, preventing further conversion,

persevering any potential imbalances created by the sphaleron process48.

The initial conditions required49 for an electroweak baryogenesis picture to make sense require a hot,

radiation heavy early Universe with a net baryon charge of zero where SU(2)XU(1) electroweak symmetry

is present (before the separation of the electroweak force). As the universe cooled to temperature below

100 GeV, the electroweak scale temperature at which neutral atoms can begin to form, the Higgs field,

which gives gauge boson mass, settles into the vacuum state. At this point, the electroweak symmetry

may be broken spontaneously into the one-dimensional U(1) subgroup. In order for baryogenesis to occur,

23


Figure 13: A potential Vo separating two vacuum states where the potential Vo , in a quantum mechanicalsense, can be thought of as the sphaleron process. The ’sphaleron potential’ mediates the conversion betweenbaryons and antileptons and the conversion between antibaryons and leptons48.

an electroweak phase transition occurs when bubbles of broken phase, discrete regions of the expanding

and cooling universe, nucleate with the surrounding plasma. Bubble nucleation describes the process of a

system moving to a lower energy state, particularly a vacuum or lower energy vacuum state. The bubbles of

discrete universe expand, collide, and coalesce until a global broken phase persists (the phase of the bubbles

is different than the phase of the surrounding plasma), resulting in a global electroweak phase transition.

Figure 14: Each circle represents a discrete ’bubble’ of universe with unique, nonzero phase. As the regionsof electroweak-broken phase expand, the surrounding plasma in the electroweak-symmetric phase,(< φ > = 0), is broken51.

Electroweak baryogenesis relies on the diffusion of charges (diffusion of particles and antiparticles) and

thus can be described using Boltzmann equations. It is believed that Baryon creation occurs near the

expanding bubble walls as the wall induces a force on the particles as it changes their phase50. This is

different for particles and antiparticles if and only if CP is broken in a specific reaction. The force of the

bubble wall can be thought of in terms of the momentum (pz) of the wall along the z-direction (the coordinate

along the wall-profile), where the collision term is a function of the following partial derivatives:

C[f ] = f(δt + zδz + pzδpz ), (32)

where the force is expressed in terms of the dispersion relation of the medium,

24


pz = −δzE(z, pz), (33)

which simply describes the dispersivity of the medium.

Morrissey et al. 2012 describes a three step process by which baryons are created. First, particles in the

plasma scatter with the bubble walls which produces C and CP asymmetries in particle density on the leading

edge of the expanding bubble. As of now, it is not clear how such C and CP asymmetries emerge, but as

further experimental work is done, probing the nature of CP asymmetry, clues to this process will hopefully

emerge. As seen in Figure 15, the produced CP asymmetries create nonzero phase behind (or within) the

bubble, and a symmetric (zero) phase ahead of the bubble. It is at this boundary where sphaleron transitions

are biased to produce baryons over antibaryons. The net baryon charge outside the wall is absorbed into the

broken phase by the expanding wall, resulting in a strongly diminished sphaleron transition rate, preserving

a higher baryon number within the bubble wall. This step is crucially important as the baryons created

in the first two steps would be ’washed out’ if the sphaleron process continues (back and forth between

baryons and antibaryons). Electroweak baryogenesis satisfies the three Sakharov conditions as the departure

from thermodynamic equilibrium is established by the bubble expansion through the cosmological plasma.

Violation of baryon number of course is a result of the rapid sphaleron transitions in the symmetric phase.

Finally, CP violating scattering produces the particle asymmetry near the bubble edge that is required to

bias baryon creation over antibaryon creation by the sphaleron process.

As was previously mentioned, all necessary components for electroweak baryogenesis are contained in the

Standard Model, however the order of magnitude is not quite correct. That is, observed baryon asymmetry is

not feasible within the Standard Model alone. The primary impediment to electroweak baryogenesis remains

that electroweak phase transitions occur at first-order where the mass of the Higgs boson is less than 70

GeV. The experimental value however for the Higgs mass is closer to 125 GeV. New physics beyond the

Standard Model are necessary to reconcile this problem. I guess its good they found a new Higgs boson (to

be discussed in a future paper).

Figure 15: Baryon production occurs on the leading (right) edge of the bubble wall. As the bubble expands,the sphaleron process no longer occurs, ceasing the biased production of baryons over antibaryons, perseveringa nonzero baryon number52.

25


8 CONCLUDING THOUGHTS

This treatise was meant to make the complex assumptions behind baryon asymmetry more accessible

to those not schooled in Quantum Field Theory. As such, a great deal of information has been left out

and may be found in the corresponding sources provided. Electroweak baryogenesis is simply one of many

possible explanations that might answer the question of, why is there something and not nothing. Each

individual theory is somehow incomplete, and a larger understanding of fundamental physics is needed in

order to reach a comprehensive theory to this most fundamental inquiry. Labs, like CERN, across the globe

are probing deeper into the fundamental structure of nature with increasingly higher energy experiments and

increased sensitivity. In the past two decades52, two investigations into the nature of B mesons, BaBar at

the Stanford Linear Accelerator Center (SLAC) and Belle at KEK laboratory in Japan, have yielded a great

deal of knowledge on CP violation. Even with these breakthroughs however, knowledge of CP violation

is incomplete to deal with the order of magnitude of CP violation necessary in theories like electroweak

baryogenesis. Most scientists believe however that the key to answering the question of antimatter lies in

CP violation, and experiments at the Large Hadron Collider (LHC) have been planned to further pursue

strange and down B meson decay in order to find further, more revealing, instances of CP violation.

The massive nature of neutrinos may yield further insight into CP violation, particularly in lepton

reactions. As neutrino mass may oscillate between matter and antimatter (helicity oscillations), experiments

are being designed to probe the oscillatory parameters of neutrinos. The MINOS experiment at Fermilab and

reactor-based experiments are on the forefront of answering these questions and defining these parameters.

Efforts to test the limits of our theoretical knowledge are crucial to understanding fundamental symmetry

breakage as most theories, like electroweak baryogenesis, conscribe to most observed phenomena. The only

solution, of course, remains to observe new physics using more sensitive experimental instruments that may

finally settle incongruities in the existing theoretical framework. While it is possible a full understanding

may be impossible, the history of science is rife with shocking results and marriages of theoretical and

experimental physics. It seems to be only a matter of time.

26


REFERENCES

1 E. Sather, The Mystery of the Matter Asymmetry, Beam Line, Spring/Summer 1996

ANNOTATION: Sather’s paper on antimatter provides an excellent overview on the problem of baryon

asymmetry in accessible terms. The text is relatively mathematically minimalist, however he presents the

text in not too abstract terms.

2 M. Dine et al., Origin of the matter-antimatter asymmetry, REVIEWS OF MODERN PHYSICS, VOL-

UME 76, JANUARY 2004

ANNOTATION: The work of Dine et al. on the origin of matter-antimatter asymmetry seems to be a

principal text on the question, describing the subtle and terrible complex details of 5 different possible

mechanisms for the asymmetry. I choose to discuss electroweak baryogenesis given Dine’s assertion that

it seemed to be one of the more probable theories as well as its close relationship with the gauge theory

previously established.

3 A.G. Cohen, A. De Rujula, and S.L. Glashow, A Matter-Antimatter Universe?, arXiv:astro-ph/9707087v2

15 Nov. 1997

ANNOTATION: Cohen et al., cited in Dine, discuss the possible locations of dense antimatter regions.

4 A. Sakharov, Violation of CP invariance, C asymmetry, and baryon asymmetry of the universe, Sov. Phys.

Usp. 34 (5), pp.392393, May 1991

ANNOTATION: Sakharov’s milestone text establishes the three conditions for baryon asymmetry produced

as a result of physical laws.

5 M. A. Markov, JETP 51:878 (1966); Sov. Phys. JETP 24:504 (1967), trans.

ANNOTATION: Markov presents one, rather exotic, explanation for what may have caused a violation in

thermal equilibrium in the early history of the universe.

6 E.D. Commins and P. H. Bucksbaum, Weak interactions of leptons and quarks, Cambridge University

Press, 1983, Chapter 1.1

ANNOTATION: Commins and Bucksbaum present a very clear and concise picture of the Standard model

and set the groundwork for larger consideration of the relationship between electromagnetism, the weak

interaction, and the strong nuclear force.

7 The Standard Model

URL:http://www.physik.uzh.ch/groups/serra/StandardModel.html

ANNOTATION: Image showing the standard model of physics.

8 Table of Baryons, Hyper Physics

URL: http://hyperphysics.phy-astr.gsu.edu/hbase/particles/baryon.html#c1

ANNOTATION: Useful table that lists composition, rest mass, symbol, as well as other characteristics of

most baryons.

9 Table of Mesons, Hyper Physics

27

http://www.physik.uzh.ch/groups/serra/StandardModel.html

http://hyperphysics.phy-astr.gsu.edu/hbase/particles/baryon.html#c1


URL:http://hyperphysics.phy-astr.gsu.edu/hbase/particles/meson.html#c1

ANNOTATION: Useful table that lists composition, rest mass, symbol, as well as other characteristics of

most mesons.

10 F. Wilczek, QCD Made Simple, American Institute of Physics, Physics Today, August 2000.

ANNOTATION: An interesting and accessible discussion of quantum chromodynamics that served a larger

discussion of the nature of fundamental particles and quarks.

11 Feynman Diagram Example

URL:https://en.wikipedia.org/wiki/Feynman_diagram#/media/File:Feynman_EP_Annihilation.svg

ANNOTATION: An example of a Feynman diagram showing pair annihilation and creation.

12 V. Icke The Force of Symmetry, Cambridge University Press, 1995, Chapter 10.3

ANNOTATION: Icke presents a very conceptually based, mathematically minimalist text that attempts

to describe topics in symmetry, particle physics, and general fundamental physics. Chapter 10.3 begins an

accessible explanation of global rotation as that which produces gauge fields, forces, and thus gauge bosons.

13 K. Moriyasu, An Elementary Primer for Gauge Theory, Wspc, 1983, Chapter 2.2

ANNOTATION: Moriyasu’s text on Gauge Theory is a complex and rich text provided a secondary approach

to Icke’s discussion of gauge fields. Chapter 2.2 builds a interplay between relativity, symmetry and the

corresponding relationship with coordinate violation and the difficulty of frame transformation.

14 Vector Rotation and Transformation, K. Moriyasu, An Elementary Primer for Gauge Theory, Wspc, 1983,

pp. 10

ANNOTATION: An image showing that certain quantities are not invariant under Lorentz transformation.

15 K. Moriyasu, An Elementary Primer for Gauge Theory, Wspc, 1983, Chapter 2.3

ANNOTATION: Chapter 2.3 furthers the discussion of frame transformation and applies the formalism

developed for general relativity and gravity to electromagnetism, and thus the weak interaction. The chapter

describes the exploits of Weyl’s gauge theory, the failures of his theory, and more importantly the distinct

features that would survive and inform much of modern gauge theory.

16 V. Icke The Force of Symmetry, Cambridge University Press, 1995, Chapter 10.4, 10,6, and Index

ANNOTATION: Icke furthers the discussion in chapter 3 towards a larger and more comprehensive picture

of gauge twists, phase, symmetry groups. The index provides succinct and accessible definitions that would

prove crucial to the development of this project.

17 V. Martin, SH Particle Physics, Symmetries in Particle Physics, University of Edinburgh, 2012, Section

13.1.1, 13.1.2.

URL: http://www2.ph.ed.ac.uk/~vjm/Lectures/SHParticlePhysics2012_files/PPNotes4.pdf

ANNOTATION: Martin provides summary notes from a series of lectures given at the University of Edin-

burgh which give explicit examples of SU(3) and U(1) groups and how they pertain to physical symmetry

described by unitary groups.

28

http://hyperphysics.phy-astr.gsu.edu/hbase/particles/meson.html#c1

https://en.wikipedia.org/wiki/Feynman_diagram#/media/File:Feynman_EP_Annihilation.svg

http://www2.ph.ed.ac.uk/~vjm/Lectures/SHParticlePhysics2012_files/PPNotes4.pdf




ANNOTATION: Chapter 2.4 provided a technical, yet still fruitful, discussion of spontaneous symmetry

breaking. While some elements were too complex for the scope of this paper, Commins and Bucksbaum

provide incredible insight into the crucially important concept of spontaneous symmetry breakage.

19 Youngs-modulus, Encyclopedia Britannica, 2014

URL: http://www.britannica.com/science/Youngs-modulus

ANNOTATION: A detailed description of Youngs-modulus.


ANNOTATION: Chapter 12.3 of Icke describes the asymmetry of the vacuum and allowed a larger and

more focused discussion of symmetry breakage, as established by Commins and Bucksbaum, Chapter 2.4, in

the context of the vacuum.

21 Spontaneous Symmetry Breakage, E.D. Commins and P. H. Bucksbaum, Weak interactions of leptons and quarks,

Cambridge University Press, 1983, pp. 45

ANNOTATION: An image showing a physical example of spontaneous symmetry breakage.

22 The Nobel Prize in Physics, 2016

URL:http://www.nobelprize.org/nobel_prizes/physics/laureates/1979/

ANNOTATION: A description of the 1979 Nobel prize.


ANNOTATION: Icke presents, in Chapter 12.6, an accessible discussion of the electroweak force, outlining

the central idea behind the symmetry and superposition arguments of Glashow, Weinberg, and Salam’s work

in electroweak unification.

24 P. Darriulat, The W and Z particles: a personal recollection, CERN Courier, April 2004

URL:http://cerncourier.com/cws/article/cern/29053

ANNOTATION: A description of the experimental history of determining the mass of the W and Z bosons.

25 Feynman Diagram W 0 Boson, V. Icke The Force of Symmetry, Cambridge University Press, 1995, pp.

237

ANNOTATION: An image showing the difference, in terms of Feynman diagrams between electroweak

interactions that produce photons or Z0 bosons.

26 Parity Violation

URL:https://www.aps.org/publications/apsnews/200112/history.cfm

ANNOTATION: A visual representation of parity violation.

27 CPT Invariance, Charge Conjugation, Time Reversal, and CP Invariance, Hyper Physics

URL:http://hyperphysics.phy-astr.gsu.edu/hbase/particles/cpt.html#c1

29

http://www.britannica.com/science/Youngs-modulus

http://www.nobelprize.org/nobel_prizes/physics/laureates/1979/

http://cerncourier.com/cws/article/cern/29053

https://www.aps.org/publications/apsnews/200112/history.cfm

http://hyperphysics.phy-astr.gsu.edu/hbase/particles/cpt.html#c1


ANNOTATION: A detailed discussion that served as a starting point for CPT invariance, charge conjuga-

tion, time reversal and CP invariance.

28 Charge, Parity, and Time Reversal (CPT) Symmetry, Guide to Nuclear Wall Chart, 2000

URL:http://www2.lbl.gov/abc/wallchart/chapters/05/2.html

ANNOTATION: An additional discussion of CPT symmetry.

29 Parity Symmetry Conservation, CPT Invariance, Charge Conjugation, Time Reversal, and CP Invariance,

Hyper Physics

URL:http://hyperphysics.phy-astr.gsu.edu/hbase/particles/cpt.html#c1

ANNOTATION: An example of a parity transformation.

30 G. Baym, Lectures on Quantum Mechanics, The Benjamin Cummings Publishing Company, 1969, Chap-

ter 2

ANNOTATION: Bayms discussion of kaon decay is more in terms of elementary quantum phenomena

which served as an excellent compliment to Commins and Bucksbaum, whose focus was more in terms of

the electroweak phenomena.

31 J.H. Christenson, J.W. Cronin, V. L. Fitch, and R. Turlay, Evidence for the 2π decay of the K0 meson,

Physical Review Letter, Volume 13, Number 4, July 27, 1964

ANNOTATION: Christenson, Cronin, and Fitch’s groundbreaking experiment that showed CP violation

by considering the decay of neutral K mesons. This was the first documented case of CP violation, recorded

in 1964.



ANNOTATION: A detailed discussion of kaon decay.

33 Kaon Decay Experimental Design, E.D. Commins and P. H. Bucksbaum, Weak interactions of leptons and quarks,

Cambridge University Press, 1983, pp. 246

ANNOTATION: An image depicting the experimental apparatus of Christenson et al. 1964.



ANNOTATION: An examination of the results and consequences of Kaon decay in terms of particle state

oscillations, beginning a larger discussion of CP violation.

35 G. Baym, Lectures on Quantum Mechanics, The Benjamin Cummings Publishing Company, 1969, Chap-

ter 2

ANNOTATION: A description of the quantum interference effects in Kaon decay.

36 Kaon Decay Oscillation, G. Baym, Lectures on Quantum Mechanics, The Benjamin Cummings Publish-

ing Company, 1969, pp. 43

ANNOTATION: An graph showing particle antiparticle oscillations of the neutral kaon.

30

http://www2.lbl.gov/abc/wallchart/chapters/05/2.html

http://hyperphysics.phy-astr.gsu.edu/hbase/particles/cpt.html#c1


37 CP Violation in Kaon Decay, Hyper Physics

URL:http://hyperphysics.phy-astr.gsu.edu/hbase/particles/cronin.html#c2

ANNOTATION: A concise summary of kaon decay in terms of particle equations.



ANNOTATION: Chapter 7.3 of Commins and Bucksbaum discuss kaon decay in terms of the mass and

decay matrices. A majority of this chapter is quite cryptic, however it provides important insight into the

Hamiltonian governing time evolution of the decay system.

39 P. Bloch, CPT Invariance Test in Neutral Kaon Decay, CERN, 2006

URL:http://pdg.lbl.gov/2006/reviews/cpt_s011254.pdf

ANNOTATION: A detailed discussion of CPT invariance tests in neutral kaon decay discussing the theo-

retical approach to measuring CPT conservation in neutral kaon decay.



ANNOTATION: A consideration of the Origins of CP violation. The chapter goes into much greater detail

of each possible origin than discussed in this paper.

41 E. Zolfagharifard, Have scientists found a huge new Higgs Boson?, 2015

URL:http://www.dailymail.co.uk/sciencetech/article-3362895/Have-scientists-huge-new-Higgs\

-Boson-Cern-discovered-tantalising-signs-mysterious-particle.html

ANNOTATION: A news article reporting on the potential discovery of a new Higgs Boson at the end of

2015.


ANNOTATION: An discussion of antimatter in terms of the Feynman description of antimatter using a

general relativistic framework.

43 Feynman, Richard. ”The Reason for Antiparticles.” The 1986 Dirac Memorial Lectures. California

Institute of Technology. 18 Dec. 2015. Lecture.

URL:https://www.youtube.com/watch?v=Z7D_tey2qpY

ANNOTATION: Feynman elaborates on the finer points of antimatter that Icke does not discuss in Chapter

8.4 of his text.

44 Antiparticle Pair Creation, V. Icke The Force of Symmetry, Cambridge University Press, 1995, pp. 122

ANNOTATION: An image showing the same experiment from different reference frames, motivating the

existence of antimatter in terms of general relativity.

45 Space and Time-Like Regions

URL:http://www.lecture-notes.co.uk/susskind/special-relativity/lecture-4/space-time-separation/

ANNOTATION: An image showing the different space and time regions with respect to the light cone.

31

http://hyperphysics.phy-astr.gsu.edu/hbase/particles/cronin.html#c2

http://pdg.lbl.gov/2006/reviews/cpt_s011254.pdf

http://www.dailymail.co.uk/sciencetech/article-3362895/Have-scientists-huge-new-Higgs \ -Boson-Cern-discovered-tantalising-signs-mysterious-particle.html

http://www.dailymail.co.uk/sciencetech/article-3362895/Have-scientists-huge-new-Higgs \ -Boson-Cern-discovered-tantalising-signs-mysterious-particle.html

https://www.youtube.com/watch?v=Z7D_tey2qpY

http://www.lecture-notes.co.uk/susskind/special-relativity/lecture-4/space-time-separation/


46 M. Dine et al., Origin of the matter-antimatter asymmetry, Reviews of Modern Physics, Volume 76, Jan-

uary 2004, Section E

ANNOTATION: Section E of Dine’s paper discusses the finer points of electroweak baryogenesis.

47 Klinkhamer, F. R.; Manton, N. S. (1984). ”A saddle-point solution in the Weinberg-Salam theory”. Phys.

Rev. D 30 (10): 22122220.

ANNOTATION: A discussion of sphaleron process in terms of its geometric interpretation.

48 P. Arnold and L. McLerran, Sphalerons, small fluctuations, and baryon-number violation in electroweak

theory, Physical review D Volume 36, Number 2, 1987

ANNOTATION: A treatment of sphalerons that provides a more intuitive description of sphalerons and

their greater context in quantum field theory.

49 D. E. Morrissey and M. J. Ramsey-Musolf, Electroweak baryogenesis, arXiv:1206.2942v1 [hep-ph], 2012

ANNOTATION: Morrissey’s text provides an excellent introduction to electroweak baryogenesis that com-

plimented Dine’s exploration well. Morrissey et al. go on to discuss a experimental tests for the validity of

electroweak baryogenesis, which will be fascinating for a future paper.

50 S. Huber, The electroweak phase transition, Sussex University, Final Colloquium, Bielefeld, September

2012

URL:https://www2.physik.uni-bielefeld.de/fileadmin/user_upload/workshops/huber.pdf

ANNOTATION: A detailed discussion of the electroweak phase transition and electroweak baryogenesis.

This lecture was helpful in fleshing out some of the specifics of the Morrissey paper.

51 Phase Discrpeancy in the Early Universe, D. E. Morrissey, et al. Electroweak baryogenesis, arXiv:1206.2942v1

[hep-ph], 2012

ANNOTATION: An image showing the phase difference between discrete bubbles of universe and the

surrounding plasma.

52 Bubble Expansion and sphaleron process, D. E. Morrissey, et al. Electroweak baryogenesis, arXiv:1206.2942v1

[hep-ph], 2012

ANNOTATION: An image showing an expanding bubble boundary as it eclipses sphaleron processes and

zero phase plasma.

53 Fermilab, What happened to the antimatter? The Birth of the Universe, Science, Fermilab, U.S. Depart-

ment of Energy, 2014

URL:http://www.fnal.gov/pub/science/questions/birth-universe-02.html

ANNOTATION: A discussion of the overall experimental future of probing matter-antimatter asymmetry.

A very accessible and concise piece out of Fermilab.

32

https://www2.physik.uni-bielefeld.de/fileadmin/user_upload/workshops/huber.pdf

http://www.fnal.gov/pub/science/questions/birth-universe-02.html

comps final

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