Master´s Dissertation IFT-D.13/2022

The muon g-2 discrepancy and

hints about new physics

Rodrigo Teixeira Aguiar

Advisor

Prof. Dr. Juan Carlos Montero Garcia

October/ 2022



 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  
               Aguiar, Rodrigo Teixeira 

 A283m            The muon g-2 discrepancy and hints about new physics / Rodrigo 
Teixeira Aguiar. – São Paulo, 2022 

96 f. 
 

Dissertação (mestrado) – Universidade Estadual Paulista (Unesp), 
Instituto de Física Teórica (IFT), São Paulo 

Orientador: Juan Carlos Montero Garcia 
 

1. Modelo padrão (Física nuclear).  2. Muons. 3. Partículas (Física 
nuclear). I. Título  

 

 

  

Sistema de geração automática de fichas catalográficas da Unesp. Biblioteca 
do Instituto de Física Teórica (IFT), São Paulo. Dados fornecidos pelo 

autor(a). 

 

 



I dedicate this thesis to the memory of my grandmother, Giselda Vieira de Aguiar,
mathematics and logic teacher.

. . .

i



Acknowledgements

To my wife and best friend, for always being by my side and supporting me in my

trajectory. To my parents and brothers that also always supported me.

To my advisor, Juan Carlos Montero, by the help and incentive in all difficulties that

arose at performing this work.

To IFT, all teachers and fellow employees, by giving me an adequate and enabling

environment for my studies.

To all the great teachers I’ve had throughout my life.

And lastly, to CAPES for the financial support that made this work possible.

ii



“I went to a bookstore and asked the saleswoman,
’Where’s the self-help section?’
She said if she told me, it would defeat the purpose.”

George Carlin

iii



Abstract

In this work, the structuring of the standard model was presented and some
issues that this model currently faces were analyzed, selecting for discussion
the question of the muon g− 2. A detailed evaluation of this factor was made,
including all the necessary tools to obtain the result, the computations up to one
loop, and the proof of the gauge invariance for the same. After that, it was shown
what is the issue with the muon g− 2 and provided a detailed explanation of the
experiment that confronts the theoretical prediction.

In the second part, a possible solution to the problem was tested: the inclusion
of a flavor-violating vertex and two new heavy particles fields, a fermion, and a
scalar. Using at first a toy model it was observed that models that contain this
type of vertex and particles could include, in the predictions, values at the right
order of magnitude as the SM discrepancy, even if the mass scale of these particles
is above the experimental exclusion. The ratio between the masses of the new
fermion over the new scalar being above 0.1 could result in values for the muon
g− 2 at the order of the current discrepancy between theory and experiment for
scalar masses in the TeV scale.

Having that in mind, a more concrete example (a 3-3-1 type model, taken from
the literature) was analysed, but the same effect was not observed. Due to the
chiral interactions of the scalar fields in this theory the contributions to the muon
g− 2 are lowered by around 10−3

(
m

MH

)
. Overall, this result seem to be more

reliable than the toy model, since the know interactions are chiral, and additions
of scalars with SM-like interactions seems to be, in general, unnatural ways of
fixing the g− 2 discrepancy.

Keywords: Standard Model; Muon; Muon g-2; Magnetic Moment; 3-3-1 Model.

Areas of Knowledge: Physics; Muon physics; Beyond the Standard Model.

iv



Resumo

Neste trabalho foi apresentado a estruturação do modelo padrão e analisados
alguns problemas que este modelo enfrenta atualmente, selecionando para dis-
cussão a questão do muon g− 2. Uma detalhada avaliação do muon g− 2 é feita,
incluindo todas ferramentas necessárias para se obter os resultados numéricos,
a conta a um loop e a prova da invariancia de gauge feitas explicitamente. Feito
isso, foi mostrado qual o problema com o muon g− 2 e foi dada uma explicação
detalhada do experimento que confronta a previsão teórica.

Na segunda parte uma possível solução do problema foi testada: a inclusão de
um vertice com violação de sabor e duas partículas pesadas novas, um fermion e
um escalar. Usando um modelo hipotético foi observado que modelos que contém
este tipo de vértice e partículas poderia incluir nas previsões valores da ordem de
magnitude necessária, mesmo que a escala de energia destas particulas esteja fora
dos limites experimentais. A relação entre as massas do novo fermion sobre a do
novo escalar sendo acima de 0.1 poderiam resultar em valores para o muon g− 2
da ordem da discrepancia entre teoria e experimento atuais para escalares com
massa na escala de TeV.

Tendo isso em mente, um modelo concreto como exemplo (um modelo do tipo
3-3-1, retirado da literatura) foi analisado, mas o mesmo efeito não foi observado.
Por causa das interações quirais dos escalares dessa teoria, as contribuições para
o g− 2 são reduzidas por volta de 10−3

(
m

MH

)
. De modo geral, este resultado

parece mais viável, devido as interações conhecidas serem normalmente quirais, e
portanto adições de escalares com interações do tipo do modelo padrão parecem
ser, em geral, maneiras não naturais de resolver a discrepância do g− 2.

Palavras-chave: Modelo Padrão; Muon; Muon g-2; Momento Magnético; Modelo
3-3-1.

Áreas de conhecimento: Física; Física de muons; Além do Modelo Padrão.

v



Contents

1 Introduction 1

2 The Standard Model of Particle Physics 4
2.1 Conventions and definitions . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.2 Feynman rules . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1.3 Other useful definitions . . . . . . . . . . . . . . . . . . . . . 9

2.2 Composition and symmetry groups of the SM . . . . . . . . . . . . 10
2.2.1 SM field description . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.2 Spontaneous Symmetry Breaking . . . . . . . . . . . . . . . 12
2.2.3 Higgs mechanism . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3 Particle mass and interactions . . . . . . . . . . . . . . . . . . . . . . 16
2.3.1 Lepton interactions . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3.2 Quark interactions . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3.3 Gauge bosons kinetic terms . . . . . . . . . . . . . . . . . . . 19

2.4 Anomalies in the SM . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.4.1 An example of gauge anomaly . . . . . . . . . . . . . . . . . 20
2.4.2 Anomaly cancellation in the SM . . . . . . . . . . . . . . . . 22

2.5 Issues of the SM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3 The Muon g− 2 28
3.1 History of the Anomalous Magnetic Moment . . . . . . . . . . . . . 28
3.2 Feynman diagram calculation of the g factor . . . . . . . . . . . . . 30

3.2.1 Leading order calculation of the Form Factors . . . . . . . . 33
3.2.2 NLO calculation of the Form Factors . . . . . . . . . . . . . . 34

3.3 Muon g− 2 in the SM . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.3.1 Higgs-boson contributions . . . . . . . . . . . . . . . . . . . 38
3.3.2 Z-boson contributions . . . . . . . . . . . . . . . . . . . . . . 39
3.3.3 W-boson contributions . . . . . . . . . . . . . . . . . . . . . . 42

3.4 Theoretical value for the muon g− 2 in the SM . . . . . . . . . . . . 44

vi



4 Muon g− 2 experiments 48
4.1 History of the muon g− 2 experiments . . . . . . . . . . . . . . . . 48
4.2 The Fermilab experiment . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.2.1 Muon decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.2.2 Fermilab Results . . . . . . . . . . . . . . . . . . . . . . . . . 56

5 Beyond the SM 60
5.1 Scalar-like and heavy lepton-like particles in a flavour violation

muon g− 2 loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.1.1 Numerical analysis . . . . . . . . . . . . . . . . . . . . . . . . 64

5.2 The 3-3-1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.2.1 Numerical analysis . . . . . . . . . . . . . . . . . . . . . . . . 75

6 Conclusion 76

A Dirac Hamiltonian 78

B One-loop gauge invariance in the muon g− 2 computation 80
B.1 Z diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
B.2 W diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

C Computation of the F1 form factor up to one loop 86

Bibliography 90

vii



Chapter 1

Introduction

The idea of field theory is described by the Lagrangian formalism in the form
of Lagrangian densities, a function of the fields, and its covariant derivatives, as:

L = L
(

φi(x), ∂µφi(x)
)

, (1.1)

where φi can represent all relevant fields. The junction of field theory, quantum
physics and the theory of relativity created what is called the Quantum Field
Theory (QFT), and the Standard Model of particle physics is the QFT that models
the current understanding of particle physics. It is one of the most successful
theories ever made since it was able to predict many experimental results at
incredible precision. For the completeness of the model different types of fields
are included, like scalars, fermionic, and vector fields. This work, at the very end,
centers around new fermionic and scalar degrees o freedom.

Scalar fields are the simplest fields possible to add to models. The only such
field at the fundamental level in our current theory is the Higgs field [1], exper-
imentally observed in 2012 [2]. Besides that, scalar fields are an active area of
research, since it is commonly used in effective field theories. For example, the use
of a pion pseudoscalar effective field that can represent interactions of the strong
forces at large scales (above 1 fermi according to R. Penco [3]). Having that in
mind, the addition of scalar fields is valuable since it can probe for new models in
simple forms.

Besides the tremendous success of the Standard Model, there are still some
questions waiting to be answered, like neutrinos oscillation, quantum gravity, and
dark matter for instance. This work focus on one of such issues, the very present
topic of the muon anomalous magnetic dipole moment, usually referred to as the
muon g− 2.

The muon was discovered in the 1930 decade and although it is not present
in everyday life interactions, muons do appear frequently in experiments and
observations. The cosmic rays observation of this particle was one of the first

1



Chapter 1. Introduction 2

confirmations for the time dilatation from relativistic theory, for example. It was
also the first particle to show that nature repeated itself, by having the same
characteristics and interactions as the electron, being the mass the only difference
[4]. Nowadays we know that there exists a threefold replica with each lepton and
quark having three families differing only in mass.

Around a decade before the recognition of the muon as a heavy electron, a
somewhat strange quantum phenomenon was being discovered, the anomalous
magnetic dipole moment of the electron. Such a phenomenon was predicted by
Schwinger due to radiative corrections in the electron interaction and became a
strong confirmation of the success of the quantum field theory. The implementa-
tion of the same idea to the muon case was immediate and came with an advantage:
heavy-particles contributions were much more significant for the muon. Since all
the heavy particle contributions at one loop is proportional to

F2 ∝
m2

f

M2
h

, (1.2)

where m f is the mass of the fermion that we want to measure the g − 2 factor,
Mheavy is the mass of the heavy particle that we are calculating the g− 2 contri-
bution, and F2 = g−2

2 is the anomalous part of the magnetic dipole moment. For
the muon specifically, this factor is often referred to as aµ. Then, using a fermion
with higher mass and the same interactions results in higher precision on the
measurement. The relation from Eq. 1.2 will be justified later in this work,

Having that in mind, many experiments, including three major ones made by
CERN, confirmed the value of the muon g− 2 predicted by the theory up to 6
orders of magnitude. In 1984 a collaboration of scientists at Brookhaven began to
increase the precision of the measure of the muon g− 2, to test even further the
theory, and so the experiment BNL E-821 began.

It would be the first to present values up to 2.7 standard deviations (σ) above
the theoretical value when the final result was released in 2004 [5]. In the following
years, experimental improvement of the measure of other parameters and better
techniques of computation improved the accuracy of the theoretical prediction for
the muon g− 2, obtaining the current theoretical value of:

aSM
µ = (116591810± 43) · 10−11. (1.3)

This higher precision theoretical value made the result of BNL experiment to



Chapter 1. Introduction 3

go up to 3.7 σ.
Although significant, this value of deviation is not decisive to assert the pres-

ence of a problem with the SM. The commonly used threshold is 5 σ which is
approximately 1 chance in 3.5 million of having the discrepancy being a statistical
fluctuation. More experimental data was needed, therefore. Hence, a new collabo-
ration began at Fermilab to make a more meticulous measure. In 2021 the release
by Fermilab [6] of the first result for the new experiment E-989 agreed with the
previous result by BNL E-821 and improved slightly its confidence in the value.
The combined result of both experiments is:

aEXP
µ = (116592061± 41) · 10−11. (1.4)

Taking the difference between this value and the theoretical one (Eq. 1.3), we
obtain:

aEXP
µ − aSM

µ = (251± 59) · 10−11, (1.5)

which gives a divergence of 4.2 σ, which boosts the discussion about new physics.
Thus we have the motivation for this work: the divergence of results from

experiments and theory for the muon g− 2. With Fermilab E-989 experiment still
running and some other experiments with different techniques of measurement
(e.g. J-PARC [7] experiment, projected to release the first result in 2023), the muon
g− 2 is a promising area of research.

The main goal of this work is to try to explain this deviation using elements
of new physics that add at least one new scalar in the theory which couples to
the muon in a non-standard way. We will start by reviewing the construction and
other aspects of the Standard Model. Then we will show how to calculate and
measure the value of the muon g− 2. Finally, discuss about hints of beyond the
Standard Model physics.



Chapter 2

The Standard Model of Particle Physics

The Standard Model of particle physics (SM) is the name given to one of the
most sophisticated physics theories, which models almost all phenomena involv-
ing elementary particles. It is the pinnacle of years of study by many scientists,
mostly made in the second half of the last century, and became experimentally
complete in 2012 with the discovery of the last piece of the model, the Higgs
boson [2]. This model became standard because of the extraordinary experimental
validation. Some authors consider this theory the one with the best experimental
precision among all physical theories [8].

The mathematical tool used for the construction of the SM is Quantum Field
Theory, and with it, we can redefine elementary particles as elementary fields. We
should emphasize that the concept of elementary is believed to be an effective
concept rather than actual elementary. The author C. Quigg [9] (2013, page 2)
explains briefly this idea:

“An elementary particle, in the time-honored sense of the term, is struc-
tureless and indivisible. Although history cautions that the physicist’s list
of elementary particles is dependent upon experimental resolution — and
thus subject to revision with the passage of time - ... . We thus have no
experimental reason but tradition to suspect that they are not the ultimate
elementary particles."

In the next sections, we will set forth some definitions and conventions used
in this work, and focus on the Electroweak part of the SM (ESM). We will start
by reviewing the composition, gauge symmetries, and symmetry breaking of the
ESM. Then we will take a look at the generation of mass for the particles, and
some interactions among them. Finally, we will explore anomalies cancellation
and some of the current issues of the SM.

4



Chapter 2. The Standard Model of Particle Physics 5

2.1 Conventions and definitions

Since many different conventions can be found in the literature, this section
will lay out the conventions used in this work as well as some useful definitions.
Most of them are the same as in M. D. Schwartz [10], and the Feynman rules were
taken from J. Romão and J. Silva [11].

2.1.1 Conventions

• All calculations are done in natural units c = h̄ = 1;

• In this work it is used the Einstein summation convention, where multipli-
cations with repeated indices are summed over as shown in the example
below:

Aµ · Bµ = A0B0 + A1B1 + A2B2 + A3B3;

• Greek letter indexes (µ, ν, σ, ρ...) are used for 4-dimension Minkowski space,
and regular letter indexes (i, j, k...) are used for 3-dimension Euclidean space.
Exceptions for this rule will be explicitly written in the text;

• The metric of spacetime used is the mostly negative metric:

gµν =


1 0 0 0
0 −1 0 0
0 0 −1 0
0 0 0 −1

 ; (2.1)

• The Pauli matrices are defined as:

σ1 =

(
0 1
1 0

)
;

σ2 =

(
0 −i
i 0

)
;

σ3 =

(
1 0
0 −1

)
;

(2.2)



Chapter 2. The Standard Model of Particle Physics 6

With which we define a Pauli vector as:

~σ = σi =
(

σ1, σ2, σ3
)

; (2.3)

And also the Pauli 4-vectors as:

σµ = (1, σi) ; σ̄µ = (1,−σi); (2.4)

• The Dirac matrices are defined as:

γ0 =

(
0 1

1 0

)
;

γi =

(
0 σi

−σi 0

)
;

γ5 = iγ0γ1γ2γ3,

(2.5)

where each entry is a 2x2 matrix and σi are the Pauli matrices. This definition
is called Weyl representation;

• The left and right projectors are defined as:

PL =
1− γ5

2
; PR =

1 + γ5

2
; (2.6)

• The Levi-Civita symbol is defined as:

εijk =


+1 for even permutations of i, j and k;
−1 for odd permutations of i, j and k;
0 if i = j, or j = k, or k = i.

(2.7)

2.1.2 Feynman rules

This subsection is devoted to showing explicitly the Feynman rules used in
this work. The formulas have the following conventions:

· p and k are always used for the momentum and, unless explicitly shown,
always have direction from left to right;

· ξ is the gauge choice;



Chapter 2. The Standard Model of Particle Physics 7

· mi and Mi are used for masses of the ‘i’ particle;

· ε is an infinitesimal increment;

· Q f is the electrical charge quantum number of the particle in question, for
example, +1 for positrons and -1 for electrons;

· I3 is the third component of the left isospin doublet and can be ±1/2 as
shown in Table 2.1;

· The slashed terms are 4-vectors contracted with gamma matrices, for exam-
ple:

/A = Aµ · γµ.

· g′ and g are the coupling constants of U(1)Y and SU(2)L, respectively.

The Feynman rules that were used are:

• Propagators

Photon:

A
µ ν = −i

[
gµν

k2 + iε
− (1− ξ)

kµkν

(k2)2

]
; (2.8)

Fermion:

f
= i /p + m f

p2 −m2
f + iε

; (2.9)

W-boson:

W
µ ν = −i

1
k2 −M2

W + iε

[
gµν − (1− ξ)

kµkν

k2 − ξM2
W

]
; (2.10)

Z-boson:

Z
µ ν = −i

1
k2 −M2

Z + iε

[
gµν − (1− ξ)

kµkν

k2 − ξM2
Z

]
; (2.11)

Higgs-boson:

H
=

i
p2 −M2

H + iε
; (2.12)



Chapter 2. The Standard Model of Particle Physics 8

Neutral Golstone-boson:

GZ
=

i
p2 − ξM2

Z + iε
; (2.13)

Charged Golstone-boson:

G±
=

i
p2 − ξM2

W + iε
. (2.14)

• Vertices:

Fermion - photon vertex

Aµ

f f = ieQ f γµ; (2.15)

Fermion - Z-boson vertex

Zµ

f f = −i
g

cosθW
γµ(

1
2

I3 −Q f sin2θW −
1
2

I3γ5); (2.16)

Fermion - W-boson vertex

Wµ

f f = −i
g√
2

γµPL; (2.17)

Fermion - Higgs-boson vertex

H
f f = −i

g
2

m f

MW
; (2.18)

Fermion - neutral Goldstone-boson vertex

GZ

f f = −gI3
m f

MW
γ5; (2.19)



Chapter 2. The Standard Model of Particle Physics 9

Fermion - charged Goldstone-boson vertex

G±

l,νi νi,l = −i
g√
2

ml
MW

PR,L; (2.20)

Charged Goldstone-boson - photon vertex

p1 p2

Aµ

G± G± = ∓ie(p1 + p2)µ; (2.21)

Charged Goldstone-boson - W-boson - photon vertex

Aµ

G± W±ν = −ieMW gµν; (2.22)

W-boson - W-boson -photon vertex

p1 p2

Aµ ↓ q

W±ρ W±ν =
−ie

[
gρν(p1 + p2)µ+

gµν(−p2 − q)ρ + gµρ(q− p1)ν

]
.

(2.23)

2.1.3 Other useful definitions

Some other useful definitions that facilitate to visualise the formulas are shown
below:

• The Weinberg angle (θW) is defined as the rotation angle between the neutral
gauge boson of the SU(2)L and the U(1)Y gauge boson:

(
Zµ

Aµ

)
=

(
cosθW −sinθW

sinθW cosθW

)(
b3

µ

A′µ

)
; (2.24)

cosθW =
g√

g2 + g′2
; sinθW =

g′√
g2 + g′2

.

This angle also relates the masses of the Z and W bosons as:

MW = cosθW MZ. (2.25)



Chapter 2. The Standard Model of Particle Physics 10

• The electric charge, which is the constant multiplying the photon field inter-
actions, is defined as:

e = gsinθW = g′cosθW

.

• The fine-structure constant is defined as:

α =
e2

4π
.

2.2 Composition and symmetry groups of the SM

The SM has two types of fermions: leptons and quarks. The leptons are
composed of three families (the families of the electron (e), muon (µ), and tau(τ),
each with its respective neutrino (ν{e,µ,τ})). The quarks also have three families,
(divided as the family of the up (u) and down (d), of the charm (c) and strange
(s), and of the top (t) and bottom (b)) but they have, additionally, three different
colours: red, green, and blue. In total, the SM describes the behaviour of 12
fermions: three doublets of leptons and three doublets of quarks [12].

Quarks are not observed free in nature but always coupled to other quarks in a
singlet of colour. These composite particles are called hadrons, and we observe
mostly two types of them: mesons (composed of a quark and an anti-quark) and
baryons (composed of three quarks or three anti-quarks). There are hundreds of
such composite particles, including the well-known protons and neutrons, as one
can find in [13].

Besides the elementary fermions, there are vector bosons that mediate the
interactions among them and which are related to the forces of nature. Leaving
aside gravity (it plays no role in the scale of particle physics because of its small
magnitude) there is one boson mediating the electromagnetic force (photon), three
bosons mediating the weak force (W+, W− and Z) and eight gluons mediating the
strong force (Ga, a={1 to 8}).

The last particle that composes the SM is the Higgs boson, the particle that
generates the masses of all the other particles through the implementation of
symmetry breaking, Higgs mechanism, and Yukawa couplings, which will be
explained shortly in this chapter. Figure 2.1 shows a summary of the classification
of particles of the SM.



Chapter 2. The Standard Model of Particle Physics 11

Figure 2.1: Summary of the Standard Model particles. Source: The Symmetry
Magazine, n. d. (https://www.symmetrymagazine.org/standard-model/)

The symmetry group of the SM particles is

SU(3)C ⊗ SU(2)L ⊗U(1)Y ,

where C, L, and Y represents the colour, the left chiral symmetry of the fermions,
and the hypercharge, respectively. Using this symmetry group and some mathe-
matical tools we are able to compile the final form of the SM, with all interactions,
mass terms, and kinetic terms.

2.2.1 SM field description

The fields for the leptons and quarks are very similar. Both contain a doublet of
SU(2)L for each family and singlets for the right component of every particle with
the exception of the neutrinos, which only have the left chiral component. This
description is exemplified below, for the case of the electron, electron´s neutrino,
quark up and quark down.



Chapter 2. The Standard Model of Particle Physics 12

PLψe = ψL =

(
νe,L

eL

)
; PRψe = ψR = eR;

PLQ = QL =

(
uL

dL

)
; PRQ = QR = {uR , dR} .

(2.26)

The gauge bosons are vector fields, therefore each of them is a 4-components
Lorentz vector (Tµ). There is also a complex scalar doublet of SU(2)L, described
as:

ϕ =

(
φ+

φ0

)
. (2.27)

The Gell-Mann-Nishijima relation [14] defines the charge operator as:

Q = I3 +
1
2

Y, (2.28)

where I3 is the third component of the SU(2)L isospin, Y is the value of the U(1)Y

hypercharge, and Q is the quantized electric charge. And with it, we can find
the quantum numbers for each particle field of the theory. The results are shown
in Table 2.1. Also included in this table is the Higgs boson field (H) quantum
numbers, which will be used in the next sections.

Particle I I3 Y Q

ν{e,µ,τ}L
1
2 +1

2 −1 0

{e, µ, τ}L
1
2 −1

2 −1 −1
{e, µ, τ}R 0 0 −2 −1
{u, c, t}L

1
2 +1

2
1
3

2
3

{d, s, b}L
1
2 −1

2
1
3 −1

3

{u, c, t}R 0 0 4
3

2
3

{d, s, b}R 0 0 −2
3 −1

3

H 1
2 −1

2 +1 0

Table 2.1: Quantum numbers for the fermion fields

2.2.2 Spontaneous Symmetry Breaking

Considering the Lagrangian for a complex scalar field:



Chapter 2. The Standard Model of Particle Physics 13

L =
1
2

∂µφ∗∂µφ−V (φ) , (2.29)

with the potential:

V (φ) = −µ2φ∗φ + λ(φ∗φ)2, (2.30)

where µ2 and λ are arbitrary positive real constants.
It is manifest that if φ→ eiθφ the Lagrangian is invariant, configuring a U(1)

symmetry. Such potential (Eq. 2.30) has a minimum where the field is different
from zero. Therefore the field φ acquires a Vacuum expected Value (VeV) in the
lowest energy state, represented by v√

2
as in:

〈φ〉vacuum =
v√
2
=

√
µ2

2λ
. (2.31)

There is a liberty to choose the direction of the VeV, so it is convenient to choose
it in the real positive axis, as shown below:

φ (x) =
1√
2

[
v + φ′1 (x) + iφ′2 (x)

]
, (2.32)

where φ′1 and φ′2 are real scalar fields.
Rewriting the potential in terms of φ′1 and φ′2 gives:

V = − µ4

4λ
+ µ2φ′1

2 + µ
√

λ φ′1
3 + µ

√
λ φ′1φ′2

2 +
λ

4

(
φ′1

2 + φ′2
2
)2

. (2.33)

This potential doesn’t have a U(1) symmetry like the original Lagrangian from
Eq. 2.29 (if φ′i → eiθφ′i the extra phase in the terms with φ′1

3 and with φ′1φ′2
2 doesn’t

cancel). Thus the symmetry is said to be broken or hidden [10]. Other consequence
is that the field φ′1 has a mass term, while φ′2 is massless. The appearance of
massless fields in theories with spontaneously broken continuous symmetry is
called Goldstone theorem. It states that for every spontaneously broken linearly
independent generator of a continuous symmetry there is a massless boson, usually
called Goldstone-boson.

For the case of the SM scalar doublet field (Eq. 2.27) with Eq. 2.30 representing
the potential (trading φ by ϕ and φ∗ by ϕ†), the broken symmetry is the SU(2)L

that has three generators. Consequently, it will have three Goldstone-bosons. We



Chapter 2. The Standard Model of Particle Physics 14

can choose a parametrization for this scalar in a clever form as in:

ϕ =
e−i σi

2
ϕi(x)

v
√

2

(
0

v + H

)
, (2.34)

where σi are the Pauli matrices, ϕi(x) are the Goldstone-bosons fields and H is
the physical scalar field of the SM, the Higgs-boson field. This formula is simply
another way to write Eq. 2.27, but with the VeV and the physical Higgs field
explicit. Remarking that the VeV choice in the second component of this doublet
is due to it being neutral. This always has to be the case, since we cannot have a
charged vacuum.

2.2.3 Higgs mechanism

After analysing the symmetry breaking, we can include local (gauge) symme-
tries for the SM and the gauge invariance concept. For that, we have to use the
concept of covariant derivatives, defined as:

∂µ → Dµ = ∂µ +
i
2

Yg′A′µ +
i
2

gσibi
µ, (2.35)

for SU(2)⊗U(1). And the gauge transformations are defined as:

ϕ→ eiα(x)+iσiβ
i(x)ϕ;{

A′µ, bµ

}
→
{

A′µ −
i
g′

∂µα(x), eiσiβ
i(x)
[

bµ −
i
g

∂µ

]
e−iσiβ

i(x)
}

.
(2.36)

It is evident that with the right choice of the gauge in the equation above we
can correctly cancel the phase term from Eq. 2.34, leaving the scalar of the SM
with the standard form of:

φ =
1√
2

(
0

v + H

)
. (2.37)

The consequence of this choice of gauge is that the kinetic term of the scalar
field (Dµφ†Dµφ) will present a product of the gauge bosons with the VeV. The
result after calculating this kinetic term is:



Chapter 2. The Standard Model of Particle Physics 15

Dµφ†Dµφ =

(
∂µH

)2

2
+

1
8

g2v2
(

b1
µ + ib2

µ

) (
b1

µ − ib2
µ

)
+

1
8

v2(g′A′µ − gb3
µ)

2 + interactions.
(2.38)

It is practical to make a rotation of the fields to the mass eigenstates using the
definitions of Eq. 2.24, rewritten below for the substitution of fields:

A′µ =
1√

g2 + g′2
(

gAµ − g′Zµ

)
;

b3
µ =

1√
g2 + g′2

(
g′Aµ + gZµ

)
;(

b1
µ ∓ ib2

µ

)
=
√

2 W±µ ,

(2.39)

to get the final result for the masses of the physical fields, including the Higgs
(from Eq. 2.33), written as:

LSSB,mass =

(
∂µH

)2

2
+ M2

WW+
µ Wµ− +

1
2

M2
Z(Zµ)

2

+
1
2

M2
H H2 + interactions,

(2.40)

where M2
W = 1

4 g2v2, M2
Z =

M2
W

cos2θW
= 1

4

(
g2 + g′2

)
v2 and M2

H = λv2. The gauge
fields have acquired mass from the disappearance of the Goldstone-bosons in the
theory. This mechanism of acquisition of mass by the gauge field caused by a
spontaneous symmetry breaking in a gauge invariant theory is called the Higgs
Mechanism [15].

We should emphasize that the degrees of freedom of the theory remain the
same since the acquisition of mass by the gauge bosons increases its degree of
freedom by one, and the Goldstone-bosons have one degree of freedom each. We
can also point out that the field Aµ has not acquired mass, as expected. Therefore it
is interpreted as the U(1)EM gauge field of the electromagnetic theory - the photon
field.



Chapter 2. The Standard Model of Particle Physics 16

2.3 Particle mass and interactions

The leptons and quarks cannot have direct mass terms in the Lagrangian,
because it would not respect the imposed symmetries. We can see this by trying to
manually introduce mass to the electron using:

meψ̄eψe = meψ̄e(
1 + γ5

2
)ψe + meψ̄e(

1− γ5

2
)ψe = me(ψ̄LψR + ψ̄RψL). (2.41)

This result does not respect the invariance under SU(2)L, therefore cannot be
inserted in the Lagrangian. This idea can be generalized to all families of quarks
and leptons.

Hence we will have to look for allowed terms of interactions that respect the
symmetries. In the next sections, a careful look at the interactions of leptons and
quarks will be exhibited together with the gauge boson kinetic terms.

2.3.1 Lepton interactions

We start by looking at the kinetic term for the leptons. Using the Dirac equation
the Lagrangian without mass is simply:

ψ̄iγµ∂µψ→ ψ̄iγµDµψ = ψ̄LiγµDµψL + ψ̄RiγµDµψR. (2.42)

Then, using Eq. 2.36, the values from Table 2.1, and the physical gauge fields
from Eq. 2.39 and Eq. 2.26 it is possible to compute the interaction between leptons
and gauge bosons. After some algebra, the resultant Lagrangian is shown in Eq.
2.43, divided in the following way: the first line is the charged lepton neutral
current, the second is the neutrino neutral current and the third is the lepton
charged current.

ψ̄eiγµ∂µψe + eψ̄eγ
µ Aµψe +

g
4cosθW

ψ̄eγ
µ
(

1− 4sen2θW − γ5
)

Zµψe;

ψ̄νiγµ∂µψν −
g

4cosθW
ψ̄νγµ

(
1− γ5

)
Zµψν; (2.43)

− g
2
√

2

(
ψ̄νγµ

(
1− γ5

)
W+

µ ψe + ψ̄eγ
µ
(

1− γ5
)

W−µ ψν

)
,

where ψe = {e, µ, τ} and ψν =
{

νe, νµ, ντ

}
.

Next, the interaction terms. The simplest allowed terms that preserve the



Chapter 2. The Standard Model of Particle Physics 17

symmetries are the Yukawa interactions between the Higgs and two lepton fields,
as shown below:

GY ,ijψ̄iφψj + H.c.. = GY ,ij

(
ψ̄L,iφψR,j + ψ̄R,iφ

†ψL,j

)
+ H.c.., (2.44)

where GY,ij is the Yukawa coupling of the Higgs field with each pair of fermion
”i, j”. This coupling is a free parameter for the theory that can be any complex
number.

After the spontaneous symmetry breaking, the Higgs VeV (Eq. 2.31) will
appear as a constant factor multiplying the left-right components, which should
be interpreted as a mass term. And because of the chosen parametrization of
the Higgs doublet (Eq. 2.37), it will only multiply the isospin (−1

2) of the left
lepton doublet, giving mass terms only for the electron, muon, and tau. We should
also note that in Eq. 2.44 the ”i” and ”j” can belong to any family of leptons,
hence mixing of mass terms will appear in the most general form of this Yukawa
Lagrangian as can be seen in:

LY ,lepton =GY ,ee
v√
2

ēe + GY ,µµ
v√
2

µ̄µ + GY ,ττ
v√
2

τ̄τ + GY ,eµ
v√
2

ēµ

+ GY ,eτ
v√
2

ēτ + GY ,µτ
v√
2

µ̄τ + interactions + H.c..
(2.45)

The next step is to diagonalize the mass matrix of this Lagrangian, eliminating
crossed terms, and finding the actual physical mass for the fields. Since one of the
suppositions of the SM is that the neutrinos have no mass, we can freely rotate the
neutrino fields to match the mass eigenstates of the charged lepton and eliminate
the complex phase resultant with a chiral rotation. The final Lagrangian with
lepton mass terms and Higgs interaction is:

LY ,lepton = me ēe + mµµ̄µ + mτ τ̄τ +
me

v
ēeH +

mµ

v
µ̄µH +

mτ

v
τ̄τH, (2.46)

where mi = G′Y ,i
v√
2
, being G′Y the eigenvalues of the mass matrix of the Lagrangian

from Eq. 2.45, therefore are reals.
With the results Eq. 2.46 and Eq. 2.43, we can generate all Feynman diagrams

and propagators for the leptons in the SM.



Chapter 2. The Standard Model of Particle Physics 18

2.3.2 Quark interactions

Ignoring at first the SU(3)C, because it will not change throughout the calcula-
tions, the quark interactions follow the same logic as the lepton one. We commence
with a massless Dirac equation term (Eq. 2.42) to derive the interactions between
quarks and gauge bosons. Again we use Eq. 2.36, Table 2.1, Eq. 2.39, and Eq.
2.26 to derive the resulting Eq. 2.47. The first and second lines show the neutral
currents and the third one shows the charged current.

ψ̄uiγµ∂µψu−
2
3

eψ̄uγµ Aµψu +
g

cos θW
ψ̄uγµ

(
4 sin2 θW − 1

2
− 1− sin2 θW

2
γ5

)
Zµψu;

ψ̄diγµ∂µψd +
1
3

eψ̄dγµ Aµψd +
g

cos θW
ψ̄dγµ

(
1
2
+

2
3

sin2 θW −
1
2

γ5
)

Zµψd; (2.47)

− g√
2

(
ψ̄uγµ

(
1− γ5

)
W+

µ ψd + ψ̄dγµ
(

1− γ5
)

W−µ ψu

)
,

where ψu = {u, s, t} and ψd = {d, c, b}.
Then we take into account the Yukawa interactions, and this is where a dif-

ference from the leptons appears. The right-handed field for the isospin (+1
2)

has to be considered as well since both quarks need to have mass terms due to
experimental evidence. Fortunately, for the SU(2) the conjugate representation is
identical to the original one and we can have both written for the scalar field in
the Lagrangian, as shown below:

φ =
1√
2

(
0

v + H

)
; φ̃ = iσ2φ =

1√
2

(
v + H

0

)
.

With that we can write the Yukawa interactions as follows:

LY ,quark = GY ,dQ̄LφQR + GY ,uQ̄Lφ̃QR. (2.48)

Remarking again that QL and QR can represent any of the families of the quarks
so we will need to make the correct rotation to the mass eigenstates. In this case,
since we do not have the liberty to freely rotate the fields like with the leptons,
it will result in a mixed state for the masses. Realizing this effect, the authors
M.Kobayashi and T.Maskawa [16] deduced the mixing matrix that is now called



Chapter 2. The Standard Model of Particle Physics 19

the Cabibbo–Kobayashi–Maskawa (CKM) matrix, commonly written as:

VCKM =

 c12c13 s12c13 s13e−iδ

−s12c23 − c12s23s13eiδ c12c23 − s12s23s13eiδ s23c13

s12s23 − c12c23s13eiδ −c12s23 − s12c23s13eiδ c23c13

 , (2.49)

where cij and sij are the cosine and sine of the mixing angle between the ith and jth

particle, respectively.
The final Yukawa Lagrangian is shown below:

LY ,quark = mqq̄q +
mq

v
q̄qH, (2.50)

where again, mq = G′Y ,q
v√
2
, being G′Y the eigenvalues of the mass matrix and

q = {u, d, s, c, t, b}.
One consequence of the mixed states will appear in the charged current, which

is not written in the physical basis in Eq. 2.47. So, we need to fix that term rotating
to the mass eigenstates (multiplying by VCKM) which gives:

− g√
2

(
ū c̄ t̄

)
γµW+

µ VCKM

d
s
b

− g√
2

(
d̄ s̄ b̄

)
γµW−µ VT

CKM

u
c
t

 . (2.51)

Another consequence that we will just remark on here is that the phase angle δ

is responsible for the CP-violation phenomena in the SM. This can be seen in Eq.
2.51 since the terms in this equation are not CP invariant. This will be important
when discussing some issues with the SM in section 2.5.

2.3.3 Gauge bosons kinetic terms

The ending piece of the SM is the kinetic terms for the gauge sector. It is known
[15] that the kinetic terms for gauge fields can be written using the antisymmetric
tensor Tµν as:

− 1
4

TµνTµν = −1
4 ∑

a

(
∂µTa

ν − ∂νTa
µ + g f abcTb

µTc
ν

)2
, (2.52)

where f abc is the structure constant of the group and the summation is over all
gauge fields.

With Eq. 2.52 we can directly write the kinetic terms for the SM in the following



Chapter 2. The Standard Model of Particle Physics 20

way:

Lgauge = −
1
4

A′µν A′µν − 1
4

BµνBµν − 1
4

GµνGµν, (2.53)

where we keep the notations that A′, B, and G are representative of U(1)Y, SU(2)L

and SU(3)C gauge fields, respectively.
Since we want to write the Lagrangian in terms of the physical fields there is a

few computation steps to do that are omitted here. Using Eq. 2.39 to modify Eq.
2.53 and some algebra, we get:

Lgauge = −
1
4

FµνFµν − 1
4

ZµνZµν − 1
4

W+
µνW−µν − 1

4
GµνGµν + interactions, (2.54)

where Fµν =
(
∂µ Aν − ∂ν Aµ

)
, Zµν =

(
∂µZν − ∂νZµ

)
and W±µν =

(
∂µW±ν − ∂νW±µ

)
.

The gluons (gauge bosons of SU(3)C) kinetic term remains intact.

2.4 Anomalies in the SM

Anomalies in quantum field theory refer to having symmetries of the classical
theory that is somehow not present in the quantum theory. Another way to say it
is that the Slavnov-Taylor or the Ward identity is violated [17], like:

Dµ Jµ = 0 ;
〈

Dµ Jµ
〉
6= 0, (2.55)

where Jµ is the Noether conserved current.
For the case of global symmetries, anomalies will only imply in the classical

rules not to be obeyed, but for gauge symmetries, anomalies have to be canceled
otherwise unitarity and renormalizability would be violated. In the following
subsections, we will look at an example of gauge anomaly and then connect it
with the SM.

2.4.1 An example of gauge anomaly

A common example of a gauge anomaly is for massless fermions with chiral
interactions in a U(1) gauge: supposing a theory with the following Lagrangian:

L = ψ̄R
(
i/∂ +Q f e /A

)
ψR + ψ̄L

(
i/∂ +Q f e /A

)
ψL. (2.56)



Chapter 2. The Standard Model of Particle Physics 21

This Lagrangian is invariant under the transformation:

ψ→ eiαψ, (2.57)

where the parameter alpha can either be local (α(x)) or global if we have the proper
gauge field transformations, and from that, we derive the relations of Eq. 2.55

In quantum field theory, the path integral is written as:

〈Θ (x1 · · · xn)〉 =
1

Z (0)

∫
DADψ̄Dψ ei

∫
d4xiψ̄ /DψΘ (x1 · · · xn) , (2.58)

where Z(0) is a renormalization factor and Θ is any operator invariant under the
symmetry.

Even though the Lagrangian is invariant under the transformation from Eq.
2.57, the measure of this integral is not always invariant. If this change of variable
contains a chiral term (α = βγ5) the Jacobian for this change of variable is:

 = Det[exp(iα)] = exp
[

i
∫

dxβTr[γ5]

]
, (2.59)

and the proper transformation for the measure is:

Dψ̄Dψ→ Dψ̄′Dψ′ ||−2 = Dψ̄′Dψ′e−i
∫

2βTr[γ5], (2.60)

where ||−2 is the inverse Jacobian square due to the anticommuting objects in the
measure.

The integral in the exponent seems to vanish due to γ5 = 0 but if β(x) does not
vanish at x → ∞ this may not be true. In that case, the integral in x is divergent
and can be regulated using an exponential of a gauge invariant term like exp[− /D2

Λ2 ],
with Λ being the regulator. For brevity only the result for the anomalous current
is shown:

∂µ Jµ5 =
e2

16π2 εµνρσFµνFρσ = A. (2.61)

If the quantum effective action is not invariant (A 6= 0), this symmetry is
anomalous and one can show that anomalies in 4 dimensions are 1-loop 3-point
function effect. Therefore it is proportional to:



Chapter 2. The Standard Model of Particle Physics 22

b

a

c

+

b

a

c

=

A ∝ Tr
[

TaTbTc
]

+ Tr
[

TaTcTb
]

,

(2.62)

where T{a,b,c} are the gauge generators being the hypercharge absorbed in this
definition.

2.4.2 Anomaly cancellation in the SM

The form of the anomaly in Eq. 2.61 is for an Abelian group but it can also
be generalized for non-Abelian groups. The traces from Eq. 2.62 can be divided
into a symmetric and an antisymmetric part, but since the antisymmetric part is
summed it will vanish. Hence:

A ∝ dabc = Tr
[

Ta
{

Tb, Tc
}]

, (2.63)

Noting that the symmetric matrix Tr[Ta {Tb, Tc}] may contain mixed repre-
sentation for the symmetries of the theory, which, for the SM, means mixed
representations of SU(3)C, SU(2)Land U(1)Y. Also, observing that in the fermion
loops any fermion can be inserted, implies that we have to sum over all of their
contributions. With the above and bearing in mind that gauge anomalies are a
sign of a flawed theory, the gauge anomalies have to cancel in the SM, as it will be
seen. If we are talking about gauge symmetry, then β = β(x) is a function of the
position.

Firstly QCD theory is non-chiral so left and right anomalies will cancel each
other in a SU(3)3

C anomaly. Also for the SU(2)3
L we can use

{
σα, σβ

}
= 1

2 δαβI

and Tr[σα] = 0 to verify that there is no anomaly. And the trace of any separated
generator of SU(N) for N ≥ 2 is also zero, therefore any mixed state containing
only one SU(N) will also give zero. Thus, the only ones that need to be checked
are U(1)3

Y , U(1)Y × SU(2)2
L, and U(1)Y × SU(3)2

C. Here we will not discuss
anomalies due to gravity, but it can also be shown that they are not present in the
SM [10, 15].



Chapter 2. The Standard Model of Particle Physics 23

For the U(1)3
Y we have:

dabc = ∑
Le f t

Y3 − ∑
Right

Y3, (2.64)

where T=Y was used for the Abelian gauge group.
Then using the values from Table 2.1:

YeL = YvL = −1 ; YeR = −2 ; YuL =
1
3

; YuR =
4
3

; YdL =
1
3

; YdR = −2
3

,

and the fact that we have 3 quark colors, we get:[
2(−1)3 + 6

(
1
3

)3
]
−
[
(−2)3 + 3

(
4
3

)3

+ 3
(
−2

3

)3
]
= 0. (2.65)

For U(1)Y × SU(2)2
L we only get contributions for left-handed fermions. Using

again
{

σα, σβ
}
= 1

2 δαβI, we have:

tr[Ta
{

σb, σc
}
] =

1
2

δbc ∑
Le f t

Y =
1
2

δbc

[
2(−1) + 6

(
1
3

)]
= 0. (2.66)

And finally, for U(1)Y × SU(3)2
C we only have the contribution from quarks:

tr
[
Y
{

Tb, Tc
}]

∝
1
2

δbc

[
Quarks

∑
Le f t

Y−
Quarks

∑
Right

Y

]
∝
[

6
(

1
3

)]
−
[

3
(

4
3

)
+ 3

(
−2

3

)]
= 0,

(2.67)
where we used Tr

{
Tb, Tc} ∝ 1

2 δbc.
Eq. 2.65, Eq. 2.66, and Eq. 2.67 shows that the SM is free of all possible

gauge anomalies, showing its robustness. An important remark is that for the
consistency of any new model it will have to be anomaly-free as well, implying in
a reevaluation of the computations above.

2.5 Issues of the SM

Besides the tremendous success of the SM, it still has some issues or fails to
explain some phenomena. In what follows, some of these issues are listed and a
brief explanation is provided.

• Number of free parameters



Chapter 2. The Standard Model of Particle Physics 24

The amount of free parameters of the SM is relatively large: 19 according
to M. Thomson [18], and when accounting for the additional masses of the
neutrinos and the respective mixing angles, it could come up to 26. This
might not seem like an issue, but the Bayesian model comparison method [19]
states that the theory that best fit the experimental data with the less amount
of input is the most complete one.

U. Wiese [20] says that "it is hard to believe that there should not be a
more fundamental theory that will be able to explain the values of these
parameters", and I. Britto and M. Trott [21] indicate that an excessive number
of parameters is a sign of an effective theory, rather than a fundamental one.
Since physicists always intend to build the most fundamental theory, the
search for a better understanding of where these free parameters come from,
especially those coming from the Higgs sector (that accounts for the majority
of them), is an ongoing one.

• Neutrino Oscillation

Since 1998, with the strong evidence of neutrinos oscillation from the Super-
Kamiokande collaboration [22], this issue has been one of the most studied
themes of particle physics. In 2006 the SNO collaboration [23] demonstrated
that solar neutrinos indeed oscillate, asserting that this is an issue for the SM.

The fact that neutrinos oscillate implies directly that they have mass, and,
therefore, a right-handed component. This contradicts one of the first as-
sumptions of the SM. Although it is possible to add a non-interacting right-
handed neutrino trivially, the addition of a very low-mass particle with the
same mass acquisition mechanism (The Higgs Mechanism) as all the other
massive particles is not very satisfactory. Hence the search for new physics.

The search is for a model that provides the low mass for neutrinos without
the use of many more free parameters, for the same reasons stated in the
first item of this section. The author E. K. Akhmedov [24] presents a more
detailed explanation of neutrino physics and the consequences of neutrino
oscillation.

• Strong CP Problem

The strong CP problem, as mentioned before, is related to the free phase
of the CKM mass matrix (Eq. 2.49). The combination of the CKM phase
with an anomalous gauge chiral phase in QCD, if large enough, would cause



Chapter 2. The Standard Model of Particle Physics 25

QCD not to conserve the CP (or T) symmetry, and would be observable, for
example, as an electric dipole in the neutron. The fact that we do not observe
such electric dipole nor strong CP violation means that this combination of
phases has a small magnitude (less than 10−10, according to E. Kolb and M.
Turner [25]).

Since we know that the phase of the CKM matrix is of order 1 by measures in
weak CP violation, the exact cancellation of the CKM phase by an anomalous
phase without any obvious correlation is not intuitive. Thus the hunt for a
mechanism that explains the smallness of the QCD phase in a natural way.

• Matter-antimatter asymmetry

This issue is related to the strong CP problem. The baryon-antibaryon
asymmetry in the universe can only be caused by a certain amount of CP
violation, that in the SM is not present (the full set of prerequisites for
explaining this effect is the same as for the baryogenesis and was laid down
by A. Sakharov in 1967 [26]). Even though some CP violation is allowed
in the SM, it is not sufficient to amount to the values from cosmological
data [27].

• Lepton Flavour Violation

In current experiments, there is no evidence of lepton flavour violation,
which agrees with the strong suppression of this phenomenon in the SM.
This means that any new physics that one adds to the current model will have
very high restrictions on its free parameters, imposed by the present-day
observations.

One of the most relevant flavour violation restrictions for this work is the
decay of the muon into an electron and a photon and the branching ratio of
this decay is limited to Π(µ→ e + γ) < 4.2 · 10−13 (90% CL) [28].

• Dark Matter

One of the main areas of physics research today is a candidate for dark
matter: an unknown matter that does not interact with the photon. Evi-
dences for such matter are quite abundant, especially from cosmological and
astrophysical data, and date back to even the first half of the 20th century. E.
Schmitz [29] makes a reasonable summary of such pieces of evidence.



Chapter 2. The Standard Model of Particle Physics 26

The only particle in the SM that have the characteristics of dark matter
(long-lived, and without interaction with a photon) are neutrinos, but the
thermal production of neutrinos gives hints that they are probably not the
correct answer to the missing matter. Therefore, the SM does not provide a
candidate particle to solve the issue, and the natural way to think is to make
an extension to the SM. In the literature, there are many such extensions (e.g.
complex scalar dark matter [30], axions [31], among others), but none has
been proved experimentally yet.

• Dark Energy

Like Dark Matter, little is known about what composes the dark energy, but
cosmological theories predict that it does exist in abundance in our universe
(around 70% of the composition of the universe). The SM does not have a
natural candidate for it [32]. Also, some models try to solve this issue, but
most of them are not in the current experimental range.

• Gravity

For many reasons, gravity has not yet been incorporated in the SM, mainly
because there is no theory of quantum gravity broadly accepted. Supersym-
metry and String theory are areas that try to solve that issue but have not yet
succeeded [8].

• Higgs hierarchy

The Higgs hierarchy problem arrives if we have new physics at a certain
scale Λ. Radiative corrections for a scalar boson determine that the mass
parameter for the Higgs should be:

M2
H = M

′2
H − Σ− δm ; Σ ∝ Λ2, (2.68)

where M′H is the bare mass and δm is the renormalization term containing
the infinities. This happens even if the new particle interacts with the Higgs
field only through loops.

Since we know that the Higgs mass is roughly 125 GeV, it is not reasonable
for the new scale to be too much above it, since it would require an unnatural
cancellation of many decimal places for the Higgs to have the known spec-
ified mass. With present experiments, we were able to probe for particles
up to a few TeV but none was found. This suggests that we need new ways



Chapter 2. The Standard Model of Particle Physics 27

to explain this phenomenon if we are to find new physics. Some examples
of a different explanation are Supersymmetry, which solves the problem
by adding a fermion counterpart for the Higgs and cancel the radiative
correction by gauge invariance, or composite Higgs models, that can raise
the allowance of the new physics scale without the need of fine-tuning the
parameters [33].

• Anomalous mass of the W boson

A recent publication by the CDF collaboration [34] exposed a precise measure
of the W boson mass to be significantly above the SM prediction (around 7
standard deviations).

The value predicted by the SM is MSM
W = 80.357± 0.006 GeV and the mea-

surement by CDF is MCDF
W = 80.433± 0.009 GeV. This result, if confirmed,

would require a correction to the SM to explain it and is in clear conflict with
some other experiments as well.

• Muon g− 2

This issue comes from an experimental measure of the muon g− 2 factor that
is in disagreement with the SM prediction. This is the focus of the present
study, therefore it will be explained in detail in the next section.



Chapter 3

The Muon g− 2

The issue with the SM selected for this study is, as mentioned before, the muon
g− 2. It refers to the anomalous magnetic dipole moment of the muon or, more
specifically, the divergence of the classical value of this factor. In recent years, after
the release by the Fermilab Muon g− 2 Collaboration [6] of the results for the
measure of this factor, it was confirmed that the discrepancy between experiments
and theory can be large enough to indicate new physics, which is the motivation
for this work.

In this chapter, some of the backgrounds for understanding how to compute,
the relevance, as well as a complete leading order (LO) and next to leading order
(NLO) calculation of the muon g− 2 will be shown.

3.1 History of the Anomalous Magnetic Moment

The non-relativistic Hamiltonian of a fermion in the presence of an external
magnetic field is:

H~µ =

[
p2

2m
+ V(x)−~µ ·~B

]
, (3.1)

where V(x) is a potential that depends only on the position, ~B is the magnetic
field, and ~µ is the magnetic moment.

The magnetic moment is known from classical physics to have a relation with
the orbital angular moment which can be derived for a classical point charge as:

~L = m~r×~v ; ~µ =
q

2
~r×~v = gL

q

2m
~L, (3.2)

where ~L is the orbital angular momentum,~r is the radius, m is the mass of the
particle, ~µ is the magnetic moment, ~v is the velocity, and q is the charge. Also
included is the gL which is called the Landé g factor, the same g as in the title of
the chapter. For this particular case of a point-like classical particle, the gL factor is
simply 1.

28



Chapter 3. The Muon g− 2 29

For elementary particles with spin angular momentum (~S), there is no particu-
lar reason to assume a value for the Landé g factor. Hence the correct formula for
it is to add the spin contribution like:

~µ =
e

2m

(
~L + gS~S

)
, (3.3)

where e is the electron charge and gS is, at first, a free parameter of the theory.
Pauli was the first to find out that the actual value for gS is 2 and using the

Dirac equation, published in 1928, it is easy to figure out this number. It can be
shown how to extract the magnetic moment as follows:

Starting with the Dirac equation:

( /D−m)ψ = 0, (3.4)

we multiply by ( /D + m), and using the relation ( /D2 = D2 + e
2 Fµνσµν), with σµν =

i/2 [γµ, γν], to obtain: (
D2 + m2 +

e
2

Fµνσµν
)

ψ = 0. (3.5)

Comparing this equation to the Klein-Gordon equation:(
D2 + m2

)
φ = 0, (3.6)

we can see that there is an extra term for particles with spin half. Expanding this
extra term using the Weyl representation we obtain:

e
2

Fµνσµν = −e

(~B + i~E
)
~σ (

~B− i~E
)
~σ

 . (3.7)

Then, rewriting Eq. 3.5 as:(
D2 + m2 − 2e~B ·~S± i2e~E ·~S

)
ψ = 0, (3.8)

we can show that the dipole magnetic moment is the term that multiplies the
magnetic field (~B) divided by 2m. The proof of this relation is shown in Appendix
A. Comparing Eq. 3.8 with Eq. 3.3, we can see that the value of the g factor is
indeed 2.

Although the experiments at the time were consistent with the value for g



Chapter 3. The Muon g− 2 30

equal to 2 within 10% accuracy [35], Schwinger [36] realized that there were still
some quantum effects unaccounted for. In 1947 he calculated the first radiative
correction to the anomalous magnetic moment, the famous α

2π . Figure 3.1 shows
the Dirac magnetic moment factor and the Schwinger one-loop correction diagram.

p1

↓ q

p2

Aµ

(a)

p1
A

↓ q

p2

Aµ

ρ ν

(b)

Figure 3.1: Feynman diagram for (a) tree-level contribution and (b) one loop
photon contribution to the muon g factor

Schwinger’s predictions were then measured with extreme precision in ex-
periments later on and it was one of the greatest confirmations of the validity of
quantum field theory. In section 3.2.2 Schwinger’s computation will be shown
explicitly, and although he calculated it for the electron, the same procedure is
valid for the muon and it will be used for the NLO computation of the muon g− 2
factor.

3.2 Feynman diagram calculation of the g factor

The first step to understanding where the magnetic moment appears is to
identify all possible terms that can contribute to it. We want to write all possible
terms with just one uncontracted Lorentz index because we know that these are
the possible terms that contract with the external magnetic field. Figure 3.2 shows
the most general Feynman diagram to be drawn and Eq. 3.9 gives the most general
form of the evaluation of this diagram.

iMµ = (−ie) ū(p2)
(

f1γµ + f3pµ
1 + f4pµ

2
)

u(p1). (3.9)

Terms with γ5 are ignored because QED is parity invariant. Also, terms with
qµ can be directly rewritten as the other two momenta by momentum conservation
(qµ = pµ

2 − pµ
1 ) so they are absorbed in the definition of f3 and f4. The next step is



Chapter 3. The Muon g− 2 31

p1

↓ q

p2

µ µ

Aµ

gS

Figure 3.2: General Feynman diagram for gS factor contribution. The symbol × at
the photon propagator indicates that it is a virtual photon.

to use the ward identity:

qµMµ = (−e) ū(p2)
(

f1qµγµ + f3qµ pµ
1 + f4qµ pµ

2
)

u(p1) = 0, (3.10)

and find more relations among the f factors.
In the first term inside the bracket, we can use the relations from the Dirac

equation

/p1u(p1) = m1u(p1);

ū(p2)/p2 = m2ū(p2),
(3.11)

and momentum conservation again to find the trivial m1 = m2. So for now on, we
will drop the index in the mass term and m will be used exclusively for the muon
mass.

For the second term in the brackets, some work needs to be done. Using

qµ pµ
i = (p2 − p1)µ pµ

i =
(

m2 − p2p1

)
(−1)i ; i = 1, 2, (3.12)

we write:
f3qµ pµ

1 + f4qµ pµ
2 =

(
p2p1 −m2

)
( f3 − f4) = 0, (3.13)

and show that f3 = f4. Another useful redefinition is to write f3 = − F2
2m so that

the F2 factor is dimensionless. The minus sign and the factor 1
2 are conventions

that simplify the notation. After these changes we have:

iMµ = (−ie) ū(p2)

[
f1γµ − F2

2m
(

pµ
1 + pµ

2
)]

u(p1). (3.14)

Using Gordon’s identity [10] for on-shell spinors:

ū(p2)
(

pµ
1 + pµ

2
)

u(p1) = ū(p2) [2mγµ − iσµν (p2 − p1)ν] u(p1), (3.15)



Chapter 3. The Muon g− 2 32

we rewrite the second term of Eq. 3.14 and use a last redefinition of f1 − F2 = F1

to acquire the final form of this equation:

iMµ = (−ie) ū(p2)

[
F1γµ + iF2

σµν

2m
qν

]
u(p1). (3.16)

The terms F1 and F2 are called the form factors and are functions of q2 and
m2. More precisely, they need to be functions of q2

m2 since they are dimensionless,
but because we will consider the non-relativistic limit (q2 → 0) they shall be
considered as constant.

The final step is to relate the form factors and the Landé g factor. Using:

u(p) =
(√

p · σ
√

p · σ̄

)
u(0), (3.17)

where u(0) are the 0 momentum solutions of the Dirac equation. Considering the
non-relativistic limit:

√
pσ =

pσ + m√
2 (E + m)

∼=
E + m− ~p~σ√

4m
∼=
√

m
(

1− ~p~σ
2m

)
, (3.18)

we can show that the first term in the brackets of Eq. 3.16 is:

F1ū(p2)γ
µu(p1) = F1 u†(0)

(√
p2σ

√
p2σ̄
)

γ0γµ

(√
p1σ
√

p1σ̄

)
u(0)

= m F1u†(0)
[(

1− ~p2~σ

2m

)
σi
(

1− ~p1~σ

2m

)
−
(

1− ~p2~̄σ

2m

)
σi
(

1− ~p1~̄σ

2m

)]
u(0)

= m F1u†(0)
(
~p1~σ

m
σi + σi ~p2~σ

m

)
u(0),

(3.19)

where in the last line we used σ̄i = −σi from Eq. 2.5 and only the crossed terms
remained. Now we can use the Pauli matrices identity σiσj = δij + iεijkσk to find:

F1ū(p2)γ
µu(p1) = F1u†(0)

(
pi

1 + pi
2 + iεijk (p2 − p1)j σk

)
u(0), (3.20)

where we can ignore the terms pi
1 and pi

2 since they are spin-independent, and
look only at the magnetic moment term, identifying iεijkqj Ak = −Bi, with qj =

(p2 − p1)j. Then we have to use the identities σi = 2Si and u†(0)Siu(0) = 1
2m 〈S〉i



Chapter 3. The Muon g− 2 33

to end up with the known format of the magnetic moment:

2F1
e

2m

〈
~S
〉

·~B. (3.21)

For the second term of Eq. 3.16, we have to do similar considerations. First,
we expand the terms using the non-relativistic approximation (Eq. 3.18) and
remembering that σµν = i

2 [γ
µ, γν], we get:

ū(p2) [σ
µνqν] u(p1) =

i
2

mu†(0)
[(

1 +
~p2~σ

2m

) [
σj, σi

] (
1− ~p1~σ

2m

)
+

(
1− ~p2~σ

2m

) [
σj, σi

] (
1 +

~p1~σ

2m

)]
u(0)qj

= −imu†(0)
[
εjikσk

]
u(0)qj + O

(
1
m

)
,

(3.22)

and again, make the identification of the magnetic moment term using the same
identities as before. The final result, obtained after multiplying by the appropriate
constants, is:

2F2
e

2m

〈
~S
〉

·~B. (3.23)

Finally, we combine Eq. 3.21 and Eq. 3.23 to have the relation between the
form factors and the Landé g factor, shown below:

gS = 2 (F1(0) + F2(0)) , (3.24)

where the zero shows explicitly that the non-relativistic limit was used.

3.2.1 Leading order calculation of the Form Factors

The tree-level Feynman diagram evaluation from Figure 3.1a gives directly:

iMµ
0 = −ieū(p2)γ

µu(p1), (3.25)

which is a term similar to the one found when computing the F1 form factor in
the last section. Comparing this equation with Eq. 3.16 it is straightforward to
conclude that F1 = 1 and F2 = 0 for this diagram. This reproduces gS = 2 precisely,
as one should expect.



Chapter 3. The Muon g− 2 34

3.2.2 NLO calculation of the Form Factors

The Feynman diagram of the loop with a photon is the most relevant one since
it has the strongest coupling. As pointed out before, this calculation was primarily
done by Schwinger for the electron but the same computation applies to the muon
since it does not depend on the fermion mass. The momentum convention for
loop diagrams used in this work is shown in Figure 3.3.

p1

k k + q

↓ q

p2
k− p1

Figure 3.3: Momentum convention for one-loop diagrams. All arrows in this
diagram are for the momentum direction used, not to indicate any kind of spin or
propagator.

With this convention we can now evaluate the diagram from Figure 3.1b as
follows:

iMµ
γ = (−ie)3

∫ d4k

(2π)4
−igνρ

[(k− p1)2 + iε]
ū(p2)

× γν i(/q + /k + m)

[(q + k)2 −m2 + iε]
γµ i(/k + m)

[k2 −m2 + iε]
γρu(p1)

= −e3ū(p2)
∫ d4k

(2π)4
γν(/q + /k + m)γµ(/k + m)γν

[(k− p1)2 + iε] [(q + k)2 −m2 + iε] [k2 −m2 + iε]
u(p1),

(3.26)

where m is the mass of the on-shell fermion.
To proceed we need to use the Feynman trick:

1
ABC

=
∫ 1

0

∫ 1

0

∫ 1

0
dxdydz

Γ(3)δ(1− x− y− z)
[xA + yB + zC]3

, (3.27)

where x, y and z are called the Feynman parameters. Separating the numerator
and denominator inside the integral to facilitate the notation of Eq. 3.26 we get:

iMµ
γ = −e3

∫ d4k

(2π)4

∫ 1

0
dx
∫ 1−x

0
dy

(
Nγ

Dγ

)
, (3.28)



Chapter 3. The Muon g− 2 35

with the denominator being:

Dγ = [k2 − 2xkp1 + 2ykq + xp2
1 + yq2 − (y + z)m2 + iε]3. (3.29)

Following the procedure of the Feynman trick, we perform the change of
variables:

kµ → lµ − yqµ + xpµ
1 ; d4k→ d4l, (3.30)

and using p2
1 = p2

2 = m2 and p1 · p2 = m2 − q2

2 , we obtain the final form of this
denominator, shown below:

Dγ = (l2 − ∆γ + iε)3;

∆γ = (y2 + xy− y)q2 + (1− x)2m2.
(3.31)

This same format of denominator will appear many times, with small changes
in the ’∆’ parameter.

For the numerator, some algebra is needed. Leaving aside, for now, the inte-
gration and the factor of −e3, we substitute for the new variable from Eq. 3.30 and
use the property of Dirac matrices γν/a/b/cγν = −2/c/b/a to get:

Nγ = (2!)(−2)ū(p2)[/l + x/p1− y/q + m]γµ[ 6 l + x/p1 + (1− y)/q + m]u(p1). (3.32)

Because of Lorentz invariance we have the following relations:

∫
d4l [lµlν] =

∫
d4l
[

1
4

gµνl2
]

;∫
d4l
[
lµ f (l2)

]
= 0,

(3.33)

and with that we can eliminate any odd powers of l inside the integral. Then use
the relations from Eq. 3.11 and γµγν = 2gµν − γνγµ, to obtain:

Nγ = 2l2 [ū(p2)γ
µu(p1)]

+ 4
[

x2 + 2x− 1
]

m2 [ū(p2)γ
µu(p1)]

+ 4 [(x + y)(y− 1)] q2 [ū(p2)γ
µu(p1)]

− 8
[

x2 + xy− 2x− 2y + 1
]

m
[
ū(p2)pµ

1 u(p1)
]

− 8 [−xy + x + 2y− 1]m
[
ū(p2)pµ

2 u(p1)
]

.

(3.34)



Chapter 3. The Muon g− 2 36

The objective now is to use the Gordon identity (Eq. 3.15). Then we have to
work on the last two lines of Eq. 3.34 to get the terms with (p2 − p1 = q) and
(p2 + p1). For that we use the simple relations:

Ap1 + Bp2 = C1(p1 + p2) + C2(p2 − p1);

C1 =
A + B

2
; C2 =

B− A
2

,
(3.35)

which gives:

− 8
[

x2 + xy− 2x− 2y + 1
]

m
[
ū(p2)pµ

1 u(p1)
]

− 8 [−xy + x + 2y− 1]m
[
ū(p2)pµ

2 u(p1)
]

=− 4(x2 − x)m [ū(p2)(p1 + p2)
µu(p1)]

− 4(x2 + 2xy− 3x− 4y + 2)m [ū(p2)qµu(p1)] .

(3.36)

Then finally we can use the Gordon identity (Eq. 3.15) and find:

Nγ = 2
[
l2 − 2m2(x2 − 4x + 1) + 2(x + y)(y− 1)q2

]
[ū(p2)γ

µu(p1)]

+ 4im(x2 − x)qν [ū(p2)σ
µνu(p1)]

− 4m(x2 + 2xy− 3x− 4y + 2)qµ [ū(p2)u(p1)] .

(3.37)

The last line of the above equation is zero because the Feynman parameters
can be rewritten as (x − 2)(z − y) when using the Feynman parameters delta
[δ(1− x− y− z)]. Since this term is asymmetric in the exchange of y and z and
everything else in the integral is symmetric over the same exchange, then it must
be zero. Therefore, we have the same structure as Eq. 3.16 from where we can
extract the form factors from the expression above as:

−ieF1,γ = −2e3
∫ 1

0
dx
∫ 1−x

0
dy
∫ d4l

(2π)4

[
l2 − 2m2(x2 − 4x + 1) + 2(x + y)(y− 1)q2]

Dγ
;

eF2,γ

2m
= −4ie3m

∫ 1

0
dx
∫ 1−x

0
dy
∫ d4l

(2π)4
(x2 − x)

Dγ
,

(3.38)

where we can see explicitly the dependence on q2 and m2 of the form factors that
were stated before.

The first integral will not be evaluated here, because it is a long calculation
that leads to zero when we take the limit of q2 → 0 and renormalize the theory.



Chapter 3. The Muon g− 2 37

This result can be found in Appendix C and is not surprising, since as q2 → 0 we
should reproduce the coulomb potential, consistent with the tree-level result. And
since this must be true for all orders in the perturbative expansion, for now on,
the terms multiplying γµ will simply be ignored in loop diagrams, and F1 = 1 will
be maintained, representing the classical part of the g factor.

With that in mind, we can recognize that the full loop effect happens when

F2 =
g− 2

2
6= 0. (3.39)

This factor is called the anomalous magnetic dipole moment, and for the muon
is often referred to by aµ (remarking that this µ is not a Lorentz index). Using the
integral identity: ∫ d4l

(2π)4
1

(l2 − ∆ + iε)3 =
−i

32π2∆
, (3.40)

we can compute the second term of Eq. 3.38 to arrive at:

F2,γ =
αm2

π

∫ 1

0

∫ 1−x

0

x(1− x)
(y2 + xy− y)q2 + (1− x)2m2 , (3.41)

where we used the definition of the fine-structure constant from section 2.1.3.
Finally, taking the limit of q2 → 0, the integration gives simply 1

2m2 , and we are
able to repeat the famous result by Schwinger [36]:

F2,γ = aµ,γ =
α

2π
. (3.42)

3.3 Muon g− 2 in the SM

So far the calculations done are the same for the muon and the electron since it
is independent of the lepton mass. So we can safely say that the NLO correction
for the muon is the same as Eq. 3.42. The differences from the electron will appear
when we consider the heavy particles inside the loop diagram. Since the muon is
approximately 200 times heavier than the electron these loop diagrams will not
be as suppressed as for the case of the electron. So we can identify the one-loop
diagrams that will contribute to the muon g− 2 shown in Figure 3.4.

We will analyze each diagram separately in the next subsections, but the mo-
mentum convention used will be the same as for the photon loop, from Figure 3.3.



Chapter 3. The Muon g− 2 38

µ±

A

µ±

µ± µ±

Aµ

ρ ν

(a)

µ±

H

µ±

µ± µ±

Aµ

(b)

µ±

Z

µ±

µ± µ±

Aµ

ρ ν

(c)

W±

νµ±

W±

µ± µ±

Aµ

ρ
σ λ

ν

(d)

Figure 3.4: One-loop Feynman diagrams for the muon g − 2 contribution. (a)
photon loop, (b) Higgs-boson loop, (c) Z-boson loop , and (d) W-boson loop.

The computations that follow will be done in less detail since we can search directly
for the terms that multiply (σµνqν), knowing that these contain the anomalous part
of the magnetic dipole moment.

3.3.1 Higgs-boson contributions

Probably the simpler of the three remaining calculations is the one with the
Higgs-boson loop, for it is a scalar field. The evaluation of the Feynman diagram
from Figure 3.4b is given by:

iMµ
H = (−ie)

(
−g2

4

)(
m

MW

)2 ∫ d4k

(2π)4 ū(p2)
i[

(k− p1)2 −M2
H + iε

]
i(/k + /q + m)

[(k + q)2 −m2 + iε]
γµ i(/k + m)

[k2 −m2 + iε]
u(p1)

= (−ie)
(

ig2

4

)(
m

MW

)2 ∫ d4k

(2π)4 ū(p2)

(/k + /q + m)γµ(/k + m)

[(k− p1)2 −MH + iε] [(q + k)2 −m2 + iε] [k2 −m2 + iε]
u(p1)

= (−ie)
(

i
g2

4

)(
m

MW

)2 ∫ d4k

(2π)4

∫ 1

0
dx
∫ 1−x

0
dy

(
NH

DH

)
.

(3.43)

The denominator is similar to the one for the photon loop, with the only



Chapter 3. The Muon g− 2 39

difference being an extra xM2
H in the delta term, as shown below:

DH = [l2 − ∆H + iε]3;

∆H = (y2 + xy− y)q2 + (1− x)2m2 + xM2
H.

(3.44)

The numerator, however, is simpler. The computation, following the same
steps as before, yields:

NH = (2!)ū(p2)(/k + /q + m)γµ(/k + m)u(p1)

= 2m(1− x2)i[ū(p2)σ
µνqνu(p1)] + O [ū(p2)γ

µu(p1)] .
(3.45)

Substituting these values in Eq. 3.43 we can calculate the F2 factor as:

eF2,H

2m
= (−ie)

(
g2

4

)(
m

MW

)2 ∫ d4l

(2π)4

∫ 1

0
dx
∫ 1−x

0
dy

[
2m(x2 − 1)

[l2 − ∆H + iε]3

]
,

(3.46)

that can be computed analytically. The result expanded in powers of
(

m
MH

)
is

shown below:

F2,H =
g2

64π2

(
m

MW

)2
[(

m
MH

)2 1
6

(
−7− 12ln

(
m

MH

))

+

(
m

MH

)4 1
4

(
−13− 24ln

(
m

MH

))]
+ O

[(
m

MH

)6
]

.

(3.47)

3.3.2 Z-boson contributions

To simplify the calculations everything will be computed using the Feynman
gauge (ξ = 1), and in appendix B.1 the gauge invariance is proven with more
details. Since it is not the unitary gauge there will be an extra diagram, the loop
with the Goldstone-boson in the place of the Z-boson. These two diagrams can be
seen in Figure 3.5.



Chapter 3. The Muon g− 2 40

µ±

Z

µ±

µ± µ±

Aµ

ρ ν

(a)

µ±

ϕZ

µ±

µ± µ±

Aµ

(b)

Figure 3.5: Feynman diagrams for the Z-boson contribution in the Feynman gauge.
(a) is the Z-boson loop, and (b) the neutral Goldstone-boson loop.

The first diagram (Figure 3.5a) is evaluated as:

iMµ
Z1

= (−ie) (−i)
(

g
4cosθw

)2 ∫ d4k

(2π)4

ū(p2)
γν(AZ + γ5)(/k + /q + m)γµ(/k + m)γν(AZ + γ5)

[(k + q)2 −m2 + iε][k2 −m2 + iε][(k− p1)2 −M2
Z + iε]

u(p1)

= (−ie) (−i)
(

g
4cosθw

)2 ∫ d4k

(2π)4

∫ 1

0
dx
∫ 1−x

0
dy

(
NZ1

DZ

)
,

(3.48)

where AZ = 4sin2θw − 1.
Once again, we divide the numerator and denominator to facilitate the compu-

tations. The denominator acquires the familiar form of:

Dz = [l2 − ∆Z + iε]3;

∆Z = (y2 + xy− y)q2 + (1− x)2m2 + xM2
Z,

(3.49)

and the numerator, after following the same steps as before, yields:

NZ1 = (2!)γν(AZ + γ5)(/k + /q + m)γµ(/k + m)γν(AZ + γ5)

= 4m[A2
Z(x2 − x) + x2 + 3x)]i[ū(p2)σ

µνqνu(p1)]

+ O
[

ū(p2)γ
µu(p1) , ū(p2)(...)γ5u(p1)

]
.

(3.50)



Chapter 3. The Muon g− 2 41

The second diagram (Figure 3.5b) is a little simpler, evaluated as:

iMµ
Z2

= (−ie) (−i)
(

g2

4

)(
m

MW

)2 ∫ d4k

(2π)4

ū(p2)
γ5(/k + /q + m)γµ(/k + m)γ5

[(k + q)2 −m2 + iε][k2 −m2 + iε][(k− p1)2 −M2
Z + iε]

u(p1)

= (−ie) (−i)
(

g
4cosθW

)2

4
(

m
MZ

)2 ∫ d4k

(2π)4

∫ 1

0
dx
∫ 1−x

0
dy

(
NZ2

DZ

)
.

(3.51)

The denominator is exactly the as (Eq. 3.49). And the numerator computation
is shown below:

NZ2 = (2!)γ5(/k + /q + m)γµ(/k + m)γ5

= [2m(1− x)2]i[ū(p2)σ
µνqνu(p1)] + O(ū(p2)γ

µu(p1)).
(3.52)

The combination of the results from Eq. 3.50 and Eq. 3.52 give the F2 factor of:

eF2,Z

2m
= (−ie)

(
g

2cosθw

)2 ∫ d4l

(2π)4

∫ 1

0
dx
∫ 1−x

0
dy

m[A2
Z(x2 − x) + x2 + 3x)] + [2m

(
m

MZ

)2
(1− x)2]

[l2 − ∆Z + iε]3
,

(3.53)

which can be integrated analytically. The result after expanding in powers of(
m

MZ

)
is:

F2,Z =
1

64π2

(
g

cosθw

)2
[(

m
MZ

)2 1
3

(
A2

Z − 5
)

+
1

12

(
m

MZ

)4(
25A2

Z + 24ln
(

m
MZ

)
A2

Z − 19− 24ln
(

m
MZ

))]

+ O

[(
m

MZ

)6
]

,

(3.54)

which agrees with R. Jackiw and S. Weinberg [37].



Chapter 3. The Muon g− 2 42

3.3.3 W-boson contributions

The W-boson loop is the most complicated, involving 4 diagrams, shown in
Figure 3.6. The muon antineutrino and muon neutrino are represented by νµ+ and
νµ− , respectively. Remarking again that here we are using the Feynman gauge,
and the gauge invariance is proven in appendix B.2.

W±

νµ±

W±

µ± µ±

Aµ

ρ
σ λ

ν

(a)

ϕ±

νµ±

W±

µ± µ±

Aµ

λ
ν

(b)

W±

νµ±

ϕ±

µ± µ±

Aµ

ρ
σ

(c)

ϕ±

νµ±

ϕ±

µ± µ±

Aµ

(d)

Figure 3.6: Feynman diagram for the calculation of the W-boson contribution in
the Feynman gauge. (a) is the pure W-boson loop, (b) and (c) the mixed loop
with one W-boson and one charged Goldstone-boson, and (d) the pure charged
Goldstone-boson loop.

The evaluation of the first diagram (Figure 3.6a) grants:

iMµ
W1

= (−ie)
(

i
g2

2

) ∫ d4k

(2π)4

ū(p2)
γνPLgνλ · Intλσµ

W · gσρ(/k − /p1)γ
ρPL

[(k + q)2 −M2
W + iε][k2 −M2

W + iε][(k− p1)2 + iε]
u(p1)

= (−ie)
(

i
g2

2

) ∫ d4k

(2π)4

∫ 1

0
dx
∫ 1−x

0
dy

(
NW1

DW

)
,

(3.55)

with
Intλσµ

W =
[

gλσ(2k + q)µ + gλµ(−k− 2q)σ + gσµ(q− k)λ
]

. (3.56)

Once again, we divide the denominator and the numerator. The denominator



Chapter 3. The Muon g− 2 43

will be the same for all four diagrams due to the choice of gauge, and is shown in:

DW = [l2 − ∆W + iε]3;

∆W = (y2 + xy− y)q2 + (x2 − x)m2 + (1− x)M2
W .

(3.57)

The numerator for the diagram evaluation in Eq. 3.55 is:

NW1 = (2!)γλ Intλσµ
W (/k − /p1)γσPL

= m[2x2 − 5x + 3]i[ū(p2)σ
µνqνu(p1)]

+ O
[

ū(p2)γ
µu(p1) , ū(p2)(...)γ5u(p1)

]
.

(3.58)

The second and third diagrams (Figures 3.6b and 3.6c) are evaluated jointly as:

iMµ
W2,3

= (−ie)
(

i
g2

2

) ∫ d4k

(2π)4

ū(p2)
m
[
γµ(/k − /p1)PR + (/k − /p1)γ

µPL
]

[(k + q)2 −M2
W + iε][k2 −M2

W + iε][(k− p1)2 + iε]
u(p1)

= (−ie)
(
−i

g2

2

) ∫ d4k

(2π)4

∫ 1

0
dx
∫ 1−x

0
dy

(NW2,3

DW

)
.

(3.59)

Having the same denominator as Eq. 3.57, all we need to do is evaluate the
numerator, computed in terms of the Feynman parameters like:

NW2,3 = (2!)mγµ(/k − /p1)PR + (/k − /p1)γ
µPL

= m[1− x]i[ū(p2)σ
µνqνu(p1)]

+ O
[

ū(p2)γ
µu(p1) , ū(p2)(...)γ5u(p1)

]
.

(3.60)

The fourth and last diagram (Figure 3.6d) for the W-boson contribution is
evaluated as:

iMµ
W4

= (−ie)
(
−i

g2

2

)(
m2

M2
W

) ∫ d4k

(2π)4

ū(p2)
(2k− q)µ(/k − /p1)PR

[(k + q)2 −M2
W + iε][k2 −M2

W + iε][(k− p1)2 + iε]
u(p1)

= (−ie)
(
−i

g2

2

)(
m2

M2
W

) ∫ d4k

(2π)4

∫ 1

0
dx
∫ 1−x

0
dy

(
NW4

DW

)
,

(3.61)



Chapter 3. The Muon g− 2 44

and the numerator computed in terms of the Feynman parameters is:

NW4 = (2!)(2k + q)µ(/k − /p1)PR

= m[x− x2]i[ū(p2)σ
µνqνu(p1)]

+ O
[

ū(p2)γ
µu(p1) , ū(p2)(...)γ5u(p1)

]
.

(3.62)

To compute the F2 factor for the W-boson, we combine Eq. 3.55, Eq. 3.59, and
Eq. 3.61 obtaining:

eF2,W

2m
= (−ie)

(
g2

2

) ∫ d4l

(2π)4

∫ 1

0
dx
∫ 1−x

0
dy[

−m(2x2 − 5x + 3)
]
− [m(1− x)] +

[
m[x− x2]

(
m2

M2
W

)]
[l2 − ∆W + iε]3

.

(3.63)

Following the previous computations, we perform the integral and then expand
in terms of

(
m

MW

)
to get:

F2,W =
g2

64π2

[
10
3

(
m

MW

)2

+
2
3

(
m

MW

)4
]
+ O

[(
m

MW

)6
]

. (3.64)

One final comment on this result is that the mass of the W boson appears
squared in the denominator, so corrections to the mass due to the measures shown

in section 2.5 would be 1−
(

MCDF
W

MSM
W

)2
∼= 0.19% which is currently outside of the

experimental resolution, as will be shown in the next chapter.

3.4 Theoretical value for the muon g− 2 in the SM

The computations shown explicitly so far are only for the one-loop contribu-
tions for the g− 2 factor, but one can go further and calculate more loop diagrams
to increase the theoretical precision of the result. This was done by a collaboration
of many researchers and they are all shown in very detail by the White Paper
report [38]. A brief explanation of the results will be provided in this section.

The diagram contribution for the muon g− 2 factor can be divided into electro-
magnetic (QED), weak (EW), and hadronic (QCD). The full SM theoretical result



Chapter 3. The Muon g− 2 45

for the muon g− 2 is the sum of all these contributions:

aSM
µ = aQED

µ + aEW
µ + aQCD

µ . (3.65)

The QED and EW multiple loops computations have features similar to the
one-loop calculations shown in earlier sections. The largest contribution by far is
the electromagnetic one. It has been calculated up to five loops (corrections up to(

α
2π

)5) which gives the target precision of around 10−11 of the current experiments.
Six loops or more would range around values of 10−12, therefore are ignored. The
leading order for this QED computation is exemplified in Figure 3.4a, as stated
before.

To precisely compute the value of the anomaly for the QED contribution a very
precise measure of the fine structure constant (α) is needed. There are two major
techniques to obtain it: atom interferometry and electron g− 2 experiments. The
result shown here [39] was obtained by using atom interferometry and has the
value of:

α = [137.035999046(27)]−1 , (3.66)

but we remark that the value found using the other technique is within the same
range of precision and agrees with this result. It can be found in [40]. The result of
the muon g− 2 for the QED contribution considering this value of α is given by:

aQED
µ = (116584718.931± 0.104) · 10−11. (3.67)

The EW contributions go up to NLO only (the leading order contribution is
already around ∼ 10−9). The leading order contribution is exemplified by Figures
3.4c and 3.4d, as stated before, and the result for this contribution is:

aEW
µ = (153.6± 1.0) · 10−11. (3.68)

Lastly the hadronic computation. This is the one that gives the most uncer-
tainties and discussion. We cannot calculate directly this contribution due to
non-perturbative computations in Quantum-Chromo Dynamics (QCD). Therefore
it requires some data-driven or lattice QCD simulation results.

There are two main diagrams to evaluate the QCD contribution. They are called
Hadronic Vacuum Polarization (HVP) and Hadronic Light-by-Light contribution
(HLbL). These diagrams are exemplified at leading order in Figure 3.7.



Chapter 3. The Muon g− 2 46

µ

A

µ

A
µ µ

Aµ

h

(a)

A A A
h

µ µ

Aµ

(b)

Figure 3.7: Feynman diagrams of the leading order hadronic (h) contribution to
the muon g− 2. (a) is the HVP contribution and (b) the HLbL contribution.

The HVP muon g− 2 value was calculated up to 3 orders using only the data-
driven methods, since lattice QCD results still have a higher order uncertainty
(±184 · 10−11). The resulting computed value is:

aHVP
µ =aHVP,LO

µ + aHVP,NLO
µ + aHVP,NNLO

µ

= (6931± 40) · 10−11 + (−98.3± 7) · 10−11 + (12.4± 1) · 10−11

= (6845± 40) · 10−11.

(3.69)

The HLbL result is a combination of data-driven and lattice QCD for the leading
order result, summed with the next leading order for the data-driven result. The
resulting contribution is:

aHLbL
µ =aHLbL,LO

µ + aHLbL,NLO
µ

= (90± 17) · 10−11 + (2± 1) · 10−11 = (92± 18) · 10−11.
(3.70)

Combining the results from Eq. 3.67, Eq. 3.68, Eq. 3.69, and Eq. 3.70 considering
their uncertainties uncorrelated, we find that the most recent theoretical value for
the muon g− 2 is:

aSM
µ = aQED

µ + aEW
µ + aHVP

µ + aHLbL
µ = (116591810± 43) · 10−11. (3.71)

This result is a combination of values found in [40–59] which should be cited
in any work that uses or cites the result above.

For completeness, we should mention that there is one lattice result, from the
BMW collaboration [60] that calculated the LO of the HVP diagrams to be:

aHVPBMW ,LO
µ = (7075± 55) · 10−11, (3.72)



Chapter 3. The Muon g− 2 47

which is in a greater agreement with the current experiments, giving:

aBMW
µ = (116591954± 58) · 10−11, (3.73)

a 1.5 σ discrepancy when considered alone. When combined with the previously
mentioned SM result (Eq. 3.71) it gives:

aSM+BMW
µ = (116591881.5± 38) · 10−11, (3.74)

which, in turn, is a 3.2 σ discrepancy. Lattice results are usually laborious to check
and take time, so for this work we will assume the result published by the White
Paper [38] to be correct (result from Eq.3.71), but keeping in mind that there is a
conflict among the current theoretical results that need further evaluation.



Chapter 4

Muon g− 2 experiments

We discussed how the anomalous magnetic moment of the muon appears in
the theory, but we have not yet justified the use of the muon for the experiments
to test the SM. If there exists new physics at a scale of Λ, then the sensibility of a
lepton anomalous magnetic moment to this new physics is of order

(ml
Λ

)2
, (4.1)

according to A. Pich [61], where ml is the lepton mass. This is in agreement with
the results obtained in the last chapter. Therefore the heavier the lepton, the
greater the sensibility to heavier particles in the g− 2 factor. In the SM we have
three massive leptons to use. The electron, the muon and the tau, in ascending
order of mass. The tau particle is very unstable and decays in around 2.9 · 10−13s,
which makes it very hard to use in experiments. In turn, electrons are very light,
implying that very high precision is needed to find a heavy unknown particle.
Consequently, the use of muons becomes the most reasonable choice.

The objective of this chapter is to give an overview of the more than 60 years
of historical experiments searching for the muon g− 2. Some characteristics of
their evolution will be laid out, and a more detailed explanation of the latest
experiment to release results, the Fermilab (E-989), will be provided, clarifying
how the measure was done.

4.1 History of the muon g− 2 experiments

The first observation of a muon-related phenomenon was made by Paul Kunze
in 1933 [62], [63] but was only recognized as an elementary particle in 1936 by C.
D. Anderson and S. H. Neddermeyer [64]. And the recognition of the muon as
heavy lepton is attributed to the 1957 experiment by R. L. Garwin, L. M. Lederman,
and M. Weinrich [65], which observed the muon spin and its precession under a
Magnetic field. Since then, many other experiments were done to measure, each

48



Chapter 4. Muon g− 2 experiments 49

time more precisely, the g− 2 for the muon in the never-ending search for new
physics.

The first measurement of the muon g factor was done by R. L. Garwin in
1957 using the decay of π+ → µ+ + νµ+ , yielding a rough measure of gS,µ =

2± 0.2, but in the same year Cassels and Garwin again would, separately, improve
this measure to gS,µ = 2

(
1.00113+0.00016

−0.00012

)
that confirmed the calculations by

Schwinger and the behaviour of the muon as a heavy lepton.
After these early measures, CERN made 3 important experiments, each improv-

ing methods and techniques, and consequently the precision of the measure. The
first experiment was done with a rectangular magnet with a small gradient that
caused the circular orbit of the muon to go to the edge of the magnet. Then another
magnet would eject the muon to a target that would stop it without destroying its
polarization. The subsequent decay of the muon contained information about the
amount it precessed, hence a measure of the aµ factor. Some technical issues, that
would be solved later, crippled the accuracy, and the final result of this experiment,
attained in 1962, was

aC1
µ = (1 162 ± 5) · 10−6.

So far all measures included stopping the muon at a target, which meant that
only µ+ could be used. Low energy µ− would be captured into atomic orbits
resulting in a lower effective lifetime for the muon, which made it very hard to
use in experiments.

The second CERN experiment was the first to use a storage ring, and the
future ones would all repeat this kind of storage for the muons that allows for
much higher energy beams of particles. Other improvements were also made,
like the use of kickers to stabilize the orbits, incident beam of particles, etc... [35].
Some technical issues hindered the accuracy of this result as well, like the use of a
magnetic gradient to trap the muons, and large noise in the detectors. The result
obtained in 1968 was:

aC2
µ = (11 661 ± 3.1) · 10−7.

The third and last CERN experiment made some considerable advances in
precision. It was the first to use an electrostatic focusing of the particles, the so-
called Penning trap, and the use of the "magic" energy of 3.09 GeV for the muon,
which importance will be discussed later in this work. The use of the Penning
trap allowed for a uniform magnetic field, which is important to guarantee that
all muons precess at the same rate. Another important innovation was the use of



Chapter 4. Muon g− 2 experiments 50

a tool to cancel the magnetic field at the pion entrance (called pulsed magnetic
inflector), diminishing the noise in the detectors and allowing a more uniform
beam to enter the storage ring. The improved result obtained in 1979 was:

aC3
µ = (1 165 924 ± 8.5) · 10−9.

So far all the results were still in accordance with the SM up to three orders of
radiative correction. The experiment that changed this was the Brookhaven BNL
E-821 experiment, located in New York, USA. BNL experiment was a collaboration
that took around 20 years, and more than 100 scientists to obtain a more accurate
measure of the muon g− 2. The technical advances made by CERN experiments
were added, including the storage ring, the penning trap, the ”magic” energy, and
others. To improve the precision some other advances were made, including a
more uniform magnetic field and incident beam energy, a higher density incident
beam, a faster muon kicker, and more small advances in technique. The accuracy
of this experiment was the first to include four orders of radiative correction,
including weak and hadronic loops, and the last measure released in 2004 [5] was:

aBNL
µ = (116 592 080 ± 63) · 10−11.

This result was the first to deviate from the theoretical result by between 2.2
and 2.7 standard deviations (σ). Thus, there is a glimpse of new physics and a
more precise experiment started to be elaborated at Fermilab to test to a higher
accuracy this phenomenon. This experiment will be described in detail in the next
section.

4.2 The Fermilab experiment

The Fermilab experiment (E-989) is an ongoing experiment to the date this
work is written which is taking place in Illinois, USA. It is a collaboration of many
scientists with measurements up to 0.46 · 10−9 precision and with still some data
to be released. To understand how this measure is made a brief description of the
experiment and its technicalities will be provided.

To measure the magnetic moment it is used the fact that a particle with a
magnetic moment inside a magnetic field precesses with a certain frequency. In
the case of polarized particles perpendicular to the magnetic field, this frequency



Chapter 4. Muon g− 2 experiments 51

is proportional to its magnetic moment times the magnetic field, as shown in:

~wL = ~µ× ~B ⊥
= g

(Q f e
2m

)
~B, (4.2)

where Q f is again ±1 depending on the charge of the fermion, and wL is called
the Larmor frequency. Looking at this formula, it becomes clear the importance of
a very uniform magnetic field for measurement accuracy: maintaining a uniform
magnetic field keeps the frequency of precession constant. The magnet used in the
Fermilab experiment is the same as the earlier experiment BNL E-821 since it has
an extremely uniform magnetic field.

This precession frequency (Larmor frequency) can be measured either by
observing electromagnetic radiation emitted or absorbed at the Larmor frequency,
or by directly observing through a characteristic of the parity violation decay of
the muon that will be discussed in section 4.2.1. This latest is the technique used
in the Fermilab experiment.

To create a perpendicular polarized beam of muons there are lots of steps.
First high energy positrons are created and collided with a target in the particle
accelerator at Fermilab, producing many different elementary particles. Among
them the charged pions (π±) are selected, using their specific mass and charge, to
enter in the pion decay chamber, a tunnel that allows pions to decay during its
course.

Pions decay into muon and neutrinos with a 99.99% branching ratio [13], and
this is a 100% parity violation weak decay (illustrated in Eq. 4.3). Therefore
the forward and backward direction of this decay can be selected to create an
extremely polarized beam of muons.

π+

{
W+

d̄

u

νµ

µ+

⇒ π+ → µ+ + ν̄µ;

π−
{

W−

d

ū

µ−

ν̄µ

⇒ π− → µ− + νµ.

(4.3)

This polarized beam of muons is then inserted into the storage ring. To enter
the ring it passes through what is called a superconducting inflector magnet [66]
that cancels the magnetic field of the ring at the entering point to facilitate the



Chapter 4. Muon g− 2 experiments 52

adjustment of the beam inside the ring. To make sure that the inflector does not
intervene in the trajectory of the muon inside the ring, it is positioned slightly to
the external side.

Then, since the muon does not have the energy necessary to maintain its
trajectory inside the ring, it spirals inwards. After about a quarter of a turn a very
quick magnetic pulse, called kicker [67], is used to boost the muon to the exact
energy necessary to be kept inside the ring. This process is done with extreme
precision equipment, and all uncertainties are measured in detail, which were
thoroughly analysed by the Fermilab Muon g− 2 Collaboration [68].

Finally, with the muon stored inside the ring, it is possible to measure the
precession frequency. Since the muons of the experiment are highly energetic, the
actual precession frequency needs to be corrected due to relativistic effects by:

~wS =−
Q f e
m

[(
g
2
− 1 +

1
γ

)
~B−

(g
2
− 1
) γ

γ + 1

(
~β ·~B

)
~β

−
(

g
2
− γ

γ + 1

)
~β× ~E

]
.

(4.4)

where β is the velocity of the particle in natural units, γ = 1√
1−β2

is the Lorentz

factor, and ~E is the electric field. The inclusion of the correction relative to the
electric field is due to the Penning trap, technique implemented in the Fermilab
experiment as well.

We can disregard the term ~β ·~B because of the perpendicular polarization of
the muon beam. Now we need to subtract the rotation frequency (wC) due to the
movement of the charged particle in the magnetic field, which is

~wC = −
Q f e
m

[
~B
γ
− γ

γ2 − 1
~β× ~E

]
, (4.5)

resulting in the frequency called waµ written as:

~waµ = ~wS − ~wC = −
Q f e
m

[
aµ~B−

(
aµ −

1
γ2 − 1

)
~β× ~E

]
, (4.6)

where we used the substitution aµ = g/2− 1.
Looking at the term that multiplies ~β× ~E we can see that there will be a value



Chapter 4. Muon g− 2 experiments 53

of γ that will make it vanish, as shown below:

aµ −
1

γ2
m − 1

= 0 ; γm ' 29.3 ; Em ' 3.09GeV. (4.7)

These values are the so-called "magic" Lorentz factor and energy mentioned in
the last section, represented by γm and Em. We should note that some iterations
are needed to obtain this value since aµ is the factor that we want to measure with
precision.

Another advantage of using this "magic" gamma is the time dilation of the
muon lifetime. It goes from 2.2 µs in the muon reference frame, to the extended
64.4 µs in the laboratory reference frame. And since one full precession takes
around 4.4 µs to occur, this elongated lifetime gives enough duration for the muon
frequency to be calculated.

After these simplifications the final form of this frequency is simply:

~waµ = −aµ
q~B
m

, (4.8)

and this result is very useful since it gives directly the anomalous part of the
magnetic moment, after a simple division, justifying the choice of the convention
for the name (~waµ).

The final step is to detect this frequency, and this was done in Fermilab with the
use of 24 calorimeters evenly disposed along the internal part of the ring that can
sense the energy of the decay electron or positron above 50MeV. To understand
this step a more detailed look at the muon decay distribution will be shown in the
following subsection.

4.2.1 Muon decay

To better comprehend the muon g− 2 experiments it is important to understand
the decay modes of the muon. Primordially there is only one decay channel, the
weak decay, shown in Figure 4.1. This mode corresponds experimentally to more
than 99.99% of the muon decays [13]. The diagram exchanging the W± with the
Goldstone-boson (ϕ±) can be ignored since it will have a factor of memµ

M2
W
∼ 10−8 to

its contribution, hence we have only one diagram to evaluate.



Chapter 4. Muon g− 2 experiments 54

p

p3W±

p1

p2µ±

νe∓

e±

νµ±

Figure 4.1: Main decay mode for the muon

Again, we will use the Feynman gauge (ξ = 1). The evaluation of this diagram
using the Feynman rules from section 2.1.2 grants:

iMµ→eνν̄ = ū(p1)

[
−i

g√
2

γµPL

]
u(p)

[
−igµν

k2 −M2
W + iε

]
ū(p2)

[
−i

g√
2

γµPL

]
u(p3).

(4.9)

The first simplification is k2 << M2
W (since k2 < m2

µ << M2
W) and we can

ignore it. Knowing that we need to calculate the modulus square of this element
for the decay width, we can also use for the neutrinos and electrons the relations:

∑
s

us(p)ūs(p) = /p + m; ∑
s

vs(p)v̄s(p) = /p −m, (4.10)

since we are not interested in their spin configuration. For the muon, we want
to keep the information about the direction of the spin, so it is useful to use the
projection operators for spin and energy [69]:

us(p)ūs(p) = (/p + m)
1 + γ5/s

2
, (4.11)

where sµ = (0,~s) and~s is the spin direction is space. Ignoring the electron and
neutrino masses, we obtain:

|M|2 =
g4

4M4
W

Tr
[
(/p + m)

1 + γ5/s
2

γν
/p1γµPL

]
Tr
[
/p3γν/p2γµPL

]
=2

g4

M4
W
[(p−ms) · p3][p1 · p2].

(4.12)

In order to calculate the decay width, according to D. Bailin and A. Love [70],



Chapter 4. Muon g− 2 experiments 55

we use:

dΓ(µ→ eνν̄) =
1

(2π)5 |M|
2δ4(p− p1 − p2 − p3)

d3p1d3p2d3p3

2m2E12E22E3
. (4.13)

Since we are interested in the energy distribution of the electron, we can
integrate over the momentum of the neutrinos, using:

∫ d3p1

2E1

d3p3

2E3
δ(p− p1 − p2 − p3)p1,µ p3,ν =

π

24
[(p− p2)

2gµν + 2(p− p2)µ(p− p2)ν],

(4.14)

and writing the integral over the electron momentum in terms of the energy
d3p2 = E2

2dE2dΩ, we have:

dΓ(µ→ eνν̄) =

(
g

MW

)4 E2dE2dΩ
96(2π)4m

{
[(p · p2)−m(s · p2)]

[
p2 − 2(p · p2) + p2

2

]
+
[

p2 − (p · p2)−m(s · p) + m(s · p2)
] [

(p · p2)− p2
2

]}
.

(4.15)

Then, we can use θ to be the angle between the electron and the spin direction,
p2 · s = −E2 cos θ (since |s| = 1), p2

2 = p · s = 0 and pµ = (m, 0, 0, 0) to obtain:

dΓ(µ→ eνν̄) =

(
g

MW

)4 m2E2
2dE2dΩ

96(2π)4

[
3− 4

E2

m
+ (1− 4

E2

m
) cos θ

]
. (4.16)

To continue, we look at the Mandelstam variable [71] s2 = (p− p2)
2 and we

can see that we can choose a frame where p = p2 so that s2 = 0 = p2− 2pp2 cos κ +

p2
2 = m2 −mE2 and the highest possible energy is:

E2(max) =
m2

µ + m2
e

2mµ
≈

mµ

2
. (4.17)

So we can define x = 2E2
mµ

which is the proportion of the maximum energy, use
dΩ = 2πd cos θ and rewrite Eq. 4.16 as:

dΓ(µ→ eνν̄)

dx d cos θ
=

(
g

MW

)4 m5

768(2π)3 x2 [3− 2x + (1− 2x) cos θ] . (4.18)

Since we know that the energy threshold for the calorimeters used in the



Chapter 4. Muon g− 2 experiments 56

experiment is around 50MeV, which is approximately the maximum energy, we
can fix x → 1 and plot the distribution, obtaining Figure 4.2, where it is clear that
most of the high energy electrons will be found in the direction directly opposite
to the muon spin. Figure 4.3 shows this relation simplified for the case of the
positron and the electron.

Figure 4.2: Angular distribution of the resulting electron with maximum possible
energy (E2 =

mµ

2 ) at a muon decay

⇐

⇒

⇒ µ+
⇒

ν̄µ

νe

e+

(a)

⇒

⇐

⇐ µ− ⇐νµ

ν̄e

e−

(b)

Figure 4.3: Relation between the muon polarization and the maximum distribution
probability of the maximum energy decayed positron [electron] (a), [(b)], respec-
tively. The double arrow represents the spin direction, and the dashed arrow the
momentum direction.

4.2.2 Fermilab Results

As a summary of the topics discussed, Figure 4.4 shows the schematics of the
storage ring used in the experiment and illustrates many of the features exposed
in this section.

Now that all the tools to calculate the muon g− 2 have been laid the results can
be understood deeply. One of the graphical results obtained is shown in Figure
4.5 that demonstrates the behaviour of the oscillation of the number of electrons
(or positrons) absorbed in the calorimeters above the threshold energy (50 MeV).



Chapter 4. Muon g− 2 experiments 57

Figure 4.4: Fermilab‘s muon g− 2 experiment schematics. In the image can be
seen the location of quadrupoles for the Penning Trap, the calorimeters (detectors),
the fast muon kickers, and the inflector. Also shown are the Tracking stations that
track where the bean is located relative to the center of the ring optimal trajectory
and the collimators that help correct this beam trajectory. Source: Fermilab Muon
g− 2 Collaboration [68].

Figure 4.5: Electron decay energy 102.5 µs after injection of muon beam into the
storage ring in Fermilab’s muon g− 2 experiment. Source: Fermilab Muon g− 2
Collaboration [6]

Using only this figure we can estimate the value of the muon g− 2: there are
approximately 645.0µs of data and 148 oscillations which results in a period of
Tµ = 4.36µs for each precession. The frequency is then ωµ = 2π

Tµ
= 1.44 · 106s−1.



Chapter 4. Muon g− 2 experiments 58

Using this result in formula 4.8 with the Magnetic field used in Fermilab of
~B = 1.45T (T = Kg · C−1 · s−1), a charge of −1.6 · 10−19C and a muon mass of
105.7MeV ∼= 188 · 10−30Kg, we obtain:

aestimate
µ =

(1.44 · 106 s−1)(188 · 10−30 kg)
(1.6 · 10−19 C)(1.45 Kg · C−1 · s−1)

≈ 1.16 · 10−3, (4.19)

which gives the expected Schwinger computation. A more detailed and accurate
measurement of these factors mentioned was done, giving the final result of:

aFNAL
µ = (116 592 040 ± 54) · 10−11, (4.20)

which agreed with the previous BNL E-821 experiment, differing from the the-
oretical result by 3.3 σ. The combined results from both of them give a value
of:

aEXP
µ = (116 592 061 ± 41) · 10−11, (4.21)

which gives a deviation of:

∆(aµ) = aEXP
µ − aSM

µ = (251± 59) · 10−11. (4.22)

These values can be found in the Fermilab Muon g− 2 Collaboration result report
[6].

Such a result is very significant because it deviates from the theoretical calcu-
lation by 4.2 σ, as can be seen in Figure 4.6. There is a consensus that around 5 σ

deviation is enough statistical confirmation to mark the result as an issue, or as a
real indication of new physics.

The experiment E-989 at Fermilab is still running. So far it only used around 6%
of its full statistical data [72]. Therefore a more complete result from the following
runs of the experiment will give a more precise measurement and could increase
the confidence that new physics is needed.



Chapter 4. Muon g− 2 experiments 59

Figure 4.6: Comparison between experimental and theoretical value of aµ. In the
image, the results from the last two experiments (BNL E-821 and FNAL E-989) are
shown, along with the combined result of both, the standard model prediction
and the discrepancy between them. The results from the BMW collaboration and
its combination with the white paper result are also shown for the demonstration
of the theoretical tension.



Chapter 5

Beyond the SM

The SM is a very successful theory with, as stated before, many precise predic-
tions. Presently, many physicists focus on discovering what lies beyond the SM. In
Section 2.4 we explored a few issues with the SM that indicates the need for new
physics in order to explain them.

New physics can be "bottom-up", meaning we add the new physics to explain
an issue and then see the implication of them, or "top-down", meaning we depar-
ture from a more fundamental theory or postulates, like supersymmetry or grand
unification theories, and attempt to find the correct values for the free parameters
when comparing with the experiment. There are many possibilities when making
extensions to the SM. For example, the addition of new types of particles, the
extension of the gauge group to higher scales, among others. Mostly used is the
addition or extension of gauge groups. This also means new free parameters and
new particles that have to obey the known experimental limits.

As explained along with the issues with the SM, additions to the theory can
cause a conflict favouring high scales to solve some of them (e.g. baryogenesis,
dark energy) and low scales to not to cause others (e.g. Higgs hierarchy, flavour
changing). This shows that whatever the new theory is, if we do not want one
more fine-tuning problem, it will have constraints coming from different angles.

One thing many of the new physic theories have in common is the extension
of the number of particles of the SM with new interactions and masses. In the
literature many such cases can be found, for example, axions [73] that add only one
particle in a singular manner, or Supersymmetric models like the MSSM (Minimal
Supersymmetric Standard Model)