Coupling constants of the Higgs boson

Anouk Geenen

10143858

July 7, 2015

Bachelorproject

Size 15 EC, conducted between March 30, 2015 and July 7, 2015.

Supervisor: prof. dr. E. L. M. P. Laenen

Second advisor: dr. W. J. Waalewijn

Institute for Theoretical PhysicsFaculteit der Natuurwetenschappen, Wiskunde en Informatica

Universiteit van Amsterdam

Abstract

A Higgslike particle with a mass of 125 GeV was discovered at CERN in 2012. Toensure this particle really is the Standard Model version of the Higgs boson, more of itsproperties need to be tested, the most crucial being its coupling constants. This thesiswill review the theory of the Higgs mechanism and its couplings, followed by how thesetheoretical predictions can be checked with experimental values. For this, the Higgsproduction from two gluons and Higgs decay to two photons will be closely examined.These processes both contain a loop of virtual particles and will be reviewed using themethod of Passarino-Veltman and dimensional regularization. Some calculations willbe performed in FORM. The latest results from CMS show that the particle which wasdiscovered in 2012 indeed behaves like the expected Standard Model Higgs.

Title: Coupling constants of the Higgs bosonAuthor: Anouk Geenen10143858, ajpgeenen@gmail.com, 10143858Supervisor: prof. dr. E. L. M. P. LaenenSecond advisor: dr. W. J. WaalewijnDate: July 7, 2015

Institute for Theoretical PhysicsUniversiteit van AmsterdamScience Park 904, 1098 XH Amsterdamhttp://iop.uva.nl/divisions/itfa

Populaire samenvatting

In 2012 kopte elke krant met het grote nieuws: het Higgs-deeltje was gevonden! Al jarenzijn natuurkundigen op zoek naar de meest elementaire bouwstenen van de natuur: watzijn de allerkleinste puzzelstukjes waaruit de wereld is opgebouwd? Het Higgs-deeltje -ook wel bekend als het God-deeltje - was het laatste ontbrekende stukje in deze puzzelder materie. De theorie die deze puzzel beschrijft heet het Standaard Model van ele-mentaire deeltjes. Het Standaard Model bestaat uit materiedeeltjes en krachtdeeltjes,die de wereld om ons heen beschrijven. De materiedeeltjes worden ook wel fermionengenoemd, en daarvan hebben we er twaalf in ons huidige model. De krachtdeeltjes gaanook door als bosonen, en hiervan kennen we er zes. Elk boson kan gekoppeld wordenaan een kracht. Zo hoort het foton bij elektromagnetisme, het gluon bij de sterke kern-kracht en wordt de zwakke kernkracht beschreven door drie bosonen: W+,W− and Z0.De oplettende lezer ziet dat we nu pas vijf bosonen hebben genoemd. De laatste ishet beroemde Higgs-deeltje of Higgs-boson. Dit deeltje representeert geen kracht, maarrepresenteert het mechanisme waarmee alle andere deeltjes massa krijgen. Zonder ditHiggsdeeltje valt de hele theorie uit elkaar.

Geen wonder dus, dat sinds de formulering ervan in 1964, de jacht op het Higgs-deeltje islosgebarsten. Experimenten ter waarde van miljarden euros zijn opgezet en wereldwijdhouden wetenschappers zich ermee bezig. De zoektocht naar het Higgs-deeltje vindtplaats in deeltjesversnellers: hierin worden deeltjes zoals bijvoorbeeld protonen metzulke hoge energie op elkaar geknald, dat er nieuwe deeltjes kunnen ontstaan. De LHC(Large Hadron Collider) in CERN in Zwisterland is hier een voorbeeld van, en werdgebouwd met het doel om het Higgs te vinden. En dat is gelukt. Althans, waarschijn-lijk. Hoewel het deeltje dat in 2012 ontdekt is dezelfde massa heeft als het theoretischvoorspelde Higgs-deeltje, kunnen we niet zomaar zeggen dat het ook echt om dit deel-tje gaat. Het zou ook zomaar een ander exotisch deeltje kunnen zijn dat we nog nietkennen! Gezien het belang van Higgs-deeltje is enige voorzichtigheid geboden. Daaromis het zaak de verschillende eigenschappen van het deeltje te checken. Een van dezeeigenschappen is de kracht waarmee het Higgs-deeltje koppelt aan andere deeltjes. Ditwordt ook wel de koppelingsconstante genoemd.

In deze scriptie wordt onderzocht hoe we theoretische voorspellingen van de koppelings-constante kunnen maken en hoe deze vervolgens door experimenten gecheckt kunnenworden. Dit wordt gedaan door eerst het Higgsmechanisme verder uit te zoeken: opwelke manier geeft dat deeltje nou eigenlijk massa aan alle andere deeltjes? Als dit dui-delijk is worden de processen die plaatsvinden in de LHC nauw onder de loep genomen:op welke wijze kunnen we voorspelde waarden voor de koppelingsconstantes vergelijkenmet de experimenteel gevonden waarden?

De LHC is sinds juni weer in werking en zal het verlossende antwoord moeten geven:hebben we het Higgs gevonden of staan we aan het begin van een tijdperk vol nieuwenatuurkunde? Hoe dan ook gaat de deeltjesfysica spannende tijden tegemoet!

Acknowledgements

I would like to take the opportunity to thank some people without whom this thesiswould not have come about. First, my supervisor Eric Laenen: I very much appreciatethe time and effort taken in helping me write this thesis, without spelling everythingout allowing me to work independently. I would also like to thank Wouter Waalewijnfor being my second advisor. Finally, a big thank you to my roommates and neighboursat Nikhef: your words of advice and continuing interest in my project were very muchwelcomed and enjoyed.

Contents

Populaire samenvatting

Acknowledgements

Introduction 2

1. The Standard Model 41.1. Quantum Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2. Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3. Gauge Bosons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2. Gauge Fields 62.1. Abelian case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.1. Global vs Local gauge invariance . . . . . . . . . . . . . . . . . . 62.2. Non-Abelian case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3. Higgs mechanism 113.1. Spontaneous Symmetry Breaking . . . . . . . . . . . . . . . . . . . . . . 113.2. Abelian case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.2.1. Unitary gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.3. Non-Abelian case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4. Electroweak theory 164.1. SU(2)L × U(1)Y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.1.1. Higgs mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.2. Particle masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.3. Higgs coupling constants . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.4. Chiral fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

5. Higgs production and decay 205.1. Higgs production gg → H . . . . . . . . . . . . . . . . . . . . . . . . . . 21

5.1.1. Passarino-Veltman reduction . . . . . . . . . . . . . . . . . . . . 225.2. Dimensional Regularization . . . . . . . . . . . . . . . . . . . . . . . . . 24

5.2.1. Determining B0(2, 3) . . . . . . . . . . . . . . . . . . . . . . . . . 255.2.2. Determining C0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

5.3. Higgs decay H → γγ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.3.1. Higgs decay H → γγ through a top-quark loop . . . . . . . . . . 275.3.2. Higgs decay H → γγ through a W -boson loop . . . . . . . . . . 28

5.4. FORM File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.5. Finalizing the calculation . . . . . . . . . . . . . . . . . . . . . . . . . . 30

6. The experiments 326.1. Higgs search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

6.2. Verifying the coupling constants . . . . . . . . . . . . . . . . . . . . . . 32

Conclusion 35

Bibliography 36

Appendices 40

A. Lagrange formalism 41A.1. Classical case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41A.2. Relativistic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

A.2.1. Klein-Gordon Lagrangian for scalar (spin-0) field . . . . . . . . . 41A.2.2. Dirac Lagrangian for spinor (spin-1/2) field . . . . . . . . . . . . 41A.2.3. Proca Lagrangian for vector (spin-1) field . . . . . . . . . . . . . 42

B. FORM script 43B.1. Computation of the matrix element . . . . . . . . . . . . . . . . . . . . . 43

B.1.1. Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43B.1.2. Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

B.2. Applications Passarino-Veltman reduction . . . . . . . . . . . . . . . . . 47B.2.1. Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47B.2.2. Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

Introduction

The curious nature of humans towards the world around us has resolved in a search forthe smallest building blocks of matter. This question of the smallest indivisible piecescan be traced back all the way to Greek philosopher Democritus and his Atomism.Thousands of years later, this question still troubles our minds and dominates the fieldof particle physics. It has led to the development of the Standard Model of particlephysics (SM), which describes all matter-interactions that we know. This elegant modelwas developed throughout the latter half of the 20th century by a combination of theo-retical insights and experimental results.

The SM is without doubt one of the great accomplishments of modern physics. Itgives an overview of the twelve fundamental fermions and three of the four fundamentalforces that we know.1 The SM in its modern form was developed by Glashow (1961),Weinberg (1967) and Salam (1969). One of the strengths of the SM was that it predictedparticles that had not been observed before. One of those particles is the Higgs boson.This key element of the SM is responsible for the mass of the other particles postulatedby the theory. Years have gone by since its theoretical formulation by Higgs (1964)and Brout & Englert (1964), and billion-euro experiments have been built and run,in order to find this microscopic particle of macroscopic importance. In 2012, physicsreached headlines. Forty years after the formulation of the SM and twelve Nobel prizesfor contributions to this field later, all predictions had come true: the Higgs boson wasdiscovered (Aad et al. 2012, Chatrchyan et al. 2012).Or at least, a particle of similar mass as the theoretically predicted one had been found.Although the SM has been successful in describing nature throughout many experi-ments, it also has some flaws. It does not include gravity for example, nor dark energyand dark matter. Theoretical issues concern the hierarchy problem and charge-parityviolation in combination with baryogenesis. Over the years, other theories have beendeveloped, termed as ‘Physics Beyond the Standard Model’. These theories also makeuse of a Higgs mechanism, but different in some aspects from the original one developedin 1964.

Physicists are on a hunt for the real description of nature, and therefore it is importantto know which of the theories are true. In order for that, more has to be known aboutthe particle that was discovered in 2012. Is it really the Higgs, and if so, what kindof Higgs? Or is it some other exotic particle that we have never heard of before? Toanswer these questions, more tests have to be run and more experimental results haveto be checked against their theoretical predictions. The couplings of the Higgs to otherparticles are generally regarded as the most crucial observables. The Large Hadron

1There are four fundamental forces at work in the universe: the gravitational force, the electromagneticforce, the strong force and the weak force. The latter three are described by the SM. The fit of gravity,which is described by General Relativity, into the SM, which is described by the quantum world, isstill an open question today. Luckily, for the field of particle physics, this is of little concern since atthe microscopic scale of particles, gravity becomes negligible and the SM predictions still hold.

2

Collider (LHC) started its second run with higher levels of energy in early June thisyear and the hopes are high. Expectations are that new physics will be revealed, whichup till now, has been hidden in the higher order quantum corrections. These next toleading order processes are described by Feynman diagrams containing loops and turnout to be of great importance in the further development of the SM.

The aim of this thesis is to calculate some of these loop diagrams associated with theHiggs production and decay. Also, we want to understand how theory and experimentregarding the Higgs-boson can be related. We will do this by starting with a theoreticaloverview of the electroweak part of the SM. Then, the process of calculating loop dia-grams will be explored, which is relevant for two reasons: the processes at LHC containloop diagrams and loop diagrams are important for future high-precision calculationsand predictions. After all, it is most likely that new physics is hidden in quantum cor-rections.In chapter one, an introduction to the SM will be given; how did it come to be, what par-ticles and forces are incorporated in the theory and which language is used for describingthis? Chapter two will elaborate further on gauge theories, the theoretical basics of theSM. What is local gauge invariance and why is it such an important concept in particlephysics? Chapter three will focus on spontaneous symmetry breaking and the Higgsmechanism. How does mass arise in our theory? Chapter four will put the theoreticalfoundations from the previous chapters together and describe the electroweak sector asit is formulated in the current SM. The coupling constants of the Higgs to other particleswill be given, which are important predictions with which we can test the theory. Next,chapter five will focus on the Feynman diagrams concerning the production and decayof the Higgs boson. These diagrams contain a loop and the method for evaluating theseloop diagrams will be discussed in detail. One of the calculations will be done usingthe program FORM. Chapter six will briefly discuss the experimental results concerningthe coupling constants of the Higgs boson. Finally, two appendices are added: the firstcontains a short review of the basics of the Lagrange formalism and the second containsthe calculations performed in FORM.

3

1. The Standard Model

The SM describes our twelve fundamental fermions and three fundamental forces. Theseforces are described by the exchange of particles called gauge bosons. A special kind ofgauge boson is the Higgs boson, which gives rise to the masses of particles in the SM.

The electroweak part of the SM was formulated by Glashow (1961), Weinberg (1967)and Salam (1969). The formulation of the Higgs mechanism was added by Brout &Englert (1964) and Higgs (1964). The theory was not complete however, as it yieldedunwanted infinities. ’t Hooft and Veltman delivered the proof for the renormalizationof the theory, which removed those infinities (’t Hooft & Veltman 1972, ’t Hooft 1971).The theory of strong interaction was finalized in the 1970’s, when experiments confirmedthe existence of quarks and Quantum Chromo Dynamics (QCD) was added to the mix.The major success of the SM is its successful prediction of various experimental results.In this chapter, the basics of the SM will be treated. First, a closer look at QuantumField Theory (QFT) will be given, the mathematical language of the SM. Then, theelementary matter particles from the SM will be discussed, which are all fermions.Next, the fundamental forces and corresponding gauge bosons belonging to the SM willbe discussed.

1.1. Quantum Field Theory

In quantum mechanics, particles are described by wavefunctions that satisfy the ap-propriate wave equation. In QFT, particles are described as excitations of a quantumfield that satisfies the appropriate quantum mechanical field equations. The dynamicsof QFT can be expressed in terms of the Lagrangian density (shortened from now toLagrangian). Some of the most common Lagrangians are given in Appendix A. Fromthe Lagrangians, the Feynman rules can be derived, which are used to construct Feyn-man diagrams. The Feynman diagrams depict transitions between particles, governedby the exchange of force-carrying gauge bosons. A Lagrangian can be divided into akinetic and a potential term. The kinetic term involves the derivatives of fields and rep-resents the propagators, where the potential term is expressed in the fields themselvesand represents the interaction terms, or the vertices in the Feynman diagrams.

One important aspect of QFT is symmetry. The SM is constructed from Lagrangiansthat obey gauge symmetry. In mathematical language, the SM is a non-abelian gaugeQFT containing the symmetries of the unitary product group SU(3)C×SU(2)L×U(1)Y .What exactly this entails will become more clear in the later chapters on gauge theoriesand the electroweak theory. Roughly, each of these three gauge symmetries give rise toone of the three fundamental forces.

1.2. Fermions

The matter particles we know are spin-12 particles, or fermions. They behave according

to the Pauli exclusion principle and obey the Dirac equation (Appendix A). From

4

this equation, the phenomenon of antiparticles arises. Each of the twelve fundamentalfermions has its own corresponding antiparticle; a particle with the same mass butopposite charge.

There are two kinds of matter particles: leptons and quarks. Each of these groupsconsists of six particles, which are related in pairs, or ‘generations’. The six leptonsconsist of three electrically charged particles (the electron, muon and tauon) and threecorresponding neutrinos, which are electrically neutral. The quarks carry color chargeand appear in the pairs (up, down), (charm, strange) and (top, bottom). Pairs ina generation exhibit similar interaction behavior. The first generation contains thelightest particles, whereas the second and third generation are made up of the heavierand less stable particles. Most of the matter in the universe consists of particles thatbelong to the first generation, simply because the others are too heavy and decay to thenext stable level.

1.3. Gauge Bosons

Each of the three forces in the SM can be described by the exchange of a spin-1 force-carrying particle, known as a gauge boson. This boson mediates the force, which effec-tively comes down to a transfer in momentum. The exchanged particle is virtual andcannot be observed.

Electromagnetism is described by Quantum Electrodynamics (QED), where it is thephoton that mediates the interaction between charged particles. QED works on chargedparticles, so only the neutrinos are excluded from this interaction. The photon is mass-less and therefore EM has infinite range.

The weak force is mediated by three kinds of bosons: two charged W± bosons andone neutral Z-boson. All three force particles have mass and thus the weak force worksonly on small distances.1 The weak force works on all twelve fermions, since the chargeof weak interaction is the isospin and every fundamental particle has non-zero isospin.

Finally, the strong interaction, which is described by Quantum Chromo Dynamics(QCD) and has the massless gluon as its force carrier. The gluon mediates interactionsbetween color charged particles, so only the six quarks take part in this interaction.There are eight types of gluons. They carry color charge themselves as well, which fun-damentally changes the interaction.2

Then there is still one very special gauge boson left, the scalar Higgs boson. It differsfrom the others since it has spin-0. Moreover, it has a very high mass. Via interactionwith the Higgs field, particles gain mass. The Higgs mechanism is subtle and will beelaborated further in Chapters 3 and 4.

1The fact that massive particles have a smaller range, is easily obtained from Heisenbergs uncertaintyrelations ∆x∆p ∼ ~, ∆E∆t ∼ ~. Combining these with ∆x = c∆t and ∆E = mc2, leaves us withthe relation ∆x = ~

mcand we see that the distance x and the mass m are inversely proportional.

2In the rest of this thesis, the QCD or SU(3) part of the SM will be ignored, since we are only interestedin the electroweak or SU(2)×U(1) sector of the SM for this topic.

5

2. Gauge Fields

This chapter explores the phenomenon of gauge fields, which underlie all elementaryparticle interactions. A gauge field is a field that can be added to the theory withoutchanging its physical outcomes. If such a field exists, the theory is called gauge invariant.There are two types of gauge invariance: global and local, where the latter is morestringent. The idea of local gauge invariance goes back to the work of Weyl (1919). In theSM, all of the fundamental interactions are generated by a local gauge invariance.1 QEDis generated by an Abelian gauge symmetry, with U(1) as its gauge group. The weakinteraction and QCD are obtained by extending the local gauge principle to respectivelySU(2) and SU(3) gauge groups; these are non-Abelian cases.

The concept of gauge fields will be explored for two different cases; the Abelian andnon-Abelian case, starting with the first. The following is based on the books by de Witet al. (2015, Chapter 7,8), Quigg (2013, Chapter 4) and Thomson (2013, Chapter 15,17). Throughout this work we use the natural units, meaning ~ = c = 1, and theconvention gµν = gµν = diag(1,−1,−1,−1).

2.1. Abelian case

For the Abelian case we consider a U(1) gauge invariance. This means that all gaugetransformations belong to the U(1) gauge group, consisting of 1×1 matrices (or rather,scalars). Abelian means that the generators of the group commute with each other.

2.1.1. Global vs Local gauge invariance

We start with the Dirac Lagrangian for spin 1/2 particles:

L = iψγµ∂µψ −mψψ. (2.1)

We apply a transformation of the form

ψ′(x) = Uψ(x), (2.2)

U being an element of the U(1) gauge group. For U = eiθ, we note that the Lagrangianremains invariant. This is called a global phase transformation, since it gives rise to aphase change that is the same everywhere, on every point in space-time.

Next, a local phase transformation is implemented. This means that the phase trans-formation is dependent of xµ and differs on each point in space-time. We thus haveU = eiθ(x), or written out fully:

ψ(x)→ ψ′(x) = eiqθ(x)ψ(x). (2.3)

1A nice introductory paper on this is written by Nobelprize winner Gerard ’t Hooft (1980), where hepassionately explains how the world around us can be described by gauge theories and how thosetheories came to be.

6

Note that the parameter q has been added to measure the strength of the phase trans-formation. When inserting ψ′ into (2.1), the derivatives now also act on the local phaseθ(x) and render an extra term in the Lagrangian:

∂µ(eiqθ(x)ψ) = iq(∂µ(θ(x)))eiqθ(x)ψ + eiqθ(x)∂µψ, (2.4)

L → L′ = L − qψγµ(∂µθ(x))ψ. (2.5)

Thus, the Dirac Lagrangian is not invariant under U(1) local phase transformations.To make it locally invariant, the extra term has to be absorbed somehow. This can beachieved by changing the derivative ∂µ into the covariant derivative Dµ:

∂µ → Dµ = ∂µ + iqAµ. (2.6)

Simultaneously, a new field Aµ is introduced. In order to cancel the extra term in (2.5),this field has to obey the following transformation:

Aµ → A′µ = Aµ − ∂µθ(x). (2.7)

When combining the transformations for ψ and Aµ, the extra terms indeed render thecovariant derivative and we’ve obtained the behaviour that we wanted:

(Dµψ)′ = eiqθ(x)Dµψ (2.8)

Putting all this together, the locally gauge-invariant Lagrangian for a spin-half particlebecomes:

L = iψγµ∂µψ −mψψ − qψγµAµψ. (2.9)

The extra term describes an interaction between the fermion and the new field Aµ.We have identified a new interaction! But we are not there yet: a Lagrangian alwaysconsists of a kinetic and potential term, and for the new gauge field Aµ we have yet toadd a kinetic term. Realizing that Aµ is a vector field, we turn to the Proca Lagrangian(Appendix A) which describes vector fields:

L = −1

4FµνFµν +

1

2m2AµAµ, (2.10)

with Fµν = ∂µAν − ∂νAµ. (2.11)

Before adding these terms to equation (2.9), we need to check whether they are as wellinvariant under a local transformation. This turns out to be problematic. The Fµν-termis invariant, but the massterm is not. We conclude that the gauge field must be massless(mA = 0), otherwise local gauge invariance will be lost. This lack of mass for the gaugefield will become more important later on, as we will see.

Our final Lagrangian thus becomes

L = iψγµ∂µψ −mψψ − qψγµAµψ −1

4FµνFµν . (2.12)

Note closely how we have added an extra field to the Lagrangian, in order to safeguardlocal invariance. This gauge field interacts with the fermion field, and thereby givesrise to a new interaction. We have added a force to the theory by demanding localinvariance! This remarkable effect of local gauge symmetries is where the SM is builtupon. Looking back at (2.12), we can see that this force is the electromagnetic force:

7

by putting in the electron charge q = −e and identifying Aµ as the photon field, werecognize the QED Lagrangian. We have thus just formulated the Lagrangian for theelectromagnetic force, simply by making the Dirac Lagrangian locally invariant.

As shown, the difference between global and local gauge transformations arises whencalculating the derivatives of the fields. A local transformations picks up an extra termin the Lagrangian, which can be canceled by the introduction of a covariant derivative.The substitution of Dµ for ∂µ is thus required for converting a global symmetry into alocal one. This is known as the minimal coupling rule. The covariant derivative intro-duces a new vector field however. This gauge field and its corresponding gauge bosonhave to be massless in order to maintain local symmetry. This is an important featureof gauge theories: the introduction of a local invariance is always accompanied by theintroduction of a massless vector field.

2.2. Non-Abelian case

Yang & Mills (1954) considered gauge invariance for the non-Abelian SU(2) group, againinsisting that global invariance holds locally. Non-Abelian means that the generators ofthe group do not commute. This leads to extra terms in the Lagrangian which describegauge boson self-interactions, as we will see later.

We again start with the Dirac Lagrangian from (2.1). Note that since we are dealingwith SU(2) gauge groups, which are represented by 2×2 matrices, ψ has to have two

components. We call this the isospin doublet and it can be expressed as: ψ =( ψpψn

).2

For the non-Abelian case we also find that the Lagrangian is invariant under globaltransformations and we move on to the local transformation. The transformation in(2.2) is now given by:

U(x) = exp[igα(x) ·T]. (2.13)

The parameter g measures the strength of the transformation, where α and T arecalled the parameters and generators of the group, respectively, and both consist ofthree elements. T is also known as the weak isospin and can be written in the followingway

Ti =1

2σi, (i = 1, 2, 3) (2.14)

with σi being the Pauli matrices:

σ1 =

(0 11 0

), σ2 =

(0 −ii 0

), σ3 =

(1 00 −1

). (2.15)

From the previous section we have learned that the recipe for achieving local gaugeinvariance is to replace the derivative ∂µ with the covariant derivative Dµ. The latter

2This notation resembles the fact that such doublets were first applied by Yang & Mills (1954) in thecase of protons and neutrons. These particles can be regarded as an isospin doublet, since they havealmost identical masses and play a similar role in the strong interaction. Yang and Mills tried thisapproach to describe the strong interaction, which later turned out to be described by a SU(3) gaugetheory.

8

has to be defined in terms of the generators of the group, which becomes more apparentfor SU(2) group:

∂µ → Dµ = ∂µ + igW iµTi

= ∂µ +1

2igW i

µσi

= ∂µ +1

2ig

(W 3µ W 1

µ − iW 2µ

W 1µ + iW 2

µ −W 3µ

).

(2.16)

Note that there is an implicit I =(

1 00 1

)in front of the ∂µ. Because SU(2) has three

generators T, the change of derivative introduces three new gauge fields W iµ, i = 1, 2, 3.

These new fields also are introduced to cancel the extra term in the Lagrangian com-ing from the derivative, just as in the Abelian case, and therefore have the followingtransformation property:

W 3µ →W

′3µ = W 3

µ − ∂µα3 − gε123α1W 2µ , (2.17)

where the last term is needed because the generators do not commute: [T1, T2] =−ε123T3. Again we have obtained the exact behaviour that we wanted, namely

(Dµψ)′ = eigα(x)·TDµψ (2.18)

We end up with the new Lagrangian:

L = iψγµ∂µψ −mψψ −1

2igW i

µσiψγµψ. (2.19)

The new gauge fields also need a kinetic term in the Lagrangian. Looking back at theAbelian case and simply translating this to Gµν = ∂µWν − ∂νWµ does not give us aninvariant term, again, because of the non-commuting generators. We can use the F -termas an example however, by noting that it can also be written as

Fµν =1

iq[Dµ, Dν ]

=1

iq[(∂µ + iqAµ), (∂ν + iqAν)]

=1

iq

(

:0[(∂µ, ∂ν ] + [∂µ, iqAν ] + [iqAµ, ∂ν ] + [iqAµ, iqAν ]

)= ∂µAν − ∂νAµ + iq[Aµ, Aν ].

(2.20)

The last term vanishes in the Abelian case. Applying the same strategy in SU(2) gives:

G1µν =

1

ig[D1

µ, D1ν ]

= ∂µW1ν − ∂νW 1

µ − igε123W2µW

3ν .

(2.21)

Note that Gµν is decomposed in terms of the group generators: Gµν = GaµνTa and againthe fact is used that the generators do not commute. The total expression for Gµνbecomes

Gµν = ∂µWν − ∂νWµ + g[Wµ,Wν ]. (2.22)

9

We then have our final Lagrangian:

L = iψγµ∂µψ −mψψ −1

2igW i

µσiψγµψ − 1

4GµνGµν . (2.23)

The extra term in the Gµν-term is needed for the kinetic term to be invariant and givesrise to additional gauge boson self-interactions. As U(1) local symmetry worked well todescribe the electromagnetic force, it seems logical to try SU(2) local symmetry as adescription for the weak force. There is one problem however: we learned that the gaugebosons associated to the gauge fields have to be massless, in order to keep invariance.The weak bosons W and Z are known to have masses of respectively 80 and 90 GeV,not even close to negligible. Something new is needed to generate masses in a symmetricLagrangian.

10

3. Higgs mechanism

The way to insert mass into a gauge theory is by the phenomenon of spontaneous sym-metry breaking (SSB). This sounds strange at first: we just demanded a symmetry andnow we are already breaking it? The fact is that the local symmetry has to be brokenin order to gain a mass-term, but this happens in a subtle way. Explicit breaking ofthe symmetry by simply adding mass-terms to the Lagrangian ruins the invariance andalso renormalizability of the theory and thus something more sophisticated is needed.Anderson (1963) was the first to suggest the method of SSB and applied it to super-conductivity. In the current formulation of the SM, this procedure is described by theHiggs mechanism, developed by Higgs (1964) and Brout & Englert (1964).

In this chapter we will look how mass can arise from a spontaneously broken symmetry.Then we will explore the Higgs mechanism: first in the Abelian case, and then in thenon-Abelian case. This is done using the books of de Wit et al. (2015, Chapter 18, 19),Quigg (2013, Chapter 5) and Thomson (2013, Chapter 17).

3.1. Spontaneous Symmetry Breaking

Although the Lagrangian shows an apparent gauge symmetry, it admits ground statesthat are not invariant. The phenomenon of SSB describes a Lagrangian that has acertain symmetry, but lowest-energy solutions that do not obey this symmetry. Thesymmetry is broken when a certain ground state is chosen from all possible degenerateones as physical ground state or vacuum. This happens spontaneously as there is nopreferred choice of ground state. The choice of the ground state hides the originalsymmetry of the theory and its accompanying dynamics.1 When we consider a realscalar field φ, we have a discrete symmetry with a finite number of ground states. It ismore interesting however to look at continuous symmetries, which occur when dealingwith complex scalar fields. When SSB happens for a continuous symmetry, masslessspinless particles pop up, called Goldstone bosons. For each generator of the brokensymmetry group, one Goldstone boson will arise. This is called Goldstone’s theorem(Goldstone 1961). This is the opposite effect of what we were looking for. Instead oflosing our massless gauge vector bosons, we gain extra massless spin-zero bosons. WhenSSB happens to a local symmetry and a smart choice of gauge is exploited however, anelegant interplay of these massless bosons results in massive gauge bosons and removesthe Goldstone bosons. This is called the Higgs mechanism.

1A good physical example of SSB is the ferromagnet. Below the Curie temperature TC and in absenceof an external magnetic field, the spins are randomly oriented. This is a rotationally invariant groundstate. However, the spin-spin interactions give rise to a spontaneous magnetization, by aligning thespins in the state of lowest energy. This ground state is not rotationally invariant. The direction ofthe magnetization is random, which translates to an infinitely degenerate ground state.

11

Figure 3.1.: The left plots shows the potential V as a function of φ1 and φ2 for the case µ2 > 0.There is just one minimum: φ = 0. The right plot shows the potential when µ2 < 0,and we see that there is a degenerate set of possible minima.

3.2. Abelian case

We consider a complex scalar field φ = 1√2(φ1 + iφ2) 2, for which the corresponding

locally invariant Lagrangian is:

L = −1

4FµνFµν + (Dµφ)†(Dµφ)− V (φ), (3.1)

with V (φ) = µ2(φ†φ) + λ(φ†φ)2. (3.2)

This is the Klein-Gordon Lagrangian for spin-0 particles, as described in appendix A.An extra term for the potential is added. This potential is known as the Higgs potential.Note that the covariant derivative is used and the rules from section 2.1 apply.

The SSB hides in the ground states, so we are looking for the minimal values of thepotential. We can obtain these by setting the derivative of V equal to zero: ∂V

∂φ = 0 In

order to have a minimum, λ has to be positive: λ > 0. For µ2, the following optionshold:

• µ2 > 0: We get a unique minimum at φ0 = 0,

• µ2 < 0: We get a continuous set of minima φ20 = φ2

1 + φ22 = −µ2

2λ = v2

2 .

This is depicted in figure 3.1. In the case µ2 < 0, the ground state does not occur atφ = 0 and the field is said to have an non-zero vacuum expectation value v. The vacuumstate is degenerate and we can choose any state as physical vacuum. The choice of thephysical vacuum state spontaneously breaks the symmetry of the Lagrangian. This iscalled spontaneous, as there is no preferred direction for the minimum: any choice ispossible without affecting the physical outcomes.

The physical vacuum state is chosen to be φ1 = v, φ2 = 0. To explore the physicalspectrum belonging to this vacuum state, we can consider the perturbations of φ around

2We switch from fermion fields ψ to scalar fields φ, as it turns out that fermions carry a certain chiralstate which makes their description in the Higgs mechanism more complicated. This will becomemore clear in chapter 4.

12

the vacuum state, which describe the excitations of the field. In order to do this, weshift the fields with η(x) in the φ1 direction and ξ(x) in the φ2 direction, as can be seenin figure 3.1. This gives us the following expression for φ in the minimum:

φ(x) =1√2

(v + η(x) + iξ(x)). (3.3)

Substituting this new φ into (3.1) we get:

L =

massive η field︷ ︸︸ ︷1

2(∂µη)(∂µη)− λv2η2 +

1

2(∂µξ)(∂

µξ)︸ ︷︷ ︸massless ξfield

−

massive gauge field︷ ︸︸ ︷1

4FµνFµν +

1

2g2v2AµA

µ−Vint. (3.4)

where Vint(η, ξ, A) contains the three- and four-point interaction terms of the fields.Notice that (3.1) and (3.4) are exactly the same, only (3.4) hides the underlying gaugesymmetry. Constant factors in front of terms quadratic in the fields can be associatedwith mass, by equalizing them to 1

2m2. We see that we have obtained a massive gauge

field Aµ, which is what we were looking for. However, in addition we have also picked

up a massive η field with mη =√

2λv2, and a massless Goldstone boson ξ, which wewere not looking for.3

3.2.1. Unitary gauge

As said in the previous section, the way to get rid of these unwanted extra masslessGoldstone bosons, is by a smart choice of gauge.4 This choice of gauge correspondsto taking θ(x) = −ξ(x)

qv in U = eiqθ(x). Remember that we were discussing a locallygauge invariant Lagrangian, so this choice of gauge does not alter the theory. Thegauge-transformation becomes:

φ(x)→ φ′(x) = e

−iqξ(x)qv φ(x) = e

−iξ(x)v φ(x), (3.5)

with the corresponding transformation for the gauge field:

Aµ(x)→ A′µ(x) = Aµ(x)− 1

gv∂µξ(x). (3.6)

By substituting the first order approximation of (3.3) in (3.5) we get

φ′(x) = e

−iξ(x)v (

1√2

(v + η(x))eiξ(x)/v) (3.7)

=1√2

(v + η(x)). (3.8)

3The excitations of η(x) are associated to the radial direction, where the potential is quadratic. Theexcitations of ξ(x) are in the angular direction, where the potential is constant, and therefore thisparticle remains massless.

4There is another physical reason for changing of gauge than simply wanting to lose the Goldstoneboson. When taking a closer look at the Lagrangian in (3.4), we notice that an extra degree of freedomhas appeared, compared to (3.1). The mass of the gauge field gives rise to an extra longitudinal degreeof freedom.

13

This choice of gauge is known as the Unitary Gauge. Putting it into our Lagrangian,we get

L =1

2(∂µη)(∂µη)− λv2η2 − 1

4FµνFµν +

1

2g2v2A

′µA′µ − Vint. (3.9)

The Goldstone bosons ξ have disappeared and only true physical fields are left.5 Thefield η(x) will later be identified as the Higgs field.

Another way of writing this is by starting with the parametrization φ = 1√2ρ(x)e

iκ(x)v .

The gauge transformation is then expressed by κ(x) → κ′(x) = κ(x) + qvξ(x). This

choice of gauge also changes our transformation of the gauge field, called B this time.We can express the Lagrangian in the new fields ρ and B, expand ρ about v, and ob-tain a mass for gauge field B. We are working in unitary gauge and therefore have noGoldstone bosons. The unitary gauge can also be achieved by setting θ(x) = 0, which

amounts to choosing φ(x) = ρ(x)√2

.

3.3. Non-Abelian case

Remember that in the non-Abelian case we are using SU(2) gauge groups, which consistof 2×2 matrices. We thus have a complex doublet of spinless fields this time: φ(x) =( φ1(x)φ2(x)

). The Lagrangian from (3.1) can be used, when exchanging Fµν for its non-

Abelian substitute Gµν and applying the transformation rules from section 2.2. Again,we have a unique minimum for µ2 > 0 and are interested in the case µ2 < 0, when a setof degenerate minima arises. We break the symmetry by choosing as physical minimum:

φ0 =1√2

(0v

). (3.10)

Retracing the steps from the Abelian case, we consider perturbations of φ(x):

φ(x) = exp

[i

v(ζi(x)T i

](0

v+η(x)√2

), (i = 1, 2, 3). (3.11)

Putting this into our Lagrangian will give three massless Goldstone bosons, one for eachgenerator of the symmetry group, as we know from to Goldstone’s theorem(Goldstone1961). In order to avoid this, we right-away take the unitary gauge, which is

U(x) = exp

[−iv

(ζi(x)T i)

]. (3.12)

Thus we have new transformed fields

φ(x)→ φ′(x) = exp

[−iv

(ζi(x)T i)

]φ(x) (3.13)

=1√2

(0

v + η(x)

). (3.14)

5Notice that the degree of freedom of the Goldstone field ξ has been replaced by the massive gaugefield Aµ, and we have four degrees of freedom, just as before symmetry breaking. It is often saidthat the gauge field has ‘eaten’ the Goldstone boson in order to acquire mass.

14

using equation (3.11). When substituting this term in the Lagrangian, we will see thatthe field η(x) has a mass term, the Goldstone fields ξ(x) are absent and the gauge fieldsW 1µ and W 2

µ have acquired the same mass, whereas W 3µ remains massless.

There is again another way of writing this, by writing the complex doublet of spin-less fields φ as 1√

2Φ(x)

( 0ρ(x)

), with Φ(x) an SU(2) matrix and a generalization of eiθ(x).

By a suitable transformation, we make the doublet φ(x) of the form( 0

ρ√2

). Again we

need to redefine the gauge fields in accordance to the new transformation. It is alsopossible to use the unitary gauge straightaway, by setting Φ equal to the identity matrix

and thus replacing the doublet field φ by( 0

ρ√2

).

It does not matter which way we choose to describe the Abelian or non-Abelian case,in either way we obtain one massive gauge field or two massive and one massless gaugefield, respectively. If we want to describe nature with the three massive gauge bosonsW+,W−, Z and the massless photon, it turns out we need a combination of the Abelianand non-Abelian theories.

15

4. Electroweak theory

The electroweak part of the SM was developed by Glashow (1961), Salam (1969) andWeinberg (1967) and consists of a SU(2)L × U(1)Y gauge theory. It unites the electro-magnetic and weak force and produces the massless photon and the massive W± andZ-bosons. The way to massive gauge bosons is again through symmetry breaking viathe Higgs mechanism (Higgs 1964, Brout & Englert 1964). The theory was finalizedwhen ’t Hooft and Veltman proved its renormalizability in the seventies (’t Hooft 1971,’t Hooft & Veltman 1972). That is, that UV-divergences can consistently be accountedfor and finite predictions can be made.

This chapter will give a short review of the electroweak part of the SM. It builds onthe previous chapters concerning gauge theories and spontaneous symmetry breaking,extending these to a case of SU(2)L × U(1)Y . We will start with the gauge invarianceand the appropriate transformations, continue with the Higgs mechanism that givesmass to the gauge bosons and then review the couplings that the Higgs has to bosonsas well as fermions. For this, the book of Quigg (2013, chapter 6) and the notes of Merket al. (2014) will be used.

4.1. SU(2)L × U(1)YWe immediately note that the previously used gauge groups have new subscripts, mean-ing a special variant of the group is used. In the SM, the U(1) that described elec-tromagenetism (as in section 2.1) is replaced by U(1)Y . This group couples to a newcharge called the weak hypercharge Y . The accompanying transformations are slightlychanged as well:

U(x) = exp

[ig′Y

2β(x)

]. (4.1)

The coupling parameter is now defined as g′2 , and a new field Bµ arises.

The relation between the weak hypercharge Y and the electromagnetic charge Q isgiven by

Q = T 3 +1

2Y, (4.2)

where T 3 is the third component of the weak isospin, as described in section 3.3. Thisrelation is called the Gell-Mann-Nishijima relation and was formulated in the 1950’s(Nishijima 1955, Gell-Mann 1956).

The SU(2) group has the subscript L since experiments showed that the charged weakbosons W± only couple to left-handed fermions.1 We therefore define a new doublet forthe SU(2):

(νee

)L

.2 The right-handed particles are grouped in singlets (ν)R and (e)R,

1Left- and right-handedness in particle physics is determined by spin and momentum: a particle isright-handed when its spin and momentum have the same direction, and left-handed when these arein opposite direction of each other. These are also called the chiral states of the fermions.

2The fact that new discoveries have showed that the neutrino does carry mass is neglected here.

16

given that they only interact with the U(1) part of the symmetry. The transformationrules from section 2.2 still apply, with coupling parameter g.

We now take the product of both groups to be our new gauge group. We again considerthe following Lagrangian:

L = (Dµφ)†(Dµφ)− V (φ), (4.3)

with V as defined in equation (3.2). For our SU(2)L × U(1)Y we have a new covariantderivative, consisting of two terms, one for each group:

Dµ = ∂µ + ig′

2BµY + igTiW

iµ. (4.4)

Here we have 1 gauge field Bµ from U(1) and three gauge fields Wµ from SU(2). Notethat for the right-handed singlet, the term with the Wµ does not contribute. In currentdescription, all these gauge fields are massless, thus we turn to the process of SSB.

4.1.1. Higgs mechanism

The process of SSB in a SU(2)L × U(1)Y gauge theory is the true Higgs mechanism asdescribed by the SM, and from which we can make real predictions which we can testat colliders such as the LHC.Following the procedure from the previous chapter, we pick µ2 < 0 and get an infinite setof possible minima, from which we choose our physical minimum to be φ = 1√

2

(0v

). With

this choice of minimum, the symmetry of the U(1)EM is preserved (since Q(φ0) = 0,but T(φ0) 6= 0 and Y (φ0) 6= 0, meaning their symmetry is broken).We again expand our fields around the minimum.

φ = exp

[iξσ

2v

](0

v+η(x)√2

). (4.5)

In order to avoid massless Goldstone bosons and only retain physical fields, we rightaway turn to the unitary gauge:

φ→ φ′ = exp

[−iξσ

2v

]︸ ︷︷ ︸Unitary gauge

(0

v+η(x)√2

)φ =

1√2

(0

v + η(x)

). (4.6)

The following expressions now hold for the physical gauge fields:

W±µ =1√2

(W 1µ ± iW 2

µ), (4.7)

Zµ =−g′Bµ + gW 3

µ√g2 + g′2

, (4.8)

Aµ =gBµ + g′W 3

µ√g2 + g′2

. (4.9)

The charged W -boson fields W±µ still behave according to SU(2). However, the Bµ andW 3µ field have formed a linear combination which results in the photon field Aµ and the

neutral Z-boson field Zµ.

17

4.2. Particle masses

From the expression for the Lagrangian that follows from the unitary gauge, the massesfor the new gauge boson fields can be identified. The term in the Lagrangian thatgenerates the masses is (Dµφ)†(Dµφ). The mass terms appear as terms quadratic in thegauge boson fields. By identifying the factor in front of the fields as 1

2m2x, we get the

following masses:

m2W =

1

4g2v2,

m2Z =

1

4v2(g2 + g

′2),

mA = 0,

mη = −2µ2 > 0.

(4.10)

The relation between the parameters g and g′ is defined as by the weak mixing angle

θW and is given by g′

g = tan(θW ). We can use this to compute the ratio of the bosonmasses:

mW

mZ=

12vg

12v√g2 + g′2

. (4.11)

From this we can conclude that the Z-boson is heavier than the W -boson, which is inaccordance to the experimentally measured values of mW ≈ 80 GeV and mZ ≈ 90 GeV.The photon field Aµ is massless (because the symmetry of U(1)EM is preserved), andwe also have a massive Higgs boson described by η.

4.3. Higgs coupling constants

Next, we turn to the interactions between the different fields and their coupling strength.These are given by terms in the Lagrangian that involve a combination of fields. Ac-cording to Ellis et al. (1976), the Higgs couplings are

gWWH =2m2

W

v,

gZZH =2m2

Z

v,

gffH =mf

v,

with λv2 = −µ and the relation m2H = −2µ2.

(4.12)

These can be retrieved by taking a smart look at the mass terms in (4.10). Rememberthat via the Higgs mechanism, the new term (v + η) entered the theory, with η theHiggs field. The masses arise from the v2 part of the new term, not involving the newη field. The interaction parts arise from the terms that do involve the η field, and in alinear way. This gives the three-point interactions between the gauge field and the Higgsfield. So, taking the linear part of (v + η), we have 2vη. If we take m2

W as an example,substituting the linear term for v2, will give us 1

2g2(2vη) (where the 1

4 became a 12 since

that is how it appears in the Lagrangian). Now, rewriting this in terms of the m2W as

formulated in (4.10), we are left with a factor2m2

W ηv , where the η is part of the fields.

18

This thus leaves us with the interaction or coupling strength gW =2m2

Wv , as shown in

(4.12).

4.4. Chiral fermions

In (4.12) the coupling for the Higgs to a fermion has also been given. In contrary to thegauge boson couplings, which are quadratically in mass, the fermion coupling is linearin its mass. The story for the fermions is slightly different, since they carry a chiralstate. Left-handed fermions appear in doublets whereas right-handed fermions comein singlets. This makes that they transform differently under the given SU(2) × U(1)transformation, and makes terms like (ψLψR +ψRψL) not gauge invariant. This can besolved however, by combining the left- and right-handed states with the complex scalar

field φ from the Higgs mechanism: (ψLφψR +ψRφψL). When using φ =( 0v+η√

2

), we will

get the right invariant terms in our Lagrangian. It turns out that the electron massterm can be identified as me = λv√

2, and the interaction term with the Higgs field as

λ√2. The new parameter λ is known as the Yukawa coupling, and is often expressed as

λ =√

2(mfv

). Comparing this with equation (4.12), we indeed see that the coupling of

fermions to the Higgs field is given by λ√2

=mfv and is proportional to the mass of the

fermion. 3

3Mind that only the very slim basics of the chiral fermions and Yukawa coupling are given here. Asthese are of no further importance for this thesis, they will not be further elaborated here and theinterested reader is referred to Merk et al. (2014), lecture 13.

19

5. Higgs production and decay

In this chapter, we will take a closer look at the Feynman diagrams associated with theHiggs production and decay processes.In a hadron collider like the LHC, the dominant Higgs production mechanism is throughgluon fusion and a virtual fermion loop. As we learned in the previous chapter, thecoupling of particles to the Higgs boson is proportional to their mass, so the mostcommon is a top-quark loop (Rainwater 2007).

When it comes to the decay, the Higgs boson can decay to all SM particles, becauseof its high mass. The largest branching ratio is to bottom quarks; this process happensabout 57.8 % of the time (Thomson 2013). Although the decay to quarks is the mostcommon, it is not easy to detect. The process will result in a set of jets, almost indistin-guishable from the background jets. The Higgs can also decay to massless particles likegluons or photons. This might sound strange at first, remembering that the Higgs onlycouples to massive particles. Decay to massless particles is possible however, namely viaan intermediate loop containing massive particles. Decaying to gluons will again lead tojets and brings us no further. The decay to two photons however, is more easy to detectas their signal-to-background ratio is larger. It occurs a lot less often though (2/1000decays, (Dittmaier et al. 2012)). It therefore takes more time and data to detect, butwhen the process does happen, one can be quite sure to detect it.

Both the production and decay of the Higgs thus involve a virtual loop, consisting ofmassive particles. Loop diagrams however, often lead to unwanted infinities. In order toavoid this, the method of renormalization is employed. There are roughly four methodsof renormalization possible, where this dimensional regularization is best when one wantsto preserve the symmetries of the theory (Kleinert & Schulte-Frohlinde 2001, Chapter8). Before we arrive at a point where this method is necessary, we first encounter somedifficult tensor integrals. To get these out of the way, the Passarino-Veltman methodwill be applied (Passarino & Veltman 1979). Both methods will be explained throughoutthe text.

Although loop diagrams are hard to solve, there are two reasons for reviewing them:first, they occur most often or are most easy to detect from background signals. Sec-ond, if the Standard Model turns out not to be correct, new physics will be hidden inquantum corrections or loops, so it is important to understand the mathematics behindthese loops for future precision calculations.

In this chapter, three process will be closely examined: Higgs production from twogluons via a top-quark loop, and Higgs decay to two photons via either a top-quark loopor a W -boson loop. The methods applied are quite involved, and will be fully writtenout for the production process. After that, the decay process will go according to thesame method, and only important differences or subtle methods will be highlighted.

20

Figure 5.1.: At lowest order there are two diagrams for the process gg → H. We see two incominggluons which form a top-quark loop from which a Higgs boson comes forth. Thecharge flow is is indicated by the arrows on the fermion lines. The momentumflow is indicated by the separate arrows and there is a loop momentum l. Source:Bentvelsen et al. (2005).

5.1. Higgs production gg → H

In this section, we will calculate the matrix element1 of the diagrams corresponding tothe Higgs production via gluon fusion, which are depicted in figure 5.1. The notes ofBentvelsen et al. (2005) will be followed closely for this purpose.

We start with the Feynman diagrams and corresponding Feynman rules. The lattercan be found in Peskin & Schroeder (1995). Since gluons are massless particles, we canwrite down the following kinematic relations:

k21 = 0,

k22 = 0,

(k1 + k2)2 = m2H .

(5.1)

Now, in order to construct the matrix element M, we have to follow the lines againstthe direction of the charge flow and use the needed Feynman rules. This will lead to thefollowing result:

M = (−igs)2

(−i yt√

2

)i3Tr[ta, tb](−1)εν(λ1, k1)εµ(λ2, k2)∫

ddl

(2π)d1

D1D2D3Tr[(6 l+ 6 k2 +m)γµ( 6 l +m)γν(6 l− 6 k1 +m)+

(− 6 l+ 6 k1 +m)γν(− 6 l +m)γµ(− 6 l− 6 k2 +m)],

(5.2)

where we have used:

D1 = l2 −m2

D2 = (l − k1)2 −m2

D3 = (l + k2)2 −m2.

(5.3)

These denominators each correspond to one of the propagators in the loop. To computethe matrix element, we have to perform the trace and then calculate the integral overthe loop momentum l. The result of the trace can be found in Bentvelsen et al. (2005).It turns out that only terms proportional to lµlν and gµν l · l remain and we are left witha tensor integral of the form Cµν .

1The matrix element is an element from the scattering matrix S, that relates the inital and final stateof a scattering process via |ψout〉 = S |ψin〉.

21

5.1.1. Passarino-Veltman reduction

A tensor integral is hard to solve but becomes a lot easier with the use of the Passarino-Veltman reduction. This method is used to decompose tensor or vector integrals intoeasier-to-solve scalar integrals. It does involve quite some steps however.We start with defining the various integrals we have to work with.

Cµν =

∫ddl

(2π)dlµlν

D1D2D3,

Cµ =

∫ddl

(2π)dlµ

D1D2D3,

C0 =

∫ddl

(2π)d1

D1D2D3,

Bµ(i, j) =

∫ddl

(2π)dlµ

DiDj,

B0(i, j) =

∫ddl

(2π)d1

DiDj.

(5.4)

Our goal is to express the tensor integral Cµν in terms of the scalar integrals C0 andB0. The first step is to decompose the tensor Cµν , using the vectors k1, k2 and gµν .

Cµν = k1,µk1,νC21 + k2,µk2,νC22 + k1, k2µνC23 + gµνC24 (5.5)

(with k1, k2µν ≡ k1,µk2,ν + k1,νk2,µ). We call C21, C22 etc. the form factors. We wantto write these form factors in terms of the scalar integrals C0 and B0. In order to achievethis, we first contract the tensor Cµν with each of the external momenta k. We will callthese new vectors vµ and wµ:

vµ = Cµνkν1 = k2,µ

m2H

2C22 + k1,µ

m2H

2C23 + k1,µC24

wµ = Cµνkν2 = k1,µ

m2H

2C21 + k2,µ

m2H

2C23 + k2,µC24.

(5.6)

Next, we contract the loop momentum with the external momenta:

k1 · l = −1

2(D2 −D1 − k2

1)

k2 · l =1

2(D3 −D1 − k2

2).

(5.7)

Note that this term also appears when contracting the integral expression of Cµν withthe external momenta. Combining (5.4) and (5.7), one denominator will disappear andthe following expressions remain:

vµ = Cµνkν1 = −1

2(Bµ(1, 3)−Bµ(2, 3))

wµ = Cµνkν2 =

1

2(Bµ(1, 2)−Bµ(2, 3)).

(5.8)

One step is done: the Cµν integrals have turned into Bµ integrals. These Bµ integralsalso have to be reduced via the same method. We thus again start with decomposing

22

the integral into its form factors:

Bµ(1, 2) = k1,µB1(1, 2), with B1(1, 2) =1

2B0(1, 2) (5.9)

Bµ(1, 3) = k2,µB1(1, 3), with B1(1, 3) = −1

2B0(1, 3) (5.10)

These are rather straightforward. The reduction to scalar integrals has been made usingequation (5.7) again. Reducing Bµ(2, 3) takes some more work, since we are dealing withboth k1 and k2 in the denominators here. We first have to apply a shift in momentum:l = l

′+k1. Integrating over l

′then leaves us with Bµ(2, 3) = B

′µ(2, 3)+k1,µB0(2, 3). The

first term has the same structure as Bµ(1, 2) and Bµ(1, 3), with momentum k = k1 +k2.Working this term out will get us the following expression for Bµ(2, 3) :

Bµ(2, 3) =1

2(k1,µ − k2,µ)B0(2, 3). (5.11)

All the vector integrals Bµ(i, j) have now been expressed in scalar integrals. We canput these scalar integrals in the expression for vµ and wµ (5.8) and obtain

vµ =1

4k2,µ(B0(1, 3) +

1

4(k1,µ − k2,νB0(2, 3))

wµ =1

4k1,µ(B0(1, 2)− 1

4(k1,µ − k2,νB0(2, 3)).

(5.12)

Looking back at what we started with, we have expressed vµ and wµ in terms of the formfactors of Cµν and in terms of the scalar integrals B0(i, j). In order to express Cµν inscalar integrals, which is our goal, we thus need to relate the form factors to the scalarintegrals. This can be done by introducing projection operators Pµk1 and Pµk2 , whichbehave like Pµkikj,µ = δij . We have to the following representation of these operators:

Pµk1 =2

m2H

kµ2

Pµk2 =2

m2H

kµ1 .

(5.13)

If we let these operators work on both expressions we obtained for vµ and wµ (equations(5.8) and (5.12)) we can define a set of scalar functions Ri to relate both equations:

R3 = Pµk1vµ =m2H

2C23 + C24

R4 = Pµk1wµ =m2H

2C21

R5 = Pµk2vµ =m2H

2C22

R6 = Pµk2wµ =m2H

2C23 + C24.

(5.14)

23

But also, we obtain

R3 =1

4B0(2, 3)

R4 = −1

4B0(2, 3) +

1

4B0(1, 2)

R5 = −1

4B0(2, 3) +

1

4B0(1, 3)

R6 =1

4B0(2, 3).

(5.15)

All that is left to do, is to combine these sets of scalar functions in such a way thatwe can express the form factors in scalar integrals. We do this by expressing the formfactors in terms of the scalar functions Ri. This takes more effort than simply reversingthe equations in (5.14) and we need the introduction of another projection operator:

Pµν =1

d− 2(gµν − Pµk1k

ν1 − P

µk2kν2 ), (5.16)

which selects the form factor C24 when acting on (5.5). We let it act on the integralexpression for Cµν . Note the use of a little trick:

gµνCµν =

∫ddl

(2π)dl2 −m2

D1D2D3+

∫ddl

(2π)dm2

D1D2D3= B0(2, 3) +m2C0. (5.17)

Now, finally, we arrived at the point where we can express the form factors in terms ofthe scalar functions Ri, which can be expressed in integrals B0(i, j).

C21 =2

m2H

R4

C22 =2

m2H

R5

C23 =2

m2H

(R6 − C24)

C24 =1

d− 2(B0(2, 3) +m2C0 −R3 −R6).

(5.18)

Putting all this together and combining equations (5.18), (5.15) and (5.5), we obtainthe following expression for Cµν :

Cµν =k1,µk1,ν [−1

4B0(2, 3) +

1

4B0(1, 2)]

+ k2,µk2,ν [−1

4B0(2, 3) +

1

4B0(1, 3)]

+ k1, k2µν2

m2H

[1

4B0(2, 3)− (

1

d− 2(B0(2, 3) +m2C0 −

1

2B0(2, 3))]

+ gµν [1

d− 2(B0(2, 3) +m2C0 −

1

2B0(2, 3)].

(5.19)

5.2. Dimensional Regularization

We are now left with a loop integral expressed in scalar integrals B0 and C0. We will nowapply dimensional regularization, which evaluates the integral in d dimensions, instead

24

of four, where d = 4 − 2ε. By lowering the dimension, originally divergent integralsbecome finite. At the end, ε can be sent to zero, and a finite four-dimensional integralis left.

First, we put the expression obtained for Cµν back into the integral for the matrixelement. Using 1

d−2 ∼12(1 + ε+ ε2), the result becomes (after some calculations):

M = (ε1 · ε2)a+ (k1 · ε2)(k2 · ε1)b

with a = 8m(ε+ ε2)B0(2, 3)− 4mm2HC0 + 16m3(1 + ε+ ε2)C0

b =−2a

m2H

(5.20)

(Bentvelsen et al. 2005). We have obtained an expression for the matrix element interms of the easier-to-solve scalar integrals, which is what we wanted. All that is left todo, is to compute these scalar integrals B0(2, 3) and C0.

5.2.1. Determining B0(2, 3)

In (5.20), it is easily spotted that B0(2, 3) only appears multiplied with the regulator εor higher orders of it. Since the regulator will be set to zero at the end, as we go backto four dimensions, the terms proportional to ε and higher order will disappear and weare only interested in poles in ε of B0 (henceforward we will drop the (2,3).We start with writing down the expression for B0:

B0 = µ2ε

∫ddl

(2π)d1

(l2 −m2)((l + k1 + k2)2 −m2). (5.21)

Note that the shift in momentum: l = l′+ k1 has been applied again. A factor µ2ε has

been added to keep the correct dimensions (this is called the dimensional regularizationmass scale µ). To compute this integral, a few steps are needed. Starting with the useof ‘Feynman’s trick’, which rewrites a fraction into an integral:

1

AB=

∫ 1

0dx

1

[xA+ (1− x)B]2. (5.22)

This leaves us with

B0 = µ2ε

∫ddl

(2π)d

∫ 1

0dx

1

[(1− x)(l2 −m2) + x(l + k1 + k2)2 −m2)]2. (5.23)

We now use κ = k1 + k2, work out the brackets and apply the shift l = l′ − xκ, so that

we obtain

B0 = µ2ε

∫ddl

(2π)d

∫ 1

0dx

1

[l2 + xm2H − x2m2

H −m2]2(5.24)

(note that we have dropped the prime and used κ2 = (k1 + k2)2 = m2H).

Next, we use the following expression for the integral over d dimensions2:∫ddl

(2π)d[l2 −M2 + iε2]−s = (−1)s

i(4π)ε

16π2

Γ(s− d2)

Γ(s)[M2 − iε]

d2−s. (5.25)

2For a more elaborate discussion of the origin and use of the formula used in this section, the readeris referred to chapter 10 of de Wit et al. (2015)

25

Combining this with equation (5.24) , we see that M2 = xm2H − x2m2

H −m2 and findthe following result for B0:

B0 = µ2ε i(4π)ε

16π2

Γ(ε)

Γ(2)

∫ 1

0dx[x2m2

H − xm2H +m2]−ε. (5.26)

Remember that we were interested in the poles in ε. To recognize these, we first haveto rewrite

Γ(ε) =1

εΓ(1 + ε). (5.27)

We have now identified the pole 1ε . For the integral over x we are only interested in the

constant term (as any other terms will vanish in the final result of the matrix element).Expanding the constant term will give (m2)−ε = 1− ε ln

(m2)

+O(ε2). It is easily seenthat the whole integral over x can thus be approximated as 1. Further, we recognizethat Γ(2) = 1 and obtain B0 = µ2ε i

16π2 (1ε − γE + ln 4π). The 1

ε term will cancel withthe other ε terms from (5.20) and we are ready to take the limit ε→ 0:

B0 =i

16π2. (5.28)

5.2.2. Determining C0

Next, we can determine C0 using a similar procedure. Since this procedure has beenwritten out in detail for B0, we will now only state the steps and some extra details ifneeded, but no calculation will be given.As C0 appears not only with a term in ε, but with a constant term as well in (5.20),we have to pay extra attention. This term of course will not disappear when settingε → 0. The first step is to write out C0, by partially working out the brackets inthe denominator. This time we do not need an extra factor to keep the dimensionscorrect. There are three terms in the denominator this time, which means we will haveto make use of Feynman’s trick twice. The second time, there is a squared term in thedenominator, for which an extended version of Feynman’s trick is needed:

1

Aα11 ...Aαnn

=Γ(α1 + ...+ αn)

Γ(α1)...Γ(αn)

∫ 1

0dx1...dxnδ(1− x1 − ...− xn)

xα1−11 ...xαn−1

n [x1A1...xnAn]−(α1+...+αn).

(5.29)

Note that we will get two integrals now, over x and over y. Then again, some rewritingand shifting in the integration parameter l is needed. The suitable shift is l

′= l + K,

with K = x(1− y)k1 − yk2. Then, C0 is written in such a way that equation (5.25) canbe applied again. This time, we will find a factor Γ(1 + ε) in the answer, which containsno pole in ε. The limit can thus be taken straightaway and will lead to a finite answer.The result becomes:

C0 = (−1)1

16π2i

∫ 1

0dx

∫ 1

0dy

y

(m2 − xym2H(1− y))3

. (5.30)

This integral is very tedious, but can be done using conventional techniques. Thisinvolves quite a lot of steps however, and therefore we refer the interested reader to

26

Bentvelsen et al. (2005) The result becomes:

C0 =i

16π2m2H

f(η) (5.31)

withf(η) =

arcsin2( 1√

η ), for η ≥ 1,

−14 [ln 1+

√1−η

1−√

1−η − iπ]2, for η < 1with η =

4m2t

m2H

. (5.32)

We can now put the values found for B0 and C0 back into equation (5.20) and obtainthe expression for the matrix element. This matrix element is needed to calculate thecross section σ, which is associated with the production rate, or how often this processof Higgs production via gluon fusion might happen. The formula used for this is

σ =1

2m2H

∫dPSΣspins,color|M|2 (5.33)

The calculation of this integral can be found in section 7 of Bentvelsen et al. (2005). Weare here only interested in the result, which is

σ0 =GFα

2

288√

2π|A(η)|2, (5.34)

with A(η) =3

2η[1 + (1− η)f(η)] (5.35)

where f(η) is defined as in (5.32) and with α the strong coupling constant and GF isFermi’s constant. This cross section can be measured in experiments and used to testtheoretical predictions with experimental results.

5.3. Higgs decay H → γγ

There are two possible decay modes for H → γγ, through either a top-quark or W-boson loop. Ellis et al. (1976) were the first to calculate this decay. However, thecorrectness of this calculation and the use of dimensional regularization was questionedby Gastmans et al. (2011a,b). A revisit of the calculation was necessary. Marciano et al.(2012) justified the use of dimensional regularization and proved the first calculation tobe correct. Their method will be closely followed in this section.

5.3.1. Higgs decay H → γγ through a top-quark loop

The diagram for this decay is depicted in figure 5.2. This decay highly resembles thepreviously described production. Again, we are dealing with a quark loop. The maindifference is that our external particles are massless photons this time. They do notcarry any color charge like the gluons from the Higgs production. This will lead to aslight change in the matrix element: the trace over the gluon colors will disappear, andthe vertex-factors for photons and quarks will be used. The needed Feynman rules canbe found in Peskin & Schroeder (1995).The same tensor integral as before will arise. Then again, one needs to follow the aboveprocedure of the Passarino-Veltman reduction, followed by dimensional regularization.Since we are working with almost the same, but reversed, diagram, the output is thesame and the calculation can be reviewed in the previous section.

27

Figure 5.2.: At lowest order there are two diagrams for the process H → γγ, where only oneis depicted here. The other one can be obtained by exchanging the photons. TheHiggs decays through a top-quark loop and then forms two photons. This decayprocess and the previously described production process very much alike.

5.3.2. Higgs decay H → γγ through a W -boson loop

The decay through a W-loop takes a closer look to the details since we are dealing withmassive particles in the loop this time. The diagrams can be found in figure 5.3.

Figure 5.3.: In unitary gauge, there are three diagrams for the process H → γγ through aW-loop. Another diagram can be obtained by exchanging the two photons in thetriangle diagram on the left. These two triangle diagrams give the same amplitudehowever, so we simply included a factor 2. Note, the loop momentum is called p inthese diagrams. The momentum flow is depicted in the separate arrows. Source:Marciano et al. (2012)

We start with constructing the matrix element. The same kinematic relations as inequation (5.1) hold. This time we are dealing with W-bosons in the loop, which aremassive particles. The slight change in particles also leads to a slight change in thematrix element. Again, the needed Feynman rules can be found in Peskin & Schroeder(1995).

The observant reader will note that the loop momentum has changed letter from lto p. This also leads to changes in the denominators. In this case, they are:

D1 = (p− k1)2 −m2W

D2 = p2 −m2W

D3 = (p− k1 − k2)2 −m2W .

(5.36)

28

This matrix element again comes down to a tensor integral, for which the method ofPassarino-Veltman will be used. It is trusted that this was explained in enough detailin the previous section, so we will work through it a bit quicker this time, mentioningonly important differences or new steps.

First, Cµν is decomposed as in (5.5). Next, we follow the same method as before:we start off with contracting the external momenta k with the decomposed tensor andwith the loop momenta p, which will result in a simpler form of the tensor integral.There is a slight change in contraction of both momenta this time

k1 · p = −1

2(D1 −D2)

k2 · p = −1

2(D3 −D1 −m2

H).

(5.37)

This leaves us with slightly different expressions for the contracted vectors, which wenow call xµ and zµ:

xµ = Cµνkν1 =

1

2(Bµ(1, 3)−Bµ(2, 3))

zµ = Cµνkν2 = −1

2(Bµ(1, 2)−Bµ(2, 3)−m2

HCµ).

(5.38)

Note the extra Cµ term in the expression for zµ. We now proceed in the same way forboth Bµ and Cµ, starting with the latter. Cµ can be decomposed as follows:

Cµ = k1,µC11 + k2,µC12. (5.39)

Contracting this expression with the external momenta gives

kµ1Cµ = m2H · C11 (5.40)

kµ2Cµ = m2H · C12. (5.41)

Applying (5.37) on the integral expression for Cµ leads to the following expressions forC11 and C12:

C11 =−1

m2H

(B0(1, 2)−B0(2, 3)−m2HC0) (5.42)

C12 =1

m2H

(B0(1, 3)−B0(2, 3)). (5.43)

And we see that only scalar integrals are left.Now we turn to the Bµ. Decomposing this gives:

Bµ(1, 2) = k1,µB1(1, 2), with B1(1, 2) =1

2B0(1, 2) (5.44)

Bµ(1, 3) = k2,µB′1(1, 3) + k1,µB0(1, 3), with B

′1(1, 3) =

1

2B0(1, 3) (5.45)

Bµ(2, 3) = (k1 + k2)µB1(2, 3), with B1(2, 3) =1

2B0(2, 3) (5.46)

The Bµ(1, 3) is a bit more complicated this time, but can be obtained the same way asBµ(2, 3) in the calculation for the production. Note that B

′1(1, 3) = B1(1, 2) under the

exchange of k1 and k2. And here too, only scalar integrals are left and we are done withthe Passarino-Veltman reduction.

29

5.4. FORM File

What we are looking for, is the expression of the matrix element in terms of the justcalculated scalar integrals. Doing this calculation by hand however, involves quite somemuscle ache. Therefore, the computational language ‘FORM’, which was developedby Jos Vermaseren at Nikhef, has been used for these calculations (Vermaseren 2000).Making this FORM file, we have closely followed the steps Marciano et al. (2012) showin their appendix. The resulting FORM files have been attached in Appendix B. Mar-ciano et al. (2012) decompose the matrix element into five separate ones and apply thePassarino-Veltman reduction on each separately. This too has been done in FORM.The first file in the Appendix shows the computation of the matrix element, and thedivision into five separate terms. The output of this file gives the same result as theintermediate steps of Marciano et al. (2012). However, the M1-term was written slightlydifferent, and in order to make sure we could keep comparing our terms to those givenin Marciano et al. (2012), a new FORM file was made where we inserted the M1 termas given in Marciano et al. (2012). We explicitly checked that out term and the termthey give were exactly equal to each other. In the second file, the application of thePassarino-Veltman reduction is shown and the computation of the matrix element interms of scalar integrals is finalized.Using FORM, the final matrix element becomes

M =e2g

(4π)2mW

1

m2H

[m2H + 6m2

W − 6m2W (m2

H − 2m2W )C0]×

× (m2Hg

µν − 2kµ2kν1 )εµ(k1)εν(k2)

with C0 =−2

m2H

f(η),

(5.47)

where f(η) is defined as in equation (5.32), with η =4m2

W

m2H

.

5.5. Finalizing the calculation

We have now computed the two possible decays separately. According to Marciano et al.(2012), the complete matrix element containing both the quark- and W -loop, can bewritten as follows:

M =e2gF

(4π)2mW(k1 · k2g

µν − kµ2kν1 )εµ(k1)εν(k2), (5.48)

where F includes the contributions from the quark- and W-loop:

F = FW (ηW ) +∑f

NcQ2qFq(ηq). (5.49)

Nc is a color factor, which is 3 in the case of quarks. The separate F -functions are:

FW (η) = 2 + 3η + 3η(2− η)f(η), (5.50)

Fq(η) = −2η[1 + (1− η)f(η)] (5.51)

30

Again, the f(η) function is defined as in (5.32). The η-functions are defined as

ηW =4m2

W

m2H

, and ηf =4m2

f

m2H

. (5.52)

This leads to the following theoretical prediction for the decay width for H → γγ, asfound in Marciano et al. (2012):

Γ(H → γγ) = |F |2( α

4π

)2 GFm3H

8√

2π. (5.53)

Again, α is the strong coupling constant and GF is Fermi’s constant. Recall the equationfor the cross section (5.34) and note the similarities. This decay can be measured inexperiments like LHC, and from there this theoretical prediction can be compared withexperimental results.

31

6. The experiments

In the previous chapters we developed the theory of the SM Higgs and looked at theprocesses that happen at LHC. Now it is time to look at real data from LHC. In orderto make sure the particle detected in 2012 is the actual SM Higgs, properties like itscouplings constants need to be checked. This chapter will investigate how theory andexperiment concerning the Higgs couplings can be related. First, a brief review of thesearch for the Higgs boson will be given, up to its discovery in 2012. Then it will bediscussed how we can extract coupling constants from detector data.

6.1. Higgs search

The search for a Higgslike particle started over two decades ago at the Large ElectronPositron (LEP) collider. The main production mechanism in e+e− collisions is throughan intermediate (above mass-shell) Z-boson production, which then decays into a Higgsand a lower energy Z-boson. This process is known as Higgsstrahlung (Rainwater 2007).No signal was observed at LEP, establishing a lower bound of 114.4 GeV on the Higgsmass (Barate et al. 2003).

Next up was the Tevatron in the United States. The dominant production mechanismshere were via gluon fusion or vector boson fusion from a W -boson (Rainwater 2007).This proton-antiproton collider was shut down in 2011 after ten years of collecting data.The data also showed no definite sign of a Higgs, but an excess of events in the bottom-quark decay-channel led to a mass range between 115 GeV and 135 GeV (Tevatron NewPhysics Higgs Working Group and CDF and D0 Collaborations 2012).

The hunt for the Higgs continued at the Large Hadron Collider (LHC), built to bea Higgs factory (amongst other scientific goals) (Rainwater 2007). The main Higgsproduction mechanism of this proton-proton collider was through gluon fusion becauseof the high energies at LHC. The most looked for decay was that to two energeticphotons. This decay only happens 2 out of every 1000 times, but is relatively easy todetect due to its greater signal/background ratio. On July 4, 2012, the releasing answercame, a Higgs-like particle was observed (Aad et al. 2012, Chatrchyan et al. 2012).Combined search led to an observed mass of mH = 125.09 GeV (Aad et al. 2015).

6.2. Verifying the coupling constants

The careful reader will have noticed that a Higgslike particle was observed. It is a newparticle, with a mass consistent with the expected SM Higgs mass. But in order to besure it really is the SM Higgs and not some other exotic variant, more properties ofthis particle have to be tested to their correspondence with the theory. The couplingsof the Higgs to other particles are generally regarded as the most crucial observables(Rainwater 2007).

A description of these coupling constants has been given in section 4.2. All have anunknown parameter v, which can be related to the Higgs boson mass through λv2 = −µ

32

and m2H = −2µ2 (note µ2 < 0). When plugging in the observed mass of 125.09 GeV, it

is possible to theoretically predict values for the couplings constants which then can befitted to experimental data.

It should be noted that one cannot measure the coupling constants directly. A de-tector measures rates, a combination of production and decay rates, to be more precise.Production and decay (σ and Γ, respectively) cannot be measured separately at a col-lider, since produced particles decay before they hit the detector. Each observed processthus involves a Higgs production and a Higgs decay, and at least two Higgs couplingconstants.

At LHC, the main Higgs boson production mechanism is through gluon fusion, fol-lowed by vector boson fusion. When it comes to decays, the LHC is most sensitive tothe processes H → γγ, ZZ∗,WW ∗, ττ and bb. For a Higgs boson with a mass of 125GeV, the SM predictions of the cross sections and decay branching ratios (BR) canbe calculated (ATLAS collaboration 2015). The BR is BR(i) = Γi

ΓH, or the ratio of a

certain decay mode to the total decay width, where it is assumed all possible decayscan be detected and there are no ‘invisible’ ones.1 The predicted values for productionand decay can then be compared to experimental results. This is done by the use of thesignal strength µ2:

µ =σ ·BR

(σ ·BR)SM. (6.1)

The signal strength measures deviations from the SM predictions. The latest mea-surement from ATLAS collaboration (2015) shows a global signal strength value ofµ = 1.18+0.15

−0.14, with an uncertainty due to statistical, systematic and theoretical uncer-tainties.

These signal strengths do not reveal the coupling strengths immediately. The zero-width assumption is used, which means

σ(i→ H → f) = σ(i→ H) ·BR(H → f) (6.2)

=σi(κj) · Γf (κj)

ΓH(κj), (6.3)

where κ is scale factor. The definition of cross section and partial width share the samematrix element and differ only by phase space factors. A value of κj = 1 corresponds toSM expectation (LHC Higgs Cross Section Working Group et al. 2012). The theoreticallypredicted coupling strengths are tested through fits to the observed data. So far, theresults found are the best fit-values and are consistent with the predictions for the SMHiggs (ATLAS collaboration 2015):

κV = κW = κZ κV = 1.09+0.07−0.07 (6.4)

κF = κt = κb = κτ = κg = κµ κF = 1.11+0.17−0.15 (6.5)

where errors due to statistical, systematic and theoretical uncertainties are shown.One may now rightly wonder how the coupling constants follow from this. They are

quite well hidden in the values for µ and κ. However, we can trace our way up fromcouplings as they were given in section 4.2. We saw here that the coupling constants

1This is a theoretically assumption. The total width cannot be determined experimentally at LHC.2Note that this is a different µ then the one used in the Higgs potential.

33

to different particles find their origin in the Lagrangian. Remembering from section1.1, the Lagrangian offers the basis for the Feynman rules. From these, the matrixelement corresponding to a certain production- or decay process can be formulated.This matrix element is then used in both the formula for the cross-section σ (associatedwith production) and partial width Γ (associated with decay). These values can bemeasured at an accelerator. How exactly the matrix element (in which the couplingconstants are absorbed) is used in the expression for production or decay is most easilyseen in the formula for a two-body decay, as given in Thomson (2013, page 68)

Γfi =p

32π2m2a

∫|M |2dΩ. (6.6)

A similar formula exists for the cross section, which only differs by phase space factors.

Figure 6.1 shows the latest results from CMS concerning the Higgs couplings. When re-calling the couplings in section 4.2, we see that they are linear in mass for the fermions,and quadratically in mass for the gauge bosons. Note that on the vertical axis, thelinear fermion-coupling is plotted, and the square root of the boson-coupling. We wouldexpect this relation between couplings strength and mass to result in a straight line,as it indeed does! The statistical errors are represented by the blue and red bars, thatcorrespond to a 95% or 68% confidence level, as also holds for the yellow and greenfields. We can see that even when considering these statistical errors, it still resemblesa straight line. The results in this graph and from equation (6.4) and (6.5) show thatfor now, the Higgslike particle indeed behaves very much like the SM Higgs.

Figure 6.1.: This graph shows the coupling constants of the Higgs to various particles, plottedagainst their respective mass. The experimentally found values seem to be in goodagreement with the SM predicted ones. Note that the Higgs couplings are linearin mass to the fermions, and quadratically in mass to the gauge bosons. On thevertical axis, the square root of the boson-couplings is given, so we indeed expect astraight line. Source: Pieri (2015).

34

Conclusion

In this thesis, a flight through the story of Higgs was taken: from the start at the theoriesthat lie beneath, to a finish at the means of verifying its existence.

We learned how the SM in its current formulation came to be by demanding varioussorts of local symmetries. Each time a local gauge invariance is demanded, it gives riseto a new massless vector field and therefore each local gauge invariance implies a newforce. These new gauge fields are massless however. The way to give mass to thesefields, is through spontaneous symmetry breaking. This applies when the Lagrangianstill preserves the symmetry, but the lowest energy solutions do not obey this symmetry.When a certain ground state is chosen, the gauge fields gain mass. By working in theUnitary gauge, the massless Goldstone fields can be avoided. When the local gauge in-variance is written in an SU(2)L × U(1)Y symmetry group, the spontaneous symmetrybreaking is formulated in the Higgs mechanism and we have found a way to the describethe electroweak part of the SM, in which the photon is the massless force-carrier ofelectromagnetism, and the two charged W -bosons and the neutral Z-boson are massiveand carry the weak force. The Higgs mechanism also gives rise to a new massive scalarfield and its accompanying Higgs boson.This Higgs boson is an important part of the SM, and forty years after its theoreticalprediction it was finally discovered in 2012. A way of verifying whether this particle isthe SM Higgs or something new, is by checking its coupling constants. These are given

by gWWH =2m2

Wv , gZZH =

2m2Zv and gffH =

mfv , where we see that he Higgs couples

linear to fermion mass and quadratically to boson mass. To check these values, moreHiggs process need to be analyzed. The most important Higgs processes at LHC con-cerning production and decay both contain a loop of virtual particles. The calculationof these loop diagrams can be done using the method of Passarino-Veltman, in whichtensor integrals are reduced to easier-to-solve scalar integrals. These scalar integralsneed to be solved in d = 4 − 2ε dimension, in order to avoid infinities. This is calleddimensional regularization. When the matrix element is computed, this can be used inthe formula for production and decay. A combination of both processes is what is mea-sured at the detectors at LHC. With some good data analysis, it is possible to substractthe coupling constants from this data. It turns out that for now, the particle discoveredin 2012 indeed behaves very much like the SM Higgs.

The hunt for the Higgs might have triumphed in 2012, the search for the smallest stillcontinues. The second run of the LHC started in early June of 2015. It now producescollisions at an energy of 13 TeV, almost twice as much as during the first run. Thisnew run is exciting, because it might prove once more the consistency of the SM, orit could reveal new physics: physics beyond the Standard Model. Various models havealready been developed, for example: extra dimensional Higgsless constructions (Sim-mons et al. 2006), little Higgs (Perelstein 2007) and the twin Higgs mechanism (Chackoet al. 2006). If there is any new physics, it will most likely appear in loops, whereit is possible for new, never before seen particles to arise as a virtual particles. Thiswould lead to different coupling strengths, and a deviation of the SM predicted values.

35

We need state-of-the-art higher order corrections for more precise predictions and thisshould be the main focus of the Higgs calculations of the coming years.No matter how it turns out, another confirmation of the theory of the Standard Model,or a glimpse into never seen new physics: the coming years will be very interesting forthe field of particle physics.

36

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39

Appendices

40

A. Lagrange formalism

In this appendix the Lagrangian formalism will be explained by the use of Thomson(2013, chapter 17.2). For a more mathematically detailed description the reader isreferred to Quigg (2013, chapter 2) or de Wit et al. (2015, Chapter 1).

A.1. Classical case

In classical mechanics, the Lagrangian L of a system is defined as the kinetic energy ofthe system T minus the potential energy V :

L = T − V (A.1)

The Lagrangian L(qi, ˙qi) of a particle is a function of its coordinates qi, and their timederivatives, qi. From the Lagrangian, one can extract the equations of motion by usingthe Euler-Lagrange equations:

d

dt

(∂L

∂qi

)=∂L

∂qi(A.2)

A.2. Relativistic fields

Where we previously had discrete particles, we now have continuous fields that extentthroughout the whole space. The Lagrangian (or technically, the Lagrangian density)L, becomes dependent of the fields φi(t, x, y, z), and their derivatives, with respect tofour-dimensional space-time.

∂µ

(∂L

∂ (∂µφi)

)=∂L∂φi

(A.3)

To go from L to L, one can simply integrate over four-dimensional space.

A.2.1. Klein-Gordon Lagrangian for scalar (spin-0) field

L =1

2(∂µφ) (∂µφ)− 1

2m2φ2 (A.4)

When applying the Euler-Lagrange equation to this Lagrangian, one obtains the Klein-Gordon equation for a free scalar field φ(x) : ∂µ∂

µφ+m2φ = 0.

A.2.2. Dirac Lagrangian for spinor (spin-1/2) field

L = iψγµ∂µψ −mψψ (A.5)

Applying the Euler-Lagrange here will deliver the Dirac equation for fermions: iγµ(∂µψ)−ψ = 0. Note: the field ψ(x) is complex spinor, consisting of 4 components. This givesrise to 8 independent real fields, which can be expressed as linear combinations of ψand its adjoint spinor ψ. When taking the partial derivaties with respect to the adjointspinor, one gets the Dirac equation for ψ and vice versa.

41

A.2.3. Proca Lagrangian for vector (spin-1) field

L = −1

4FµνFµν +

1

2m2AµAµ (A.6)

Plugging this Lagrangian in the Euler-Lagrange equations gives rise to the field equationsfor a massive spin-1 particle. To obtain the Lagrangian for the free photon, which is amassless spin-1 particle, we simply lose the massterm and obtain L = −1

4FµνFµν .

42

B. FORM script

In this Appendix the FORM files resulting from the calculation of the Higgs decay aregiven. There are two separate files: the first contains the construction of the matrix ele-ment, which is done by decomposing the matrix element in five separate terms, accordingto their Lorentz indices, just as in Marciano et al. (2012). The second file contains theapplication of the Passarino-Veltman reduction on this matrix element, and providesthe final matrix element that only contains scalar integrals.

The actual script is printed here, where comments that clarify the steps are givenbetween asterisks (*). The output rendered by FORM is also printed.

B.1. Computation of the matrix element

B.1.1. Input

** Higgs decay through W loop**

*****Declarations*****

S mW2 mH2 D1 D2 D3 v;

S AAA BBB CCC;

S C11 C12 C21 C22 C23 C24;

S R3 R4 R5 R6;

S [B0(1,3)] [B0(1,2)] [B0(2,3)] [B1(2p,3p)] [B1(1,3)] [C0(1,2,3)];

S [B1(2,3)];

S A0 B1 B2 C0;

V p q l k1 k2 epsk1 epsk2 r eps;

I mu=n nu=n ro=n si=n mup=n nup=n rop=n sip=n;

T V1 V2 V3 V4;

T prop1 prop2 prop3;

.global

*****Set up Matrix element*****

g Diag1 = AAA*epsk1(mu) * epsk2(mup)* (2* i_* mW2/v) * d_(si,sip)

* prop1(nup,ro)*V1(mu,nu,ro) * prop2(nu,si)

* prop3(sip,rop)*V2(mup,nup,rop) ;

g Diag2 = AAA*epsk1(mu) * epsk2(mup)* (2* i_* mW2/v) * d_(si,sip)

* prop1(nup,ro)*V3(mup,nu,ro) * prop3(nu,si)

* prop2(sip,rop)*V4(mu,nup,rop) ;

g Diag3 = CCC*epsk1(mu) * epsk2(mup)* (2* i_* mW2/v) * d_(si,sip)

* prop2(nu,si) * prop3(sip,ro)*

((2*d_(mu,mup)*d_(nu,ro) - d_(mu,nu)*d_(mup,ro)

- d_(mu,ro)*d_(mup,nu))) ;

43

g Diff12 = Diag1-Diag2; **If this is zero, we can use Diagram 1

twice in the total**

g Totaal = 2*Diag1 + Diag3;

Contract;

.sort

*****Feynman rules needed for the Matrix element*****

id prop1(mu?,nu?) = (d_(mu,nu)-(p(mu)-k1(mu))*(p(nu)-k1(nu))/mW2)/D1;

id prop2(mu?,nu?) = (d_(mu,nu)-p(mu)*p(nu)/mW2)/D2;

id prop3(mu?,nu?) = (d_(mu,nu)-(p(mu)-k1(mu)-k2(mu))*(p(nu)-k1(nu)-k2(nu))

/mW2)/D3;

id V1(mu?,nu?,ro?)=d_(nu,mu)*(-p(ro)-k1(ro)) +

d_(nu,ro)*(2*p(mu)-k1(mu)) + d_(ro,mu)*(2*k1(nu)-p(nu));

id V2(mup?,nup?,rop?)=d_(nup,mup)*(-p(rop)+k1(rop)-k2(rop)) +

d_(nup,rop)*(2*p(mup)-2*k1(mup)-k2(mup)) + d_(rop,mup)*(2*k2(nup)-p(nup)+

k1(nup));

id V3(mu?,nu?,ro?)=d_(nu,mu)*(p(ro)-k1(ro)-2*k2(ro)) +

d_(nu,ro)*(-2*p(mu)+2*k1(mu)+k2(mu)) + d_(ro,mu)*(k2(nu)+p(nu)-k1(nu));

id V4(mup?,nup?,rop?)=d_(nup,mup)*(p(rop)-2*k1(rop)) +

d_(nup,rop)*(-2*p(mup)+k1(mup)) + d_(rop,mup)*(k1(nup)+p(nup));

.sort

*****Start simplifying*****

id k1.k1=0;

id k2.k2=0;

id epsk1.k1 = 0;

id epsk2.k2 = 0;

id p.p = D2 + mW2;

id k1.k2 = mH2/2;

.sort

*** M1 term ***

if (match(epsk1.epsk2)>0);

multiply D1*D2*D3*mW2/2;

id D1 = D2 -2*p.k1;

id D2 = p.p - mW2;

id p.k1 = (D1-D2)/(-2);

id p.k2 = (D3-D1-mH2)/(-2);

id p.p = D2 + mW2;

multiply 1/D1/D2/D3;

id D1 = D2 -2*p.k1;

id D3 = D2 - 2*k2.p-2*k1.p + mH2;

id D2/D3 = 1-(mH2-2*p.k1-2*p.k2)/D3;

endif;

*** M2 term ***

if (match(p.epsk1*p.epsk2)>0);

multiply D1*D2*D3*mW2/2;

44

id D1 = D2 -2*p.k1;

id D2 = p.p - mW2;

multiply 1/D1/D2/D3;

endif;

*** M3 term ***

if (match(p.epsk1*k1.epsk2)>0);

multiply D1*D2*D3*mW2/2;

id D1 = D2 -2*p.k1;

id D2 = p.p - mW2;

id p.k1 = (D1-D2)/(-2);

id p.k2 = (D3-D1-mH2)/(-2);

id p.p = D2 + mW2;

multiply 1/D1/D2/D3;

endif;

*** M4 term ***

if (match(p.epsk2*k2.epsk1)>0);

multiply D1*D2*D3*mW2/2;

id D1 = D2 -2*p.k1;

id D2 = p.p - mW2;

multiply 1/D1/D2/D3;

id p.k1 = (D1-D2)/(-2);

id p.k2 = (D3-D1-mH2)/(-2);

id p.p = D2 + mW2;

endif;

*** M5 term ***

if (match(k1.epsk2*k2.epsk1)>0);

multiply D1*D2*D3*mW2/2;

id D1 = D2 -2*p.k1;

id D2 = p.p - mW2;

multiply 1/D1/D2/D3/mW2;

id p.p = D2 + mW2;

endif;

.sort

id AAA=1*v*(-1)*i_;

id CCC=-1*v*(-1)*i_;

b i_ v epsk1 epsk2 D1 D2 D3 ;

print Diff12;

print +s Totaal;

.sort

.end;

45

B.1.2. Output

Totaal =

+ p.epsk1*p.epsk2*D1^-1*D2^-1*D3^-1 * (

+ 4*mW2*mH2

- 8*mW2^2

+ 8*mW2^2*n

)

+ p.epsk1*k1.epsk2*D1^-1*D2^-1*D3^-1 * (

- 4*mW2*mH2

+ 8*mW2^2

- 8*mW2^2*n

)

+ p.epsk1*k1.epsk2*D1^-1*D2^-1 * (

+ 4*mW2

)

+ p.epsk1*k1.epsk2*D2^-1*D3^-1 * (

- 1/2*mH2

- 7*mW2

)

+ p.epsk1*k1.epsk2*D2^-1 * (

+ 1/2

)

+ p.epsk1*k1.epsk2*D3^-1 * (

+ 1/2

)

+ p.epsk2*k2.epsk1*D1^-1*D3^-1 * (

- 4*mW2

)

+ p.epsk2*k2.epsk1*D2^-1*D3^-1 * (

+ 1/2*mH2

+ 7*mW2

)

+ p.epsk2*k2.epsk1*D2^-1 * (

- 1/2

)

+ p.epsk2*k2.epsk1*D3^-1 * (

46

- 1/2

)

+ k1.epsk2*k2.epsk1*D1^-1*D2^-1*D3^-1 * (

+ 16*mW2

)

+ k1.epsk2*k2.epsk1*D1^-1*D3^-1 * (

+ 4

)

+ epsk1.epsk2*D1^-1*D2^-1*D3^-1 * (

- 8*mW2^2*mH2

)

+ epsk1.epsk2*D1^-1 * (

- 2*mW2

)

+ epsk1.epsk2*D2^-1*D3^-1 * (

- 1/2*mH2^2

- 2*mW2*mH2

+ 2*mW2^2

- 2*mW2^2*n

+ 2*p.k1*mH2

+ 4*p.k1*mW2

)

+ epsk1.epsk2*D2^-1 * (

+ 1/2*mH2

+ 2*mW2

- p.k1

+ p.k2

)

+ epsk1.epsk2*D3^-1 * (

- 1/2*mH2

- p.k1

+ p.k2

);

.end;

B.2. Applications Passarino-Veltman reduction

B.2.1. Input

** Higgs decay through W loop**

47

*****Declarations*****

S mW2 mH2 D1 D2 D3 ;

S C11 C12 C21 C22 C23 C24;

S R3 R4 R5 R6;

S [B0(1,2)] [B0(1,3)] [B0(2,3)];

S [B1(1,2)] [B1(1,3)] [B1(2,3)];

S [B1(1p,3p)] [B1(2p,3p)];

S A0 [C0(1,2,3)] n;

V p k1 k2 epsk1 epsk2 r eps;

V r1 r2 ;

.global

****Now follow the terms M1 - M5, obtained from the previous FORM file. ****

** Note: for consistency reasons, the M1 term given in **

** (Marciano, Zhang, Willenbrock, 2012) is used. **

g M1 = epsk1.epsk2*

( D1^-1*D2^-1*D3^-1 * (- 8*mW2^2*mH2)

+ D1^-1 * (- 2*mW2 )

+ D2^-1*D3^-1 * (+ 1/2*mH2^2+ 2*mW2^2

- 2*mW2^2*n - 2*p.k2*mH2 - 4*p.k2*mW2 )

+ D2^-1 * (- 1/2*mH2- p.k1+ p.k2 )

+ D3^-1 * ( + 1/2*mH2 + 2*mW2 - p.k1 + p.k2));

g M2 = + p.epsk1*p.epsk2*

D1^-1*D2^-1*D3^-1 * (+ 4*mW2*mH2- 8*mW2^2 + 8*mW2^2*n);

g M3 = p.epsk1*k1.epsk2*D1^-1*D2^-1*D3^-1 * (- 4*mW2*mH2 + 8*mW2^2-

8*mW2^2*n )

+ p.epsk1*k1.epsk2*D1^-1*D2^-1 * ( + 4*mW2 )

+ p.epsk1*k1.epsk2*D2^-1*D3^-1 * (- 1/2*mH2- 7*mW2 )

+ p.epsk1*k1.epsk2*D2^-1 * ( + 1/2 )

+ p.epsk1*k1.epsk2*D3^-1 * ( + 1/2 );

g M4 = + p.epsk2*k2.epsk1*D1^-1*D3^-1 * (- 4*mW2 )

+ p.epsk2*k2.epsk1*D2^-1*D3^-1 * (+ 1/2*mH2 + 7*mW2)

+ p.epsk2*k2.epsk1*D2^-1 * (- 1/2 )

+ p.epsk2*k2.epsk1*D3^-1 * (- 1/2 );

g M5 = k1.epsk2*k2.epsk1*D1^-1*D2^-1*D3^-1 * ( + 16*mW2 )

+ k1.epsk2*k2.epsk1*D1^-1*D3^-1 * ( + 4 );

** When all terms are added for the total, it is necessarry to give them

all the same dimension **

g Tot = (M1/mW2+M2/mW2+M3/mW2+M4/mW2+M5)*mH2;

.sort

***PASSARINO VELTMAN***

** Decompose tensor integrals**

48

id r1?.p*r2?.p*D1^-1*D2^-1*D3^-1=(r1.k1*r2.k1*C21

+r1.k2*r2.k2*C22+(r1.k2*r2.k1+r1.k1*r2.k2)*C23 +r1.r2*C24); **Cmunu**

id r1?.p*D1^-1*D2^-1*D3^-1= r1.k1*C11+r1.k2*C12; **Cmu**

id r1?.p*D1^-1*D2^-1 = r1.k1*[B1(1,2)]; **Bmu(1,2)**

id r1?.p*D1^-1*D3^-1 = r1.k1*[B0(1,3)]+r1.k2*[B1(1p,3p)]; **Bmu(1,3)**

id r1?.p*D2^-1*D3^-1 = (r1.k1+r1.k2)*[B1(2,3)]; **Bmu(2,3)**

id r1?.p*D3^-1 = (r1.k1+r1.k2)*A0; **shift p=p’+k1+k2**

id r1?.p*D2^-1 = 0; **uneven integral=0**

id 1/D1/D2/D3 = [C0(1,2,3)];

id 1/D1/D2 = [B0(1,2)];

id 1/D2/D3 = [B0(2,3)];

id 1/D1/D3 = [B0(1,3)];

id 1/D1 = A0;

id 1/D2 = A0;

id 1/D3 = A0;

id k1.k1=0; **Make use of the kinematic relations**

id k2.k2=0;

id epsk1.k1 = 0;

id epsk2.k2 = 0;

id k1.k2 = mH2/2;

.sort

** Invert form factors to scalar functions **

id C21 = 2*R4/mH2;

id C22 = 2*R5/mH2;

id C23 = 2*(R6-C24)/mH2;

id C24 = ([B0(1,3)]+mW2*[C0(1,2,3)]-R3-R6)*((1+eps)/2);

** Write scalar functions in terms of scalar integrals **

id R3 = (-1/2)*([B1(2,3)]-[B0(1,3)]);

id R4 = (-1/2)*([B1(1,2)]-[B1(2,3)]-mH2*C11);

id R5 = (-1/2)*([B1(2,3)]-[B1(1p,3p)]);

id R6 = (1/2)*([B1(2,3)]+mH2*C12);

id C11 = ([B0(1,2)]-[B0(2,3)]-mH2*[C0(1,2,3)])/(-mH2);

id C12 = ([B0(1,3)]-[B0(2,3)])/mH2;

id [B1(1,2)]= [B0(1,2)]/2;

id [B1(2,3)]= 1/2*[B0(2,3)];

id [B1(1p,3p)]= [B0(1,3)]/2;

** A closer look on the dimension and resulting poles **

* The use of these poles renders the constant terms in the *

* expressions for M1-M5 in (Marciano, Zhang, Willenbrock, 2012)*

id n = 4-2*eps;

id eps*[C0(1,2,3)] = 0; ** Since C0 is finite, setting eps->0 will make

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C0 disappear**

id eps*[B0(1,3)] = 1; ** Since the B0 terms are not finite,**

**they will not disappear **

id eps*[B0(2,3)] = 1;

* The following equation is used on some of the terms to make the *

* constant terms appear. Because it is not used on all, it is showed here *

* not turned on, this equation should be applied by hand. *

*id [B0(1,3)] = A0/(mW2-1);

.sort

b epsk1 epsk2 eps;

print + s M1 M2 M3 M4 M5 Tot ;

.sort

.end;

B.2.2. Output

M1 =

+ epsk1.epsk2 * (

- mW2*mH2*[B0(2,3)]

+ 4*mW2^2

- 6*mW2^2*[B0(2,3)]

- 8*mW2^2*mH2*[C0(1,2,3)]

);

M2 =

+ k1.epsk2*k2.epsk1 * (

- 2*mW2

- 4*mW2*[B0(2,3)]

+ 4*mW2*[B0(1,3)]

- 12*mW2^2*mH2^-1

- 24*mW2^2*mH2^-1*[B0(2,3)]

+ 24*mW2^2*mH2^-1*[B0(1,3)]

- 4*mW2^2*[C0(1,2,3)]

- 24*mW2^3*mH2^-1*[C0(1,2,3)]

)

+ epsk1.epsk2 * (

+ mW2*mH2

+ mW2*mH2*[B0(2,3)]

+ 2*mW2^2

+ 6*mW2^2*[B0(2,3)]

+ 2*mW2^2*mH2*[C0(1,2,3)]

50

+ 12*mW2^3*[C0(1,2,3)]

)

+ eps*k1.epsk2*k2.epsk1 * (

+ 8*mW2^2*mH2^-1

)

+ eps*epsk1.epsk2 * (

- 4*mW2^2

);

M3 =

+ k1.epsk2*k2.epsk1 * (

+ 1/2*A0

- 1/4*mH2*[B0(2,3)]

+ 1/2*mW2*[B0(2,3)]

- 4*mW2*[B0(1,3)]

+ 24*mW2^2*mH2^-1*[B0(2,3)]

- 24*mW2^2*mH2^-1*[B0(1,3)]

);

M4 =

+ k1.epsk2*k2.epsk1 * (

- 1/2*A0

+ 1/4*mH2*[B0(2,3)]

+ 7/2*mW2*[B0(2,3)]

- 4*mW2*[B0(1,3)]

);

M5 =

+ k1.epsk2*k2.epsk1 * (

+ 4*[B0(1,3)]

+ 16*mW2*[C0(1,2,3)]

);

Tot =

+ k1.epsk2*k2.epsk1 * (

- 2*mH2

- 12*mW2

+ 12*mW2*mH2*[C0(1,2,3)]

- 24*mW2^2*[C0(1,2,3)]

)

+ epsk1.epsk2 * (

+ mH2^2

51

+ 6*mW2*mH2

- 6*mW2*mH2^2*[C0(1,2,3)]

+ 12*mW2^2*mH2*[C0(1,2,3)]

)

+ eps*k1.epsk2*k2.epsk1 * (

+ 8*mW2

)

+ eps*epsk1.epsk2 * (

- 4*mW2*mH2

);

.end;

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