Coupling constants of the Higgs boson

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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 Physics

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Abstract

A Higgslike particle with a mass of 125 GeV was discovered at CERN in 2012. To ensure this particle really is the Standard Model version of the Higgs boson, more of its properties need to be tested, the most crucial being its coupling constants. This thesis will review the theory of the Higgs mechanism and its couplings, followed by how these theoretical predictions can be checked with experimental values. For this, the Higgs production 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 the method of Passarino-Veltman and dimensional regularization. Some calculations will be performed in FORM. The latest results from CMS show that the particle which was discovered in 2012 indeed behaves like the expected Standard Model Higgs.

Title: Coupling constants of the Higgs boson Author: Anouk Geenen

10143858, ajpgeenen@gmail.com, 10143858 Supervisor: prof. dr. E. L. M. P. Laenen Second advisor: dr. W. J. Waalewijn Date: July 7, 2015

Institute for Theoretical Physics Universiteit van Amsterdam

Science Park 904, 1098 XH Amsterdam http://iop.uva.nl/divisions/itfa

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Populaire samenvatting

In 2012 kopte elke krant met het grote nieuws: het Higgs-deeltje was gevonden! Al jaren zijn natuurkundigen op zoek naar de meest elementaire bouwstenen van de natuur: wat zijn de allerkleinste puzzelstukjes waaruit de wereld is opgebouwd? Het Higgsdeeltje -ook wel bekend als het God-deeltje - was het laatste ontbrekende stukje in deze puzzel der 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 fermionen genoemd, en daarvan hebben we er twaalf in ons huidige model. De krachtdeeltjes gaan ook door als bosonen, en hiervan kennen we er zes. Elk boson kan gekoppeld worden aan een kracht. Zo hoort het foton bij elektromagnetisme, het gluon bij de sterke kern-kracht en wordt de zwakke kernkern-kracht beschreven door drie bosonen: W+, W−and Z0. De oplettende lezer ziet dat we nu pas vijf bosonen hebben genoemd. De laatste is het beroemde Higgs-deeltje of Higgs-boson. Dit deeltje representeert geen kracht, maar representeert het mechanisme waarmee alle andere deeltjes massa krijgen. Zonder dit Higgsdeeltje valt de hele theorie uit elkaar.

Geen wonder dus, dat sinds de formulering ervan in 1964, de jacht op het Higgs-deeltje is losgebarsten. Experimenten ter waarde van miljarden euros zijn opgezet en wereldwijd houden wetenschappers zich ermee bezig. De zoektocht naar het Higgs-deeltje vindt plaats in deeltjesversnellers: hierin worden deeltjes zoals bijvoorbeeld protonen met zúlke 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 werd gebouwd 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 theoretisch voorspelde Higgs-deeltje, kunnen we niet zomaar zeggen dat het ook echt om dit deel-tje gaat. Het zou ook zomaar een ander exotisch deeldeel-tje kunnen zijn dat we nog niet kennen! Gezien het belang van Higgs-deeltje is enige voorzichtigheid geboden. Daarom is het zaak de verschillende eigenschappen van het deeltje te checken. Één van deze eigenschappen is de kracht waarmee het Higgs-deeltje koppelt aan andere deeltjes. Dit wordt 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 kunnen worden. Dit wordt gedaan door eerst het Higgsmechanisme verder uit te zoeken: op welke 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 vergelijken met 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 nieuwe natuurkunde? Hoe dan ook gaat de deeltjesfysica spannende tijden tegemoet!

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Acknowledgements

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

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Introduction

The curious nature of humans towards the world around us has resolved in a search for the smallest building blocks of matter. This question of the smallest indivisible pieces can 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 field of particle physics. It has led to the development of the Standard Model of particle physics (SM), which describes all matter-interactions that we know. This elegant model was 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. It gives an overview of the twelve fundamental fermions and three of the four fundamental forces 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 predicted particles 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 postulated by 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, physics reached headlines. Forty years after the formulation of the SM and twelve Nobel prizes for contributions to this field later, all predictions had come true: the Higgs boson was discovered (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 energy and dark matter. Theoretical issues concern the hierarchy problem and charge-parity violation in combination with baryogenesis. Over the years, other theories have been developed, termed as ‘Physics Beyond the Standard Model’. These theories also make use of a Higgs mechanism, but different in some aspects from the original one developed in 1964.

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

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There are four fundamental forces at work in the universe: the gravitational force, the electromagnetic force, 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, is still an open question today. Luckily, for the field of particle physics, this is of little concern since at the microscopic scale of particles, gravity becomes negligible and the SM predictions still hold.

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Collider (LHC) started its second run with higher levels of energy in early June this year and the hopes are high. Expectations are that new physics will be revealed, which up till now, has been hidden in the higher order quantum corrections. These next to leading order processes are described by Feynman diagrams containing loops and turn out 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 the Higgs production and decay. Also, we want to understand how theory and experiment regarding the Higgs-boson can be related. We will do this by starting with a theoretical overview 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 contain loop diagrams and loop diagrams are important for future high-precision calculations and 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 describing this? Chapter two will elaborate further on gauge theories, the theoretical basics of the SM. What is local gauge invariance and why is it such an important concept in particle physics? Chapter three will focus on spontaneous symmetry breaking and the Higgs mechanism. How does mass arise in our theory? Chapter four will put the theoretical foundations from the previous chapters together and describe the electroweak sector as it is formulated in the current SM. The coupling constants of the Higgs to other particles will 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 decay of the Higgs boson. These diagrams contain a loop and the method for evaluating these loop diagrams will be discussed in detail. One of the calculations will be done using the program FORM. Chapter six will briefly discuss the experimental results concerning the coupling constants of the Higgs boson. Finally, two appendices are added: the first contains a short review of the basics of the Lagrange formalism and the second contains the calculations performed in FORM.

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1. The Standard Model

The SM describes our twelve fundamental fermions and three fundamental forces. These forces are described by the exchange of particles called gauge bosons. A special kind of gauge 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 yielded unwanted infinities. ’t Hooft and Veltman delivered the proof for the renormalization of 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 confirmed the 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 Quantum Field Theory (QFT) will be given, the mathematical language of the SM. Then, the elementary matter particles from the SM will be discussed, which are all fermions. Next, the fundamental forces and corresponding gauge bosons belonging to the SM will be 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 quantum field that satisfies the appropriate quantum mechanical field equations. The dynamics of QFT can be expressed in terms of the Lagrangian density (shortened from now to Lagrangian). Some of the most common Lagrangians are given in Appendix A. From the Lagrangians, the Feynman rules can be derived, which are used to construct Feyn-man diagrams. The FeynFeyn-man diagrams depict transitions between particles, governed by the exchange of force-carrying gauge bosons. A Lagrangian can be divided into a kinetic 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 themselves and represents the interaction terms, or the vertices in the Feynman diagrams.

One important aspect of QFT is symmetry. The SM is constructed from Lagrangians that obey gauge symmetry. In mathematical language, the SM is a non-abelian gauge QFT 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 theories and the electroweak theory. Roughly, each of these three gauge symmetries give rise to one 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

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this equation, the phenomenon of antiparticles arises. Each of the twelve fundamental fermions has its own corresponding antiparticle; a particle with the same mass but opposite charge.

There are two kinds of matter particles: leptons and quarks. Each of these groups consists of six particles, which are related in pairs, or ‘generations’. The six leptons consist of three electrically charged particles (the electron, muon and tauon) and three corresponding neutrinos, which are electrically neutral. The quarks carry color charge and appear in the pairs (up, down), (charm, strange) and (top, bottom). Pairs in a generation exhibit similar interaction behavior. The first generation contains the lightest particles, whereas the second and third generation are made up of the heavier and less stable particles. Most of the matter in the universe consists of particles that belong to the first generation, simply because the others are too heavy and decay to the next 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 and cannot be observed.

Electromagnetism is described by Quantum Electrodynamics (QED), where it is the photon that mediates the interaction between charged particles. QED works on charged particles, 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 and one neutral Z-boson. All three force particles have mass and thus the weak force works only on small distances.1 The weak force works on all twelve fermions, since the charge of 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 interactions between 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 differs from the others since it has spin-0. Moreover, it has a very high mass. Via interaction with the Higgs field, particles gain mass. The Higgs mechanism is subtle and will be elaborated further in Chapters 3 and 4.

1

The fact that massive particles have a smaller range, is easily obtained from Heisenbergs uncertainty relations ∆x∆p ∼ ~, ∆E∆t ∼ ~. Combining these with ∆x = c∆t and ∆E = mc2_{, leaves us with}

the relation ∆x = ~

mc and we see that the distance x and the mass m are inversely proportional. 2

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

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2. Gauge Fields

This chapter explores the phenomenon of gauge fields, which underlie all elementary particle interactions. A gauge field is a field that can be added to the theory without changing 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 more stringent. The idea of local gauge invariance goes back to the work of Weyl (1919). In the SM, all of the fundamental interactions are generated by a local gauge invariance.1 QED is generated by an Abelian gauge symmetry, with U (1) as its gauge group. The weak interaction and QCD are obtained by extending the local gauge principle to respectively SU (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 and non-Abelian case, starting with the first. The following is based on the books by de Wit et 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 the convention 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 gauge transformations 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 = i ¯ψγµ∂µψ − m ¯ψψ. (2.1)

We apply a transformation of the form

ψ0(x) = U ψ(x), (2.2)

U being an element of the U (1) gauge group. For U = eiθ, we note that the Lagrangian remains invariant. This is called a global phase transformation, since it gives rise to a phase 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 have U = eiθ(x)_{, or written out fully:}

ψ(x) → ψ0(x) = eiqθ(x)ψ(x). (2.3)

1

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

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Note that the parameter q has been added to measure the strength of the phase trans-formation. When inserting ψ0 into (2.1), the derivatives now also act on the local phase θ(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 → L0= L − q ¯ψγµ(∂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 be achieved by changing the derivative ∂µ into the covariant derivative Dµ:

∂µ→ Dµ= ∂µ+ iqAµ. (2.6)

Simultaneously, a new field Aµis introduced. In order to cancel the extra term in (2.5),

this field has to obey the following transformation:

Aµ→ A0µ= Aµ− ∂µθ(x). (2.7)

When combining the transformations for ψ and Aµ, the extra terms indeed render the

covariant derivative and we’ve obtained the behaviour that we wanted:

(Dµψ)0 = eiqθ(x)Dµψ (2.8)

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

L = i ¯ψγµ∂µψ − m ¯ψψ − q ¯ψγµAµψ. (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 always consists of a kinetic and potential term, and for the new gauge field Aµ we have yet to

add a kinetic term. Realizing that Aµis a vector field, we turn to the Proca Lagrangian

(Appendix A) which describes vector fields: L = −1 4F µν_F µν+ 1 2m 2_Aµ_A µ, (2.10) with Fµν= ∂µAν− ∂νAµ. (2.11)

Before adding these terms to equation (2.9), we need to check whether they are as well invariant under a local transformation. This turns out to be problematic. The Fµν-term

is 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 gauge

field will become more important later on, as we will see. Our final Lagrangian thus becomes

L = i ¯ψγµ∂µψ − m ¯ψψ − q ¯ψγµAµψ −

1 4F

µν_F

µν. (2.12)

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

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by putting in the electron charge q = −e and identifying Aµ as the photon field, we

recognize the QED Lagrangian. We have thus just formulated the Lagrangian for the electromagnetic force, simply by making the Dirac Lagrangian locally invariant.

As shown, the difference between global and local gauge transformations arises when calculating the derivatives of the fields. A local transformations picks up an extra term in 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 a

local 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 boson have to be massless in order to maintain local symmetry. This is an important feature of gauge theories: the introduction of a local invariance is always accompanied by the introduction of a massless vector field.

2.2. Non-Abelian case

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

We again start with the Dirac Lagrangian from (2.1). Note that since we are dealing with 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 = ψp

ψn.

2

For the non-Abelian case we also find that the Lagrangian is invariant under global transformations 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 are called the parameters and generators of the group, respectively, and both consist of three elements. T is also known as the weak isospin and can be written in the following way

Ti =

1

2σi, (i = 1, 2, 3) (2.14)

with σi being the Pauli matrices:

σ1 = 0 1 1 0 , σ2 = 0 −i i 0 , σ3 = 1 0 0 −1 . (2.15)

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

2

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

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has to be defined in terms of the generators of the group, which becomes more apparent for SU (2) group: ∂µ→ Dµ= ∂µ+ igWµiTi = ∂µ+ 1 2igW i µσi = ∂µ+ 1 2ig W3 µ Wµ1− iWµ2 W_µ1+ iW_µ2 −W3 µ . (2.16)

Note that there is an implicit I = 1 0

0 1 in front of the ∂µ. Because SU (2) has three

generators T, the change of derivative introduces three new gauge fields W_µi, 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 followcom-ing transformation property:

W_µ3→ W03

µ = Wµ3− ∂µα3− gε123α1Wµ2, (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)0 = eigα(x)·TDµψ (2.18)

We end up with the new Lagrangian:

L = i ¯ψγµ∂µψ − m ¯ψψ −

1 2igW

i

µσiψγ¯ µψ. (2.19)

The new gauge fields also need a kinetic term in the Lagrangian. Looking back at the Abelian case and simply translating this to Gµν = ∂µWν − ∂νWµ does not give us an

invariant term, again, because of the non-commuting generators. We can use the F -term as 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: G1_µν = 1 ig[D 1 µ, D1ν] = ∂µWν1− ∂νWµ1− igε123Wµ2Wν3. (2.21)

Note that Gµν is decomposed in terms of the group generators: Gµν = GaµνTa and again

the fact is used that the generators do not commute. The total expression for Gµν

becomes

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We then have our final Lagrangian: L = i ¯ψγµ∂µψ − m ¯ψψ − 1 2igW i µσiψγ¯ µψ − 1 4G µν_G µν. (2.23)

The extra term in the Gµν-term is needed for the kinetic term to be invariant and gives

rise to additional gauge boson self-interactions. As U (1) local symmetry worked well to describe the electromagnetic force, it seems logical to try SU (2) local symmetry as a description for the weak force. There is one problem however: we learned that the gauge bosons 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 symmetric Lagrangian.

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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 symsym-metry and now we are already breaking it? The fact is that the local symmetry has to be broken in order to gain a mass-term, but this happens in a subtle way. Explicit breaking of the symmetry by simply adding mass-terms to the Lagrangian ruins the invariance and also 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 the Higgs 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 the non-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 states that are not invariant. The phenomenon of SSB describes a Lagrangian that has a certain symmetry, but lowest-energy solutions that do not obey this symmetry. The symmetry is broken when a certain ground state is chosen from all possible degenerate ones as physical ground state or vacuum. This happens spontaneously as there is no preferred choice of ground state. The choice of the ground state hides the original symmetry of the theory and its accompanying dynamics.1 _{When we consider a real}

scalar field φ, we have a discrete symmetry with a finite number of ground states. It is more interesting however to look at continuous symmetries, which occur when dealing with complex scalar fields. When SSB happens for a continuous symmetry, massless spinless particles pop up, called Goldstone bosons. For each generator of the broken symmetry 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 of losing our massless gauge vector bosons, we gain extra massless spin-zero bosons. When SSB happens to a local symmetry and a smart choice of gauge is exploited however, an elegant interplay of these massless bosons results in massive gauge bosons and removes the Goldstone bosons. This is called the Higgs mechanism.

1

A good physical example of SSB is the ferromagnet. Below the Curie temperature TCand in absence

of an external magnetic field, the spins are randomly oriented. This is a rotationally invariant ground state. However, the spin-spin interactions give rise to a spontaneous magnetization, by aligning the spins in the state of lowest energy. This ground state is not rotationally invariant. The direction of the magnetization is random, which translates to an infinitely degenerate ground state.

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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

4F

µν_F

µν+ (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 the potential. 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 options hold:

• µ2_{> 0: We get a unique minimum at φ} 0 = 0,

• µ2_{< 0: We get a continuous set of minima φ}2

0= φ21+ φ22 = −µ2

2λ = v2

2 .

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

The physical vacuum state is chosen to be φ1 = v, φ2 = 0. To explore the physical

spectrum belonging to this vacuum state, we can consider the perturbations of φ around

2

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

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the vacuum state, which describe the excitations of the field. In order to do this, we shift the fields with η(x) in the φ1 direction and ξ(x) in the φ2 direction, as can be seen

in 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 z }| { 1 2(∂µη)(∂ µ_{η) − λv}2_η2₊1 2(∂µξ)(∂ µ_ξ) | {z } massless ξf ield −

massive gauge field

z }| { 1 4F µν_F µν+ 1 2g 2_v2_A µAµ−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 gauge symmetry. Constant factors in front of terms quadratic in the fields can be associated with mass, by equalizing them to 1₂m2. 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 we}

were not looking for.3

3.2.1. Unitary gauge

As said in the previous section, the way to get rid of these unwanted extra massless Goldstone bosons, is by a smart choice of gauge.4 This choice of gauge corresponds to taking θ(x) = −ξ(x)_qv in U = eiqθ(x). Remember that we were discussing a locally gauge invariant Lagrangian, so this choice of gauge does not alter the theory. The gauge-transformation becomes: φ(x) → φ0(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

0

µ(x) = Aµ(x) −

1

gv∂µξ(x). (3.6)

By substituting the first order approximation of (3.3) in (3.5) we get φ0(x) = e−iξ(x)v (√1 2(v + η(x))e iξ(x)/v₎ _(3.7) = √1 2(v + η(x)). (3.8) 3

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

4_{There is another physical reason for changing of gauge than simply wanting to lose the Goldstone}

boson. When taking a closer look at the Lagrangian in (3.4), we notice that an extra degree of freedom has appeared, compared to (3.1). The mass of the gauge field gives rise to an extra longitudinal degree of freedom.

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This choice of gauge is known as the Unitary Gauge. Putting it into our Lagrangian, we get L = 1 2(∂µη)(∂ µ_{η) − λv}2_η2₋1 4F µν_F µν+ 1 2g 2_v2_A0 µA 0_µ − V_int. (3.9) The Goldstone bosons ξ have disappeared and only true physical fields are left.5 The field 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) → κ0(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 no Goldstone 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 consist of 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 F

µν _{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 set of degenerate minima arises. We break the symmetry by choosing as physical minimum:

φ0 = 1 √ 2 0 v . (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 each generator of the symmetry group, as we know from to Goldstone’s theorem(Goldstone 1961). In order to avoid this, we right-away take the unitary gauge, which is

U (x) = exp −i

v (ζi(x)T

i₎

. (3.12)

Thus we have new transformed fields

φ(x) → φ0(x) = exp −i v (ζi(x)T i₎ φ(x) (3.13) = √1 2 0 v + η(x) . (3.14) 5

Notice that the degree of freedom of the Goldstone field ξ has been replaced by the massive gauge field Aµ, and we have four degrees of freedom, just as before symmetry breaking. It is often said

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using equation (3.11). When substituting this term in the Lagrangian, we will see that the field η(x) has a mass term, the Goldstone fields ξ(x) are absent and the gauge fields W_µ1 and W_µ2 have acquired the same mass, whereas W_µ3 remains massless.

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

2Φ(x) 0

ρ(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 _√0ρ

2. Again we

need to redefine the gauge fields in accordance to the new transformation. It is also possible to use the unitary gauge straightaway, by setting Φ equal to the identity matrix and thus replacing the doublet field φ by _√0ρ

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 gauge field, respectively. If we want to describe nature with the three massive gauge bosons W+, W−, Z and the massless photon, it turns out we need a combination of the Abelian and non-Abelian theories.

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4. Electroweak theory

The electroweak part of the SM was developed by Glashow (1961), Salam (1969) and Weinberg (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± and Z-bosons. The way to massive gauge bosons is again through symmetry breaking via the Higgs mechanism (Higgs 1964, Brout & Englert 1964). The theory was finalized when ’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 accounted for and finite predictions can be made.

This chapter will give a short review of the electroweak part of the SM. It builds on the 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 invariance

and the appropriate transformations, continue with the Higgs mechanism that gives mass to the gauge bosons and then review the couplings that the Higgs has to bosons as well as fermions. For this, the book of Quigg (2013, chapter 6) and the notes of Merk et al. (2014) will be used.

4.1. SU (2)

L

× U (1)

Y

We 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 new

charge called the weak hypercharge Y . The accompanying transformations are slightly changed as well: U (x) = exp ig0Y 2β(x) . (4.1)

The coupling parameter is now defined as g0₂, and a new field Bµ arises.

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

Q = T3+1

2Y, (4.2)

where T3 is the third component of the weak isospin, as described in section 3.3. This relation 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 weak bosons W±only couple to left-handed fermions.1 We therefore define a new doublet for the SU (2): νe

e

L.

2 _{The right-handed particles are grouped in singlets (ν)}

R and (e)R,

1_{Left- and right-handedness in particle physics is determined by spin and momentum: a particle is}

right-handed when its spin and momentum have the same direction, and left-handed when these are in opposite direction of each other. These are also called the chiral states of the fermions.

2

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given that they only interact with the U (1) part of the symmetry. The transformation rules 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 consider the 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 covariant

derivative, consisting of two terms, one for each group: Dµ= ∂µ+ i

g0

2BµY + igTiW

i

µ. (4.4)

Here we have 1 gauge field Bµ from U (1) and three gauge fields Wµ from SU (2). Note

that for the right-handed singlet, the term with the Wµ does not contribute. In current

description, 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 as

described by the SM, and from which we can make real predictions which we can test at colliders such as the LHC.

Following the procedure from the previous chapter, we pick µ2< 0 and get an infinite set of possible minima, from which we choose our physical minimum to be φ = √1

2 0

v. 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 right away turn to the unitary gauge:

φ → φ0= exp −iξσ 2v | {z } 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 1 µ± iWµ2), (4.7) Zµ= −g0_B µ+ gWµ3 p g2_{+ g}02 , (4.8) Aµ= gBµ+ g0Wµ3 p g2_{+ g}02 . (4.9)

The charged W -boson fields W_µ± still behave according to SU (2). However, the Bµ and

W_µ3 field have formed a linear combination which results in the photon field Aµ and the

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4.2. Particle masses

From the expression for the Lagrangian that follows from the unitary gauge, the masses for the new gauge boson fields can be identified. The term in the Lagrangian that generates the masses is (Dµφ)†(Dµφ). The mass terms appear as terms quadratic in the

gauge boson fields. By identifying the factor in front of the fields as 1₂m2_x, we get the following masses: m2_W = 1 4g 2_v2_, m2_Z = 1 4v 2_(g2_{+ g}02_), mA= 0, mη = −2µ2> 0. (4.10)

The relation between the parameters g and g0 is defined as by the weak mixing angle θW and is given by g

0

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

masses: mW mZ = 1 2vg 1 2v p g2_{+ g}02. (4.11)

From this we can conclude that the Z-boson is heavier than the W -boson, which is in accordance 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), and

we 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

gW W H = 2m2_W v , gZZH = 2m2_Z v , gf f H = mf v ,

with λv2= −µ and the relation m2_H = −2µ2.

(4.12)

These can be retrieved by taking a smart look at the mass terms in (4.10). Remember that via the Higgs mechanism, the new term (v + η) entered the theory, with η the Higgs field. The masses arise from the v2 _{part of the new term, not involving the new}

η field. The interaction parts arise from the terms that d´o involve the η field, and in a linear way. This gives the three-point interactions between the gauge field and the Higgs field. 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 1₂g2(2vη) (where the 1₄ became a 1₂ since that is how it appears in the Lagrangian). Now, rewriting this in terms of the m2_W as formulated in (4.10), we are left with a factor 2m2Wη

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This thus leaves us with the interaction or coupling strength gW = 2m 2 W v , 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 the gauge boson couplings, which are quadratically in mass, the fermion coupling is linear in its mass. The story for the fermions is slightly different, since they carry a chiral state. Left-handed fermions appear in doublets whereas right-handed fermions come in singlets. This makes that they transform differently under the given SU (2) × U (1) transformation, and makes terms like (ψ_LψR+ ψRψL) not gauge invariant. This can be

solved however, by combining the left- and right-handed states with the complex scalar field φ from the Higgs mechanism: (ψ_LφψR+ ψRφψL). When using φ =

0

v+η_√

2 , we will

get the right invariant terms in our Lagrangian. It turns out that the electron mass term can be identified as me = √λv₂, 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 mf

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

fermions to the Higgs field is given by √λ 2 =

mf

v and is proportional to the mass of the

fermion. 3

3

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

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5. Higgs production and decay

In this chapter, we will take a closer look at the Feynman diagrams associated with the Higgs production and decay processes.

In a hadron collider like the LHC, the dominant Higgs production mechanism is through gluon fusion and a virtual fermion loop. As we learned in the previous chapter, the coupling of particles to the Higgs boson is proportional to their mass, so the most common is a top-quark loop (Rainwater 2007).

When it comes to the decay, the Higgs boson can decay to all SM particles, because of its high mass. The largest branching ratio is to bottom quarks; this process happens about 57.8 % of the time (Thomson 2013). Although the decay to quarks is the most common, 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 like gluons or photons. This might sound strange at first, remembering that the Higgs only couples to massive particles. Decay to massless particles is possible however, namely via an intermediate loop containing massive particles. Decaying to gluons will again lead to jets and brings us no further. The decay to two photons however, is more easy to detect as their signal-to-background ratio is larger. It occurs a lot less often though (2/1000 decays, (Dittmaier et al. 2012)). It therefore takes more time and data to detect, but when 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 of massive particles. Loop diagrams however, often lead to unwanted infinities. In order to avoid this, the method of renormalization is employed. There are roughly four methods of renormalization possible, where this dimensional regularization is best when one wants to preserve the symmetries of the theory (Kleinert & Schulte-Frohlinde 2001, Chapter 8). Before we arrive at a point where this method is necessary, we first encounter some difficult tensor integrals. To get these out of the way, the Passarino-Veltman method will be applied (Passarino & Veltman 1979). Both methods will be explained throughout the 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 in quantum corrections or loops, so it is important to understand the mathematics behind these loops for future precision calculations.

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

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Figure 5.1.: At lowest order there are two diagrams for the process gg → H. We see two incoming gluons which form a top-quark loop from which a Higgs boson comes forth. The charge flow is is indicated by the arrows on the fermion lines. The momentum flow 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 to the Higgs production via gluon fusion, which are depicted in figure 5.1. The notes of Bentvelsen et al. (2005) will be followed closely for this purpose.

We start with the Feynman diagrams and corresponding Feynman rules. The latter can be found in Peskin & Schroeder (1995). Since gluons are massless particles, we can write 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 against the direction of the charge flow and use the needed Feynman rules. This will lead to the following result: M = (−igs)2 −i√yt 2 i3Tr[ta, tb](−1)ν(λ1, k1)µ(λ2, k2) Z _dd_l (2π)d 1 D1D2D3 Tr[(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 compute the matrix element, we have to perform the trace and then calculate the integral over the 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 gµνl · l remain and we are left with a tensor integral of the form Cµν.

1

The matrix element is an element from the scattering matrix S, that relates the inital and final state of a scattering process via |ψouti = S |ψini.

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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 into easier-to-solve scalar integrals. It does involve quite some steps however.

We start with defining the various integrals we have to work with.

Cµν = Z _dd_l (2π)d lµlν D1D2D3 , Cµ= Z ddl (2π)d lµ D1D2D3 , C0 = Z ddl (2π)d 1 D1D2D3 , Bµ(i, j) = Z _dd_l (2π)d lµ DiDj , B0(i, j) = Z _dd_l (2π)d 1 DiDj . (5.4)

Our goal is to express the tensor integral Cµν in terms of the scalar integrals C0 and

B0. The first step is to decompose the tensor Cµν, using the vectors k1, k2 and gµν.

Cµν = k1,µk1,νC21+ k2,µk2,νC22+ {k1, k2}µνC23+ gµνC24 (5.5)

(with {k1, k2}µν ≡ k1,µk2,ν+ k1,νk2,µ). We call C21, C22 etc. the form factors. We want

to write these form factors in terms of the scalar integrals C0and B0. In order to achieve

this, we first contract the tensor Cµν with each of the external momenta k. We will call

these new vectors vµ and wµ:

vµ= Cµνkν1 = k2,µ m2_H 2 C22+ k1,µ m2_H 2 C23+ k1,µC24 wµ= Cµνkν2 = k1,µ m2 H 2 C21+ k2,µ m2 H 2 C23+ k2,µC24. (5.6)

Next, we contract the loop momentum with the external momenta: k1· l = − 1 2(D2− D1− k 2 1) k2· l = 1 2(D3− D1− k 2 2). (5.7)

Note that this term also appears when contracting the integral expression of Cµν with

the external momenta. Combining (5.4) and (5.7), one denominator will disappear and the following expressions remain:

vµ= Cµνk1ν = − 1 2(Bµ(1, 3) − Bµ(2, 3)) wµ= Cµνk2ν = 1 2(Bµ(1, 2) − Bµ(2, 3)). (5.8)

One step is done: the Cµν integrals have turned into Bµ integrals. These Bµ integrals

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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 using equation (5.7) again. Reducing Bµ(2, 3) takes some more work, since we are dealing with

both k1 and k2 in the denominators here. We first have to apply a shift in momentum:

l = l0+k1. Integrating over l

0

then leaves us with Bµ(2, 3) = B

0

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 can

put these scalar integrals in the expression for vµand 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 vµand wµin terms of the form

factors of Cµν and in terms of the scalar integrals B0(i, j). In order to express Cµν in

scalar integrals, which is our goal, we thus need to relate the form factors to the scalar integrals. This can be done by introducing projection operators P_kµ

1 and P

µ

k2, which

behave like P_kµ

ikj,µ= δij. We have to the following representation of these operators:

P_kµ 1 = 2 m2_Hk µ 2 P_kµ 2 = 2 m2 H kµ₁. (5.13)

If we let these operators work on both expressions we obtained for vµand wµ(equations

(5.8) and (5.12)) we can define a set of scalar functions Ri to relate both equations:

R3= P_kµ₁vµ= m2_H 2 C23+ C24 R4= P_kµ₁wµ= m2_H 2 C21 R5= Pkµ2vµ= m2_H 2 C22 R6= Pkµ2wµ= m2_H 2 C23+ C24. (5.14)

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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 that we can express the form factors in scalar integrals. We do this by expressing the form factors in terms of the scalar functions Ri. This takes more effort than simply reversing

the equations in (5.14) and we need the introduction of another projection operator: Pµν = 1 d − 2(g µν_{− P}µ k1k ν 1 − P µ k2k ν 2), (5.16)

which selects the form factor C24 when acting on (5.5). We let it act on the integral

expression for Cµν. Note the use of a little trick:

gµνCµν = Z _dd_l (2π)d l2− m2 D1D2D3 + Z _dd_l (2π)d m2 D1D2D3 = B0(2, 3) + m2C0. (5.17)

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

C21= 2 m2_HR4 C22= 2 m2 H R5 C23= 2 m2_H(R6− C24) C24= 1 d − 2(B0(2, 3) + m 2_C 0− R3− R6). (5.18)

Putting all this together and combining equations (5.18), (5.15) and (5.5), we obtain the 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, k2}µν 2 m2_H[ 1 4B0(2, 3) − ( 1 d − 2(B0(2, 3) + m 2_C 0− 1 2B0(2, 3))] + gµν[ 1 d − 2(B0(2, 3) + m 2_C 0− 1 2B0(2, 3)]. (5.19)

5.2. Dimensional Regularization

We are now left with a loop integral expressed in scalar integrals B0and C0. We will now

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

First, we put the expression obtained for Cµν back into the integral for the matrix

element. Using _d−21 ∼ 1₂(1 + ε + ε2), the result becomes (after some calculations): M = (1· 2)a + (k1· 2)(k2· 1)b

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

b = −2a m2_H

(5.20)

(Bentvelsen et al. 2005). We have obtained an expression for the matrix element in terms of the easier-to-solve scalar integrals, which is what we wanted. All that is left to do, 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 ε

or higher orders of it. Since the regulator will be set to zero at the end, as we go back to four dimensions, the terms proportional to ε and higher order will disappear and we are 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 Z ddl (2π)d 1 (l2_{− m}2_{)((l + k} 1+ k2)2− m2) . (5.21)

Note that the shift in momentum: l = l0 + k1 has been applied again. A factor µ2ε has

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

1 AB = Z 1 0 dx 1 [xA + (1 − x)B]2. (5.22)

This leaves us with B0 = µ2ε Z ddl (2π)d Z 1 0 dx 1 [(1 − x)(l2_{− m}2_{) + x(l + k} 1+ k2)2− m2)]2 . (5.23)

We now use κ = k1+ k2, work out the brackets and apply the shift l = l

0 − xκ, so that we obtain B0 = µ2ε Z ddl (2π)d Z 1 0 dx 1 [l2_{+ xm}2 H − x2m2H − 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: Z _dd_l (2π)d[l 2_{− M}2_{+ iε}2_]−s = (−1)si(4π) ε 16π2 Γ(s −d₂) Γ(s) [M 2_{− iε]}d₂−s . (5.25) 2

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

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Combining this with equation (5.24) , we see that M2 = xm2_H − x2_m2

H − m2 and find

the following result for B0:

B0= µ2ε i(4π)ε 16π2 Γ(ε) Γ(2) Z 1 0 dx[x2m2_H − xm2_H+ m2]−ε. (5.26) Remember that we were interested in the poles in ε. To recognize these, we first have to rewrite

Γ(ε) = 1

εΓ(1 + ε). (5.27)

We have now identified the pole 1_ε. 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 seen}

that the whole integral over x can thus be approximated as 1. Further, we recognize that Γ(2) = 1 and obtain B0 = µ2ε_16πi2(

1

ε − γE + ln 4π). The 1

ε term will cancel with

the 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 been

written out in detail for B0, we will now only state the steps and some extra details if

needed, 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 setting ε → 0. The first step is to write out C0, by partially working out the brackets in

the denominator. This time we do not need an extra factor to keep the dimensions correct. There are three terms in the denominator this time, which means we will have to make use of Feynman’s trick twice. The second time, there is a squared term in the denominator, for which an extended version of Feynman’s trick is needed:

1 Aα1 1 ...A αn n =Γ(α1+ ... + αn) Γ(α1)...Γ(αn) Z 1 0 dx1...dxnδ(1 − x1− ... − xn) xα1−1 1 ...x αn−1 n [x1A1...xnAn]−(α1+...+αn). (5.29)

Note that we will get two integrals now, over x and over y. Then again, some rewriting and shifting in the integration parameter l is needed. The suitable shift is l0 = l + K, with K = x(1 − y)k1− yk2. Then, C0 is written in such a way that equation (5.25) can

be applied again. This time, we will find a factor Γ(1 + ε) in the answer, which contains no pole in ε. The limit can thus be taken straightaway and will lead to a finite answer. The result becomes:

C0= (−1) 1 16π2i Z 1 0 dx Z 1 0 dy y (m2_{− xym}2 H(1 − y))3 . (5.30)

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

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Bentvelsen et al. (2005) The result becomes: C0 = i 16π2_m2 H f (η) (5.31) withf (η) =(arcsin 2₍_√1 η), for η ≥ 1, −1₄[ln1+ √ 1−η 1−√1−η− iπ] 2_, _{for η < 1} with η = 4m2 t m2 H . (5.32)

We can now put the values found for B0 and C0 back into equation (5.20) and obtain

the expression for the matrix element. This matrix element is needed to calculate the cross section σ, which is associated with the production rate, or how often this process of Higgs production via gluon fusion might happen. The formula used for this is

σ = 1

2m2 H

Z

dP SΣspins,color|M|2 (5.33)

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

σ0 = GFα2 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 is

Fermi’s constant. This cross section can be measured in experiments and used to test theoretical 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, the correctness of this calculation and the use of dimensional regularization was questioned by 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 to be 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 the previously described production. Again, we are dealing with a quark loop. The main difference is that our external particles are massless photons this time. They do not carry any color charge like the gluons from the Higgs production. This will lead to a slight change in the matrix element: the trace over the gluon colors will disappear, and the vertex-factors for photons and quarks will be used. The needed Feynman rules can be found in Peskin & Schroeder (1995).

The same tensor integral as before will arise. Then again, one needs to follow the above procedure of the Passarino-Veltman reduction, followed by dimensional regularization. Since we are working with almost the same, but reversed, diagram, the output is the same and the calculation can be reviewed in the previous section.

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Figure 5.2.: At lowest order there are two diagrams for the process H → γγ, where only one is depicted here. The other one can be obtained by exchanging the photons. The Higgs decays through a top-quark loop and then forms two photons. This decay process 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 with massive 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 a W-loop. Another diagram can be obtained by exchanging the two photons in the triangle diagram on the left. These two triangle diagrams give the same amplitude however, so we simply included a factor 2. Note, the loop momentum is called p in these 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 in equation (5.1) hold. This time we are dealing with W-bosons in the loop, which are massive particles. The slight change in particles also leads to a slight change in the matrix 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 l to 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.

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This matrix element again comes down to a tensor integral, for which the method of Passarino-Veltman will be used. It is trusted that this was explained in enough detail in the previous section, so we will work through it a bit quicker this time, mentioning only 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 and with 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− m 2 H). (5.37)

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

xµ= Cµνkν1 = 1 2(Bµ(1, 3) − Bµ(2, 3)) zµ= Cµνkν2 = − 1 2(Bµ(1, 2) − Bµ(2, 3) − m 2 HCµ). (5.38)

Note the extra Cµ term in the expression for zµ. We now proceed in the same way for

both 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)

k₂µCµ= m2H · C12. (5.41)

Applying (5.37) on the integral expression for Cµ leads to the following expressions for

C11 and C12: C11= −1 m2_H(B0(1, 2) − B0(2, 3) − m 2 HC0) (5.42) C12= 1 m2 H (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 0 1(1, 3) + k1,µB0(1, 3), with B 0 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 as

Bµ(2, 3) in the calculation for the production. Note that B

0

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 with

Coupling constants of the Higgs boson