The Higgs boson coupling to polarised W bosons in vector boson fusion

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University of Amsterdam

MSc Physics

GRAPPA/particle

Master Thesis

The Higgs boson coupling to polarised W bosons

in vector boson fusion

A study of cross sections and jet kinematics

by

Martijn Pronk 10191739 June 2016

54 ECTS project + 6 ECTS thesis September 2015 - June 2016 Supervisors:

Prof. dr. ing. Bob van Eijk Magdalena Slawinska PhD.

Examiner: Prof. dr. ir. Paul de Jong Prof. dr. Robert Fleischer

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Abstract

We study the W+_W−_{→ W}+_W− _{scattering process theoretically. We calculate the matrix elements}

for the longitudinal part of this process and show that without the Higgs boson the process diverges for high centre of mass energies, which violates unitarity. Then we show that the Higgs boson propagator diagrams exactly cancel this divergence in the standard model. The main motivation for the study of the Higgs boson couplings to polarised W bosons is to verify whether this cancellation is exact. We investigate the Higgs boson coupling to polarised W bosons in vector boson fusion (VBF) pp → Hjj

where the Higgs boson decays into W bosons which decay leptonically H → W+_W− _{→ e + ν}

eµ−ν¯µ.

We define the longitudinal coupling aL =

g_HWLWL

gHW W and transverse coupling aT =

g_{HWT WT}

gHW W . In the

standard model aL= 1.0, aT = 1.0. We test how varying these parameters changes the jet kinematics

and the cross section of the signal process. The study is performed at parton level. We apply the VBF selection criteria presented in ref. [1] and include irreducible background processes. We show

that the t¯t background makes up a significant part of the total background. From the cross sections

we calculate the number of events at an integrated luminosity of 300 fb−1. Given these conditions we

conclude that in order to achieve a median significance for the discovery of a signal of 5.0, we find that we are sensitive to the parameter region where the signal cross section is larger than 0.295 fb if the background is known with a negligible uncertainty and larger than 0.440 if the uncertainty is not negligible and the background is a nuisance parameter for the test statistic. We test the region with

aL and aT within 30% off the standard model. We find that in this region we are sensitive mostly to

the longitudinal coupling. We also calculated the exclusion regions for aL and aT at 95% CL given

the standard model signal hypothesis. Based on the exclusion regions for the norm of the vectorial sum of the jet transverse momenta we place the limits on aL and aT at respectively 0.92 < aL< 1.08

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Introduction

Before the discovery of the Higgs boson, one of the deficiencies of the Standard Model of Par-ticle Physics (SM), was the lack of ability to explain the presence of the vector boson masses. The particles in the SM and their properties are described by the SM Lagrangian. One of the features of Lagrangian theories is the capability of explaining fundamental forces by requiring some invariance of the Lagrangian. In order to have mass generating terms in the Lagrangian it is necessary to break the electroweak SU(2)L×U(1)Y symmetry. The minimal model in

which one can break this symmetry without breaking the invariance of the Lagrangian is the Higgs Mechanism, in which the electroweak symmetry is broken spontaneously by the Higgs doublet, a scalar doublet with a potential that has a non-zero expectation value. By introducing this Higgs doublet the presence of at least one scalar boson emerges. And indeed in the year 2012 the observation of a new particle compatible with the SM Higgs boson was announced [2] [3].

As is said, the SM Higgs mechanism is the minimal model to induce spontaneous electroweak symmetry breaking (EWSB). Although it is expected that physical laws in general prefer to be simple and aesthetic, it is not excluded that a more involved mechanism causes EWSB. For example there may be not one but two Higgs doublets, resulting in five different Higgs bosons, of which the discovered particle is one. In order to detect deviations from the SM Higgs Mechanism one has to probe the properties of the observed particle, like its production and decay modes.

In this thesis we study the coupling of the Higgs boson to polarised W bosons, the vector bosons inducing the charged electroweak currents. The massive W bosons have three po-larisation states: two transverse and one longitudinal. The transverse popo-larisation states correspond to the electroweak gauge bosons before EWSB. The longitudinal polarisation is generated by the EWSB inducing Higgs Mechanism and it is expected that possible

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devia-tions from the SM Higgs Mechanism are visible in the coupling of longitudinally polarised W bosons to the Higgs boson. Therefore we propose to research the Higgs boson coupling to polarised W bosons (HWW). This study serves as preparation for analysing the HWW coupling in data obtained by the ATLAS detector.

The next chapter will treat the theoretical framework from which the HWW coupling emerges and will introduce the channel for probing the HWW coupling. We will define two param-eters aL and aT, which represent the coupling strength of respectively the longitudinal and

transverse polarisation states. The detector chapter will give a general review of the Large Hadron Collider and the ATLAS detector. A small chapter will be dedicated to Monte Carlo event generation and will be followed by two analysis chapters. The first chapter contains a study in cross sections and the second one a study in jet kinematics, which are expected to be characteristic in probing the HWW coupling. These chapters research how aL and aT affect

the kinematics of the signal channel. In the last chapter we will summarize the findings in the analysis chapters and the conclusions drawn throughout this thesis.

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Chapter 1

From Lagrangian to vector boson

fusion

The cross sections for Standard Model (SM) interactions can be calculated from the SM Lagrangian. In this and following sections the focus lies on the electroweak (EW) and the scalar part of the Lagrangian. We will see that these Lagrangians are

L_EW = iX

ψ

¯

ψγµDµψ (1.1)

L_H = (Dµφ)†(Dµφ) + µ2φ†φλ(φ†φ)2. (1.2)

The sum over ψ is the sum over the weak isospin doublets for the spinors ψ. The scalar field φ is the weak isospin Higgs doublet and Dµ is the covariant derivative for the EW SU(2)L×

U(1)Y symmetry group

Dµ= ∂µ+ igW 2 σiW i µ+ ig0 Y 2Bµ, (1.3)

in which ∂µ is the partial four-derivative, σi, i ∈ {1, 2, 3} are the three Pauli spin matrices,

gW is the coupling strengths for the SU(2)L weak interaction with the fields Wµ1, Wµ2, Wµ3 and

g0 is the coupling strength for the U(1)Y hypercharge with the field Bµ. Before arriving at

these equations, we will first summarize the basic Lagrangian formalism and show how SU(2)L

invariance provides a natural way to explain the weak interaction. In the following section it is illustrated how the physical W and Z bosons are constructed after EW unification and how EW unification emerges from the Higgs mechanism. In the section thereafter the polarisation

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of vector bosons is discussed. In section 1.4 it is illustrated how the existence of the Higgs boson repairs the unitarity violation in longitudinal W boson scattering and a glimpse of the calculation of the full W boson scattering will be given. In the next to last section the channel for probing the Higgs boson coupling to two W bosons is introduced and in the last section we define how we probe the channel for beyond the standard model properties.

1.1 Gauge invariance in the Lagrangian

In order to show how Electroweak Symmetry Breaking (EWSB) emerges from the Higgs mech-anism in a natural way, we have to turn to Quantum Field Theory (QFT) and particularly to the Lagrangian formalism in QFT.

1.1.1 Quantum Field Theory and Lagrangian formalism

Unlike classical quantum dynamics, relativistic QFT uses Lagrangian densities, in which the generalised coordinates qi are replaced by fields φi(xµ)

L qi, dqi dt → L φi, ∂φi ∂xµ = L (φi, ∂µφi) , (1.4)

where xµ is the four-vector (t, x, y, z) and ∂µφi the covariant derivative of the field φi. The

Lagrangian and the Lagrangian density are related by L =R Ld3x. It is customary in particle physics to refer to the Lagrangian density as simply the Lagrangian. Analogously to classical quantum dynamics, the Euler-Lagrange equation for the fields φiare obtained (see for example

Schwartz for derivation [4])

∂µ ∂L ∂(∂µφi) − ∂L ∂φi = 0. (1.5)

Hence, qi is replaced by φi, ˙qi by ∂µφi and _dtd by ∂µ. In QFT, scalar particles are considered

to be excitations of a scalar field. The general form of the Lagrangian for a free massive scalar field is given by

L_S = 1

2 (∂µφ)(∂

µ_{φ) − m}2_φ2_, _(1.6)

where the first term represents the kinetic energy and the second term is the mass term for the field φ. Its Euler-Lagrange equation is the well-known Klein-Gordon equation. Likewise,

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for a free massive spinor field described by the Lagrangian

LD = i ¯ψγµ∂µψ − m ¯ψψ (1.7)

the Dirac equation is obtained.

1.1.2 SU(2)L gauge invariance

The strength of the Lagrangian formalism is that invariance of the Lagrangian under a trans-formation implies the presence of a symmetry. The fundamental forces in the SM can be linked to symmetries that follow from invariance of the Lagrangian under local phase trans-formations. In order to illustrate how the properties of the SM weak interaction emerge from the Lagrangian formalism, the invariance under SU(2) local phase transformations is considered ψ(x) → ψ0(x) = exp igW 2 αi(x)σ i ψ(x), (1.8)

in which the functions αidescribe the local phase as a function of space-time. The generators

σi of the SU(2) group are the three Pauli spin matrices, which satisfy the commutation relations

[σi, σj] = ijkσk, (1.9)

with i, j, k ∈ {1, 2, 3}. Since the Pauli matrices do not commute, the SU(2) group is not Abelian, as is for example the U(1) group. Substituting equation (1.8) into the kinematic term of the Dirac Lagrangian, we obtain

L0₁ = i exp −igw 2 αi(x)σ i ¯ ψγµ∂µ exp igW 2 αi(x)σ i ψ(x) = i exp −igw 2 αi(x)σ i ¯ ψγµ exp igW 2 αi(x)σ i ∂µψ + igW 2 (∂µαi(x))σ i_exp igW 2 αi(x)σ i ψ = L1− igW 2 ψγ¯ µ_(∂ µαi(x))ψ. (1.10)

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Substituting equation (1.8) into the mass term of the Dirac Lagrangian gives L0₂= m exp −igw 2 αi(x)σ i ¯ ψ exp igW 2 αi(x)σ i ψ = m exp −igw 2 αi(x)σ i exp igW 2 αi(x)σ i ¯ ψψ = m ¯ψψ = L2. (1.11)

The L2 term is invariant, but the free Dirac Lagrangian as a whole is not invariant under an

SU(2) local phase transformation

L → L0= L − igW 2 ψγ¯

µ_(∂

µαi(x))ψ. (1.12)

In order to satisfy invariance of the Lagrangian, the covariant derivative Dµ is introduced.

The covariant derivative for the SU(2) transformation is

Dµ= ∂µ+

igW

2 σiW

i

µ(x), (1.13)

where Wi= {W1, W2, W3} are three new fields. Later on it is shown that ψ in equation (1.8) has two components. Therefore, strictly we have to write Dµδab and ∂µδab in order to let the

derivatives work on both components of ψ. For readability δab is omitted, but it is implicitly

present. The gauge transformation of the covariant derivative of equation (1.13) is given by

D_µ0 = ∂µ+

igW

2 σi· W

0i

µ(x). (1.14)

The transformation properties of the new fields can be derived by using the gauge transfor-mation of the covariant derivative and the the gauge invariance of equation (1.7)

W_µk→ W_µ0k = W_µk− ∂µαk− gWijkαiWj,µ. (1.15)

The first two terms of equation (1.15) are the same as for a U(1) transformation. As stated before, the SU(2) group is non-Abelian. Therefore, to satisfy the gauge invariance, the third term in equation (1.15) is necessary. This term gives rise to self interactions between the newly introduced fields Wi, just as the second term in equation (1.13) gives rise to interaction terms of the form ¯ψW_µiψ, which represent interactions between a gauge boson, a fermion and an anti-fermion.

The last step is adding a kinetic term for the fields Wi to the Lagrangian. For the U(1) group the term −1₄FµνFµν would suffice, where the field strength tensor Fµν is defined as

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Fµν = ∂µAν − ∂νAµ if Aµ would be the gauge field. However, for the SU(2) group such a

term would not be gauge invariant. To restore the gauge invariance a self interaction term similar to the third term of equation (1.15) needs to be added to the field strength tensor, such that it becomes

W_µνk = ∂µWνk− ∂νWµk− gWijkWi,µWj,ν. (1.16)

In the end, the appropriate Lagrangian that is invariant under SU(2) gauge transformations is L_{SU (2)} = i ¯ψγµDµψ + m ¯ψψ − 1 4W i µνW µν i , (1.17)

where Dµ is defined as in equation (1.13) and Wiµν as in equation (1.16).

With the left- and right-handed (LH and RH) chiral projection operators PR = 1₂(1 + γ5)

and PL = 1₂(1 − γ5), fermions can be decomposed into LH (with PL) and RH (with PR)

chiral components. For anti-fermions PR ’selects’ the LH component and PL the RH

com-ponent. From experiments it is known that W boson interactions maximally violate parity [5]. Therefore, the weak interaction has a pure V − A structure, which means that the weak interaction current has an equally large vector component (V ) and axial vector component (A). A vector-like interaction is of the form ¯ψγµψ and changes sign under a parity transfor-mation, while an axial vector-like interaction is of the form ¯ψγµγ5ψ and does not change sign under a parity transformation. Therefore, the weak interaction has a term γµ(1 − γ5). The coupling of a W boson with RH fermions then contains the term

jµ∝ ¯ψRγµ(1 − γ5)ψR ∝ ¯ψPLγµ(1 − γ5)PRψ ∝ ¯ψ1 2(1 − γ 5_)γµ_{(1 − γ}5₎1 2(1 + γ 5_)ψ ∝ ¯ψ1 4(1 − γ 5_)γµ_{(1 − γ}5_{+ γ}5_{− (γ}5₎2_{) = 0,} _(1.18)

since (γ5)2 = 1. Hence, W bosons do not couple to RH particles and we write the subscript L: SU(2)L.

1.1.3 Mass terms in Lagrangians

The Lagrangian from which the Dirac equation can be derived (equation (1.7)) contains a mass term for the fermionic field ψ. For the field Wi the mass term is absent. Therefore, the

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bosonic fields Wi should be massless. However, from experiments we know that the W± and Z boson propagators do have a large mass, so if these propagators are linked to the fields Wi, there should be a mass term present in the Lagrangian for the fields Wi. However, adding a term like 1₂m2W_iµW_µi would break the gauge invariance.

Another broken gauge invariance due to mass terms can be found in the mass term of equation (1.17). We stated that this Lagrangian is invariant under an SU(2)Ltransformation, but this

is only true if the field ψ is massless, such that the mass term m ¯ψψ disappears. We can see that the gauge invariance is broken by decomposing the massive field into its chiral states. For example the mass term for the electron field would be decomposed into

meee = m¯ e(¯eReL+ ¯eLeR), (1.19)

Under an SU(2)L weak interaction LH particles and RH anti-particles transform as weak

isospin doublets, and right-handed particles and left-handed anti-particles as singlets (section 1.2.1). Due to the difference in transformation of the two parts in equation (1.19), such a mass term breaks the gauge invariance.

1.2 Electroweak unification & EW symmetry breaking in the

Higgs mechanism

1.2.1 Electroweak unification

In section 1.1.2 it was illustrated that the weak interaction is associated with SU(2)L gauge

invariance. To satisfy the gauge invariance the gauge fields W_µ1, W_µ2 and W_µ3 were introduced, which correspond to the gauge bosons W1, W2 and W3. As stated before, the generators of the SU(2)L group are the Pauli spin-matrices. For these 2 × 2-matrices to be able to act

on the field ψ, ψ has to be a two-component object, the weak isospin doublet. The weak interaction couples same-flavour fermions via exchange of a W± boson. Since the W± boson has an electric charge of ±1, the doublet must contain two same-flavour fermions differing by one unit of electric charge. In the SM these doublets are

  νe e−   L ,   νµ µ−   L ,   ντ τ−   L ,   u d0   L ,   c s0   L ,   t b0   L . (1.20)

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The doublets live in the weak isospin space. Analogous to the intrinsic spin of a particle, both components of a weak isospin doublet have total weak isospin IW = 1₂, with a third

component projection of I_W3 = +1₂ for the upper component and I_W3 = −1₂ for the lower component. Since the weak interaction only couples to LH particles and RH anti-particles, only these particles form an isospin doublet. The RH particles and LH anti-particles form isospin singlets with isospin IW = I_W3 = 0. At this point we have to note that the particles

in the doublets are the weak eigenstates, which are not necessarily the mass eigenstates. In fact, it is already known that for example mass eigenstates of down-type quarks are linear combinations of the down-type flavour eigenstates, linked together via the CKM-matrix (see [6] for currently best values). Due to the interaction term introduced in equation (1.13) the doublets couple to the fields W_µ1, W_µ2 and W_µ3

igW

2 ψ¯Lσiγ

µ_Wi

µψL, (1.21)

where ψL is a LH doublet. These interactions give rise to three weak currents, corresponding

to the three Pauli matrices

j_iµ= gW 2 ψ¯Lγ

µ_σ

iψL. (1.22)

From the Pauli matrices the well-known raising and lowering operators σ±= 1₂(σ1± iσ2) are

constructed, from which the physical weak charged currents are obtained

jµ_±= g√W 2

¯

ψLγµσ±ψL (1.23)

and on comparing equations (1.21), (1.22) and (1.23) we find that the physical W± bosons are linear combinations of W1 and W2

W_µ±= √1

2 W

1

µ∓ iWµ2 . (1.24)

With the doublets as in equation (1.20), the matrix representations of σ± are

σ+=   0 1 0 0  , σ− =   0 0 1 0  . (1.25)

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For the third current we write out the components of the doublet and σ3 for the example

ψL= (νeLL). With σ± as given it follows that σ3 = ( 1 0

0 −1) and the third current is

j₃µ= gW 2 ¯ νe,L ¯eL γµ   1 0 0 −1     νe,L eL   = gW 2 (¯νe,Lγ µ_ν e,L− ¯eLγµeL) . (1.26)

Apparently this neutral current carried by the W3 boson couples a LH fermion to its LH anti-fermion. The positive sign of the first term in equation (1.26) corresponds with the isospin projection I_W3 = +1₂, and likewise the second term with the negative sign corresponds with I_W3 = −1₂. With the use of the aforementioned chiral projection operator PLequation ((1.26))

is rewritten to

j₃µ= I_W3 gWf γ¯ µ

1 2(1 − γ

5_)f, _(1.27)

in which f and ¯f are any fermion and its anti-fermionic partner. In equations (1.26) and (1.27) we see that the W3 boson only couples to LH particles. W1 and W2 were already linked to W+ _{and W}−_{. In that line of thought one would identify W}3 _{as the Z boson, but}

from experiments it is known that, although not equally, the Z boson does couple to both LH and RH particles [7]. Therefore, the W3 _{boson is not the same as the Z boson.}

To link the W3 boson with the Z boson, the SU(2)L gauge symmetry is united with the

U(1)Y gauge symmetry. This symmetry is completely similar to the U(1) gauge symmetry of

Quantum Electro Dynamics (QED), except that the charge Qe is replaced with Y₂g0, where Y is the so called hypercharge and g0 the corresponding coupling factor. The interaction term of this symmetry is of the form g0 Y₂ψγ¯ µBµψ, where Bµ is a new field, equivalent to the photon

field Aµ and g0 is the coupling strength. It appears that the fields Aµ and Zµ are linear

combinations of Bµ and Wµ3. The unification of the weak interaction and the hypercharge

is called electroweak unification (EWU). The electroweak mixing is described by a unitary matrix, which can be parametrized like the usual rotation matrix, such that

  Zµ Aµ  =   cos θW − sin θW sin θW cos θW     W_µ3 Bµ  , (1.28)

where θW is the weak mixing angle.

Since QED on its own already is a consistent theory the quantum numbers Y and IW should

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current as given by electroweak unification and set it equal to the EM current as given by QED. As in the previous part we continue with the example ψL = (νeLL). In EWU the EM

current is

j_EMµ = j_Yµcos θW + j3µsin θW (1.29)

where jµ_Y is the hypercharge current. From QED we have

j_EMµ = Qe ¯ψγµψ (1.30)

We already wrote out j₃µin equation (1.26) and the hypercharge current j_Yµ is similar, except that it takes the coupling g0 instead of gW and it is not restricted to LH particles

j_Yµ = g 0 2 Yνe,Lν¯e,Lγ µ_ν e,L+ YeL¯eLγ µ_e L+ Yνe,Rν¯e,Rγ µ_ν e,R+ YeR¯eRγ µ_e R . (1.31)

Likewise, the QED EM current in terms of LH and RH particles becomes

j_EMµ = Qe¯eLγµeL+ Qe¯eRγµeR. (1.32)

By filling in equations (1.26), (1.31) and (1.32) into (1.29), we can break up equation (1.29) into four equations, one for each combination of particles.

g0 2YeLcos θW − gW 2 YeLsin θW = Qe (1.33) g0 2Yνe,Lcos θW + gW 2 Yνe,Lsin θW = 0 (1.34) g0 2YeRcos θW = Qe (1.35) g0 2Yνe,Rcos θW = 0. (1.36)

With the introduction of the U(1)Y gauge symmetry, we also have to introduce invariance

under an U(1)Y local gauge transformation. In order to satisfy the required invariance under

both the U(1)Y and SU(2)Ltransformations the hypercharge of the particles in a weak isospin

doublet have to be the same. A difference in hypercharge within a doublet would after a U(1)Y local gauge transformation result in a phase difference, which would break the SU(2)L

invariance. Therefore, we conclude that Yνe,L = YeL. Q can be written as a linear combination

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and I_W3 = 1₂. Therefore, Q = aY 2 + bI 3 W (1.37) −1 = aYe,L 2 − 1 2b 0 = aYνe,L 2 + 1 2b and since Yνe,L = YeL it follows that a = −b, so that

Q = Y 2 − I 3 W (1.38) ⇒      YeL = 2(−1 + 1 2) = −1 Yνe,L = 2(0 − 1 2) = −1 (1.39)

and indeed Yνe,L = YeL= −1. Now we can subtract equation (1.34) from (1.33)

−g 0 2 cos θW + gW 2 sin θW = −e −g 0 2 cos θW − gW 2 sin θW = 0 −gWsin θW = −e − (1.40)

and their sum is

−g 0 2 cos θW + gW 2 sin θW = −e −g 0 2 cos θW − gW 2 sin θW = 0 −g0_{cos θ} W = −e + . (1.41)

For the RH particles with IW = 0 we get from equation (1.38) YeR = −2 and Yνe,R = 0.

Filling in YeR = −2 into equation (1.35) gives equation 1.41 and with Yνe,R = 0 also equation

(1.36) is consistent.

After electroweak unification the EW EM current is consistent with QED and we continue with writing down the weak neutral current. We have

j_Zµ = −j_Yµsin θW + j3µcos θW. (1.42)

All LH fermions in the SM can be placed in weak isospin doublets. Therefore, the value for I_W3 is known for all LH SM fermions, and for the RH fermions we know I_W3 = 0. Together with the already known charge for each fermion the hypercharge can be determined (table 1.1). For the electromagnetic current an electron and electron neutrino are used as an example, but

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fermion Q I_W3 Y LH RH LH RH e−, µ−, τ− −1 −1 2 0 −1 −2 νe, νµ, ντ 0 +1₂ 0 -1 0 u, c, t +2₃ +1₂ 0 +1₃ +4₃ d, s, b −1 3 − 1 2 0 + 1 3 − 2 3

Table 1.1: Quantum numbers of fermions in the standard model. LH particles are particles in the left handed chiral state and RH particles are particles in the right handed chiral state. The three quantum numbers are interrelated via Q = 1₂Y − I_W3 .

since the three quantum numbers are now known for all fermions, the weak neutral current is easily generalized to any fermion f

jµ_Z= −g 0 2 YfLf¯Lγ µ_f L+ YfRf¯Rγ µ_f R sin θW + gWIW3 f¯LγµfLcos θW. (1.43)

Using equation (1.38) and sorting terms based on chiral states gives

j_Zµ = −g0(Q − I_W3 ) sin θW + gWIW3 cos θW

_¯

fLγµfL− g0Q ¯fRγµfRsin θW. (1.44)

From equations 1.40 and 1.41 follows

gWsin θW = g0cos θW ⇒ g0 = gW

sin θW

cos θW

(1.45)

such that jZ is related solely to the coupling gW

j_Zµ = −gW(Q − IW3 ) sin2θW cos θW + gWIW3 cos θW ¯ fLγµfL− gWQ ¯fRγµfR sin2θW cos θW = gW (I_W3 − Q)sin 2_θ W cos θW + I_W3 cos θW ¯ fLγµfL− gWQ sin2θW cos θW ¯ fRγµfR (1.46)

and with the definition gZ ≡ _{cos θ}gW_W the weak neutral current in terms of the weak neutral

coupling gZ becomes

j_Zµ = gZ IW3 − Q sin2θW

_¯

fLγµfL− gZQ sin2θWf¯RγµfR. (1.47)

In the end we see that indeed electroweak unification allows for the Z boson to couple to both LH particles and RH particles, albeit not equally.

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The QED EM current does not distinguish between LH and RH particles. Therefore, to check whether the EW EM current is consistent with the QED one, jµ_EM is constructed analogously to j_Zµ. j_EMµ = g 0 2 YfLf¯Lγ µ_f L+ YfRf¯Rγ µ_f R cos θW + gWIW3 f¯LγµfLsin θW =g0(Q − I_W3 ) cos θW + gWIW3 sin θW _¯ fLγµfL+ g0Q ¯fRγµfRcos θW = gW(Q − IW3 ) sin θW + IW3 sin θW _¯ fLγµfL+ gWQ sin θWf¯RγµfR = gWQ sin θW f¯LγµfL+ ¯fRγµfR (1.48)

and indeed the EWU EM current couples to LH and RH particles equally.

1.2.2 Higgs Mechanism

In section 1.1.3 it was stated that adding simple mass terms to the Lagrangian breaks the gauge invariance of the Lagrangian.and that experiments have shown that the W+, W− and Z bosons are not massless. Therefore, the Lagrangian that describes these bosons should somehow feature a mass term for these bosons, while remaining gauge invariant. The Higgs mechanism achieves this invariance by adding a new field φ to the Lagrangian with a kinetic term and a potential. The massless W+, W− and Z bosons each have two transverse polar-izations. Particles with mass also have a longitudinal polarization. Therefore, the new field should at least have four degrees of freedom. One for its own physical field, and three for the longitudinal polarizations of the three gauge bosons. In order to allow this field to couple to the SU(2)L weak isospin doublets, the minimal possible model consists of two complex

scalar fields, which form a weak isospin doublet together. Because the model needs to gener-ate masses of both neutral and charged gauge bosons, one of the scalar fields is taken to be neutral φ0, and the other one to be charged φ+ in a way such that (φ+)∗= φ−. The doublet will then have the form

φ =   φ+ φ0  = 1 √ 2   φ1+ iφ2 φ3+ iφ4  , (1.49)

where φi represent the degrees of freedom. The Higgs potential

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is part of the Lagrangian for a complex scalar field

L = (∂_µφ)†(∂µφ) − V (φ). (1.51)

Later on the shape of the potential is justified. For the potential to have a finite minimum we need λ > 0. On the other hand, µ2 can be both positive and negative. For positive µ2 the potential has a global minimum at the origin of the four-dimensional field. However, for negative µ2_{, this global minimum becomes a local maximum and the potential has a new set}

of degenerate minima, which satisfy

φ†φ =   φ+ φ0   φ− φ0 = 1 2(φ 2 1+ φ22+ φ23+ φ24) = − µ2 2λ (1.52)

and 1₂(φ2₁+ φ2₂+ φ2₃+ φ₄2) describes a hypersphere of radius v₂2, so that v₂2 = −µ_2λ2. The value v corresponds to the non-zero vacuum expectation value of the field φ. Since the system is totally symmetric the choice of the direction of the vacuum state is free. Therefore the electroweak symmetry is spontaneously broken. It is customary to choose the vacuum point at φ0 = v₂2. By taking φ3 ≈ v + η(x), the fields can be expanded around this vacuum point.

This results in one massive scalar field η(x) and three Goldstone bosons. These Goldstone bosons provide the necessary third degree of freedom for the massive gauge bosons. After the gauge-transformation that absorbs the Goldstone bosons with their degrees of freedom into the gauge bosons the following doublet is obtained

φ = √1 2   0 v + h(x)  , (1.53)

where h(x) now represents the physical scalar field, the Higgs field. Of the initial degrees of freedom, only one remains in the doublet φ.

In section 1.1.2 it was shown that for the Lagrangian to be invariant under an SU(2)L

trans-formation the introduction of the covariant derivative

Dµ= ∂µ+

igW

2 σiW

i

µ (1.54)

was necessary. In the electroweak united model the Lagrangian also has to be invariant under the U(1)Y local phase transformation. As mentioned in section 1.1.2, the U(1) group is

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contain gauge boson self interactions. Simply adding the term ig0 Y₂Bµis sufficient. Therefore

the covariant derivative in the SU(2)L × U(1)Y symmetry becomes

Dµ= ∂µ+ igW 2 σiW i µ+ ig0 Y 2Bµ. (1.55)

With this covariant derivative not only the Lagrangian for fermions, but also the Higgs La-grangian becomes gauge invariant. Now, both the Dirac and Higgs LaLa-grangians are gauge invariant, but the EW symmetry is broken. Since the Higgs doublet is a weak isospin doublet φ+ has I_w3 = 1₂, while φ0 has I_w3 = −1₂, and since the charge of the doublet components are known, the hypercharge of the Higgs doublet equals Y = 1, which can be filled in in equation (1.55). With equations (1.53) and (1.55) Dµφ and its Hermitian conjugate (Dµφ)† become

(Dµφ)†(Dµφ) = 1 2(∂µh)(∂ µ_{h) +} gW2 8 W 1 µW1 µ+ Wµ2W2 µ (v + h)2 +1 8(gWW 3 µ− g 0 Bµ)(gWW3 µ− g0Bµ)(v + h)2. (1.56)

The first term gives the kinetic energy of the Higgs field h. By writing out the product (v +h)2 in the second term terms with v2 appear, which can be identified as the mass terms for the fields W1 and W2

v2g2_W

8 W

1

µW1 µ+ Wµ2W2 µ . (1.57)

From the mass of the W boson follows that

MW =

gWv

2 . (1.58)

The W boson mass is clearly proportional to the vacuum expectation value v. Therefore the Higgs potential has to have a non-zero expectation value. The minimal polynomial potential to achieve this is a fourth order polynomial with µ2 negative and λ positive. The term with h2 gives the quartic coupling of two W bosons with two Higgs bosons with coupling strength

gHHW W =

g_W2

4 , (1.59)

and the cross term with 2vh gives the triple coupling of two W bosons with one Higgs boson with coupling strength

gHW W =

g2_Wv

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Instead of writing out the product in the third term of equation (1.56) we rewrite it in matrix form (v + h)2 8 W_µ3 Bµ   g_W2 −g0gW −g0gW g02     W3 µ Bµ  . (1.61)

The mass terms for the gauge fields are again given by the v2 term from (v + h)2. These masses belong to the mass eigenstates of the gauge bosons. Therefore, we diagonalize the mass matrix in the expression above, which induces a mixing of the fields W_µ3 and Bµ. The

eigenvalues are λ = 0 and λ = g2

W + g02 and in terms of the physical fields Aµ and Zµ the

equation becomes (v + h)2 8 Aµ Zµ   0 0 0 g2_W + g02     Aµ Zµ  . (1.62)

From this expression we find that the mass of the photon field is

mA= 0 (1.63)

and the mass of the Z boson field

mZ= v 2 q g2 W + g02= MW gW q g2 W + g02= MW cos θW , (1.64)

where we used gW sin θW = g0cos θW from the previous section. From the diagonalisation

of the mass matrix gives the physical fields Aµ and Zµ as a linear combination of the gauge

fields W_µ3 and Bµ Aµ= g0W_µ3+ gWBµ q g_W2 + g02 = W_µ3sin θW + Bµcos θW (1.65) Zµ= g0W3 µ− gWBµ q g2_W + g02 = W_µ3cos θW − Bµsin θW. (1.66)

These two eigenvectors are exactly the same linear combinations of W_µ3 and Bµ as the ones

from equation (1.28). Therefore, we see that the electroweak unification and the mixing of fields as described in section 1.2.1 emerges from the Higgs mechanism in a natural way. Furthermore, the Higgs mechanism provides mass terms for the massive gauge bosons. In section 1.1.3 it was shown that the fermion mass terms in the Dirac Lagrangian breaks the SU(2)Lsymmetry. The Higgs isospin doublet φ transforms under an infinitesimal SU(2)Llike

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equation (1.8), and so do the LH fermion isospin doublets fL. Since ¯fL = f_L†γ0, the adjoint

isospin doublet ¯fL has this transformation property with a minus sign in the exponential.

We construct the quantity ¯fLφ and see that it is invariant under SU(2)L transformations.

However, this combination is not invariant under a U(1)Y transformation. The combination

with the RH singlet does preserve this invariance. Therefore, the term ¯fLφfR is SU(2)L ×

U(1)Y gauge invariant and a term in the form

Lf = −gf ¯fLφfR+ ¯fRφ†fL

(1.67)

may be added to the Lagrangian, where fL are the isospin doublets from equation (1.20).

Writing out the Higgs doublet after spontaneous symmetry breaking gives for example for the muon

Lf = −

gµ

√

2(¯µLµR+ ¯µRµL) (v + h). (1.68) The term proportional to v is identified as the mass term for the muon field, from which follows that the mass of the muon is mµ = √gµ₂v. The term proportional to h represents

the coupling of the muon to the Higgs boson. One might mention that the Lagrangian from equation (1.67) only generates mass terms for the lower component of the weak isospin fermion doublets, since the upper component of the Higgs doublet is zero after spontaneous symmetry breaking. By introducing the conjugate Higgs doublet mass terms for the upper components of the fermion doublets can be constructed

L_f = gf ¯fLφcfR+ ¯fRφ†cfL

(1.69)

where the conjugate Higgs doublet φcis φc= −iσ2φ∗ = −i(0 −i_{i 0} )(v+h0 ) = (−v−h0 ). In general,

the coupling of the fermions to the Higgs boson is proportional to the fermion mass

gf =

√ 2mf

v . (1.70)

Consequently, the Higgs boson prefers to couple to the most massive fermions within its kinematically accessible region.

The most heavy fermion in the SM to which the Higgs boson can decay is the bottom quark. Because the Higgs boson prefers to couple to massive particles, the Higgs decay into bottom quarks has the largest branching fraction (see table 1.2). Although 2MW > MH, the decay

of a Higgs boson into two W bosons is allowed when one of the two W bosons is produced off its mass shell.

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Decay: Higgs to Branching ratio b¯b 57.7% W±W∓ 21.5% τ+τ− 6.3% gg 8.6% c¯c 2.9% ZZ 2.7% γγ 0.2%

Table 1.2: Predicted branching ratios for the decay modes of the MH = 125 GeV Higgs boson.

Note that the for the decay modes with the W and the Z boson, at least one of the two bosons has to be off shell in order to enable these decay modes. Numbers from [8].

1.3 Vector boson polarisation

The previous section mentioned that the massless gauge bosons in the EWU have two trans-verse polarisations and that the Higgs mechanism provides an extra degree of freedom for a longitudinal polarisation. The wave function for the massless spin-1 field Aµ can be written in terms of a transverse polarisation four-vector and a plane wave Aµ= µαe−ip·x where it is

customary to take the transverse polarisation to be

µ₋= √1 2 0 1 −i 0 (1.71) µ₊= √1 2 0 1 i 0 . (1.72)

These two polarisations correspond with the two degrees of freedom of the massless photon field. Via the Higgs mechanism the W and Z fields acquired a third degree of freedom by absorbing the Goldstone bosons. This third degree of freedom leads to a third polarisation state, the longitudinal polarisation. For illustration we use some massive spin-1 boson Wµ. From the Klein-Gordon equation

(∂ν∂ν+ m2)Wµ= 0 (1.73)

the plane wave solutions for a spin-1 boson come forth

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From the Lorenz condition ∂µWµ = 0 follows that µ satisfies pµµ = 0. The first two

polarisations of Wµ are identical to the transverse polarisations of the massless spin-1 boson Aµ. The third polarisation has to be orthogonal to the first two in order to be linearly independent. Therefore this polarisation is of the form

µ₃ =a 0 0 b ⇒ pµµ₃ = aE − bpz = 0 ⇒ µ_L= 1 m pz 0 0 E , (1.75)

where the factor _m1 is included for normalisation and µ₃ is identified as being the longitudinal polarisation.In the rest frame of the massive boson, the longitudinal polarisation reduces to (0, 0, 0, 1).

For on shell massive gauge bosons, the polarisation states as given are sufficient. However, in the decay of a Higgs boson to W or Z bosons at least one of the two gauge bosons is off shell. In order to account for both on shell and off shell gauge bosons the V ∈ {W, Z} fields are separated into a transverse and a longitudinal part

V_Tµ= Pµ_{T ν}Vν V_Lµ= Pµ_{L ν}Vν. (1.76)

PµT ν is the operator that projects the field onto the plane perpendicular to the direction of

propagation, the z-axis in this case

P0T ν= P µ T 0= 0 (1.77) PiT j = δji− pipj p2 (1.78)

with p the three-momentum of the field and i, j ∈ {1, 2, 3}. The longitudinal projection operator then becomes

Pµ_{L ν} = (I − PT)µν, (1.79)

such that the sum of the two projections is the physical field Vµ [9]. The projections of Vµ are not Lorentz invariant. Consider a W boson propagating in the z direction. When the W boson gains a boost orthogonal to the z direction and parallel to the W boson momentum, the longitudinal and transverse projections become mixed. The W boson can be longitudi-nally polarised in one frame, while it has a both a longitudinal and a transverse polarisation component in another frame.

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1.4 W

+

W

−

→ W

+

_W

−

_scattering

γ, Z W+ W− W+ W−

(a) t-channel γ and Z propa-gator γ, Z W+ W− W+ W−

(b) s-channel γ and Z propa-gator W+ W− W+ W− (c) Four-point interaction H W+ W− W+ W−

(d) t-channel Higgs propagator

H

W+ W−

(e) s-channel Higgs propagator

Figure 1.1: Feynman diagrams for the process W+W−→ W+_W−

Although the Higgs mechanism provides a solid theory for the gauge boson masses, it brings at first sight a problem with it as well. The Higgs mechanism generates the masses of the gauge bosons by providing the necessary degrees of freedom for the longitudinal polarisation of the gauge bosons. Equation (1.75) and later equation (1.80) show that the longitudinal polarisation of the W boson is proportional to the centre of mass energy. This energy depen-dence may lead to diverging cross sections at high centre of mass energies. Such divergences can be found in W+W−→ W+_W− _{scattering. This process consists of the seven Feynman}

diagrams in figure 1.1. In the following only the longitudinal part of this process will be taken into consideration, in which we are guided by ref.[10]. Later on we will give an example of scattering with both a transverse and a longitudinal part.

1.4.1 Longitudinal scattering

From the Feynman rules that follow from the Lagrangians in equation (1.1) and 1.2 the expressions for the gauge boson self-couplings, the Higgs boson couplings to gauge bosons and the gauge boson and Higgs boson propagators can be found. They are listed in table A.1. Since the incoming and outgoing particles are vector bosons, the polarisation states for

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p1

p2

q1

q2

θ

Figure 1.2: Momenta in the CM frame, in which β = q

1 −4MW2 s .

the external legs in the Feynman diagrams have to be taken into account. The polarisation state in equation (1.75) is defined along the momentum of the particle. In any given frame in which the particle moves with a velocity ~β(= _E~p) this generalizes to µ_L(p) = γ(β,~β_β), where γ = √1

1−β2. The calculations are performed in the centre of mass frame as shown in figure

1.2. For this frame the momenta of the incoming W bosons, p1 and p2, and the momenta

of the outgoing W bosons, q1 and q2 are given in appendix A. The incoming particles are

chosen along the z-axis, β = q

1 −4MW2

s and θ is the scattering angle from the z-axis. From

the definition of these momentum vectors the longitudinal polarisations for the four legs in the diagrams become

µ_L(p1) = √ s 2MW β 0 0 1 µ_L(p2) = √ s 2MW β 0 0 −1 µ_L(q1) = √ s 2MW β sin θ 0 cos θ µ_L(q2) = √ s 2MW β − sin θ 0 − cos θ . (1.80)

It is straightforward to show that these polarisation vectors are indeed orthonormal to the momentum vector of the corresponding W boson. From equation (1.80) and table A.1 the matrix elements for the seven diagrams in figure 1.1 can be found

Ms_H,L = − g 2 WMW2 s − M2 H + iΓHMH α_L(p1)β_L(p2)_Lγ(q1)δL(q2)gαβgγδ Mt_H,L = − g 2 WMW2 t − M2 H + iΓHMH α_L(p1)β_L(p2)_Lγ(q1)δL(q2)gαγgβδ M_4W,L = g2_Wα_L(p1)β_L(p2)γ_L(q1)δL(q2) [2gαδgβγ− gαβgγδ− gαγgβδ] (1.81)

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Ms_γ,L = g 2 W sin2θW s α L(p1)βL(p2)Lγ(q1)δL(q2)Γαβµ(p1, p2, −q1− q2)Γδγν(−q2, −q1, p1+ p2)gµν Mt_γ,L = g 2 W sin 2_θ W t α L(p1)β_L(p2)_Lγ(q1)δL(q2)Γαγµ(p1, −q1, p2− q2)Γδβν(−q2, p2, p1− q1)gµν Ms Z,L = g_W2 cos2θW s − M_Z2 + iMZΓZ gµν−(p1+ p2) µ_(p 1+ p2)ν M_W2 / cos2_θ W α_L(p1)β_L(p2)γ_L(q1)δL(q2) × Γ_αβµ(p1, p2, −q1− q2)Γδγν(−q2, −q1, p1+ p2)gµν Mt Z,L = g_W2 cos2θW t − M_Z2 + iMZΓZ gµν− (p1− q1) µ_(p 1− q1)ν M_W2 / cos2_θ W α_L(p1)β_L(p2)γ_L(q1)δL(q2) × Γαγµ(p1, −q1, p2− q2)Γδβν(−q2, p2, p1− q1)gµν.

The definition of the Γ-symbol can also be found in A.1 and s and t are the Mandelstam variables. In the following the matrix elements are calculated one by one. By contracting the repeated indices the relevant combinations of polarisation vectors are obtained. First consider the Higgs s-channel. Contracting indices and filling in the appropriate ’s gives the combinations α_L(p1)L,α(p2) = _4Ms2 W (β2+ 1) and γ_L(q1)L,γ(q2) = _4Ms2 W (β2 + 1). With the definition β = q 1 −4MW2 s in mind we find Ms_H,L= − g 2 W s − M2 H + iΓHMH (s − 2M_W2 )2 4M2 W . (1.82)

For the Higgs t-channel the combinations α_L(p1)L,α(q1) and γ_L(p2)L,γ(q2) are present, which

give the matrix element

Mt_H,L= − g 2 W t − M_H2 + iΓHMH (8M_W4 − (t + 2M2 W)s)2 2M_W2 (s − 4M_W2 ) . (1.83) Both the polarisation combinations of the Higgs s-channel and the Higgs t-channel occur in the W boson four-point interaction, along with the u-channel-like terms α_L(p1)L,α(q2) and

γ_L(p2)L,γ(q1). Combining terms results in

M4W,L = g_W2 s 4M4 W(−4MW2 + s)2 −64M6 W + 48MW4 (s + t) − 4MW2 s(3s + 7t) + s(s2+ 4st + t2) . (1.84) For the Z boson and photon propagator channels combinations of polarisation vectors with momentum vectors occur. The numerator of the Z boson propagator contains two terms. The first one equals the numerator for the photon propagator. It can be shown that the second

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term evaluates to zero for both the s-channel and the t-channel. Therefore the only difference between the Z boson matrix elements and the photon matrix elements is the denominator of the propagator and the coupling constant at the vertex. Bearing this in mind, we carry out the contraction of indices and fill in the expressions for the polarisation vectors and the momentum vectors. For the Z boson and photon s-channel this results in (2MW2 +s)2(4MW2 −s−2t)

4M4 W

such that the matrix elements become

Ms_γ,L= g 2 W sin2θW s (2M2 W + s)2(4MW2 − s − 2t) 4M_W4 (1.85) Ms_Z,L= g 2 Wcos2θW s − M_Z2 + iMZΓZ (2M2 W + s)2(4MW2 − s − 2t) 4M_W4 (1.86)

and on the same token the t-channel matrix elements become

Mt γ,L = g_W2 sin2θW 4M_W4 t(−4M_W2 + s)2 256M 10 W − s2t2(2s + t) − 64MW8 (4s + t) + 8MW2 s2t(s + 3t) +16M_W6 s(5s + 14t) − 4M_W4 s(2s2+ 21st + 20t2) (1.87) Mt_Z,L= g 2 W cos2θW 4M4 Wt(−4MW2 + s)2(s − MZ2 + iMZΓZ) 256M10 W − s2t2(2s + t) − 64MW8 (4s + t) +8M_W2 s2t(s + 3t) + 16M_W6 s(5s + 14t) − 4M_W4 s(2s2+ 21st + 20t2) . (1.88)

Consider the W boson four-point interaction matrix element (equation (1.84)). This matrix element is proportional to the centre of mass energy to the power four (√s4 = s2). Therefore, the matrix element quadratically diverges for large s. The s-channel and t-channel photon propagator diagrams do reduce this divergence, but since the coupling constant for the four-point interaction and the coupling constant for the photon diagrams differ a factor sin θW,

the s2-divergence is not entirely cancelled. To fully cancel the s2-divergence also the Z boson diagrams are necessary. The four-point coupling constant and the Z boson coupling constant differ a factor cos θW. The matrix elements for the different diagrams are proportional to the

coupling constants squared. Therefore, due to the relative minus sign between the photon and Z boson diagrams and the four-point diagram, we get g_W2 − g2

Wsin2θW− g2Wcos2θW = 0

and the s2-divergence is fully cancelled.

Let us now revert to equation (1.84) again. The s2-divergence is taken care of by including the photon and Z boson diagrams. However, there is also a s-divergence present this divergence is not cancelled by the photon and Z boson diagrams. The Higgs boson matrix elements

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(equations (1.82) and (1.83)) are proportional to s, and by taking the Taylor expansion of the matrix elements for s → ∞, one can show that this s-dependence cancels the s-divergence. Therefore, one may conclude that the Higgs boson diagrams are indeed necessary.

1.4.2 Scattering with both transverse and longitudinal polarisations

In the previous section the matrix elements for the longitudinal part of the scattering pro-cess W+_W− _{→ W}+_W− _{were calculated. The participating polarisation combinations are}

summed in appendix A. In this appendix the calculations of terms is expanded to transverse polarisation states. For completeness one could attempt to calculate matrix elements for all 81 possible polarisation configurations (four external legs and three polarisation states give 34 _{= 81 configurations). Although a number of the configurations give similar results, the}

entire calculation is lengthy and tedious. Therefore, we will give only one more example in order to establish the procedure. We choose a matrix element for a combination of transverse and longitudinal polarisations in order to simultaneously check that the divergence cancella-tion applies in such terms as well. We calculate MH,L+L+ where the indices indicate that

the used polarisation vectors are L(p1), +(p2), L(q1) and +(q2) (definitions in appendix

A). The calculations are in appendix A.5 and the results are

Ms_H,mix= 0 Mt_H,mix= − g 2 WMW2 t − M_H2 + iΓHMH (4M_W2 − s − t)(8M4 W − 2MW2 s − st) 2M_W2 (s − 4M W2₎2 M4W,mix= g2W −32M6 W + st(s + t) + 8MW4 (2s + t) − 2MW2 s(s + 3t) 2M2 W(s − 4MW2 )2 Ms_γ,mix= g 2 W sin2θW s s(4M_W2 − s − t) M_W2 (1.89) Mt γ,mix= g_W2 sin2θW 2M_W2 t(s − 4M_W2 )2 128M 8 W − 128MW6 s + st(2s2+ st − t2) +8M_W4 (5s2+ 7st − t2) − 2M_W2 s(2s2+ 11st + 3t2) Ms_Z,mix= g 2 Wcos2θW s − M2 Z+ iΓZMZ s(4M_W2 − s − t) M2 W Mt_Z,mix= g 2 Wcos2θW (t − M_Z2 + iΓZMZ)(2M_W2 (s − 4M_W2 )2) 128M8 W − 128MW6 s +st(2s2+ st − t2) + 8M_W4 (5s2+ 7st − t2) − 2M_W2 s(2s2+ 11st + 3t2) .

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The Taylor expansion for s → ∞ shows that the largest order of s in each individual term is of order O(s), which implies that no s2 divergence is present. Summing up the Taylor expanded matrix elements results in the desired cancellation of the linear s divergence.

1.5 Vector Boson Fusion

Probing the Higgs boson coupling to two W bosons requires selecting signal events in the ATLAS detector (chapter 2.2) in which a Higgs boson is produced. In this section the processes in which such a Higgs boson is produced are discussed, as well as the so called background processes.

1.5.1 Description VBF signal process

In the Large Hadron Collider (chapter 2.1) the main Higgs production modes are gluon fusion (ggF) (fig. 1.3a), vector boson fusion (VBF) (fig. 1.3b) and associated production (VH) (fig. 1.3c). Although ggF has at a centre of mass energy of√s = 13 TeV a larger production cross

t, b t, b t, b g g h

(a) Gluon fusion (ggF)

V V q q q0 h q0

(b) Vector boson fusion (VBF)

V V

q ¯

q _h

(c) Associated production (VH)

Figure 1.3: Dominant Higgs boson production channels. The incoming quarks q and gluons g are partons from the colliding protons in the LHC. V = W±, Z and in VBF q0 is either the antiparticle of q if V = Z or a different flavour quark if V = W±.

section (43.92 pb at NNLO+NNLL QCD and NLO EW) than VBF (3.748 pb at NNLO QCD and NLO EW) [11], the production channel of choice is VBF. This choice will be justified in a moment. In VBF, the Higgs boson is produced by fusing two vector bosons, either W+W− → h or ZZ → h. The vector bosons themselves are radiated from the incoming partons, which in general are either quarks or gluons. Since gluons do not couple to W or Z boson, the incoming partons in VBF are quarks. By radiating a vector boson V , the quarks change flavour if V = W or becomes an anti-quark if V = Z. The changed quarks are denoted

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by the prime in q0 in figure 1.3b and are part of the final state. Section 2.3 will show that in the detector there is no such thing as a final state quark, but that these quarks form jets of particles. In this thesis however, the processes will be treated at parton level and effects like showering and hadronisation are not taken into account. Therefore we do not need to handle actual jets, but we do refer to these final state quarks as jets. Although we are actually interested in Higgs boson production with W bosons, we cannot distinguish between W and Z bosons in the production mode. Therefore the two vector bosons will be treated on equal footing.

For the decay modes we refer back to table 1.2. Although the Higgs decay to b¯b is the dominant decay mode, we again choose for the second dominant mode in order to be able to probe the HWW coupling. By choosing the process as given, we are provided with two ways of measuring the HWW coupling, both via the production the decay mode. Although the chosen production and decay modes do not result in the largest cross sections, the process does contain the W+_W−_{→ W}+_W−_{scattering that we discussed before, while ggF and Higgs}

decay in to b¯b do not.

The W bosons in the decay mode are unstable and will therefore decay further into final state particles, being four leptons, of which two are charged leptons and two are neutrinos. The two charged leptons in the final state are required to differ in flavour (i.e. µ+e−or e+µ−) in order to exclude the Higgs decay into Z bosons and four leptons H → ZZ, Z → l+l−, Z → νlν¯l,

where the Z boson decays into two same flavour leptons or neutrinos. The resulting signal process is displayed in figure 1.4.

It is expected that the coupling structure at the HWW vertex directly affects the kininematics of the W bosons in both production and decay. The production W bosons are directly related to the jets via the splitting of the incoming partons. Therefore, in the production process we aim at probing the HWW polarisation structure by studying the jet kinematics. In the decay process, the decay W bosons split into charged leptons and neutrinos. Although the neutrinos leave the detector unnoticed, we expect that the kinematics of the charged leptons give a good handle to probe the HWW polarisation structure in the decay process. In this thesis we specifically look at events with e+νeµ−ν¯µ in the final state. The branching ratio for

h → W+W− → e+_ν

eµ−ν¯µ is 0.25%[8]. Combining the VBF production cross section and the

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W± W∓ H W+ W− q q j e+, µ+ νe, νµ ¯ νµ, ¯νe µ−, e− j

Figure 1.4: Lowest order tree level Feynman diagram for signal channel. q is an incoming parton and j is an outgoing quark. The dotted vertices indicate the important HWW coupling. The final state charged leptons are required to be of opposite flavour.

Longitudinal and transverse parts

The longitudinal an transverse projection operators from section 1.3 are not Lorentz invariant. Therefore it is necessary to choose a reference frame. In the rest frame of the Higgs boson the projection operators evaluate such that the HWW coupling is separated in a purely longitudinal and a purely transverse part. This is the reference frame of choice. In the Lagrangian the Higgs boson coupling of interest then is written as

1 2gWMWW − µW+µH = 1 2gWMW(W − L,µW +µ L H + W − T ,µW +µ T H) (1.90)

without any mixed polarisation terms.

1.5.2 Background processes

In the detector the signal process is selected by searching for the characteristic final state particles jje+νeµ−ν¯µ. However, other processes mimic this final state. Such processes are

considered as backgrounds and can be divided into different categories. Irreducible back-grounds are the background processes which truly have the same final state as the signal process. Reducible backgrounds are processes in which a final state particle fakes one of the final state particles we select, like for example a high energy photon that in the detector is

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misreconstructed as an electron. In the following only the irreducible backgrounds are con-sidered, since detector influences are not taken into account. The background properties are discussed in the rest of this section.

Gluon fusion and associated production

Other Higgs production processes than VBF are obviously part of the background. In gluon fusion (figure 1.3a) the partons are gluons, which via a fermionic loop interact and produce a Higgs boson. Since the Higgs boson favours coupling to heavy fermions, the fermionic loop is either a top quark loop or a bottom quark loop (< 10%[12]). The ggF production cross section was given in the previous section. With the given branching ratio for h → W+W− → e+νeµ−ν¯µ the total cross section for the ggF background is 110 fb. In associated production

(figure 1.3c) at leading order, the two partons fuse to form a vector boson, which then radiates a Higgs boson. Associated production with a W boson gives a cross section of 1.380 pb and with a Z boson gives 0.8696 pb. Both cross sections are at NNLO QCD and NLO EW [11]. Combining the two cross sections with the branching ratio h → W+W−→ e+_ν

eµ−ν¯µ gives a

total cross section of 5.62 fb. Both the given cross sections of ggF and VH do not include the

W+ W+ H W+ W− q q j j e+ νe ¯ νµ µ− (a) W+ W+ H W+ W− q q e+ νe j j ¯ νµ µ− (b)

Figure 1.5: Lowest order tree level Feynman diagrams for Higgs associated production.

calculation of jet production. In ggF these jets will originate from the production partons or intermediate particles in the Higgs production. In the VH process, after the Higgs decay into two vector bosons, there are three vector bosons present. The two jets and four leptons can originate from either of the three vector bosons. Therefore two leptons may also originate from the associated vector boson (figure 1.5). In chapter 4 we will calculate the cross sections

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for these processes with the jets included, but only fore the phase space region defined therein. EW background W+ νµ W− µ+ W+ q q j ¯ νµ e+ νe µ− j (a) Z W Z q q j ¯ νµ µ− e+ νe j (b) γ, Z µ+ W + e + γ q q µ− ¯ νµ νe e+ j j (c)

Figure 1.6: Lowest order tree level Feynman diagrams for irreducible EW background

The electroweak (EW) background contains all the processes that just as the signal process are of O(g_W6 ), the leading order amplitude. A first contribution to this background are the W W → W W scattering processes other than the one with the s-channel Higgs boson from figure 1.1. Each of these diagrams can replace the W W → W W scattering part in figure 1.4. An EW process without W W → W W scattering is for example one in which the vector bosons radiating from the partons directly decay into leptons. In a more involved process the W bosons radiated from the partons interchange a neutrino and both decay in the same-flavour but opposite-charge lepton. One of the leptons then radiate another W boson and changes into a neutrino The extra W boson then decays into the different-flavour lepton and neutrino (upper left panel figure 1.6). Both upper diagrams in figure 1.6 are VBF-like in that the partons radiate a vector boson leave the scattering as jets. Just as for the VH background, the partons may also fuse and form a vector boson. This vector boson then forms a decay

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chain of leptons and vector bosons, and eventually jets, such that the characteristic final state is constructed (lower panel figure 1.6).

QCD background

The next background of importance is the quantum-chromodynamic (QCD) background. While the EW background only contained electroweak processes, the processes making up the QCD background also contain quark-gluon interactions. We consider the lowest order processes which contain at most two quark-gluon interactions. The left panel of figure 1.7 shows an example in which the incoming partons are two quarks. In this case the QCD part is responsible for the jet production via the radiation of a gluon from one of the partons, followed by the splitting of the gluon in two more gluons. The final state leptons are produced via the radiation of one or two vector bosons from the initial quarks, which then decay into the final state leptons. In the right panel of figure 1.7 the initial quarks interchange a gluon. Both quarks then form the two jets. The final state leptons are produced via the radiation of a vector boson from the initial quarks before or after the interchange of the gluon and the subsequent decay of the vector boson.

q Z g νµ W+ q q ¯ νµ µ− e+ νe g g g Z, γ µ+ W+ q q j e+ νe ¯ νµ µ− j

Figure 1.7: Lowest order tree level Feynman diagrams for irreducible QCD background

t¯t background

The last background is the t¯t background. It has three kinds of background processes, of which two are shown in figure 1.8. In the left diagram the partons, being two gluons, fuse to a single gluon, which decays into t¯t. In the right panel one of the incoming gluons produces a t¯t pair, of which either the t or the ¯t interacts with the other incoming gluon. The third

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diagram is similar to the first one, but with quarks in the initial state. At 13 TeV centre of mass energy gluon production dominates over quark production. The contribution of the third process to the t¯t background is therefore small. For all three diagrams, the top quark splits into a W+ boson and a bottom quark t → bW+ and the anti-top splits into ¯bW−. The W bosons decay leptonically and the b¯b in the final state form the tagging jets.

g t W+ ¯ t _W− g g b e+ νe ¯ νµ µ− ¯ b t t ¯ t W− W+ g g ¯ b ¯ νµ µ− e+ νe b

Figure 1.8: Lowest order tree level Feynman diagrams for irreducible t¯t background

1.6 Beyond the Standard Model

In section 1.4 we showed how in WLWL→ WLWL scattering the two Higgs boson diagrams

cancel the divergences present in the W boson four point coupling diagram. In the SM this cancellation is exact. Any deviations from the SM HWW coupling disturb this cancellation and can hint towards beyond SM physics (BSM). At the start of this chapter the possibility of the existence of two separate Higgs doublets was mentioned. In this BSM theory, the discovered Higgs boson would be the lightest of five different Higgs bosons. The SM coupling constant gHW W is then a composition of the coupling constants of this light Higgs h0 and the

heavier H0: g_{HW W}2 = g2_h0_{W W} + g_H20_{W W} [13]. In the event that the discovered Higgs boson is

actually h0, then the cancellation of divergences is not complete and two additional diagrams, featuring the H0 as a propagator, are necessary to complete the cancellation.

Another possibility is a more complex structure of the HWW coupling than the linear SM coupling from table A.1. Higher order operators in the Lagrangian may for example add terms to the SM HWW coupling which are proportional to the momenta of the W bosons and the Higgs boson [9].

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parameters aL =

g_HWLWL

gHW W and aT =

g_{HWT WT}

gHW W , where gHW W, gHWLWL and gHWTWT are the

coupling strengths for respectively the full HWW coupling, the longitudinal part, and the transverse part. In equation (1.90) the longitudinal part the coupling gW is multiplied with

aL: gW → aLgW, where aL= 1 is the SM value. Likewise for the transverse part gW → aTgW

and aT = 1 gives the SM. The parameters aL and aT are allowed to take on any real value.

In the analysis of this thesis we aim at investigating how changes in aL and aT affect the

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Chapter 2

The ATLAS detector @ the LHC

Although the work in this thesis is based on Monte Carlo simulations and no real data will be used, we give an overview of the Large Hadron Collider (LHC) and the ATLAS detector (A Toroidal LHC ApparatuS), since the work in this thesis serves as a preparation for analysing ATLAS data and therefore the data obtained from Monte Carlo simulations have to mimic and be compatible with the data as it is recorded with the ATLAS detector.

2.1 LHC

The LHC is a proton-proton collider with the capacity to collide protons with the centre of mass energy up to 14 TeV[14][15]. The LHC accelerates protons in a 27 km circular tunnel containing two beam pipes surrounded by magnets. Dipole magnets cause the protons to fol-low the circular path of the beam pipes, while quadrupole magnets keep the proton bunches focussed. In the two beam pipes the magnetic field strength is equal but in opposite direc-tion, forcing the two proton beams to go in opposite direction. The superconducting dipole magnets produce a magnetic field of 8.36 T in order to keep the protons in their circular path. At four positions in the tunnel the beam pipes cross and the two opposite beams interact. At these four interaction points, particle detectors are placed, of which one is the ATLAS detector. Before protons enter the LHC they are pre-accelerated in four stages (figure 2.1). In the first stage the protons are accelerated in the Linac2 linear accelerated to an energy of 50 MeV. Then in the Proton Synchrotron Booster and the Proton Synchrotron the protons are accelerated to respectively 1.4 GeV and 26 GeV. In the final stage before the protons

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Figure 2.1: Before protons enter the LHC, they are pre-accelerated in the injection chain. Although not mentioned in the text, the LHC is also able to accelerate ions, as one may infer from the figure [15].

enter the LHC the Super Proton Synchrotron accelerates the protons to 450 GeV. At that energy the protons are divided in two beams and injected into the two beam pipes of the main ring. In the current setting the LHC accelerates both beams to 6.5 TeV, giving a total centre of mass energy of 13 TeV. During the pre-acceleration the protons are arranged into bunches of 1.1 × 1011protons per bunch with a time separation of 25 ns between two bunches.

2.2 ATLAS detector

The ATLAS detector [16][17] is with a length of 44 m and a diameter of 25 m the largest detector at the LHC. It consists of different components of which most are shown in figure 2.2. The detection components are divided in three detector parts. Surrounding the particle interaction point and the beam axis (z axis) at the centre of the detector is the inner detector, enclosed by a solenoidal magnet. The solenoid creates a magnetic field of 2 T in the inner detector, parallel to the beam pipe. Charged particles follow a curved path in the inner detector due to this magnetic field. The next layer contains the electromagnetic and hadronic calorimeter, of which the central parts are arranged cylindrically symmetrically around the solenoid and the forward parts are placed perpendicular to the beam axis. The outer layer is

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Figure 2.2: ATLAS detector overview [18].

the muon spectrometer and follows the common layout. Throughout the muon spectrometer toroidal magnets create a magnetic field up to 3.9 T. The field lines form a cylinder around the beam pipe, such that the muons in the muon spectrometer are deflected in perpendicular to the deflection direction in the inner detector. Combining the two deflections gives the deflection in two dimensions. The components mentioned in figure 2.2 are either part of one of the detector layers or part of the magnet system. In the sections below each component is shortly highlighted. The coverage of each detector component is given in terms of the pseudorapidity η, which is defined by η = − ln tanθ₂, where θ is the polar angle from the beam axis. At θ = π₂, i.e. perpendicular to the beam axis η equals zero. The angle in the plane transverse to the beam pipe is denoted φ and the described detector parts cover the entire 2π range in φ.

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Figure 2.3: Inner detector, consisting of pixel detectors, semiconductor trackers and transition radiation trackers [19].

2.2.1 Inner Detector

The inner detector is shown in figure 2.3. It is 6.2 m long, has a diameter of 2.1 m and covers a range up to |η| < 2.5. The purpose of the inner detector is to measure the tracks of charged particles close to the interaction point in order to determine the momentum and charge of charged particles and to reconstruct the primary interaction vertex. Three detector types make up the inner detector, being the pixel detector, the semiconductor tracker (SCT) and the transition radiation tracker (TRT). The pixel detector and the SCT are both solid-state detectors. Charged particles that traverse these detectors create electron-hole pairs by ionising the semiconducting material in the detectors. Due to the present electric field, the electron-hole pairs produce signal currents which can be read out. The difference between the two detectors is that the pixel detector consists of pixels of 50 µm × 400 µm and has a very high resolution, while the SCT consists of strips of varying lengths of order O(10 − 100) mm.

The Higgs boson coupling to polarised W bosons in vector boson fusion

University of Amsterdam

MSc Physics

GRAPPA/particle

Master Thesis

The Higgs boson coupling to polarised W bosons

in vector boson fusion

Contents

Introduction

Chapter 1

From Lagrangian to vector boson

fusion

1.1

Gauge invariance in the Lagrangian

1.2

Electroweak unification & EW symmetry breaking in the

Higgs mechanism

1.3

Vector boson polarisation

1.4

W

W

→ W

W

scattering

1.5

Vector Boson Fusion

1.6

Beyond the Standard Model

Chapter 2

The ATLAS detector @ the LHC

2.1

LHC

2.2

ATLAS detector

_W

_scattering