A Multivariate analysis of Higgs to τhadτhad for Vector Boson

Fusion production.

Dana van der Wende

Supervisors:Stan BentvelsenPamela FerrariStefania Xella

Abstract

At Juli 2012 the announcement has been made that the Higgs particle has been found in several channels.For exploration of unknown properties of the Higgs, the fermionic channel can be considered. In this thesisresearch is done to the Higgs decaying into two hadronic taus. Multi Variate Analysis and efficiency curvesare used to optimize the separation between the Vector Boson Fusion signal and the background processes.Various sets of variables are considered. This method gives a 0.92 sigma result for the existence of a 125GeV Higgs in the VBF production mode. Further optimization of the methods and increase of statisticsboth can decrease the error-bars, thereby increasing the significance.

Contents

1 Theoretical Background 31.1 Introduction to the Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Gauge Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.1 Local Invariance of Lagrangians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.2 Standard Model Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Limits on the Higgs boson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3.1 Unitarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3.2 Triviality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.3.3 Vacuum stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 The Detector 122.1 LHC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2 ATLAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3 The ATLAS coordinate system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.4 Magnets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.5 Inner Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.5.1 Pixel detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.5.2 SCT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.5.3 TRT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.6 Calorimeters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.6.1 Electromagnetic Calorimeter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.6.2 Hadronic Calorimeter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.6.3 Forward Calorimeter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.7 Muon spectrometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.8 Trigger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.9 Event Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.9.1 Track reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.9.2 Muon and Electron reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.9.3 Hadronic Tau reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.9.4 Jet reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.9.5 MET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3 Analysis 223.1 The data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.2 Event generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.3 Higgs to ττ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.3.1 the τ particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.3.2 Higgs production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.3.3 Higgs to τhadτhad decays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.3.4 Background of τhadτhad decays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

1

3.4 Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.4.1 Mass calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.5 Event cleaning and selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.6 MC/data comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.7 Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.8 MVA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.8.1 Boosted Decision Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.8.2 ROC curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.8.3 cutting on BDT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.9 Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.9.1 Statistical error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.9.2 Systematic error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.9.3 Significance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.10 Discussion and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

A Data/MC comparison after the cut of 1 tight tau 54

B Data/MC comparison after the cut Ptτ1 > 40GeV, P tτ2 > 25GeV 57

2

Chapter 11

Theoretical Background2

1.1 Introduction to the Standard Model3

The Greeks tried to explain the world around them and created gods to hold responsible for things they did4

not understand. Such a explanation of the world is outdated. The 20th century was the time in which a5

new model was created; the Standard Model. Exploration for this model began with the discovery of the6

electron in 1897 by J. J. Thomson. A simple experiment showed that the electron was a light and negatively7

charged particle. From his assumption that electrons were the components of atoms, which are neutral and8

heavier than electrons, some questions arose. Thomson presumed the electrons were evenly distributed in a9

positive charged sphere. Rutherford proved this was a false assumption with a now well known scattering10

experiment. He showed that the mass and the positive charge were packed together in the centre of the11

atom, the nucleus. The lightest atom, hydrogen, was discovered to have a proton as its core. Building on12

this further, Niels Bohr stated that an electron orbits around the proton, kept there by the electric attraction13

of the proton. Investigation of the second lightest atom yielded even more questions, as this particle was14

four times heavier than hydrogen. This problem was solved when the neutron was discovered, a neutral15

particle that is similar to the proton. This discovery meant that the periodic table could be explained. This16

was only the beginning of a model that seemed to be simple, but it turned out to include a whole zoo of17

different particles; the Standard Model. Over the course of the last decades a major part in high energy18

physics research was addressed towards finding the Higgs boson. This recently discovered particle has the19

purpose of giving mass to the other particles and was the missing piece in the Standard Model.20

1.2 Gauge Theories21

Physics is bounded by symmetries. These symmetries can be used to predict interactions and motions; local22

gauge symmetries dictate all particle interactions. Particles are described by a Lagrangian. Since the Euler23

Lagrange equation (1) determines the equations of motion from the Lagrangian, field equations such as the24

Klein Gordon equation (2), the Dirac equation (3) and the Proca equations (4) can be deduced [4]:25

∂µ

(∂L

∂µφ

)− ∂L

∂φ= 0 (1.1)

3

Lscalar =1

2(∂µφ)(∂µ)− 1

2m2φ2 → (∂µ∂

µ +m2)φ = 0, (spin 0) (1.2)

Lfermion = iψγµ∂µψ −mψψ → (iγµ∂

µ −m)ψ = 0, (spin1

2) (1.3)

LProca =−1

16FµνFµν +

1

8πm2AνAν → ∂µF

µν +m2Aν = 0, (spin 1) (1.4)

Fµν = ∂νAµ − ∂µAν (1.5)

φ stands for the scalar field (eq. 2) and m stands for its mass. Likewise ψ stands for a spinor field in eq. 3,26

Aµ stands for the gauge field (eq. 4) and m in both cases for their masses. γµ stands for the Dirac matrices27

[4].28

1.2.1 Local Invariance of Lagrangians29

Since physics has to obey symmetries, Lagrangians have to be invariant under gauge transformations like30

phase transformations ψ → eiθψ. For example the Dirac Lagrangian shows that the Lagrangian stays31

invariant under the global gauge transformation. Nevertheless if θ is not constant (θ → θ(x)) invariance32

does not hold anymore [1].33

L → L+ (qψγµψ)δµλ, λ(x) ≡ −1

qθ(x) (1.6)

An extra term has to be added to keep the Lagrangian invariant.34

35

L = [ihcψγµ∂µψ −mψψ]− (qψγµψ)Aµ, Aµ → Aµ + ∂µλ (1.7)

Now a new field Aµ is created so the Dirac Lagrangian is invariant under local transformations. This36

Lagrangian is not the complete description of a particle. The Lagrangian also has to include a free term.37

Free massive spin-1 fields require a Proca-Lagrangian, this Lagrangian has to be added [3][1]:38

LProca = − 1

16πFµνFµν +

1

8πm2AνAν (1.8)

Here unlike the first term (− 116πF

µνFµν), the last term ( 18πm

2AνAν) is not invariant under local gauge39

transformations, therefore the mass of this field has to be 0 in order to be invariant. The introduced new40

field Aµ is massless. The new whole Lagrangian is:41

L =[ihcψγµ∂µψ −mc2ψψ

]+

[−1

16πFµνFµν

]−[(qψγµψ)Aµ

](1.9)

Where the field Aµ is the electromagnetic potential and the last two terms are identical to the Maxwell42

Lagrangian with a current density of Jµ = cq(ψγµψ). The addition of new terms can also be accomplished43

by replacing the derivative by a covariant derivative.44

Dµ ≡ ∂µ + iq

hcAµ, so Dµψ → e−qλ/hcDµψ (1.10)

This is a covariant derivative for a simplified Lagrangian, that is invariant under U(1))EM symmetries in45

QED [4].46

4

1.2.2 Standard Model Lagrangian47

The Standard Model is based on a SU(2)×U(1) Lagrangian that consists of four parts; a Yang Mills La-48

grangian, a fermion Lagrangian, a Higgs Lagrangian and a Yukawa Lagrangian [5]:49

LSM = Lgauge + Lferm + LH + LY (1.11)

This Lagrangian combines the electromagnetic and weak interactions. The different parts of this Lagrangianwill be explained below beginning with Lferm. The fermion Lagrangian shows the interaction of the gaugefields with the fermions:

Lferm = iΨL /DΨL + iψR /DψR (1.12)

In this formula left handed fields of charged leptons and neutrinos (ΨL) are SU(2) doublets:

`L = (ν`L , `L)TL, ` = e, µ, τ (1.13)

qL = (u, d)TL, u = u, c, t d = d, s, b (1.14)

while the right handed fields are singlets:

uR, dR, νl, l−R (1.15)

The interaction terms are hidden in the covariant derivative:50

Dµ = ∂µ + igWµ + ig′YLBµ (1.16)

The L and R represent the left- and right-handed projections of the field. In this covariant derivative51

YL/YR is a generator of its gauge groups in a suitable representation. Bµ and Wµ are gauge fields that are52

introduced to keep the Lagrangian invariant. Field Wµ only works on left-handed fields and is equal to 0 for53

the right-handed part of the Lagrangian. The gauge transformations of the fields are given as:54

ΨL → Ψ′L = eiYLθ(x)ULΨL, UL = eiTiβi(x) (1.17)

ψR → ψ′R = eiYRθ(x)ψR (1.18)

Here T i = τ i

2 are the generators of the representation of SU(2) Lie algebra [2].55

[T i, T j

]= iεijkT k (1.19)

Here i,j,k, run from 1 to 3. The symmetries of the fermion Lagrangian are causing the transformation56

properties of Bµ and Wµ to be [21]:57

Bµ → B′µ = Bµ −1

g′∂µθ (1.20)

Wµ →W ′µ = ULWµU†L +

1

g(∂)µUL)U†L (1.21)

The gauge parameters βi implicate the three gauge bosons (SU(2)L). Those gauge bosons couple to weak58

isospin while θ implicates gauge boson B that couples to hypercharge. Later it will be shown that by breaking59

the electroweak symmetry the gauge bosons W± and Z are created. The hypercharge and weak isospin are60

related in the following way [21]:61

5

Q = T 3 + Y, T 3L =

τ3

2, T 3

R = 0 (1.22)

Hence it can be deduced that the hypercharge for the leptons and quarks are:

YL(l) = −1

2, YR(l) = −1 (1.23)

YL(q) =1

6(1.24)

The gauge part of the Lagrangian contains the dynamics of the gauge fields and their interactions [3][21]62

and is written as:63

Lgauge = −1

4W iµνW

i,µν − 1

4BµνB

µν − 1

4GaµνG

a,µν (1.25)

W iµν = δµW

iν − δνW i

µ − gεijkW jµW

kν , i, j, k = 1, 2, 3 (1.26)

Bµν = δµBν − δνBµ (1.27)

Gaµν = δµGaν − δνGaµ − gsfabcGbµGcν , a, b, c = 1, ..., 8 (1.28)

Here W iµ are three SU(2) gauge bosons in the group of weak isospin and Bµ is a U(1)Y gauge boson in

the group of weak hypercharge. For the gauge bosons it is not simple to add a mass term to the equation,as it contains couplings to right- and left handed fields. As the right and left handed transformations aredifferent, the gauge symmetry will be broken in this term. The mass of either the bosons as the fermionshave to be added trough a new mechanism; spontaneous symmetry breaking [4].

For simplicity, to look at LHiggs only local (U(1)) invariant theory in QED is considered. This is onlya small part of the Standard Model theory. In QED, the Lagrangian can be kept invariant by taking asimplified covariant derivative:

L = (Dµφ)†(Dµφ)− 1

4FµνF

µν − µ2(φ∗φ)− λ(φ∗φ)2 (1.29)

Dµ = δµ − ieAµ (1.30)

A′µ = Aµ +1

eδµα (1.31)

If µ2 > 0, the vacuum of the system is at 0 as the potential looks like a parabola. This does not hold for64

the case where µ2 < 0. Now the potential does not have a vacuum at 0, the potential looks like a mexican65

hat where there are infinite vacua. For all φ21 +φ2

2 = −µ2/λ. The symmetry is broken. To illustrate this the66

kinetic part of the Lagrangian has to be considered.67

LHkin = (DµΦ)†(DµΦ), Φ =

(φ+

φ0

)=

1√2

(φ1 + iφ2

φ3 + iφ4

)(1.32)

where Φ is the complex scalar SU(2)I doublet, with weak hypercharge of 1. Now φ+ has a charge of e+ and68

φ0 is neutral. Now a vacuum can be chosen, the easiest way of doing that is setting φ1 = φ2 = φ4 = 0 and69

φ3 = v. To see what happens, small oscillations around this vacuum are considered. The oscillations can be70

considered in two directions, so71

φvac =1√2

(0

v + η + iξ

)(1.33)

Now with rotation of φ, the Lagrangian must stay the same.72

6

φrot =1√2e−iξ/vφ =

1√2e−iξ/v(v + η + iξ) =

1√2e−iξ/v(v + η)eiξ/v =

1√2

(v + h) (1.34)

Writing out LHkin gives

L = (δµ + ieAµ)1

2(v + h)(δµ − ieAµ)

1√2

(v + h)− V (φ†φ) (1.35)

=1

2(δµh)2 − λv2h2 +

1

2e2v2A2

µ + e2vA2µh+

1

2e2A2

µh2 − λvh3 − 1

4λh4 +

1

4λv4 (1.36)

This introduces a new particle h, a massive gauge field Aµ, interactions between h and gauge fields and73

self interactions of the new h particle. This new particle is called the Higgs boson. To also see the other74

particles, the Lagrangian has to be rewritten even more. Here eq. (1.16) is used for the covariant derivative.75

This chosen vacuum breaks the SU(2)L×U(1)Y , but not U(1)EM . This will give mass to the gauge bosons,76

but will keep the photon massless as will be shown later. To look at the masses of the gauge bosons only77

the v2 terms are considered. Considering this, the kinetic part of the Higgs Lagrangian can be written as:78

(Dµφ)†(Dµφ) =1

8v2[g2(W 2

1 +W 22 ) + (−gW3 + g′Yφ0Bµ)2

]+ (termswith igsT

ac G

aµ) (1.37)

This formula can be rewritten in terms of the gauge bosons. As the charge raising and lowering operators79

τ± = 1/2(τ1± iτ2) are associated with gauge bosons W+ and W−, W1 and W2 can be rewritten in terms of80

W+ and W−.81

W± =1√2

(W1 ∓W2) (1.38)

Also the second expression of (1.37) can be rewritten in a clearer way. This can be done by deconstructing82

the expression into a matrix and then diagonalizing the following formula83

(−gW3 + g′Yψ0Bµ)2 = (W3, Bµ)

(g2 − gg′Yφ0

−gg′Yφ0g′2

)(W3

Bµ

)(1.39)

For the particular vacuum that is chosen, Yφ0= 1. After diagonalization two eigenvectors are found (λ = 0

and λ = (g2 + g′2)) and two eigenvectors, which are taken to be Aµ and Zµ, the photon and the Z boson.

Zµ ≡1√

g2 + g′2(gW 3

µ − g′Bµ) (1.40)

Aµ ≡1√

g2 + g′2(g′W 3

µ + gBµ) (1.41)

84

MW =gv

2, MZ =

v√g2 + g′2

2(1.42)

85

(Dµφ)†(Dµφ) =1

8v2[g2(W+)2 + g2(W−)2 + (g2 + g′2)Z2

µ + 0×A2µ

](1.43)

This is only the part of the gauge bosons (terms with v2) in the Lagrangian. The Higgs mass (terms with86

h2) and the interactions between the bosons and Higgs (terms with vh) are not shown yet in the former87

Lagrangian. The complete Higgs Lagrangian with all the terms included is88

LH =1

2(∂h)2 +

g2

8(v + h)2((W+)2 + (W−)2) +

1

8(v + h)2(g2 + g′2)Z2

µ +µ2

2(v + h)2 − λ

16(v + h)2 (1.44)

LH =1

2(∂h)2 − 1

2M2hh

2 +1

2M2W ((W+)2 + (W−)2) +

1

2M2ZZ

2µ + gMWhW

+W−g2

4h2W+W− (1.45)

+MZ

√g2 + g′2

2Z2µ +

g2 + g′2

4h2Z2

µ −gM2

H

4MWh3 − g2M2

h

32M2W

h4 + const. (1.46)

7

89

Mh =√

2µ2 (1.47)

Now only the fermions are missing from the model. Those enter with the Yukawa Lagrangian. The mostgeneral form is:

LY = Γumnqm,Lφun,R + Γdmnqm,Lφdn,R + Γemn lm,Lφen,R + Γνmn lm,Lφνn,R + h.c. (1.48)

The couplings between the Higgs doublet and the fermions are described by the matrices Γmn. These arethe Yukawa couplings. This part of the Lagrangian is gauge invariant. As the mass terms should haveno hypercharge, the Higgs field should have two representations (Y = 1

2 , Y = − 12 ) to give mass to all

the fermions except the neutrino. The neutrino does not have a right handed counterpart, which makes itimpossible to give it mass through the Yukawa coupling. The representations and their transformations are:

φ =

(φ+

φ0

)for Y =

1

2(1.49)

φ =

(φ0∗

−φ−)

for Y = −1

2(φi = εijφ

∗j ) (1.50)

Only the first family is considered here for simplicity, now the Yukawa Lagrangian is:

LY = fe lLφeR + fuqLφuR + fdqLφdR + h.c. (1.51)

here φ is chosen such that

φ =1√2

(0

v + h

)→ 1√

2

(0v

), φ =

1√2

(v0

)(1.52)

LY becomes:

LY =fev√

2(eLeR + eReL) +

fuv√2

(uLuR + uRuL) +fdv√

2(dLdR + dRdL) (1.53)

mi = −fiv√2

(1.54)

This gives masses to all of the fermions, completing the Standard Model Lagrangian.90

1.3 Limits on the Higgs boson91

As seen in the previous section, the Standard Model suggested a new massive particle, the Higgs boson.92

Hints for the existence of this particle have recently been found. The mass of this particle can not directly93

be deduced from the theory, however the theory can put limits on the range the mass can be in. The Higgs94

mass can be limited by a few principles; unitarity, triviality and vacuum stability. Those principles will be95

clarified in the following sections. From this section it can be seen that the the mass at which the Higgs has96

been found is within the boundaries of the theories.97

1.3.1 Unitarity98

The first constraint is unitarity [4]. In the previous section the W and Z boson acquired mass by breaking99

the symmetry. Those mass terms create an inconsistency in the theory at higher energies. At high energies100

the amplitude for elastic scattering of longitudinal massive gauge bosons diverges. The longitudinal W boson101

(WL) scattering amplitude grows as the CM energy increases, violating unitarity.102

A(W+LW

−L →W+

LW−L ) =

g2

4M2W

(s+ t) (1.55)

8

At an centre of mass energy of 1.2 GeV this violates unitarity [33]. This problem can be solved by the103

existence of a particle lighter than the unitarity bound. Introducing a Higgs particle introduces the following104

term105

A(χχ→ χχ) =1

v2

[as2

s−m2h

+at2

t−m2h

](1.56)

The left part in the brackets represents the scalar exchange via a Higgs, the right part represents the inelastic106

channels. If a equals 1 and b equals 1 the model is renormalisable. Including this Higgs contribution in the107

W boson scattering gives:108

A(W+LW

−L →W+

LW−L ) =

g22m

2h

4M2W

[s

s−m2h

+t

t−m2h

](1.57)

Unitarity of WW requires the upper limit of the Higgs mass to be mh ≤ 780GeV .109

1.3.2 Triviality110

Triviality bounds the mass of the Higgs even more [3]. In a scalar theory, the potential is111

V (Φ) = µ2|Φ†Φ|+ λ(|Φ†Φ|)2 (1.58)

After spontaneous symmetry breaking λ is related to the Higgs mass:112

λ =m2H

2v2(1.59)

the quartic coupling (λ) is a running parameter [34]:113

dλ

dt=

3

4π2

[λ2 +

1

2λh2

t −1

4h4t +B(g, g′)

], t ≡ log(Q2/Q2

0) (1.60)

with Q as the effective energy scale and Q0 a reference scale. ht is the Yukawa coupling of the top to the114

Higgs. B(g,g’) is the contribution from gauge bosons:115

B(g, g′) =1

8λ(3g2 + g′2) +

1

64(2g4 + (g2 + g′2)2) (1.61)

Considering a heavy Higgs, only the first term dominates and will contribute to the Higgs mass.116

dλ

dt=

3λ2

4π2(1.62)

The solution for this equation is:117

λ(Λ) =λ(v)

1− 3λ(v)4π2 log Λ2

v2

, mH =√−2λv2 (1.63)

m2H <

8π2v2

3 log Λ2/v2(1.64)

If there is no new physics up to 1016GeV, the upper bound of the Higgs mass is 160 GeV. For a lower cut-off118

scale, the upper bound will be higher. This limit is only accurate if the equation for λ is valid for the whole119

range of λ. However, at large λ, higher order corrections have to be included, which makes the calculation120

more complicated.121

9

1.3.3 Vacuum stability122

Also a lower limit on the Higgs mass can be set. Now only the second part of equation 1.60 is considered,123

as the contributions of the gauge bosons are bigger than the contribution of the Higgs.124

dλ

dt=

1

16π2

[−12g4

t +3

16(2g4 + (g2 + g′2)2)

](1.65)

which gives125

λ(Λ) = λ(v) +1

16π2

[−12g4

t +3

16(2g4 + (g2 + g′2)2)

]log

(Λ2

v2

)(1.66)

Taken that λ(Λ) >0, the lower bound of the Higgs is:126

m2H >

v2

8π2

[−12g4

t +3

16(2g4 + (g2 + g′2)2)

]log

(Λ2

v2

)(1.67)

Using the 2 loop renormalization group effective potential and the development of all couplings, the require-127

ment of vacuum stability can be calculated for different cut-off scales. Taking the cut-off energy to be the128

Planck scale (1016 GeV);129

mH > 130.5 + 2.1(mt − 174) ≈ 132.5 (1.68)

for a cut-off of 1 TeV, the Higgs mass is bound by:130

mH > 71 + 0.74(mt − 174) ≈ 71.8 (1.69)

In Figure 1.1 is shown how the mass bounds relate to the cut-off scale.131

10

Figure 1.1: Dependence of the bounds on the Higgs mass on the cut-off scale. Outside the bands the Higgsis excluded.

11

Chapter 2132

The Detector133

2.1 LHC134

The Large Hadron Collider (LHC) is a proton-proton and nucleus accelerator [10]. This collider is build in135

the former LEP [27] tunnel in Geneva and is designed to reach a centre of mass energy of 14 TeV and a136

luminosity of 1034cm−2s−1. This high luminosity is needed to have a higher chance to observe the processes137

that are hoped to be found at the LHC. Processes such as Higgs production and new physics are expected138

to have small cross-sections. In order to achieve this high luminosity and to reduce inelastic collisions that139

occur simultaneously in the detectors (pile-up), the beam crossings occur 25 ns apart. At the intended140

luminosity, this would on average result in 23 inelastic interactions per beam-crossing. There are several141

stages to accelerate the protons before they collide. The protons are acquired by stripping off the electron142

of a hydrogen source. The protons are then accelerated up to 50 MeV by LINAC2, a linear accelerator.143

Then the protons are inserted in the Proton Synchrotron Booster, then in the Proton Synchrotron and then144

in the Super Proton Synchrotron. Those are circular accelerators and accelerate the particles to 1.4 GeV,145

26 GeV and 450 GeV, respectively. Then the particles are introduced into the LHC, which accelerates the146

protons up to the collision energy. The detectors at the LHC are ATLAS, CMS, ALICE and LHCb. ATLAS147

(the detector of interest here) and CMS are comparable detectors that aim to test physics processes in the148

Standard Model. For this purpose all outgoing particles have to be measured, so those detectors cover almost149

the whole angular range around the interaction point. ALICE is designed to measure heavy ion collisions,150

to look for the substructure of nuclei, quark confinement and the quark-gluon plasma. LHCb is built to look151

at some SM parameters and CP-violation. The placing of the LHC with its detectors is shown in Figure 2.1.152

153

2.2 ATLAS154

ATLAS (A large Toroidal LHC ApparatuS) is one of the general purpose detectors at the LHC. This detector155

is 44m long and has a diameter of 25m. In total this detector weights 77000 tonnes. ATLAS is designed156

to look at p-p and heavy ion collisions. The detector is designed to cover as many physics processes as157

possible. Known SM processes are used for checking the performance of the subdetectors. Afterwards those158

detectors are used for finding the Higgs. For finding the Higgs, multiple production and decay mechanisms159

that are depending on the Higgs-mass can be considered. The subdetectors are arranged in several layers160

around the interaction point. This assures that a particle always has to travel through several layers of the161

detector, if the particle has enough energy to leave the beam-pipe. Also, the different subdetectors measure162

complementary properties of the particles that are going through, covering the expected and sought particles.163

From the interaction point to the outside of the detector the subdetectors are:164

165

- Inner Detector (ID), consisting of: the Pixel detector, the SCT tracker and the TRT tracker;166

12

CERNfaqLHCthe guide

Figure 2.1: The LHC experiment including the placing of its detectors and its pre-accelerators. Figure takenfrom [46].

- Electromagnetic and hadronic calorimeters;167

- Muon spectrometer;168

169

The magnet system is used to acquire more information on the momentum and charge of the particles.170

The information recorded by the subdetectors is triggered almost simultaneously, reducing the data that has171

been saved on the disk space. The placement of the subdetectors and magnets is shown in Figure 2.2.172

2.3 The ATLAS coordinate system173

To be able to define tracks, a coordinate system has to be chosen. In the coordinate system the nominal174

interaction point is taken as the origin [10]. The z-axis is defined to be the direction of the beam and the175

x-y plane is transverse to the beam. The positive x axis is pointing towards the centre of the LHC ring176

and the positive y-axis is pointing upwards. A peudo-spherical coordinate system has been chosen. The177

azimuthal angle φ is the angle around the beam axis. Polar angle θ is the angle from the beam axis, but178

usually the Pseudorapidity η is used, as this variable is invariant under Lorentz boosts. η is defined as179

− ln tan (θ/2), but for massive objects η=1/2 ln[(E + pz)/(E − pz)]. The distance between two objects is180

defines as ∆R=√

∆η2 + ∆φ2. The transverse variables are defined to be in the x-y plane (pT , ET and181

ETmiss).182

13

Figure 2.2: The ATLAS detector and its subdetectors. Figure taken from [28].

2.4 Magnets183

An important part of the detector is the magnet system. The magnets make it possible to determine the184

momentum and charge of high energetic particles [9], since electrically charged particles are bent when going185

through a magnetic field. The bending is stronger for less energetic particles giving information about the186

momentum. ATLAS has two different magnet systems: A superconducting solenoid encloses the ID and187

three superconducting toroids are surrounding the calorimeters. The solenoid magnet system is 1 radiation188

length thick. Since this magnet is located in the detector, its size is limited.189

2.5 Inner Detector190

The ID combines high resolution detectors and continuous tracking elements [10] and is required to have191

a good momentum resolution for |η| < 2.5. This detector has the purpose to determine momentum and192

orientation of tracks very precisely, identify primary and secondary vertices and is used for electron identi-193

fication. To be able to meet this purpose, high requirements of momentum and vertex resolution have to194

be met. Pixel trackers, Silicon microstrip Trackers (SCT) and the Transition Radiation Tracker (TRT) are195

designed to meet these standards. Together they form the ID. The ID operates in a 2 Tesla magnetic field196

that is generated by the central solenoid. The combination of the Pixel, the SCT and the TRT results in a197

solid pattern recognition and precise measurements in both R-φ and z. The ID also measures the transverse198

momentum. Typically each track crosses three pixel layers. For the SCT, 8 layers are crossed resulting in 4199

space points. Together a robust pattern recognition is achieved in both φ and z. Also the straw tubes of the200

TRT provide a large number of hits, whereby they contribute significantly to the momentum measurement,201

even though they have a lower precision per point. This gets compensated by the measured track length and202

the large number of measurements.203

2.5.1 Pixel detector204

The Pixel detector consists of silicon detectors with a high granularity and is positioned as close as possible205

to the interaction point. Pixel has three barrel layers and three endcap layers at each side. The layers206

are formed of slightly overlapping modules that are made of silicon pixels, the overlaps assure that every207

particle typically crosses three Pixel layers. The Pixel detector gives three measurements with high spatial208

14

Figure 2.3: The inner detector with its subdetectors. Figure taken from [28].

granularity, enabling the separation of the large number of tracks that is going through. The most inner209

barrel layer is important for vertex identification. The silicon detectors work through the principle of doping.210

Here the silicon is changed so it either has an excess of electrons (n-type material) or an excess of holes (p-211

type material). A hole is the lack of an electron in a lattice. This hole can travel through the lattice as212

an positive charged particle would. If the n-type and p-type material are placed next to each other, the213

electrons travel to the p-type and the holes to the n-type material creating a ’depletion zone’, two opposite214

charged layers. Hereby an electric field is created, stopping the flow of the electrons and holes, and creating215

a static situation. A charged particle going through the silicon will create electron-hole pairs. When it is216

going through the depletion zone the electrons will go to the p-side and the hole will go to the n-side creating217

a current, that is read out.218

2.5.2 SCT219

The semi-conductor tracker (SCT) consists of a barrel part and two end-caps. The SCT contributes to the220

identification of charged particles; measured are four two dimensional measurements of those particles. The221

SCT is composed of silicon sensors, like the Pixel detector. However, for this subdetector the sensors are222

strips since the flux of the particles going through is lower. The detection principle is the same as is explained223

for the Pixel detector. The barrel of the SCT measures up to |η| < 1.4, while the end-caps extend this to224

|η| < 2.5.225

2.5.3 TRT226

The Transition Radiation Tracker (TRT) consists of straw detectors. The straw detectors are drift tubes227

that are 4mm thick, filled with gas. If a charged particle comes through a tube, the gas ionizes. A potential228

is made between the tube and an anode wire at the centre of the tube. This potential makes the ions made229

at the ionization move toward the tube and the electron move towards the wire, which induces a small230

current. This current is read out at the end of the tube. In the material of the tubes there are materials231

of different refractive indices, causing high energetic particles that are going through to radiate off photons232

with an energy of order of the KeV: transition radiation photons. The gas in the tubes contains xenon,233

which absorbs those photons. The amount of radiation depends on the mass of the particle: an electron234

emits much more transition radiation than a pion, which makes it possible to distinguish between those two235

particles.236

15

2.6 Calorimeters237

The inner detector is surrounded by calorimeters. The Calorimeters in ATLAS are designed to give precision238

measurements of photons, electrons, jets and missing energy (MET). This part of the detector is the only part239

that can measure neutral objects. The calorimeters can be divided between the electromagnetic calorimeters240

(ECAL) and the hadronic calorimeters (HCAL), measuring up to |η| =3.2. Additional coverage 3.1 < |η| <4.9241

is reached by the forward calorimeter (FCAL). The calorimeters are sampling detectors, which uses layers242

of absorbing material and active medium. The absorbing material induces the particle traveling through to243

shower, losing some of its energy. The active medium measures the energy of the charged particles in the244

shower.245

Figure 2.4: The Calorimeter. Figure taken from [28].

2.6.1 Electromagnetic Calorimeter246

The Electromagnetic Calorimeter (EMCAL) measures the energy of the electrons and photons. The Elec-247

tromagnetic calorimeter consists of a barrel component and two endcap components. The barrel calorimeter248

is divided in two halves which are separated by 4mm at z=0 (interaction point). The endcap calorimeters249

consist of two parts, an outer (1.375 < |η| < 2.5) and an inner wheel (2.5 < |η| < 3.2). The passive material250

in this calorimeter, lead, induces the particles to shower. Those showers are observed in the active material,251

liquid argon (LAr). In this material the charged particles ionize the argon. Like the TRT a potential induces252

a signal that can be read out. The potential is caused by electrodes in the liquid argon. The plates are253

structured in a zigzag method to ensure enough radiation lengths of material (22 for barrel, 38 for endcap)254

to absorb EM showers and to ensure a complete coverage in φ. In the overlap region between the barrel and255

endcap (1.37 < |η| < 1.52), the resolution is limited, electrons in this region are taken out of the analysis.256

The depth of this calorimeter is optimized to stop the electromagnetic showers, while letting through most257

hadronic showers.258

2.6.2 Hadronic Calorimeter259

The Hadronic Calorimeter (HCAL) measures the hadronic particle jets caused by particles that are strongly260

interacting, like taus, quarks and gluons. Those jets are not absorbed by the EMCAL, so they can be261

measured by HCAL. HCal consists a barrel, two extended barrels and two endcaps, covering a range up262

to η < 3.2. Those detectors are also sampling detectors, but their materials vary. The barrel parts are263

16

made of steel (passive) and scintillators (active), the scintillators induce the particles to produce light. This264

light is transported to, and measured by optical fibers in photomultiplier tubes. The endcap hadronic265

calorimeter is made from copper (passive) and liquid argon (active) for the readout. The thickness of the266

hadronic calorimeter is optimized to assure a low possibility of punch-through of the hadronic showers to267

the muon-system.268

2.6.3 Forward Calorimeter269

The Forward Calorimeter (FCAL) covers the region 3.1 < η < 4.9 to also include the particles in the forward270

direction. The large coverage in η and the overlap in coverage of the detectors makes sure that the energy271

of all particles going out and therefore the missing transverse energy (energy imbalance) can be measured.272

FCAL consists of three copper and liquid argon modules: one measuring the electromagnetic deposits and273

two measuring the hadronic energy deposits.274

2.7 Muon spectrometer275

It is important to measure high energetic muons well, as those appear frequently in processes of interest at276

the LHC. Since of the particles produced by the interaction only muons reach the muon spectrometer, this277

detector is very effective for muon identification. The muon system is based on the bending of muon tracks278

in the magnetic field of the toroid magnets [28], which is measured by the high precision tracking chambers279

[9]. Muons with a high momentum have a smaller curvature. To acquire the momentum of the muons280

correctly, long muon tracks are needed, hence the large volume of this subdetector [38]. Monitored drift281

tubes (MDTs) are tubes with a diameter of 30mm that are 1 to 6 m in length. Those tubes are filled with282

Ar/CO2 gas and have a tungsten-rhenium anode wire in the centre that has a potential of 3080 V. MDTs use283

the technique of the multi-wire proportional chamber, where the charged particle going through will ionize284

the gas, creating an avalanche of electrons. This avalanche induces a signal in the wire. The MDTs provide285

the high precision track coordinates in the z and R direction. Those drift tubes are assembled in layers of286

3-4 tubes thick and 1-2m width. Two of those assemblies form a MDT chamber. The muon spectrometer287

includes more that 1000 MDT chambers. The track region < 2.7 is covered. In the region 2.0 < η < 2.7 the288

inners layers are cathode strip chambers (CSC), multi-wire proportional chambers. Those chambers have a289

better time resolution and can process higher signal rates. The CSC has two cathode strips and multiple290

wires per chamber. In the muon spectrometer, muons get triggered by the resistive plate chambers (RPC)291

at the barrel and the thin gap chambers (TGC) at the end-caps. To trigger on muons those chambers should292

process information very fast. The RPCs are chambers filled with gas where two parallel resistive plates293

with a distance of 2mm induce a field of 9.8kV. It provides 6 points in η and φ. This chamber works like294

a multi-wire chamber, but are wireless. The gas gets ionized by charged particles and an avalanche gets295

created traveling to the anode. The time resolution of the RPC is 1 ns. The TGCs in the end-caps are296

similar to the CSCs, but for the TGC the distance between the wires is optimized so the drift time is faster297

than the time between bunch crossings [44]. The drift time in the TGCs is shorten than 25ns and provides298

information about φ and η.299

2.8 Trigger300

As the information produced by the LHC is far too much to all record, a selection has to be applied almost301

simultaneously with the measurements. This preselection is done by the trigger. The Trigger at ATLAS is302

divided into three stages: L1, L2 and the event filter (EF) [9], where the L1 and L2 are online triggers and303

EF is an offline trigger. L1 is a hardware trigger. This trigger only uses information from the muon trigger304

(RPC and TGC) and the calorimeters. Here the aim is to select high energetic events. Muon signatures305

and large energy depositions indicating photons, electrons or hadrons are taken into consideration, even as306

the total transverse energy and MET. When an event is selected on those consideration it is passed to the307

17

Figure 2.5: The Muon system. Figure taken from [45].

L2 trigger. L2 is a software trigger. At this level the complete detector data is used to select events of308

interest. After an improved selection, the selected events are passed on the event filter. At the event filter309

the complete detector data is also used. Here also, reconstruction and analysis algorithms are used to select310

events. Only an event that passes L1, L2 and EF as the same object goes through to the analysis. The event311

filter EF tau29Ti medium1 tau20Ti medium1 [30] will be used in the analysis discussed in this thesis. This312

trigger selects events with a minimum of 2 taus, where the first tau has a momentum higher than 29 GeV313

and the second tau a momentum higher than 20 GeV. Only taus that satisfy medium tau ID and having314

less than 5 tracks are taken into account.315

2.9 Event Reconstruction316

After triggering, objects can be reconstructed by multiple algorithms. The identification performed by317

ATLAS and the decisions for object reconstruction in this analysis, will be shown. This reconstruction is318

challenging as it has to deal with pile-up [42]. Due to the high luminosity of the LHC multiple collisions319

occur simultaneously, which implies difficulties distinguishing separate events. The additional collisions are320

called pile-up. More collisions cause more charged tracks at the ID and more energy measurements at321

the calorimeters. To resolve this primary vertex measurement and tracking have to be very precise. The322

following sections describe the objects that will be used in the analysis discussed in this thesis, this includes323

the constraints on the selection of the tracks, particles, jets and missing transverse energy (MET).324

2.9.1 Track reconstruction325

Usually the particle identification begins with the reconstruction of tracks, using the measurements of the326

ID [43]. The hits in the pixel detectors and SCT are used for space point measurements in three dimensions.327

Now the direction from this track is followed to search for expected hits, extending the track into the TRT.328

Now more requirements are set on the tracks to minimize mis-identification of tracks. The tracks are required329

to have 9 hits in pixel, and no holes, which is a space where a hit should have been, but is not detected.330

High momentum tracks have a reconstruction efficiency of 90%. Now the primary and secondary vertex can331

be reconstructed. A vertex finding algorithm and vertex fitting algorithm are used for this. The primary332

vertex is taken to be the vertex with the highest scalar momentum (scalar sum of the momentum of all333

tracks associated to the vertex) in the event.334

18

2.9.2 Muon and Electron reconstruction335

Muons and electrons can be reconstructed combining the data from the subdetectors. Electron candidates336

are detected by a combination from a track in the inner detector and a cluster in the electromagnetic337

calorimeter, where they are stopped. For this analysis they need to be identified as MediumPP [41]. The338

requirement mediumPP uses information from the middle layer of the calorimeter, energy leakage into the339

hadronic calorimeter, tracking information from the whole inner detector, use the b-layer and track-to-cluster340

matching in η. Electrons in the region η > 1.37 and η <1.52 are neglected due to measuring difficulties341

because of the transition between the barrel and end-cap calorimeter. As the electron has to have charge,342

the required absolute value of the charge has to be equal to 1. Moreover Quality criteria have to be applied343

[40]. Muons are the only direct detectable particles that reach the muon spectrometer, which reconstruct the344

muons, with supplementary information of the ID and calorimeters. Muons have to be reconstructed by the345

STACO algorithm [23]. The muons are to be found in the region |η| < 2.5. Furthermore quality criteria on346

the inner detector track [39] are applied. These quality criteria take the amount of hits of the BLayer, hits347

of the Pixel detector, hits of the SCT and hits of the TRT into account. The electrons and muons used for348

this analysis described in the rest of the thesis are requested to be pT > 15 GeV, pT > 10 GeV, respectively.349

2.9.3 Hadronic Tau reconstruction350

In this thesis only hadronically decaying taus are considered. Those taus are reconstructed by the clus-351

ters in the hadronic and electromagnetic calorimeters [12]. The tau is identified with the Boosted De-352

cision Tree (BDT) identification method [22], which uses a Boosted decision tree to identify the tau.353

Tight/Medium/Loose BDT Scores are respectively about 40%/60%/70% signal efficient [13]. In order to354

be selected, the tau candidate has to pass Tau BDT Medium. Also the tau has to have a momentum above355

20 GeV. The track with the highest momentum has to be at either |η| < 1.37 or 1.52 < |η| < 2.5, this ex-356

cludes the transition region the barrel and end-cap of the calorimeter. As the tau gets identified differently357

in data and Monte Carlo, the scale factors for 1- and 3- prong taus differ: 0.98±0.02 for 1-prong taus and358

1.05±0.05 for 3-prong taus. Furthermore, the taus need to have an absolute charge of 1, also they can only359

have 1 or three tracks at a distance of 0.2 around the cone axis. In the end, the tau has to match the tau360

that has been firing the trigger at the Event Filter. The matching has to be within ∆R = 0.2.361

2.9.4 Jet reconstruction362

In ATLAS, the calorimeter and the ID are used for measuring jets. Jets are induced by the quarks and363

gluons that come from the interaction. Those quarks and gluons have color, which makes them hadronize.364

This hadronization leads to a shower of hadrons around the concerning quark or gluon. This shower is365

measured and is called a jet. Jets are reconstructed with the anti-kt algorithm [29]. Moreover, the jets need366

to have a pseudorapidity |η| < 4.5. In this thesis only tagging jets are considered, those jets have a few367

more requirements. The jets with |η| <2.4, have the requirement |JVF| >0.5 and pjetT >30 GeV. JVF stands368

for the Jet Vertex Fraction [26], which is the sum of all the tracks within the jet that are matched to the369

primary vertex divided by the momentum of all tracks of the jet. For a jet with |η| >2.4 only pjetT >35 GeV370

is required.371

2.9.5 MET372

Neutrino’s can not directly be measured since they do not interact with the detector. Indirectly some infor-373

mation can be collected though. For this the momentum imbalance in the transverse direction is used. The374

protons before the collision do not have any momentum in the transverse direction. Since momentum should375

be conserved, the vector sum of all objects should have a transverse component that is equal to zero. If376

there are neutrinos or other not-detectable particles (beyond the SM) in the event, the measured transverse377

vector momentum will be non-zero. This is the missing transverse momentum, which is equal to the sum of378

not-measured particles [43]. This missing transverse energy can be reconstructed using objects measured by379

19

calorimeter cells [12]. The x and y component are the sum of the missing transverse energy of all objects,380

which is the negative sum of the calibrated cell energies in the calorimeter. Jet objects used in this calcula-381

tion are weighted with the Soft Vertex Fraction since a higher pile-up has to be taken into account. The Soft382

term Vertex Fraction [25] is the summed momenta of all tracks matched with the primary vertex divided383

by the total jet-matched track momenta. If there are tracks that are not from the primary vertex in the jet384

the STVF has to be less than 1. For MET only the tracks that come from the primary vertex should be385

considered. The tracks that are not from the primary vertex are likely to come from another process than386

the process of interest, usually pile-up, which implies this re-weighting will improve the MET measurement387

in the data. To check if the re-weighting of MET is needed, MET RefFinal STVF and MET RefFinal are388

compared to see for which variable the data/MC agreement is the best. Those plots are made after the389

selection explained in section 3.5.390

391

As can be seen in figure 2.6 the Missing transverse energy is modelled better in the low energy region using392

the variable MET RefFinal STVF, which is expected since the pile-up is in the low-energy range. The error393

bars show the combined systematic and statistical error.394

20

Missing Transverse Energy /GeV20 40 60 80 100 120 140 160

even

ts

0

50

100

150

200

250

300

350

400

MET VBFggHttbar

ντ →W ττ →Z

QCDdatastat+syst err

20 40 60 80 100 120 140 160

ratio

dat

a/M

C

00.20.40.60.8

11.21.41.61.8

2

(a) MET RefFinal.

Missing Transverse Energy /GeV20 40 60 80 100 120 140 160

even

ts

0

50

100

150

200

250

300

350

400

MET VBFggHttbar

ντ →W ττ →Z

QCDdatastat+syst err

20 40 60 80 100 120 140 160

ratio

dat

a/M

C

00.20.40.60.8

11.21.41.61.8

2

(b) MET RefFinal STVF.

Figure 2.6: Comparison of the MET without (a) and with STVF weighting (b) after the selection explainedin section 3.5.

21

Chapter 3395

Analysis396

The analysis has a few steps that will be described in this thesis. First the data used in this analysis is397

explained (section 3.1), then the event generation is set out in section 3.2. In section 3.3 the signal and398

background processes are illustrated and analyzed. Then in section 3.4 all variables that are used are shown399

and explained. Section 3.5 set the constraints on the selection of the events out. For a few variables it has400

been checked in section 3.6 whether the Monte Carlo’s added up compare well with the data. Section 3.7401

discloses a comparison of the shapes of the variable distributions of both the signal and the background402

variables where the shape of the signal and background distribution differ are likely to be good variables for403

the Multi Variate Analysis (MVA). MVA and BDT’s (Boosted Decision Trees) are explained and optimized404

in section 3.8. In section 3.8.3 a cut on the BDT Output is made, giving the results in section 3.10 using405

the statistics explained in 3.9.406

3.1 The data407

In this analysis data from proton-proton collisions is used, collected by ATLAS at the LHC running at 8408

TeV in 2012. The data has a luminosity of about 20.343fb−1 after the ATLAS quality checks.409

3.2 Event generation410

The LHC is built to check models and to discover or exclude new particles. In this analysis the Standard411

Model is being checked. If the Standard Model is valid, the Higgs boson has to be found. To see whether a412

model is true, samples of all (significant) processes in this model have be made to compare with the data.413

Those samples are called generated samples and are made with a Monte Carlo generator. Because of the414

knowledge of these models the particles that come out after a proton-proton collisions and their features can415

be predicted. Monte Carlo samples have to be made of all processes that ought to be of significance in the416

data. If the predicted model is valid, the Monte Carlo’s of all processes added up should be in agreement417

with the data. The event generation is done in phases: hard scattering, parton shower and hadronization.418

Also the underlying events are generated. Those will be discussed in this order.419

420

The hard scattering is the part of the event that is of interest, where the protons collide effectively. An421

event generator takes a random value for the possible positions. The incoming and outgoing particles have422

a defined four-momentum. For the event the cross-sections, the weight can be calculated. The weight is423

related to the probability this event would occur. The result of the hard scattering generation is a set of424

simulated events that behave like real events, behaving according to the theory that is chosen. Examples425

of Monte Carlo generators simulating hard scattering are POWHEG [16] and ALPGEN [17]. The processes426

that are calculated with those generators are set out later. The next step is parton showering. After the427

hard scattering the partons can radiate a quark, gluon or photon or divide into other partons. This creates a428

22

large quantity of partons. Practically is it not possible to calculate such processes for a large number of tree429

splittings, also virtual emission and absorption gives problems. Therefore an approximation of this step is430

made with models of parton showering. After the parton shower there are colored partons. The transition of431

the colored partons to colorless hadrons is called hadronization. Hadrons are particles that can be detected432

in the detector. Since hadronization is non-perturbative, this step can’t be calculated exactly, it has to be433

modeled. The hadronization ratio is tuned to former experiments. Hadronization and parton showers can434

be simulated with HERWIG [15]. The underlying event should also be calculated. The underlying event435

covers the interaction of the particles that are not included in the hard scattering, this could be interactions436

of proton remnants, gluons and quarks. Secondary interactions are important as they can be detected by437

ATLAS. Usually those processes are soft interactions. The underlying event is followed by parton showering438

and hadronization, like hard scattering and it can affect the measured number of particles. An example of439

an underlying event generator is JIMMY [18]. The response of the detector to the simulations is done by440

GEANT4 [32]441

442

Which generators are used to simulate the signal and background processes is described later in this chapter.443

First the decay process of interest in this analysis, H→ ττ (VBF), is explained.444

3.3 Higgs to ττ445

Before going further into the analysis it is important to understand the processes of interest. Higgs to tau446

tau where the taus decay hadronically ought to be found. Here only the Higgs created by Vector Boson447

fusion is considered to be signal. The tau particle, Higgs production, Higgs decay and the main background448

processes will be explained in this section.449

450

The Higgs particle can not be seen directly from the data as it decays quickly, the Standard Model predicts451

a mean lifetime of 1.6×10−22s. In a percentage of the cases the Higgs particle decays into two τ particles.452

The decay width of the Higgs to two τ particles can be calculated by the following formula [6].453

ΓH→ττ =1

8π

(g

2

mτ

mW

)2

mH

(1− 4

(mf

mH

)2) 3

2

(3.1)

Here the term 1 − 4 (mf/mH)2

can be approximated by one since the τ particle is much lighter than the454

Higgs. The formula shows that the branching ratio scales to the square of the τ mass. For a Higgs boson455

mass of 125 GeV, the decay width of Higgs to ττ is 2.6× 10−4 GeV. Besides H→bb (branching ratio ∼70%),456

the ττ channel (branching ratio ∼8%) is the only fermionic channel that is measurable in this run of the457

LHC.458

3.3.1 the τ particle459

The tau particle is the heaviest lepton in the Standard Model (mτ = 1.777 GeV), which implies its’ very460

short lifetime of (290.6±1.0) ×10−15s (87 µm). For this reason it can not be detected directly in the detector,461

the decay products have to be detected. Tau decays can be classified as 1-prong or 3-prong, which means462

that the tau particle decays to respectively 1 or 3 charged particles in the final state. Tau decays with more463

than 3 particles are rare [6]. There are two algorithms used in ATLAS for identification and reconstruction of464

tau leptons, tauRec and tau1P3P [14]. Since hadronic tau leptons look like a narrow jet in the detector it is465

very important to have a good tau/jet separation, which is a cornerstone of the algorithms. The tau particle466

can decay in two different ways; hadronically or leptonically, as can be seen in figure 3.1. The hadronic467

decays have a branching fraction of approximately 65% and the leptonic decays have a branching fraction of468

35%, so H→ τhτh has a branching ratio of 8%×42,3%.469

470

23

Figure 3.1: The decay of a tau lepton.

Table 1 shows the most common decays of the tau lepton and the branching fractions. For hadronic decays,471

one or more pions are formed.472

h/l prong channel BRleptonic 1-prong τ− → e−νeντ 17.8%leptonic 1-prong τ− → µ−νµντ 17.4%hadronic 1-prong τ− → π−ντ 10.8%hadronic 1-prong τ− → K−ντ 0.7%hadronic 1-prong τ− → π−π0ντ 25.5%hadronic 1-prong τ− → K−π0ντ 0.4%hadronic 1-prong τ− → π−2π0ντ 9.4%hadronic 1-prong τ− → π−3π0ντ 1.1%hadronic 3-prong τ− → π−π−π+ντ 9.3 %hadronic 3-prong τ− → π−π−π+π0ντ 4.6%

Table 3.1: The decay channels of the τ particle [7].

3.3.2 Higgs production473

There are four main ways for the Higgs boson to be created; by gluon gluon fusion (ggH), vector boson474

fusion (VBF), associated decay (ZH or WH), or associated top anti top production with a Higgs (ttH).475

These processes are shown in Figure 3.2.476

477

478

The last two have a cross-sections smaller than the first two, so those are not added to this analysis. VBF479

has a very distinct signature. In this process, two forward jets are formed, while the Higgs decays into two480

taus. Simulations of VBF and ggH are performed by POWHEG [16], while then underlying event, parton481

shower and hadronization is done though an interface with Pythia8 [48]. Because of this distinct features,482

VBF is a good channel to use for analysis. ggH has a higher cross-section than VBF, so in this thesis it is483

added as a background.484

3.3.3 Higgs to τhadτhad decays485

The ratio of Higgs particles that decay into two taus compared to other end products can be seen in Figure486

3.3. The branching ratio of the Higgs to two taus is about 8% for a Higgs with a mass around 125 GeV.487

488

24

Figure 3.2: The main Higgs production processes (a) ggH, (b) VBF, (c) ZH/WH and (d) ttH.

3.3.4 Background of τhadτhad decays489

The main backgrounds for Higgs decaying into an hadronic tau, are QCD multi-jets and Z→ ττ . Further-490

more there are some contributions from W→ τν, tt, single-top and WW, from which the last two are left out491

of this analysis since their contribution is very small. Also ggH is taken to be background as the analysis is492

optimized to find the Higgs via VBF.493

494

One of the main background is Z/γ*. Here there is one real Z boson and a virtual off-shell photon. The495

two taus are formed by the decay of the Z. This background has the same final state as the signal, this is496

called irreducible background. If a Z is produced in association with two jets, it is only possible to reduce497

this background considering the mass of the Z, which is lower than the Higgs, or the fact that the two jets498

are not necessarily forwards. The Monte Carlo samples for Z→ ττ are made with event generator ALPGEN499

[17]. The parton shower and hadronization are simulated with HERWIG [15] and the underlying events500

calculated with JIMMY [18].501

502

In W→ τν, the W that is formed does only decay into one tau. The reason W still appears as a background,503

where only two taus get selected, is that a second tau is selected misinterpreting another particle or jet.504

The second tau is very likely to be a mis-identified QCD jet, those jets are typically produced in the hard505

scattering. This misidentified tau is likely to have another relative direction to the real tau than would be506

expected from a tau pair from the Higgs decay. Also the directions of the jets and invariant mass seem to507

be good variables to separate this background from the signal.508

509

The Monte Carlo samples for W→ τν are, like for Z→ τν, made with event generator ALPGEN, HERWIG510

and JIMMY.511

512

Another irreducible background is the contribution of tt. Also for tt the relative directions of the tau leptons513

and the jets could reduce this background.514

515

The tt Monte Carlo is produced by MC@NLO [19], like the background processes above, the hadronization516

25

Figure 3.3: Branching ratios depending on Higgs mass.

Figure 3.4: Decay of Z/γ.

Figure 3.5: Decay of W.

and parton shower are produced by HERWIG and the underlying event with JIMMY.517

518

The probability that two jets are misidentified as taus is very small, but as the QCD multijet cross-section519

is extremely large, the QCD background has, a big contribution to the background. The QCD background520

is derived from the data. QCD is assumed to be the same in the same sign (SS) region, as the opposite sign521

(OS) region. The OS region is the region where the normal selection that is used in this analysis is applied522

(see section 3.5), with the requirement the two taus have an opposite charge. The SS region requires the523

taus to have the same charge. In the OS region, there are a lot of other processes than QCD, while the SS524

region mainly consists of QCD. A scale factor F is applied on the SS region to model differences in the OS525

26

Figure 3.6: Decay of tt.

region in respect to the SS region. The factor F is derived by first considering four regions;526

527

A) signal region (OS region): opposite sign data, both taus pass medium tau ID. The signal region is528

what we will be comparing the predictions with, it mainly contains Z → ττ , QCD and the signal.529

530

B) control region 1 (SS region): same sign data, both taus pass medium tau ID. The first control re-531

gion mainly contains QCD. The shape of the distributions should be the same as the QCD for the opposite532

sign QCD.533

534

C) control region 2: opposite sign data, both taus do not pass medium tau ID. Apart from those cri-535

teria the same criteria as the signal region apply.536

537

D) control region 3: same sign data, both taus do not pass medium tau ID. Apart from those crite-538

ria the same criteria as the signal region apply.539

540

Aside from QCD, the control regions could have a contribution from Z→ ττ , W→ τν and tt. For con-541

trol region 1 this is shown in Table 3.5. To have clean QCD samples it is needed to correct for those542

contributions. Per control region, the Z→ ττ , W→ τν and tt passing the requirements of the concerning543

region are subtracted from the control region.544

545

The number of events selected in region C are divided with the events in region D to get the ratio of546

OS to SS events, this is the scale factor F. To get the QCD background the events from control region 1 (SS547

region) are re-weighted with this factor.548

549

To illustrate the MC/data agrees well after this correction on the QCD, see Figure 3.7. The dashed bands550

are the combined statistical error of the backgrounds. The error-bars on the data are also only the statistical551

error.552

553

27

Leading Tau Momentum /GeV40 60 80 100 120 140 160 180 200

even

ts

0

100

200

300

400

500

Leading Tau Pt VBFggHttbar

ντ →W ττ →Z

QCDdatastat+syst err

40 60 80 100 120 140 160 180 200

ratio

dat

a/M

C

0

0.5

1

1.5

2

Figure 3.7: Distribution of the momentum of the leading tau after the selection explained in section 3.5.

28

3.4 Variables554

To see which variables can be used in the Multivariate Analysis [31], we look at their distributions. If the555

histogram of the signal (VBF) looks very different than one or more background distributions, it is very likely556

the regarding variable is a good addition to the analysis. Variables can be studied at either truth level or557

reconstructed level. Truth level means that the values that are seen are really from the concerning processes558

and are not misidentified events from another proces. Events need to be reconstructed in order to see how559

they would be measured with the detector. Here is taken into account that an event can be misidentified or560

missed by the detector. The detector simulation is done with GEANT4 [32]. For analysis the reconstructed561

variables are used. The variables that are considered in the analysis are:562

Variable explanationPtτ1/τ2 leading/subleading tau Ptητ1/τ2 leading/subleading tau ηφτ1/τ2 leading/subleading tau φ∆ητ distance between the leading- and subleading tau in η∆φτ distance between the leading- and subleading tau in φ∆Rτ distance between the leading- and subleading tau in RPtj1/j2 leading/subleading jet Ptηj1/j2 leading/subleading jet η∆ηj distance between the leading- and subleading jet in η∆φj distance between the leading- and subleading jet in φ∆Rj distance between the leading- and subleading jet in RMTMET,τ1 Transverse Mass of MET and the leading tauηj1 × ηj2 the η angles of the two jets multipliedMET Missing Transverse EnergyMjj Invariant mass of the the leading- and subleading jet (JJEta)ScPt Scalar momentum, the scalar sum of all objects and METV Pt Vector momentum, the vector sum of all objects and METMETτ1 ∆φ Distance between MET and the leading tau in φMETτ2 ∆φ Distance between MET and the subleading tau in φMcol Collinear MassMMC[20] invariant mass of the two taus (Missing mass calculator, explained in 3.4.1)

Table 3.2: The variables that are considered in this thesis.

3.4.1 Mass calculations563

The possibilities to reconstruct the mass of the parent of the two taus precisely is explored. This mass would564

be a valuable variable, since the Higgs has another mass than the Z and W. This mass can not be calculated565

in an exact way, as variables like the angle of the neutrinos are not known. However, there are a few ways to566

approach this mass. The Transverse mass method, the Collinear mass approximation and the Missing Mass567

calculator technique are described [20].568

569

Transverse massSince the neutrinos that come from the taus escape the detector it is hard to reconstruct the τ invariantmass. One method to reconstruct the mass of the Higgs is calculating the transverse mass, the invariantmass of the visible τ decay products and the missing energy.

M2T = PµPµ, Pµ = Pµ(τvis1) + Pµ(τvis2) + Pµ(�ET ) (3.2)

29

An advantage of this method is that this partial mass can be defined for every event. No events have to beleft out of the analysis. However this method is not ideal. The neutrino momenta are not fully accountedfor creating a reduced sensitivity in the measurements. This becomes a big problem at the low Higgs-massregimes where it is hard to separate the Higgs to ττ from the Z to ττ background.

Collinear massAnother way to make an approximation of the mass of the decayed particle is the collinear approximationtechnique. In this method the Higgs mass is reconstructed by the invariant mass of the ττ decays. In thisapproximation it is assumed that the tau lepton and its decay products are collinear and the missing trans-verse energy only comes from neutrinos. The missing energy created by the neutrinos can now be calculatedby:

�ETx = pmis1 sin θvis1 cosφvis1 + pmis2 sin θvis2 cosφvis2 (3.3)

�ETy = pmis1 sin θvis1 sinφvis1 + pmis2 sin θvis2 sinφvis2 (3.4)

This makes it possible to calculate the invisible momenta of the taus:

pmis1 =�ETx sinφvis2 −�ETy cosψvis2

sin θvis1(cosφvis1 sinφvis2 − sinφvis1 cosφvis2)(3.5)

pmis2 =�ETx sinφvis1 −�ETy cosψvis1

sin θvis2(cosφvis2 sinφvis1 − sinφvis2 cosφvis1)(3.6)

The invariant mass of the system can be calculated with570

Mττ = mvis/√x1x2, x1,2 = pvis1,2/(pvis1,2 + pmis1,2) (3.7)

In this method the full mass of the ττ system can be reconstructed instead of the partial mass that thetransverse mass calculation provides. However this method has its shortcomings. It works well with a jet ofhigh transverse energy where the τ decay products are not back to back. If the decay products of the twotaus are back to back (φvis1 = φvis2 + π), the solutions of the missing momentum diverge. The method isalso sensitive to the resolution of the missing momentum leading to an overestimation of the ττ mass. Thiseffect creates a long tail at the high end of the mass distribution.

The Missing Mass CalculatorThe Missing Mass Calculator (MMC) does not have the limitations the other two methods have. The MMCcan reconstruct the complete event kinematics and an improved invariant mass of the di-tau system. Unlikethe collinear mass calculation the MMC does not degrade the reconstructed mass resolution. In MMC thereare less assumptions than the former methods; it assumes that the missing energy only comes from theneutrinos of the τ decays. In the case of two hadronic taus, there are 6 unknown variables; the x-,y- and z-components of the neutrino’s of both taus. Those unknowns have to be solved by the following 4 equations:

�ETx = pmis1 sin θmis1 cosφmis1 + pmis2 sin θmis2 cosφmis2 (3.8)

�ETy = pmis1 sin θmis1 sinφmis1 + pmis2 sin θmis2 sinφmis2 (3.9)

M2τ1 = m2

mis1 +m2vis1 + 2

√p2vis1

+m2vis1

√p2mis1

+m2mis1

− 2pvis1pmis1 cos ∆θνm1(3.10)

M2τ2 = m2

mis2 +m2vis2 + 2

√p2vis2

+m2vis2

√p2mis2

+m2mis2

− 2pvis2pmis2 cos ∆θνm2 (3.11)

In these equations Mτ is the tau-lepton mass (Mτ = 1.777GeV/c2) and pmiss,θmiss and φmiss are the un-571

knowns. mmiss is 0 for hadronic decays. As there are only 4 equations and 6 unknowns this can not be572

done in an exact way. However not all solutions to these equations are equally likely. Knowledge of τ decays573

can be used to filter the likely solutions from the unlikely ones. An example is the expected distance of the574

30

neutrinos and the other decay products. For hadronic decays each solution can be solved for any point of575

(φvis1 , φvis2). In every case the extra kinematic information (as the ∆R) can be taken into account. Some576

∆R values are more likely than others. Per event the most likely configuration is chosen. ALso the likelihood577

of the momenta of the taus is taken into account.578

579

Comparing The Missing Mass Calculator to the Collinear mass580

581

To see which method of calculating the mass is better, the results of the collinear mass are compared582

to the results of the Missing Mass calculator. The transverse mass is not taken into account because of its583

reduced sensitivity, which makes it hard to separate Higgs to ττ from Z to ττ . In Figure 3.8 is shown how584

the two methods compare. The Missing Mass calculator gives better results as well as more results. As can585

be seen the collinear mass has the following term: Mττ = mvis/√x1x2. if x1×x2 is a negative number there586

is no outcome to the equation, so the event can not contribute. This happens in about 10% of the events587

(see Table 3.3).588

Mcol events MMC events Mcol/MMC ratioData 2837 3115 0.911VBF 18332 19077 0.961ggH 3012 3170 0.950Z→ ττ 490 526 0.932W→ τν 31 38 0.816tt 463 539 0.859same-sign data 1003 1129 0.889same-sign Z→ ττ 10 11 0.909same-sign W→ τν 3 4 0.750

Table 3.3: The ratio of events in the sample for the Mcol- or MMC variable. The events are not weighted.

31

Mass, GeV0 50 100 150 200 250 300 350 400

even

ts

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Reconstructed invariant di-tau massMMC

Collinear M

Reconstructed invariant di-tau mass

(a) VBF

Mass, GeV0 50 100 150 200 250 300 350 400

even

ts

0

0.5

1

1.5

2

2.5

Reconstructed invariant di-tau massMMC

Collinear M

Reconstructed invariant di-tau mass

(b) ggH

Mass, GeV0 50 100 150 200 250 300 350 400

even

ts

0

50

100

150

200

250

300

Reconstructed invariant di-tau massMMC

Collinear M

Reconstructed invariant di-tau mass

(c) Z→ ττ

Mass, GeV0 50 100 150 200 250 300 350 400

even

ts

0

2

4

6

8

10

12

14

Reconstructed invariant di-tau massMMC

Collinear M

Reconstructed invariant di-tau mass

(d) W→ τν

Mass, GeV0 50 100 150 200 250 300 350 400

even

ts

0

1

2

3

4

5

6

7

8

Reconstructed invariant di-tau massMMC

Collinear M

Reconstructed invariant di-tau mass

(e) tt

Figure 3.8: Reconstructed mass of the signal and backgrounds (MC samples). The Missing Mass Calculator(MMC) and Collinear method (Mcol) are compared after the selection from section 3.5, except the cut onMMC itself.

32

3.5 Event cleaning and selection589

Before the selection of the events on basis of the signal and background the event samples have to be590

’cleaned’. This cleaning contains the selection of good-quality data events and other requirements to reject591

non-colliding events (cosmic rays or beam halo). One of those requirements is that the event has to have at592

least one primary vertex with at least four associated tracks. Furthermore, both taus have to come from the593

same primary vertex. Then a set of jet cleaning cuts is applied, prescribed by the JetEtMiss Combined Per-594

formance group [47]. Additionally, jets within dR=0.5 of a tau are removed, so a tau can not be interpreted595

as a jet. For taus that are within dR=0.5 to another tau, the tau with the lowest momentum is removed.596

The same applies for jets.597

598

In order to select relevant events, a selection with various cuts is applied. Those cuts are:599

600

- No muons or electrons in the event:601

Since only hadronically decaying taus are considered, there will not be muons or electrons in the events. A602

loose electron veto (efficiency of 95%) and standard muon veto (reduction of fake muons of around 40%)[24]603

are applied. Muons and electrons are defined in section 2.9.604

605

- In the events exactly 2 taus are measured in which both taus have 1- or 3- tracks in ∆R < 0.6):606

As the taus decay with 1 or 3 prong, this amount of tracks is looked for. The tau requirements are set out607

in section 2.9.608

609

- The two taus have an opposite charge:610

Since the taus in the signal process are expected to be the decay products of a neutral Higgs, they should611

have an opposite charge considering charge conservation.612

613

- At least one of the taus passes BDT tauID Tight:614

70% certainty that one of the taus is real (60% for Medium). See appendix A for the data/MC comparison615

after this cut.616

617

- MMC>80 GeV (see section 3.4.1 for explanation for MMC).618

At low mass regions MMC does not match the data well.619

620

621

- Distance in η between the taus has to be smaller than 1.5:622

Reducing W and QCD.623

624

- 0.8 < ∆R(τ1τ2) < 2.8:625

The lower cut is made to reject events where the two taus overlap. The upper cut is primarily made to626

reduce the QCD background for which the direction of the two taus is randomized.627

628

- Ptτ1 > 40GeV ′,Ptτ2 > 25GeV :629

Those momentum cuts are set to cope with the scale factor uncertainties that occur at lower values, which630

induces MC/data disagreement. See appendix B for the data/MC comparison after this cut.631

632

- MET should be in-between the taus, else the distances of either τ1 and τ2 with met in φ should be633

considered. The lowest of the two numbers should be less than 0.2π (min{∆φ(EmissT , τ1,2)} < 0.2π):634

For the signal process the missing energy comes from the two. For QCD, Z→ ττ and W the directions of635

the missing energy is randomized, as are the relative directions of the taus for the QCD and W samples.636

637

- MET>20GeV:638

At lower energies the missing transverse energy appears to be mis-modelled, so this is left out of the analysis.639

33

Also this cut reduces background with less MET than the signal.640

641

- Exactly 2 tagging jets in the events:642

The two jets are a distinct feature of VBF. QCD, Z and W events do not necessarily have 2 jets. Also only643

a small percentage of ggH has two precisely 2 jets in the event.644

645

Table 3.4 shows the cutflow of the different samples. In Table 3.5 the cutflow of the different contribu-646

tions of QCD are shown. The composition of the QCD is explained in section 3.3.4. The errors in those647

tables are only the statistical errors. Plots of some distributions after either all cuts, after the cut of the648

leading and subleading tau Pt, and after the selection of two taus are shown in section 3.6.649

650

VBF ggH Z→ ττ W → τν tt QCD Total Databackground

no cut 89.51 932.02 255579 47864.3 4550.85 9.82245×106

±0.20 ±2.25 ±843.20 ±744.84 ±28.25 ± 3134.08

muon and 88.65 925.77 248283 47116 3875.15 9.7838×106

electron veto ±0.19 ±2.24 ±831.07 ±737.68 ±26.07 ± 3127.9

2 taus 61.32 646.99 172560 25581.8 1964.31 4.24731×106

±0.16 ±1.87 ±692.53 ±549.01 ±18.56 ± 2060.9

opposite charge 60.02 639.48 167984 20487.4 1604.03 2.42162×106

(QCD SS) ±0.16 ±1.86 ±682.87 ±482.97 ±16.77 ± 1556.16

1 tight tau 55.68 592.60 154715 17296.1 1381.72 1.47116×106 1.64515×106 1.70775×106

±0.15 ±1.79 ±655.307 ±439.42 ±15.56 ±3.90×103 ±3.98×103 ± 1.31×103

MMC 52.79 559.55 118160 12870.3 1057.19 1.06509×106 1.19779×106 1.23241×106

±0.15 ±1.74 ±572.00 ±376.25 ±13.61 ±2.965×103 ±3.043×103 ±1.110×103

∆ητ <1.5 47.24 500.29 110014 8337.39 735.30 439450 559036.98 575937±0.15 ±1.65 ±551.75 ±297.59 ±11.35 ±1932.17 ±2031.35 ±758.91

0.8< ∆Rτ <2.8 36.80 194.2 24254.5 4224.62 516.15 89649.08 118838.55 122162±0.13 1.02 ±266.28 ±204.03 ±9.51 ±865.77 ±928.54 ±349.52

Ptτ1 > 40GeV 32.42 170.49 17986.4 3081.18 407.27 43464.51 65109.85 66497Ptτ2 > 25GeV ±0.12 0.96 ±229.57 ±168.77 ±8.45 ±576.70 ±643.30 ±257.87

MET 26.34 105.36 9329.59 673.06 229.15 13854.52 24191.68 24327between taus ±0.11 0.75 ±166.40 ±53.67 ±6.34 ±325.85 ±369.85 ±155.971

MET>20GeV 21.13 76.00 5460.28 548.106 213.89 5370.02 11668.30 11742±0.09 0.64 ±127.72 ±47.98 ±6.12 ±175.19 ±222.13 ±108.36

2 jets 7.97 16.69 1135 97.84 81.70 799.83 2131.18 2076±0.06 ±0.30 ±59.13 ±17.87 ±3.78 ±76.25 ±98.20 ±45.56

Table 3.4: Selection Cutflow of the signal (VBF) and background processes. Only statistical errors areconsidered here. QCD only comes in at the ’tight tau’ criteria, as the scale factor FQCD only makes senseafter this cut. The ’Total background’ is the sum of all backgrounds (ggH, Z→ ττ , W → τν, tt and QCD).

34

dataSS Z→ ττ SS W→ τν SS tt SS SS sum FQCD QCD

1 tight tau 1.24982×106 3888.1 4156.2 288.23 1.24149×106 1.185 1.47116×106

±1117.9 ± 106.21 ±233.30 ±7.11 ±1146.9 ±0.003 ±3.96 ×103

MMC 917964 2782.11 3070.06 218.28 911893.55 1.168 1.06509×106

±958.10 ±89.92 ±184.75 ±6.19 ±979.90 ±0.003 ±2.965×103

∆η(τ) <1.5 358837 2013.60 1721.33 134.12 354967.95 1.238 439450.32±599.03 ±76.27 ±125.91 ±4.85 ±616.87 ±0.005 ±1932.17

0.8< ∆Rτ <2.8 73366 804.20 857.87 99.23 71604.70 1.252 89649.08±270.86 ±48.90 ±81.39 ±4.17 ±287.05 ±0.011 ±865.77

Ptτ1 > 40GeV 35586 514.90 501.3 74.16 34495.64 1.260 43464.51Ptτ2 > 25GeV ±188.64 ±40.21 ±60.29 ±3.61 ±202.11 ±0.015 ±576.70

MET 11741 212.86 141.13 40.15 11346.86 1.221 13854.52between taus ±108.36 ±25.28 ±21.32 ±2.65 ±113.32 ±0.026 ±325.85

MET >20GeV 5039 117.17 93.93 37.52 4790.38 1.121 5370.02±70.99 ±18.77 ±18.00 ±2.58 ±75.65 ±0.032 ±175.19

2 jets 749 15.49 9.48 12.44 711.59 1.124 799.83±27.37 ±6.93 ±5.47 ±1.48 ±28.80 ±0.097 ±76.25

Table 3.5: QCD calculated from same sign data minus the same sign contributions of the backgrounds. Onlystatistical errors are considered here.

35

3.6 MC/data comparison651

In the selection table (Table 3.4) it can be seen that the amount of Monte Carlo events agree with the data652

after the selection. To see whether the variables are modelled well it is good to compare the monte Carlo’s653

with the data. If those do not agree it could be possible that the variable is mis-modelled. In section 3.8 will654

be explained that the variables are used in the MVA. Using a mis-modelled variable in the MVA leads to655

misinterpreted data. A few variables are chosen based on their correlation, since highly correlated variables656

do not add information. The variables that are chosen are the leading tau momentum (T1Pt), invariant657

mass of the two jets (Mjj), invariant mass of the di-tau system (MMC), multiplication of the (eta)angles of658

the two jets (JJEta) and the R angle between the two taus (TdR). For all variables the right figure the y659

axis is logarithmic to see the VBF contribution. Figure 3.9, 3.10, 3.11 and 3.12 are the distributions after660

the full selection. In appendix A, respectively the data/MC comparison distributions after the selection of661

two taus with an opposite charge where one of the taus is identified as tight and the cut on the (Sub)Leading662

Tau Pt can be seen.663

664

Leading Tau Momentum /GeV40 60 80 100 120 140 160 180 200

even

ts

0

100

200

300

400

500

Leading Tau Pt VBFggHttbar

ντ →W ττ →Z

QCDdatastat+syst err

40 60 80 100 120 140 160 180 200

ratio

dat

a/M

C

0

0.5

1

1.5

2

Leading Tau Momentum /GeV40 60 80 100 120 140 160 180 200

even

ts

-210

-110

1

10

210

Leading Tau Pt VBF

ggH

ttbar

ντ →W

ττ →Z

QCD

data

stat+syst err

Leading Tau Pt

Figure 3.9: Leading Tau Momentum (Ptτ1).

The striped error bars on the MC stack is the combined statistical and systematic error on the background665

processes, explained in section 3.9. The main contributions to this error are from Zττ and QCD. The red666

crosses are the data points and their statistical error.667

668

It can be seen that the Z→ ττ sample has not very much statistics. This can be seen by the non-smooth669

distributions, especially for Tau∆R. Even so, it shows that the data and Monte Carlo samples are in good670

agreement, as the data/MC ratio is within the errors.671

672

Appendix A shows some distributions after the opposite charge cut in the selection. Here it can be seen that673

the data and MCs do not agree very well for low Ptτ1 , MET, ∆Rτ and MMC. In Appendix B (after the cut674

Ptτ1 > 40GeV , Ptτ2 > 25GeV ), the low, not-fitting regions of MMC, ∆Rτ and Ptτ1 are taken out, giving a675

data/MC ratio that is a bit better. The cut on MET comes in later in the selection. The final results (the676

plots in the current chapter) agree well with the data because of those cuts.677

36

invariant mass jets /GeV0 200 400 600 800 1000 1200

even

ts

0

100

200

300

400

500

mjj VBFggHttbar

ντ →W ττ →Z

QCDdatastat+syst err

0 200 400 600 800 1000 1200

ratio

dat

a/M

C

00.20.40.60.8

11.21.41.61.8

2

invariant mass jets /GeV0 200 400 600 800 1000 1200

even

ts

-110

1

10

210

mjj VBF

ggH

ttbar

ντ →W

ττ →Z

QCD

data

stat+syst err

mjj

Figure 3.10: Invariant Mass of the two jets (Mjj).

MMC /GeV80 100 120 140 160 180 200 220 240

even

ts

0

50

100

150

200

250

300

350

400

mass VBFggHttbar

ντ →W ττ →Z

QCDdatastat+syst err

80 100 120 140 160 180 200 220 240

ratio

dat

a/M

C

0

0.5

1

1.5

2

MMC /GeV80 100 120 140 160 180 200 220 240

even

ts

1

10

210

mass VBF

ggH

ttbar

ντ →W

ττ →Z

QCD

data

stat+syst err

mass

Figure 3.11: Invariant mass of the di-tau system (MMC).

37

Jet delta Eta-6 -4 -2 0 2 4 6

even

ts

0

50

100

150

200

250

300

350

400

450

Jet deltaEta VBFggHttbar

ντ →W ττ →Z

QCDdatastat+syst err

-6 -4 -2 0 2 4 6

ratio

dat

a/M

C

00.20.40.60.8

11.21.41.61.8

2

Jet delta Eta-6 -4 -2 0 2 4 6

even

ts

-210

-110

1

10

210

Jet deltaEta VBF

ggH

ttbar

ντ →W

ττ →Z

QCD

data

stat+syst err

Jet deltaEta

Figure 3.12: The η angle between the two selected jets (ηj1 × ηj2).

Tau delta R0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8

even

ts

0

50

100

150

200

250

Tau deltaR VBF

ggH

ttbar

ντ →W

ττ →Z

QCD

data

stat+syst err

0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8

ratio

dat

a/M

C

0.70.80.9

11.11.21.31.41.5

Tau delta R0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8

even

ts

1

10

210

Tau deltaR VBF

ggH

ttbar

ντ →W

ττ →Z

QCD

data

stat+syst err

Tau deltaR

Figure 3.13: Distance of the two taus in R (∆Rτ ).

38

3.7 Shapes678

In order to separate the signal from the background, the shapes of the variables are important. Possibly a679

cut can be made where the background gets reduced, while the signal does not lose a lot of events. Most680

of those cuts are already applied in the Preselection. As the Multi Variate Analysis (explained in the next681

section), in contrary to Cut-based analysis, uses the shapes of the variables, those should also be considered:682

if the signal distribution is very different from the background distributions, the variable is very likely to be683

a good input for MVA. In thi section, the signal and background distributions are renormalized to one.684

685

LeadTauPt

50 100 150 200 250 300

310×

1.4

6e

+0

4

/ (1

/N)

dN

0

5

10

15

20

25

30

35

­610×

Signal

BackgroundU

/O­f

low

(S

,B):

(0.0

, 0.0

)% / (

0.0

, 0.0

)%

Input variable: LeadTauPt

LeadTauPhi

­3 ­2 ­1 0 1 2 3

0.3

3

/ (1

/N)

dN

0

0.05

0.1

0.15

0.2

0.25

U/O

­flo

w (

S,B

): (

0.0

, 0.0

)% / (

0.0

, 0.0

)%

Input variable: LeadTauPhi

LeadTauEta

­2 ­1 0 1 2

0.2

61

/

(1/N

) d

N

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

U/O

­flo

w (

S,B

): (

0.0

, 0.0

)% / (

0.0

, 0.0

)%

Input variable: LeadTauEta

SubLeadTauPt

40 60 80 100 120 140

310×

6.7

1e

+0

3

/ (1

/N)

dN

0

0.01

0.02

0.03

0.04

0.05

0.06

­310×

U/O

­flo

w (

S,B

): (

0.0

, 0.0

)% / (

0.0

, 0.0

)%

Input variable: SubLeadTauPt

SubLeadTauPhi

­3 ­2 ­1 0 1 2 3

0.3

3

/ (1

/N)

dN

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

U/O

­flo

w (

S,B

): (

0.0

, 0.0

)% / (

0.0

, 0.0

)%

Input variable: SubLeadTauPhi

SubLeadTauEta

­2 ­1 0 1 2

0.2

62

/

(1/N

) d

N

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

U/O

­flo

w (

S,B

): (

0.0

, 0.0

)% / (

0.0

, 0.0

)%

Input variable: SubLeadTauEta

Figure 3.14: Variables Ptτ1/τ2 , ητ1/τ2 and φτ1/τ2 . Signal and background renormalized to one.

As can be seen in Figure 3.14, for the leading and subleading tau, the variables φ and η are not expected686

to be of value in the MVA. For those variables the signal and background are too similar. It is still pos-687

sible that those variables are good for excluding one of the backgrounds (QCD/Z→ ττ/W/tt), but for688

these variables that is unlikely, due to the wide spread of the signal distributions. The momentum of either689

of the taus is more likely to be of value in the analysis, as the signal momentum for both is on average higher.690

691

Figure 3.15 shows that the relative distance between the taus is not very different for signal and background.692

∆Rτ1,τ2 might add a bit information to the analysis, but this is uncertain. The momenta of the jets also do693

not seem to be good variables to add in the analysis. However, the distributions of ηj1,j2 (Figure 3.16) are694

very different for signal and background. Those are hard to cut on, as the signal lies in-between two back-695

ground peaks. Therefore those variables can not be added to Cut-based analysis, but for the multivariate696

analysis those can be very good.697

698

Also the relative η between the jets and the η of both jets multiplied, seem to be good variables to use,699

probably even better than either of the jet-η variables. Later in the analysis I will compare those variables700

and explore which is better to use in the MVA. The relative φ between the jets is not a very good variable701

and therefore it is likely that the ∆Rj1,j2 of the jets mainly looks good due to the good ∆ηj1,j2 separation.702

703

Figure 3.17 shows that either the transverse mass of the leading tau and MET, the vector sum of all objects704

39

tau_dphinew

­2 ­1 0 1 2 3

0.2

93

/

(1/N

) d

N

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

U/O

­flo

w (

S,B

): (

0.0

, 0.0

)% / (

0.0

, 0.0

)%

Input variable: tau_dphinew

tau_detanew

­1 ­0.5 0 0.5 1 1.5

0.1

57

/

(1/N

) d

N

0

0.1

0.2

0.3

0.4

0.5

0.6

U/O

­flo

w (

S,B

): (

0.0

, 0.0

)% / (

0.0

, 0.0

)%

Input variable: tau_detanew

tau_dRnew

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8

0.1

05

/

(1/N

) d

N

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

U/O

­flo

w (

S,B

): (

0.0

, 0.0

)% / (

0.0

, 0.0

)%

Input variable: tau_dRnew

LeadJetPt

100 200 300 400 500

310×

2.5

5e

+0

4

/ (1

/N)

dN

0

2

4

6

8

10

­610×

U/O

­flo

w (

S,B

): (

0.0

, 0.0

)% / (

0.0

, 0.0

)%

Input variable: LeadJetPt

LeadJetEta

­2 ­1 0 1 2 3 4

0.3

85

/

(1/N

) d

N

0

0.05

0.1

0.15

0.2

0.25

U/O

­flo

w (

S,B

): (

0.0

, 0.0

)% / (

0.0

, 0.0

)%

Input variable: LeadJetEta

SubLeadJetPt

50 100 150 200 250

310×

1.2

8e

+0

4

/ (1

/N)

dN

0

5

10

15

20

25

30

35

40

­610×

U/O

­flo

w (

S,B

): (

0.0

, 0.0

)% / (

0.0

, 0.0

)%

Input variable: SubLeadJetPt

Figure 3.15: Variables ∆φτ1,τ2 , ∆ητ1,τ2 , ∆Rτ1,τ2 , Ptj1/j2 and ηj1.

SubLeadJetEta

­2 ­1 0 1 2 3 4

0.3

88

/

(1/N

) d

N

0

0.05

0.1

0.15

0.2

0.25

0.3

U/O

­flo

w (

S,B

): (

0.0

, 0.0

)% / (

0.0

, 0.0

)%

Input variable: SubLeadJetEta

jet_dphinew

­3 ­2 ­1 0 1 2 3

0.3

3

/ (1

/N)

dN

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

U/O

­flo

w (

S,B

): (

0.0

, 0.0

)% / (

0.0

, 0.0

)%

Input variable: jet_dphinew

jet_detanew

­6 ­4 ­2 0 2 4 6

0.7

27

/

(1/N

) d

N

0

0.05

0.1

0.15

0.2

0.25

U/O

­flo

w (

S,B

): (

0.0

, 0.0

)% / (

0.0

, 0.0

)%

Input variable: jet_detanew

jet_dRnew

1 2 3 4 5 6 7

0.3

57

/

(1/N

) d

N

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

U/O

­flo

w (

S,B

): (

0.0

, 0.0

)% / (

0.0

, 0.0

)%

Input variable: jet_dRnew

jetjet_etanew

­10 ­5 0 5 10 15

1.3

6

/ (1

/N)

dN

0

0.05

0.1

0.15

0.2

0.25

0.3

U/O

­flo

w (

S,B

): (

0.0

, 0.0

)% / (

0.0

, 0.0

)%

Input variable: jetjet_etanew

MET

50 100 150 200 250 300

310×

1.5

7e

+0

4

/ (1

/N)

dN

0

5

10

15

20

25

­610×

U/O

­flo

w (

S,B

): (

0.0

, 0.0

)% / (

0.0

, 0.0

)%

Input variable: MET

Figure 3.16: Variables ηj2, jet ∆φj1,j2, ∆ηj1,j2, ∆Rj1,j2, ηj1 × ηj2 and MET.

and the distance in φ between MET and either of the taus, are variables that do not seem useful for MVA.705

The scalar sum of all momenta could however be useful. Most important is here the invariant mass of the706

two jets, which seems to be a very discriminating variable.707

708

The invariant mass of the two taus, calculated by the Missing Mass Calculator, appears to be a good vari-709

40

invjj

500 1000 1500 2000 2500 3000

310×

1.7

4e

+0

5

/ (1

/N)

dN

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

­610×

U/O

­flo

w (

S,B

): (

0.0

, 0.0

)% / (

0.0

, 0.0

)%

Input variable: invjj

MT

50 100 150 200 250

310×

1.2

9e

+0

4

/ (1

/N)

dN

0

2

4

6

8

10

12

14

16

18

20

22

­610×

U/O

­flo

w (

S,B

): (

0.0

, 0.0

)% / (

0.0

, 0.0

)%

Input variable: MT

ScPT

200 400 600 800 1000 1200

310×

5.2

5e

+0

4

/ (1

/N)

dN

0

1

2

3

4

5

6

­610×

U/O

­flo

w (

S,B

): (

0.0

, 0.0

)% / (

0.0

, 0.0

)%

Input variable: ScPT

ScPTxy

0 20 40 60 80 100 120 140 160 180 200

310×

1e

+0

4

/ (1

/N)

dN

0

0.01

0.02

0.03

0.04

0.05

0.06

­310×

U/O

­flo

w (

S,B

): (

0.0

, 0.0

)% / (

0.1

, 0.0

)%

Input variable: ScPTxy

METtau0_phi

­3 ­2 ­1 0 1 2 3

0.3

28

/

(1/N

) d

N

0

0.1

0.2

0.3

0.4

0.5

0.6

U/O

­flo

w (

S,B

): (

0.0

, 0.0

)% / (

0.0

, 0.0

)%

Input variable: METtau0_phi

METtau1_phi

­3 ­2 ­1 0 1 2 3

0.3

29

/

(1/N

) d

N

0

0.1

0.2

0.3

0.4

0.5

U/O

­flo

w (

S,B

): (

0.0

, 0.0

)% / (

0.0

, 0.0

)%

Input variable: METtau1_phi

Figure 3.17: Variables Mjj , MT, ScPt, VPt, METτ1 ∆φ and METτ2 ∆φ.

MMC_mass

100 120 140 160 180 200 220 240 260

9.1

7

/ (1

/N)

dN

0

0.005

0.01

0.015

0.02

0.025

U/O

­flo

w (

S,B

): (

0.0

, 0.0

)% / (

0.0

, 0.1

)%

Input variable: MMC_mass

Figure 3.18: Variable MMC.

able even though there is already a cut on this variable. This is expected, as it reconstructs the mass of the710

particle that decays into the two taus. For QCD this mass is a broad range and for Z the mass lies beneath711

the expected Higgs-mass. It could be either good to make a cut even above the 80 GeV, or add this variable712

to the BDT as a variable.713

714

Based on the signal and background shapes the variables that are not very likely to contribute to the715

analysis are φτ1,τ2 and ητ1,τ2, ∆φτ1,τ2, ∆ητ1,τ2, ∆φj1,j2, MET, MT, VPt, METτ1 ∆φ and METτ2 ∆φ.716

41

3.8 MVA717

To obtain the best separation between signal and background, it would be preferable to combine the variables718

to one. This is done in MVA techniques. Multiple variables and their dependencies are used to distinguish719

the signal. There are several different MVA methods. In this analysis only Boosted Decision Trees are used.720

3.8.1 Boosted Decision Trees721

The boosted decision algorithm is a very effective learning technique. This algorithm combines multiple722

variables to one more powerful variable. The ultimate goal is to divide the signal from the background.723

Therefore the Monte Carlo samples first have to be divided in two; a training sample and a testing sample.724

Those will be used to train the tree and to test whether the sample was overtrained. Overtraining is caused725

by too few datapoints combined with too many restrictions. This results in too few degrees of freedom to726

separate signal and background well: an equivalent sample, which should give te same result, gives another727

result. An example of overtraining can be seen in Figure 3.19 a.728

729

BDT response

­0.3 ­0.2 ­0.1 0 0.1 0.2 0.3 0.4 0.5

dx

/ (1

/N)

dN

0

1

2

3

4

5

6

7

8

9Signal (test sample)

Background (test sample)

Signal (training sample)

Background (training sample)

Kolmogorov­Smirnov test: signal (background) probability = 0.005 ( 0)

U/O

­flo

w (

S,B

): (

0.0

, 0

.0)%

/ (

0.0

, 0

.0)%

TMVA overtraining check for classifier: BDT

(a) Adaptive Boosting

BDTG response

­0.8 ­0.6 ­0.4 ­0.2 0 0.2 0.4 0.6 0.8

dx

/ (1

/N)

dN

0

1

2

3

4

5

6

7

8

9Signal (test sample)

Background (test sample)

Signal (training sample)

Background (training sample)

Kolmogorov­Smirnov test: signal (background) probability = 0.645 ( 0.31)

U/O

­flo

w (

S,B

): (

0.0

, 0

.0)%

/ (

0.0

, 0

.0)%

TMVA overtraining check for classifier: BDTG

(b) Gradient Boosting

Figure 3.19: BDT Output for Set 1 (25 variables), comparing the overtraining of Adaptive Boosting andGradient Boosting.

The tree consists of several variables. For those variables the events get ordered by the value of that variable.730

Per event, the tree repeatedly makes yes/no decisions on one variable, this is repeated with the following731

variable until a stopcriterium is fulfilled. In the end the tree is filled with events that are either more likely732

to be signal, or more likely to be background. Signal events get a score of 1, background events get a score733

of -1. Some events get misclassified, they end up in a signal leaf being a background event, or the other734

way around. To minimize this problem, boosting is used: a new tree with re-weighted events is made for735

the same sample. Misidentified events get a higher weight. Typically a few hundred to a thousand trees are736

used to boost the wrong events. In the end all events are combined into a weighted average of the trees.737

738

There are several boosting options. To choose the best boosting method ROC curves can be used, those are739

explained in the next section. Gradient Boosting and Adaptive Boosting [31] seem to give the best results in740

this analysis for the whole set of variables (Table 3.6, Set 1) as can be seen in Figure 3.20. However gradient741

boosting is known to give better results for overtraining. Figure 3.19 shows the train and test sample of742

the BDTG Output. The test and train tree compare better for the gradient boosting than for Adaptive743

boosting (Figure 3.19), as can be seen from the shapes in the figures and the Kolmogorow-Smirnov tests.744

42

The adaptive boosting has overtrained the samples. This overtraining can be caused by either the amount745

of variables (25), or low statistics of one or more of the samples. BDTG is explored further in this thesis.746

For reference the result of the variables with BDTA is also shown.747

748

Signal efficiency

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Backg

rou

nd

reje

cti

on

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

MVA Method:

BDT

BDTG

BDTD

BDTB

Background rejection versus Signal efficiency

Figure 3.20: ROC curves, testing multiple boosting methods.

3.8.2 ROC curves749

The ROC curve compares the signal efficiency to the background rejection. The signal efficiency is calcu-750

lated by taking the fraction of signal events after a cut on the BDToutput divided by all the signal events.751

Background efficiency is the fraction of background events after the same BDToutput cut as is done for the752

signal efficiency, divided by all the background events. The background rejection is acquired subtracting753

1 by the background efficiency. For multiple cuts on the BDToutput the signal efficiency and background754

rejection is calculated so it can be set against each-other. The aim is to get the signal efficiency and the755

background rejection as high as possible with the least amount of variables. A large amount of variables will756

induce overtraining, since their correlations can give a bias.757

758

Various sets of variables are trained and tested. Variables not used in the analysis on basis of the Shape759

analysis, correlations and their ranking by TMVA. In this ranking the importance of the variables separating760

the signal and the background are ordered.761

762

The aim is to get a set with a minimum amount of variables that does not degrade the ROC curve drasti-763

cally. Various sets are tried for gradient boosting and adaptive boosting. The ROC curves of a few sets with764

gradient boosting are shown in Figure 3.22.765

766

The sets of variables and the integrals of their ROC curves are shown in Table 3.6. Adaptive boosting767

seems like the better option for most of those sets. However, even only 6 variables induces overtraining for768

adaptive boosting, so this method will not be used. For BDTG either Set 1 or Set 5 give the best ROC769

curve. However, those sets have a lot of variables which induces overtraining. Therefore Set 10 is chosen;770

43

this set only has 6 variables and the ROC integral is still comparable with the Set 1. Reducing the number771

of variables further degrades the ROC curve drastically.772

Set Variables Nvar ROC integral ROC integralBDTG BDTA

1 Ptτ1/τ2 , φτ1/τ2 , ητ1,τ2 , ∆φ(τ1,τ2), ∆η(τ1,τ2) 25 0.92532 0.929042∆R(τ1,τ2), Ptj1/j2, ηj1/j2, ∆φ(j1,j2), ∆η(j1,j2),∆R(j1,j2), ηj1 × ηj2, MET, Mjj , MT, ScPT,

VPT, METτ1∆φ, METτ2∆φ, MMC2 Ptτ1/τ2 , ητ1,τ2 , ∆η(τ1,τ2), ∆R(τ1,τ2), Ptj1/j2, 20 0.92524 0.93020

ηj1/j2, ∆φ(j1,j2), ∆η(j1,j2), ∆R(j1,j2), ηj1 × ηj2,MET, Mjj , MT, ScPt, VPt, MMC

3 Ptτ1 , ητ1,τ2 , ∆η(τ1,τ2), ∆R(τ1,τ2), Ptj1, 18 0.92487 0.93330ηj2, ∆φ(j1,j2), ∆η(j1,j2), ∆R(j1,j2), ηj1 × ηj2,

MET, Mjj , MT, ScPt, VPt, MMC4 Ptτ1 , ητ1,τ2 , ∆η(τ1,τ2), ∆R(τ1,τ2), ηj1/j2, ∆φ(j1,j2), 16 0.92507 0.93464

∆η(j1,j2), ∆R(j1,j2), ηj1 × ηj2, MET, Mjj , MT,ScPt, VPt, MMC

5 ητ1,τ2 , ∆η(τ1,τ2), ∆R(τ1,τ2), ηj2, ∆φ(j1,j2), ∆η(j1,j2), 14 0.92558 0.93392ηj1 × ηj2, MET, Mjj , MT, ScPt, VPt, MMC

6 ητ1,τ2 , ∆η(τ1,τ2), ∆R(τ1,τ2), ηj2, ∆φ(j1,j2), ∆η(j1,j2), 12 0.92437 0.93189ηj1 × ηj2, Mjj , ScPt, VPt, MMC

7 ητ1,τ2 , ∆η(τ1,τ2), ∆R(τ1,τ2), ∆φ(j1,j2), ηj1 × ηj2, 10 0.92337 0.93025Mjj, ScPt, VPt, MMC

8 ητ2 , ∆R(τ1,τ2), ∆φ(j1,j2), ηj1 × ηj2, Mjj , ScPt, 8 0.92380 0.93186VPt, MMC

9 ητ2 , ∆R(τ1,τ2), ∆φ(j1,j2), ηj1 × ηj2, Mjj , VPt, MMC 7 0.92381 0.9295410 ∆R(τ1,τ2), ∆φ(j1,j2), ηj1 × ηj2, Mjj , VPt, MMC 6 0.92429 0.92771

Table 3.6: Sets with their variables, the number of variables (Nvar) and the integral of their ROC curve.

BDTG response

­0.8 ­0.6 ­0.4 ­0.2 0 0.2 0.4 0.6 0.8

dx

/ (1

/N)

dN

0

2

4

6

8

10 Signal (test sample)

Background (test sample)

Signal (training sample)

Background (training sample)

Kolmogorov­Smirnov test: signal (background) probability = 0.336 (0.498)

U/O

­flo

w (

S,B

): (

0.0

, 0

.0)%

/ (

0.0

, 0

.0)%

TMVA overtraining check for classifier: BDTG

(a) BDTG

BDT response

­0.2 0 0.2 0.4 0.6

dx

/ (1

/N)

dN

0

1

2

3

4

5

6

Signal (test sample)

Background (test sample)

Signal (training sample)

Background (training sample)

Kolmogorov­Smirnov test: signal (background) probability = 0.239 ( 0)

U/O

­flo

w (

S,B

): (

0.0

, 0

.0)%

/ (

0.0

, 0

.0)%

TMVA overtraining check for classifier: BDT

(b) BDTA

Figure 3.21: BDT Output for Set 10 using Gradient Boosting (a) and Adaptive Boosting (b).

44

Signal eff0.5 0.6 0.7 0.8 0.9 1

Bac

kgr

reje

ctio

n (1

-eff)

0.5

0.6

0.7

0.8

0.9

1

MVA_BDTG Set1

Set2

Set3

Set4

Set5

Set6

Set7

Set8

Set9

Set10

MVA_BDTG

Figure 3.22: ROC curve of different sets of variables (Table 3.6), for clarity the range is set from 0.5 to 1.

45

3.8.3 cutting on BDT773

The BDT output separates the signal and background very effectively. By cutting on this variable a lot of774

the background can be cut away keeping a large part of the signal. The variable can bee seen at Figure775

3.21 and 3.23. Different cuts on this output can be considered. The amount of events per sample and the776

signal-over-background ratio (S/B) are shown in Table 3.7.

BDT Output-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

even

ts

0

100

200

300

400

500

600

700

800

BDT ttbarντ→W

ττ→ZQCDggHVBFx10datastat+syst err

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

ratio

dat

a/M

C

00.20.40.60.8

11.21.41.61.8

2

(a) Range -1 t0 1.

BDT Output0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

even

ts

0

10

20

30

40

50

60

70

80

BDT ttbarντ→W

ττ→ZQCDggHVBFx10datastat+syst err

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

ratio

dat

a/M

C

0.8

1

1.2

1.4

1.6

1.8

(b) Range 0 to 1.

Figure 3.23: BDT Output

777

cut on VBF ggH Z→ ττ W→ τν tt QCD Total data S/BBDTout (signal) background

0.2 6.28 5.93 98.99 6.65 7.54 71.18 190.30 221 0.033±0.05 ±0.18 ±20.62 ±4.71 ±1.15 ±14.49 ±29.66 ±14.87

0.3 6.11 5.42 89.49 6.65 5.26 61.61 168.44 195 0.036±0.05 ±0.17 ±19.33 ±4.71 ±0.96 ±13.23 ±27.69 ±13.96

0.4 5.81 4.20 70.71 0 4.73 49.57 129.22 160 0.045±0.05 ±0.15 ±16.68 ±0 ±0.91 ±11.68 ±23.99 ±12.65

0.5 5.58 3.64 55.02 0 2.98 43.24 104.88 138 0.053±0.05 ±0.14 ±14.32 ±0 ±0.72 ±10.05 ±21.08 ±11.75

0.6 5.32 3.09 48.99 0 2.81 32.14 87.02 118 0.061±0.05 ±0.13 ±13.39 ±0 ±0.70 ±8.54 ±19.25 ±10.86

0.7 4.98 2.39 33.26 0 1.93 29.18 66.76 87 0.075±0.05 ±0.11 ±10.68 ±0 ±0.58 ±6.98 ±15.82 ±9.33

0.8 4.51 1.86 24.32 0 1.58 14.49 42.25 64 0.107±0.04 ±0.10 ±9.01 ±0 ±0.53 ±4.52 ±12.88 ±8

0.9 3.75 1.13 9.25 0 0.70 5.45 16.64 27 0.225±0.04 ±0.08 ±5.44 ±0 ±0.35 ±2.64 ±7.98 ±5.20

Table 3.7: Amount of events for the different samples after a bDT cut varying from 0.2 to 0.9.

46

The error on the background is composed from the statistical error of all backgrounds and the systematic778

error from QCD and Z→ ττ . The systematic errors are taken from the ATLAS Note from October 30th,779

2012 [12]. The systematic and statistic errors are explained in section 3.9.780

781

It is not desirable having little statistics for VBF. Taking this and the S/B ratio into consideration, a cut at782

BDToutput of 0.7 is chosen.783

784

The distributions of the variables that are used and the BDT output can be seen in Figure 3.24, 3.25, 3.26,785

3.27, 3.28, 3.29 and 3.30. Those figures show that the data Monte Carlo comparison is still reasonably within786

the error bars after the cut on the BDT output.787

788

Invariant Mass Jets /GeV200 300 400 500 600 700 800 900 1000

even

ts

0

5

10

15

20

25

30

35

40

45

mjj ttbarντ→W

ττ→ZQCDggHVBFx5datastat+syst err

200 300 400 500 600 700 800 900 1000

ratio

dat

a/M

C

0.8

1

1.2

1.4

1.6

1.8

2

Invariant Mass Jets /GeV200 300 400 500 600 700 800 900 1000

even

ts

-110

1

10

mjj ttbar

ντ→W

ττ→Z

QCD

ggH

VBFx5

data

stat+syst err

mjj

Figure 3.24: Invariant mass of the two jets (Mjj).

47

Jet delta Phi-4 -3 -2 -1 0 1 2 3 4

even

ts

0

5

10

15

20

25

30

35

40

45

Jet dPhi ttbarντ→W

ττ→ZQCDggHVBFx5datastat+syst err

-4 -3 -2 -1 0 1 2 3 4

ratio

dat

a/M

C

11.21.41.61.8

22.22.4

Jet delta Phi-4 -3 -2 -1 0 1 2 3 4

even

ts

-110

1

10

Jet dPhi ttbar

ντ→W

ττ→Z

QCD

ggH

VBFx5

data

stat+syst err

Jet dPhi

Figure 3.25: Distance in φ between jets (∆φj).

MMC Mtautau /GeV80 90 100 110 120 130 140 150 160 170 180

even

ts

0

10

20

30

40

50

MMC ttbarντ→W

ττ→ZQCDggHVBFx5datastat+syst err

80 90 100 110 120 130 140 150 160 170 180

ratio

dat

a/M

C

00.20.40.60.8

11.21.41.61.8

2

MMC Mtautau /GeV80 90 100 110 120 130 140 150 160 170 180

even

ts

-110

1

10

MMC ttbar

ντ→W

ττ→Z

QCD

ggH

VBFx5

data

stat+syst err

MMC

Figure 3.26: mass of the di-tau system (MMC).

48

Tau delta R0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8

even

ts

0

5

10

15

20

25

30

35

40

Tau deltaR ttbarντ→W

ττ→ZQCDggHVBFx5datastat+syst err

0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8

ratio

dat

a/M

C

0.5

1

1.5

2

2.5

Tau delta R0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8

even

ts

-110

1

10

Tau deltaR ttbar

ντ→W

ττ→Z

QCD

ggH

VBFx5

data

stat+syst err

Tau deltaR

Figure 3.27: distance between the two taus in R (∆Rτ ).

J1 eta * J2 eta-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1

even

ts

0

10

20

30

40

50

60

JetJet Eta ttbarντ→W

ττ→ZQCDggHVBFx5datastat+syst err

-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1

ratio

dat

a/M

C

0.5

1

1.5

2

2.5

J1 eta * J2 eta-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1

even

ts

-110

1

10

JetJet Eta ttbar

ντ→W

ττ→Z

QCD

ggH

VBFx5

data

stat+syst err

JetJet Eta

Figure 3.28: Eta angles of the two jets multiplied (ητ1 × ητ2).

49

Vector Momentum /GeV0 10 20 30 40 50 60 70 80 90 100

even

ts

10

20

30

40

50

60

70

Vector Momentum ttbarντ→W

ττ→ZQCDggHVBFx5datastat+syst err

Vector Momentum

0 10 20 30 40 50 60 70 80 90 100

ratio

dat

a/M

C

0

0.5

1

1.5

2

0 10 20 30 40 50 60 70 80 90 100-110

1

10

210

Vector Momentum ttbar

ντ→W

ττ→Z

QCD

ggH

VBFx5

data

stat+syst err

Figure 3.29: Vector Momentum (VPt).

BDT Output0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

even

ts

0

10

20

30

40

50

BDTttbarντ→W

ττ→ZQCDggHVBFx5datastat+syst err

0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

ratio

dat

a/M

C

00.20.40.60.8

11.21.41.61.8

2

BDT Output0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

even

ts

-110

1

10

BDTttbar

ντ→W

ττ→Z

QCD

ggH

VBFx5

data

stat+syst err

BDT

Figure 3.30: BDT output

50

3.9 Statistics789

To test the hypothesis of the existence of the Higgs particle, the significance can be calculated. This790

calculation is often used in High Energy Physics [35]. The significance is a test whether an excess in the791

data is due to signal events or fluctuations of the background source. Even without an uncertainty on the792

background hypothesis test can already be challenging because of conceptual issues. In this analysis the793

background uncertainty is nowhere near negligible: it is about four times the sought signal and can be794

divided between the statistical error and the systematic error.795

3.9.1 Statistical error796

As the production and decay of particles is subject to chance, calculating the statistical error is very im-797

portant. For a small sample it is very well possible that the measured number of particles deviates from798

the predicted one. The statistical error takes this into account. This error is relatively easy to calculate.799

For the data this error is just the square root of the selected events, since the data does not have to be800

re-weighted. The Monte Carlos samples and same-sign data do need to be re-weighted. The Monte Carlo801

samples do not have the same luminosity as the data, so their re-weighting is the division of their luminosity802

while multiplying with the luminosity of the data. The re-weighting of the same-sign data is explained in803

section 3.3.4. The errors for the weighted samples are calculated as:804

err2statN =

Nevents∑0

W 2 (3.12)

Here W is the weight of the event and N is the number of selected events.805

806

The statistic error on the same-sign data is more complicated. This is due to the factor F. Since this807

factor is calculated by dividing control region 2 by control region 3, the statistical errors on those regions808

also have to be added. The error on the factor F gets calculated as:809

err2F QCD =

Nevents∑0

((∆C2

C2

)2

+

(∆C3

C3

)2)× F (3.13)

Where ∆C2 and ∆C3 represent the statistical errors on control group 2 and 3, C2 and C3 are the number810

of events in those control groups and F is the calculated factor described in section 3.3.4. This error has to811

be added to the weighting error, so for QCD:812

err2stat = err2

statN + err2F QCD (3.14)

3.9.2 Systematic error813

Systematic errors are biases in the measurements caused by the experiment itself. The systematic error814

in this analysis can be divided in different parts: the luminosity measurement uncertainty, the energy815

scale uncertainty, the jet energy resolution uncertainty, the MET uncertainty, hadronic tau identification816

uncertainty and the trigger efficiency uncertainty. Those uncertainties are explained in [12] and result in an817

uncertainty of 11% for Z→ ττ and 9.4% for QCD.818

51

3.9.3 Significance819

The aim of this analysis is to identify the Higgs particle via VBF where the Higgs decays into two hadronictaus. To see how well the method works the observation of the signal has to be quantified. This quantifi-cation is done via the p-value. The p-value is the probability that the observations are caused by statisticalfluctuations of the background processes.

p = P (s ≥ observed | assume only background) (3.15)

As is shown before in this chapter the uncertainties on the backgrounds are not negligible. There are several820

ways to calculate the p-value [36]. The Poissonian p-value, for example, does not include the uncertainty on821

the background. The binomial p-value does include this uncertainty, so this calculation is used.822

pBi = PBi(≥ x | w, k) =

k∑j=x

k!

j!(k − j)!wj(1− w)k−j (3.16)

Here x is the total measured amount of events, k is the expected amount of background events andw=α/(1 + α), the expected deviation of the Poisson means of the background-and-signal and the only-background hypothesis.

From the p-value the standard deviation (Z) can be calculated. The standard deviation shows how muchvariation exists on the background-only hypothesis. The standard deviation (Z) is calculated as:

Z = Φ−1(p), Φ(z) =1√2π

∫ z

−∞e−t

2/2dt (3.17)

For large Z (Z≥1.5) the relation can be written as Z≈√u− Ln(u), where u = -2Ln(p

√2π). LHC searches823

are primarily interested in 5σ deviation from the background-only hypothesis, which is a discovery. 5σ im-824

plies there is a chance of p=2.85×10−7 the excess does not originate from the signal.825

826

The expected and observed probability from the numbers in section 3.8.3 are:827

828

expected p-value =0.448925, Z value (Gaussian sigma) = 0.128378829

observed p-value =0.176946, Z value (Gaussian sigma) = 0.927067830

831

This result is still compatible with a theory without the Higgs. The reason for this compatibility is the832

background that survive the cuts, which have large errors.833

3.10 Discussion and Conclusion834

The method is very effective as it brought back the background from 3111.5 to 66.8, while the signal only835

got reduced from 8.1 to 5.0. The results show the method is very promising. Further investigation is needed836

though as the error bars on the background are large (about 3 times as large as the signal). This large error837

is mainly due to the statistical error, but also to the systematic errors of QCD and Z→ ττ . The (relative)838

statistical error can be reduced by having more statistics. More data will be taken in the next run of the839

LHC, so the statistics of the data and QCD will be much better after this run. Also the other Monte Carlo840

samples have to be expanded, especially Z→ ττ . In other analyses the Z→ ττ sample is an embedded sample,841

made from Z→ µµ events where the muons are replaced by taus. As this sample is data-driven, there would842

be no need for a larger Monte Carlo sample of Z→ ττ .843

844

52

The conclusion is that this method is powerful bringing the number of background events back, keeping845

most of the signal events. However, more data and simulation events are needed to pull back the statis-846

tic error on the background. Moreover, new additional variables could reduce the background even more,847

obtaining a better result.848

53

Appendix A849

Data/MC comparison after the cut of850

1 tight tau851

Leading Tau Momentum /GeV0 20 40 60 80 100 120 140 160 180 200

even

ts

100

200

300

400

500

600

700

310×Leading Tau Pt VBF

ggHttbar

ντ →W ττ →Z

QCDdatastat+syst err

0 20 40 60 80 100 120 140 160 180 200

ratio

dat

a/M

C

00.20.40.60.8

11.21.41.61.8

Leading Tau Momentum /GeV0 20 40 60 80 100 120 140 160 180 200

even

ts

1

10

210

310

410

510

610

Leading Tau Pt VBF

ggH

ttbar

ντ →W

ττ →Z

QCD

data

stat+syst err

Leading Tau Pt

Figure A.1: Leading Tau Momentum

54

Missing Transverse Energy /GeV0 20 40 60 80 100 120 140 160 180 200

even

ts

100

200

300

400

500

600

310×MET VBF

ggHttbar

ντ →W ττ →Z

QCDdatastat+syst err

0 20 40 60 80 100 120 140 160 180 200

ratio

dat

a/M

C

0.40.60.8

11.21.41.61.8

2

Missing Transverse Energy /GeV0 20 40 60 80 100 120 140 160 180 200

even

ts

1

10

210

310

410

510

610

MET VBF

ggH

ttbar

ντ →W

ττ →Z

QCD

data

stat+syst err

MET

Figure A.2: MET

Scalar Momentum /GeV100 200 300 400 500 600 700

even

ts

50

100

150

200

250

310×Scalar Momentum VBF

ggHttbar

ντ →W ττ →Z

QCDdatastat+syst err

100 200 300 400 500 600 700

ratio

dat

a/M

C

0.6

0.8

1

1.2

1.4

1.6

Scalar Momentum /GeV100 200 300 400 500 600 700

even

ts

1

10

210

310

410

510

Scalar Momentum VBF

ggH

ttbar

ντ →W

ττ →Z

QCD

data

stat+syst err

Scalar Momentum

Figure A.3: MET

55

Tau delta Phi-4 -3 -2 -1 0 1 2 3 4

even

ts

50

100

150

200

250

300

350

310×Tau deltaPhi VBF

ggHttbar

ντ →W ττ →Z

QCDdatastat+syst err

-4 -3 -2 -1 0 1 2 3 4

ratio

dat

a/M

C

0.80.85

0.90.95

11.05

1.11.15

1.2

Tau delta Phi-4 -3 -2 -1 0 1 2 3 4

even

ts

1

10

210

310

410

510

Tau deltaPhi VBF

ggH

ttbar

ντ →W

ττ →Z

QCD

data

stat+syst err

Tau deltaPhi

Figure A.4: TdPhi

Tau delta R0 1 2 3 4 5 6

even

ts

50

100

150

200

250

300

350

400

310×Tau deltaR VBF

ggHttbar

ντ →W ττ →Z

QCDdatastat+syst err

0 1 2 3 4 5 6

ratio

dat

a/M

C

0.20.40.60.8

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Tau delta R0 1 2 3 4 5 6

even

ts

1

10

210

310

410

510

Tau deltaR VBF

ggH

ttbar

ντ →W

ττ →Z

QCD

data

stat+syst err

Tau deltaR

Figure A.5: TdR

56

Appendix B852

Data/MC comparison after the cut853

Ptτ1 > 40GeV, P tτ2 > 25GeV854

Leading Tau Momentum /GeV40 60 80 100 120 140 160 180 200

even

ts

0

5000

10000

15000

20000

25000

Leading Tau Pt VBFggHttbar

ντ →W ττ →Z

QCDdatastat+syst err

40 60 80 100 120 140 160 180 200

ratio

dat

a/M

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0.20.40.60.8

11.21.41.61.8

2

Leading Tau Momentum /GeV40 60 80 100 120 140 160 180 200

even

ts

1

10

210

310

410

Leading Tau Pt VBF

ggH

ttbar

ντ →W

ττ →Z

QCD

data

stat+syst err

Leading Tau Pt

Figure B.1: Leading Tau Momentum

57

Missing Transverse Energy /GeV0 20 40 60 80 100 120 140 160 180 200

even

ts

0

2000

4000

6000

8000

10000

12000

14000

16000

MET VBFggHttbar

ντ →W ττ →Z

QCDdatastat+syst err

0 20 40 60 80 100 120 140 160 180 200

ratio

dat

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2

Missing Transverse Energy /GeV0 20 40 60 80 100 120 140 160 180 200

even

ts

1

10

210

310

410

MET VBF

ggH

ttbar

ντ →W

ττ →Z

QCD

data

stat+syst err

MET

Figure B.2: MET

Scalar Momentum /GeV100 200 300 400 500 600 700

even

ts

2000

4000

6000

8000

10000

12000

14000

16000

Scalar Momentum VBFggHttbar

ντ →W ττ →Z

QCDdatastat+syst err

100 200 300 400 500 600 700

ratio

dat

a/M

C

0.51

1.5

2

2.5

3

3.5

Scalar Momentum /GeV100 200 300 400 500 600 700

even

ts

10

210

310

410

Scalar Momentum VBF

ggH

ttbar

ντ →W

ττ →Z

QCD

data

stat+syst err

Scalar Momentum

Figure B.3: Scalar Momentum

58

Tau delta Phi-4 -3 -2 -1 0 1 2 3 4

even

ts

2000

4000

6000

8000

10000

12000

14000

16000

18000

Tau deltaPhi VBFggHttbar

ντ →W ττ →Z

QCDdatastat+syst err

-4 -3 -2 -1 0 1 2 3 4

ratio

dat

a/M

C

0.80.85

0.90.95

1

1.051.1

1.15

Tau delta Phi-4 -3 -2 -1 0 1 2 3 4

even

ts

10

210

310

410

Tau deltaPhi VBF

ggH

ttbar

ντ →W

ττ →Z

QCD

data

stat+syst err

Tau deltaPhi

Figure B.4: TdPhi

Tau delta R0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8

even

ts

2000

4000

6000

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10000

12000

14000

16000

Tau deltaRVBFggHttbar

ντ →W ττ →Z

QCDdatastat+syst err

0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8

ratio

dat

a/M

C

0.850.9

0.951

1.051.1

1.151.2

Tau delta R0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8

even

ts

10

210

310

410

Tau deltaRVBF

ggH

ttbar

ντ →W

ττ →Z

QCD

data

stat+syst err

Tau deltaR

Figure B.5: TdR

59

MMC /GeV80 100 120 140 160 180 200 220 240

even

ts

2000

4000

6000

8000

10000

12000

14000

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20000

mass VBFggHttbar

ντ →W ττ →Z

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80 100 120 140 160 180 200 220 240

ratio

dat

a/M

C

0.20.4

0.6

0.81

1.21.4

1.6

MMC /GeV80 100 120 140 160 180 200 220 240

even

ts

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mass VBF

ggH

ttbar

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ττ →Z

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data

stat+syst err

mass

Figure B.6: Invariant mass of the di-tau system

60

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