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Page 1: Separating the process gg HH b b from irreducible background at the LHC · 2020-06-08 · Separating the process gg!HH!b b from irreducible background at the LHC Martijn Pronk Student

Separating the process gg → HH → bbγγ from irreduciblebackground at the LHC

Martijn Pronk

Student ID: 10191739

July 6, 2014

H

H

H

Verslag van Bachelorproject Natuur- en Sterrenkunde, omvang 15 EC,

uitgevoerd tussen 03-04-2014 en 27-06-2014

Naam begeleider: Magdalena Slawinska, Nikhef

Tweede beoordelaar: Stan Bentvelsen, Nikhef & UvA

Faculteit Natuurkunde, Wiskunde en Informatica, Universiteit van Amsterdam

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Abstract

We rederive the Higgs mechanism for the U(1) unitary group and find a value of λ3H ≈190 GeV for the strength of the triple Higgs coupling. We study the channels for double Higgs

production and the decay of the Higgs boson at the LHC at 14 TeV centre of mass energy.

For a Higgs boson mass of 125 GeV we choose the channel gg → HH → bbγγ that has a cross

section of 4.27 ∗ 10−2 fb. We start our analysis of the signal and irreducible b¯γγ background

processes from LO matrix elements and included the effect of Initial State Radiation (ISR),

Final State Radiation (FSR) and hadronisation on the variables ∆R(γ, γ) and min(∆R(γ, b))

We investigate the cuts proposed by an earlier ATLAS study [1]. Our analysis is based on

Monte Carlo simulations. We use MadGraph5 aMC@NLO for generating gg → HH matrix

elements and Pythia 8 for Higgs decay, ISR, FSR and hadronisation. We calculate numbers

of events as expected at a luminosity of L = 3000fb−1. We find that the proposed cuts give a

S/√B-ratio of 4.22, but our optimized values for these cuts improve the S/

√B-ratio to 4.69.

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Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Theory of the Higgs boson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1 Higgs mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Higgs boson production . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Higgs boson decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Di-Higgs kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4 bbγγ final state and irreducible background . . . . . . . . . . . . . . . . . . . 16

4.1 Hard process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.2 Initial state radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

5 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

5.1 Jets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

5.2 Isolated photon selection . . . . . . . . . . . . . . . . . . . . . . . . . . 21

5.3 Tagging b-quarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

5.4 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

6 Conclusion & Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3

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4

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1 Introduction

The Higgs mechanism was introduced in 1964 to explain why gauge bosons have mass. Con-

sequently however, the theory predicted a new scalar boson, its interactions with itself (triple

and quadruple Higgs coupling) and with massive fermions and bosons. In 2012 the announce-

ment was made by ATLAS (A Toroidal LHC Apparatus) & CMS(Compact Muon Solenoid)

experiments, both working at the LHC (Large Hadron Collider) that a particle has been dis-

covered which was probably the Higgs boson [2, 3]. If this new particle truly is the Standard

Model Higgs boson, then according to the theory, also the triple and quadruple Higgs coupling

exist, with values predicted by the theory. Therefore the search for these interactions is a

good way to test the Higgs mechanism in the Standard Model.

In this bachelor thesis a study has been made on measuring the Higgs pair production,

which is important for measuring the triple Higgs coupling. In the first part of section 2 the

Higgs mechanism is described for simplicity as a U(1) symmetry group. This captures the

essence of the Higgs mechanism, although the unification of the electromagnetic interaction

and the weak interaction is a U(1)×SU(2)L symmetry group. The second part of the section

2 will describe the Higgs boson pair production, and the third part its decay modes. In these

last two sections we explain the choice of experimental signature.

In section 3 then we investigate the kinematics of the Higgs pair production with some

typical variables of the process, such as the transverse momentum and the separation angle.

A good understanding of the kinematics is necessary for separating the Higgs production and

decay from the background processes, which are discussed in section 4, along with kinematic

cuts. In section 4 we follow earlier studies at ATLAS [1] and analyse the hard process signal

and irreducible background processes. This is followed by including initial state radiation

(ISR). In section 5 we will make a more realistic approach and we will take final state radiation

(FSR) and hadronisation into account. We conclude with section 6 in which we will summarize

our findings and discuss them.

Note that in this writing we will take c = 1, so masses and momenta are measured in

GeV.

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2 Theory of the Higgs boson

2.1 Higgs mechanism

In quantum field theory particles are described by excitations of a quantum field which can

be expressed in terms of the Lagrangian density [4]. This Lagrangian density we will just call

the Lagrangian. Using the principle of least action, the Euler-Lagrange equation for fields φi

becomes

∂µ

(∂L

∂(∂µφi)

)− ∂L∂φi

= 0 (1)

According to Noether’s theorem a symmetry of the Lagrangian corresponds to a conserved

quantity. In field theory this quantity is a conserved current. For example if we transform

the Lagrangian for the free Dirac field

L = iψγµ∂µψ −mψψ (2)

under the global U(1) phase transformation

ψ → ψ′ = eiθψ ψ → ψ′ = e−iθψ (3)

we get

L → L′ = i(e−iθψ)γµ∂µ(eiθψ)−me−iθψeiθψ

= i(e−iθeiθ)ψγµ∂µψ −m(e−iθeiθ)ψψ

= iψγµ∂µψ −mψψ = L (4)

so the Dirac Lagrangian is invariant under this transformation and we have the corresponding

current

jµ = ψγµψ (5)

which is conserved, according to the continuity equation.

If we now consider a scalar field φ with the so called ’Higgs’-Lagrangian [4]

L = (∂µφ)∗(∂µφ)− V (φ)

V (φ) = µ2φ∗φ+ λ(φ∗φ)2 (6)

where φ is a complex scalar field of the form

φ =1√2

(φ1 + iφ2) (7)

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ΦV HΦL

Figure 1: λ < 0

Φ

V HΦL

Figure 2: λ > 0, µ2 < 0

Φ

V HΦL

Figure 3: λ > 0, µ2 > 0

We fill in equation (7) into equation (6) and get

L =1

2(∂µφ1)∗(∂µφ1) +

1

2(∂µφ2)∗(∂µφ2)− 1

2µ2(φ2

1 + φ22)− 1

4λ(φ2

1 + φ22)2 (8)

For the potential (density) V (φ) to have a stable minimum, λ > 0, but µ2 can be both positive

or negative, as can be seen in figure 1,2 and 3 (for a simplified two dimensional potential).

We find the extrema by calculating ∂∂φ1

V (φ1, φ2) = 0 and ∂∂φ1

V (φ1, φ2) = 0.

∂φ1V (φ1, φ2) = µ2φ1 + λ(φ2

1 + φ22)φ1 = 0 (9)

∂φ2V (φ1, φ2) = µ2φ2 + λ(φ2

1 + φ22)φ2 = 0 (10)

For µ2 ≥ 0 the minimum of the potential occurs when both φ1 and φ2 are zero and the

vacuum state corresponds to the fields being zero, but when µ2 < 0, then the extremum at

φ1 = φ2 = 0 is a local maximum instead of a global minimum, and the potential now has

minima for all values of the fields which satisfy

φ21 + φ2

2 =−µ2

λ≡ v2 (11)

with v being the vacuum expectation value. We will show in the following that v 6= 0 is a

necessary condition under which the Higgs boson can generate the masses of the W and Z

boson. There is an infinite amount of possible vacuum states, but because there is only one

actual vacuum state, the system has to ’choose’ one of the vacuum states, a process known as

spontaneous symmetry breaking. Without loss of generality, we can choose the vacuum state

to be the state at which φ2 = 0 and φ1 = v. Because we want to expand the field about the

vacuum state we define φ1(x) ≡ η(x) + v and φ2(x) = ξ(x) and thus φ = 1√2(η + v + iξ). ξ

describes the excitation along the minima and η the excitation perpendicular to the minima.

The essential difference between the two fields is that ξ is expanded around zero, while η is

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Figure 4: The minima of the potential satisfy φ21 + φ2

2 = v2, which is the equation for a circle

with radius v.

expanded around v, which is a non-zero value. Substituting this in equation (6) we get

L(η, ξ) =1

2(∂µ(η + v + iξ)∗(∂µ(η + v + iξ)− 1

2µ2(η + v − iξ)(η + v + iξ)

− 1

4λ(η + v − iξ)(η + v + iξ)2

=1

2(∂µη)(∂µη) +

1

2(∂µξ)(∂µξ)− V (η, ξ)

V (η, ξ) = −1

2λv2[(η + v)2 + ξ2] +

1

4λ[(η + v)2 + ξ2]2

= −1

2λv2η2 − 1

2λv4 − λv3η − 1

2λv2ξ2 +

1

4λv4 + λv3η +

3

2λv2η2

+ λvη3 +1

4λη4 +

1

4λξ4 +

1

2λv2ξ2 +

1

2λη2ξ2 + λvηξ2

= −1

4λv4 + λv2η2 + λvη3 +

1

4λη4 +

1

4λξ4 +

1

2λη2ξ2 + λvηξ2 (12)

Because the Lagrangian in equation (6) contains derivatives, it is not invariant under the local

U(1) gauge transformation φ→ φ′ = eiqχ(x)φ

L → L′ = (∂µe−iqχ(x)φ∗)(∂µeiqχ(x)φ)− µ2e−iqχ(x)eiqχ(x)φ∗φ− λ(e−iqχ(x)eiqχ(x)φ∗φ)2

= [−iq(∂µχ(x))φ∗ + (∂µφ∗)]e−iqχ(x)[iq(∂µχ(x))φ+ (∂µφ)]eiqχ(x) − µ2φ∗φ− λ(φ∗φ)2

= q2(∂µχ(x))(∂µχ(x))− iq(∂µχ(x))(∂µφ) + iq(∂µχ(x))(∂µφ∗) + (∂µφ∗)(∂µφ)

− µ2φ∗φ− λ(φ∗φ)2

= L+ q2(∂µχ(x))(∂µχ(x)) 6= L (13)

To make the Lagrangian invariant we change the ∂µ into the covariant derivative defined

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as Dµ = ∂µ + iqAµ. Substituting this change into equation (13) we see that the extra

term q2(∂µχ(x))(∂µχ(x)) is cancelled and Lagrangian is invariant again under local gauge

transformation. By introducing the covariant derivative we also introduced the field A, which

can be associated with gauge bosons. This field is required to be massless, because its mass

term 12mAA

µAµ would break the gauge invariance. On the contrary the kinetic term of the

gauge field −14F

µνFµν with Fµν = ∂µAν−∂νAµ is gauge invariant and can therefore be added

to the Lagrangian. The full Lagrangian now becomes

L = −1

4FµνFµν + (∂µφ)∗(∂µφ)− µ2φ∗φ− λ(φ∗φ)2

− iqAµφ∗(∂µφ) + iq(∂µφ∗)Aµφ+ q2AµAµφ∗φ (14)

Expanding this around the vacuum state we get (using equation (12))

L =1

2(∂µη)(∂µη) + λv2η2︸ ︷︷ ︸

massive η

+1

2(∂µξ)(∂µξ)︸ ︷︷ ︸massless ξ

− 1

4FµνFµν +

1

2q2AµAµv

2︸ ︷︷ ︸massive A-field

+ λvη3 +1

4λη4 +

1

4λξ4 +

1

2λη2ξ2 + λvηξ2︸ ︷︷ ︸

self interactions of η and ξ and interactions between them

− 1

4λv4︸ ︷︷ ︸

constant term

+1

2q2AµAµη

2 + q2AµAµvη +1

2q2AµAµξ

2 + qAµη(∂µξ)︸ ︷︷ ︸interactions of A with η and ξ

+qAµv(∂µξ) (15)

We see that we have a massive η-field with 12m

2η = λv2 → mη =

√2λv2, a massless ξ-

field, the so called Goldstone boson. Additionally, the previously massless gauge field A

now has a mass term 12q

2v2AµAµ and various interaction terms involving η, ξ and A have

appeared in the Lagrangian. But now there appear to be some problems. At first there a

term appears in equation (15) which represents a direct coupling between A and ξ. The

gauge-field A represents a spin-1 particle, but the Goldstone boson ξ is a spin-0 particle,

so this interaction term suggests that a spin-1 particle can transform into a spin-0 particle.

Secondly, the original Lagrangian contained one degree of freedom for both φ1 and φ2 and two

for a massless A-field, but in the Lagrangian in equation (15) there is an additional degree of

freedom, because the A-field now has a longitudinal polarisation state due to the mass term.

The solution lies in eliminating the Goldstone boson by making the gauge transformation

Aµ → Aµ′ = Aµ + 1qv∂

µξ. This corresponds to taking χ(x) = − ξ(x)qv in the transformation in

equation (13). The expansion of φ about the vacuum expectation value can approximately

be expressed as φ = 1√2(v + η(x))eiξ(x)/v. If we now use the gauge transformation as we did

before, we get

φ→ φ′ =1√2e−iξ(x)/v(v + η(x))eiξ(x)/v =

1√2

(v + η(x)) (16)

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So we have eliminated ξ(x) and the complex scalar field φ(x) is now entirely real. The field η

can now be identified as being the Higgs field h(x). We can write the Lagrangian of equation

(15) as

L =1

2(∂µh)(∂µh) + λv2h2︸ ︷︷ ︸

massive h

− 1

4FµνFµν +

1

2q2AµAµv

2︸ ︷︷ ︸massive A-field

λvh3 +1

4λh4︸ ︷︷ ︸

self interactions of h

+1

2q2AµAµh

2 + q2AµAµvh︸ ︷︷ ︸interactions of A with h

−1

4λv4 (17)

We have a Lagrangian describing the Higgs field and the massive gauge boson A. The mass

of the Higgs boson can be identified as mH =√

2λv2 and the mass of the gauge boson as

mB = qv, so we see that for the gauge boson to have mass it is essential that the potential has

a non-zero expectation value and for that reason the potential of equation (6) is used. If we

would repeat the above calculations, this time breaking a more general U(1)× SU(2)L local

gauge symmetry we would obtain three gauge fields that acquire mass. There will be three

Goldstone bosons, which provide the longitudinal degrees of freedom for the W+ the W− and

the Z boson [4]. For the W-boson the Higgs-mechanism predicts its mass to be mW = 12gW v

with g2W =

8m2WGF√

2and GF the Fermi constant. With the values GF = 1.16638×10−5 GeV−2

and mW = 80.385 GeV [4] we find gW = 0.426. These values for mW and gW give a vacuum

expectation value of v = 246 GeV and also the mass of the Higgs particle is known (mH =

125 GeV). Taking from equation 17 the terms which involve only the Higgs boson we have

V (h) = λv2h2 + λvh3 +1

4λh4 (18)

which we rewrite:

V (h) =1

2m2Hh

2 +1

3!λ3Hh

3 +1

4!λ4Hh

4 (19)

with λ3H = 3!λv =3m2

Hv and λ4H = 6λ =

3m2H

v2. Apparently both λ3H and λ4H only depend

on mH and v, so both of them have fixed values (λ3H ≈ 190 GeV and λ4H ≈ 0.775). The term

λ3H determines the strength of the triple Higgs boson coupling. By measuring this coupling

it is possible to test the Higgs mechanism in the Standard Model.

2.2 Higgs boson production

The SM predicts several ways to produce a single Higgs boson. The four most frequently

occurring processes at the LHC at a centre of mass energy√s = 14 TeV are gluon fusion

with a production cross-section σ = 49.85 pb at NNLO QCD + NLO EW, Higgs associated

production with W and Z bosons, with a cross-section σ = 1.504 pb for the W boson and

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σ = 0.8830 pb for the Z boson, both at NNLO QCD + NLO EW, vector boson fusion (VBF)

(σ = 4.180 pb) at NNLO QCD + NLO EW and associated top quark fusion (σ = 0.6113 pb)

at NLO QCD [5].

In section 2.1 we defined λ3H , the Higgs triple coupling. This coupling is directly accessible

only in double Higgs production. Therefore we can use the same production processes as for

the production of a single Higgs boson but now the initial Higgs boson has to be off shell, to

produce two on-shell secondary Higgs bosons. This is because the Higgs boson decay width

Γ = 6.1+7.7−2.9 MeV [6] is much smaller than its mass. The production processes at the LHC

are then gluon fusion gg → H∗ → HH, Higgs associated production qaqb → (W ∗, Z∗) →(W,Z)H∗ → (W,Z)HH, VBF qaqb → (W,Z)(W,Z)qcqd → H∗qcqd → HHqcqd and top

fusion gg → tttt → H∗tt → HHtt. However, there are also processes which do not involve

triple Higgs coupling, but which lead to the same final states with two Higgs bosons. These

extra double Higgs production processes make the measurement of the triple Higgs coupling

extremely difficult.

Feynman diagrams describing the amplitude of Higgs pair production in the four channels

above are shown in figures 5 to 12. The production cross-sections are calculated to be (at

a centre of mass energy of√s = 14 TeV) σ = 33.89 fb with a total theoretical uncertainty

of +37.2% and −29.8% at NLO for gluon fusion, σ = 2.01 fb with an uncertainty of 7.6%

and −5.1% at NLO for VBF, σ = 0.57 fb at NNLO for Higgs associated production with W

bosons, σ = 0.42 fb with an uncertainty of +7.0% and −5.5% at NNLO for Higgs with Z

bosons and σ = 1.02 fb at LO for associated top fusion [7]. Note that these production cross-

sections are calculated for all processes with two Higgs bosons in the final state, including the

processes without triple Higgs coupling. In the following we will consider the dominant gluon

fusion process described by the loop diagrams in figures 5 and 6. Since there are three and

four fermion lines in these loops, respectively, the two diagrams have a relative minus sign.

Therefore, the cross-section involving these diagrams will feature a destructive interference.

t

t

tH∗

g

g

H

H

Figure 5: Gluon fusion: Higgs pair

production via triple Higgs coupling

t

t

t

t

g

g

H

H

Figure 6: Gluon fusion: Higgs pair

production without triple Higgs cou-

pling

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W ∗, Z∗H∗

fb

fa

H

H

W,Z

Figure 7: Higgs associated produc-

tion with W and Z bosons: Higgs pair

production via triple Higgs coupling

W ∗, Z∗

fb

fa

H

H

W,Z

Figure 8: Higgs associated produc-

tion with W and Z bosons: Higgs pair

production without triple Higgs cou-

pling

H∗

fa

fb

fc

H

H

fd

W,Z

Figure 9: Vector boson fusion: Higgs

pair production via triple Higgs cou-

pling

fa

fb

fc

H

H

fd

W,Z

Figure 10: Vector boson fusion:

Higgs pair production without triple

Higgs coupling

t

t

H∗

g

g

t

H

H

t

Figure 11: Associated top fusion:

Higgs pair production via triple Higgs

coupling

g

g

t

H

H

t

Figure 12: Associated top fusion:

Higgs pair production without triple

Higgs coupling

2.3 Higgs boson decay

In figure 13 the branching ratios for the Higgs boson decay are displayed. For a Higgs boson

of the mass equal to 125 GeV the most frequent decay is H → bb with a branching ratio

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of 57.7+1.85−1.89% [5]. The bottom quarks will decay and form jets that can be reconstructed,

but there is some probability that b-jets are misidentified as photons, charm quarks or light

jets. Because of the small cross-section for gg → HH final states with only one bb-pair are

considered. The second most frequent decay is H → τ+τ− (brancing ratio of 6.32+0.36−0.34%). Tau

leptons however have a very short life time (290.6×10−15s [8]) and decay into leptons or light

hadrons. The latter decay channel is very challenging to reconstruct experimentally. Charm

quarks (brancing ratio 2.91+0.35−0.36%) are even more hard to distinguish from jets produced by

light quarks or gluons than bottom quarks, because of the lower mass of the charm quark and

thus a lower energy in the jet belonging to the charm quark. For the same reasons all of the

other quarks are too difficult to identify experimentally. According to figure 13 weak boson

pairs e.g. W+W− (branching ratio 2.15+0.09−0.09%) or ZZ (branching ratio 2.64+0.11

−0.11%) are both

good candidates for decay products, but one has to take into account their decays as well

and these decays effectively reduce the branching ratio. Almost all of the decay products in

[GeV]HM80 100 120 140 160 180 200

Hig

gs B

R +

Tot

al U

ncer

t

-410

-310

-210

-110

1

LHC

HIG

GS

XS W

G 2

013

bb

oo

µµ

cc

gg

aa aZ

WW

ZZ

Figure 13: Branching ratios for Higgs boson decay. Note that for W+W− and ZZ one of the

bosons is off shell [1].

figure 13 are now discounted, except for the diphoton (branching ratio 2.28+0.11−0.11×10−1%) and

dimuon (branching ratio 2.19+0.13−0.13 × 10−2%) production channels, which are both very clean

channels, because both photons and muons are relatively easy to detect. Their branching

ratios, however, are too small to allow for HH → 4µ or HH → 4γ searches, so one has to

compromise between large event yields and clean signatures and therefore look forHH → bbγγ

or HH → bbµ+µ− instead. Since the branching ratio of the latter is about a factor 10 smaller,

only the final states with one b-pair and two photons are considered (figures 14 and 15).

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Naturally, one has to take into account background processes as well while making such a

selection. We will note at this point theoretical analyses of [1] and [9]

H

b

b

Figure 14: A Higgs boson

decaying in a bb pair.

t

t

tH

γ

γ

Figure 15: A Higgs boson

decays into two photons via

a top quark loop.

W,Z

W,Z

W,ZH

γ

γ

Figure 16: A Higgs boson

decays into two photons via

a W or Z boson loop.

3 Di-Higgs kinematics

In the previous section we mentioned the presence of background processes. To distinguish the

signal from those processes one has to have a good understanding of the kinematics. Typical

kinematic variables are displayed in figures 17 to 20. The data forming these histograms

are produced with MadGraph5 aMC@NLO, which gives a gg → HH cross section of

σ = 16.22 fb at LO. If we would for example produce the Higgs boson pair out of a head-on

collision of an electron and a positron with equal energy, we would see that the Higgs bosons

have the same but opposite momenta. At the LHC, however, the Higgs bosons are produced

out of two colliding gluons. The gluons have no transverse momenta, hence the sum of the

transverse momenta of the two Higgs bosons will be zero as well. In general those gluons,

both coming from protons, do not have the same energy. The energy of a gluon is expressed

as a fraction x of the energy of the originating proton, which is in the case of the LHC

7 TeV. The fraction x is distributed according to a parton distribution function (PDF). The

PDF used by MadGraph5 aMC@NLO is CTEQ6L1. The gluons in general have different

energies, leading to different distributions of x. Therefore the collision products have net

momenta in the z direction (along the beam axis). This can be seen in figure 20 in which the

pseudorapidity η is plotted. η is defined as

η = − ln tanθ

2(20)

14

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CME in GeV0 100 200 300 400 500 600 700 800 900 1000

Num

ber

of e

vent

s

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

Figure 17: Centre of mass energy in gg → HH

events normalised to the number of events ex-

pected at a luminosity of L = 3000fb−1

PT in GeV0 50 100 150 200 250 300 350 400 450 500

Num

ber

of e

vent

s

0

2000

4000

6000

8000

10000

12000

Figure 18: Transverse momentum of Higgs

bosons in gg → HH events normalised to the

number of events expected at a luminosity of

L = 3000fb−1

Angle in radians0 0.5 1 1.5 2 2.5 3

Num

ber

of e

vent

s

0

500

1000

1500

2000

2500

Figure 19: Angle between the two Higgs

bosons in gg → HH events normalised to the

number of events expected at a luminosity of

L = 3000fb−1

η-5 -4 -3 -2 -1 0 1 2 3 4 5

Num

ber

of e

vent

s

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

Figure 20: Pseudorapidity of Higgs bosons in

gg → HH events normalised to the number

of events expected at a luminosity of L =

3000fb−1

with θ the angle from the transverse plane in the z direction. In fact the pseudorapidity is a

special case of the rapidity

y =1

2ln

(E + pzE − pz

)(21)

with pz ≈ E cos θ. For θ = 0 ⇒ η → ∞ and the particles go along the beam axis and for

θ = π2 ⇒ η = 0 and the particles go perpendicular to the beam axis. To be able to create two

on shell Higgs bosons at rest in the first place the centre of mass energy CME has to be equal

to twice the mass of the Higgs boson, i.e. 250 GeV. Since gluons are massless, the CME is

equal to√s =√x1x2s, with x1 and x2 the fractions for the two gluons and s the CME of

the originating protons (at the LHC√s = 14 TeV). Any excess of energy will go into the

15

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momenta of the Higgs bosons. Indeed in figure 17 can be seen that there is a threshold in the

CME at 250 GeV and the mean value of the CME lies above that value. The CME can also

be calculated from the energy and momenta of the Higgs bosons via

s = (E1 + E2)2 − (pT,1 + pT,2)2 − (pz,1 + pz,2)2 (22)

For the energies we can use the invariant mass for the separate Higgs bosons, and if we would

set the momenta to zero, we see that the CME is exactly twice the mass of the Higgs boson.

In formula 22 we used pT the transverse momentum of the Higgs bosons. In figure 18 the

distribution the pT is displayed. Since the pz is missing the relation between the CME and pT

is not trivial, but from figures 17 and 18 can be deduced that it is correct that for a CME of

twice the Higgs mass the produced Higgs bosons should not have any pT , and the maximum

value of the CME is about 500 GeV, which corresponds with an excess in energy of about

250 GeV and indeed for the pT we see the maximum lies about 250 GeV. Another variable

to look at is the angle between the two Higgs bosons. In the rest frame of those particles the

angle is exactly π radians, because they lie back to back. But we saw that the Higgs bosons

have a boost along the beam axis. Because of this boost the angle between the two Higgs

bosons is smaller than π radians and the larger the boost, the smaller the angle (figure 19).

4 bbγγ final state and irreducible background

4.1 Hard process

In experiments in high energy physics, all interesting processes are accompanied by so called

backgrounds. Background processes are different than the signal, yet have the same final

state. Since we look at the process HH → bbγγ we have to take into account all the processes

that decay to bbγγ. These are irreducible background processes. There are also reducible

backgrounds that mimic the signal after taking into account detector inefficiencies but we

will not consider them here.

Those non-signal processes can be generated with the program MadGraph5 aMC@NLO.

This program gives 422 different Feynman diagrams including QCD(Quantum Chrome Dy-

namic) processes, single Higgs production etc. With MadGraph5 aMC@NLO we find the

total cross section of those processes to be σ = 55.65 fb at LO. The total cross section of

the Higgs pair signal (the Higgs decay into bb and γγ is simulated by Pythia 8) is equal to

σ(gg → HH)×BR(H → bb)×BR(H → γγ)× 2. The factor two appears because the com-

binations bbγγ and γγbb are indistinguishable. This gives a total cross section for the signal

of 16.22 ∗ 0.577 ∗ 0.00228 ∗ 2 = 4.27 ∗ 10−2 fb. Because the cross section for the background is

much larger than the cross section of the signal, we must find differences in the kinematics of

16

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the two in order to separate the signal from background. We aim at finding variables, like the

ones in section 3, defining a region with many signal events and only a few background events.

By applying cuts one tries to reduce the amount of background events while maintaining the

amount of signal events as much as possible. Equation 23 shows some basic kinematic cuts.

The cuts on η are due to detector coverage and the cuts in pT are motivated by the use of

detector triggers [10, 11].

pT (b) > 45 GeV, |η(b)| < 2.5, ∆R(b, b) > 0.4, 105 GeV < Mb,b < 145 GeV

pT (γ) > 20 GeV, |η(γ)| < 2.5, ∆R(γ, γ) > 0.4, 122.7 GeV < Mγ,γ < 127.3 GeV (23)

in which ∆R is defined as ∆R =√|∆φ|2 + |∆η|2. Reference [1] also mentions a cut ∆R(γ, b) >

0.4, but later we will be looking at the minimum value of this variable, and therefore we do

not apply this cut. The cuts on masses are based on the fact the decay products should have

)γ, γ R (∆0 0.5 1 1.5 2 2.5 3 3.5 4

Num

ber

of e

vent

s

0

10

20

30

40

50

60

70 Background

Signal

Figure 21: Distribution of ∆R between two

photons with the cuts of equation 23 nor-

malised to the number of events expected at

a luminosity of L = 3000fb−1

, b))γ R (∆min(0 0.5 1 1.5 2 2.5 3 3.5 4

Num

ber

of e

vent

s

0

10

20

30

40

50

60

70

80 Background

Signal

Figure 22: Distribution of the minimal ∆R

between a photon and a b-quark with the

cuts of equation 23 normalised to the num-

ber of events expected at a luminosity of

L = 3000fb−1

an invariant mass equal to the mass of the Higgs boson. Reference [1] describes two variables

that offer a good discrimination between signal and background. We will follow their choice

but replace the mass of the Higgs boson from 120 GeV to 125 GeV [12]. These variables are

∆R(γ, γ) and min(∆R(γ, b)) > 0.4. By the latter we mean the minimal value of the ∆R

separation among all combinations of a b-type quark and a photon in the event. If there is

more than one γγ-combination or bb-combination in an event, then the combination with the

invariant mass closest to the Higgs mass MH is chosen, since this combination is most likely

to come directly from the Higgs boson. The ∆R(b, γ) is typically much smaller for the back-

grounds than for the signal, because b-quarks can radiate photons almost collinear with itself.

17

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For the ∆R(γ, γ), however, we expect the signal to have small values, since the photons come

directly from the heavy Higgs boson decay (The opening between the two photons depends

on the boost of the Higgs boson, but in figure 17 we saw that most events have a centre of

mass energy of more than 250 GeV, twice the Higgs mass. Therefore we expect the Higgs

bosons to be highly boosted), while for the background, the photons do not have to have a

direct physical relation amongst them.

)γ, γ R (∆0 0.5 1 1.5 2 2.5 3 3.5 4

Num

ber

of e

vent

s

0

10

20

30

40

50

Background

Signal

Figure 23: Distribution of ∆R between two

photons with the cuts of equation 23 af-

ter including ISR normalised to the number

of events expected at a luminosity of L =

3000fb−1

, b))γ R (∆min(0 0.5 1 1.5 2 2.5 3 3.5 4

Num

ber

of e

vent

s0

10

20

30

40

50

60 Background

Signal

Figure 24: Distribution of the minimal ∆R

between a photon and a b-quark with the cuts

of equation 23 after including ISR normalised

to the number of events expected at a lumi-

nosity of L = 3000fb−1

In figures 21 and 22 the distributions of min(∆R(γ, b)) and ∆R(γ, γ) are displayed for the

signal and the background. These distributions resemble the ones of [1] very much, except

for a small shift in the distribution due to the difference in Higgs boson mass. The higher

Higgs mass causes the Higgs boson to be less boosted, whereby the opening between the two

photons coming from the Higgs boson is larger. The same is true for the b-quarks coming

from the other Higgs 1. But because these ∆R values are larger, the value of min(∆R(γ, b))

will be smaller. Note that reference [1] used scale factors to account for NLO corrections,

whereas we did not. The reason we did not account for NLO corrections is because we only

look at the LO cross sections. As we can see in figures 21 and 22 a large excess of background

occurs in a region where little signal is expected. Therefore reference [1] suggests to apply

cuts on those variables at

∆R(γ, b) > 1.0, ∆R(γ, γ) < 2.0 (24)

The authors mention that these cuts are not optimal. We will revisit this selection based

on a more realistic modelling of signal and background. We will model the hard process at

1Also the angle between the Higgs bosons is larger since the initial Higgs boson is less boosted

18

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leading order and include the effects of Initial state radiation (ISR) Final State Radiation

(FSR) and hadronisation.

4.2 Initial state radiation

So far we looked at the hard process of gg → HH → bbγγ, where we used a PDF which

assumes that the initial gluons have zero transverse momentum. However, since we have high

energy proton collisions the value of the coupling constant of the strong interaction is relatively

small with respect to the value of this coupling at low energy (αS(MZ) = 0.1185(6)[8] <

αS(1 GeV) ≈ 1 ). Therefore, due to this low value of the coupling constant, various kinds of

QCD-interactions occur before the gluons interact and form Higgs bosons. These interactions

are called Initial State Radiation (ISR).

)γ, γ R (∆0 0.5 1 1.5 2 2.5 3 3.5 4

Num

ber

of e

vent

s

0

1

2

3

4

5

Signal

Signal+ISR

+hadronisation +ISR+FSR Signal

-1) for signal as expected at a luminosity of 3000 fbγ, γ R (∆

Figure 25: Signal distribution of ∆R between

two photons with the cuts of equation 23 nor-

malised to the number of events expected at

a luminosity of L = 3000fb−1

, b))γ R (∆min(0 0.5 1 1.5 2 2.5 3 3.5 4

Num

ber

of e

vent

s

0

1

2

3

4

5 Signal

Signal+ISR

+hadronisation +ISR+FSR Signal

-1 for signal as expected at a luminosity of 3000 fbmin

, b)γ R (∆

Figure 26: Signal distribution of the min-

imal ∆R between a photon and a b-quark

with the cuts of equation 23 normalised to the

number of events expected at a luminosity of

L = 3000fb−1

Due to this ISR it the gluons taking part in the hard process interaction might no longer

have zero transverse momentum, because they radiate other gluons and quarks. The quarks

that come from ISR could be b-quarks which can affect min(∆R(γ, b)), or they could radiate

photons which can affect ∆R(γ, γ). In the following we will look at how much this ISR will

affect our distributions of ∆R(γ, γ) and min(∆R(γ, b)). We use the program Pythia 8 to

simulate the ISR. Note that the data sets in which the ISR-simulations are included are the

same as for the distributions without ISR. The distributions with ISR are displayed in figures

23 and 24. In these figures we see that the distributions haven’t changed much with respect

to figures 21 and 22, but it is hard to say how much they have actually changed. Therefore

we present in figures 25 to 28 the background and signal of both ∆R(γ, γ) and ∆R(γ, b) are

19

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)γ, γ R (∆0 0.5 1 1.5 2 2.5 3 3.5 4

Num

ber

of e

vent

s

0

10

20

30

40

50

60

70Background

+ISRBackground

+hadronisation +ISR+FSR Background

-1) for background as expected at a luminosity of 3000 fbγ, γ R (∆

Figure 27: Background distribution of ∆R

between two photons with the cuts of equa-

tion 23 normalised to the number of events

expected at a luminosity of L = 3000fb−1

, b))γ R (∆min(0 0.5 1 1.5 2 2.5 3 3.5 4

Num

ber

of e

vent

s

0

10

20

30

40

50

60

70

80Background

+ISRBackground

+hadronisation +ISR+FSR Background

-1 for background as expected at a luminosity of 3000 fbmin

, b)γ R (∆

Figure 28: Background distribution of the

minimal ∆R between a photon and a b-quark

with the cuts of equation 23 normalised to the

number of events expected at a luminosity of

L = 3000fb−1

displayed separately. As we can see in figures 27 and 28 the peak in the background of both

∆R(γ, γ) and ∆R(γ, b) is less high and the distribution is wider with respect to the peaks

and the width of the distributions in background without ISR. For the signal we see in figures

25 and 26 that for ∆R(γ, γ) the signal with ISR has a smaller peak and its width is bigger,

just as in case of the background, but for ∆R(γ, b) the distribution is shifted towards lower

values.

5 Analysis

After investigating the signal and background properties on the level of partonic cross sections

and including ISR we proceed with a more realistic analysis that also takes into account final

state radiation (FSR) and hadronisation. FSR is like ISR, only for final state particles instead

of initial state particles. Hadronisation is the forming of hadrons (mesons and baryons) out

of quarks and gluons due to colour confinement. As particles loose energy, they experience

the strong coupling being larger and eventually they form hadrons. This is called colour con-

finement and means that colour charged particles cannot be isolated singularly, and therefore

cannot be directly observed. The combination of ISR and FSR cause the forming of jets out

of unstable particles. These unstable particles decay and form a shower of particles called

jets. The final state particles in such a jet are the particles which are detected in the detector.

In the case of bbγγ the two b-quarks are unstable, so they will form jets. From now on we

will use the term showering for the combination of ISR, FSR and hadronisation.

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5.1 Jets

A jet is a narrow cone of particles produced by radiating off a quark or gluon. In the detector

they are observed as collimated streams of hadrons and leptons. Typically, jets are not

elementary particles, so in the Monte Carlo simulations we need a prescription in order to

cluster particles into jets and reconstruct the particles out of which the showering forms.

Such a prescription is called a jet algorithm. We will use the anti-kt algorithm [13] provided

by the fastjet 3.0.6 package [14]. Fastjet requires a list of particles to be clustered, the

clustering algorithm to be used and the radius parameter ∆R, as defined in section 4, for

which we will take the value ∆R(jet) < 0.4. The list of particles to be clustered are all final

state particles, except for all the neutrinos, which we remove from the list since the detector

cannot detect them, and all particles with |η| > 2.5, because the particles that do not suffice

this equality cannot be detected by the ATLAS detector. The anti-kt algorithm clusters soft

particles that lie in a cone of radius ∆R around a hard particle i.e. a particle with high pT .

If another hard particle lies within this cone, the algorithm will reconstructed two hard jets,

one of them or both not perfectly conical. In this situation particles can be clustered to the

wrong jet, causing some energy losses. Due to showering all kind of extra unstable particles

are radiated, which also will form jets. Therefore, one needs a prescription to identify the

jets coming from the b-quark and the b-quark and one needs to trace the γγ coming from the

Higgs boson.

5.2 Isolated photon selection

When we look at figure 23 we see that for values of ∆R(γ, γ) < 0.4 almost no signal events

are present. The same can be said about figure 24. Therefore we call photons coming from

the Higgs boson isolated. We find isolated photons in our MC simulations in case one of

the two following conditions are met. At first we treat a photon as being isolated if it has

no jet activity occurring within the direct region of the photon (∆R(γ, any jet) > 0.4). The

jet algorithm however, also clusters the isolated photons with soft particles around them.

Photons are added to the clustering algorithm since in advance we don’t know which photons

are isolated. Therefore we define a second case of isolated photons in the following. We make

a selection of the possible hard isolated photons and match those photons with the jets after

clustering. The selected photons suffice |η| < 2.5, to be able to detect them, and pT > 20 GeV,

to suppress photons originating from soft QED radiation. As is said, also the isolated photons

are clustered, so in this second case the photon is not isolated as described before, because now

there is jet activity within the direct region of the photon (∆R(γ, any jet) < 0.4). However,

since we expect that the photons coming from the Higgs boson are isolated from jets, the

21

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jet which contains the isolated photon contains most likely just this photon and maybe some

soft radiation. Therefore we compare the list of possible isolated photons with all jets and

check if there is some jet-photon combination which has ∆R(γ, jet) < 0.1 and a jet energy

compatible within 15% to that of the photon. Such a photon is then also called an isolated

photon and the jet belonging to this photon is removed from the list of jets.2

5.3 Tagging b-quarks

The prescription we will use to identify jets coming from b-quarks is b-tagging. For real

data coming from the ATLAS detector b-tagging is a tedious and challenging job including

difficult algorithms. We, however, do not have to use these difficult algorithms, since we are

dealing with Monte Carlo simulations of the signal and background and therefore have access

to all intermediate particles. The b-quark can decay into up-quarks and charm-quarks with

emission of a W-boson, but because these decays are suppressed by the small values in the

CKM-matrix [4] the b-quark will more often form a b-hadron than decay. b-hadrons have

a relatively long lifetime, it is possible to find b-hadrons by looking at the vertex of the jet

coming from b-hadrons. For the Monte Carlo simulation we just have to select all b-hadrons

form the list of intermediate particles. We will identify a jet coming from a b-hadron, a

so called b-jet, when it has a b-hadron within its radius, so ∆R(jet, b-hadron) < 0.4. Of

course for b-jets we use b-hadrons, and for b-jets we use b-hadrons. In this way our b-tagging

prescription is 100% efficient.

5.4 Analysis

Now that b-jets have been reconstructed and isolated photons are selected one can do the same

analysis as for the situation with only ISR included. Because the signal contains two isolated

photons, a b-quark and a b-quark, we accept only events that have at least two isolated

photons, one b-jet and one b-jet. The new distributions of ∆R(γ, γ) and min(∆R(γ, b))

with showering included are shown in figures 29 and 30. We use the same cuts as before

(equation 23), but instead of pT (b) > 45 GeV we take pT (b-jet) > 30 GeV, because the

jet reconstruction is not 100% efficient, so some energy losses will occur. The total event

reconstruction efficiency, with pT -cuts and η-cuts taken into account is 40.1% for the signal

and 68.5% for the background, as can be seen in table 1 (bold numbers).

Referring back to figures 25 to 28 one can now see the effect showering on both the

background and the signal. In table 1 we can see that after including showering the total

2In a first approach, these photons were excluded from the clustering algorithm, making it easier to find

the isolated jets. Not excluding these photons from the clustering algorithm, however, turned out to be better

at reconstruction level.

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)γ, γ R (∆0 0.5 1 1.5 2 2.5 3 3.5 4

Num

ber

of e

vent

s

0

5

10

15

20

25

30

35

40

45Background

Signal

Figure 29: Distribution of ∆R between two

photons with the cuts of equation 23 after

including showering normalised to the num-

ber of events expected at a luminosity of

L = 3000fb−1

, b))γ R (∆min(0 0.5 1 1.5 2 2.5 3 3.5 4

Num

ber

of e

vent

s

0

10

20

30

40

50

60

70 Background

Signal

Figure 30: Distribution of the minimal ∆R

between a photon and a b-quark with the cuts

of equation 23 after including showering nor-

malised to the number of events expected at

a luminosity of L = 3000fb−1

number of background events is 70.9% of the number of background events before including

showering. For the signal 69.2% of the events remains. In figure 27 we can see that for

∆R(γ, γ) the shape of the background distribution has not changed and is not shifted. Also

we see the decrease in number of events. In figure 25 we can see the same for the signal. The

distribution has the same shape but the number of events has decreased. In figures 26 and

28 we can see a difference for min(∆R(γ, b)). For the signal the top of the distribution is

flattened and for the background the distribution looks almost the same, except for values

min(∆R(γ, b)) < 0.5. A big decrease in events can be found in that region. In table 1 one

can also see what the acceptance of each cut is relative to the previous cut. For example in

the last column one can see that for the signal 92.5% of the events is accepted after the cut

on the invariant mass of the two photons, while for the background only 3.16% is accepted.

To see what is the discriminating power of each cut we calculate the signal to square root

of background ratio S/√B after each cut. The calculated values can be found in table 2.

Looking at this table we see that the cut on M(γ, γ) improves the S/√B-ratio for the hard

process with showering included by about a factor 5, while the cut on ∆R(b, b) has no effect

at all. And that the cut on M(b, b) improves the S/√B-ratio by a factor 2. We see also that

this last cut is reduced the most by including showering.

Finally we aim at optimizing the cuts of equation 24. In the last two rows for signal

and the last two rows for background in table 1 one can see the effect of those cuts. The

combination of the two cuts have an acceptance of 74.1% ∗ 88.0% = 62.8% for the signal and

10.1% ∗ 79.4% = 8.02% for the background, so it appears that these cuts are very effective

23

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Used cuts

Hard

process

Nevents

Accep-

tance

(%)

+ISR

Nevents

Accep-

tance

(%)

+ISR

+FSR

+hadr.

Nevents

Accep-

tance

(%)

Signal

generated bbγγ-events 136 136 136

pT (γ) > 20 GeV

pT (b) > 45 GeV (pT (b-jet) > 30 GeV)

|η| < 2.5

51 37.1 54 39.7 55 40.4

∆R(γ, γ) > 0.4 51 99.9 54 99.5 55 100

122.7 GeV < M(γ, γ) < 128.3 GeV 51 100 54 100 51 92.5

∆R(b, b) > 0.4 50 99.9 54 99.5 51 100

105.0 GeV < M(b, b) < 145.0 GeV 50 100 54 100 35 68.3

∆R(γ, γ) < 2.0 25 71.4

min(∆R(γ, b)) > 1.0 22 88.0

Background

generated bbγγ-events 166947 166947 166947

pT (γ) > 20 GeV

pT (b) > 45 GeV (pT (b-jet) > 30 GeV)

|η| < 2.5

166849 99.9 129466 77.5 114372 68.5

∆R(γ, γ) > 0.4 166849 100 128989 99.6 113561 99.3

122.7 GeV < M(γ, γ) < 128.3 GeV 3461 2.07 2870 2.23 3585 3.16

∆R(b, b) > 0.4 3461 100 2868 99.9 3585 100

105.0 GeV < M(b, b) < 145.0 GeV 468 13.5 366 12.8 332 9.26

∆R(γ, γ) < 2.0 34 10.1

min(∆R(γ, b)) > 1.0 27 79.4

Table 1: Cutflow diagram for hard process, hard process + ISR and hard process + showering.

Note that for the latter the cut on pT (b) > 45 GeV replaced by pT (b-jet) > 30 GeV. Numbers

of events are numbers as expected at a luminosity of L = 3000fb−1. The bold numbers are the

event reconstruction efficiencies for respectively signal and background. The cuts in equation

24 are only calculated for the hard process + showering.

and indeed if we look at the last two rows of table 2 one can see that the S/√B-ratio has

improved significantly with respect to the S/√B-ratio with only the cuts of equation 23

applied. Reference [1] mentions that the cuts of equation 24 roughly optimize the S/√B-

ratio. To see if this is still the case we let the value on which the cut is made variate slightly.

We look at cuts of ∆R(γ, γ) < 1.8, ∆R(γ, γ) < 2.0 and ∆R(γ, γ) < 2.2 and to cuts of

24

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Used cutsS/

√B

Hard process

+ISR+FSR

+hadronisa-

tion

generated bbγγ-events 0,334 0,334

pT (γ) > 20 GeV

pT (b) > 45 GeV (pT (b-jet) > 30 GeV)

|η| < 2.5

0,124 0,163

∆R(γ, γ) > 0.4 0,124 0,164

122.7 GeV < M(γ, γ) < 128.3 GeV 0,858 0,853

∆R(b, b) > 0.4 0,858 0,853

105.0 GeV < M(b, b) < 145.0 GeV 2,33 1,92

∆R(γ, γ) < 2.0 4,29

min(∆R(γ, b)) > 1.0 4,22

Table 2: Signal to square root of background ratio S/√B for hard process and hard process +

showering. Note that for the latter the cut on pT (b) > 45 GeV replaced by pT (b-jet) > 30 GeV.

min(∆R(γ, b)) > 0.8, min(∆R(γ, b)) > 1.0 and min(∆R(γ, b)) > 1.2. Numbers of events are

displayed in table 3. As we can see the higher values of both cuts reduces the S/√B-ratio,

while the lower values increase the S/√B-ratio.

Cuts Signal Background S/√B

∆R(γ, γ) < 1.8

min(∆R(γ, b)) > 0.8 20 19 4,69

min(∆R(γ, b)) > 1.0 19 18 4,52

min(∆R(γ, b)) > 1.2 18 17 4,40

∆R(γ, γ) < 2.0

min(∆R(γ, b)) > 0.8 23 28 4,35

min(∆R(γ, b)) > 1.0 22 27 4,22

min(∆R(γ, b)) > 1.2 20 25 4,11

∆R(γ, γ) < 2.2

min(∆R(γ, b)) > 0.8 25 38 4,06

min(∆R(γ, b)) > 1.0 24 35 4,00

min(∆R(γ, b)) > 1.2 22 32 3,94

Table 3: Number of events for signal and background after including ISR, FSR and hadroni-

sation as expected at a luminosity of L = 3000fb−1, and signal to square root of background

ratio for variations on the cuts of equation 24. The cuts of equation 23 are applied as well.

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6 Conclusion & Discussion

In the theory section we rederived the Higgs mechanism for the U(1) unitary group and got the

terms describing the Higgs mass, the triple Higgs coupling and the quadruple Higgs coupling.

We calculated the strength of the triple Higgs coupling to be λ3H ≈ 190 GeV. We selected

the channel gg → HH → bbγγ, because H → bb has the highest branching ratio, and H → γγ

has a very clean signal. In this channel often after applying all cuts we are left with about

20 signal events (3). This is a very optimistic result, because we did not include detector

inefficiencies in the present study. It might be interesting to look at the HH → bbbb- channel,

since the branching ratio would then be 0.5772 = 0.333 instead of 0.577 ∗ 0.00228 = 0.00132,

but that requires further, more realistic research on jet reconstruction.

When we looked at figures 25 and 27 we saw that the distributions of ∆R(γ, γ) for both

signal and background don’t change very much after including ISR or after including ISR, FSR

and hadronisation, though after including the latter the distribution is somewhat smeared due

to energy losses in the reconstruction. In table 2 we saw that the S/√B-ratio is significantly

enhanced after applying the cuts of equation 23. ISR, FSR and hadronisation decrease the

S/√B-ratio compared to the hard process only. But when we then applied the cut ∆R(γ, γ) <

2.0 we saw that the S/√B-ratio is significantly enhanced to a value of S/

√B = 4.29.

The second distribution we investigated was min(∆R(γ, b)). The distributions for this

variable are displayed in figures 26 and 28) After including ISR the background still looked

the same, but the signal was slightly shifted towards lower values. Therefore the signal and

background distributions with ISR included overlap more than the distributions without ISR.

But when we included ISR, FSR and hadronisation we saw that the signal was flattened at

the top. This is due to energy losses in the event reconstruction. When we compared the

pseudorapidity the transverse momentum, the angle and the CME (as described in section 3)

for the truth Higgs bosons and the reconstructed Higgs bosons nd we concluded that energy

losses occur after reconstruction. Also for the background distribution in figure 28 we saw

that the distribution with showering included resembled the distribution without showering,

except for the region min(∆R(γ, b)) < 0.5. There we saw a big decrease in events. This can

be explained by the fact that we defined isolated photons as photons having no jet activities

within ∆R(γ, jet) < 0.4. Therefore also the opening between the photon and the b− jet has

to be larger than ∆R = 0.4. So it is not the decrease in number of events that is strange, but

the large peak near ∆R = 0. This peak is probably caused by either some internal bug in the

selection procedure or some unknown misreconstruction. We checked the truth background

and the distribution should look like the one for the hard process, only a little bit smeared

due to ISR and FSR. The proposed cut on min(∆R(γ, b)), however, includes this region as

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well, so this region will be cut away. Further research should point out where this peak is

coming from and if it affects the distribution in the interesting region. Looking at table 2 we

see that the S/√B-ratio slightly decreases with respect to the S/

√B-ratio of only the cut on

∆R(γ, jet).

Reference [1] mentions that the cuts in equation 24 roughly optimize the value of S/√B

while retaining a significant portion of the signal, but when we varied the values of the two

cuts we saw (table 3) that the S/√B-ratio enhances for cuts on lower values. The best value

was obtained for cuts on ∆R(γ, γ) < 1.8 and min(∆R(γ, b)) > 0.8. Note at this point that

we varied the variables only to see if different values would enhance the S/√B-ratio. Also we

saw in table 2 that the cut on the two photons invariant mass M(γ, γ) and the invariant mass

of the b-quark (jet) and the b-quark (jet) M(b, b) enhances the value of S/√B best of all cuts

mentioned in equation 23. Again, this study was made on the MC hard process with ISR,

FSR and hadronisation included. To make an even more realistic study one should include

detector effects and misreconstructed reducible backgrounds. Therefore we would suggest to

further research these cuts find values that optimize the value of S/√B best with the detector

effects and reducible backgrounds included.

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