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
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.
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
4
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.
5
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)
6
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
7
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
8
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)
9
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
10
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
11
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
12
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).
13
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
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
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
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
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
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
)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.
20
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
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.
22
)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
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
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.
25
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
26
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.
27
Bibliography
[1] U. Baur, T. Plehn, and D. Rainwater, Probing the higgs self-coupling at hadron
colliders using rare decays, Phys. Rev. D 69 (Mar, 2004) 053004.
[2] ATLAS Collaboration Collaboration, G. Aad et. al., Observation of a new particle
in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC,
Phys.Lett. B716 (2012) 1–29, [arXiv:1207.7214].
[3] CMS Collaboration Collaboration, S. Chatrchyan et. al., Observation of a new boson
at a mass of 125 GeV with the CMS experiment at the LHC, Phys.Lett. B716 (2012)
30–61, [arXiv:1207.7235].
[4] M. Thomson, Modern Particle Physics. University of Cambridge, 2013.
[5] LHC Higgs Cross Section Working Group, Sm higgs production cross sections at√s = 14 TeV, 2014.
[6] V. Barger, M. Ishida, and W.-Y. Keung, Total width of 125gev higgs boson, Phys. Rev.
Lett. 108 (Jun, 2012) 261801.
[7] J. Baglio, A. Djouadi, R. Grber, M. Mhlleitner, J. Quevillon, and M. Spira, The
measurement of the higgs self-coupling at the lhc: theoretical status, hep-ph 2013
(2013), no. 4 1–40.
[8] J. Beringer et al. (Particle Data Group), Review of particle physics, Phys. Rev. D 86
(2012) 010001.
[9] U. Baur, T. Plehn, and D. Rainwater, Examining the higgs boson potential at lepton
and hadron colliders: A comparative analysis, Phys. Rev. D 68 (Aug, 2003) 033001.
[10] ATLAS Collaboration, ATLAS detector and physics performance: Technical Design
Report, 2. CERN, 1999.
28
[11] CMS Collaboration, CMS, the Compact Muon Solenoid : technical proposal. Cern,
1994.
[12] ATLAS Collaboration Collaboration, G. Aad et. al., Measurement of the Higgs
boson mass from the H → γγ and H → ZZ∗ → 4` channels with the ATLAS detector
using 25 fb−1 of pp collision data, arXiv:1406.3827.
[13] M. Cacciari, G. P. Salam, and G. Soyez, The anti- k t jet clustering algorithm, Journal
of High Energy Physics 2008 (2008), no. 04 063.
[14] M. Cacciari, G. P. Salam, and G. Soyez, FastJet User Manual, Eur.Phys.J. C72 (2012)
1896, [arXiv:1111.6097].
29