# Electroweak experimental constraints on the Higgs boson mass

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## The bosons and the fundamental interactions

The bosons are responsible of the SM interactions. We can distinguish three kinds of bosons:
• the photon (γ): this is the well known light’s particle. It is the mediator of the electromagnetic interaction. The photon is massless and stable there-fore the electromagnetic interaction has an infinite range. The photon can interact with all the particles carrying an electric charge.
• the W ± and Z bosons: these are the mediators of the weak interaction between particles of different flavors. The range of the weak force is finite since the W and Z bosons have a mass and a finite lifetime. They couple to all particles which carry a weak isospin charge.
• the gluons: there are 8 gluons differing by their colour charge. These parti-cles are the mediators of the strong interaction. One of the most important properties of QCD is the confinement  which forbids the presence of free coloured particles. Therefore the gluons have a small lifetime as free states, although they are massless, and the range of the strong interaction is finite. The gluons can interact with all coloured particles.
Table 1.3 summarises the fundamental interactions along with the correspond-ing bosons and some of their properties. The gravitational interaction is 34 orders of magnitude weaker than the weak interaction and can be totally neglected in the range of energies relevant for particle physics.

### Chiral Yang-Mills theory

The SM is a chiral Yang-Mills theory based on the group of symmetry SU(2)L × U(1)Y × SU(3)c. In a Yang-Mills theory, the Lagrangian is invariant under the symmetry group independently in each space-time point. This is the so-called local gauge symmetry. This symmetry provides an accurate description of the physical interactions. The Dirac action for massless particles can be written as follows: Z SDirac[Ψ, Ψ] = dV Ψ∂ Ψ where ∂ = iγ µ ∂µ and Ψ = Ψ†γ0.
To insure the gauge symmetry, one should replace the normal derivative,∂ , by the covariant derivative, D = iγ µ (∂µ + Aµ ). Aµ is a member of a base in the Lie algebra of the corresponding symmetry group and will represent the bosonic fields. The group SU(2)L ×U(1)Y , of dimension four, will generate the W 1,2,3 and the B bosons. With an appropriate change of base, these bosons will mix to give the physical W ± , Z bosons and the photon. The group SU(3)c is the generator of the 8 gluons.
A group is an abstract object following some mathematical rules. One needs to find a concrete mathematical object, describing the group, to be able to do useful calculation for physics. This is called the representation of the group. The choice of the representation is mandatory as it will dictates the interaction form, as explained in what follows. The actual representation of Aµ will be noted ρ˜(Aµ ) and thus the Dirac action can be rewritten as: SDirac[Ψ, A] = dV Ψ(iγ µ (∂µ + ρ˜(Aµ )))Ψ Ψ represents the fermionic fields that are elements of the following Hilbert space: HP ⊗ HI. HI is a space of representation of the interaction symmetry group. HP is a space of representation of the Poincar´e group, e.g. the spinioral representation. The spinorial representation was introduced by Dirac to represent the fermions. The spinor four components can be viewed as the two spins of each of the fermions and their corresponding anti-fermions. This representation can be divided into two irreducible representations of dimension two: HP = HL ⊕ HR. In this representation, a particle has two components of respectively left and right chirality. This representation is introduced to account for the observed parity violation in the weak interactions.
The covariant derivative, D, introduces an interaction term (Ψρ˜(Aµ )Ψ) be-tween bosons and fermions, to the Lagrangian. The choice of the representation dictates the action of the boson operator on the fermions and therefore dictates the interaction form. As an example the left-handed fermions are represented as dou-blets in the representation space of the SU(2) group. The defining representation is chosen and the weak bosons operators, in the corresponding Lie algebra, su(2)1, will be 2 × 2 matrices. The right-handed fermions are SU(2) singlets, therefore only a representation of dimension one is needed. The trace of the matrix is chosen to represent the weak boson. Since the trace of su(2) matrices is zero the right-handed fermions do not interact weakly. The weak charge or the weak isospin is defined as the eigen value of the third generator I3 of the su(2) algebra.

#### Our current knowledge of the Higgs boson

The Higgs boson mass is the only unknown parameter of the SM. The determina-tion of the Higgs boson mass is very important to constrain the Higgs sector of the electroweak theory. Several indications on the Higgs boson mass can be derived from some theoretical assumptions as well as from electroweak experimental data and direct searches.

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Theoretical limits on the Higgs boson mass

Even if the Higgs boson mass is not predicted by the SM, different theoretical limits can be derived using different assumptions and approximations. We will not list all these in detail here but we will just focus on the general approach to compute a lower and an upper limit on the Higgs mass. For further details the readers are referred to [22, 23, 24].
As seen in section 1.3.1, the Higgs mass is given by v 2λ . This is given at tree level and quantum corrections will appear at higher orders. These corrections lead to some divergences and a renormalisation procedure is needed. Using the renormalisation group equation, the running λ coupling is given by: ∂λ(Q) 1 12λ 2 + 6λCtop2 − 3Ctop4 + extra terms = ∂ ln Q2 16π2 Q02.
where λ is the self coupling and Ctop the top Yukawa coupling. If the Higgs self-coupling is larger than the top Yukawa coupling, we can neglect the terms with Ctop. The Higgs self-coupling, λ (Q), will diverge with energy as λ 2ln(Q2). If we want the SM to stay valid at all scales (Q → ∞) one should set λ to 0. This implies a trivial non-interacting theory. Even if we chose λ , at the electroweak scale, very small, λ (Q) would eventually blow up at a certain scale given by the Landau pole extracted from: λew λ(Q) = 3λew Q2.

Introduction
1 The Standard Model of particle physics
1.1 The elementary particles and their interactions
1.1.1 The fermions
1.1.2 The bosons and the fundamental interactions
1.2 The Standard Model formalism
1.2.1 Chiral Yang-Mills theory
1.2.2 The Standard Model case
1.3 The Higgs boson
1.3.1 The Higgs mechanism
1.3.2 The fermionic masses
1.4 Summary on the Standard Model
1.5 Our current knowledge of the Higgs boson
1.5.1 Theoretical limits on the Higgs boson mass
1.5.2 Direct limits on the Higgs boson mass
1.5.3 Electroweak experimental constraints on the Higgs boson mass
1.6 Beyond the Standard Model
2 The ATLAS experiment
2.1 The LHC
2.1.1 Proton-proton phenomenology
2.1.2 Physics goals of the general-purpose experiments at the LHC
2.2 The ATLAS detector
2.2.1 The inner detector
2.2.1.1 The pixel detector
2.2.1.2 The silicon microstrip tracker
2.2.1.4 Inner detector environmental services
2.2.2 The calorimeters
2.2.2.1 The electromagnetic calorimeter
2.2.2.3 Forward calorimeter
2.2.3 The muon spectrometer
2.3 The ATLAS trigger system
3 The pixel detector
3.1 General description
3.1.1 Layout
3.1.2 Pixel detector sensor and frontend electronics
3.1.3 Overview of the pixel detector data acquisition system
3.1.4 Overview of the pixel detector slow-control system
3.2 Detector calibration and conditions
3.2.1 Description of calibration scans
3.2.2 Calibration results
3.2.3 Special pixel map
3.3 Noise measurements with the pixel detector
3.3.1 Noise measurements on surface with a partial detector
3.3.3 Noise studies in-situ with the full detector
3.3.3.1 Noise properties and noise stability
3.3.3.2 Update frequency for the online noise mask
3.4 Pixel hit efciency with cosmic rays
3.4.1 Basic selection of tracks and clusters
3.4.2 Underestimation of the number of holes
3.4.3 Inefciency due to cluster-track association
3.4.4 Inefciencies and biases due to problematic pixels
3.4.5 Summary of tracking-dependent efciency and various corrections
3.4.6 Cluster intrinsic efciency
3.4.7 Efciency for various detector components
3.4.8 Systematic uncertainties
3.5 Conclusion
4 The t¯tH(H !b¯b) channel
4.1 The Higgs boson at the LHC
4.2 The light Higgs boson scenario with ATLAS
4.3 The t¯tH(H !b¯b) channel
4.4 Monte Carlo generation and detector simulation
4.5 Previous studies
4.6 Scope of the analysis
4.7 Important denitions
4.8 Event pre-selection
4.8.1 Trigger efciency
4.8.2 Lepton pre-selection
4.8.2.1 Electron pre-selection
4.8.2.2 Muon pre-selection
4.8.3 Jet pre-selection and calibration
4.8.3.1 Overlaps with electrons
4.8.3.2 Jet four-momentum corrections
4.8.4 B-tagging
4.8.5 Results of pre-selection on signal and background
4.9 Reconstruction of the leptonically decayingW boson
4.10 Reconstruction of the hadronically decayingW boson
4.11 Top-quark pair reconstruction
4.11.1 The cut-based approach
4.11.2 The multivariate approaches
4.12 Higgs boson reconstruction
4.13 Signal purity
4.13.1 Maximal achievable purity after preselection
4.13.2 Jet resolution effects
4.13.3 Closer look at jet combinatorics
4.14 Background extraction from data
4.15 Systematic errors
4.15.1 Final signicance estimate
4.16 Conclusion
Conclusion

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