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The Standard Model and beyond
The Glashow-Salam-Weinberg (GSW) model of the electroweak interaction was pro-posed by Glashow [1], Weinberg [2] and Salam [3] for leptons and extended to the hadronic degrees of freedom by Glashow, Iliopoulos and Maiani [38]. The GSW model is a Yang-Mills theory [39] based on the symmetry group SU (2)L ×U (1)Y . It describes the electromagnetic and weak interactions of the known 6 leptons and 6 quarks. The electromagnetic interaction is mediated by a massless gauge boson, the photon (γ). The short-range weak interaction is carried by 2 massive gauge bosons, Z and W . The strong interaction, mediated by the massless gluon, is also a Yang-Mills the-ory based on the gauge group SU (3)C . This is known as Quantum chromodynamics (abbreviated as QCD) [4, 5, 6, 7]. The Standard Model of particle physics is just a trivial combination of GSW model and QCD. The particle content of the SM is listed in Table. 1.1. There is an additional scalar field called the Higgs boson (H), the only remnant of the spontaneous symmetry breaking (SSB) mechanism invented by Brout, Englert, Guralnik, Hagen, Higgs and Kibble [40, 41, 42, 43, 44]. The SSB mechanism is responsible for explaining the mass spectrum of the SM.
To date, almost all experimental tests of the three forces described by the Standard Model agree with its predictions [8, 9, 45]. The measurements of MW and MZ together with the fact that their relation MW2 = MZ2 c2W (with c2W ≈ 0.77 defined in Eq. (1.10)) has been experimentally proven imply two things. First, the existence of massive gauge bosons means that the local gauge symmetry is broken. Second, the mass relation indicates that the effective Higgs (be it fundamental or composite) is isospin doublet [45]. Experiments have also confirmed that couplings that are mass-independent like the ones of quarks and leptons to the W ± and Z gauge bosons or triple couplings among electroweak gauge bosons agree with those described by the gauge symmetry [45]. It means that the only sector which remains untested is the mass couplings or in other words the nature of SSB mechanism.
The primary goal of the LHC is to find the scalar Higgs boson and to understand its properties. The main drawback here is that we do not know the value of the Higgs mass which uniquely defines the Higgs profile. The LEP direct searches for the Higgs and precision electroweak measurements lead to the conclusion that 114GeV < MH < 190GeV [9]. The most prominent property of the Higgs is that it couples mainly to heavy particles at tree level. This has two consequences at the LHC: the Higgs production cross section is small and the Higgs decay product is very complicated and usually suffers from huge QCD background. Thus, it is completely understandable that searching for the Higgs is not an easy task, even at the LHC.
We have adopted the Feynman rules of [46, 47] (derived by using L) which differ from the normal Feynman rules (derived by using iL) by a factor i. One can use those Feynman rules to calculate tree-level QCD processes or QED-like processes by keeping in mind that the gluon has only two transverse polarisation components. However, in a general situation where a loop calculation is involved one needs to quantize the classical Lagrangian (1.1). The covariant quantization following the Faddeev-Popov method [48] introduces unphysical scalar Faddeev-Popov ghosts with additional Feynman rules: −δab
The main difference between QCD and QED is that the gluon couples to itself while the photon does not. In QED, only the transverse photon can couple to the electron hence the unphysical components (longitudinal and scalar polarisations) decouple from the theory and the Faddeev-Popov ghosts do not appear. The same thing hap-pens for the gluon-quark coupling. However, an external transverse gluon can couple to its unphysical states via its triple and quartic self couplings. Those unphysical states, in some situation, can propagate as internal particles without coupling to any quarks and give an unphysical contribution to the final result. In that situation, one has to take into account also the ghost contribution for compensation.
Indeed, there is another way to calculate QCD processes by taking into account only the physical contribution, i.e. only the transverse gluon components involve and no ghosts appear. This is called the axial (non-covariant) gauge [49]. The main differ-ence compared to the above covariant gauge is with the form of the gluon propagator. The covariant propagator includes the unphysical polarisation states via1 gν = ǫ−ǫν+∗ + ǫ+ǫν−∗ − ǫi ǫiν∗, (1.3) i=1 where ǫ± are two unphysical polarisation states and ǫi with i = 1, 2 are the two transverse polarisation states. In the axial gauge,
with n2 = 0 and n.k = 0, which includes only the transverse polarisation states. The main drawback of this axial gauge is that the propagator’s numerator becomes very complicated.
The main calculation of this thesis is to compute the one-loop electroweak corrections to the process gg → bbH. Though the triple gluon coupling does appear in various Feynman diagrams, it always couples to a fermion line hence the virtu-ally unphysical polarisation states cannot contribute and the ghosts do not show up. We will therefore use the covariant Feynman rules and take into account only the contribution of the transverse polarisation states of the initial gluons2.
Fermionic gauge sector
Left-handed fermions L of each generation belong to SU (2)L doublets while right-handed fermions R are in SU (2)L singlets. The fermionic gauge Lagrangian is just LF = i Lγ D L + i RγDR, where the sum is assumed over all doublets and singlets of the three generations. Note that in the covariant derivative D acting on right-handed fermions the term involving g is absent since they are SU (2)L singlets. Neutrinos are left-handed in the SM. Fermionic mass terms are forbidden by gauge invariance. They are introduced through the interaction with the scalar Higgs doublet.
One-loop renormalisation
Given the full Lagrangian LGSW above, one proceeds to calculate the cross sec-tion of some physical process. In the framework of perturbative theory this can be done order by order. At tree level, the cross section is a function of a set of input parameters which appear in LGSW . These parameters can be chosen to be O = {e, MW , MZ , MH , MU,Dij } which have to be determined experimentally. There are direct relations between these parameters and physical observables at tree level.
Table of contents :
Introduction
1 The Standard Model and beyond
1.1 QCD
1.2 The Glashow-Salam-Weinberg Model
1.2.1 Gauge sector
1.2.2 Fermionic gauge sector
1.2.3 Higgs sector
1.2.4 Fermionic scalar sector
1.2.5 Quantisation: Gauge-fixing and Ghost Lagrangian
1.2.6 One-loop renormalisation
1.3 Higgs Feynman Rules
1.4 Problems of the Standard Model
1.5 Minimal Supersymmetric Standard Model
1.5.1 The Higgs sector of the MSSM
1.5.2 Higgs couplings to gauge bosons and heavy quarks
2 Standard Model Higgs production at the LHC
2.1 The Large Hadron Collider
2.2 SM Higgs production at the LHC
2.3 Experimental signatures of the SM Higgs
2.4 Summary and outlook
3 Standard Model b¯bH production at the LHC
3.1 Motivation
3.2 General considerations
3.2.1 Leading order considerations
3.2.2 Electroweak Yukawa-type contributions, novel characteristics
3.2.3 Three classes of diagrams and the chiral structure at one-loop
3.3 Renormalisation
3.4 Calculation details
3.4.1 Loop integrals, Gram determinants and phase space integrals
3.4.2 Checks on the results
3.5 Results: MH < 2MW
3.5.1 Input parameters and kinematical cuts
3.5.2 NLO EW correction with λbbH 6= 0
3.5.3 EW correction in the limit of vanishing λbbH
3.6 Summary
4 Landau singularities
4.1 Singularities of complex integrals
4.2 Landau equations for one-loop integrals
4.3 Necessary and sufficient conditions for Landau singularities
4.4 Nature of Landau singularities
4.4.1 Nature of leading Landau singularities
4.4.2 Nature of sub-LLS
4.5 Conditions for leading Landau singularities to terminate
4.6 Special solutions of Landau equations
4.6.1 Infrared and collinear divergences
4.6.2 Double parton scattering singularity
5 SM b¯bH production at the LHC: MH ≥ 2MW
5.1 Motivation
5.2 Landau singularities in gg → b¯bH
5.2.1 Three point function
5.2.2 Four point function
5.2.3 Conditions on external parameters to have LLS
5.3 The width as a regulator of Landau singularities
5.4 Calculation and checks
5.5 Results in the limit of vanishing λbbH
5.5.1 Total cross section
5.5.2 Distributions
5.6 Results at NLO with λbbH 6= 0
5.6.1 Width effect at NLO
5.6.2 NLO corrections with mb 6= 0
5.7 Summary
6 Conclusions
A The helicity amplitude method
A.1 The method
A.2 Transversality and gauge invariance
B Optimization with FORM
B.1 Optimization
B.2 Technical details
B.3 Automation with FORM
C Phase space integral
C.1 2 → 3 phase space integral
C.2 Numerical integration with BASES
D Mathematics
D.1 Logarithms and Powers
D.2 Dilogarithms
D.3 Gamma and Beta functions
D.4 Integrals
E Scalar box integrals with complex masses
E.1 Integral with two opposite lightlike external momenta
E.2 Integral with two adjacent lightlike external momenta
Bibliography
