Hadronic production of a Higgs boson in association with two jets at ...

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1.1. The Standard Model of Particle Physics 7Figure 1.1: A simple one-loop Feynman diagramWe have now described all the pieces needed to construct guage-invariant Lagrangiansand hence build the SM. All that remains to do is to define the particulargauge group in which represents the various theories of nature. As we have shown,QED arises naturally from a U(1) gauge group where g, the coupling of matter tophotons, is given by the electric charge of the fermions. The strong force describedby Quantum Chromodynamics (QCD) was over time shown to be described by anSU(3) non-Abelian gauge theory 2 . The weak force is more subtle. Over time itwas established that the weak force was chiral in nature (i.e. it coupled to particlesdepending on their spin orientation relative to the direction of motion) and that thedesired gauge group to describe the theory was SU(2). The problem of assigning amass to the W and Z vector bosons in a gauge-invariant way resulted in the conceptof electroweak symmetry breaking, which we will discuss in section 1.2. First we reviewa couple of other topics which are relevant to the work performed in this thesis,the regularisation of loop amplitudes and the kinematics of a hadronic collision.1.1.2 Regularisation of UV and IR divergencesIn this section we briefly describe the concepts regarding the regularisation of loopamplitudes in quantum field theories. The need arises for regularisation when onemoves beyond the calculation of tree-level (0-loop) amplitudes. When one considersloop diagrams it is simple to see that one can assign any momentum to a loop particlein the diagram. This results in the need to integrate over all allowed momenta whenone considers a loop diagram. One of the simplest non-trivial loop diagrams is the2 For a nice historical overview of QCD see [15].
1.1. The Standard Model of Particle Physics 8bubble diagram shown in Fig 1.1. Application of the Feynman rules and reductionof intermediate tensor integrals [16] would ultimately lead to the following sort ofterm,∫I2 4D =d 4 l(2π) 4 1(l 2 )(l − P) 2 (1.21)This diagram diverges as l → ∞ (known as UV divergence) and to expose thesingularity structure of the integral we wish to regularise the integral at intermediatestages. By far the most popular method of regularisation is that of dimensionalregularisation, first proposed by ’t Hooft and Veltman [17]. In this approach onealters the number of spacetime dimensions to 4 − 2ǫ. Singularities then revealthemselves as inverse powers of ǫ. This method has numerous advantages, includingmaintaining gauge-invariance and regularising both UV (l → ∞) and IR (l →0) singularities at the same time. This point needs some clarification since thesesingularities arise from different sources, a UV singularity occurs when the powersof l in the numerator dominate as l → ∞. To regularise these divergences wewould wish to define ǫ > 0. Clearly the situation is reversed for IR singularities,where the denominator dominates and we would wish that ǫ < 0. However, inpractical calculations one can define ǫ > 0, reguarlise and renormalise (which willbe explained shortly) the UV singularities and then analytically continue to ǫ < 0,which regulates the remaining IR singularities.How one treats external particles is up to the discretion of the calculator, andseveral schemes exist and are related to each other by predictable quantities. In thisthesis unless stated we will work in the four-dimensional helicity scheme (FDH). Thisallows us to keep external particles strictly in four-dimensions, whilst only the loopmomenta (and the metric) are D-dimensional. The t’Hooft-Veltman scheme [17]defines γ µ in d dimensions with γ 5 = iγ 0 γ 1 γ 2 γ 3 defined such that it anticommuteswith γ µ for µ ∈ {0, 1, 2, 3} and commutes with γ µ for all other µ.Of course one does not expect to predict infinite cross sections, and this certainlyis not what is observed at colliders! Ultimately we wish to remove the singularities inǫ and there are systematic ways of doing this. Ultraviolet divergences can be removedby a process known as renormalisation. Basically the physical quantities written in
Page 7 and 8: Contentsvii1.4.1 Effective Lagrangi
Page 9 and 10: Contentsix4 One-loop Higgs plus fou
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Page 18 and 19: 1.1. The Standard Model of Particle
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Page 47 and 48: 2.1. Unitarity 32of two tree level
Page 49 and 50: 2.2. Quadruple cuts 34the rational
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Page 53 and 54: 2.2. Quadruple cuts 38We now must s
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2.6. The unitarity-bootstrap 56Next
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2.6. The unitarity-bootstrap 58Figu
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2.6. The unitarity-bootstrap 60of o
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2.6. The unitarity-bootstrap 62rati
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2.6. The unitarity-bootstrap 64Afte
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2.8. Recent progress: One-loop auto
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2.9. Summary 68cluding the recent c
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3.2. The cut-constructible parts 70
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eplacemen3.2. The cut-constructible
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3.3. The rational pieces 98m−1∑
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3.3. The rational pieces 100+m−1
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3.3. The rational pieces 102+R(4 +
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3.3. The rational pieces 104The cal
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3.4. Cross Checks and Limits 1063.4
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3.4. Cross Checks and Limits 108The
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3.4. Cross Checks and Limits 110The
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3.4. Cross Checks and Limits 112−
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3.5. Summary 114The rational terms
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Chapter 4One-loop Higgs plus four-g
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4.2. Cut-Constructible Contribution
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4.3. Rational Terms 124contains two
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4.4. Higgs plus four gluon amplitud
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4.4. Higgs plus four gluon amplitud
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4.6. Summary 130p µ 2 = (+0.344450
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Chapter 5The φqqgg- NMHV amplitude
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5.1. Introduction 134Figure 5.1: Sa
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5.1. Introduction 136H amplitude φ
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5.2. One-loop results 138A L 4 (φ,
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5.2. One-loop results 140with K 1 =
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5.2. One-loop results 142Relation f
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5.2. One-loop results 144+ 1s 123[
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5.4. Summary 146(φ, 1 −¯q , 2 +
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6.2. Improvements from the semi-num
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6.4. Tevatron results 150The W mass
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6.4. Tevatron results 152m H [GeV]
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6.5. LHC results 154Eq. (6.11) and
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6.5. LHC results 156m H [GeV] 120 1
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6.5. LHC results 158Figure 6.3: The
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6.5. LHC results 160of describing T
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6.5. LHC results 162Figure 6.6: p T
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6.6. Summary 164at LO and the effec
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6.6. Summary 166of Higgs masses, 15
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Chapter 7. Conclusions 168one can d
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Appendix ASpinor Helicity Formalism
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A.1. Spinor Helicity Formalism nota
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A.3. Pure QCD amplitudes 174Further
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Appendix BOne-loop Basis IntegralsI
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B.2. Box Integral Functions 178boxe
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B.2. Box Integral Functions 180L 2
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Bibliography 182[10] R. Feynman, Ma
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Bibliography 184[36] M. Bahr et. al
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Bibliography 186[63] M. Shifman, A.
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Bibliography 188[85] R. V. Harlande
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Bibliography 190[104] R. K. Ellis,
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Bibliography 192[125] G. Ossola, C.
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Bibliography 194[148] R. Boels and
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Bibliography 196[170] Z. Bern and A
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Bibliography 198anti-t + 2 jets at
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Bibliography 200[214] C. Anastasiou
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