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

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1.1. The Standard Model of Particle Physics 9the Lagrangian such as charges and masses are not the true physical quantitiesobserved in nature. At each loop order one must calculate counterterms which areabsorbed into the masses and charges. When combined with divergent integrals thesecounter terms lead to (UV) finite amplitudes at each order in perturbation theory.Infra-red divergences arise from two sources, when l 2 → 0 in a loop amplitude andwhen an external (spin-1) particle becomes soft (E 0 → 0) or collinear to anotherexternal particle 3 . In the second case, an (n+1) parton amplitude is observationallyequivalent to an n parton amplitude. In the first case the loop particle does notaffect the momentum flow of the diagram and the n parton m loop amplitude tendstowards an n parton (m − 1) loop amplitude. Therefore we see that in an IR regionone can combine (n + 1) parton (m − 1)-loop amplitudes with n-parton m loopamplitudes, resulting in IR pole cancellation. This procedure works systematicallyat all loop orders [18–20] and in our case we will need to combine (n + 1) partontree level amplitudes with n-parton one-loop amplitudes.1.1.3 An overview of a hadronic collisionIn this section we provide an extremely brief overview of a particle collision inan hadronic environment. We discuss factorisation of QCD amplitudes, jets andhadronisation.Factorisation and cross sectionsA hadronic collider such as the Tevatron or the LHC collides composite objectsrather than fundamental particles (such as electrons and positrons at LEP). Ingeneral a cross-section for a physical observable can be obtained from the followingformula [7,21],σ(S) = ∑ i,j∫dx 1 dx 2 f i (x 1 , µ 2 )f j (x 2 , µ 2 )ˆσ ij (ŝ = x 1 x 2 S, α S (µ 2 ), Q2µ 2 ) (1.22)3 Quark pairs can also produce collinear singularities through qq → g.
1.1. The Standard Model of Particle Physics 10and explaining the various terms in this formula is the goal of this section. Calculatingquantities in QCD is considerably more difficult than those in QED, primarilybecause we do not observe isolated coloured particles, but instead we observe colourlessbound states (hadrons). At a certain scale (≈ Λ QCD ∼ 1 GeV) hadronisationoccurs resulting in the varied spectrum of baryons and mesons observed in detectors.The problem is exacerbated by the fact that we initially collide hadrons! How canwe make predictions using perturbation theory when we are colliding bound states?Fortunately, the situation is not as bad as may be first thought. Factorisationallows us to split up the various problems associated with the different physicalscales in the problem. This factorisation is apparent in eq. (1.22). The scatteringwhich occurs at high energies (hard scale) is calculated using perturbation theory(ˆσ), the scale µ represents the factorisation scale below which interactions are absorbedinto f i , the parton-distribution functions (PDFs). The PDFs incorporateboth perturbative and non-perturbative physics and can be thought of as the probabilityof extracting a parton of type i with a momentum fraction x 1 of the totalproton momentum. PDF’s are calculated from both experimental data and theoreticalpredictions [22–26] and as result are quoted in terms of the order of perturbationtheory with which they are matched to. Currently Next-to-Next-to Leading Order(NNLO) PDF’s are available, in this thesis we use the (LO or NLO) MSTW08 PDFsets [22].The strategy to generate a partonic cross section is now clear. One firsts calculatesthe partonic cross section for the hard process of interest. Then to produce across section which can be compared with experiment one must convolve the partoniccross section with the PDFs and integrate over x i , the partonic momentafraction. This deals with the issue of colliding bound states. However the finalstates produced in a hadronic collision are predominately coloured objects. Theissues associated with hadronisation are considered in the next section.
Page 7 and 8: Contentsvii1.4.1 Effective Lagrangi
Page 9 and 10: Contentsix4 One-loop Higgs plus fou
Page 11: ContentsxiB.2 Box Integral Function
Page 18 and 19: 1.1. The Standard Model of Particle
Page 20: 1.1. The Standard Model of Particle
Page 23: 1.1. The Standard Model of Particle
Page 27 and 28: 1.2. Electroweak Symmetry Breaking
Page 33 and 34: 1.3. Higgs searches at colliders 18
Page 39 and 40: 1.4. Effective coupling between glu
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Page 43 and 44: 1.5. Higgs plus two jets: Its pheno
Page 45 and 46: 1.5. Higgs plus two jets: Its pheno
Page 47 and 48: 2.1. Unitarity 32of two tree level
Page 49 and 50: 2.2. Quadruple cuts 34the rational
Page 51 and 52: 2.2. Quadruple cuts 360000000111111
Page 53 and 54: 2.2. Quadruple cuts 38We now must s
Page 55 and 56: 2.3. Forde’s Laurent expansion me
Page 61 and 62: 2.4. Spinor integration 46Here we h
Page 63 and 64: 2.4. Spinor integration 48The expli
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Page 67 and 68: 2.5. MHV rules and BCFW recursion r
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Page 71 and 72: 2.6. The unitarity-bootstrap 56Next
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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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