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2.4. Spinor integration 47+ 2π[˜λη] (1− δ(〈λ|P |η])g(λ) +〈λ|P |˜λ]〈λ|P |η]k∑j=1)δ(〈λB j 〉)g(λ)〈λB j 〉 . (2.56)That is we sum over the poles coming from g(λ) and the denominator 〈λ|P |λ]〈λ|P |η].At first glance one might expect to see a piece proportional to δ(〈λ|P |λ]), howeverone cannot satisfy the vanishing of the 〈λ| and the conjugation relation simultaneously,so there is no pole here. Upon integration over λ the first term vanishes suchthat the integral is localised by the remaining δ functions. This allows to write thespinor integral of our function g(λ) as,∫P 2I = 〈λ, dλ〉[˜λ, d˜λ]〈λ|P |λ] 2g(λ)= − 1k∑P 2g(λ P) +j=1[B j η]〈B j |P |B j ]〈B j |P |η]∏ ki=1 〈B jA i 〉∏l≠j 〈B jB l 〉(2.57)where |λ P 〉 = |P |η]. Knowledge of the integral of the function g(λ) allows us totrivially determine the double cut of a bubble integral, g(λ) = 1,∆I 2 = −1. (2.58)We can also investigate the cut of a three mass triangle,∫∆I 3 = d 4 lδ(l 2 )δ(l − K 1 ) 2 1(l + K 3 ) 2∫ ∞∫= t dt 〈λ, dλ〉[˜λ, d˜λ] δ(K2 1 − t〈λ|K 1 |˜λ])0K3 2 + t〈λ|K 3|˜λ]∫K12 〈λ|K 1 |λ]= 〈λ, dλ〉[˜λ, d˜λ]〈λ|K 1 |˜λ] 2 K3 2〈λ|K 1|˜λ] + K∫1 2〈λ|K 3|˜λ]1= 〈λ, dλ〉[˜λ, d˜λ]〈λ|K 1 |˜λ]〈λ|Q|˜λ] , (2.59)where Q µ = K2 3K µ K1 2 1 + Kµ 3 . We can further simplify this by introducing a Feynmanparameter x which combines the two denominators at the cost of one extraintegration,∫11〈λ|K 1 |˜λ]〈λ|Q|˜λ] = 1dx(2.60)〈λ|R|˜λ] 2with R = (1 − x)K + xQ. As a result of this transformation the integral over λ isequal to that of the scalar bubble which leaves only the x integration,∆I 3= −0∫ 10dx 1 R2. (2.61)
2.4. Spinor integration 48The explicit value of this integral is not of interest here (it being dependent onwhether the external momenta Ki 2 are positive or negative), the point of interest isthat it integrates to a logarithmic function. In fact, all basis integrals other thanthe scalar bubble integrate in the double cut regime to a logarithmic function of thekinematics. This is a very useful observation, since if one can separate the integralinto pieces which integrate to logarithmic funcitons, and those that do not, thenthe task of extracting the coefficient of the scalar bubble becomes simpler. It wasshown in [123] that one can always separate these contributions and determine thecoefficient of the scalar bubble using the holomorphic anomaly eq. (2.56). Since theoriginal paper, this approach has been extended to D dimensions [133–135], andclosed forms for all basis integrals appear in the double cuts have been provided[136,137].In this thesis we used a more recent variant of spinor integration [132], which usesStokes’ Theorem rather than the holomorphic anomaly to perform the integrationsin λ. This is described in the following section.2.4.2 Spinor integration via Stokes’ TheoremIn this section we describe the application of spinor integration via Stokes’ Theorem[132]. Firstly of course, we start with the double cut measure,∫∫ ∞d 4 l 1 δ(l 2 1)δ((l 1 − P) 2 ) = t dt δ(t − P ) 2 ∫ 〈l dl〉[ldl]〈l|P |l] 〈l|P |l]0(2.62)Here as in eq. (2.49) we have rescaled l 1 (the loop momenta appearing in the treeproducts) as l µ 1 = t〈l|γµ |l]/2, which is equivalent to the following spinor shifts,|l 1 〉 = √ t|l〉 and |l 1 ] = √ t|l]. (2.63)At first glance there appears in eq. (2.62) an inverse factor of 〈l|P |l] relative toeq. (2.49). This however is expected, since in eq. (2.49) the second δ function(which is of the form δ(f(t)) has not been applied, whereas in the above equationthe δ function is of the form δ(t − a).To proceed one notices that with two free parameters one can write the loopmomentum in terms of two vectors p µ and η µ , such that the sum of p and η is equal
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Contentsvii1.4.1 Effective Lagrangi
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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 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: 2.4. Spinor integration 46Here we h
Page 65 and 66: 2.4. Spinor integration 50with F z
Page 67 and 68: 2.5. MHV rules and BCFW recursion r
Page 69 and 70: 2.5. MHV rules and BCFW recursion r
Page 71 and 72: 2.6. The unitarity-bootstrap 56Next
Page 73 and 74: 2.6. The unitarity-bootstrap 58Figu
Page 75 and 76: 2.6. The unitarity-bootstrap 60of o
Page 77 and 78: 2.6. The unitarity-bootstrap 62rati
Page 79 and 80: 2.6. The unitarity-bootstrap 64Afte
Page 81 and 82: 2.8. Recent progress: One-loop auto
Page 83 and 84: 2.9. Summary 68cluding the recent c
Page 85 and 86: 3.2. The cut-constructible parts 70
Page 87 and 88: 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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