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

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2.6. The unitarity-bootstrap 572.6.1 BCFW at one-loopSince the rational pieces of one-loop amplitudes contain no discontinuities they havethe same kinematic structure as tree-level amplitudes. This lead to the realisationthat they could be calculated, in principle, using BCFW recursion relations [161].In the previous section the proof of BCFW recursion relations was sketched, herewe consider the integral over the shifted amplitude A(z),I = 1 ∮A(z)2πi z . (2.96)CThe contour C is taken around the circle at infinity, if, as was required in the proof,A(z) → 0 as z → ∞, we find the following result,0 = A(0) + ∑poles αRes z=zαA(z)z . (2.97)If, on the other hand there is a contribution from A(z) as z → ∞ (equal to C ∞ ),the relation would beA(0) = C ∞ − ∑poles αRes z=zαA(z)z . (2.98)For one-loop amplitudes many of the properties of the shifted amplitude A(z) changefrom those at tree-level. For example the proof in the previous section required thatthe amplitude contained only simple poles which arose from internal propagatorsgoing on-shell. At one-loop the situation changes and logarithms appear whichpossess discontinuities and are defined on the complex plane with a branch cut. Thedifferences between the complex planes for tree-level and one-loop amplitudes areshown schematically in Fig. 2.4 and Fig. 2.5. The BCFW recursion relations mustbe altered to work at one-loop [161–163]. We begin by assuming once again thatfor a specific shift A(z) → 0 as z → ∞. We then consider the following vanishingintegral which is taken along the circle at ∞ and deformed such that it moves aroundthe branch cuts.0 = 1 ∮A(z)2πi C z . (2.99)The integral, although it still vanishes, is no longer equal to the sum of residuesof the simple poles. We must also include the contribution from the line integrals
2.6. The unitarity-bootstrap 58Figure 2.4: Tree-level amplitudes contain only simple poles in the shift parameterz, these are each associated with an internal propagator going on-shellFigure 2.5: One-loop amplitudes contain both simple poles and discontinuous functions(illustrated by the branch cut) in the shift parameter z.
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Contentsvii1.4.1 Effective Lagrangi
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Contentsix4 One-loop Higgs plus fou
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ContentsxiB.2 Box Integral Function
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1.1. The Standard Model of Particle
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1.1. The Standard Model of Particle
Page 23 and 24: 1.1. The Standard Model of Particle
Page 25 and 26: 1.1. The Standard Model of Particle
Page 27 and 28: 1.2. Electroweak Symmetry Breaking
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Page 39 and 40: 1.4. Effective coupling between glu
Page 41 and 42: 1.4. Effective coupling between glu
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
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: 2.6. The unitarity-bootstrap 56Next
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
Page 113 and 114: 3.3. The rational pieces 98m−1∑
Page 115 and 116: 3.3. The rational pieces 100+m−1
Page 117 and 118: 3.3. The rational pieces 102+R(4 +
Page 119 and 120: 3.3. The rational pieces 104The cal
Page 121 and 122: 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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