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

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2.2. Quadruple cuts 35Here the LHS of eq. (2.6) represents the application of the four on-shell constraintsδ(l 2 i ) to the LHS of eq. (2.5), shown schematically in Fig. 2.1, whilst the RHSrepresents the application of the four cuts to the basis integral. In four-dimensionsthe loop momenta has the same number of components as there are cut propagatorsand as such is frozen by the cuts. Since the integral over the cut phase space merelycontributes a Jacobian, which is identical on both sides of the equation, we canrelate the coefficient of a box (with a given topology P) to the solution of the fourproducts of treesA (0)1 (l 1 , l 2 )A (0)2 (l 2 , l 3 )A (0)3 (l 3 , l 4 )A (0)4 (l 4 , l 1 ) = c 4,{P } , (2.7)where the loop momenta are solved via the on-shell constraints.There is a subtlety, however, regarding the application of four cuts to a one-loopamplitude. There are six classes of box integral which can be classified in terms ofthe number of legs bunched at each corner. Two or more legs at a specific cornerresults in a “mass” since the squared sum of corner momenta is no longer zero. Forfour external (massless) particles it is thus possible to have a zero-mass box. As thenumber of external particles increase so can the number of masses, one can drawa one-mass box, two two-mass boxes 1 , a three-mass and a four-mass box. For allbut the the four-mass box, after the application of the four-cut, one finds a cornercontaining a three-point vertex A (0)3 (1λ 1, 2 λ 2, 3 λ 3). Since this amplitude is Lorentzinvariant it must depend only on squares of the individual momenta p 2 i, or invariantproducts between them p i · p j . For massless momenta p 2 i = 0 so the amplitudecan only depend on invariant products between momenta. However, conservation ofmomenta forces each product to be zero,(p i + p j ) 2 = p 2 k = 0 =⇒ p i · p j = 0, (2.8)which implies A (0)3 (1 λ 1, 2 λ 2, 3 λ 3) = 0. For real momenta this appears to be a majorflaw in the quadruple cut approach. The crucial observation in [120] is that ifone switches to complex momenta this obstacle can be overcome. Specifically, in1 The integral in which the massive corners are opposite is called the easy configuration. Theother configuration in which the masses are adjacent is referred to as the hard configuration
2.2. Quadruple cuts 360000000111111100000001111111000000011111110000000111111100000001111111000000011111110000000111111100000001111111000000011111110000000111111100000001111111000000011111110000000111111100000001111111000000011111110000000111111100000001111111000000011111110000000111111100000001111111000000011111110000000111111100000001111111000000011111110000000111111100000001111111000000011111110000000111111100000001111111000000011111110000000111111100000001111111000000011111110000000111111100000001111111000000011111110000000111111100000001111111000000011111110000000111111100000001111111000000011111110000000111111100000001111111Figure 2.1: The application of four cuts to a one-loop amplitude results in thefactorisation onto the product of four tree amplitudes. Since boxes are the onlyone-loop basis function to contain four propagators each four cut selects a uniquebox function from the basis. In addition for four-dimensional cuts the loop momentais frozen by the cutting procedure.
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 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
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: 2.2. Quadruple cuts 34the rational
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 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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3.2. The cut-constructible parts 86
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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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