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

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3.4. Cross Checks and Limits 1133.4.5 Soft limit of A (1)4 (φ, 1− , 2 + , 3 − , 4 + )The final test is to take the limit as the φ momentum becomes soft, this limit occurswhen p φ → 0 such that m 2 φ → 0. Our naive expectation is that in this limit, the φfield is essentially constant so thatCφtrG SD µν G µ,νSD → trG SD µνG µ,νSD . (3.185)In other words, the amplitude should collapse onto the gluon-only amplitude. Inref [106], it was postulated that the amplitude should factorise in following form,A (1)n (φ, n − g − , n + g + ) p φ→0→ n − A (1)n (n − g − , n + g + ), (3.186)whileA (1)n (φ† , n − g − , n + g + ) p† φ →0→ n + A (1)n (n −g − , n + g + ). (3.187)We first consider the cut constructible contributions. These factorise onto thefour gluon amplitude in rather trivial manner since in our construction we separatedgluon-only like diagrams and those which require a non-vanishing φ-momentum. Inthe soft limit, the one and two mass easy box and triangle functions have smoothlimits so that,( µ2−m 2 φ) ǫ pφ →0→ 0, (3.188)( ) µ2 ǫp φ →0→ 0. (3.189)−s φiFurthermore, in the soft limit the L k functions become the massless bubble functions,L k (s 234 , s 23 ) = Bub(s 234) − Bub(s 23 ) p φ →0→(s 234 − s 23 ) kAltogether, we find that( )(−1) k µ2 ǫ. (3.190)s k 23ǫ(1 − 2ǫ) −s 23C 4 (φ, 1 − , 2 + , 3 − , 4 + ) p φ→0→ 2C 4 (1 − , 2 + , 3 − , 4 + ), (3.191)where C 4 (1 − , 2 + , 3 − , 4 + ) is the cut-constructible pieces of the four-gluon amplitude.This confirms that the cut-constructible terms of the amplitude do follow the naivefactorisation of eq. (3.186)
3.5. Summary 114The rational terms of eqs. (4.45) and (3.117), are each apparently singular inthis limit. However, careful combination reveals the soft behaviour,ˆR 4 (φ, 1 − , 2 + , 3 − , 4 + ) + CR 4 (φ, 1 − , 2 + , 3 − , 4 + ) p φ→0→N P3 A(0) (1 − , 2 + , 3 − , 4 + ).(3.192)This is similar to the soft limit found in ref. [108,205] for the MHV amplitudes withadjacent negative helicities, but, as anticipated in ref. [106], is not consistent withthe naive limit of eq. (3.186).3.5 SummaryIn this chapter we have investigated the φ-MHV amplitude with general helicities.Detailed descriptions of the unitarity method used to generate the cut-constructiblepieces for all n have been given. The rational pieces have also been studied, howeverto limit the number and complexity of the equations we have focused on thefour-gluon amplitude for the overlap and recursive pieces. We have performed severalchecks on our results, including soft Higgs and collinear checks. In the nextchapter we will use the methods described in this chapter to generate the φ-NMHVamplitude.
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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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1.2. Electroweak Symmetry Breaking
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1.3. Higgs searches at colliders 18
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1.4. Effective coupling between glu
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1.4. Effective coupling between glu
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1.5. Higgs plus two jets: Its pheno
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1.5. Higgs plus two jets: Its pheno
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2.1. Unitarity 32of two tree level
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2.2. Quadruple cuts 34the rational
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2.2. Quadruple cuts 360000000111111
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2.2. Quadruple cuts 38We now must s
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2.3. Forde’s Laurent expansion me
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2.4. Spinor integration 46Here we h
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2.4. Spinor integration 48The expli
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2.4. Spinor integration 50with F z
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2.5. MHV rules and BCFW recursion r
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2.5. MHV rules and BCFW recursion r
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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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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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Page 119 and 120: 3.3. The rational pieces 104The cal
Page 121 and 122: 3.4. Cross Checks and Limits 1063.4
Page 123 and 124: 3.4. Cross Checks and Limits 108The
Page 125 and 126: 3.4. Cross Checks and Limits 110The
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Page 131 and 132: Chapter 4One-loop Higgs plus four-g
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Page 139 and 140: 4.3. Rational Terms 124contains two
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Page 145 and 146: 4.6. Summary 130p µ 2 = (+0.344450
Page 147 and 148: Chapter 5The φqqgg- NMHV amplitude
Page 149 and 150: 5.1. Introduction 134Figure 5.1: Sa
Page 151 and 152: 5.1. Introduction 136H amplitude φ
Page 153 and 154: 5.2. One-loop results 138A L 4 (φ,
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Page 157 and 158: 5.2. One-loop results 142Relation f
Page 159 and 160: 5.2. One-loop results 144+ 1s 123[
Page 161 and 162: 5.4. Summary 146(φ, 1 −¯q , 2 +
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Page 165 and 166: 6.4. Tevatron results 150The W mass
Page 167 and 168: 6.4. Tevatron results 152m H [GeV]
Page 169 and 170: 6.5. LHC results 154Eq. (6.11) and
Page 171 and 172: 6.5. LHC results 156m H [GeV] 120 1
Page 173 and 174: 6.5. LHC results 158Figure 6.3: The
Page 175 and 176: 6.5. LHC results 160of describing T
Page 177 and 178: 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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