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

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A.4. φ plus parton amplitudes 175A.4 φ plus parton amplitudesFor the MHV configurations the φ plus parton amplitudes have the same structureas the pure QCD ones (although momentum is now no longer conserved amongstthe partons only).A (0)n (φ, 1+ , . . .,i − , . . .,j − , . . .,n + ) =〈ij〉 4∏ n−1α=1 〈α(α + 1)〉〈n1〉 (A.4.31)A (0)n (φ, 1− q , 2+ , . . .,j − , . . .,n + q ) = 〈1j〉 3 〈nj〉∏ n−1α=1 〈α(α + 1)〉〈n1〉 (A.4.32)A (0)n (φ, 1+ q , . . .,i−, . . Q .j+ Q , . . .,n− q ) = − 〈1i〉〈in〉 2 〈nj〉∏ n−1α=1 〈α(α + 1)〉〈n1〉 (A.4.33)A (0)n (φ, 1− q , . . .,i− Q , . . .j+ Q , . . .,n+ q ) =The all plus and single minus φ amplitudes also vanish,〈1i〉 3 〈jn〉∏ n−1α=1 〈α(α + 1)〉〈n1〉 (A.4.34)A (0)n (φ, 1± , 2 + , . . .,n + ) = 0A (0)n (φ, 1− q , 2+ , . . .,n + q ) = 0(A.4.35)(A.4.36)The φ all-minus amplitude does not vanish however,A (0)n (φ, 1− , . . ., n − ) =We will also need the following φ-NMHV amplitudes,A (0)n (φ, 1+ , 2 − , 3 − , 4 − ) =−(−1) n m 4 φ∏ n−1α=1 [α(α + 1)][n1] (A.4.37)m 4 φ 〈24〉4s 124 〈12〉〈14〉〈2|p φ |3]〈4|p φ |3] + 〈4|p φ |1] 3s 123 〈4|p φ |3][12][23] − 〈2|p φ |1] 3s 134 〈2|p φ |3][14][34] .and the φqq-NMHV amplitude,(A.4.38)A (0)n (φ, 1+ q , 2− , 3 − , 4 −¯q ) =〈24〉 3 m 4 φs 124 〈14〉〈2|p φ |3]〈4|p φ |3] − 〈4|p φ |1] 2〈4|p φ |3][12][23] + 〈2|p φ|1] 2 〈2|p φ |4]s 134 〈2|p φ |3][14][34] .(A.4.39)Finally for the subleading colour amplitudes we will need the following amplitude,A (0)4 (φ, 1− q , 2− , 3 + q , 4− ) = − 〈4|p φ|3] 2[12][23]s 123+ {2 ↔ 4}(A.4.40)
Appendix BOne-loop Basis IntegralsIn this appendix we present the basis integral functions used to construct the variouscontributions to the Higgs plus four parton helicity amplitudes.B.1 Extraction of kinematic factorsIn general a one-loop n point basis integral appearing in the basis expansion usedthroughout this thesis has the following structure,∫d d l 1I n [{P ji }] = ∏(2π) d n−1α=0 (l2 α)(B.1.1)where l i = l − P ji is a combination of a loop momenta and some subset of theexternal momenta. In general we find it convenient to work with the following basisintegrals,I 1m4 (s, t; P 2 ) = 1 st F1m 4 (s, t; P 2 ) (B.1.2)I 2me4 (s, t; P 2 Q 2 ) =I 2mh1(st − P 2 Q 2 ) F2me 4 (s, t; P 2 , Q 2 ) (B.1.3)4 (s, t; P 2 , Q 2 ) = 1 st F2mh 4 (s, t; P 2 , Q 2 ) (B.1.4)()I3 2m 1(s, t) = F 1m3 (s) − F1m 3(s − t)(t) (B.1.5)I 1m3 (s) = 1 s F1m (s) (B.1.6)and in this appendix we give explicit formulae for the F functions.3176
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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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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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Page 147 and 148: Chapter 5The φqqgg- NMHV amplitude
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Page 151 and 152: 5.1. Introduction 136H amplitude φ
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Page 157 and 158: 5.2. One-loop results 142Relation f
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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
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Page 175 and 176: 6.5. LHC results 160of describing T
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Page 181 and 182: 6.6. Summary 166of Higgs masses, 15
Page 183 and 184: Chapter 7. Conclusions 168one can d
Page 185 and 186: Appendix ASpinor Helicity Formalism
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Page 189: A.3. Pure QCD amplitudes 174Further
Page 193 and 194: B.2. Box Integral Functions 178boxe
Page 195 and 196: B.2. Box Integral Functions 180L 2
Page 197 and 198: Bibliography 182[10] R. Feynman, Ma
Page 199 and 200: Bibliography 184[36] M. Bahr et. al
Page 201 and 202: Bibliography 186[63] M. Shifman, A.
Page 203 and 204: Bibliography 188[85] R. V. Harlande
Page 205 and 206: Bibliography 190[104] R. K. Ellis,
Page 207 and 208: Bibliography 192[125] G. Ossola, C.
Page 209 and 210: Bibliography 194[148] R. Boels and
Page 211 and 212: Bibliography 196[170] Z. Bern and A
Page 213 and 214: Bibliography 198anti-t + 2 jets at
Page 215: Bibliography 200[214] C. Anastasiou
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