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

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4.2. Cut-Constructible Contributions 121There are four three-mass triangles, which satisfy,C 3;φ|34|12 (φ, 1 + , 2 − , 3 − , 4 − ) = C 3;φ|12|34 (φ, 1 + , 2 − , 3 − , 4 − ) (4.19)C 3;φ|41|23 (φ, 1 + , 2 − , 3 − , 4 − ) = C 3;φ|23|41 (φ, 1 + , 2 − , 3 − , 4 − ). (4.20)The symmetry under the exchange of gluons with momenta p 2 and p 4 relates theremaining two coefficients,C 3;φ|23|41 (φ, 1 + , 2 − , 3 − , 4 − ) = C 3;φ|12|34 (φ, 1 + , 4 − , 3 − , 2 − ). (4.21)To compute C 3;φ|23|41 we use both Forde’s method [122] and the spinor integrationtechnique [128]. For a given triangle coefficient C 3;K1 |K 2 |K 3(φ, 1 + , 2 − , 3 − , 4 − ) withoff-shell momenta K 1 , K 2 and K 3 , we introduce the following massless projectionvectorsIn terms of these quantities we find,C 3;φ|12|34 (φ, 1 + , 2 − , 3 − , 4 − ) =K ♭µ1 = γ γKµ 1 − K1K 2 µ 2,γ 2 − K1 2K2 2K ♭µ2 = γ γKµ 2 − K2 2Kµ 1,γ 2 − K1K 2 22 √γ ± (K 1 , K 2 ) = K 1 · K 2 ± K 1 · K2 2 − K1K 2 2. 2 (4.22)∑γ=γ ± (p φ ,p 1 +p 2 )m 4 φ−〈K♭ 1 2〉3 〈34〉 32γ(γ + m 2 φ )〈K♭ 11〉〈K13〉〈K ♭ 14〉〈12〉 , ♭which, as expected, correctly vanishes in the soft Higgs limit (p φ → 0).(4.23)4.2.3 Bubble Integral CoefficientsThe non-vanishing bubble topologies for the φ-NMHV amplitude are shown inFig. 4.3. We find that the double-cuts associated with Fig. 4.3(a) contain onlycontributions from boxes and triangles, and therefore the coefficient of log(s 1234 ) iszero. In a similar fashion, the double cuts associated with diagram Fig. 4.3(c) withtwo external gluons with negative helicity emitted from the right hand vertex haveonly box and triangle contributions, so that the coefficients of log(s 23 ) and log(s 34 )are also zero.
4.2. Cut-Constructible Contributions 122φ1φ1φ122 43C 2;φ 4 4 C32;φ4 3 C 2;φ342(a) (b) (c)Figure 4.3: The three bubble integral topologies that appear for A (1)4 (φ, 1, 2, 3, 4).We must also include cyclic permutations of the four gluons.The leading singularity of the bubble integral is O(1/ǫ),( )1 µ2 ǫI 2 (s) ∝. (4.24)(1 − 2ǫ)ǫ −sHowever for the total amplitude there is no overall ǫ pole, and this implies a relationamongst the bubble coefficients such that,4∑(C 2;φk + C 2;φkk+1 ) = 0. (4.25)k=1It is therefore most natural to work with log’s of ratios of kinematic scales (ratherthan log(s/µ 2 )), since the coefficients of individual logarithms must cancel pairwise.To this end, as in the last chapter, we express our result in terms of the followingfunctions,L k (s, t) =log (s/t)(s − t) k . (4.26)Using the Stokes’ theorem method [132], we generated compact analytic expressionsfor the coefficients of each bubble-function, which we also checked numerically withForde’s method [122]. The combination of all double-cuts is given by,(C 2 (φ, 1 + , 2 − , 3 − , 4 − ) = 4 − N ) (fC (1)2 + 1 − N )fC (2)2 (4.27)NN cwith{C (1) 〈24〉〈3|pφ |1] 22 = − L 1 (s 124 , s 12 ) − 〈23〉〈4|p } { }φ|1] 2L 1 (s 123 , s 12 ) − (2 ↔ 4)s 124 [42]s 123 [32](4.28)
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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
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 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
Page 127 and 128: 3.4. Cross Checks and Limits 112−
Page 129 and 130: 3.5. Summary 114The rational terms
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 +
Page 163 and 164: 6.2. Improvements from the semi-num
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
Page 179 and 180: 6.6. Summary 164at LO and the effec
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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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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