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

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4.2. Cut-Constructible Contributions 119We also find that three of the two-mass hard boxes (Fig. 4.1(d)) have coefficientsrelated to the coefficients of eqs. (4.5), (4.10) and (4.11),C 4;φ|12|3|4 (φ, 1 + , 2 − , 3 − , 4 − ) = s 123s 34s 23 s 12C 4;φ4|1|2|3 (φ, 1 + , 2 − , 3 − , 4 − ), (4.12)C 4;φ|23|4|1 (φ, 1 + , 2 − , 3 − , 4 − ) = s 234s 41s 23 s 34C 4;φ1|2|3|4 (φ, 1 + , 2 − , 3 − , 4 − ), (4.13)C 4;φ|34|1|2 (φ, 1 + , 2 − , 3 − , 4 − ) = s 134s 12s 41 s 34C 4;φ2|3|4|1 (φ, 1 + , 2 − , 3 − , 4 − ). (4.14)The final two-mass hard box coefficient is,C 4;φ|3|4|12 (φ, 1 + , 2 − , 3 − , 4 − ) = s (34 〈3|p φ |1] 42 〈3|p φ |2]〈3|p φ |4][21][41]〈24〉 4 m 4 )φ+〈12〉〈14〉〈2|p φ |3]〈4|p φ |3](4.15)The remaining one-mass box configuration C 4;φ3|4|1|2 is the only one which receivescontributions from N f fermionic loops,( 1C 4;φ3|4|1|2 (φ, 1 + , 2 − , 3 − , 4 − ) = s 41 s 12 C 4;φ|3|4|12 (φ, 1 + , 2 − , 3 − , 4 − )s 124 s 34(− 1 − N ) (f 2〈3|pφ |1] 24N c s 124 [24] − 1 − N ) )f [12][41]〈3|pφ |2]〈3|p φ |4]. (4.16)2 N c s 124 [24] 4Each of the coefficients (4.11), (4.10), (4.15) and (4.16) correctly tends to zero inthe soft Higgs limit (p φ → 0).4.2.2 Triangle Integral CoefficientsAltogether, there are twenty-four triangle topologies, which can be obtained fromcyclic permutations of those shown in Fig. 4.2. The different topologies can be characterisedby the number of off-shell legs. Fig. 4.2(a) has one off-shell leg, Figs. 4.2(b)-(e) have two off-shell legs while for Fig. 4.2(f) all legs are off-shell. We refer to thetriangle integrals with one- and two-off-shell legs as one-mass and two-mass respectively.They have the following form,I3 1m (s) ∝ 1 ( )1 µ2 ǫ, Iǫ 2 3 2m (s, t) ∝ 1 (( )1 µ2 ǫ ( ) µ2 ǫ )− (4.17)s −sǫ 2 (s − t) −s −tand therefore only contribute pole pieces in ǫ to the overall amplitude (as was discussedin detail in chapter 3. In fact, the sole role of these functions is to ensure
4.2. Cut-Constructible Contributions 120211123φφφ4 C 3;φ34|1|2 C 3;φ4|12|3 C 3;φ4|1|23344(a) (b) 2 (c)12 31 123442334φC 3;φ|1|234C 3;φ|123|4C 3;φ|12|34φφ(d) (e) (f)Figure 4.2: The various triangle integral topologies that appear for A (1)4 (φ, 1, 2, 3, 4).From the six topologies we must also include cyclic permutations of the four gluons.(a) has one off-shell leg, (b)-(e) have two off-shell legs while in (f) all legs are off-shell.the correct infrared behaviour by combining with the box pieces to generate thefollowing pole structure,A (1) (φ, 1 + , 2 − , 3 − , 4 − ) = −A (0) (φ, 1 + , 2 − , 3 − , 4 − ) c Γǫ 24∑( µ2i=1−s ii+1) ǫ+ O(ǫ 0 ). (4.18)This relation holds for arbitrary external helicities [107,108,206]. We computed thecoefficients of all one- and two-mass triangles and explicitly verified eq. (4.18). Thenon-trivial relationship between the triangle and box coefficients provides a strongcheck of our calculation. However, since we now wish to obtain compact results forthe four gluon amplitude, we find it more compact to present the final answer ina basis free of one- and two-mass triangles. That is, we choose to expand the boxintegral functions into divergent and finite pieces, combining the divergent pieceswith the one- and two- mass triangles to form (4.18) and giving explicit results forthe finite pieces of the box functions.A new feature in the φ-NMHV amplitudes is the presence of three-mass triangles,shown in Fig. 4.2(f). In previous calculations [106–109, 206] the external gluonhelicities prevented these contributions from occurring.
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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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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∑
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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
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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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