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4.3. Rational Terms 123and{C (2) 2s124 〈24〉〈34〉 2 [41] 22 = −L 3 (s 124 , s 12 )3[42]+ 〈34〉[41] (3s 124〈34〉[41] + 〈24〉〈3|p φ |1][42])L3[42] 2 2 (s 124 , s 12 )(2s124 〈34〉 2 [41] 2+− 〈24〉〈3|p )φ|1] 2L〈24〉[42] 3 1 (s 124 , s 12 )3s 124 [42]+ 〈3|p φ|1] (4s 124 〈34〉[41] + 〈3|p φ |1](2s 14 + s 24 ))Ls 124 〈24〉[42] 3 0 (s 124 , s 12 )− 2s 123〈23〉〈34〉 2 [31] 23[32]+ 〈23〉〈4|p φ|1] 2L 1 (s 123 , s 12 )3s 123 [32]L 3 (s 123 , s 12 ) + 〈23〉〈34〉[31]〈4|p φ|1]L 2 (s 123 , s 12 )3[32]} { }− (2 ↔ 4) . (4.29)In the above formulae (and those following) we stress that the symmetrising actionapplies to the entire formula, and also acts on the kinematic invariants of the basisfunctions. We see that C 2 (φ, 1 + , 2 − , 3 − , 4 − ) vanishes in the soft Higgs limit p φ → 0.4.2.4 The Cut-Completion termsThe basis functions L 3 (s, t) and L 2 (s, t) are singular as s → t. Since this is anunphysical limit one expects to find some cut-predictable rational pieces which ensurethe correct behaviour of the amplitude as these quantities approach each other.These rational pieces are called the cut-completion terms and are obtained by makingthe following replacements in (4.29)(L 3 (s, t) → ˆL 1 13 (s, t) = L 3 (s, t) −2(s − t) 2 s + 1 ),tL 2 (s, t) → ˆL 1 12 (s, t) = L 2 (s, t) −2(s − t)(s + 1 ),tL 1 (s, t) → ˆL 1 (s, t) = L 1 (s, t),L 0 (s, t) → ˆL 0 (s, t) = L 0 (s, t). (4.30)4.3 Rational TermsWe now turn our attention to the calculation of the remaining rational part of theamplitude. In general the cut-unpredictable rational part of φ plus gluon amplitudes
4.3. Rational Terms 124contains two types of pieces, a homogeneous piece, which is insensitive to the numberof active flavours and a piece proportional to (1 − N f /N c ),(R 4 (φ, 1 + , 2 − , 3 − , 4 − ) = R4(φ, h 1 + , 2 − , 3 − , 4 − ) + 1 − N )fR N P4 (φ, 1 + , 2 − , 3 − , 4 − ).N c(4.31)The homogeneous term R h 4(φ, 1 + , 2 − , 3 − , 4 − ) can be simply calculated using theBCFW recursion relations [140,141],R h 4(φ, 1 + , 2 − , 3 − , 4 − ) = 2A (0) (φ, 1 + , 2 − , 3 − , 4 − ). (4.32)This contribution cancels against a similar homogeneous term for the φ † amplitudewhen combining the φ and φ † amplitudes to form the Higgs amplitude.The N P piece allows the propagation of quarks in the loop, and can be completelyreconstructed by considering only the fermion loop contribution. Furthermore, onecan extract the φ contribution to R N P4 by considering the full Higgs amplitude and removingthe fully rational φ † contribution calculated in [106]. Since there is no directHqq coupling in the effective theory, the most complicated structure is a secondranktensor box configuration. Of the 739 diagrams contributing to the Hggggamplitude 1 , only 136 contain fermion loops and are straightforward to evaluate.After subtracting the cut-completion and homogeneous rational terms from theexplicit Feynman diagram calculation the following rational pieces remain.{ (R N P14 (H, 1 + , 2 − , 3 − , 4 − 〈23〉〈34〉〈4|pH |1][31]) =− 〈3|p H|1] 22 3s 123 〈12〉[21][32] s 124 [42] 2+ 〈24〉〈34〉〈3|p H|1][41]− [12]2 〈23〉 2− 〈24〉(s 23s 24 + s 23 s 34 + s 24 s 34 )3s 124 s 12 [42] s 14 [42] 2 3〈12〉〈14〉[23][34][42]+ 〈2|p H|1]〈4|p H |1]3s 234 [23][34]− 2[12]〈23〉[31]23[23] 2 [41][34])}+{(2 ↔ 4)}. (4.33)The last line in the above equation is the one-loop rational expression for the φ †contribution [106]. We can thus define the rational terms for the φ contribution.{ (R N P14 (φ, 1 + , 2 − , 3 − , 4 − 〈23〉〈34〉〈4|pH |1][31]) =− 〈3|p H|1] 22 3s 123 〈12〉[21][32] s 124 [42] 21 Feynman diagrams were generated with the aid of QGRAF [207].
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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
Page 87 and 88: eplacemen3.2. The cut-constructible
Page 89 and 90: 3.2. The cut-constructible parts 74
Page 113 and 114: 3.3. The rational pieces 98m−1∑
Page 115 and 116: 3.3. The rational pieces 100+m−1
Page 117 and 118: 3.3. The rational pieces 102+R(4 +
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
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
Page 133 and 134: 4.2. Cut-Constructible Contribution
Page 135 and 136: 4.2. Cut-Constructible Contribution
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Page 141 and 142: 4.4. Higgs plus four gluon amplitud
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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,
Page 155 and 156: 5.2. One-loop results 140with K 1 =
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.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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