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

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Chapter 7ConclusionsIn this thesis we have studied the hadronic production of a Higgs boson in associationwith two jets. At hadron colliders the Higgs is produced copiously through gluonfusion, therefore amplitudes containing a Higgs and additional QCD radiation areimportant backgrounds to Higgs search channels such as vector boson fusion. Henceknowledge of these amplitudes at Next-to-Leading Order (NLO) in a perturbativeexpansion in the strong coupling constant is an essential requirement for the LHCand Tevatron. The NLO calculation of Higgs plus two jets has previously beenperformed semi-numerically [104], however to improve the speed of the code analyticcalculations of the amplitudes were desired.In obtaining compact analytic expressions for the various Higgs plus parton helicityamplitudes we used various ideas from the recent advances in on-shell techniques.These techniques use on-shell tree-level amplitudes to construct one-loopamplitudes. Since tree-level amplitudes are sums of Feynman diagrams, gauge cancellationsoccur at the beginning of a calculation rather than at the end. Also thefactorial growth of the number of Feynman diagrams is severely curtailed leading toa polynomial growth in complexity with increasing multiplicities.The fundamental concept in generalised unitarity methods is that of multiplecuts together with the use of complex momenta. Multiple cuts allow the isolation ofspecific coefficients which enter the one-loop basis expansion, resulting in simplificationsfrom the older double-cut analyses. Complex momenta are necessary so that167
Chapter 7. Conclusions 168one can define a non-vanishing three parton amplitude. For real momenta p i ·p j = 0requires that 〈ij〉 = [ij] = 0. For complex momenta however, the conjugate spinorsare independent variables such that either 〈ij〉 = 0 or [ij] = 0.This idea was first applied to four-cuts in Ref. [120] which allowed entire amplitudesin N = 4 SYM to be calculated. In four-dimenions a quadruple cut freezesthe loop momenta, such that extracting the coefficient of a box integral becomes analgebraic operation. These methods have since been extended to include the extractionof triangle coefficients from triple cuts [122]. Here one cannot completely freezethe loop momentum since there are now only three constraints. As such the loopmomenta becomes dependent on a single parameter, cut box diagrams contributeextra propagators in this parameter and therefore enter the Laurent expansion asresidues. The remaining pieces have a polynomial dependence in the parameter and,with suitable definitions, the extraction of the triangle coefficient is algorithmic.Although complex momenta are not strictly necessary for double cuts (sincea three vertex which vanishes for real momenta will correspond to the cut of amassless bubble) ideas from multiple cuts filtered down to the two cut level. TheLaurent expansion method [122] again works in an algorithmic way to extract bubblecoefficients. However, the formulae from this method often are more complicatedthan those obtained from spinor integration [123, 132]. This method reduces theextraction of bubble coefficients to taking residues.When four-dimensional cuts are applied a vital piece of the amplitude is missed.These cut-unconstructible pieces are called rational pieces, since they possess nodiscontinuities in physical invariants. On-shell methods have been developed toobtain these pieces, such as the unitarity bootstrap [124,161–165] , which uses theBCFW recursion relations [140,141]. We used the unitarity bootstrap to obtain aformula for the rational pieces for the φ-MHV amplitude in chapter 3. However, forthe calculation of φ-NMHV amplitudes (chapters 4 and 5), we found it easiest towork directly with Feynman diagrams.Upon completing the analytic calculation of Higgs plus four parton amplitudesthese were implemented into MCFM (chapter 6). This program, which is pub-
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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
Page 131 and 132: Chapter 4One-loop Higgs plus four-g
Page 133 and 134: 4.2. Cut-Constructible Contribution
Page 139 and 140: 4.3. Rational Terms 124contains two
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
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Page 185 and 186: Appendix ASpinor Helicity Formalism
Page 187 and 188: A.1. Spinor Helicity Formalism nota
Page 189 and 190: A.3. Pure QCD amplitudes 174Further
Page 191 and 192: Appendix BOne-loop Basis IntegralsI
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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