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

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2.8. Recent progress: One-loop automatisation 67Blackhat [126, 178–181] is a program which implements the four-dimensionalmethods described in this chapter. The Laurent expansion method [122] is appliedin four dimensions to determine the coefficients of the cut-constructible pieces. Therational pieces are then calculated using the unitarity-bootstrap approach with multipleshifts [124,161–165]. The program also includes the D-dimensional applicationof the Laurent expansion [172] for an alternate method of generating the rationalterms. This program has been successfully integrated with the Monte Carlo eventgenerator Sherpa [33,37,38], which provides an efficient mechanism for generatingthe real matrix elements and performing the integration over phase space. Blackhathas been successfully applied to the calculation of V +3 jets (at NLO) [126,178–181]where V is massive vector boson.The program Rocket also uses unitarity techniques to numerically generate oneloopamplitudes [182]. Rocket uses a numerical implementation of D-dimensionalunitarity [183–185], which can be applied to massive and massless particles. Thisprogram also uses the QCDLoop package [186] which calculates the scalar basisintegrals. Rocket has also calculated W + 3 jets [187,188] and the results betweenthe two groups are in agreement. Very recently, Rocket has been used with MCFMto compute pp → W + W + + 2j [189].In addition to the unitarity based programs described above the Helac-Phegascollaboration [190] have calculated ttbb production and tt + 2j [191] using the OPPreduction technique [125,192–195]. This reduction algorithm is similar to unitaritytechniques in that it is four-dimensional and solves for coefficients of basis integralsusing on-shell constraints, however the OPP method can be applied equallyto products of amplitudes (as in the unitarity case) or to individual Feynman diagrams.The rational pieces are generated differently however, with a separation intopieces which can be generated recursively and those which can be deduced from thereduction. The OPP method has also been applied to the calculation of tri-bosonproduction [196,197].Efforts have also been made to automate 2 → 4 Feynman diagram calculationsand the GOLEM collaboration [198, 199] has made progress in this direction, in-
2.9. Summary 68cluding the recent calculation of qq → bbbb [177]. This program uses improvedFeynman diagram reduction techniques [200–202] and has been used to calculatethe six photon amplitude [129].2.9 SummaryIn this chapter on-shell methods for calculating one-loop amplitudes have been describedin detail. Several unitarity methods which will be used in later chaptershave been discussed. Inspired by the use of complex momenta generalised unitarityand the BCFW recursion relations have been detailed with examples given ofthe techniques we will apply in the following chapters. Other on-shell techniques,the MHV rules and D-dimensional unitarity have also been included for completeness,although they will play a limited role in the remainder of this thesis. A shortoverview of the recent progress towards automisation has been given.In the next chapter we will use the techniques described in the previous sectionsto calculate the φ-MHV amplitude. We will calculate the cut-constructible piecesfor all multiplicities, the rational pieces for the amplitude A (1)4 (φ, 1 − , 2 + , 3 − , 4 + ) willalso be calculated using the BCFW recursion relations.
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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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1.2. Electroweak Symmetry Breaking
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Page 47 and 48: 2.1. Unitarity 32of two tree level
Page 49 and 50: 2.2. Quadruple cuts 34the rational
Page 51 and 52: 2.2. Quadruple cuts 360000000111111
Page 53 and 54: 2.2. Quadruple cuts 38We now must s
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Page 61 and 62: 2.4. Spinor integration 46Here we h
Page 63 and 64: 2.4. Spinor integration 48The expli
Page 65 and 66: 2.4. Spinor integration 50with F z
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Page 71 and 72: 2.6. The unitarity-bootstrap 56Next
Page 73 and 74: 2.6. The unitarity-bootstrap 58Figu
Page 75 and 76: 2.6. The unitarity-bootstrap 60of o
Page 77 and 78: 2.6. The unitarity-bootstrap 62rati
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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 119 and 120: 3.3. The rational pieces 104The cal
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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
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4.2. Cut-Constructible Contribution
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4.3. Rational Terms 124contains two
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4.4. Higgs plus four gluon amplitud
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4.4. Higgs plus four gluon amplitud
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4.6. Summary 130p µ 2 = (+0.344450
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Chapter 5The φqqgg- NMHV amplitude
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5.1. Introduction 134Figure 5.1: Sa
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5.1. Introduction 136H amplitude φ
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5.2. One-loop results 138A L 4 (φ,
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5.2. One-loop results 140with K 1 =
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5.2. One-loop results 142Relation f
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5.2. One-loop results 144+ 1s 123[
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5.4. Summary 146(φ, 1 −¯q , 2 +
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6.2. Improvements from the semi-num
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6.4. Tevatron results 150The W mass
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6.4. Tevatron results 152m H [GeV]
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6.5. LHC results 154Eq. (6.11) and
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6.5. LHC results 156m H [GeV] 120 1
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6.5. LHC results 158Figure 6.3: The
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6.5. LHC results 160of describing T
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6.5. LHC results 162Figure 6.6: p T
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6.6. Summary 164at LO and the effec
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6.6. Summary 166of Higgs masses, 15
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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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