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4.5. Numerical Evaluation 129+ 〈23〉〈4|p H|1] 23s 123 [32])}ˆL 1 (s 123 , s 12 ){ }+ (2 ↔ 4) . (4.46)For convenience we have introduced the following combinations of the finite piecesof one-mass (F 1m4F) and two-mass hard (F2mh 4F ) box functions (see Appendix B),W (1)W (2)W (3)= F 1m4F (s 23, s 34 ; s 234 ) + F 2mh4F (s 41, s 234 ; m 2 H , s 23) + F 2mh4F (s 12, s 234 ; s 34 , m 2 H )= F 1m4F (s 14 , s 34 ; s 134 ) + F 2mh4F (s 12 , s 134 ; m 2 H, s 34 ) + F 2mh4F (s 23 , s 134 ; s 14 , m 2 H)= F 1m4F (s 12, s 14 ; s 124 ) + F 2mh4F (s 23, s 124 ; m 2 H , s 14) + F 2mh4F (s 34, s 124 ; s 12 , m 2 H ).In addition, to simplify the coefficients of the three-mass triangle F 3m3 (K2 1 , K2 2 , K2 3 )with three off-shell legs K 2 1, K 2 2, K 2 3 ≠ 0, we use the notation of eq. (4.22). The rationalpart of the Higgs NMHV amplitude is given by eq. (4.33) (which incorporatesthe rational A (1)4 (φ † , 1 + , 2 − , 3 − , 4 − ) amplitude derived in [106]),{(R 4 (H, 1 + , 2 − , 3 − , 4 − ) = 1 − N ) (f 1 〈23〉〈34〉〈4|pH |1][31]N c 2 3s 123 〈12〉[21][32]+ 〈24〉〈34〉〈3|p H|1][41]3s 124 s 12 [42]+ 〈2|p H|1]〈4|p H |1]3s 234 [23][34]− 〈3|p H|1] 2s 124 [42] 2− [12]2 〈23〉 2− 〈24〉(s 23s 24 + s 23 s 34 + s 24 s 34 )s 14 [42] 2 3〈12〉〈14〉[23][34][42])} { }− 2[12]〈23〉[31]2 + (2 ↔ 4) . (4.47)3[23] 2 [41][34]4.5 Numerical EvaluationIn this section we provide numerical values for the helicity amplitudes given in theprevious section at a particular phase space point. To this end, we redefine the finitepart of the Higgs amplitude as:A (1)4 (H, 1 λ 1, 2 λ 2, 3 λ 3, 4 λ 4) = c Γ A (0) (H, 1 λ 1, 2 λ 2, 3 λ 3, 4 λ 4)(− 1 ǫ 24∑( ) −µ2 ǫ(4.48)s i,i+1).+M F,g4 (λ 1 , λ 2 , λ 3 , λ 4 ) + N fN cM F,f4 (λ 1 , λ 2 , λ 3 , λ 4 ) + N sN cM F,s4 (λ 1 , λ 2 , λ 3 , λ 4 )We evaluate the amplitudes at the phase space point used by Ellis et al. [104],i=1p µ Hp µ 1= (−1.00000000000, 0.00000000000, 0.00000000000, 0.00000000000),= (+0.30674037867, −0.17738694693, −0.01664472021, −0.24969277974),
4.6. Summary 130p µ 2 = (+0.34445032281, +0.14635282800, −0.10707762397, +0.29285022975),p µ 3 = (+0.22091667641, +0.08911915938, +0.19733901856, +0.04380941793),p µ 4= (+0.12789262211, −0.05808504045, −0.07361667438, −0.08696686795).(4.49)The results are presented in table 4.1 where we have chosen the renormalisationscale to be µ 2 = m 2 H .2Helicity A (0) M F,g4 M F,f4 M F,s4− − −− -116.526220-18.681775 i -9.540396-0.001010 i -0.176850+0.001010 i 0.176850-0.001010 i+ − −− 10.308088-0.824204 i -10.809925+0.056646 i -0.388288+0.198369 i 0.296783-0.155132 i− − ++ 20.511457-0.888525 i -10.991033+0.320009 i 0.268501-0.068414 i 0.066595-0.015451 i− + −+ 4.683784+4.242678 i -10.332320+0.149216 i 0.028668-0.066437 i 0.166800+0.038844 iTable 4.1: Numerical values for the finite parts of the Higgs + 4 gluon helicityamplitudes at the phase space point given in eq. (6.2).4.6 SummaryIn this chapter we have calculated the last (analytically) unknown building block ofthe Higgs plus four gluon amplitude, the φ-NMHV amplitude A (1)4 (φ, 1+ , 2 − , 3 − , 4 − ).We chose to split the calculation into two parts, one being cut-constructible (to whichwe applied the techniques of four-dimensional unitarity) and a rational part, whichis insensitive to four-dimensional cuts. We used the unitarity methods describedin chapters 2 and 3 to calculate the cut-constructible pieces. We used Feynmandiagrams to calculate the rational pieces proportional to N f and BCFW recursionrelations to calculate the remaining pieces. We checked our results for the φ-MHVand φ-NMHV amplitudes against the semi-numerical code of Ref. [104].In the next chapter we will compute the remaining helicity amplitude, the φqq-NMHV amplitude, completing the analytic calculation of φ + parton amplitudes.We shall observe that although similar to the calculation carried out in this chapter,2 We have been informed by John Campbell, that the entries for M F,g4 and M F,q4 in Table 4.1are in agreement with results obtained using the seminumerical code described in Ref. [104].
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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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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 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 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
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
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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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