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

More documents

Recommendations

Info

2.4. Spinor integration 512.4.3 A double cut exampleAs a more concrete example of the above method, we describe the calculation of abubble coefficient in the s 12 channel for the one-loop amplitude A (1)4 (φ, 1− , 2 + , 3 − , 4 + ).The tree products have the following form,∑i=±A (0)L (li 1, φ, 3 − , 4 + , l −i2 )A (0)R (li 2, 1 − , 2 + , l −i1 ) =−〈l 1 3〉 4 〈l 2 1〉 4 − 〈l 1 1〉 4 〈l 2 3〉 4〈l 1 3〉〈34〉〈4l 2 〉〈l 2 1〉〈12〉〈2l 1 〉〈l 1 l 2 〉 2.Using the Schouten identity we can re-write the numerator in terms of three functions(which will be defined in chapter 3), for now, we concentrate on one of these threefunctions,(2.74)F s12 = − 〈13〉2 〈l 1 1〉〈l 2 3〉〈34〉〈l 2 4〉〈12〉〈l 1 2〉 . (2.75)Next we use momentum conservation to remove l 1 in favour of l 2 ,F s12 = − 〈13〉2 [l 2 |l 1 |1〉〈l 2 3〉〈34〉〈l 2 4〉〈12〉[l 2 |l 1 |2〉= − 〈13〉2 [l 2 |P 12 |1〉〈l 2 3〉〈34〉〈l 2 4〉〈12〉[l 2 |P 12 |2〉We rescale l 2 = tl and perform the trivial t integration,F s12 =∆F s12 ==〈13〉 2 [l 2 2]〈l 2 3〉〈34〉〈l 2 4〉〈12〉[l 2 1]∫t 〈13〉 2 [l2]〈l3〉〈l|P 12 |l] 〈34〉〈l4〉〈12〉[l1]∫s 12 〈13〉 2 [l2]〈l3〉〈l|P 12 |l] 2 〈34〉〈l4〉〈12〉[l1](2.76)(2.77)Here we use ∆F s12 to represent the double cut integral (for simplicity and to highlightthe operations on the integrand we have suppressed the details of the measure). Nextwe wish to rewrite l in terms of p and η, recall that p + η = P, here P = P 12 so anobvious choice for p and η is p = p 1 and η = p 2 , so we write that |l] = |1] + z|2] andintegrate in z dropping log terms.∫〈13〉 2 〈3l〉∆F s12 =〈34〉〈4l〉〈12〉〈2l〉(〈2l〉 − z〈1l〉)Finally we replace |l〉 and take residues in z,∫〈13〉 2 (〈13〉 + z〈23〉)∆F s12 =〈34〉(〈14〉 − z〈24〉)〈12〉(1 − zz)(2.78)
2.5. MHV rules and BCFW recursion relations 52∆F s12 =〈13〉 2 [42]〈24〉〈4|P 12 |4] . (2.79)We observe that there was no residue at z = 0, meaning that the coefficient ofthe two point function for this integrand came solely from the reduction of tensortriangles, in fact 〈4|P 12 |4], which appears in the denominator is a signature of a firstrank tensor triangle.In this thesis we will use the above method to construct the coefficients of scalarbubbles for the various Higgs plus four parton one-loop amplitudes that we study.2.5 MHV rules and BCFW recursion relationsIn this section we discuss two important on-shell techniques for the generation oftree-level amplitudes, the MHV rules [130,131,139] and BCFW recursion relations[140,141]. These techniques have had many important applications, some of whichwill be described in the following sections.2.5.1 The MHV/CSW rulesThe MHV (or Parke-Taylor) amplitudes have long been known to possess a remarkablysimple structure for all gluon multiplicities [142],A (0)n (1+ , . . .,i − , . . ., j − , . . .,n + ) =〈ij〉 4∏ n−1(2.80)α=1〈α(α + 1)〉〈n1〉.MHV stands for Maximally-Helicity-Violating because if one classes amplitudes interms of the number of negative helicity gluons present these are the first which arenon-zero, i.e:A (0)n (1+ , . . .,i + , . . .,j + , . . .,n + ) = A (0)n (1+ , . . ., i + , . . .,j − , . . .,n + ) = 0. (2.81)MHV amplitudes are the conjugates of eq. (2.80),A (0)n (1 − , . . .,i + , . . .,j + , . . .,n − ) =[ij] 4∏ n−1(2.82)α=1[α(α + 1)][n1].All relevant tree amplitudes (with definitions of spinor products) for this thesis arecollected together in Appendix A.
Page 7 and 8:
Contentsvii1.4.1 Effective Lagrangi
Page 9 and 10:
Contentsix4 One-loop Higgs plus fou
Page 11:
ContentsxiB.2 Box Integral Function
Page 18 and 19: 1.1. The Standard Model of Particle
Page 20: 1.1. The Standard Model of Particle
Page 27 and 28: 1.2. Electroweak Symmetry Breaking
Page 33 and 34: 1.3. Higgs searches at colliders 18
Page 39 and 40: 1.4. Effective coupling between glu
Page 41 and 42: 1.4. Effective coupling between glu
Page 43 and 44: 1.5. Higgs plus two jets: Its pheno
Page 45 and 46: 1.5. Higgs plus two jets: Its pheno
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
Page 55 and 56: 2.3. Forde’s Laurent expansion me
Page 61 and 62: 2.4. Spinor integration 46Here we h
Page 63 and 64: 2.4. Spinor integration 48The expli
Page 65: 2.4. Spinor integration 50with F z
Page 69 and 70: 2.5. MHV rules and BCFW recursion r
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
Page 79 and 80: 2.6. The unitarity-bootstrap 64Afte
Page 81 and 82: 2.8. Recent progress: One-loop auto
Page 83 and 84: 2.9. Summary 68cluding the recent c
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∑
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:
Page 137 and 138:
Page 139 and 140:
4.3. Rational Terms 124contains two
Page 141 and 142:
4.4. Higgs plus four gluon amplitud
Page 143 and 144:
4.4. Higgs plus four gluon amplitud
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
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
show all

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

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?