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

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B.2. Box Integral Functions 177B.2 Box Integral FunctionssssP 2ttF 1m4FF 2mh4FF 2meP 2 Q 2 P 2 Q 2 4FtFigure B.1: Conventions for labelling the three scalar box integrals appearing in theone-loop H plus parton amplitudes.Figure B.1 sets our labelling conventions. We express our results in terms ofbasis functions which are related to the scalar integral I by a kinematic factor,which cancels against the same factor in the coefficient. The zero-,one- and twomasseasy box have representations in terms of hypergeometric series to all ordersin ǫ,F4 0m (s, t) = 2 [( ) µ2 ǫ (2Fǫ 2 1 1, −ǫ; 1 − ǫ; − u )−st( ) µ2 ǫ (+ 2F 1 1, −ǫ; 1 − ǫ; − u )], (B.2.7)−tsF4 1m (s, t; P 2 ) = 2 [( ) µ2 ǫ (2Fǫ 2 1 1, −ǫ; 1 − ǫ; − u )−st( ) µ2 ǫ (+ 2F 1 1, −ǫ; 1 − ǫ; − u )−ts( ) µ2 ǫ (− 2F−P 2 1 1, −ǫ; 1 − ǫ; − uP )]2, (B.2.8)stF4 2me (s, t; P 2 , Q 2 ) = 2 [( ) µ2 ǫ ()us2Fǫ 2 1 1, −ǫ; 1 − ǫ;−sP 2 Q 2 − st( ) µ2 ǫ ()ut+ 2F 1 1, −ǫ; 1 − ǫ;−tP 2 Q 2 − st( ) µ2 ǫ ()uP 2− 2F−P 2 1 1, −ǫ; 1 − ǫ;P 2 Q 2 − st( ) µ2 ǫ ()]uQ 2− 2F−Q 2 1 1, −ǫ; 1 − ǫ; , (B.2.9)P 2 Q 2 − stwhen expanded in ǫ through to order ǫ 0 we find that the one- and two-mass easy
B.2. Box Integral Functions 178boxes relevant for this thesis have the following expansion,F 1m4 (s, t, : P 2 ) = 2 [( ) µ2 ǫ ( ) µ2 ǫ ( ) µ2 ǫ ]+ −ǫ 2 −s −t −P 2+ F 1m4F(s, t; P 2 )F 2me4 (s, t, : P 2 , Q 2 ) = 2 [( ) µ2 ǫ ( ) µ2 ǫ ( ) µ2 ǫ+ − −ǫ 2 −s −t −P 2+ F 2me4F (s, t; P 2 , Q 2 )Whilst the two-mass hard box has the following ǫ expansion,F 2mh4 (s, t; P 2 , Q 2 ) = 2 [( ) µ2 ǫ ( ) µ2 ǫ ( ) µ2 ǫ+ − −ǫ 2 −s −t −P 2+ F 2mh4F (s, t; P 2 , Q 2 )( µ2−Q 2 ) ǫ ]( µ2−Q 2 ) ǫ ](B.2.10)(B.2.11)(B.2.12)The finite parts of the one-mass and two-mass (easy and hard) have the followingforms,F 1m4F (s, t; P 2 ) = −2(Li 2(1 − P ) 2+ Li 2(1 − P )2+ 1 ( ) ) ss t 2 ln2 + π2,t 6(B.2.13)F 2mh4F (s, t; P 2 , Q 2 ) = −2(Li 2(1 − P ) )2+ Li 2(1 − Q2+ 1 ( ) stt 2 ln2 t− 1 ( ) ( )) s s2 ln ln , (B.2.14)P 2 Q 2F 2me4F (s, t; P 2 , Q 2 ) = −2(Li 2(1 − P ) 2+ Li 2(1 − P ) )2+ Li 2(1 − Q2+Li 2(1 − Q2ts t)− Li 2(1 − P )2 Q 2+ 1 ( sst 2 ln2 ts)).(B.2.15)B.2.1Triangle basis integralsOne- and two-mass triangles can be expressed in terms of the following function,F3 1m (s) = 1 ( ) µ2 ǫ, (B.2.16)ǫ 2 −sThis function is used in chapter 3 explicitly, in latter chapters we expand the boxfunctions and combine the ǫ −2 for the amplitude. Therefore, the above function isnot seen in the formulae in chapters 4 and 5. The finite three-mass triangle is givenby [121,219],F 3m3 (M2 1 , M2 2 , M2 3 ) = i√∆3∑k=1( ( )) ( )))1 + iδk1 − iδk(Li 2 −− Li 2(− ,1 − iδ k 1 + iδ k
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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
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Chapter 4One-loop Higgs plus four-g
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4.2. Cut-Constructible Contribution
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4.3. Rational Terms 124contains two
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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,
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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: Appendix BOne-loop Basis IntegralsI
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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