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

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3.2. The cut-constructible parts 89As expected we see that G factors onto the tree-level amplitude multiplying ahelicity blind function. The denominators in the above equations have exactlythe form we expect from our four-cut calculations. Spinor products of the form〈lj〉 ∝ (l − p j ) 2 /[lj] are linked to Feynman propagators and can be associated withbox diagrams. Indeed the residues of these propagators correspond to setting afurther propagator on-shell and as such correspond to a four-cut. At first glancewe observe three spinor products associated with inserting additional propagators(l − p j ) 2 , (l − p j+1 ) 2 and (l − p i+1 ) 2 . Of these the first two correspond to two-masseasy boxes and the third corresponds to a two-mass hard topology. We observed inthe previous section that there are no such contributions to the φ-MHV amplitude,implying that somehow this residue must not contribute to a box-coefficient. Uponcloser inspection we see that there is indeed no residue associated with this termsince the non-vanishing three-point vertex in the triangle requires that |l〉 ∝ |i〉 andas such when the solution for the loop-momenta is inserted there is a cancellationbetween 〈i(i + 1)〉 in the denominator and numerator.To determine F and S one merely has to insert the parameterisation for the loopmomentum in terms of eq. (3.62) and take the t 0 coefficient in a series expansionaround t = ∞. We find,F 2,1m (1, i, j) = A (0)n(〈j(j − 1)〉〈im〉〈i1〉 2 〈m|P j;i |i]+ 〈im〉2 〈i1〉〈1|P j;i |i]〈1m〉 2 〈ij〉〈i(j − 1)〉〈ij〉〈i(j − 1)〉− 〈im〉2 〈i1〉 2 〈j|P j;i |i]〈ij〉 2 〈i(j − 1)〉)− 〈im〉2 〈i1〉 2 〈(j − 1)|P j;i |i]. (3.70)〈ij〉〈i(j − 1)〉 2After using the Schouten identity to simplify the above formula we findF 2,1m (1, i, j) = A (0)n (−bij 1m + b i(j−1)1m )〈i|P j;i |i], (3.71)where b ij1m is defined as in eq. (3.46). The calculation for S is identical to that ofF (although here the intermediate formulae are more complicated so we quote onlythe final result)S 2,1m (1, i, j) = A (0)n ((bij 1m )2 − (b i(j−1)1m )2 )〈i|P j;i |i]. (3.72)With these solutions in hand we are now able to calculate the coefficient of anyone- or two-mass triangle appearing in the φ-MHV amplitude. All that remains is
3.2. The cut-constructible parts 90iβjαβ−iαii αβj − + j +αjiβjFigure 3.10: The above combination of four two-mass triangles with the two-masseasy box is IR finite. The above combinations uses the F definitions of Appendix B,these are related to the scalar basis integrals I by a kinematic factor. The definitionsα = (j − 1) and β = (i + 1) are used to simplify the Figure.to correctly define the sum over allowed triangles and check the IR safety of theformula.3.2.3 Cancellation of N f ǫ −2 polesWe show how our results for the two-mass triangles result in the cancellation ofthe N f ǫ −2 poles which arise from the two-mass boxes. We consider two-mass boxeswhich are proportional to (1 − N f /4N c ) (i.e F terms) for simplicity, however theproof for the (1 − N f /N c ) boxes proceeds identically.A given two-mass box in A φF,4−cutn;1 has a coefficient b ij1m, we find four two-masstriangles which have a term proportional to b ij1m . The combination of all of thecontributions with a piece proportional to b ij1m (W(b ij1m)) is given by,W(b ij1m ) = −bij 1m F2me 4 (s i+1,j , s i,j−1 ; s i,j , s i+1,j−1 )(+ (−b ij1m + b i(j+1)1m )(F 1m3 (Pi,j) 2 − F 1m3 (Pi+1,j))2+ (−b ij+ (−b i(j−1)1m+ (−b (i+1)j1m1m + b(i−1)j) 1m)(F 1m3 (P i,j 2 ) − F1m 3 (P i,j−1 2 ))+ b ij1m)(F 1m3 (P 2i,j−1) − F 1m3 (P 2i+1,j−1))+ b ij)1m)(F 1m3 (P i+1,j 2 ) − F1m 3 (P i+1,j−1). 2 )) (3.73)When we expand the above we find using the definitions in Appendix BW(b ij1m ) = −bij 1m F2me 4F (s i,j, s i+1,j−1 ; s i+1,j , s i,j−1 ) + . . ., (3.74)
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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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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
Page 67 and 68: 2.5. MHV rules and BCFW recursion r
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
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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 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,
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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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