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

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3.2. The cut-constructible parts 71decomposition,C n (φ, 1 − , 2 + , . . .,m − , . . .,n + ) = ∑ iC 4;i I 4;i + ∑ iC 3;i I 3;i + ∑ iC 2;i I 2;i . (3.5)Here C j;i represents the coefficient of a j-point scalar basis integral (I j;i ) with adistribution of momenta {i}. Using the methods described in Chapter 2 we willisolate each coefficient separately using a dedicated cut for that integral. We workwith a top down approach and calculate the box integral coefficients first [120], thenthe triangle coefficients [122] finally using a double cut to determine the coefficientassociated with the two-point functions [132].3.2.1 Box coefficients from four-cutsWe begin by discussing the boxes which do not contribute to the φ-MHV amplitude.Specifically these are the four-, three- and two-mass hard boxes, which are shownin Figs. 3.1-3.3. The three- and four-mass box configurations vanish trivially, sincehowever one assigns the helicities to the loop momenta one always finds at least onezero-tree amplitude at one of the corners. Many of the two-mass hard topologiesvanish for a more subtle reason. When two MHV or MHV three-point amplitudesare adjacent in a box topology the corresponding coefficient is zero. This can beillustrated by considering the following product of two on-shell MHV amplitude,A 2×MHV = A (0)3 (l + 1 , 2 − , l − 2 )A (0)3 (l + 2 , 3 − , l − 3 )〈l 2 2〉 3 〈l 3 3〉 3=〈l 1 2〉〈l 2 l 1 〉〈l 2 3〉〈l 3 l 2 〉 . (3.6)The complex solution of the on-shell constraints ensure that [l 2 2] = 0 from (l 2 +p 2 ) 2 = 0. However, the constraint at the second vertex implies that (l 2 + p 3 ) 2 = 0,for which the complex solution is that [l 2 3] = 0. This then implies that [23] = 0, or|2] ∝ |3]. This solution is unphysical and as such we must throw it away.Therefore we have established that the only box functions which can appear inthe general one-loop φ-MHV amplitude are one and two-mass easy box functions.We will now classify the boxes and their solutions for general gluon multiplicities.
eplacemen3.2. The cut-constructible parts 72++−−±∓±−−++−−++0000 1111∓±∓+−+−(a) (b) (c)Figure 3.1: Configurations where the coefficient of the four-mass box vanishes, externalparticles are either; positive helicity gluons (solid black lines) of which therecan be an arbitrary number at any vertex, negative helicity gluons (shown in green)of which there can be only two and one φ (dashed black line). In each configuration(a) − (c) we find one zero tree-level amplitude, these are highlighted by the redboxes.One-mass boxesThe one-mass box is simplest and we begin by classifying the two solutions forthe loop momenta l with the general kinematics shown in Fig.3.4. The on-shellconditions for the various loop momenta are as follows,(l + m 1 ) 2 = 0 (3.7)l 2 = 0 (3.8)(l − m 2 ) 2 = 0 (3.9)(l − M 23 ) 2 = 0 (3.10)where we have used the shorthand notation M ij = m i +m j . We choose to expand l interms of two massless basis vectors m 1 and m 2 with four free parameters {α, β, ρ, δ}which are fixed by the on-shell constraints.l µ = αm µ 1 + βmµ 2 + ρ 2 〈m 2|γ µ |m 1 ] + δ 2 〈m 1|γ µ |m 2 ]. (3.11)The first and third on-shell constraint require that, α = β = 0. Then setting l onshell requires that either ρ or δ = 0. We begin by choosing δ = 0. Then the final
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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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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 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: 3.2. The cut-constructible parts 70
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Page 115 and 116: 3.3. The rational pieces 100+m−1
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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
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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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