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

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3.2. The cut-constructible parts 77i0100000001111111 0100000001111111 010000000111111101+j000001111100000111110000011111++−−0000 11110000 11110000 11110000 1111+−0000 11110000 11110000 111100000011111101000000111111010000001111110100000011111101+−0000 11110000 11110000 11110000 1111− ++ −0000 11110000 11110000 11110000 1111+0000 11110000 11110000 11110000 111100000001111111 0100000001111111 0100000001111111 011111111000000001±0000011111000001111100000111110000011111± ∓(a) (b) (c)−∓∓±0000011111000001111100000111110000011111±∓00000111110000011111000001111100000111110000001111110000001111110000001111111111110 000000000 1 111−−+0000 11110000 11110000 11110000 1111+00000001111111000000011111110000000111111100000001111111 1 −−+000001111100000111110000011111+000000011111110000000111111100000001111111000000011111110 1 −−+0000 11110000 11110000 11110000 1111−+0000 11110000 11110000 11110000 1111−− + − ++ + −− +0000 11110000 11110000 11110000 11110000011111000001111100000111110000011111000001111100000111110000011111000001111100000111110000011111(d) (e) (f)+0000 11110000 11110000 11110000 1111Figure 3.5: One-mass boxes for the φ-MHV amplitudes are defined by the positionof the two negative helicity gluons, there can be an arbitrary number of positivehelicity gluons at the massive vertex of which the end points are denoted i and j.Diagrams (a) − (e) are non-zero whilst diagram (f) which is a special case for the5-point amplitude, is zero. Diagram (c) is of interest since it allows fermions (andscalars) to propagate in the loop.
3.2. The cut-constructible parts 78This leaves Fig.3.5(c), which has a far richer structure than those previouslystudied. Each loop helicity is unconstrained, which means that fermions as well asgluons can propagate around the loop. In addition, there are now two contributingtopologies which we label D (c)g,+ and D (c)g,− the subscript indicating the species andhelicity of l 1 in the φ containing amplitude.D (c)g,+ = A (0)n−1(φ, l + 1 , (n − 1)+ , . . .,m − , . . .,3 + , l − 2 )A(0) 3 (l + 2 , 2+ , l − 3 )×A (0)3 (l+ 3 , 1− , l − 4 )A(0) 3 (l+ 4 , n+ , l − 1 ), (3.26)D (c)g,− = A (0)n−1(φ, l − 1 , (n − 1) + , . . .,m − , . . .,3 + , l + 2 )A (0)3 (l − 2 , 2 + , l + 3 )×A (0)3 (l− 3 , 1− , l + 4 )A(0) 3 (l− 4 , n+ , l + 1 ). (3.27)There are two diagrams which have the same topology as (c), one with p 1 ∈ P i,j andone with p m ∈ P i,j (P i,j represents the massive leg). For convenience we consideronly the case where p m ∈ P i,j explicitly, since it is trivial to obtain the remainingdiagram from this one. The momentum constraints are as follows,and the two specific solutions for l arel µ (1) = [1n]2[n2] 〈1|γµ |2] andl 2 = l + p 2 (3.28)l 3 = l (3.29)l 4 = l − p 1 (3.30)l 1 = l 4 − p n (3.31)l µ (2) = 〈1n〉2〈2n〉〈2|γµ |1]. (3.32)Once again the solution associated with l (1) does not contribute and we are left withonly l → l (2) . We combine the two helicity solutions (eqs. (3.26)-(3.27))D (c)g =〈m|l 1 |n] 4 〈1|l 3 |2] 4 + 〈m|l 2 |2] 4 〈1|l 4 |n] 4γ〈l 2 l 1 〉〈l 1 (n − 1)〉〈l 2 3〉〈l 3 l 4 〉〈l 3 1〉〈l 4 1〉[l 3 l 2 ][l 4 l 1 ][nl 1 ][nl 4 ][2l 2 ][2l 3 ] . (3.33)Here we have introduced γ = ∏ 4α=n−1〈α(α − 1)〉 to simplify the formula. Next weuse the momentum constraints to remove l 1 and l 2 from the numerator.D (c)g =〈m|l 4 |n] 4 〈1|l 3 |2] 4 + 〈m|l 3 |2] 4 〈1|l 4 |n] 4γ〈l 2 l 1 〉〈l 1 (n − 1)〉〈l 2 3〉〈l 3 l 4 〉〈l 3 1〉〈l 4 1〉[l 3 l 2 ][l 4 l 1 ][nl 1 ][nl 4 ][2l 2 ][2l 3 ] .
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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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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 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
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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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