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

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2.5. MHV rules and BCFW recursion relations 55momentum conservation is retained,p µ i → p µ i + z 2 〈j|γµ |i] and p µ j → pµ j − z 2 〈j|γµ |i] (2.87)We note that p i (z) + p j (z) = p i + p j such that local momentum conservation is alsopreserved. Furthermorep 2 i(z) = (p µ i + z 2 〈j|γµ |i])(p i,µ + z 2 〈j|γ µ|i]) = 0 (2.88)which implies that p i (z) is also on-shell. The shifts we have made are equivalent tothe following spinor shifts,|i〉 → |i〉 + z|j〉 and |j] → |j] − z|i], (2.89)whilst leaving |i] and |j〉 unaltered. Immediately we see that we are no longer dealingwith real momenta in Minkowski space (where λ = ±˜λ), and just as in the quadruplecuts of section 2.2 we work with complex momenta.The shifted amplitude A (0)n (z) is thus also an on-shell tree amplitude and inparticular will share properties associated with physical on-shell tree amplitudes.Firstly we note that A (0)n (z) is a rational function in z. Indeed A (0)n (z) only has simplepoles in z. This can be seen by considering how poles of tree-level amplitudes occur.Tree-level amplitudes can only develop a pole when an internal propagator goes onshelli.e. P 2 abc → 0, where P abc is some generic combination of external momenta.Since propagators are quadratic one may expect that double poles in z could arise,however if the propagator P α...β goes on-shell and i ∈ {α, . . ., β}, j /∈ {α, . . ., β}then we have,P(z) 2 α...β = (p α + · · · + p i + z|j〉[i| + · · · + p β ) 2 = P 2 α...β + z〈j|P α...β |i] (2.90)If both i ∈ {α, . . ., β} and j ∈ {α, . . ., β} then the z dependence drops out,P(z) 2 α...β = (p α + · · · + p i + z|j〉[i| + . . .p j − z|j〉[i] + · · · + p β ) 2 = P 2 α...β . (2.91)So only simple poles can occur and a residue occurs when z is of the form z =P 2 α...β /〈j|P α...β|i].
2.6. The unitarity-bootstrap 56Next we assume that A(z) → 0 as z → ∞, this means that one can write A(z)in the following formA(z) = ∑ ijc ijz − z ij. (2.92)Ultimately we wish to know the physical amplitude A(0),A(0) = ∑ ijc ijz ij(2.93)All that needs to be determined are the coefficients of the residues at z ij , this is asimple task however, since the only source of poles in a tree amplitude occurs whenan internal propagator goes on-shell. When this happens the amplitude factors onto two lower point amplitudes (the so-called left and right amplitudes). This meansthat we can write A(z) asA(z) = ∑ h=±∑ijA L,−h (z ij )A R,h (z ij )P 2ij (z) , (2.94)the sum over h represents the two helicity orientations of the propagator. We requireA(0) where,A(0) = ∑ h=±∑Eq. (2.95) is the BCFW recursion relation.ijA L,−h (z ij )A R,h (z ij ). (2.95)Pij22.6 The unitarity-bootstrapSo far in this chapter we have described methods for calculating the cut-constructiblepieces of one-loop amplitudes for massless theories. Eq. (2.5) shows that by workingin four dimensions we lose information associated with higher order pieces in ǫ of thecoefficient of the basis integrals. These missing pieces are referred to as the rationalpieces. In this section we describe a method which obtains these rational pieces andis still four-dimensional, the unitarity-bootstrap [124,161–165].
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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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Page 23 and 24: 1.1. The Standard Model of Particle
Page 25 and 26: 1.1. The Standard Model of Particle
Page 27 and 28: 1.2. Electroweak Symmetry Breaking
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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 and 66: 2.4. Spinor integration 50with F z
Page 67 and 68: 2.5. MHV rules and BCFW recursion r
Page 69: 2.5. MHV rules and BCFW recursion r
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
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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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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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