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

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2.3. Forde’s Laurent expansion method 4100000001111111 0000000111111100000001111111 0000000111111100000001111111 0000000111111100000001111111 0000000111111100000001111111 0000000111111100000001111111000000000000001111111 000000011111111111111 0000000111111100000001111111 0000000111111100000001111111 0000000111111100000001111111 0000000111111100000001111111 0000000111111100000001111111 0000000111111100000001111111000 111000000111111000000111111000000111111 000000111111000000111111 000000111111000000111111 000000111111000000111111 000000111111000000011111110000000111111100000001111111000000011111110000000111111100000001111111000000011111110000000111111101010100 1100 1100 1100 1100 110000000111111100000001111111000000011111110000000111111100000001111111 0000000111111100000001111111 00000001111111 0000000111111100000001111111 00000001111111 0000000111111100000001111111 00000001111111 0000000111111100000001111111 00000001111111 0000000111111100000001111111 00000001111111 0000000111111100000001111111 00000001111111 0000000111111100000001111111 00000001111111Figure 2.2: The application of three cuts to a one-loop amplitude results in thefactorisation onto the product of three tree amplitudes. Both boxes and triangleintegral functions appear in a triple cut, since multiple boxes can share a specificset of three propagators more than one box can enter. Each cut selects a specifictriangle integral function. For four-dimensional cuts the loop momenta has one freeparameter.0000000111111100000001111111000000011111110000000111111100000001111111000000011111110000000111111100000001111111000000011111110000000111111100000001111111000000011111110000000111111100000001111111000000011111110000000111111100000001111111000000011111110000000111111100000001111111000000011111110000000111111100000001111111Figure 2.3: The application of two cuts to a one-loop amplitude results in thefactorisation onto the product of two tree amplitudes. Boxes, triangle and bubbleintegral functions appear in a double cut, again with multiple triangle and boxcontributions with an individual bubble term. For four-dimensional cuts the loopmomenta has two free parameters.
2.3. Forde’s Laurent expansion method 42∫=d 4 l2∏i=0δ(l 2 i ) ([Inf t A 1 A 2 A 3 ](t) + ∑poles j)Res t=tj A 1 A 2 A 3. (2.31)t − t jIn the above equation l i = l − K i where K i is one of the two kinematic invariantswhich appear in the triple cut, l 0 = l. Inf t A 1 A 2 A 3 contains all the pieces of theintegrand which contain no poles 2 in t, i.e.()lim [Inf t A 1 A 2 A 3 ](t) − A 1 (t)A 2 (t)A 3 (t)) = 0. (2.32)t→∞We can decompose the Inf piece as follows[Inf t A 1 A 2 A 3 ](t) =m∑f i t i . (2.33)i=0The residue contributions in eq. (2.31) arise from propagators of the form (l − P) 2going on-shell. Since this equation is quadratic we expect two solutions (labelledt = t ± ) and this is indeed what we observed when calculating solutions to the fouron-shell constraints in the previous section.∫d 4 l2∏i=0δ(l 2 i ) 1(l − P) 2 ∼ 1t + − t −( ∫∫−2∏d 4 l δ(l 2 i ) 1t − ti=0 +2∏)δ(l 2 i ) 1t − t −d 4 li=0(2.34)Therefore, after suitable partial fractioning we can write the triple cut integrand asfollows,∫d 4 l2∏∫δ(l 2 i)A 1 A 2 A 3 =i=0(∑ m )dtJ t f i t i + ∑i=0boxes{l}d l D cut0 . (2.35)Here we sum over the various triple cut boxes in which the t dependence has beeneliminated by evaluating the loop momenta at the specific residue values. Theremaining piece of the equation is a sum over the positive powers of t. J t representsa Jacobian factor obtained by the transformation of the integration variable from lto t. Since there is a freedom in how we define the loop expansion (i.e. a freedom inhow we choose a i ), we can choose a specific basis in which integrals over non-zero2 This operation will also be useful when calculating the rational terms [124]
Page 7 and 8: Contentsvii1.4.1 Effective Lagrangi
Page 9 and 10: Contentsix4 One-loop Higgs plus fou
Page 11: ContentsxiB.2 Box Integral Function
Page 18 and 19: 1.1. The Standard Model of Particle
Page 20: 1.1. The Standard Model of Particle
Page 27 and 28: 1.2. Electroweak Symmetry Breaking
Page 33 and 34: 1.3. Higgs searches at colliders 18
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: 2.3. Forde’s Laurent expansion me
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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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3.2. The cut-constructible parts 92
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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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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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