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

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3.4. Cross Checks and Limits 109the momentum P (i,j) always contains both m − 1 and m, while P (j,i) never includeseither m − 1 or m, and we find relations such as,tr − (m, P (i,j) , i, 1)s 2 1mA ij m1tr − (1, P (j,i) , i, m)A i(j−1)s 2 1m1mWe note that for the special case i = m − 1,m−1‖m tr −(K, P (i,j) , i, 1)−−−−−→ A ijs 2 K1 ,1Km−1‖m tr −(1, P (j,i) , i, K)−−−−−→ A i(j−1)s 2 1K . (3.169)1KA m−1,jm1 = tr −(m, j, m − 1, 1)s m−1,j− tr −(m, j, m, 1)s m,jA m−1,j1mA i,m−1m1m−1‖m−−−−−→ 0,m−1‖m−−−−−→ 0,m−1‖m−−−−−→ 0. (3.170)Similar relations hold for the terms involving K and I. Therefore, all terms in the n-gluon version of A φFn;1 and AφS n;1 therefore either collapse onto similar terms, or vanishin such a way that the reduced summation precisely matches onto the correspondingA φFn−1;1 and AφS n−1;1 .Two positive collinear limitNext we consider the limit when two positive helicity gluons become collinear. Wefocus on the specific example where l −1 ‖ l with 3 ≤ l ≤ m −1. As in the previoussubsection, let first consider the ranges 2 ≤ i ≤ m − 1 and m ≤ j ≤ n. We notethat,b l−1j1mb lj1ml−1‖l−−−→l−1‖l−−−→b Kj1m ,b Kj1m. (3.171)The collinear factorisation of box functions has been well studied [110,111,166] andin this case, the relation,(b l−1j1m) n (F2me4F (s l−1,j , s l,j−1 ; s l,j , s l−1,j−1 ) +(l−1‖l−−−→b lj1mb Kj1m) nF2me4F (s l,j , s l+1,j−1 ; s l+1,j , s l,j−1 )) nF2me4F (s K,j, s l+1,j−1 ; s K,j , s l+1,j−1 )ensures the box terms correctly factorise onto the lower point amplitude.(3.172)
3.4. Cross Checks and Limits 110The next set of functions we consider are the triangle functions which have j asthe second index, these functions possess the general form:l∑i=l−1tr − (m, P (i,j) , j, 1) n F j(i−1)m1 L n (P (i,j−1) , P (i,j) ). (3.173)There is no contribution when i = l, because F j(l−1)m1= F j(l−1)1mremaining i = l − 1 contribution collapses onto the correct term,= 0, while theSimilarly, when we considertr − (m, P (K,j) , l − 1, 1) n F j(K−1)m1 L n (P (K,j−1) , P (K,j) ). (3.174)l∑i=l−1tr − (m, P (j,i) , j, 1) n F ji1mL n (P (j+1,i) , P (j,i) ), (3.175)there is no contribution when i = l − 1, while for i = l, we recover the correctcontribution.The remaining types of triangle function are of the formSince F ljm1 = F(l−1)j m1show that,l∑i=l−1tr − (m, P (i,j) , i, 1) n F ijm1L n (P (i+1,j) , P (i,j) ). (3.176)we have contributions from both terms, it is straightforward totr − (m, P (l−1,j) , l − 1, 1) n L n (P (l,j) , P (l−1,j) ) + tr − (m, P (l+1,j) , l, 1) n L n (P (l+1,j) , P (l,j) )Similar considerations apply tol∑i=l−1l−1‖−−→ l tr − (m, P (l+1,j) , K, 1) n L n (P (l+1,j) , P (K,j) ).(3.177)tr − (1, P (j,i) , i, m) n F i(j−1)1m L n(P (j,i−1) , P (j,i) ), (3.178)thus ensuring the correct collinear factorisation.3.4.3 The cancellation of unphysical singularitiesThe cut constructible terms eq. (3.103) - (3.105) contain poles in 〈i j〉. For the mostpart, i and j are non-adjacent gluons and as such there should be no singularity
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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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2.2. Quadruple cuts 38We now must s
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2.3. Forde’s Laurent expansion me
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2.4. Spinor integration 46Here we h
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2.4. Spinor integration 48The expli
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2.4. Spinor integration 50with F z
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2.5. MHV rules and BCFW recursion r
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2.5. MHV rules and BCFW recursion r
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2.6. The unitarity-bootstrap 56Next
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Page 77 and 78: 2.6. The unitarity-bootstrap 62rati
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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 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: 3.4. Cross Checks and Limits 108The
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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Page 139 and 140: 4.3. Rational Terms 124contains two
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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,
Page 155 and 156: 5.2. One-loop results 140with K 1 =
Page 157 and 158: 5.2. One-loop results 142Relation f
Page 159 and 160: 5.2. One-loop results 144+ 1s 123[
Page 161 and 162: 5.4. Summary 146(φ, 1 −¯q , 2 +
Page 163 and 164: 6.2. Improvements from the semi-num
Page 165 and 166: 6.4. Tevatron results 150The W mass
Page 167 and 168: 6.4. Tevatron results 152m H [GeV]
Page 169 and 170: 6.5. LHC results 154Eq. (6.11) and
Page 171 and 172: 6.5. LHC results 156m H [GeV] 120 1
Page 173 and 174: 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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