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

More documents

Recommendations

Info

3.2. The cut-constructible parts 95We can compare F to G by using the same technique that was applied to I ij tosplit the integrand into four simpler pieces,F 2−cut (a, i, j) = −f i,j + f i,j+1 + f i−1,j − f i−1,j+1 , (3.93)withf ij = 〈il 1〉〈jl 2 〉〈l 1 1〉〈l 2 m〉〈l 2 1〉〈ml 1 〉〈il 2 〉〈jl 1 〉〈l 1 l 2 〉 2 〈1m〉 2 (3.94)We note that f ij ∝ G ij 〈l 1 1〉〈l 2 m〉〈l 2 1〉〈ml 1 〉 and that G ij contained only a purebubble function which cancelled when the four G functions were combined. Herewe see that there are products of spinors in the numerator, which will contribute tohigher-rank tensor integrals some of which will reduce to 2-point functions. Therefore,from f we expect to find a less trivial 2-point coefficient than from G ij .F is actually simple enough to apply the method of [132] directly. We firstremove l 1 in favour of l 2 = tλ and after integrating t with the second delta funtionwe obtain the following integrand∫F 2−cut (1, i, j) =s Pi,j 〈1m〉 2 〈1λ〉〈4λ〉〈1|P i;j |λ]〈4|P i,j |λ]dλ〈(i − 1)λ〉〈iλ〉〈j|P i,j |λ]〈(j + 1)|P i,j |λ]〈λ|P i,j |λ] 2.(3.95)Next we define |λ〉 = |p〉 + z|η〉 and integrate in z discarding the logarithmic pieces(which as described earlier contribute only to box and triangle coefficients). Inthe remaining rational integral we define |λ] = |p] + z|η] which leaves us with thefollowing term∮F 2−point (1, i, j) = −〈1m〉 2 β(1) 2 β(m) 2dzz(1 − zz)β(i − 1)β(i)β(j)β(j + 1)(3.96)where β(x) = (z〈px〉 − 〈ηx〉). To obtain the 2-point coefficient we are interested in,we use Cauchy’s residue theorem to perform the integral. After some simplificationusing the Schouten identity we find,F 2−point (1, i, j) = 〈1(i − 1)〉〈1m〉2 〈(i − 1)m〉 2 〈j(j + 1)〉〈1|P i,j |(i − 1)]〈(i − 1)j〉〈(i − 1)(j + 1)〉〈(i − 1)|P i,j |(i − 1)]− 〈1i〉〈1m〉2 〈im〉 2 〈j(j + 1)〉〈1|P i,j |i]+ 〈1j〉〈1m〉2 〈jm〉 2 〈(i − 1)i〉〈1|P i,j |j]〈ij〉〈i(j + 1)〉〈i|P i,j |i]〈(i − 1)j〉〈(i − 1)j〉〈j|P i,j |j]− 〈1(j + 1)〉〈1m〉2 〈(j + 1)m〉 2 〈i(i − 1)〉〈1|P i,j |(j + 1)]〈(i − 1)(j + 1)〉〈(i − 1)(j + 1)〉〈(j + 1)|P i,j |(j + 1)] . (3.97)
3.2. The cut-constructible parts 96Each of the terms in the above equation has an inverse power of the form 〈i|P i,j |i] =s i,j − s i+1,j and will match up with a term from a cut with a different i and j toform the coefficient of the L i functions defined in Chapter 2. Schematically,=⇒F 2−point (1, i, j) log(s i,j ) + F 2−point (1, i + 1, j) log(s i+1,j )( ) ()c〈i|P i,j |i] + . . . clog (s i,j ) + −〈i|P i,j |i] + . . . log (s i+1,j )= cL 1 (s i,j , s i+1,j ) + . . . (3.98)Here the dots represent the other pieces which are not paired up from this particularcut combination. We find that when all cuts are considered every single log pairs toform an L 1 . Since the bubble integral has the following ǫ expansionI 2 (s) ∝ 1 ǫ+ 2 − log (s) + O(ǫ), (3.99)the pairing of all the logarithms is equivalent to the disappearance of all ǫ −1 polesin the amplitude.The technique for obtaining S is identical to that described above, however herethe intermediate integrands are more complicated. We use the following form of theSchouten identity [123] to simplify the intermediate integrands, before we do the zintegration,〈aλ〉〈bλ〉〈cλ〉 = 1 ( 〈ab〉〈bc〉〈bλ〉 − 〈ac〉 )〈cλ〉(3.100)and this has the effect of separating poles and simplifying the resulting residues.To avoid repetition we delay explicitly writing out S until we combine all the cutstogether in the following section.3.2.5 Combined cuts: The cut-constructible pieces of theφ-MHV amplitudeWe are finally ready to piece together the combination of four-, three- and two-cutsto obtain the full cut-constructible piece of the φ-MHV amplitude,C n (φ, 1 − , . . ., m − , . . .,n + ). In general we found that we could write C n in the followingwayC n (φ, 1 − , 2 + , . . .,m − , . . .,n + )
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:
Page 23 and 24:
Page 25 and 26:
Page 27 and 28:
1.2. Electroweak Symmetry Breaking
Page 29 and 30:
Page 31 and 32:
Page 33 and 34:
1.3. Higgs searches at colliders 18
Page 35 and 36:
Page 37 and 38:
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 57 and 58:
2.3. Forde’s Laurent expansion me
Page 59 and 60: 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
Page 109: 3.2. The cut-constructible parts 94
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
Page 143 and 144: 4.4. Higgs plus four gluon amplitud
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
Page 175 and 176:
6.5. LHC results 160of describing T
Page 177 and 178:
6.5. LHC results 162Figure 6.6: p T
Page 179 and 180:
6.6. Summary 164at LO and the effec
Page 181 and 182:
6.6. Summary 166of Higgs masses, 15
Page 183 and 184:
Chapter 7. Conclusions 168one can d
Page 185 and 186:
Appendix ASpinor Helicity Formalism
Page 187 and 188:
A.1. Spinor Helicity Formalism nota
Page 189 and 190:
A.3. Pure QCD amplitudes 174Further
Page 191 and 192:
Appendix BOne-loop Basis IntegralsI
Page 193 and 194:
B.2. Box Integral Functions 178boxe
Page 195 and 196:
B.2. Box Integral Functions 180L 2
Page 197 and 198:
Bibliography 182[10] R. Feynman, Ma
Page 199 and 200:
Bibliography 184[36] M. Bahr et. al
Page 201 and 202:
Bibliography 186[63] M. Shifman, A.
Page 203 and 204:
Bibliography 188[85] R. V. Harlande
Page 205 and 206:
Bibliography 190[104] R. K. Ellis,
Page 207 and 208:
Bibliography 192[125] G. Ossola, C.
Page 209 and 210:
Bibliography 194[148] R. Boels and
Page 211 and 212:
Bibliography 196[170] Z. Bern and A
Page 213 and 214:
Bibliography 198anti-t + 2 jets at
Page 215:
Bibliography 200[214] C. Anastasiou
show all

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

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?