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

4.2. Cut-Constructible Contributions 119We also find that three of the two-mass hard boxes (Fig. 4.1(d)) have coefficientsrelated to the coefficients of eqs. (4.5), (4.10) and (4.11),C 4;φ|12|3|4 (φ, 1 + , 2 − , 3 − , 4 − ) = s 123s 34s 23 s 12C 4;φ4|1|2|3 (φ, 1 + , 2 − , 3 − , 4 − ), (4.12)C 4;φ|23|4|1 (φ, 1 + , 2 − , 3 − , 4 − ) = s 234s 41s 23 s 34C 4;φ1|2|3|4 (φ, 1 + , 2 − , 3 − , 4 − ), (4.13)C 4;φ|34|1|2 (φ, 1 + , 2 − , 3 − , 4 − ) = s 134s 12s 41 s 34C 4;φ2|3|4|1 (φ, 1 + , 2 − , 3 − , 4 − ). (4.14)The final two-mass hard box coefficient is,C 4;φ|3|4|12 (φ, 1 + , 2 − , 3 − , 4 − ) = s (34 〈3|p φ |1] 42 〈3|p φ |2]〈3|p φ |4][21][41]〈24〉 4 m 4 )φ+〈12〉〈14〉〈2|p φ |3]〈4|p φ |3](4.15)The remaining one-mass box configuration C 4;φ3|4|1|2 is the only one which receivescontributions from N f fermionic loops,( 1C 4;φ3|4|1|2 (φ, 1 + , 2 − , 3 − , 4 − ) = s 41 s 12 C 4;φ|3|4|12 (φ, 1 + , 2 − , 3 − , 4 − )s 124 s 34(− 1 − N ) (f 2〈3|pφ |1] 24N c s 124 [24] − 1 − N ) )f [12][41]〈3|pφ |2]〈3|p φ |4]. (4.16)2 N c s 124 [24] 4Each of the coefficients (4.11), (4.10), (4.15) and (4.16) correctly tends to zero inthe soft Higgs limit (p φ → 0).4.2.2 Triangle Integral CoefficientsAltogether, there are twenty-four triangle topologies, which can be obtained fromcyclic permutations of those shown in Fig. 4.2. The different topologies can be characterisedby the number of off-shell legs. Fig. 4.2(a) has one off-shell leg, Figs. 4.2(b)-(e) have two off-shell legs while for Fig. 4.2(f) all legs are off-shell. We refer to thetriangle integrals with one- and two-off-shell legs as one-mass and two-mass respectively.They have the following form,I3 1m (s) ∝ 1 ( )1 µ2 ǫ, Iǫ 2 3 2m (s, t) ∝ 1 (( )1 µ2 ǫ ( ) µ2 ǫ )− (4.17)s −sǫ 2 (s − t) −s −tand therefore only contribute pole pieces in ǫ to the overall amplitude (as was discussedin detail in chapter 3. In fact, the sole role of these functions is to ensure