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

4.2. Cut-Constructible Contributions 1181 2 1 2 1 2 4 133φ34 C 4;φ4|1|2|3 3 φ C 4;φ|1|23|4 4 φ C 4;φ|1|2|34 4 φ C 4;φ|34|1|2(a) (b) (c) (d)2Figure 4.1: The various box integral topologies that appear for A (1)4 (φ, 1, 2, 3, 4).From the four topologies we must also include cyclic permutations of the four gluons.Here (a) has one off-shell leg (one-mass) whilst (b)-(d) have two off-shell legs. In (b)the two off-shell legs are not adjacent and we refer to this configuration to as thetwo-mass easy box, while in (c) and (d) the two off-shell legs are adjacent and welabel them as two-mass hard boxes.there are sixteen box topologies, which can be obtained from cyclic permutationsof those shown in Fig. 4.1. We find, after application of the solutions of the loopmomenta, that the coefficients of all the two-mass easy box configurations are zero.Of the remaining 12 coefficients, a further 5 are related to each other by the {2 ↔ 4}symmetry,C 4;φ4|1|2|3 (φ, 1 + , 2 − , 3 − , 4 − ) = C 4;φ2|3|4|1 (φ, 1 + , 4 − , 3 − , 2 − ), (4.5)C 4;φ|23|4|1 (φ, 1 + , 2 − , 3 − , 4 − ) = C 4;φ|1|2|34 (φ, 1 + , 4 − , 3 − , 2 − ), (4.6)C 4;φ|34|1|2 (φ, 1 + , 2 − , 3 − , 4 − ) = C 4;φ|4|1|23 (φ, 1 + , 4 − , 3 − , 2 − ), (4.7)C 4;φ|12|3|4 (φ, 1 + , 2 − , 3 − , 4 − ) = C 4;φ|2|3|41 (φ, 1 + , 4 − , 3 − , 2 − ), (4.8)C 4;φ|3|4|12 (φ, 1 + , 2 − , 3 − , 4 − ) = C 4;φ|41|2|3 (φ, 1 + , 4 − , 3 − , 2 − ). (4.9)We find that two of the one-mass box coefficients (Fig 4.1(a)) are given by,C 4;φ1|2|3|4 (φ, 1 + , 2 − , 3 − , 4 − ) =C 4;φ2|3|4|1 (φ, 1 + , 2 − , 3 − , 4 − ) =s 23 s 34 s 3 2342〈1|p φ |2]〈1|p φ |4][23][34] , (4.10)s 34 s 41 〈2|p φ |1] 32s 134 〈2|p φ |3][34][41] + s 34 s 41 〈34〉 3 m 4 φ2s 134 〈1|p φ |2]〈3|p φ |2]〈41〉 .(4.11)