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

3.3. The rational pieces 103They can be obtained by evaluating the residue of the cut completion term CR ngiven in eq. (3.111) in each of the physical channels. Each of the rational pieces inthe previous section contributes an overlap piece,O 4 (φ, 1 − , 2 + , 3 − , 4 + ) = O 2344 + O 234 + O34 4 + O41 4 + O12 4 + O412 4 . (3.134)Evaluating each term explicitly,O 2344 =(N P 1 〈3| P 234 P 1234 | 2〉 2 [4 2]2s 234 3 〈2 4〉 〈2 |P 234 | 1] 2+ 1 〈3 2〉 〈3 |P 234 P 1234 | 2〉 〈3 |P 234 P 1234 | 4〉[4 2]2 〈2 4〉 2 〈2 |P 234 | 1] 〈3 |P 234 | 1])+ (2 ↔ 4) . (3.135)The overlap pieces in the s 23 and s 34 channels are given by,O 41 and O 12 both vanish, whilstO4 23 = − N (P− 〈3 2〉2 〈4 |P 123 | 2] 2 [2 4]2s 23 3 〈4 |P 123 | 1] 2 〈4 2〉+ 〈3 2〉 〈3 4〉[2 4] 〈2 |P )123| 2] 〈4 |P 123 | 2]2 [1 2] 〈4 2〉 2 , (3.136)〈4 |P 123 | 1]O4 34 = O4 23 (4 ↔ 2). (3.137)O4 412 = − N (P 1 〈2 3〉 〈4 3〉 〈3 |P 412 | 4][4 2] 2− 1 〈2 3〉 2 [4 2] 3 )2s 412 2 〈2 4〉 〈3 |P 412 | 1][4 1] 3 〈2 4〉[4 1] 2 + (2 ↔ 4) . (3.138)3.3.4 The large z behaviour of the completion termsIn order for the direct recursive contribution to correctly generate the rational terms,the shifted amplitude A (1)n (z) must vanish as z → ∞. However, the cut-completionterm CR n (z) introduced in eq. (3.111) to ensure that the cut constructible partdoes not have any spurious poles, does not vanish as z → ∞. We therefore haveto explicitly remove the contribution at infinity from the rational part, which nowbecomes [124,165],ˆR n = R D n + O n − Inf CR n , (3.139)whereInf CR n = limz→∞CR n (z). (3.140)