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

3.3. The rational pieces 100+m−1∑∑n+1i=2 j=m+1( 1ρ i,j−11m (P (j,i−1)) + 1 ) m−1∑−s j,i−1 s j,in∑i=1 j=m+1( 1ρ j,i1m (P (j+1,i)) + 1 )].s j+1,i s j,i(3.111)The factor Γ n is given by,andwithρ a,bm1(P (i,j) ) =Γ n =2Π n α=1〈m| P(i,j) a| 1 〉 3N P3 〈 a |P (i,j) | a ] 2 Aab m1 +〈α α + 1〉,(3.112)〈m |P(i,j) a| 1 〉 22 〈 a |P (i,j) | a ] Kab m1, (3.113)A ab 〈m a〉 〈b 1〉m1 = − (b → b + 1),〈a b〉(3.114)Km1 ab 〈b 1〉 2〈a b〉 2 − (b → b + 1). (3.115)We have also introduced the short-hand notation,(N P = 2 1 − N )f. (3.116)N cUltimately we will require the cut-completion terms for the four parton amplitudeA (1)4 (φ, 1 − , 2 + , 3 − , 4 + ),CR 4 (φ, 1 − , 2 + , 3 − , 4 + ) = N P 12 〈1 2〉 〈2 3〉 〈3 4〉 〈4 1〉[(〈3| 2 4 |1〉3 〈3 4〉 〈2 1〉 〈3| 2 4 |1〉2 〈3 4〉 2 〈2 1〉 2 )( 1× − −3(s 234 − s 23 ) 2 〈4 2〉 2(s 234 − s 23 ) 〈4 2〉 2 + 1 )]s 23 s 234+(2 ↔ 4) + (1 ↔ 3) + (1 ↔ 3, 2 ↔ 4). (3.117)3.3.2 The recursive termsWe make a complex shift [106, 140, 141, 161–163] of the two negative gluons suchthat|ˆ1〉 = |1〉 + z|3〉, |ˆ3] = |3] − z|1], (3.118)ensuring that overall momentum is conserved sincep µ 1 (z) = pµ 1 + z 2 〈3 |γµ | 1], p µ 3 (z) = pµ 3 − z 2 〈3 |γµ | 1]. (3.119)