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

4.3. Rational Terms 123and{C (2) 2s124 〈24〉〈34〉 2 [41] 22 = −L 3 (s 124 , s 12 )3[42]+ 〈34〉[41] (3s 124〈34〉[41] + 〈24〉〈3|p φ |1][42])L3[42] 2 2 (s 124 , s 12 )(2s124 〈34〉 2 [41] 2+− 〈24〉〈3|p )φ|1] 2L〈24〉[42] 3 1 (s 124 , s 12 )3s 124 [42]+ 〈3|p φ|1] (4s 124 〈34〉[41] + 〈3|p φ |1](2s 14 + s 24 ))Ls 124 〈24〉[42] 3 0 (s 124 , s 12 )− 2s 123〈23〉〈34〉 2 [31] 23[32]+ 〈23〉〈4|p φ|1] 2L 1 (s 123 , s 12 )3s 123 [32]L 3 (s 123 , s 12 ) + 〈23〉〈34〉[31]〈4|p φ|1]L 2 (s 123 , s 12 )3[32]} { }− (2 ↔ 4) . (4.29)In the above formulae (and those following) we stress that the symmetrising actionapplies to the entire formula, and also acts on the kinematic invariants of the basisfunctions. We see that C 2 (φ, 1 + , 2 − , 3 − , 4 − ) vanishes in the soft Higgs limit p φ → 0.4.2.4 The Cut-Completion termsThe basis functions L 3 (s, t) and L 2 (s, t) are singular as s → t. Since this is anunphysical limit one expects to find some cut-predictable rational pieces which ensurethe correct behaviour of the amplitude as these quantities approach each other.These rational pieces are called the cut-completion terms and are obtained by makingthe following replacements in (4.29)(L 3 (s, t) → ˆL 1 13 (s, t) = L 3 (s, t) −2(s − t) 2 s + 1 ),tL 2 (s, t) → ˆL 1 12 (s, t) = L 2 (s, t) −2(s − t)(s + 1 ),tL 1 (s, t) → ˆL 1 (s, t) = L 1 (s, t),L 0 (s, t) → ˆL 0 (s, t) = L 0 (s, t). (4.30)4.3 Rational TermsWe now turn our attention to the calculation of the remaining rational part of theamplitude. In general the cut-unpredictable rational part of φ plus gluon amplitudes