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

4.5. Numerical Evaluation 129+ 〈23〉〈4|p H|1] 23s 123 [32])}ˆL 1 (s 123 , s 12 ){ }+ (2 ↔ 4) . (4.46)For convenience we have introduced the following combinations of the finite piecesof one-mass (F 1m4F) and two-mass hard (F2mh 4F ) box functions (see Appendix B),W (1)W (2)W (3)= F 1m4F (s 23, s 34 ; s 234 ) + F 2mh4F (s 41, s 234 ; m 2 H , s 23) + F 2mh4F (s 12, s 234 ; s 34 , m 2 H )= F 1m4F (s 14 , s 34 ; s 134 ) + F 2mh4F (s 12 , s 134 ; m 2 H, s 34 ) + F 2mh4F (s 23 , s 134 ; s 14 , m 2 H)= F 1m4F (s 12, s 14 ; s 124 ) + F 2mh4F (s 23, s 124 ; m 2 H , s 14) + F 2mh4F (s 34, s 124 ; s 12 , m 2 H ).In addition, to simplify the coefficients of the three-mass triangle F 3m3 (K2 1 , K2 2 , K2 3 )with three off-shell legs K 2 1, K 2 2, K 2 3 ≠ 0, we use the notation of eq. (4.22). The rationalpart of the Higgs NMHV amplitude is given by eq. (4.33) (which incorporatesthe rational A (1)4 (φ † , 1 + , 2 − , 3 − , 4 − ) amplitude derived in [106]),{(R 4 (H, 1 + , 2 − , 3 − , 4 − ) = 1 − N ) (f 1 〈23〉〈34〉〈4|pH |1][31]N c 2 3s 123 〈12〉[21][32]+ 〈24〉〈34〉〈3|p H|1][41]3s 124 s 12 [42]+ 〈2|p H|1]〈4|p H |1]3s 234 [23][34]− 〈3|p H|1] 2s 124 [42] 2− [12]2 〈23〉 2− 〈24〉(s 23s 24 + s 23 s 34 + s 24 s 34 )s 14 [42] 2 3〈12〉〈14〉[23][34][42])} { }− 2[12]〈23〉[31]2 + (2 ↔ 4) . (4.47)3[23] 2 [41][34]4.5 Numerical EvaluationIn this section we provide numerical values for the helicity amplitudes given in theprevious section at a particular phase space point. To this end, we redefine the finitepart of the Higgs amplitude as:A (1)4 (H, 1 λ 1, 2 λ 2, 3 λ 3, 4 λ 4) = c Γ A (0) (H, 1 λ 1, 2 λ 2, 3 λ 3, 4 λ 4)(− 1 ǫ 24∑( ) −µ2 ǫ(4.48)s i,i+1).+M F,g4 (λ 1 , λ 2 , λ 3 , λ 4 ) + N fN cM F,f4 (λ 1 , λ 2 , λ 3 , λ 4 ) + N sN cM F,s4 (λ 1 , λ 2 , λ 3 , λ 4 )We evaluate the amplitudes at the phase space point used by Ellis et al. [104],i=1p µ Hp µ 1= (−1.00000000000, 0.00000000000, 0.00000000000, 0.00000000000),= (+0.30674037867, −0.17738694693, −0.01664472021, −0.24969277974),