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

3.2. The cut-constructible parts 78This leaves Fig.3.5(c), which has a far richer structure than those previouslystudied. Each loop helicity is unconstrained, which means that fermions as well asgluons can propagate around the loop. In addition, there are now two contributingtopologies which we label D (c)g,+ and D (c)g,− the subscript indicating the species andhelicity of l 1 in the φ containing amplitude.D (c)g,+ = A (0)n−1(φ, l + 1 , (n − 1)+ , . . .,m − , . . .,3 + , l − 2 )A(0) 3 (l + 2 , 2+ , l − 3 )×A (0)3 (l+ 3 , 1− , l − 4 )A(0) 3 (l+ 4 , n+ , l − 1 ), (3.26)D (c)g,− = A (0)n−1(φ, l − 1 , (n − 1) + , . . .,m − , . . .,3 + , l + 2 )A (0)3 (l − 2 , 2 + , l + 3 )×A (0)3 (l− 3 , 1− , l + 4 )A(0) 3 (l− 4 , n+ , l + 1 ). (3.27)There are two diagrams which have the same topology as (c), one with p 1 ∈ P i,j andone with p m ∈ P i,j (P i,j represents the massive leg). For convenience we consideronly the case where p m ∈ P i,j explicitly, since it is trivial to obtain the remainingdiagram from this one. The momentum constraints are as follows,and the two specific solutions for l arel µ (1) = [1n]2[n2] 〈1|γµ |2] andl 2 = l + p 2 (3.28)l 3 = l (3.29)l 4 = l − p 1 (3.30)l 1 = l 4 − p n (3.31)l µ (2) = 〈1n〉2〈2n〉 〈2|γµ |1]. (3.32)Once again the solution associated with l (1) does not contribute and we are left withonly l → l (2) . We combine the two helicity solutions (eqs. (3.26)-(3.27))D (c)g =〈m|l 1 |n] 4 〈1|l 3 |2] 4 + 〈m|l 2 |2] 4 〈1|l 4 |n] 4γ〈l 2 l 1 〉〈l 1 (n − 1)〉〈l 2 3〉〈l 3 l 4 〉〈l 3 1〉〈l 4 1〉[l 3 l 2 ][l 4 l 1 ][nl 1 ][nl 4 ][2l 2 ][2l 3 ] . (3.33)Here we have introduced γ = ∏ 4α=n−1〈α(α − 1)〉 to simplify the formula. Next weuse the momentum constraints to remove l 1 and l 2 from the numerator.D (c)g =〈m|l 4 |n] 4 〈1|l 3 |2] 4 + 〈m|l 3 |2] 4 〈1|l 4 |n] 4γ〈l 2 l 1 〉〈l 1 (n − 1)〉〈l 2 3〉〈l 3 l 4 〉〈l 3 1〉〈l 4 1〉[l 3 l 2 ][l 4 l 1 ][nl 1 ][nl 4 ][2l 2 ][2l 3 ] .