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

3.2. The cut-constructible parts 94two helicity solutions and two species can contribute. In previous sections we foundthat these types of terms had the following breakdown,(D ∝ G + 1 − N ) (fF + 1 − N )fS (3.85)4N c N cwith G, F and S becoming increasingly more complicated. With this in mind weinspect the integrand of diagram (b),D (b)g,+ = A (0)n+2−(j−i) (φ, l+ 1 , (j + 1)+ , . . .,1 − , . . .,(i − 1) + , l − 2 )×A (0)(j−i)+2 (l+ 2 , i + , . . .,m − , . . .,j + , l − 1 )= −A (0) 〈1l 2 〉 4 〈ml 1 〉 4 〈(i − 1)i〉〈j(j + 1)〉n〈l 1 (j + 1)〉〈(i − 1)l 2 〉〈l 2 l 1 〉 2 〈l 2 i〉〈jl 1 〉〈1m〉4, (3.86)D (b)g,− = A (0)n+2−(j−i) (φ, l− 1 , (j + 1)+ , . . .,1 − , . . .,(i − 1) + , l + 2 )×A (0)(j−i)+2 (l− 2 , i+ , . . .,m − , . . .,j + , l + 1 )= −A (0) 〈ml 2 〉 4 〈1l 1 〉 4 〈(i − 1)i〉〈j(j + 1)〉n〈l 1 (j + 1)〉〈(i − 1)l 2 〉〈l 2 l 1 〉 2 〈l 2 i〉〈jl 1 〉〈1m〉 4.(3.87)When we combine the two contributions we find the following,g = −A (0) (〈1l 2 〉 4 〈ml 1 〉 4 + 〈ml 2 〉 4 〈1l 1 〉 4 )〈(i − 1)i〉〈j(j + 1)〉n(3.88)〈l 1 (j + 1)〉〈(i − 1)l 2 〉〈l 2 l 1 〉 2 〈l 2 i〉〈jl 1 〉〈1m〉 4D (b)The Schouten identity now can be applied in the same manner as previous casesand produces the following integrands,((D (b) = −A (0)n G 2−cut (a, i, j) + 4 1 − N )fF 2−cut (a, i, j)4N( c+2 1 − N ) )fS 2−cut (a, i, j)N cwith,G 2−cut (a, i, j) =(3.89)〈(i − 1)i〉〈j(j + 1)〉〈l 1 l 2 〉 2〈l 1 (j + 1)〉〈(i − 1)l 2 〉〈l 2 i〉〈jl 1 〉 , (3.90)F 2−cut (a, i, j) = 〈1l 2〉〈ml 1 〉〈ml 2 〉〈1l 1 〉〈(i − 1)i〉〈j(j + 1)〉〈l 1 (j + 1)〉〈(i − 1)l 2 〉〈l 2 i〉〈jl 1 〉〈1m〉 2 , (3.91)S 2−cut (a, i, j) = 〈1l 2〉 2 〈ml 1 〉 2 〈ml 2 〉 2 〈1l 1 〉 2 〈(i − 1)i〉〈j(j + 1)〉〈l 1 (j + 1)〉〈(i − 1)l 2 〉〈l 2 i〉〈jl 1 〉〈1m〉 4 〈l 1 l 2 〉 2 . (3.92)In these equations a ∈ {1, m} refers to the negative helicity leg which is not pairedwith φ. We note that G 2−cut = I ij and as such has no 2-point coefficient associatedwith it. This leaves us with F and S which we will proceed to integrate.