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

3.4. Cross Checks and Limits 108Therefore to check the collinear behaviour of the general φ-MHV amplitude, wesimply need to check that the fermionic and scalar contributions satisfy the followingrelation,Cnφ{F,S}∑h=±(. . .,i λ i, i + 1 λ i+1, . . .) i‖i+−−→ 1C φ{F,S}n−1 (. . .,i − 1 λ i−1, K h , i + 2 λ i+2, . . .) Split (0) (−K −h ; i λ i, i + 1 λ i+1).(3.165)In other words, the F and S contributions should factorise onto the tree-level splittingamplitude for the helicity of the gluons considered. According to the definitionof C n in eq. (3.75), there is an overall factor A (0)n , which in the collinear limit producesthe correct tree-level splitting function. It therefore remains to show that,A φF,φSn;1 → A φF,φSn−1;1 (3.166)in the collinear limit with A φFn;1(m, n) and A φSn;1(m, n) given in eqs. (3.103) and (3.105)respectively.Collinear behaviour of mixed helicity gluonsWe first consider the limit where two adjacent gluons become collinear, one of whichhas negative helicity. For definiteness, we take the limit (m − 1) ‖ m.The coefficient of the box function b ij m1 enters both A φS and A φF . In this limit,b ij m1m−1‖m tr −(K, i, j, 1) tr − (K, j, i, 1)−−−−−→ ≡ b ijs 2 ij s2 K1 . (3.167)1KFor the special cases, i = m − 1 and j = m − 1, we have,b m−1,jm1= b i,m−1m1 = 0 (3.168)so that the box contribution correctly factorises onto the lower point amplitude.The remaining terms in the sub-amplitudes are proportional to one of the auxiliaryfunctions F ijm1 with F = A, K and I and which are defined in eqs. (3.104), (3.106)and (3.107). We shall see that these too have the correct factorisation properties.Let us first consider the ranges 2 ≤ i ≤ m − 1 and m ≤ j ≤ n. When i ≤ m − 2,