Hadronic production of a Higgs boson in association with two jets at ...
Hadronic production of a Higgs boson in association with two jets at ...
Hadronic production of a Higgs boson in association with two jets at ...
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3.4. Cross Checks and Limits 108Therefore to check the coll<strong>in</strong>ear behaviour <strong>of</strong> the general φ-MHV amplitude, wesimply need to check th<strong>at</strong> the fermionic and scalar contributions s<strong>at</strong>isfy the follow<strong>in</strong>grel<strong>at</strong>ion,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 splitt<strong>in</strong>gamplitude for the helicity <strong>of</strong> the gluons considered. Accord<strong>in</strong>g to the def<strong>in</strong>ition<strong>of</strong> C n <strong>in</strong> eq. (3.75), there is an overall factor A (0)n , which <strong>in</strong> the coll<strong>in</strong>ear limit producesthe correct tree-level splitt<strong>in</strong>g function. It therefore rema<strong>in</strong>s to show th<strong>at</strong>,A φF,φSn;1 → A φF,φSn−1;1 (3.166)<strong>in</strong> the coll<strong>in</strong>ear limit <strong>with</strong> A φFn;1(m, n) and A φSn;1(m, n) given <strong>in</strong> eqs. (3.103) and (3.105)respectively.Coll<strong>in</strong>ear behaviour <strong>of</strong> mixed helicity gluonsWe first consider the limit where <strong>two</strong> adjacent gluons become coll<strong>in</strong>ear, one <strong>of</strong> whichhas neg<strong>at</strong>ive helicity. For def<strong>in</strong>iteness, we take the limit (m − 1) ‖ m.The coefficient <strong>of</strong> 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 th<strong>at</strong> the box contribution correctly factorises onto the lower po<strong>in</strong>t amplitude.The rema<strong>in</strong><strong>in</strong>g terms <strong>in</strong> the sub-amplitudes are proportional to one <strong>of</strong> the auxiliaryfunctions F ijm1 <strong>with</strong> F = A, K and I and which are def<strong>in</strong>ed <strong>in</strong> eqs. (3.104), (3.106)and (3.107). We shall see th<strong>at</strong> these too have the correct factoris<strong>at</strong>ion properties.Let us first consider the ranges 2 ≤ i ≤ m − 1 and m ≤ j ≤ n. When i ≤ m − 2,