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

3.4. Cross Checks and Limits 1063.4 Cross Checks and Limits3.4.1 Collinear limitsThe general behaviour of a one-loop amplitude when gluons i and (i + 1) becomecollinear, such that p i → zK and p i+1 → (1 − z)K, is well known,∑h=±[A (1)n (. . .,i λ i, i + 1 λ i+1, . . .) i‖i+−−→ 1A (1)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)+A (0)n−1 (. . .,i − 1λ i−1, K h , i + 2 λ i+2, . . .)Split (1) (−K −h ; i λ i, i + 1 λ i+1) ] .(3.150)The universal splitting functions are given by [110,111,166],Split (0) (−K + ; 1 − , 2 + ) =Split (0) (−K + ; 1 + , 2 − ) =Split (0) (−K − ; 1 + , 2 + ) =z 2√z(1 − z) 〈1 2〉, (3.151)(1 − z) 2√z(1 − z) 〈1 2〉, (3.152)1√z(1 − z) 〈1 2〉, (3.153)Split (0) (−K − ; 1 − , 2 − ) = 0. (3.154)The one-loop splitting function can be written in terms of cut-constructible andrational components,Split (1) (−K −h , 1 λ 1, 2 λ 2) = Split (1),C (−K −h , 1 λ 1, 2 λ 2) + Split (1),R (−K −h , 1 λ 1, 2 λ 2)whereSplit (1),C (−K ± , 1 − , 2 + ) = Split (0) (−K ± , 1 − , 2 + ) c Γǫ ×( ) 2µ2 ǫ (z1 − 2 F 1(1, −ǫ; 1 − ǫ;−s 12 z − 1))− 2 F 1(1, −ǫ; 1 − ǫ;Split (1),C (−K + , 1 − , 2 − ) = Split (0) (−K + , 1 − , 2 − ) c Γǫ ×( ) 2µ2 ǫ (z1 − 2 F 1(1, −ǫ; 1 − ǫ;−s 12 z − 1))− 2 F 1(1, −ǫ; 1 − ǫ;(3.155))))z,z − 1(3.156))))z,z − 1