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

2.3. Forde’s Laurent expansion method 42∫=d 4 l2∏i=0δ(l 2 i ) ([Inf t A 1 A 2 A 3 ](t) + ∑poles j)Res t=tj A 1 A 2 A 3. (2.31)t − t jIn the above equation l i = l − K i where K i is one of the two kinematic invariantswhich appear in the triple cut, l 0 = l. Inf t A 1 A 2 A 3 contains all the pieces of theintegrand which contain no poles 2 in t, i.e.()lim [Inf t A 1 A 2 A 3 ](t) − A 1 (t)A 2 (t)A 3 (t)) = 0. (2.32)t→∞We can decompose the Inf piece as follows[Inf t A 1 A 2 A 3 ](t) =m∑f i t i . (2.33)i=0The residue contributions in eq. (2.31) arise from propagators of the form (l − P) 2going on-shell. Since this equation is quadratic we expect two solutions (labelledt = t ± ) and this is indeed what we observed when calculating solutions to the fouron-shell constraints in the previous section.∫d 4 l2∏i=0δ(l 2 i ) 1(l − P) 2 ∼ 1t + − t −( ∫∫−2∏d 4 l δ(l 2 i ) 1t − ti=0 +2∏)δ(l 2 i ) 1t − t −d 4 li=0(2.34)Therefore, after suitable partial fractioning we can write the triple cut integrand asfollows,∫d 4 l2∏∫δ(l 2 i)A 1 A 2 A 3 =i=0(∑ m )dtJ t f i t i + ∑i=0boxes{l}d l D cut0 . (2.35)Here we sum over the various triple cut boxes in which the t dependence has beeneliminated by evaluating the loop momenta at the specific residue values. Theremaining piece of the equation is a sum over the positive powers of t. J t representsa Jacobian factor obtained by the transformation of the integration variable from lto t. Since there is a freedom in how we define the loop expansion (i.e. a freedom inhow we choose a i ), we can choose a specific basis in which integrals over non-zero2 This operation will also be useful when calculating the rational terms [124]