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

2.4. Spinor integration 47+ 2π[˜λη] (1− δ(〈λ|P |η])g(λ) +〈λ|P |˜λ]〈λ|P |η]k∑j=1)δ(〈λB j 〉)g(λ)〈λB j 〉 . (2.56)That is we sum over the poles coming from g(λ) and the denominator 〈λ|P |λ]〈λ|P |η].At first glance one might expect to see a piece proportional to δ(〈λ|P |λ]), howeverone cannot satisfy the vanishing of the 〈λ| and the conjugation relation simultaneously,so there is no pole here. Upon integration over λ the first term vanishes suchthat the integral is localised by the remaining δ functions. This allows to write thespinor integral of our function g(λ) as,∫P 2I = 〈λ, dλ〉[˜λ, d˜λ]〈λ|P |λ] 2g(λ)= − 1k∑P 2g(λ P) +j=1[B j η]〈B j |P |B j ]〈B j |P |η]∏ ki=1 〈B jA i 〉∏l≠j 〈B jB l 〉(2.57)where |λ P 〉 = |P |η]. Knowledge of the integral of the function g(λ) allows us totrivially determine the double cut of a bubble integral, g(λ) = 1,∆I 2 = −1. (2.58)We can also investigate the cut of a three mass triangle,∫∆I 3 = d 4 lδ(l 2 )δ(l − K 1 ) 2 1(l + K 3 ) 2∫ ∞∫= t dt 〈λ, dλ〉[˜λ, d˜λ] δ(K2 1 − t〈λ|K 1 |˜λ])0K3 2 + t〈λ|K 3|˜λ]∫K12 〈λ|K 1 |λ]= 〈λ, dλ〉[˜λ, d˜λ]〈λ|K 1 |˜λ] 2 K3 2〈λ|K 1|˜λ] + K∫1 2〈λ|K 3|˜λ]1= 〈λ, dλ〉[˜λ, d˜λ]〈λ|K 1 |˜λ]〈λ|Q|˜λ] , (2.59)where Q µ = K2 3K µ K1 2 1 + Kµ 3 . We can further simplify this by introducing a Feynmanparameter x which combines the two denominators at the cost of one extraintegration,∫11〈λ|K 1 |˜λ]〈λ|Q|˜λ] = 1dx(2.60)〈λ|R|˜λ] 2with R = (1 − x)K + xQ. As a result of this transformation the integral over λ isequal to that of the scalar bubble which leaves only the x integration,∆I 3= −0∫ 10dx 1 R2. (2.61)