3.2. The cut-constructible parts 810000000111111100000001111111000000011111110000000111111100000001111111 0000 1 111i+−0000 11110000 11110000 1111++−−0000011111000001111100000111110000011111010101+−j00000011111101000000111111010000001111110100000011111101+−0000 11110000 11110000 11110000 1111− ++ −0000 11110000 11110000 11110000 1111010101+00000001111111000000011111110000000111111100000001111111 1 −±00000 1111100000 1111100000 11111± ∓(a) (b) (c)∓00000 1111100000 1111100000 11111±∓∓ ±00000011111101000000111111111111000100000011111110000001−−+0000 11110000 11110000 11110000 1111+00000001111111 010000000111111100000001111111 0 0100000001111111 1−−+00000 1111100000 1111100000 11111+00000001111111 01000000011111110000000111111100000001111111000111−−+00000 1111100000 1111100000 1111100000 11111−+0000 11110000 11110000 11110000 1111−− + − ++ + −− +0000000111111101010100000 1111100000 1111100000 11111000000011111110101010000 11110000 11110000 11110000 1111(d) (e) (f)+010000001111110101Figure 3.6: Two-mass easy boxes for the φ-MHV amplitudes are def<strong>in</strong>ed by theposition <strong>of</strong> the <strong>two</strong> neg<strong>at</strong>ive helicity gluons, there can be an arbitrary number <strong>of</strong>positive helicity gluons <strong>at</strong> the massive vertices. The <strong>two</strong> massless legs are denotedi and j. Diagrams (a) − (e) are non-zero whilst diagram (f) which is a special casefor the 5-po<strong>in</strong>t amplitude, is zero. Diagram (c) is <strong>of</strong> <strong>in</strong>terest s<strong>in</strong>ce it allows fermions(and scalars) to propag<strong>at</strong>e <strong>in</strong> the loop.
3.2. The cut-constructible parts 82Here a ∈ {1, m} <strong>in</strong>dic<strong>at</strong>es which neg<strong>at</strong>ive helicity gluon is not paired <strong>with</strong> φ <strong>at</strong> themassive vertex. The <strong>in</strong>tegr<strong>at</strong>ion is almost identical to the one-mass case the onlydifference be<strong>in</strong>g th<strong>at</strong> i and j now play the roles <strong>of</strong> p 2 and n (or (m ± 1)). Thecoefficients are constructed from the follow<strong>in</strong>g function,b ij1m = 〈mi〉〈j1〉〈mj〉〈i1〉〈1m〉 2 〈ij〉 2 = tr −(m, i, j, 1) tr − (m, j, i, 1)s 2 ij s2 1m(3.46)where we have <strong>in</strong>troduced the not<strong>at</strong>ion tr − (a, b, c, d) = 〈ab〉[bc]〈cd〉[da]. In terms <strong>of</strong>this quantity we have the general resultsG 2m (a, i, j) = A(0) n2 (s i,j−1s j,i−1 − s i,j s i+1,j−1 ) (3.47)F 2m (a, i, j) = b ij A (0)n1m2 (s i,j−1s j,i−1 − s i,j s i+1,j−1 ) (3.48)S 2m (a, i, j) = −(b ij1m )2A(0)n2 (s i,j−1s j,i−1 − s i,j s i+1,j−1 ) (3.49)We stress a crucial not<strong>at</strong>ion subtlety <strong>in</strong> the above sets <strong>of</strong> formula when we refer tos ij we mean s ij = 〈ij〉[ji] and is the <strong>in</strong>variant formed between the pair <strong>of</strong> partons p iand p j . When we refer to s i,j we refer to s i,j = (p i + p i+1 + · · · + p j−1 + p j ) 2 whichis a mass associ<strong>at</strong>ed <strong>with</strong> the <strong>two</strong>-mass box.We observe th<strong>at</strong> we can obta<strong>in</strong> the one-mass box coefficients from the s<strong>of</strong>t-limit<strong>of</strong> the <strong>two</strong> mass boxes. This means th<strong>at</strong> to f<strong>in</strong>alise the box coefficients we merelyhave to def<strong>in</strong>e the summ<strong>at</strong>ion over the allowed boxes. In total we f<strong>in</strong>d th<strong>at</strong> the boxcoefficients associ<strong>at</strong>ed <strong>with</strong> the φ-MHV amplitude equal(−4Cn;1 4−cut (φ, 1 − , 2 + , . . ., m − , . . ., n + ) = A (0)n(1 − N f4N c)A φF,4−cutn;1 (m, n) − 2(A φG,4−cut1 − N fN c)A φS,4−cutn;1 (m, n)n;1 (m, n)), (3.50)where we def<strong>in</strong>ed A φ{G,F,S},4−cutn;1 to be the tree-factored comb<strong>in</strong><strong>at</strong>ions <strong>of</strong> box <strong>in</strong>tegralsmultiplied by their relevant coefficient. ExplicitlyA φG,4−cutn;1 (m, n) = − 1 2− 1 2n∑n+i−1∑i=1 j=i+3n∑i=1F 2me4 (s i+1,j , s i,j−1 ; s i,j , s i+1,j−1 )F 1m4 (s i,i+1, s i+1,i+2 ; s i,i+2 ) (3.51)