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 ...
Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
3.2. The cut-constructible parts 80S 1m (1) = A (0) 〈l 3 m〉 2 〈l 4 m〉 2 〈l 3 1〉〈l 4 1〉〈(n − 1)n〉3〈n1〉〈12〉〈23〉[nl 4 ] 3 [2l 3 ] 3n〈l 1 l 2 〉〈l 1 (n − 1)〉〈l 2 3〉〈1m〉 4 〈l 3 l 4 〉[l 3 l 2 ][l 4 l 1 ][nl 1 ][2l 2 ](3.41)The procedure to calcul<strong>at</strong>e these quantities is straightforward, remove l 1 , l 2 and l 4<strong>in</strong> terms <strong>of</strong> l 3 and substitute <strong>in</strong> the solution l (2) . This yields the follow<strong>in</strong>gG 1m (1) = A(0) n2 s 12s 1n , (3.42)( (0) ) AF 1m n 〈mn〉〈2m〉〈1n〉〈12〉(1) =2 s 12s 1n , (3.43)〈1m〉 2 〈2n〉 2( (0) )( ) 2 AS 1m n 〈mn〉〈2m〉〈1n〉〈12〉(1) = −2 s 12s 1n . (3.44)〈1m〉 2 〈2n〉 2This completes the analysis for one-mass boxes, we will observe <strong>in</strong> the next sectionth<strong>at</strong> the calcul<strong>at</strong>ion <strong>of</strong> the <strong>two</strong>-mass box coefficients proceeds <strong>in</strong> an identical fashion.Two-mass boxesWhen calcul<strong>at</strong><strong>in</strong>g the coefficients <strong>of</strong> one-mass boxes which appear <strong>in</strong> φ-MHV amplitudeswe noted th<strong>at</strong> there were <strong>two</strong> sorts <strong>of</strong> contributions. Diagrams <strong>in</strong> whichthe loop particle was constra<strong>in</strong>ed to be a gluon had a coefficient <strong>of</strong> the follow<strong>in</strong>gform, A (0)n s 1 s 2 where s 1 and s 2 are the <strong>in</strong>variants associ<strong>at</strong>ed <strong>with</strong> pairs <strong>of</strong> (adjacent)massless legs. The diagrams which allowed fermions to propag<strong>at</strong>e <strong>in</strong> the loop havea more complic<strong>at</strong>ed structure, they also conta<strong>in</strong> a piece <strong>of</strong> the form A (0)n s 1 s 2 but <strong>in</strong>addition conta<strong>in</strong> pieces proportional to N f , the number <strong>of</strong> light flavours.We f<strong>in</strong>d an identical situ<strong>at</strong>ion <strong>with</strong> the <strong>two</strong>-mass boxes which are depicted <strong>in</strong>Fig. 3.6. Diagrams (a), (b), (d) and (e) which conta<strong>in</strong> a gluon loop alone havecoefficients <strong>of</strong> the form A (0)n (s 1 s 2 − P 2 Q 2 ) where P and Q are the momenta <strong>of</strong> themassive legs. An example <strong>of</strong> this type <strong>of</strong> term was given <strong>in</strong> the previous chapter.We also f<strong>in</strong>d th<strong>at</strong> the diagrams which allow a fermion to propag<strong>at</strong>e <strong>in</strong> the loop havethe same breakdown as the one-mass boxes,(D (c) = G 2m (a, i, j) + 4 1 − N ) (fF 2m (a, i, j) + 2 1 − N )fS 2m (a, i, j).4N c N c(3.45)