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

2.8. Recent progress: One-loop automatisation 66generic integral becomes a polynomial in this parameter. We can use the resultsof [170] to relate these integrals with factors of µ 2 in the numerator to higherdimensionalscalar integrals.∫ dIn D [(µ2 ) r −2ǫ ∫µ d 4 li(−1) n+1 (4π) 2−ǫ (µ 2r )] =(2π) −2ǫ (2π) 4 (l 2 − µ 2 )((l − p 1 ) 2 − µ 2 ) . . .((l − ∑ n−1i=1 p i) 2 − µ 2 )= −ǫ(1 − ǫ) . . .(r − 1 − ǫ)In D+2r [1]. (2.127)Over the past few years many authors have investigated D-dimensional unitaritysetting varying numbers of denominators on-shell with delta functions. These includeD-dimensional generalised unitarity, [171,172] where 4-3 and 2-cuts were usedto determine amplitudes. A D-dimensional version of the triple cut was derivedin [128]. Double cuts using spinor integration in D-dimensions have been studiedextensively in [133–137, 173]. An implementation of the Laurent expansion in Ddimensions has also been developed [172].2.8 Recent progress: One-loop automatisationBefore closing this chapter the culmination of recent years work on automatisationshould be reviewed. For a long time the bottleneck in multi-leg processes was thecalculation of virtual corrections to 2 → 4 processes. The number of Feynmandiagrams associated with 2 → n processes grows factorially and as n increases newtensor integrals arise which must be reduced to the known set of basis integrals[16]. Until recently no 2 → 4 processes had been completed at one-loop usingFeynman diagram techniques, however over the last couple of years great strides inthis direction have been made, including the complete calculation of pp → tt + bb[174–176] and qq → bbbb [177]. Despite the recent progress each new parton in thefinal state brings with it many complications for the older approach. The unitaritymethods described in this chapter are well suited to multi-leg processes since thegrowth in complexity is only linked to the growth in complexity of the (on-shell) treeamplitudes from which the loops are made. Over the last couple of years severalgroups have implemented these methods into programs which can automaticallygenerate one-loop amplitudes.