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 ...
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2.2. Quadruple cuts 34the r<strong>at</strong>ional part) was used to calcul<strong>at</strong>e several phenomenologically important processes[117–119]. The ma<strong>in</strong> drawback <strong>of</strong> the method be<strong>in</strong>g th<strong>at</strong> for a given cut onewould f<strong>in</strong>d multiple basis <strong>in</strong>tegrals (for example a cut could conta<strong>in</strong> box, triangle andbubble contributions and all can enter simultaneously) and the appearance <strong>of</strong> terms<strong>in</strong> the reduction <strong>of</strong> tensors which do not conta<strong>in</strong> the appropri<strong>at</strong>e cut propag<strong>at</strong>ors.In 2004 unitarity methods were re<strong>in</strong>vigor<strong>at</strong>ed by several developments <strong>in</strong> onshelltechniques. New methods for gener<strong>at</strong><strong>in</strong>g compact helicity expressions werediscovered, the BCFW recursion rel<strong>at</strong>ions, and the MHV rules. At the same timethe idea <strong>of</strong> us<strong>in</strong>g multiple cuts was established as a useful technique, known asgeneralised unitarity.2.2 Quadruple cuts2.2.1 The quadruple cut methodAs discussed previously, massless one-loop amplitudes can be written <strong>in</strong> the follow<strong>in</strong>gbasis,A (1)n= ∑ i(c 4;i I 4,i + c 3;i I 3;i + c 2;i I 2;i ) + R n , (2.5)where I i;j is a basis <strong>in</strong>tegral <strong>with</strong> i propag<strong>at</strong>ors <strong>with</strong> a topology denoted by j (<strong>in</strong>the above equ<strong>at</strong>ion the summ<strong>at</strong>ion is over the various allowed topologies given theexternal momenta). S<strong>in</strong>ce the basis is process <strong>in</strong>dependent, the calcul<strong>at</strong>ion <strong>of</strong> oneloopamplitudes is essentially reduced to extract<strong>in</strong>g the coefficients <strong>of</strong> each basis<strong>in</strong>tegral as efficiently as possible. With this <strong>in</strong> m<strong>in</strong>d it was proposed [120] th<strong>at</strong> onecould isol<strong>at</strong>e a given box coefficient by apply<strong>in</strong>g four (r<strong>at</strong>her than <strong>two</strong>) cuts. S<strong>in</strong>ceonly the box terms <strong>in</strong> eq. (2.5) conta<strong>in</strong> four propag<strong>at</strong>ors each four-cut should isol<strong>at</strong>ean <strong>in</strong>dividual box,∫d 4 l4∏i=1δ(l 2 i)A (0)1 (l 1 , l 2 )A (0)2 (l 2 , l 3 )A (0)3 (l 3 , l 4 )A (0)4 (l 4 , l 1 )= c 4,{P }∫d 4 l4∏δ(l 2 i) (2.6)i=1