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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2.5. MHV rules and BCFW recursion rel<strong>at</strong>ions 52∆F s12 =〈13〉 2 [42]〈24〉〈4|P 12 |4] . (2.79)We observe th<strong>at</strong> there was no residue <strong>at</strong> z = 0, mean<strong>in</strong>g th<strong>at</strong> the coefficient <strong>of</strong>the <strong>two</strong> po<strong>in</strong>t function for this <strong>in</strong>tegrand came solely from the reduction <strong>of</strong> tensortriangles, <strong>in</strong> fact 〈4|P 12 |4], which appears <strong>in</strong> the denom<strong>in</strong><strong>at</strong>or is a sign<strong>at</strong>ure <strong>of</strong> a firstrank tensor triangle.In this thesis we will use the above method to construct the coefficients <strong>of</strong> scalarbubbles for the various <strong>Higgs</strong> plus four parton one-loop amplitudes th<strong>at</strong> we study.2.5 MHV rules and BCFW recursion rel<strong>at</strong>ionsIn this section we discuss <strong>two</strong> important on-shell techniques for the gener<strong>at</strong>ion <strong>of</strong>tree-level amplitudes, the MHV rules [130,131,139] and BCFW recursion rel<strong>at</strong>ions[140,141]. These techniques have had many important applic<strong>at</strong>ions, some <strong>of</strong> whichwill be described <strong>in</strong> the follow<strong>in</strong>g sections.2.5.1 The MHV/CSW rulesThe MHV (or Parke-Taylor) amplitudes have long been known to possess a remarkablysimple structure for all gluon multiplicities [142],A (0)n (1+ , . . .,i − , . . ., j − , . . .,n + ) =〈ij〉 4∏ n−1(2.80)α=1〈α(α + 1)〉〈n1〉.MHV stands for Maximally-Helicity-Viol<strong>at</strong><strong>in</strong>g because if one classes amplitudes <strong>in</strong>terms <strong>of</strong> the number <strong>of</strong> neg<strong>at</strong>ive helicity gluons present these are the first which arenon-zero, i.e:A (0)n (1+ , . . .,i + , . . .,j + , . . .,n + ) = A (0)n (1+ , . . ., i + , . . .,j − , . . .,n + ) = 0. (2.81)MHV amplitudes are the conjug<strong>at</strong>es <strong>of</strong> eq. (2.80),A (0)n (1 − , . . .,i + , . . .,j + , . . .,n − ) =[ij] 4∏ n−1(2.82)α=1[α(α + 1)][n1].All relevant tree amplitudes (<strong>with</strong> def<strong>in</strong>itions <strong>of</strong> sp<strong>in</strong>or products) for this thesis arecollected together <strong>in</strong> Appendix A.