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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2.6. The unitarity-bootstrap 572.6.1 BCFW <strong>at</strong> one-loopS<strong>in</strong>ce the r<strong>at</strong>ional pieces <strong>of</strong> one-loop amplitudes conta<strong>in</strong> no discont<strong>in</strong>uities they havethe same k<strong>in</strong>em<strong>at</strong>ic structure as tree-level amplitudes. This lead to the realis<strong>at</strong>ionth<strong>at</strong> they could be calcul<strong>at</strong>ed, <strong>in</strong> pr<strong>in</strong>ciple, us<strong>in</strong>g BCFW recursion rel<strong>at</strong>ions [161].In the previous section the pro<strong>of</strong> <strong>of</strong> BCFW recursion rel<strong>at</strong>ions was sketched, herewe consider the <strong>in</strong>tegral over the shifted amplitude A(z),I = 1 ∮A(z)2πi z . (2.96)CThe contour C is taken around the circle <strong>at</strong> <strong>in</strong>f<strong>in</strong>ity, if, as was required <strong>in</strong> the pro<strong>of</strong>,A(z) → 0 as z → ∞, we f<strong>in</strong>d the follow<strong>in</strong>g result,0 = A(0) + ∑poles αRes z=zαA(z)z . (2.97)If, on the other hand there is a contribution from A(z) as z → ∞ (equal to C ∞ ),the rel<strong>at</strong>ion would beA(0) = C ∞ − ∑poles αRes z=zαA(z)z . (2.98)For one-loop amplitudes many <strong>of</strong> the properties <strong>of</strong> the shifted amplitude A(z) changefrom those <strong>at</strong> tree-level. For example the pro<strong>of</strong> <strong>in</strong> the previous section required th<strong>at</strong>the amplitude conta<strong>in</strong>ed only simple poles which arose from <strong>in</strong>ternal propag<strong>at</strong>orsgo<strong>in</strong>g on-shell. At one-loop the situ<strong>at</strong>ion changes and logarithms appear whichpossess discont<strong>in</strong>uities and are def<strong>in</strong>ed on the complex plane <strong>with</strong> a branch cut. Thedifferences between the complex planes for tree-level and one-loop amplitudes areshown schem<strong>at</strong>ically <strong>in</strong> Fig. 2.4 and Fig. 2.5. The BCFW recursion rel<strong>at</strong>ions mustbe altered to work <strong>at</strong> one-loop [161–163]. We beg<strong>in</strong> by assum<strong>in</strong>g once aga<strong>in</strong> th<strong>at</strong>for a specific shift A(z) → 0 as z → ∞. We then consider the follow<strong>in</strong>g vanish<strong>in</strong>g<strong>in</strong>tegral which is taken along the circle <strong>at</strong> ∞ and deformed such th<strong>at</strong> it moves aroundthe branch cuts.0 = 1 ∮A(z)2πi C z . (2.99)The <strong>in</strong>tegral, although it still vanishes, is no longer equal to the sum <strong>of</strong> residues<strong>of</strong> the simple poles. We must also <strong>in</strong>clude the contribution from the l<strong>in</strong>e <strong>in</strong>tegrals