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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3.2. The cut-constructible parts 96Each <strong>of</strong> the terms <strong>in</strong> the above equ<strong>at</strong>ion has an <strong>in</strong>verse power <strong>of</strong> the form 〈i|P i,j |i] =s i,j − s i+1,j and will m<strong>at</strong>ch up <strong>with</strong> a term from a cut <strong>with</strong> a different i and j t<strong>of</strong>orm the coefficient <strong>of</strong> the L i functions def<strong>in</strong>ed <strong>in</strong> Chapter 2. Schem<strong>at</strong>ically,=⇒F 2−po<strong>in</strong>t (1, i, j) log(s i,j ) + F 2−po<strong>in</strong>t (1, i + 1, j) log(s i+1,j )( ) ()c〈i|P i,j |i] + . . . clog (s i,j ) + −〈i|P i,j |i] + . . . log (s i+1,j )= cL 1 (s i,j , s i+1,j ) + . . . (3.98)Here the dots represent the other pieces which are not paired up from this particularcut comb<strong>in</strong><strong>at</strong>ion. We f<strong>in</strong>d th<strong>at</strong> when all cuts are considered every s<strong>in</strong>gle log pairs t<strong>of</strong>orm an L 1 . S<strong>in</strong>ce the bubble <strong>in</strong>tegral has the follow<strong>in</strong>g ǫ expansionI 2 (s) ∝ 1 ǫ+ 2 − log (s) + O(ǫ), (3.99)the pair<strong>in</strong>g <strong>of</strong> all the logarithms is equivalent to the disappearance <strong>of</strong> all ǫ −1 poles<strong>in</strong> the amplitude.The technique for obta<strong>in</strong><strong>in</strong>g S is identical to th<strong>at</strong> described above, however herethe <strong>in</strong>termedi<strong>at</strong>e <strong>in</strong>tegrands are more complic<strong>at</strong>ed. We use the follow<strong>in</strong>g form <strong>of</strong> theSchouten identity [123] to simplify the <strong>in</strong>termedi<strong>at</strong>e <strong>in</strong>tegrands, before we do the z<strong>in</strong>tegr<strong>at</strong>ion,〈aλ〉〈bλ〉〈cλ〉 = 1 ( 〈ab〉〈bc〉 〈bλ〉 − 〈ac〉 )〈cλ〉(3.100)and this has the effect <strong>of</strong> separ<strong>at</strong><strong>in</strong>g poles and simplify<strong>in</strong>g the result<strong>in</strong>g residues.To avoid repetition we delay explicitly writ<strong>in</strong>g out S until we comb<strong>in</strong>e all the cutstogether <strong>in</strong> the follow<strong>in</strong>g section.3.2.5 Comb<strong>in</strong>ed cuts: The cut-constructible pieces <strong>of</strong> theφ-MHV amplitudeWe are f<strong>in</strong>ally ready to piece together the comb<strong>in</strong><strong>at</strong>ion <strong>of</strong> four-, three- and <strong>two</strong>-cutsto obta<strong>in</strong> the full cut-constructible piece <strong>of</strong> the φ-MHV amplitude,C n (φ, 1 − , . . ., m − , . . .,n + ). In general we found th<strong>at</strong> we could write C n <strong>in</strong> the follow<strong>in</strong>gwayC n (φ, 1 − , 2 + , . . .,m − , . . .,n + )