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

1.4. Effective coupling between gluons and a Higgs in the limit of aheavy top quark 25theory one must deal with massive loops and massive external particles. Amplitudeswith a Higgs boson and up to four gluons have been calculated at LO in the fulltheory [66–68]. Processes which allow large amounts of colour annihilation typicallyhave large K factors, and as such we expect NLO contributions to gluon fusion tobe large. Since NLO calculations in the full theory involve two loop diagrams witha massive loop, these calculations are formidable. To simplify the problem we canwork in an effective theory in which the top mass is sent to infinity [63–65]. Thisapproximation will work well provided that m H < 2m t . In this effective theory thetop loops are integrated out to produce vertices. These vertices arise from higherdimensionalterms in the Lagrangian which directly couple Higgs bosons and gluonfield strengths. The first of these terms is five dimensional, successive terms, whichare higher dimensional, contain higher powers of gluon field strengths,L inteff = 1 2 CH trGµν G µν + C ′ H trG µ ν Gν ρ Gρ µ + . . .. (1.49)Since each term in the Lagrangian is ultimately four dimensional we observe thatC ′ ∼ C/m 2 t, i.e. each of the higher dimensional Lagrangian pieces are suppressedby powers of m t . Therefore in the m t → ∞ limit only the first term contributes toHiggs plus gluon amplitudes. O(m 2 H /m2 t ) corrections can be included by calculatingamplitudes using the higher-dimensional pieces of the Lagrangian. One can use thesehigher-dimensional effective operators to calculate O(1/m 2 t ) corrections to Higgs plusjet amplitudes in the effective theory [69].To make predictions using the effective theory Lagrangian we must obtain theWilson coefficient C, this can be done by matching to fixed order calculations in thefull theory. In this way one obtains C as a perturbation series in α S , for exampleat leading order the (colour stripped) matrix element for H → gg in the effectivetheory is,M Eff (H → gg) = −iCg µν p 1 · p 2 ǫ ∗µ1 ǫ ∗ν2 (1.50)where p i and ǫ ∗ i represent the momentum and polarisation vector of gluon i. Wecan also calculate the H → gg amplitude in the full theory,where there is only onediagram, the triangle diagram shown in Fig 1.9. In the m t → ∞ limit the matrix