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 ...
Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
1.4. Effective coupl<strong>in</strong>g between gluons and a <strong>Higgs</strong> <strong>in</strong> the limit <strong>of</strong> aheavy top quark 25theory one must deal <strong>with</strong> massive loops and massive external particles. Amplitudes<strong>with</strong> a <strong>Higgs</strong> <strong>boson</strong> and up to four gluons have been calcul<strong>at</strong>ed <strong>at</strong> LO <strong>in</strong> the fulltheory [66–68]. Processes which allow large amounts <strong>of</strong> colour annihil<strong>at</strong>ion typicallyhave large K factors, and as such we expect NLO contributions to gluon fusion tobe large. S<strong>in</strong>ce NLO calcul<strong>at</strong>ions <strong>in</strong> the full theory <strong>in</strong>volve <strong>two</strong> loop diagrams <strong>with</strong>a massive loop, these calcul<strong>at</strong>ions are formidable. To simplify the problem we canwork <strong>in</strong> an effective theory <strong>in</strong> which the top mass is sent to <strong>in</strong>f<strong>in</strong>ity [63–65]. Thisapproxim<strong>at</strong>ion will work well provided th<strong>at</strong> m H < 2m t . In this effective theory thetop loops are <strong>in</strong>tegr<strong>at</strong>ed out to produce vertices. These vertices arise from higherdimensionalterms <strong>in</strong> the Lagrangian which directly couple <strong>Higgs</strong> <strong>boson</strong>s and gluonfield strengths. The first <strong>of</strong> these terms is five dimensional, successive terms, whichare higher dimensional, conta<strong>in</strong> higher powers <strong>of</strong> gluon field strengths,L <strong>in</strong>teff = 1 2 CH trGµν G µν + C ′ H trG µ ν Gν ρ Gρ µ + . . .. (1.49)S<strong>in</strong>ce each term <strong>in</strong> the Lagrangian is ultim<strong>at</strong>ely four dimensional we observe th<strong>at</strong>C ′ ∼ C/m 2 t, i.e. each <strong>of</strong> the higher dimensional Lagrangian pieces are suppressedby powers <strong>of</strong> m t . Therefore <strong>in</strong> the m t → ∞ limit only the first term contributes to<strong>Higgs</strong> plus gluon amplitudes. O(m 2 H /m2 t ) corrections can be <strong>in</strong>cluded by calcul<strong>at</strong><strong>in</strong>gamplitudes us<strong>in</strong>g the higher-dimensional pieces <strong>of</strong> the Lagrangian. One can use thesehigher-dimensional effective oper<strong>at</strong>ors to calcul<strong>at</strong>e O(1/m 2 t ) corrections to <strong>Higgs</strong> plusjet amplitudes <strong>in</strong> the effective theory [69].To make predictions us<strong>in</strong>g the effective theory Lagrangian we must obta<strong>in</strong> theWilson coefficient C, this can be done by m<strong>at</strong>ch<strong>in</strong>g to fixed order calcul<strong>at</strong>ions <strong>in</strong> thefull theory. In this way one obta<strong>in</strong>s C as a perturb<strong>at</strong>ion series <strong>in</strong> α S , for example<strong>at</strong> lead<strong>in</strong>g order the (colour stripped) m<strong>at</strong>rix element for H → gg <strong>in</strong> 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 polaris<strong>at</strong>ion vector <strong>of</strong> gluon i. Wecan also calcul<strong>at</strong>e the H → gg amplitude <strong>in</strong> the full theory,where there is only onediagram, the triangle diagram shown <strong>in</strong> Fig 1.9. In the m t → ∞ limit the m<strong>at</strong>rix