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Effective fermion-Higgs interactions at an $e^+e^-$ collider with polarized beams

We consider the possibility of new physics giving rise to effective interactions of the form $e^+e^-Hf \bar f$, where $f$ represents a charged lepton $\ell$ or a (light) quark $q$, and $H$ the recently discovered Higgs boson. Such vertices would give contributions beyond the standard model to the Higgs production processes $e^+e^- \to H\ell^+\ell^-$ and $e^+e^- \to H q \bar q$ at a future $e^+e^-$ collider. We write the most general form for these vertices allowed by Lorentz symmetry. Assuming that such interactions contribute in addition to the standard model production processes, where the final-state fermion pair comes from the decay of the $Z$ boson, we obtain the differential cross section for the processes $e^+e^- \to H\ell^+\ell^-$ and $e^+e^- \to Hq\bar q$ to linear order in the effective interactions. We propose several observables with differing CP and T properties which, if measured, can be used to constrain the couplings occurring in interaction vertices. We derive possible limits on these couplings that may be obtained at a collider with centre-of-mass energy of 500 GeV and an integrated luminosity of 500 fb$^{-1}$. We also carry out the analysis assuming that both the electron and positron beams can be longitudinally polarized, and find that the sensitivity to the couplings can be improved by a factor of 2-4 by a specific choice of the signs of the polarizations of both the electron and positron beams for the same integrated luminosity.