The production of lepton pairs in the Higgs boson decay $h to ell^+ ell^- gamma$ is studied. The emphasis is put on the structure of the Higgs boson interaction with the fermions. This interaction is chosen as a mixture of the scalar and pseudo-scalar couplings, and, in addition, it is supposed to be non-Hermitian. We study prediction of this model for the observables in the $h to ell^+ ell^- gamma$ decay for the $e^+ e^-$, $mu^+ mu^-$ and $tau^+ tau^-$ pairs. The differential decay width and lepton forward-backward asymmetry are calculated as functions of the dilepton invariant mass for several sets of $h f bar{f}$ coupling constants. The influence of non-Hermitian $h f bar{f}$ interaction on the forward-backward asymmetry is studied, and large influence of a possible non-Hermiticity of the Higgs interaction with the top quarks on forward-backward asymmetry for $e^+ e^-$ and $mu^+ mu^-$ pairs is stressed.
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