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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 bosoninteraction with the fermions. This interaction is chosen as a mixture of thescalar and pseudo-scalar couplings, and, in addition, it is supposed to benon-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-backwardasymmetry are calculated as functions of the dilepton invariant mass forseveral sets of $h f \bar{f}$ coupling constants. The influence ofnon-Hermitian $h f \bar{f}$ interaction on the forward-backward asymmetry isstudied, and large influence of a possible non-Hermiticity of the Higgsinteraction with the top quarks on forward-backward asymmetry for $e^+ e^-$ and$\mu^+ \mu^-$ pairs is stressed.

Higgs boson decay to lepton pair and photon and possible non-Hermiticity of the Yukawa interaction