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Radiatively corrected lepton energy distributions in top quark decays $t \to bW^+ \to b(\ell^+ ν_\ell)$ and $t \to bH^+ \to b (τ^+ ν_τ)$ and single charged prong energy distributions from subsequent $τ^+$ decays

We calculate the QED and QCD radiative corrections to the charged lepton energy distributions in the dominant semileptonic decays of the top quark $t \to b W^+ \to b(\ell^+ ν_\ell)$ $(\ell=e, μ, τ)$ in the standard model(SM), and for the decay $t \to b H^+ \to b(τ^+ ν_τ)$ in an extension of the SM having a charged Higgs boson $H^\pm$ with $m_{H^\pm} < m_t -m_b$. The QCD corrections are calculated in the leading and next-to-leading logarithmic approximations, but the QED corrections are considered in the leading logarithmic approximation only. These corrections are numerically important for precisely testing the universality of the charged current weak interactions in $t$-quark decays. As the $τ^+$ leptons arising from the decays $W^+ \to τ^+ ν_τ$ and $H^+\to τ^+ ν_τ$ are predominantly left- and right-polarised, respectively, influencing the energy distributions of the decay products in the subsequent decays of the $τ^+$, we work out the effect of the radiative corrections on such distributions in the dominant (one-charged prong) decay channels $ τ^+ \to π^+ \barν_τ, ρ^+ \barν_τ, a_1^+ \barν_τ$ and $\ell^+ ν_\ell \barν_τ$. The inclusive $π^+$ energy spectra in the decay chains $t \to b(W^+,H^+) \to b (τ^+ ν_τ) \to b (π^+ \barν_τν_τ+X)$ are calculated, which can help in searching for the induced $H^\pm$ effects at the Tevatron and the LHC.