Simple formulas for ${\sin }^2\theta {}_{\mathrm{eff}}^{\mathrm{lept}},$ $M_W,$ ${\Gamma }_l,$ and their physical applications

The calculation of the partial widths of the Z boson in the effective scheme is discussed. Simple formulas for ${\sin }^2\theta {}_{\mathrm{eff}}^{\mathrm{lept}},$ $M_W,$ and ${\Gamma }_l$ in the on-shell, $\overline{\mathrm{MS}},$ and effective renormalization schemes are presented. Their domain of validity extends to low $M_H$ values probed by current experiments, and to the range of $\Delta {\alpha }_h^{\mathrm{}\left(5\right)\mathrm{}}$ results from recent calculations. Physical applications of these formulas are illustrated. A comparative analysis of the above-mentioned schemes is presented. The paper includes a discussion of the discrepancy of the ${\sin }^2\theta {}_{\mathrm{eff}}^{\mathrm{lept}}$ values derived from the leptonic and hadronic sectors and of the shortcomings of theoretical error estimates of ${\sin }^2\theta {}_{\mathrm{eff}}^{\mathrm{lept}}$ based solely on those affecting $M_W.$