2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/207623We present the analytical calculation of the contribution of order m_q^2 α_s^4 n_f^2 to the absorptive part of the vacuum polarization function of vector currents. This term constitutes an important gauge-invariant part of the full O(m_q^2 alpha_s^4) correction to the total cross-section of e^+ e^- annihilation into hadrons. The results are compared to predictions following from various optimization schemes.The complete paper is also available via the www at http://www-ttp.physik.uni-karlsruhe.de/Preprints/High Energy Physics - Phenomenologytext