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Using the massive helicity formalism we calculate the five-body averagesquare amplitude of the decays $\tau^{-}\to \ell_i^{-} \ell_i^{+} \ell_j^{-}\bar{\nu}_{j} \nu_{\tau}$ $(\ell=e,\,\mu)$ within the Standard Model (SM), wethen introduce a dimension-five effective vertex $\Gamma^{\tau\tau \gamma}$ inorder to determine the feasibility of imposing limits on the tau anomalousmagnetic dipole moment ($a_\tau$) via the current or future experimentalmeasurements of the branching ratio for the decay $\tau^{-}\to e^{-} e^{+}e^{-} \bar{\nu}_{e} \nu_{\tau}$.

$\tau^{-}\to \ell_i^{-} \ell_i^{+} \ell_j^{-} \bar{\nu}_{j} \nu_{\tau}$ decays with a magnetic dipole term