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The width of the neutrino decay into the electron and $W$ boson in a strongexternal magnetic field is obtained from the imaginary part of the neutrinoself-energy. This result corrects the formulae existing in the literature. Themean free path of an ultra-high energy neutrino in a strong magnetic field iscalculated. An energy cutoff for neutrinos propagating in a strong field isdefined.

A decay of the ultra-high-energy neutrino $\nu_e \to e^- W^+$ in a magnetic field and its influence on the shape of the neutrino spectrum