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You can estimate the neutron lifetime using dimensional analysis. Beta decay is correctly described by the well known four-fermion Fermi theory, so the amplitude must be proportional to the coupling $G_F\approx10^{-5}\text{GeV}^{-2}$ (the Fermi constant). The decay rate is proportional to the squared amplitude: $$\Gamma\propto G_F^2\thinspace.$$ $\Gamma$ ...