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The decay rate $\gam$ of an excited dipole molecule inside a waveguide isevaluated for the strongly coupled matter-field case near a cutoff frequency$\ome_c$ without using perturbation analysis. Due to the singularity in thedensity of photon states at the cutoff frequency, we find that $\gam$ dependsnon-analytically on the coupling constant $\ggg$ as $\ggg^{4/3}$. In contrastto the ordinary evaluation of $\gam$ which relies on the Fermi golden rule(itself based on perturbation analysis), $\gam$ has an upper bound and does notdiverge at $\ome_c$ even if we assume perfect conductance in the waveguidewalls. As a result, again in contrast to the statement found in the literature,the speed of emitted light from the molecule does not vanish at $\ome_c$ and isproportional to $c\ggg^{2/3}$ which is on the order of $10^3 \sim 10^4$ m/s fortypical dipole molecules.Comment: 4 pages, 2 figure

Strongly Coupled Matter Field and Nonanalytic Decay Rate of Dipole Molecules in a Waveguide