Lifetimes of Quasi-Particles and Phonons in a Superconductor at Zero Temperature

General expressions for decay rates of a quasi-particle and a phonon in a superconductor at zero temperature are obtained from Nambu's nonlocal self-consistency conditions. The valid coherence factors in the transition probabilities are encountered for the different processes we consider. The decay rate of a quasi-particle due to emission of single phonons is found to vanish with the $\frac{7}{2}$ power of its excess excitation energy above the gap. The contribution to decay arising from emission of single pairs of particles via Coulomb interaction turns out to be at the most a few percent of that coming from phonon emission. The level width of a phonon caused by decay into single pairs of particles is shown to rise like a step function from zero to $\frac{\pi }{2}$ times the normal state value when the phonon energy reaches twice the gap energy. Additional contributions to particle decay arising from two-phonon emission or, from single pair emission, are obtained from the second-order self-energy diagram. The two-phonon emission effect is found to be completely negligible in comparison to that due to single phonon emission.