I have recently discussed this issue at length in response to another question: If a non-interacting particle behaves like an undamped wave, can an interacting particle behave like a damped wave?. In a nutshell, electrons and holes are not free particles, but excitations of a many-particle system, which stop looking particle-like after a while. This is somewhat analogous to the waves that appear on the surface of water, if
we throw a stone in it - they eventually vanish, the energy and the momentum of the stone having spread over a large volume of water.
There are also more mundane reasons why a particle-like excitation might cease to exist - electrons and hole recombine via a variety of mechanisms, notably via emission of photons and phonons, Auger process, etc.
Also, it is written that the inverse of the lifetime of an excitation is proportional to $\epsilon^2=(E-\mu)^2$ where $E$ is the energy of the particle and $\mu$ the chemical potential. Where does this formula come from ?
This statement is actually taken out of context - here one speaks a metal with partially filled conductance band (filled up to the chemical potential/Fermi level, see also Understanding the Fermi level and the Fermi-Dirac distribution.) Electrons and holes in this case are the excitations resulting from moving electrons from below the Fermi surface, to above the Fermi surface (unlike the case of semiconductors, where holes are vacancies in a filled valence band, while conduction electrons are electrons added to the otherwise empty conduction band.)
Landau Fermi liquid theory shows that a Fermi sphere, filled with electrons, can be described as a system of non-interacting particles (Landau quasiparticles), even though the actual electrons obviously strongly interact with each other. In particular, these quasiparticles turn out to have a rather long life-time, due to the phase space constraints on the Coulomb scattering (the mechanism that is responsible for the destruction of these quasiparticles.) The original argument is due to Landau, and should be found somewhere in Landau&Livshits. I propose here the version taken from these notes:
The physical basis of the Landau theory rests on the surprising ineffectiveness of electron-
electron scattering to change the momentum distribution of quasiparticles near the Fermi level.
What happens is that most of the states into which two quasiparticles near the Fermi surface
might end up after a collision are already occupied by other electrons, and therefore, according
to the Pauli exclusion principle, unavailable. Because of this “Pauli blocking” effect, which operates irrespective of the strength of the interaction, the rate at which a quasiparticle is scattered
out of a state of momentum $k\approx k_F$ vanishes for $k \longrightarrow k_F$. This result can be obtained from a simple phase space argument.
Consider a quasiparticle with initial wave vector $\vec{k}$ with $k > k_F$. At zero temperature the empty states into which the quasiparticle can decay lie within a shell of thickness $|k−k_F |$ just above the Fermi surface. The number of states in this region is clearly proportional to $|k−kF |$
– a result valid in one, two and three dimensions. Now, through the Coulomb interaction, the momentum and energy change of the quasiparticle will be offset by the momentum and energy of an electron-hole pair. In two and three dimensions, the state of the hole must lie within a shell of thickness $|k − kF |$ below the Fermi surface (see Fig. 1). This contributes another factor
of $|k−k_F |$ to the probability of decay, which, as anticipated, is thus found to be proportional to
$(k−k_F )^2$ in three dimensions.
(emphasis is mine)
Since $|k-k_F|\ll k_F$, we can linearize the dispersion relation at the Fermi surface and replace the momenta by the particle energy:
$$
\epsilon(k)\approx \epsilon(k_F) +\frac{\partial \epsilon(k)}{\partial k}|_{k=k_F}(k-k_F)=\epsilon_F + v_F(k-k_F),
$$
that is $(\epsilon(k)-\epsilon_F)^2=v_F^2(k-k_F)^2)$($=(E-\mu)^2$ in the OP notation.)
