Effects of saddle-point singularities on the electron lifetime

We calculate the imaginary part of the response function (Im {χ}) and the imaginary part of the electron self-energy (Im {Σ}) for a two-dimensional energy band with a saddle point close to the Fermi energy EF, using lowest-order perturbation theory in the screened Coulomb interaction. We find that Im {χ}(q,{ω}) remains finite for arbitrarily small frequencies {ω} for certain directions of q if the saddle point is at EF, leading to a rich behavior of Im{Σ}. For an electron far from the saddle point, Im{Σ}(k,E(k)){~}[E(k)-EF{]}1.5$, except in specific directions of k where the power is only 4/3. For an electron close to a saddle point, a linear dependence on the energy is obtained. These results are compared with numerical calculations for the Emery model of the CuO₂ planes in high-Tc superconductors.