We reconsider the self-energy of a nodal (Dirac) fermion in a two-dimensional $d$-wave superconductor. A conventional belief is that $\mathrm{Im}\Sigma (\omega ,T)\sim \max ({\omega }^3,T^3)$. We show that $\Sigma (\omega ,k,T)$ for $k$ along the nodal direction is actually a complex function of $\omega ,T$, and the deviation from the mass shell. In particular, the second-order self-energy diverges at a finite $T$ when either $\omega$ or $k-k_F$ vanish. We show that the full summation of infinite diagrammatic series recovers a finite result for $\Sigma$, but the full angle-resolved photoemission spectroscopy spectral function is nonmonotonic and has a kink whose location compared to the mass shell differs qualitatively for spin-and charge-mediated interactions.

• Received 29 November 2005

DOI:https://doi.org/10.1103/PhysRevB.73.220503