  In the weak-coupling theory of superconductivity, the diagonal self-energy
term is usually disregarded so that this term is already included in the
renormalized chemical potential. Using the bulk solution, we can easily see
that the term vanishes in the quasiclassical level. However, the validity of
this treatment is obscured in non-uniform systems, such as quantized vortices.
  In this paper, we study an isolated vortex both analytically and numerically
by the quasiclassical theory and demonstrate that the finite magnitude of the
self-energy can emerge within a vortex in some odd-parity superconductors. We
also find that the existence of diagonal self-energy can induce the breaking of
axisymmetry of vortices in chiral p-wave superconductors. This implies that the
diagonal self-energy is not negligible within a vortex in odd-parity
superconductors in general, even in the weak-coupling limit.
