Particle-hole symmetry of charge excitation spectra in the paramagnetic phase of the Hubbard model

The Kotliar and Ruckenstein slave-boson representation of the Hubbard model allows to obtain an approximation of the charge dynamical response function resulting from the Gaussian fluctuations around the paramagnetic saddle-point in analytical form. Numerical evaluation in the thermodynamical limit yields charge excitation spectra consisting of a continuum, a gapless collective mode with anisotropic zero-sound velocity, and a correlation induced high-frequency mode at $\omega\approx U$. In this work we show that this analytical expression obeys the particle-hole symmetry of the model. Other formal aspects of the approach are also addressed.