  This study of the one dimensional Su-Schrieffer-Heeger model in a weak
coupling perturbative regime points out the effective mass behavior as a
function of the adiabatic parameter $\omega_{\pi}/J$, $\omega_{\pi}$ is the
zone boundary phonon energy and $J$ is the electron band hopping integral.
Computation of low order diagrams shows that two phonons scattering processes
become appreciable in the intermediate regime in which zone boundary phonons
energetically compete with band electrons. Consistently, in the intermediate
(and also moderately antiadiabatic) range the relevant mass renormalization
signals the onset of a polaronic crossover whereas the electrons are
essentially undressed in the fully adiabatic and antiadiabatic systems. The
effective mass is roughly twice as much the bare band value in the intermediate
regime while an abrupt increase (mainly related to the peculiar 1D dispersion
relations) is obtained at $\omega_{\pi}\sim \sqrt{2}J$.
