  We study an undoped t-J model with electron-phonon interaction using the
self-consistent Born approximation (SCBA). By neglecting vertex corrections,
the SCBA solves a boson-holon model, where a holon couples to phonons and
magnons. Comparison with exact diagonalization results for the t-J model
suggests that the SCBA describes the electron-phonon interaction fairly
accurately over a substantial range of J/t values. Exact diagonalization of the
boson-holon model shows that the deviations are mainly due to the neglect of
vertex corrections for small J/t and due to the replacement of the t-J model by
the boson-holon model for large J/t. For typical values of J/t, the
electron-phonon part Sigma_ep of the electron self-energy has comparable
contributions from the second order diagram in the electron-phonon interaction
and a phonon induced change of magnon diagrams. A very simple approximation to
Sigma_ep gives a rather accurate effective mass. Using this approximation, we
study the factors influencing the electron-phonon interaction. Typically, we
find that the magnons nominally have a stronger coupling to the holon than the
phonons. The phonons, nevertheless, drive the formation of small polarons
(self-localization) due to important differences between the character of the
phonon and magnon couplings.
