  In condensed-matter systems, electrons are subjected to two different
interactions under certain conditions. Even if both interactions are weak, it
is difficult to perform perturbative calculations due to the complexity caused
by the interplay of two interactions. When one or two interactions are strong,
ordinary perturbation theory may become invalid. Here we consider undoped
graphene as an example and provide a non-perturbative quantum-field-theoretic
analysis of the interplay of electron-phonon interaction and Coulomb
interaction. We treat these two interactions on an equal footing and derive the
exact Dyson-Schwinger integral equation of the full Dirac-fermion propagator.
This equation depends on several complicated correlation functions and thus is
difficult to handle. Fortunately, we find that these correlation functions obey
a number of exact identities, which allows us to prove that the Dyson-Schwinger
equation of the full fermion propagator is self-closed. After solving this
self-closed equation, we obtain the renormalized velocity of Dirac fermions and
show that its energy (momentum) dependence is dominantly determined by the
electron-phonon (Coulomb) interaction. In particular, the renormalized velocity
exhibits a logarithmic momentum dependence and a non-monotonic energy
dependence.
