  Studying the strong correlation effects in interacting Dirac fermion systems
is one of the most challenging problems in modern condensed matter physics. The
long-range Coulomb interaction and the fermion-phonon interaction can lead to a
variety of intriguing properties. In the strong-coupling regime, weak-coupling
perturbation theory breaks down. The validity of $1/N$ expansion with $N$ being
the fermion flavor is also in doubt since $N$ equals to $2$ or $4$ in realistic
systems. Here, we investigate the interaction between (1+2)- and
(1+3)-dimensional massless Dirac fermions and a generic scalar boson, and
develop an efficient non-perturbative approach to access the strong-coupling
regime. We first derive a number of self-consistently coupled Ward-Takahashi
identities based on a careful symmetry analysis and then use these identities
to show that the full fermion-boson vertex function is solely determined by the
full fermion propagator. Making use of this result, we rigorously prove that
the full fermion propagator satisfies an exact and self-closed Dyson-Schwinger
integral equation, which can be solved by employing numerical methods. A major
advantage of our non-perturbative approach is that there is no need to employ
any small expansion parameter. Our approach provides a unified theoretical
framework for studying strong Coulomb and fermion-phonon interactions. It may
also be used to approximately handle the Yukawa coupling between fermions and
order-parameter fluctuations around continuous quantum critical points. Our
approach is applied to treat the Coulomb interaction in undoped graphene. We
find that the renormalized fermion velocity exhibits a logarithmic
momentum-dependence but is nearly energy independent, and that no excitonic gap
is generated by the Coulomb interaction. These theoretical results are
consistent with experiments in graphene.
