  Generic higher character Lifshitz critical behaviors are described using
field theory and $\epsilon_{L}$-expansion renormalization group methods. These
critical behaviors describe systems with arbitrary competing interactions. We
derive the scaling relations and the critical exponents at the two-loop level
for anisotropic and isotropic points of arbitrary higher character. The
framework is illustrated for the $N$-vector $\phi^{4}$ model describing a
$d$-dimensional system. The anisotropic behaviors are derived in terms of many
independent renormalization group transformations, each one characterized by
independent correlation lengths. The isotropic behaviors can be understood
using only one renormalization group transformation. Feynman diagrams are
solved for the anisotropic behaviors using a new dimensional regularization
associated to a generalized orthogonal approximation. The isotropic diagrams
are treated using this approximation as well as with a new exact technique to
compute the integrals. The entire procedure leads to the analytical solution of
generic loop order integrals with arbitrary external momenta. The property of
universality class reduction is also satisfied when the competing interactions
are turned off. We show how the results presented here reduce to the usual
$m$-fold Lifshitz critical behaviors for both isotropic and anisotropic
criticalities.
