  We present the Callan-Symanzik-Lifshitz method to approaching the critical
behaviors of systems with arbitrary competing interactions. Every distinct
competition subspace in the anisotropic cases define an independent set of
renormalized vertex parts via normalization conditions with nonvanishing
distinct masses at zero external momenta. Otherwise, only one mass scale is
required in the isotropic behaviors. At the critical dimension, we prove: i)
the existence of the Callan-Symanzik-Lifshitz equations and ii) the
multiplicative renormalizability of the vertex functions using the inductive
method. Away from the critical dimension, we utilize the orthogonal
approximation to compute higher loop Feynman integrals, anisotropic as well as
isotropic, necessary to get the exponents $\eta_{n}$ and $\nu_{n}$ at least up
to two-loop level. Moreover, we calculate the latter exactly for isotropic
behaviors at the same perturbative order. Similarly to the computation in the
massless formalism, the orthogonal approximation is found to be exact at
one-loop order. The outcome for all critical exponents matches exactly with
those computed using the zero mass field-theoretic description renormalized at
nonvanishing external momenta.
