Seeking Fixed Points in Multiple Coupling Scalar Theories in the $\epsilon$ Expansion

Fixed points for scalar theories in $4-\epsilon$, $6-\epsilon$ and $3-\epsilon$ dimensions are discussed. It is shown how a large range of known fixed points for the four dimensional case can be obtained by using a general framework with two couplings. The original maximal symmetry, $O(N)$, is broken to various subgroups, both discrete and continuous. A similar discussion is applied to the six dimensional case. Perturbative applications of the $a$-theorem are used to help classify potential fixed points. At lowest order in the $\epsilon$-expansion it is shown that at fixed points there is a lower bound for $a$ which is saturated at bifurcation points.