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

Fixed points for scalar theories in 4 − ε, 6 − ε and 3 − ε 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 ε-expansion it is shown that at fixed points there is a lower bound for a which is saturated at bifurcation points.