A search is begun for renormalizable Lagrangians in which the bare interaction vertices of several different fields are all related to the value of a single coupling constant. Such models have great predictive power. A study is made of the five ${\phi }^4$-type interactions of two real scalar fields. We search for renormalizable ways of imposing four relations on the five bare coupling constants. Specifically, we look for all renormalizable models in which the five bare coupling constants lie on a line in coupling-constant space, which is linearly, quadratically, or cubically parameterized by a single coupling constant. It is shown that the only theories of this type either have decoupled ${\phi }_1$, ${\phi }_2$ interactions or have the O(2)-invariant form, which is known to be renormalizable. Therefore, in this example all renormalizable interactions with a single coupling constant are symmetric under a group of field transformations; no "new" renormalizable Lagrangians are found.

• Received 10 November 1975

DOI:https://doi.org/10.1103/PhysRevD.13.832