Mixed Models: Combining Incompatible Scalar Models in Any Spacetime Dimension

Abstract

Traditionally, covariant scalar field theory models are either super renormalizable, strictly renormalizable, or nonrenormalizable. The goal of `Mixed Models' is to make sense of sums of these distinct examples, e.g., $g\varphi^4_3+g'\varphi^6_3+g''\varphi^8_3$, which includes an example of each kind for spacetime dimension $n=3$. We show how the several interactions such mixed models have may be turned on and off in any order without any difficulties. Analogous results are shown for $g\varphi^4_n+g'\varphi^{138}_n$, etc., for all $n\ge3$. Different categories hold for $n=2$ such as, e.g., ${g P(\varphi)_2+g' N\!P}(\varphi)_2$, that involve polynomial ($P$) and suitable nonpolynomial ($N\!P$) interactions, etc. Analogous situations for $n=1$ (time alone) offer simple `toy' examples of how such mixed models may be constructed.