Higher dimensional higher derivative $\phi^4$ theory

We construct several towers of scalar quantum field theories with an $O(N)$ symmetry which have higher derivative kinetic terms. The Lagrangians in each tower are connected by lying in the same universality class at the $d$-dimensional Wilson-Fisher fixed point. Moreover the universal theory is studied using the large $N$ expansion and we determine $d$-dimensional critical exponents to $O(1/N^2)$. We show that these new universality classes emerge naturally as solutions to the linear relation of the dimensions of the fields deduced from the underlying force-matter interaction of the universal critical theory. To substantiate the equivalence of the Lagrangians in each tower we renormalize each to several loop orders and show that the renormalization group functions are consistent with the large $N$ critical exponents. While we focus on the first two new towers of theories and renormalize the respective Lagrangians to $16$ and $18$ dimensions there are an infinite number of such towers. We also briefly discuss the conformal windows and the extension of the ideas to theories with spin-$\frac{1}{2}$ and spin-$1$ fields as well as the idea of lower dimension completeness.