$\phi^{n}$ quantum scalar field theories where $n$ is not an integer Consider a quantum scalar field theory with interaction terms of the form $\phi^{n}$, where $n$ is not an integer.
Where are some examples of widely-studied quantum field theories which involve such interaction terms?
 A: A perturbation theory in which is the difference of the $\phi^n$ exponent from 2 is taken as the perturbation parameter has been proposed by Bender, Milton, Moshe, Pinsky, and Simmons in their article: Novel perturbative scheme in quantum field theory.
This method is called the $\delta$- expansion. 
Please see the following article where the basics of the method is explained in section 2.
The basic construction of the method is based on the following Taylor decomposition of the interaction term:
$$(\phi^2)^{1+\delta} = \phi^2 + \sum_{n=1}^{\infty} \frac{\delta^n}{n!}\phi^2 (\mathrm{log}(\phi^2))^n $$
One can observe that when $\delta = 0$, the interaction term is quadratic and the theory is free, while when $\delta = 1$, the theory becomes the usual $\phi^4$
theory. In the delta expansion analysis, the effective quantum potential is computed order by order around the free theory corresponding to $\delta = 0$. The obtained theory
is perturbative in $\delta$ but nonperturbative in the coupling constant and the mass. 
Each order computation involves a logarithmic interaction term. This term is treated as a limit of a derivative of of a power
$$ \mathrm{log}(x) = \lim_{k \to 0} \frac{d}{dk}x^k$$
Thus the computations are performed for a polynomial theory $(\phi^2)^k$ but only the leading terms in k for $k \rightarrow 0$ need to be computed. For such a theory, the vertices of the Feynman diagrams will be of $2k$ lines. 
Also, it is possible to perform variational calculations order by order in $\delta$.
This expansion gave excellent results for problems in quantum mechanics and solvable models in 1+1 dimensions. 
The $\delta$ expansion was used to argue the triviality of $\phi^4$ theory, but did not provide a final proof. Some work was performed at its adaptation to theories with fermions (linear $\delta$ expansion), gauge theory and models in statistical physics.
