I'm wondering if $\phi^5$ is a descendant in $\phi^4$-theory in $d = 4 - \epsilon$ at the conformal Wilson-Fisher fixed point, where the coupling constant is $\lambda$. The e.o.m. tells us that $\phi^3$ is a total derivative (and therefore also a descendant) of $\phi$ (let us not care about the numerical factors, and suppress the spacetime indices on the derivatives that is summed over) \begin{equation} \partial^2\phi \sim \lambda\phi^3 \ . \end{equation} Now regarding $\phi^5$, we find from $\partial^2\phi^3$ and the e.o.m. that $\phi(\partial\phi)^2$ is related to $\phi^5$ \begin{equation} \partial^2\phi^3 \sim \phi(\partial\phi)^2 + \lambda\phi^5 \quad\text{or}\quad \phi(\partial\phi)^2 \sim \lambda\phi^5 + \partial^2\phi^3 \ . \end{equation} Since $\partial^2\phi^3$ is a descendant (which in turn is a descendant of $\phi$ due to the e.o.m.), we should only treat either $\phi^5$ or $\phi(\partial\phi)^2$ as a primary.
However, if we now study $\partial^4\phi$ using the equation above, we find \begin{equation} \partial^4\phi \sim \lambda^2\phi^5 + \lambda\partial^2\phi^3 \quad\text{or}\quad \lambda^2\phi^5 \sim \lambda\partial^2\phi^3 + \partial^4\phi \ . \end{equation} From this equation it looks like $\phi^5$ is a descendant of $\phi$. Is this correct?
If so, we can use the reasoning for $\phi^6$ (by studying $\partial^2\phi^4$ and $\partial^4\phi^2$) and find that $\phi^6$ is a descendant.
If my reasoning is incorrect, which ones are the primaries with scaling dimension $\Delta = 5 + \mathcal{O}(\epsilon)$?
