Critical dimension of a monomial scalar interaction ($D_c=2n/(n-2)$)

With a canonical kinetic term, a real scalar field has mass dimension $(D-2)/2$. A coupling constant multiplying ${\varphi }^n$ therefore has mass dimension $D-n(D-2)/2$. For $n>2$, it is dimensionless precisely at $D_c=2n/(n-2)$, giving six for a cubic interaction and four for a quartic interaction. This is a power counting statement about perturbation theory, not a claim that the corresponding quantum beta function vanishes.