= Critical dimension of a monomial scalar interaction
{title2=$D_c=2n/(n-2)$}
With a canonical <kinetic term>, a <real scalar field> has <mass dimension> $(D-2)/2$. A <coupling constant (physics)> multiplying $\phi^n$ 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.
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