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I try to show the following relation

$$ [D,:\phi^n:]=-i\left(x^{\mu} \partial_{\mu} + \frac{n}{2}\right):\phi^n: $$

where $D$ is the Dilatation operator which is given by $$ D = -~i \sum_{l,m} \left(l +\frac{n}{2}\right) a_{l,m}^{\dagger}a_{l,m},$$ $\phi$ is a free massless field in 3 dimensions and $:\phi^n:$ means the normal ordered product of the fields.

My question is: Is there a compact nice expression for $:\phi^n:$ ? Because I only have the definition $$ :\phi^2:(x) = \lim_{y \to x} \phi(y) \phi(x) - \lim_{y \to x} \frac {1}{(d-2) \omega_d} |x-y|^{2-d}$$ with $d=3$ (dimension) and $\omega_d$ the volume of the sphere in $d$ dimensions. When I try to generalize this to the $n$-th power, I can't find a compact expression for this object.

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