On-shell divergence of a quartic scalar dilation current (${\partial }_{\mu }{J}^{\mu }=(4-n)\mathcal{L}$)

For $\mathcal{L}=(\partial \varphi {)}^2/2-\lambda {\varphi }^4$ and the displayed-weight dilation variation $\delta \varphi =\varphi +x\cdot \partial \varphi$, the full divergence identity is $(4-n)\mathcal{L}+\delta \varphi (\square \varphi +4\lambda {\varphi }^3)$. The field equation removes the second term. In four spacetime dimensions the density changes by a total divergence, so the Noether current includes the subtraction $-x^{\mu }\mathcal{L}$. This is a classical statement about generic solutions and this particular current.