The divergence term in a density transformation

For an orientation-preserving diffeomorphism, $F_{\mathrm{dens}}^{\ast }r=(\det DF)(r\circ F)$. Along the flow of $\xi$, $\frac{\mathrm{d}}{\mathrm{d}s}{|}_{s=0}{F}_{s,\mathrm{dens}}^{\ast }r={\xi }^{\mu }{\partial }_{\mu }r+({\partial }_{\mu }{\xi }^{\mu })r=D_{\xi }r$. Omitting the divergence term would replace the off-shell identity (eq:QK) by an incorrect commutator; the term disappears on a free scalar only after imposing $(\square +m^2){\varphi }_0=0$.