= Scale-breaking sources in derivative field theories
For a first-derivative real <scalar field> in one space and one time dimension, put $j_D^\mu=\mathcal L_{\partial_\mu\psi}(z\psi'+t\dot\psi)-x^\mu\mathcal L$. Its on-shell divergence is
$$
\partial_\mu j_D^\mu=\mathcal L_{\partial_\mu\psi}\partial_\mu\psi-2\mathcal L-z\left.\frac{\partial\mathcal L}{\partial z}\right|_{\psi,\partial\psi}-t\left.\frac{\partial\mathcal L}{\partial t}\right|_{\psi,\partial\psi}.
$$
A quadratic derivative density with arbitrary field-dependent coefficients has zero derivative-homogeneity term. An extra term $-\alpha\psi'^4$ instead gives $-2\alpha\psi'^4$. A position-dependent kinetic coefficient $A\rho(z)/2$ gives $-Az\rho'(z)\dot\psi^2/2$. These source terms distinguish <scale invariance> from internal field-shift symmetry.
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