I am reading the yellow book Conformal Field Theory by Francesco et al, and am deeply confused by the authors' derivation of the current associated with symmetry transformations (pages 40-41). This is described in another SE question.
Keeping $O(\omega)$ terms (without any integration by parts), we can show that
$$ S' = S + \int d^dx \, \omega_a f_a - \int d^dx \, (\partial_\mu \omega_a) j_a^\mu $$
where $j_a^\mu$ is the current, given by eq (2.141) (I have verified it), and $f_a$ is some weird quantity found to be (hope that there is no mistake)
$$ \begin{aligned} f_a &\equiv \mathcal{L} \, \partial_\mu \frac{\partial x'^\mu}{\partial \omega_a} + \frac{\partial F}{\partial \omega_a} \frac{\partial \mathcal{L}}{\partial \phi} \\ &\quad + \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi)} \left[ \partial_\mu \frac{\partial F}{\partial \omega_a} - (\partial_\nu \phi) \, \partial_\mu \frac{\partial x'^{\nu}}{\partial \omega_a} \right] \end{aligned} $$
In the first answer (and also in the book) where global uniform transformation is considered, I understand their argument that $f_a$ should vanish identically (in that answer, $f_a$ is denoted by $[...]_2$). But it seems that this statement is not trivial to understand:
(Edit: According to @Qmechanic, the scaling symmetry holds in $d = 2$ only; but I think it does not prevent us to consider it as an example)
For example consider the infinitesimal uniform scaling for scalar fields (here we have only one controlling parameter $\omega$, which is the same everywhere)
$$ x' = (1 + \omega) x \qquad \phi'(x') = F(\phi(x),\omega) = \phi(x) $$
Then obviously
$$ \frac{\partial x'^\nu}{\partial \omega} = x^\nu \qquad \frac{\partial F}{\partial \omega} = 0 $$
By direct calculation, we find
$$ \begin{aligned} f &= \mathcal{L} \, \partial_\mu x^\mu + \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi)} \left[ - (\partial_\nu \phi) \, \partial_\mu x^\nu \right] \\ &= \mathcal{L} \times d - \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi)} \partial_\mu \phi \end{aligned} $$
where $d$ is the spacetime dimension. It does not look like zero at all.
So my question is: Is there any concrete example supporting that $f_a$ vanishes identically? (instead of just give some vague "physical argument")
