As I understand it, scaling transformation are universally defined by the scaling parameter ($\lambda > 0$) such that spacetime transforms as

\(D(\lambda): \mathbb{R}^n \rightarrow \mathbb{R}^n\)

\(x \rightarrow \lambda x\)

and (components of) fields present in the theory transform according to their scaling dimension $d_{\phi}$ as

\((D(\lambda)\phi) (x )= \lambda^{-d_{\phi}} \phi(\lambda^{-1}x)\)

As Arnold said, the scaling dimension of the fields (components) can be determined by the requirement that the exponent of the path integral has scaling dimension zero.