How to find scaling dimensions of the scalar and gauge vector fields

Question

The problem is

"Find the scaling dimensions of the scalar and gauge vector fields."

As I understand, a scalar field is a field with lagrangian: $$ \mathcal{L}=\partial_{\mu} \phi^{*} \partial_{\mu} \phi-m^{2} \phi^{*} \phi \tag{1} $$ And gauge field has lagrangian: $$ \mathcal{L}=-\frac{1}{4} B_{\mu \nu} B_{\mu \nu}+\frac{m^{2}}{2} B_{\mu} B_{\mu} \tag{2} $$ So, I need to find the $\Delta$ parameter after changing the scale: $$ \begin{array}{c} x \mapsto x^{\prime}=\lambda x \\ \varphi(x) \mapsto \varphi^{\prime}\left(x^{\prime}\right)=\lambda^{-\Delta} \varphi(x) \end{array} \tag{3} $$ But I have no idea how to do that, and I am also not sure that these fields have scaling symmetry.

Answer 1

You have written the Lagrangian for a massive complex scalar field in (1). The classical scaling dimension of $\phi$ is just the mass dimension $[\phi]$. Considering d=4 dimensions, the action $S=\int d^4x L$ is dimensionless. $[S]=0$ and $[d^4x]=-4$ (length has inverse dimension of mass) mean that $[L]=4$. Therefore, (1) implies that $[\phi]=1$ and so the classical scaling dimension of the scalar field is 1.

This is also known as the engineering dimension, since when we turn on interaction terms (e.g. adding in $\phi^4$), the classical scaling dimension recieves a correction when the theory is renormalised. This correction is called the anomalous dimension. A similar story is true for the theory of a gauge vector field.