# Scale transformation of the scalar field and gauge field

I am reading this paper: "Magnetic monopoles in gauge field theories", by Goddard and Olive. I don't understand some scale transformations that appear in Page 1427.

Start from the energy expression in page 1426, the energy in weak interaction theory can be written as $$H=T_\phi+T_W+V, \tag{7.1}$$ where $$T_\phi, T_W$$ and $$V$$ represent scalar field kinetic part (scalar field $$\phi$$), Yang-Mills term (gauge field $$W$$), and scalar potential, respectively. With \begin{aligned} T_\phi[\phi, W] & =\int \mathrm{d}^D x F(\phi)\left(\mathscr{D}^i \phi\right)^{\dagger} \mathscr{D}^i \phi, \\ T_W[W] & =\frac{1}{4} \int \mathrm{d}^D x G_a^{i j} G_{a i j}, \\ V[\phi] & =\int \mathrm{d}^D x U(\phi) . \end{aligned} \tag{7.2-7.4} It is assumed that $$F$$ and $$U$$ are positive functions of $$\phi$$ involving no derivatives.

Under the scale transformation $$\begin{gathered} \phi(\boldsymbol{x}) \rightarrow \phi_\lambda(\boldsymbol{x})=\phi(\lambda \boldsymbol{x}), \\ W(\boldsymbol{x}) \rightarrow W_\lambda(\boldsymbol{x})=\lambda W(\lambda \boldsymbol{x}), \end{gathered} \tag{7.5}$$ we find that $$\mathcal{D}_\mu \phi(\mathbf{x})\rightarrow \lambda \mathcal{D}_\mu \phi(\lambda\mathbf{x}), \mathbf{G}_{\mu \nu}(\mathbf{x})\rightarrow \lambda^2\mathbf{G}_{\mu \nu}(\lambda \mathbf{x})$$, then \begin{aligned} T_\phi\left[\phi_\lambda, W_\lambda\right] & =\lambda^{2-D} T_\phi[\phi, W], \\ T_W\left[W_\lambda\right] & =\lambda^{4-D} T_W[W], \\ V\left[\phi_\lambda\right] & =\lambda^{-D} V[\phi] . \end{aligned} \tag{7.6}

One conclusion of such transformation is that, if $$D=3$$ and $$\phi$$ denotes the Higgs, the energy has a unique minimum with respect to $$\nu$$ at some finite value, since the energy diverges when $$\nu\rightarrow 0$$ or $$\nu \rightarrow \infty$$. While for pure Yang-Mills theory, we don't have such minimum, since the energy varies monotonically with $$\nu$$.

$$My\ question\ is:$$ why the fields are scaled the way shown in (7.5)? Why there is a $$\lambda$$ factor before $$W$$ and not $$\phi$$? Is this $$\lambda$$ a dimensional quantity?

1. The specific scaling (7.5) originates from a dilation/dilatation of space $$x\to \lambda x$$ (and similarly for space-derivatives) used in the proof of a generalized Derrick's No-Go theorem, cf. e.g. this Phys.SE post.
2. There is no dilation/dilatation of target space the scalar field $$\phi$$, however the gauge field $$W$$ should scale since the different terms in the gauge covariant derivative should scale homogeneously.