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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?

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  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.

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  • $\begingroup$ Thanks, cool! Each term in the gauge covariant derivative should scale homogeneously! Works. $\endgroup$
    – Daren
    Jul 17, 2023 at 1:20

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