Physical interpretation of changing phase around the Higgs potential Disclaimer: I don't know much detail about the Higgs except for a few basic points. The field can be written as
$$\phi(x) = \Phi  e^{i\Theta(x)}$$
Which I took from the Wiki. I think I can see how the motion in the radial direction $\Phi$ works to give mass to a field. However, I do not understand what the phase rotation $\Theta(x)$ on the complex plane does to a field.
Is there a physical interpretation for this phase rotation? What does it do in the real world rather than in this abstract mathematical space? Is there even a physical interpretation for this?

 A: First of all, I think the idea of motion in this potential may be a bit misleading. The origin of the mass (or absence of mass) comes from that you can study perturbative quantum effects (there are also effects that are non-perturbative), by expanding the action around the minimum of the potential.
Moreover, since we can "only" solve around Gaussian models (which we use to establish the particle content of the theory), the first step is to expand the potential up to second-order around the minimum yielding a matrix
\begin{equation}
V^{\left(2\right)}=\left(\begin{array}{cc}
\dfrac{\partial^{2}V}{\partial\Phi\partial\Phi} & \dfrac{\partial^{2}V}{\partial\Theta\partial\Phi}\\
\dfrac{\partial^{2}V}{\partial\Phi\partial\Theta} & \dfrac{\partial^{2}V}{\partial\Theta\partial\Theta}
\end{array}\right).
\end{equation}
Such that the lagrangian becomes
\begin{equation}
\mathcal{L}=\mathcal{L}_{\text{kin}}+\left(\begin{array}{cc}
\Phi & \Theta\end{array}\right)V^{\left(2\right)}\left(\begin{array}{c}
\Phi\\
\Theta
\end{array}\right), 
\end{equation}
where $\mathcal{L_{\text{kin}}}$ is the kinetic part of the Lagrangian. Can you read of the mass of the fields from these terms?
To answer your question, there is relation between the properties of the potential under transformations and the physical properties of your model. Can you see which from the previous argument? What happens to the derivatives when the potential is invariant under changes in one of the fields?
