= Continuous vacuum selection by an external source
{title2=$U_h(\boldsymbol\phi)=\lambda(\boldsymbol\phi^2-v^2)^2/4-h\phi_1$}
Consider two <real scalar fields> $\boldsymbol\phi=(\phi_1,\phi_2)$ with positive $\lambda,v$, standard <kinetic terms>, and the displayed <scalar potential>. At $h=0$, the <circle group> rotates one field into the other: this is a <global symmetry in field theory>, with the same rotation at every point in space. The <vacuum manifold> is the circle $\boldsymbol\phi^2=v^2$. Choose $(v,0)$ and differentiate the <scalar potential> twice:
$$
M^2=\begin{pmatrix}2\lambda v^2&0\\0&0\end{pmatrix}.
$$
The <scalar mass matrix> assigns a restoring force to the radial displacement, whereas its tangent is one of the <Goldstone directions in the scalar mass matrix>. Moving around the circle costs no <potential energy>; moving between neighbouring points in space can still cost <gradient energy>. Flat angular <potential energy> therefore means a soft collective excitation, not that every spatial pattern is free.
A positive source $h$ explicitly selects the positive $\phi_1$ direction. The minimum is $(w,0)$, where $w>v$ is the positive solution of $\lambda w(w^2-v^2)=h$. The two <eigenvalues> of the <scalar mass matrix> there are
$$
m_{\rm radial}^2=2\lambda w^2+\frac hw,\qquad m_{\rm angular}^2=\frac hw.
$$
For example, $\lambda=v=1$ and $h=0.15$ give $w\simeq1.06792$, $m_{\rm radial}^2\simeq2.42138$ and $m_{\rm angular}^2\simeq0.140459$. The light angular excitation becomes a <Goldstone boson> as $h\downarrow0$, while the radial <mass> stays finite. The source changes both the preferred direction and the radius; expanding about the old point $(v,0)$ at nonzero $h$ would miss that shift.
\Image[/continuous-symmetry-radial-angular-curvature.png]
{title=An external source selects an orientation and lifts the angular flat direction}
{description=Original dimensionless <scalar potential> example with $\lambda=v=1$. Dashed circles mark the zero-source <vacuum manifold>. The right panel follows a radial line or a circular angular path, with <potential energy> measured from the minimum appropriate to that path. A circular path has zero angular <potential energy> at $h=0$; a straight tangent displacement agrees only to quadratic order.}
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