Continuous vacuum selection by an external source ($U_h(\mathbf{\varphi })=\lambda ({\mathbf{\varphi }}^2-v^2{)}^2/4-h{\varphi }_1$)

Consider two real scalar fields $\mathbf{\varphi }=({\varphi }_1,{\varphi }_2)$ 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 ${\mathbf{\varphi }}^2=v^2$. Choose $(v,0)$ and differentiate the scalar potential twice: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 ${\varphi }_1$ 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 areFor example, $\lambda =v=1$ and $h=0.15$ give $w\simeq 1.06792$, $m_{\mathrm{radial}}^2\simeq 2.42138$ and $m_{\mathrm{angular}}^2\simeq 0.140459$. The light angular excitation becomes a Goldstone boson as $h\downarrow 0$, 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.