= Solution
For this <three-site Abelian Higgs chain>, the potential is minimized by $|\phi_1|=|\phi_2|=v$. Choose their phases real in a local <unitary gauge>. The scalar <kinetic terms> then give
$$
\mathcal L_{\mathrm{mass}}=g^2v^2\left[(B^1-qB^2)^2+(B^2-qB^3)^2\right]
=g^2v^2B^{\mathsf T}KB,
$$
where Lorentz contractions are understood and
$$
K=R^{\mathsf T}R=\begin{pmatrix}1&-q&0\\-q&1+q^2&-q\\0&-q&q^2\end{pmatrix},\qquad
R=\begin{pmatrix}1&-q&0\\0&1&-q\end{pmatrix}.
$$
The nonzero <eigenvalues> of $K$ equal those of
$$
RR^{\mathsf T}=\begin{pmatrix}1+q^2&-q\\-q&1+q^2\end{pmatrix},
$$
namely $1+q^2-q$ and $1+q^2+q$. The third <eigenvalue> is zero because $R$ has rank two. Both nonzero <eigenvalues> are positive for $q>0$.
\b[Mass-normalization correction:] the canonical vector <mass> term is $\tfrac12B^{\mathsf T}\mathsf M^2B$, so the <Lagrangian> actually gives $\mathsf M^2=2g^2v^2K$. Its physical <masses> are therefore
$$
\boxed{m_1=0,\qquad m_2=\sqrt2gv\sqrt{q^2-q+1},\qquad
m_3=\sqrt2gv\sqrt{q^2+q+1}.}
$$
The printed dimensionless <eigenvalue> pattern is correct, but its common scale must be $m=\sqrt2gv$, not $gv$, for the written <kinetic terms> and <vacuum> value. For a concrete counterexample set $q=1$ and $B=(A,0,-A)/\sqrt2$. This is a canonically normalized eigenmode and its <mass> term is $g^2v^2A_\mu A^\mu$, giving physical squared <mass> $2g^2v^2$ rather than the printed $g^2v^2$. The printed scale $gv$ would require a <vacuum> modulus $v/\sqrt2$ or a corresponding change in scalar normalization, neither of which is the specified potential minimum.
The unbroken transformations preserve both <vacuum> phases:
$$
\alpha_1-q\alpha_2=0,\qquad\alpha_2-q\alpha_3=0,
\qquad (\alpha_1,\alpha_2,\alpha_3)=\alpha(q^2,q,1).
$$
Thus the local breaking pattern is
$$
\boxed{U(1)^3\longrightarrow U(1).}
$$
Two scalar phases supply the longitudinal <polarizations> of the two massive <vector bosons>; one massless vector remains. This is the <Higgs mechanism>, with its <mass> <matrix> fixed by the scalar charge <matrix> and the <vacuum> normalization.<Gauge Theory by David Tong>[Section 2.5.2, expanding scalar kinetic terms around a condensate to obtain a vector mass]{ref}
Back to article page
