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For this three-site Abelian Higgs chain, the potential is minimized by $|{\varphi }_1|=|{\varphi }_2|=v$. Choose their phases real in a local unitary gauge. The scalar kinetic terms then givewhere Lorentz contractions are understood andThe nonzero eigenvalues of $K$ equal those ofnamely $1+q^2-q$ and $1+q^2+q$. The third eigenvalue is zero because $R$ has rank two. Both nonzero eigenvalues are positive for $q>0$.

Mass-normalization correction: the canonical vector mass term is $\frac{1}{2}{B}^{\mathsf{T}}{\mathsf{M}}^{2}B$, so the Lagrangian actually gives ${\mathsf{M}}^2=2g^2{v}^{2}K$. Its physical masses are thereforeThe printed dimensionless eigenvalue pattern is correct, but its common scale must be $m=\sqrt{2}gv$, not $gv$, for the written kinetic terms and vacuum value. For a concrete counterexample set $q=1$ and $B=(A,0,-A)/\sqrt{2}$. This is a canonically normalized eigenmode and its mass term is $g^2{v}^2{A}_{\mu }{A}^{\mu }$, giving physical squared mass $2g^2{v}^2$ rather than the printed $g^2{v}^2$. The printed scale $gv$ would require a vacuum modulus $v/\sqrt{2}$ or a corresponding change in scalar normalization, neither of which is the specified potential minimum.

The unbroken transformations preserve both vacuum phases:Thus the local breaking pattern isTwo 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.^[ref]

Set $C_{\mu \nu }=A_{\mu }{B}_{\nu }-A_{\nu }{B}_{\mu }$, so $\mathcal{G}=G+ieC$. Expanding the minimally coupled Lagrangian density givesThus there is a derivative cubic $AB{B}^{\ast }$ interaction vertex of order $e$, and a quartic $AAB{B}^{\ast }$ interaction vertex of order $e^2$. There is no pure $B^4$ contact term in this minimal coupling theory, nor an Abelian three-photon interaction vertex.

The straight arrowed lines in the original schematic denote charged massive vector bosons; the wavy lines denote photons. For two identical positively charged vector bosons, the leading scattering amplitude has the order-$e^2$ $t$-channel photon-exchange Feynman diagram, together with its $u$-channel interchange required by identical bosons. A quartic $AAB{B}^{\ast }$ interaction vertex cannot directly attach four external massive vector bosons. For a particle-antiparticle incoming pair, the leading channels instead are $t$-channel exchange and $s$-channel photon annihilation. The shown arrows track charge flow; an outgoing charged particle corresponds to the conjugate field at a vertex.

Two canonically normalized complex scalar fields of vacuum modulus $v$ and charge rows $R_1=(1,-q,0)$, $R_2=(0,1,-q)$ couple to three canonically normalized Abelian vector fields. The scalar kinetic terms give $g^2{v}^2|RB{|}^2$, so the physical squared-mass matrix includes the factor two shown. Its nonzero eigenvalues are $2g^2{v}^2(1+q^2\pm q)$ and its normalized null vector is $(q^2,q,1)/\sqrt{q^4+q^2+1}$. The two scalar phases are absorbed into the massive vector bosons, leaving one unbroken local Abelian generator. A compact global gauge-group interpretation additionally requires a compatible charge lattice.

A boson with spin one. A massive vector boson has three physical particle polarizations. A complex Proca field describes a charged particle and its antiparticle, each with three particle polarizations; the free classical field has three complex dynamical components after its constraints are enforced.