--- title: Dark Abelian Higgs Model url: https://www.emergentmind.com/topics/dark-abelian-higgs-model type: topic --- # Dark Abelian Higgs Model Searching arXiv for the provided Dark Abelian Higgs Model papers to ground the article in current paper records. arXiv search query: 1409.3590 Higgs Model Coupled to Dark Photons The Dark Abelian Higgs Model denotes a class of hidden-sector extensions in which an extra Abelian gauge symmetry, usually written \(U(1)_D\) or \(U(1)_{\text d}\), is spontaneously broken by a complex scalar charged under that symmetry, producing a massive dark gauge boson and a physical dark Higgs scalar. In its minimal renormalizable form, the dark sector is coupled to the Standard Model through the two familiar singlet portals—gauge kinetic mixing and the Higgs portal—while more elaborate realizations add supersymmetry, classically scale-invariant symmetry breaking, neutrino portals, or non-Abelian ultraviolet completions [1409.3590] [2209.03383] [1308.6071] [2308.07845]. ## 1. Core field content and minimal formulations Across the literature, the same basic structure appears in different notations. In the hidden \(U(1)\) model with visible–dark kinetic mixing, the dark sector contains a complex scalar \(\phi\) and a dark gauge field \(A_\mu\), with \[ \mathcal L_{\text{Dark}} = |D(A)\phi|^2 - m^2 |\phi|^2 -\frac{\lambda}{2}(|\phi|^2)^2 -\frac{1}{4e_A^2}F_{\mu\nu}(A)F^{\mu\nu}(A), \] while the visible photon \(B_\mu\) couples only through \[ \mathcal L_I = -\frac{1}{4e_B^2}F_{\mu\nu}(B)F^{\mu\nu}(B) +\frac{\gamma}{2}F_{\mu\nu}(A)F^{\mu\nu}(B). \] The gauge symmetry is therefore \[ U(1)_{\text{dark}} \times U(1)_{\text{vis}}, \] and the hidden scalar is charged only under the dark factor [1409.3590]. A closely related “minimal dark abelian gauge sector” writes the dark symmetry as \(U(1)_D\), with a complex SM-singlet scalar \(S\), dark gauge boson \(Z_D\), kinetic mixing with hypercharge, and Higgs portal coupling. In that notation the scalar potential is \[ V(H,S) = -\mu^2 |H|^2 + \lambda |H|^4 - \mu_S^2 |S|^2 + \lambda_S |S|^4 + \kappa |H|^2 |S|^2, \] with \[ D_\mu S = \partial_\mu S - i g_D \hat Z_{D \mu} S. \] The paper fixes the dark charge to \(q_S=1\) and treats all SM fields as neutral under \(U(1)_D\) [2209.03383]. A minimal dark-portal formulation uses one dark Higgs boson \(h_D\) and one dark photon \(\gamma_D\), with \[ {\cal L}_{\rm gauge} = -\frac{1}{4} {\vec W}_{\mu \nu} \cdot {\vec W}^{\mu \nu} -\frac{1}{4} B_{\mu \nu} B^{\mu \nu} -\frac{1}{4} C_{\mu \nu} C^{\mu \nu} -\frac{\epsilon}{2} B_{\mu \nu} C^{\mu \nu}, \] and \[ {\cal L}_{\rm scalar} = \vert D_\mu \Phi \vert^2 + \vert D_\mu \chi \vert^2 - V _{\rm scalar}(\Phi, \chi), \] where \(\chi\) is the complex dark Higgs field and \(C_\mu\) the dark gauge boson [1308.6071]. Several representative realizations differ mainly by the portal content and the symmetry-breaking implementation. | Realization | Dark-sector content | Distinctive feature | |---|---|---| | Minimal hidden \(U(1)\) [1409.3590] | \(\phi\), \(A_\mu\), visible photon \(B_\mu\) | Kinetic mixing rescales the dark gauge coupling | | Hidden Abelian Higgs Model [2209.03383] | \(S\), \(Z_D\) | Minimal renormalizable UV completion of a dark-photon model | | \(\mathcal N=2\) SUSY two-\(U(1)\) model [1410.7701] | visible and hidden gauge-Higgs sectors | SUSY fixes a Higgs-portal-type interaction | | Dark Abelian Sector Model [2308.07845] | \(\rho\), \(C_\mu\), \(f'_{\rm d}\), \(\nu_R\) | Three portals and full 1-loop renormalization | This common structure supports a useful synthesis: the Dark Abelian Higgs Model is not one unique Lagrangian, but a family of broken-\(U(1)\) hidden sectors whose defining ingredients are a dark Abelian gauge field, a complex symmetry-breaking scalar, and one or more renormalizable portals to the SM. ## 2. Spontaneous symmetry breaking and the physical spectrum The symmetry-breaking mechanism is the standard Abelian Higgs mechanism, modified only by the portal structure. In the hidden-\(U(1)\) toy model one assumes \[ m^2<0,\qquad \lambda>0, \] so that \[ V(\phi) = m^2 |\phi|^2+\frac{\lambda}{2}(|\phi|^2)^2 \] has the Mexican-hat form with \[ v=\sqrt{-\frac{2m^2}{\lambda}}. \] Using the Kibble parametrization, \[ \phi(x)=\frac{1}{\sqrt2}\,(v+h(x))\,e^{i\xi(x)/v}, \] and going to unitary gauge removes the Goldstone mode \(\xi\), leaving one real dark Higgs fluctuation \(h\) and one massive dark vector [1409.3590]. In the minimal \(U(1)_D\) realization, the field expansions are \[ H \to \begin{pmatrix} 0 \\ (v + h_0)/\sqrt{2} \end{pmatrix}, \qquad S \to \frac{(v_S + S_0)}{\sqrt{2}}, \] and the dark gauge boson mass is approximately \[ m_{Z_D}=g_D v_S \] in the regime \(\epsilon\ll 1\) and \(m_{Z_D}\ll m_Z\). The CP-even scalar mass matrix is \[ \mathcal{M}_{hs}^2= \begin{pmatrix} 2\, \lambda v^2 & \kappa \, v \, v_S \\ \kappa \, v \, v_S & 2 \, \lambda_S v_S^2 \end{pmatrix}, \] diagonalized by a mixing angle \(\theta_h\) through \[ \begin{pmatrix} h \\ S \end{pmatrix} = \begin{pmatrix} \cos \theta_h & - \sin \theta_h\\ \sin \theta_h & \cos \theta_h \end{pmatrix} \begin{pmatrix} h_0 \\ S_0 \end{pmatrix}. \] For small portal coupling, \[ s_h \equiv \sin\theta_h \simeq \frac{\kappa v v_S}{m_S^2-m_h^2}. \] Thus the physical spectrum is a mostly-SM Higgs plus a mostly-dark Higgs, together with a massive dark photon [2209.03383]. The counting of physical degrees of freedom follows the usual Abelian Higgs pattern. Before spontaneous symmetry breaking one has a massless Abelian gauge boson with two physical polarizations and a complex scalar with two real degrees of freedom; afterward, the Goldstone is eaten, the vector becomes massive with three polarizations, and one real scalar remains [1409.3590]. A recurrent misconception is that the dark Higgs mechanism in these models is intrinsically exotic. In the simple hidden-\(U(1)\) construction it is instead “best described as a standard Abelian Higgs model with mixing-induced parameter rescaling” [1409.3590]. ## 3. Portal structure and mixing mechanisms The two standard renormalizable portals are gauge kinetic mixing and the Higgs portal, and much of the model dependence is the manner in which these two structures are combined. In the simplest visible–dark \(U(1)\times U(1)\) model, the mixed kinetic term is removed by \[ A'_\mu=A_\mu,\qquad B'_\mu=B_\mu-\gamma e_B^2 A_\mu, \] which leads to the effective dark coupling \[ \tilde e = \frac{e_A}{\sqrt{1-(\gamma e_B e_A)^2}}. \] After rescaling the dark gauge field to canonical normalization, the low-energy theory is an Abelian Higgs model with \[ D_\mu[A]=\partial_\mu-i\tilde e\,A_\mu, \] so kinetic mixing acts as a rescaling of the dark gauge coupling and therefore of the dark gauge-boson mass, \[ m_A=\tilde e\,v. \] In that construction the main effect of mixing is not a new symmetry-breaking pattern but stronger dark-sector interactions [1409.3590]. In the light hidden Abelian Higgs model, kinetic mixing with hypercharge is written as \[ \mathcal{L}_{ZB} = - \frac{1}{4} \hat B_{\mu \nu} \hat B^{\mu \nu} - \frac{1}{4} \hat Z_{D \mu \nu} \hat Z_D^{\mu \nu} + \frac{\epsilon}{2 c_W} \hat Z_{D \mu \nu} \hat B^{\mu \nu}, \] and after diagonalization the low-mass limit gives the familiar photon-like coupling \[ \mathcal{L}_\text{NC} \simeq eA_\mu J^\mu_\text{EM} - \epsilon e Z_{D\mu}J^\mu_\text{EM}. \] This separates production and decay in a characteristic way: dark-Higgs production via scalar mixing depends on \(s_h^2\), dark-photon decay and detector acceptance depend on \(\epsilon^2\), and the dominance of \(S\to Z_DZ_D\) depends on \(g_D\) relative to \(s_h\) [2209.03383]. Supersymmetric completions make the portal structure more rigid. In the \(\mathcal N=2\) two-\(U(1)\) model, the bosonic Lagrangian is \[ \mathcal L = -\frac14 F_{\mu\nu}F^{\mu\nu} -\frac14 G_{\mu\nu}G^{\mu\nu} +\frac{\xi}{2}F_{\mu\nu}G^{\mu\nu} +\frac12 |D_\mu(A)s|^2 +\frac12 |D_\mu(C)t|^2 - V(s,t), \] with \[ V[s,t] = \frac{1}{2(1-\xi^2)} \left[ \frac{e^2}{4}\left(|s|^2-s_0^2\right)^2 +\frac{g^2}{4}\left(|t|^2-t_0^2\right)^2 +\frac{eg\xi}{2} \left(|s|^2-s_0^2\right)\left(|t|^2-t_0^2\right) \right]. \] The visible-hidden Higgs portal term \[ V_{\rm portal} = \frac{eg\xi}{4(1-\xi^2)} \left(|s|^2-s_0^2\right)\left(|t|^2-t_0^2\right) \] is therefore not added by hand: it is generated automatically by the supersymmetric completion of gauge kinetic mixing [1410.7701]. A more general portal classification appears in the Dark Abelian Sector Model, where the dark sector is opened by three renormalizable portals: \[ -\frac a2 B^{\mu\nu}C_{\mu\nu},\qquad 2\lambda_{12}\Phi^\dagger\Phi\,\rho^\dagger\rho,\qquad -y_{\rho,j}\rho\,\bar f_\text d^{\prime L}\nu_j^{\prime R}+\text{h.c.} \] This field-strength portal, Higgs portal, and neutrino portal define a broader but still renormalizable dark Abelian Higgs framework [2308.07845]. ## 4. Dynamical realizations and constrained parameter spaces One important variant imposes classical scale invariance. In the Abelian \(U(1)_{\rm CW}\times{\rm SM}\) model there are no explicit mass terms, and the hidden scalar vev is generated radiatively by Coleman–Weinberg dynamics. The one-loop effective potential gives \[ V_1 (\phi;\mu)= \frac{\lambda_\phi(\mu) \phi^4}{4}+ \frac{3e_{\rm CW}(\mu)^4}{64 \pi ^2} \phi^4 \left(\log \left(\frac{\phi^2}{\mu^2}\right)-\frac{25}{6}\right), \] with matching condition \[ \lambda_\phi = \frac{11}{16\pi^2} \,e_{\rm CW}^4 +\lambda_{\rm P}\frac{v^2}{2\langle\phi\rangle^2} \qquad {\rm at} \quad \mu=\langle \phi\rangle. \] The dark gauge boson mass is \[ M_{Z'}^2 = e_{\rm CW}^2 \langle \phi \rangle^2, \] while the hidden scalar mass is loop-suppressed. This differs dynamically from the textbook broken-\(U(1)\) model with explicit tachyonic mass terms. The same study concludes that the minimal Abelian model without an extra singlet has no viable dark matter candidate and that the viable Abelian scenario is the singlet-extended version, in which the singlet stabilizes the Higgs potential and supplies the dark matter [1403.4953]. Supersymmetric realizations isolate a special BPS point rather than a generic parameter scan. In the \(\mathcal N=2\) hidden/visible two-\(U(1)\) construction, the quartic couplings are locked to gauge couplings, the Fayet–Iliopoulos terms set the symmetry-breaking scales, and a consistency condition \[ |\xi|<1 \] ensures positivity of the kinetic form and regularity of the potential [1410.7701]. This is not the generic non-SUSY Dark Abelian Higgs Model, but a constrained subspace in which gauge mixing, portal strength, and topological sectors are analytically controlled. At the precision level, the Dark Abelian Sector Model parameterizes the scalar extension by the second Higgs mass, Higgs mixing angle, and a Higgs self-coupling, the gauge extension by the \(Z'\) mass and a gauge-boson mixing angle, and the fermion sector by a heavy neutral fermion mass and mixing angle. Its gauge structure implies \[ M_W \neq c_w M_Z \] already at tree level, so the dark Abelian sector can shift electroweak precision observables appreciably. In the \(M_W\) analysis, the paper finds that for \(M_{Z'}>M_Z\) the predicted \(M_W\) increases with \(|\gamma|\), whereas for \(M_{Z'}