Hidden gauge-sector relic and portal model classes

Let $(\mathcal{X},g)$ be a declared smooth four-dimensional Lorentzian spacetime with signature $(-,+,+,+)$, and let $U\subseteq \mathcal{X}$ be a coordinate patch and local trivialization of the hidden gauge bundle. The group is compact $U(1{)}_D$. The gauge potential is a real $C^1$ one-form $A_D=A_{\mu }^{D}d{x}^{\mu }$ on $U$, and $F_D=d{A}_D$ is its $C^0$ curvature with components $F_{\mu \nu }^D={\mathrm{\partial }}_{\mu }{A}_{\nu }^D-{\mathrm{\partial }}_{\nu }{A}_{\mu }^D$; $g_D>0$ is its dimensionless kinetic normalization. Let $S,\mathrm{\Phi }\in C^2(U,\mathbb{R})$ be real scalar fields of mass dimension one and singlets under $U(1{)}_D$ and the visible gauge group. Here $\mathrm{\Phi }$ is called a modulus only if a separate compactification model identifies it as one. Let $H:U\to {\mathbb{C}}^2$ be a $C^2$ field in the visible Standard Model representation $(\mathbf{1},\mathbf{2}{)}_{Y=1\mathrm{/}2}$, using $Q=T_3+Y$, and a $U(1{)}_D$ singlet; thus $H^{†}H$ is a visible and hidden gauge singlet. $F_D^{\mu \nu }:=g^{\mu \alpha }{g}^{\nu \beta }{F}_{\alpha \beta }^D$.

Choose a vacuum invariant under the exact $Z_2$ transformation $S\mapsto -S$, with $⟨S⟩=0$, while $\mathrm{\Phi }$, $H$ and all gauge fields are even. Assume the full EFT below cutoff $\mathrm{\Lambda }$ preserves this symmetry and that $S$ is the lightest $Z_2$-odd excitation, with no lighter odd final state kinematically available. Include $\mathcal{L}\supset -\frac{1}{4g_D^2}{F}_{\mu \nu }^D{F}_D^{\mu \nu }-\frac{1}{2}{g}^{\mu \nu }{\mathrm{\partial }}_{\mu }S{\mathrm{\partial }}_{\nu }S-\frac{1}{2}{m}_S^2{S}^2-\frac{{\lambda }_S}{4!}{S}^4-\frac{{\lambda }_{HS}}{2}{S}^2{H}^{†}H-\frac{\kappa }{2{\mathrm{\Lambda }}^2}{S}^2{F}_{\mu \nu }^D{F}_D^{\mu \nu }-\frac{1}{2}{\mu }_{\mathrm{\Phi }}\mathrm{\Phi }{S}^2\mathrm{.}$ Here $m_S>0$, $m_{\mathrm{\Phi }}>0$, and ${\mu }_{\mathrm{\Phi }}\mathrm{\ne }0$; ${\lambda }_S,{\lambda }_{HS},\kappa ,g_D$ are real dimensionless couplings, ${\mu }_{\mathrm{\Phi }}$ and $\mathrm{\Lambda }$ have mass dimension one, and $F_D$ has mass dimension two. The even scalar sector supplies a canonical kinetic term and a potential term $m_{\mathrm{\Phi }}^2{\mathrm{\Phi }}^2\mathrm{/}2$. The parameters $m_S,m_{\mathrm{\Phi }}$ are potential coefficients, not automatically excitation masses. The remaining interactions are chosen so the full smooth scalar potential $V$ is bounded below over the declared cutoff range; in particular take ${\lambda }_S>0$. The declared scalar range is contained in the open patch ${\mathcal{T}}_0=\{(S,\mathrm{\Phi },H):K(S)>0\}$, where $K(S)=g_D^{-2}+2\kappa S^2\mathrm{/}{\mathrm{\Lambda }}^2$ is the full coefficient in $-K(S)F_D^2\mathrm{/}4$. If $\kappa \ge 0$ this imposes no restriction on $S$; if $\kappa <0$ it requires $\mathrm{\mid }S\mathrm{\mid }<\mathrm{\Lambda }\mathrm{/}(g_D\sqrt{-2\kappa })$. This patch contains the stipulated $S=0$ vacuum and is invariant under the exact $Z_2$ and gauge actions.

For the decay branch, adopt weakly coupled, zero-temperature Minkowski-vacuum particle kinematics, neglecting curvature, medium and loop corrections. Fix constant vacuum values $\bar{q}=(0,v_{\mathrm{\Phi }},H_0)$ with vanishing gauge backgrounds and $dV(\bar{q})=0$. Assume the canonically normalized fluctuations $s=S$ and $\phi =\mathrm{\Phi }-v_{\mathrm{\Phi }}$ are mass eigenmodes: their quadratic kinetic terms have unit normalization and no mixing, and the Hessian of the complete potential has no mixed quadratic terms involving either mode. Define their positive tree-level excitation masses by $M_S^2:={\mathrm{\partial }}_S^{2}V(\bar{q})$ and $M_{\mathrm{\Phi }}^2:={\mathrm{\partial }}_{\mathrm{\Phi }}^{2}V(\bar{q})$, with $0<M_S,M_{\mathrm{\Phi }}<\mathrm{\Lambda }$. For the displayed $S$-dependent potential terms alone, $M_S^2=m_S^2+{\lambda }_{HS}{H}_0^{†}{H}_0+{\mu }_{\mathrm{\Phi }}{v}_{\mathrm{\Phi }}$; any additional interactions contribute their own second derivatives. Define $g_{\phi SS}:={\mathrm{\partial }}_{\mathrm{\Phi }}{\mathrm{\partial }}_S^{2}V(\bar{q})$, of mass dimension one. Assume the complete cubic interaction with one $\phi$ and two $s$ fields at this order is $-g_{\phi SS}\phi s^2\mathrm{/}2$, with $g_{\phi SS}\mathrm{\ne }0$. The displayed trilinear contributes ${\mu }_{\mathrm{\Phi }}$ to this coefficient; its nonzero value alone need not exclude cancellations from additional interactions. These are branch assumptions about the complete vacuum expansion, not consequences of $m_S,m_{\mathrm{\Phi }}>0$.

For a thermal production branch, take a spatially flat FLRW chart on $I_t\times \mathrm{\Sigma }$, where $I_t\subseteq \mathbb{R}$ is an open cosmic-time interval, $\mathrm{\Sigma }$ is a spatial three-manifold with Euclidean metric ${\delta }_{ij}$ in the chart, $a\in C^2(I_t,(0,\mathrm{\infty }))$, and $g=-d{t}^2+a(t{)}^2{\delta }_{ij}d{x}^{i}d{x}^j$. Specify radiation-bath fields ${\rho }_r,p_r,T_r\in C^1(I_t)$ with ${\rho }_r,T_r>0$ and $p_r={\rho }_r\mathrm{/}3$; in this example “radiation dominated” means this bath is the dominant energy-density component on the stated interval. Its expansion rate is $\mathcal{H}(t)=\dot{a}(t)\mathrm{/}a(t)$. Let ${\mathrm{\Gamma }}_{\mathrm{c}\mathrm{h}\mathrm{e}\mathrm{m}}:I_t\to [0,\mathrm{\infty })$ be the number-changing interaction rate per $S$. If ${\mathrm{\Gamma }}_{\mathrm{c}\mathrm{h}\mathrm{e}\mathrm{m}}\gg \mathcal{H}$ for an interval so that chemical equilibrium is maintained and later becomes $\lesssim \mathcal{H}$, call the conditional production branch freeze-out. If bath reactions produce $S$ while chemical equilibrium is never attained, call it freeze-in. A distinct nonthermal branch is possible under the decay-branch assumptions above when $M_{\mathrm{\Phi }}>2M_S$: the even excitation $\phi$ then has an open $\phi \to ss$ channel. Each branch needs its own rates, initial state and entropy history; none is calculated here. In a separate non-Abelian model, a confining sector means a specified gauge theory whose low-energy physical spectrum consists of gauge-invariant color-singlet states with a mass gap; a glueball is a color-singlet bound state made from its gauge fields. Its lightest state is a relic only if its portal-induced lifetime exceeds a stated cosmological interval. This illustrative EFT supplies no abundance, observed constraint or generic string-theory prediction.