Asymmetric SM Portal

• Asymmetric Standard Model portals are effective interactions that couple the SM to hidden sectors via spurion insertions, breaking chiral and flavor symmetries.
• The framework systematically classifies portal operators by messenger spin and dimension, enabling matching to low-energy EFT and chiral perturbation theory.
• Experimental implications include altered meson decay rates and CP-odd signatures, with predictions refined through lattice QCD and minimal flavor violation constraints.

An asymmetric Standard Model portal is any effective interaction that couples the SM to a hidden sector in a way that violates left–right or gauge–representation symmetry on the SM side, or involves distinct SM sectors, particularly through independent spurion insertions (e.g., $\epsilon_L \neq \epsilon_R$, or flavor/gauge-dependent couplings). These portals often control communication of asymmetries, entropy, or exotic signals between the SM and hidden sectors, and are critical for realistic baryogenesis, dark-sector cosmology, and collider phenomenology. This article provides a comprehensive review of their construction, low-energy matching, key operator structures, and experimental implications within current effective-field-theory and portal frameworks.

1. Portal Effective Theory Construction and Operator Catalog

Portal Effective Theories (PETs) offer a systematic method to classify all couplings of SM fields to light hidden-sector messengers $\Phi$ with spin 0, 1/2, or 1 at the electroweak scale, organizing the effective Lagrangian as

$$\mathcal{L}_\mathrm{portal} = \sum_i \epsilon_i\, \mathcal{O}^{(d_i)}_\mathrm{SM}\, \mathcal{O}^{(d'_i)}_\mathrm{hidden},$$

with $d_i + d'_i \leq 5$ for the leading contributions (Arina et al., 2021). Portal operators are classified by messenger spin:

Spin-0 ("scalar portal"):

• Dimension-4: $\mu_S S (H^\dagger H)$
• Dimension-5: Couplings to SM gauge field strengths, e.g., $(\alpha_S/\Lambda)\, S\, G^a_{\mu\nu}G^{a\mu\nu}$; Yukawa-like $S\,\bar\psi_L\psi_R$; CP-odd portals with dual field strengths, e.g., $a\,G^a_{\mu\nu}\tilde{G}^{a\mu\nu}$.

Spin-1/2 ("neutrino portal/HNL"):

• Dimension-4: $y_N \bar{L} \tilde{H} N + h.c.$
• Dimension-5: Magnetic dipole $(1/\Lambda)\bar{L}\sigma^{\mu\nu} N B_{\mu\nu}$; four-fermion portals.

Spin-1 ("vector/dark photon portal"):

• Dimension-4: Kinetic mixing $(\epsilon_X/2) B_{\mu\nu} X^{\mu\nu}$; current portal $$X_\mu[\epsilon_L^f \bar{f}_L\gamma^\mu f_L + \epsilon_R^f \bar{f}_R\gamma^\mu f_R]$$.
• Dimension-5: Higgs-derivative coupling; mixed gauge–fermion.

Asymmetric portal couplings are introduced by setting $\epsilon_L^f \neq \epsilon_R^f$, or by spurion-dependent choices for gauge/flavor representation ($\epsilon_Q \neq \epsilon_u \neq \epsilon_d$), which explicitly break chiral or flavor symmetries, leading to non-universal or CP-violating structures. These parameters propagate through the matching procedure to low-energy observables and effective operators.

2. Matching to Strong-Scale EFT and Chiral Perturbation Theory (χPT)

Below the electroweak scale, portal operators are mapped onto strong-scale EFTs by integrating out $W$, $Z$, and $H$, expressing the portals in terms of light quarks, QCD, QED, and $\Phi$ (Arina et al., 2021). SM-side currents and densities take the form:

• Vector: $$J^\mu_V = \sum_{q} (\epsilon_L^q \bar{q}\gamma^\mu P_L q + \epsilon_R^q \bar{q}\gamma^\mu P_R q)$$
• Axial: $$J^\mu_A = \sum_{q} (\epsilon_L^q \bar{q}\gamma^\mu P_L q - \epsilon_R^q \bar{q}\gamma^\mu P_R q)$$
• Scalar/pseudoscalar densities: $S_q = \sum_{q} y_S^q \bar{q} q$, $P_q = \sum_{q} y_P^q \bar{q} i\gamma_5 q$

In the portal $\chi$PT matching, these currents are encoded as external spurions for $SU(3)_L \times SU(3)_R$ chiral symmetry. The resulting leading-order $\chi$PT Lagrangian contains external field insertions, producing analytic Feynman rules for meson–messenger interactions. The asymmetry in the portal couplings directly affects the meson transition amplitudes and decay widths, e.g., $K^+ \to \pi^+ X$ with coupling strength proportional to $(\epsilon_L + \epsilon_R)$ for vector, or $(\epsilon_L - \epsilon_R)$ for axial channels.

3. Nonperturbative Determination of Low-Energy Constants

The portal-modified chiral Lagrangian introduces new low-energy constants (LECs), notably $c_V$ (vector) or $c_S$ (scalar), which cannot be extracted from SM data and must be determined nonperturbatively:

• Resonance saturation: Matching two-point functions using vector-meson dominance, $$c_V \simeq \epsilon_V \cdot \frac{f_V g_{V\pi\pi}}{m_V^2}$$.
• QCD sum rules/lattice QCD: Direct computation of correlators with external currents, extracting residues for the lightest poles (Arina et al., 2021).

Asymmetric portal structures with $\epsilon_L^q \neq \epsilon_R^q$ generate independent vector and axial LECs ($c_V$, $c_A$), leading to enhanced CP-odd transitions when the axial component dominates. The uncertainty in these LECs is typically $\mathcal{O}(30$–$50\%)$, with improved precision expected from future lattice QCD studies.

4. Explicit Calculation: Meson Decay via Asymmetric Portal

For vector portal coupling to $s \to d$ quark transitions, the $\chi$PT-level K$^+ \to$ π$^+ X$ decay width is

$$\Gamma(K^+ \to \pi^+ X) = |c_V|^2\, \frac{|\epsilon_L + \epsilon_R|^2}{4\pi f_\pi^2 m_K^3} \lambda^{3/2}(m_K^2, m_\pi^2, m_X^2),$$

where $\lambda$ is the Källén function. In the purely left-handed limit, the vector-driven amplitude doubles, while in the purely axial case, it vanishes and the transition proceeds via a pseudoscalar density (kinematically suppressed by $m_K^2 - m_\pi^2$). The analytic dependence facilitates phenomenological studies in flavor-violating rare decay searches.

5. Guidelines, Theoretical Nuances, and Experimental Constraints

Design of phenomenologically viable asymmetric portals demands careful attention to:

• Gauge invariance: External sources for portal operators must respect $SU(2)_L \times U(1)_Y$ structure above $m_W$; incomplete operators require Higgs insertions for completion at higher dimension.
• Flavor structure: Off-diagonal portal couplings induce potentially large FCNCs; enforcement of Minimal Flavor Violation (MFV) or tight constraints on flavor off-diagonal entries is required.
• Chiral counting: Scalar spurions enter at $\mathcal{O}(p^2)$, vector at $\mathcal{O}(p)$—subleading effects must be tracked in higher-order matching.
• Nonperturbative uncertainties: LECs $c_S$, $c_V$ carry substantial uncertainty; lattice studies can improve predictions.
• Experimental limits: Hidden sector searches exploit channels such as $K\to \pi +$ invisible, $\pi^0 \to \gamma +$ invisible, and precision measurements constrain portal couplings in $e^+e^-$, beam-dump, and meson decay experiments.

RG running and mixing between portal operators (vector, axial, scalar) from high to low scale must be carried out to ensure accurate matching and consistency with experimental bounds.

6. Portal Generalizations and Extensions

Asymmetric portals are realized beyond the scalar/vector/axial constructions reviewed above. Portalinos, singleton fields, and multi-sector baryogenesis models (twin sector, composite ADM) introduce new field content and symmetry-breaking structures; these often exploit additional symmetries (Z$_2$, B$-$L) and allow entropy and asymmetry transfer, as illustrated in portalino and composite dark matter models (Liu et al., 2019, Ibe et al., 2018, Ibe et al., 2018, Das et al., 2024). Their low-energy effects are mapped onto analogous portal operators, with phenomenology controlled by the same EFT principles and matching procedures.

Furthermore, the framework extends naturally to "di-boson" portals involving gluons, $W$ bosons, or quark/lepton combinations as in the W-quark, quark-lepton, lepton-gluon, and W-gluon portals (Carpenter et al., 5 Dec 2025, Murphy, 2024, Carpenter et al., 26 Dec 2025, Carpenter et al., 2024). These access exotic high-dimension SU(3)$\times$SU(2)$\times$U(1) representations and produce novel collider signatures, such as single production of multi-charged fermions or scalars, nonstandard cascade topologies, and flavor-violating decays.

7. Summary and Outlook

The asymmetric Standard Model portal is a general paradigm for communicating chiral, flavor, or gauge-representation asymmetries between the Standard Model and hidden sectors. PET provides a systematic tool for classification, matching, and prediction; chiral symmetry breaking and nonperturbative QCD dynamics must be treated rigorously in determining low-energy coefficients. Asymmetric portal couplings impart distinctive phenomenological fingerprints—from enhanced rare meson decays to altered chiral LECs and collider-accessible high-dimensional exotic states. Future progress in lattice computations, minimal flavor violation, and high-luminosity experimental searches is expected to test the structure and viability of asymmetric portals in the quest for physics beyond the Standard Model.

Key references: Portal Effective Theories (Arina et al., 2021), composite ADM dark photon models (Ibe et al., 2018, Ibe et al., 2018, Das et al., 2024), portalinos (Liu et al., 2019), twin/singleton sector portals (Bishara et al., 2018), extended portal operator catalogs (Carpenter et al., 5 Dec 2025, Carpenter et al., 2024, Murphy, 2024, Carpenter et al., 26 Dec 2025).