Dark-Photon Kinetic-Mixing Parameter

• Dark-photon kinetic mixing is a dimensionless parameter that defines the bilinear gauge-invariant coupling between SM and dark photon fields.
• It is radiatively generated via portal matter loops, with typical values ranging from 10⁻⁵ to 10⁻³ depending on mass splittings and coupling strengths.
• The parameter governs dark photon production and decay rates in collider, fixed-target, and astrophysical experiments, constraining dark-sector models.

A dark-photon kinetic-mixing parameter quantifies the bilinear coupling between the field strengths of the Standard Model (SM) photon or hypercharge boson and an additional abelian gauge boson, the “dark photon.” This mixing is a dimensionless parameter—commonly denoted $\epsilon$, $\kappa$, or $\chi$—appearing in the gauge kinetic terms, and provides the leading renormalizable, gauge-invariant portal between the SM and hidden-sector gauge fields. Its value controls the induced, effective coupling of the dark photon to SM electromagnetic currents, determines production and decay rates, and sets the visibility of dark-sector signatures in laboratory, astrophysical, and cosmological contexts. Theoretical predictions, renormalization properties, radiative generation, and phenomenological consequences of the dark-photon kinetic-mixing parameter are central to a broad swath of dark-sector physics.

1. Lagrangian Origin and Gauge Structure

The kinetic mixing parameter arises from the unique gauge- and Lorentz-invariant, dimension-4 operator built from two abelian field strengths, present in any theory with $U(1)_1\times U(1)_2$: $$\mathcal{L}_{\rm kin} = -\frac14\ F_{\mu\nu} F^{\mu\nu} - \frac14\ V_{\mu\nu} V^{\mu\nu} - \frac{\epsilon}{2}\ F_{\mu\nu} V^{\mu\nu}$$ where $F_{\mu\nu}$ is the SM photon or hypercharge field strength, $V_{\mu\nu}$ that of the “dark photon,” and $\epsilon$ is the kinetic-mixing parameter (Rueter et al., 2020, Davoudiasl, 2015, Brahmachari et al., 2014).

• The structure generalizes to SM gauge extensions such as $SU(2)_L\times U(1)_Y\times U(1)_X$ (Bento et al., 2023, Sun et al., 2023, Rizzo, 2022), where mixing may occur with the $Y$ (hypercharge) or even the $W_3$ ($SU(2)_L$ neutral component) via non-abelian loops.
• $\epsilon$ is basis dependent at the level of the bare Lagrangian but physical after canonical normalization of kinetic terms and mass diagonalization.

Upon diagonalization, the low-energy consequence is an induced coupling of the “dark photon” to the SM electromagnetic current: $$\mathcal{L}\supset -\epsilon\, e\, Q\, V_\mu \bar f \gamma^\mu f$$ where $Q$ is electric charge, $e$ is the electromagnetic gauge coupling, and $V_\mu$ is the massive dark photon (Rueter et al., 2020, Davoudiasl, 2015, Cárcamo et al., 2014, Compagnin et al., 2022).

2. Radiative Generation and Natural Size

Kinetic mixing can originate at tree level (a fundamental parameter), but, in most ultraviolet completions, it is radiatively induced by integrating out “portal matter” (PM) fields charged under both $U(1)$ factors: $$\epsilon \simeq \frac{e\,g_D}{16\pi^2}\ \sum_i Q_{i}^{\rm SM} Q_{i}^{D} \, f(m_i/\mu)$$ where $g_D$ is the dark gauge coupling, $Q_{i}^{\rm SM}$ and $Q_{i}^{D}$ are the SM and dark charges of the portal matter, $m_i$ their masses, and $f$ encodes logarithmic or threshold-dependent factors (Rueter et al., 2020, Davoudiasl, 2015, Rizzo, 7 May 2025, Rizzo, 2022, Brahmachari et al., 2014).

Scalar portal matter scenario: For scalar SU(2) doublet PM (as in (Rueter et al., 2020)), the one-loop, UV-finite kinetic mixing is: $$\epsilon = \frac{g_D e}{48\,\pi^2} \ln\frac{m_2^2}{m_1^2}$$ with $m_{1,2}$ the charged scalar masses. The cancellation $\sum Q_D Q_{\rm EM} = 0$ ensures finiteness.

Scaling and parametrics:

• For mass splittings $\Delta m \ll m$, $\epsilon \propto (\Delta m)/m$; the loop factor suppresses $\epsilon$.
• For large hierarchies ($m_2 \gg m_1$), $\epsilon \sim 10^{-3} g_D e$ for $m_2/m_1 \sim 10$ (Rueter et al., 2020, Rizzo, 7 May 2025, Rizzo, 2022).

Natural sizes: For $\mathcal{O}$(few–100 GeV) portal matter, typical loop-generated values are

$\epsilon \sim 10^{-5} - 10^{-3}$

depending on $g_D$, scalar mass ratios, and vev ratios.

3. Electroweak Symmetry Breaking and Mixing Structure

The character of the kinetic mixing—whether between photon–dark photon or hypercharge–dark gauge boson—depends on the phase of electroweak symmetry:

• Before EWSB: Only portal-matter fields contribute to $B$-$V$ (hypercharge–dark photon) mixing; the mixing is unphysical before symmetry breaking if portal matter gets its mass after EWSB (Rueter et al., 2020).
• After EWSB: Mixing is realized between the physical photon and dark photon. Additional small $Z$–$V$ mass mixing arises at order $\epsilon$, suppressed by $v^2/M^2$ or ratios of vevs and couplings (Rueter et al., 2020, Bento et al., 2023, Sun et al., 2023, Rizzo, 2022).

After diagonalization, the physical couplings of the dark photon to SM currents can be written as

$$\mathcal{L}\supset -\epsilon\,e\,Q\, V_\mu \bar f \gamma^\mu f - g_D\,\sin\theta_{ZV}\,(T^3 - s_w^2 Q)\, V_\mu \bar f\gamma^\mu f + g_D\, Q_D\, V_\mu \bar\chi\gamma^\mu\chi$$

where the first is the “Holdom”-style photon–dark photon mixing term and the second term represents $Z$–$V$ mixing corrections (Rueter et al., 2020).

The effective low-energy coupling for physical processes therefore receives both pure-dark-photon ($\epsilon$) and $Z$/$V$-mixing ($\sin\theta_{ZV}$) contributions.

4. Renormalization Group and Momentum Dependence

The kinetic mixing parameter inherits scale dependence through running of the dark gauge coupling: $\epsilon^2(q) \propto \alpha_d(q)$ where $\alpha_d(q)$ is the running dark-sector fine structure constant; the renormalization-group equation (RGE) at two loops is

$$q \frac{d\alpha_d}{dq} = \frac{\alpha_d^2}{2\pi} \left[ \frac{4}{3}(n_F + n_S/4) + \frac{\alpha_d}{\pi}(n_F + n_S) \right]$$

with $n_F, n_S$ the numbers of dark fermions and scalars lighter than $q$ (Davoudiasl, 2015). Accordingly, $\epsilon(q)$ grows with energy if the dark gauge group is abelian and the PM spectrum is light, potentially leading to observable effects in dark-matter beam experiments (Davoudiasl, 2015).

There is an upper bound on the portal coupling imposed by perturbativity (i.e., avoidance of a Landau pole) up to a high scale $q^*$: $$\alpha_d(m_{Z_d}) \lesssim \frac{3\pi}{(2 n_F + n_S / 2) \ln(q^*/m_{Z_d})}$$ which translates into a maximal $\epsilon$ (Davoudiasl, 2015).

5. Phenomenological Implications and Experimental Limits

The kinetic-mixing parameter governs all leading production and detection rates of dark photons coupled to SM currents:

• Collider and Fixed-Target Experiments: Rates for $e^+ e^- \to \gamma A'$ and subsequent $A'\to \ell^+\ell^-$ scale as $\epsilon^2$ (Rueter et al., 2020, Davoudiasl, 2015, Compagnin et al., 2022, Bento et al., 2023, Jorge et al., 15 Jul 2025, Jorge et al., 3 Dec 2024).
• Cosmology and Astrophysics:
  • In the early universe, $\epsilon$ determines dark photon production via freeze-in/inverse decay and thus constraints from BBN and the CMB (Fradette et al., 2014).
  • Stellar energy-loss limits rely on plasmon–dark-photon mixing, scaling as $\kappa^2 m_V^2$ for small masses (An et al., 2013).
  • Limits from CMB birefringence, circular polarization, and spectral distortion depend strongly and simply on $\varepsilon$ (Lee et al., 2023).

Current experimental and cosmological bounds have achieved

$$\epsilon \lesssim 10^{-3} \ (10\,\mathrm{MeV}<m_{A'}<1\,\mathrm{GeV}),$$

with future searches aiming for $\epsilon \sim 10^{-6}$ or lower (Rueter et al., 2020, Lee et al., 2023, Yin, 20 Aug 2025, Jorge et al., 3 Dec 2024, Fradette et al., 2014).

6. Model Dependencies and Extensions

The value and implications of $\epsilon$ depend on the UV structure and the embedding:

• Minimal Models: Single $U(1)_D$ with abelian portal matter (either fermionic or scalar) leads to the standard loop-suppressed $\epsilon$ (Rueter et al., 2020, Brahmachari et al., 2014).
• Extended Gauge Sectors: Non-abelian extensions ($SU(2)_L \times U(1)_Y \times U(1)_{Y'}$ or $SU(2)_I \times U(1)_{Y_I}$) lead to kinetic and mass mixings involving $Z$ and $V$ bosons, introducing additional suppression, interplay, and constraints from electroweak precision data such as the $\rho$ parameter (Bento et al., 2023, Rizzo, 2022, Rizzo, 7 May 2025, Cárcamo et al., 2014).
• Higher-Dimensional Operators: Additional operators, such as dark-dipole terms, can modify decay widths, emission rates, and lift or change some traditional $\epsilon$-only constraints (Barducci et al., 2021).

In summary, the kinetic-mixing parameter $\epsilon$ encapsulates the leading, gauge-invariant interaction between a dark abelian gauge sector and the SM electromagnetic sector. Its UV origin, renormalization properties, and mixing structure after symmetry breaking dictate the allowed parameter space, experimental signatures, and constraints on viable dark photon models. Scalar-portal-matter models naturally yield $\epsilon$ values just below present experimental limits in much of the motivated mass range, making them an active target for the next generation of laboratory and cosmological searches (Rueter et al., 2020).