Dark Photon Production Cross Section

• Dark Photon Production Cross Section is a measure of the probability to generate dark photons, relying on kinetic mixing and precise incorporation of electromagnetic and hadronic structure effects.
• The methodology includes full matrix-element calculations, loop-induced processes, and refined bremsstrahlung treatments that overcome the limitations of common approximations like the Weizsäcker–Williams method.
• Accurate cross section estimates are pivotal for experimental sensitivity, impacting exclusion limits and guiding search strategies across collider, fixed-target, and beam-dump experiments.

A dark photon is a hypothetical vector boson arising from an additional U(1)′ gauge symmetry in extensions of the Standard Model, coupling to ordinary matter via kinetic mixing with the photon. The production cross section of dark photons is a central quantity in both theoretical predictions and the interpretation of searches at high-energy colliders, fixed-target experiments, and beam dumps. The value and structure of this cross section depend sensitively on the underlying production process, the associated mass scales, the kinetic mixing parameter (ξ or ε), and the correct incorporation of electromagnetic and hadronic structure effects. An accurate calculation is therefore essential for assessing experimental reach and constraining dark sector models.

1. Field-Theoretic Framework and Kinetic Mixing

The most general treatment of dark photon production begins by extending the QED Lagrangian to include a new U(1)′ field B_μ kinetically mixed with the photon A_μ: $$\mathcal{L}_\mathrm{int} = -\frac{1}{4} F_{\mu\nu}(B) F^{\mu\nu}(B) + \frac{\xi}{2} F_{\mu\nu}(B) F^{\mu\nu}(A) + \frac{1}{2} m_\gamma^2 B_\mu B^\mu$$ with standard QED and matter terms. Diagonalization yields modified couplings for matter to both photon and dark photon: the standard electromagnetic vertex remains $-i \gamma_\mu$, while the dark photon vertex is suppressed by ξ, i.e., $-i \xi \gamma_\mu$ (Arza et al., 2016). The resulting interactions are central to evaluating all dark photon production rates.

2. Loop-Induced Light-by-Light/Conversion Processes

A distinctive probe of dark photon production arises from processes analogous to the Breit–Wheeler mechanism (γγ → e⁺e⁻), with conversion of dark photons into Standard Model photons via a loop: $$\gamma' + \gamma' \to e^- e^+ \ (\text{loop}) \to \gamma + \gamma$$ The differential cross section in the low-energy limit (E ≪ m, where m is the electron mass) is found to be

$$\left.\frac{d\sigma}{d\Omega}\right|_{\gamma'\gamma'\to\gamma\gamma} = \frac{\kappa \alpha^4 \xi^4}{64\pi^2 m^8} s^3 (3+\cos^2\theta)^2$$

with α the fine structure constant, s = (p₁+p₂)², and κ a constant fixed by the loop calculation (Arza et al., 2016). In the CM frame, $s=4E^2$, leading to a cross section scaling as (E/m)^6, highlighting rapid growth at low energy.

In the high-energy regime (E ≫ m), the differential cross section is instead suppressed: $$\left.\frac{d\sigma}{d\Omega}\right|_{\theta=0} \sim \frac{\alpha^4 \xi^4}{\pi^2 m^2} \left(\frac{m}{E}\right)^2 \ln^4\left(\frac{E}{m}\right)$$

$$\left.\frac{d\sigma}{d\Omega}\right|_{\theta=\pi/2} \sim \frac{\alpha^4 \xi^4}{\pi^2 m^2} \left(\frac{m}{E}\right)^2$$

Thus, asymptotically, the cross section decays as (m/E)^2 (modulo logarithmic enhancements), a direct consequence of the loop structure (Arza et al., 2016). The suppression by ξ^4 persists throughout, distinguishing dark photon modes from their visible analogs.

3. Bremsstrahlung and Inelastic/Elastic Scattering Channels

The dominant production mechanisms at fixed-target and collider facilities are bremsstrahlung processes in electron, muon, or proton scattering off target nuclei: $e^- (p) + N (P_i) \to e^- (p') + N (P_f) + A' (k)$ as well as analogous processes for muons and protons. The differential cross section is generally written as

$$\frac{d\sigma}{dx\, d\cos\theta} = \epsilon^2 \left(\frac{\alpha^3}{8M^2}\right) \frac{|\vec{k}|E}{|\vec{p}| V} \int_{t_min}^{t_max} \frac{dt}{t^2} F(t)^2 \int_{0}^{2\pi} \frac{d\phi_q}{2\pi} \mathcal{A}^{(23)}$$

where $x = E_k / E$, and $F(t)$ encodes nuclear form factor effects (1705.01633). Exact, analytic calculation of the full 2→3 kinematics reveals that the widely used Weizsäcker–Williams (WW) and improved WW approximations may introduce errors up to 100% at MeV-scale dark photon masses, especially in the non-collinear (off-shell) regime (1705.01633, Gninenko et al., 2017).

Elastic proton bremsstrahlung, relevant for the sub-GeV–GeV mass range, requires explicit treatment of the nonzero momentum transfer (t ≠ 0), as the dominant virtuality scale is set by QCD, not by the electron mass: $$d^2\sigma/dk_\perp^2 dz = (\epsilon^2 \alpha/32(2\pi)^2 zP\tilde{\mathcal{S}}^2 \sqrt{P^2(1-z)^2+k_\perp^2}) \times [\dots]$$ where the bracket contains an integral over t that includes full kinematic coefficients, crucial for reproducing results within 3–9% of the WW approximation for protons (Gorbunov et al., 2023, Kriukova, 6 Apr 2024). In contrast, formerly common splitting function-based approaches (e.g., the Blumlein–Brunner form) can dramatically overestimate rates at small z (Gorbunov et al., 2023).

4. Structure Functions, Form Factors, and New Contributions

In the proton sector (both inelastic and elastic), the inclusion of the full electromagnetic structure—Dirac (F₁) and Pauli (F₂) form factors—is essential. In inelastic scattering, the emission rate factorizes: $$\frac{d^2\sigma(pp\rightarrow \gamma' X)}{dz\,dk_\perp^2} \simeq w(z, k_\perp^2) \cdot \sigma(pp\to X)$$ with the splitting function

$$w(z,k_\perp^2) = w_{11}|F_1|^2 + w_{22}|F_2|^2 + w_{12}(F_1 F_2^* + F_2 F_1^*)$$

where the cross terms $w_{22}$, $w_{12}$ (Pauli form factor) play a particularly pronounced role for $m_{\gamma'} \in [0.9,\,1.8]$ GeV, with |F₂| ≫ |F₁| (Gorbunov et al., 17 Sep 2024, Gorbunov et al., 17 Sep 2024). Neglecting the Pauli term would underestimate the cross section, especially for visible decay searches at SHiP and for the time-like region below the p-p̄ threshold.

5. Comparison With Meson Decays and Exotic Channels

Production via meson decays (e.g., π⁰, η → γA′) in ultraperipheral heavy-ion collisions or pp collisions provides additional cross section channels. The cross section is given by

$$\sigma_\mathrm{total}(A') = \sigma_\mathrm{prod}(M) \times \mathrm{BR}(M \to \gamma A') \times \mathrm{BR}(A' \to \ell^+\ell^-)$$

where

$$\mathrm{BR}(M \to \gamma A') = 2 \epsilon^2 \mathrm{BR}(M \to \gamma\gamma) \left(1 - \frac{m_{A'}^2}{m_M^2}\right)^3$$

The cross section is dominated at low dark photon masses ($m_{A'} \ll m_{M}$) and inherits the dependence on ε² (Goncalves et al., 2020, Kou et al., 7 Mar 2024). These channels are experimentally advantageous due to high yields and clean kinematics in ultra-peripheral collisions.

6. Experimental Sensitivity and Exclusion Limits

The dark photon production cross section sets the achievable sensitivity for visible and invisible search strategies. At high-energy colliders (e.g., the LHC), the cross section for loop-induced conversion processes γ′γ′ → γγ can approach values ($\sim 10^{-50}$ m²) similar to weak-scale neutrino cross sections, placing them at the edge of detectability (Arza et al., 2016).

Table: Scaling of Differential Cross Section in Representative Processes

Process	Low-Energy Scaling	High-Energy/Asymptotic Scaling
γ′γ′ → γγ (Breit–Wheeler analog)	(E/m)^6 × ξ⁴ × α⁴	(m/E)^2 × ξ⁴ × α⁴ (with log enhancements)
Leptonic/Nuclear Bremsstrahlung (elastic)	ε² × α³ × flux factors × form factors	ε² × α³ × flux × [WW limit]
Inelastic pp Bremsstrahlung (Pauli term)	—	w₂₂

Here, E is the initial energy, m is the loop fermion mass, ξ or ε parameterizes kinetic mixing, and α is the fine structure constant. "flux factors" and "form factors" encode process-specific details, including nuclear/t channel exchanges.

The cross section enters directly into the estimated number of signal events N = σ × L, with L the integrated luminosity. Exclusion limits in the (m_{A′}, ε) or (m_{A′}, ξ) plane depend strongly on the accuracy of cross section calculations, especially in regions dominated by form factor effects or in the MeV–GeV mass interval where new splitting functions drive the rate (Gorbunov et al., 17 Sep 2024, Gorbunov et al., 17 Sep 2024).

7. Theoretical Uncertainties, Limitations, and Future Directions

Uncertainties arise primarily from:

• The accuracy of hadronic form factors and off-shell corrections, especially in the time-like region
• The reliability and applicability of the WW and splitting-function approximations as compared to full matrix-element calculations
• The treatment of higher-order QED and QCD corrections
• The treatment of interference between initial and final state radiation in proton channels (Foroughi-Abari et al., 2021)

Dedicated measurements of proton electromagnetic form factors in the time-like region (such as planned at PANDA/FAIR) are expected to reduce theoretical uncertainties and improve predictions for inelastic bremsstrahlung channels (Gorbunov et al., 17 Sep 2024).

Emerging directions include incorporating higher electromagnetic moments (dipole, anapole, charge radius operators) in the coupling structure, leading to distinct kinematic dependencies in the recoil spectra (Catena et al., 19 Feb 2025), and accounting for dark sector showering effects when the dark photon mass is generated via a dark Higgs mechanism (Li et al., 25 Jun 2025).

8. Summary

The dark photon production cross section is process and energy dependent, governed by kinetic mixing parameters and loop-induced or bremsstrahlung diagrams, with nuclear and hadronic structure encoded via form factors. Accurate predictions require full matrix-element evaluation beyond naive splitting-function or WW approximations, explicit inclusion of Dirac and Pauli terms, and careful treatment of experimental kinematics. These refinements provide a robust basis for interpreting results and guiding searches for dark photons and related dark sector phenomena in current and future experiments.