The Positivity Geometry of Photon--Dark-Photon Effective Field Theories

Technical Summary: The Positivity Geometry of Photon–Dark-Photon Effective Field Theories

Problem Statement
Hidden-sector extensions of the Standard Model (SM), particularly those involving light, weakly coupled particles like dark photons, are central targets of current theoretical and experimental searches. While low-energy Effective Field Theories (EFTs) for pure photons are well-understood, the EFT describing the interaction between massless photons ($\gamma$) and massless dark photons ($\gamma'$) remains qualitatively and quantitatively richer due to the presence of two massless spin-1 species. This complexity leads to a larger number of particle and helicity channels, crossing relations between visible and dark states, and a Wilson coefficient space that does not reduce to the familiar two-parameter Euler-Heisenberg subspace. Prior to this work, a general and detailed analysis of the positivity geometry for the photon–massless-dark-photon EFT had not been explored. The paper addresses the need to derive model-independent positivity bounds on the dimension-eight Wilson coefficients of this theory to constrain the space of viable infrared (IR) theories and map them to possible ultraviolet (UV) completions.

Methodology
The authors employ a systematic approach combining helicity amplitude analysis, dispersion relations, and algebraic geometry:

1. Helicity Amplitude Enumeration: The authors systematically apply parity-reversal, time-reversal, and crossing ($s-t$, $s-u$, $t-u$) equations to 2$\to$2 scattering processes involving photons and dark photons. They identify 19 independent helicity amplitudes, a significant increase from the 3 independent amplitudes found in pure photon scattering.
2. Operator Basis Construction: Working in a canonically normalized operator basis where kinetic mixing is eliminated via field redefinition, they construct the complete CP-even operator basis up to dimension-8. This yields 12 independent Wilson coefficients ($c_1$ through $c_{12}$), compared to only 2 in the pure photon EFT.
3. Dispersion Relations: Using a modified forward-limit dispersion relation, the authors define an analytic function $M(s)$ in the upper-half $s$-plane. By assuming Regge boundedness and neglecting graviton exchange (via a strict decoupling limit), they utilize the optical theorem to establish that the imaginary part of the scattering amplitude is non-negative.
4. Positivity Constraints: This leads to an elastic positivity condition expressed as a convex cone constraint on the Wilson coefficients. The authors analyze this condition in analytically tractable subspaces (specifically $C^2 \otimes C^4$ and $C^2 \otimes C^2$) to derive explicit linear, quadratic, and non-linear inequalities.
5. Geometric Interpretation: The resulting constraints are interpreted as defining a "spectrahedral geometry." The authors utilize concepts such as Gram matrices, spectrahedra (convex shapes defined by linear matrix inequalities), and elliptopes to visualize the allowed regions of the Wilson coefficient space.

Key Contributions and Results

• Complete Enumeration: The paper provides the first complete enumeration of independent helicity channels (19) and the full CP-even operator basis (12 dimension-8 operators) for the photon–dark-photon EFT.
• Derivation of Bounds: The authors derive non-trivial linear and non-linear elastic positivity constraints. These include:
  • Simple positivity bounds: $c_1, c_2, c_3, c_4, c_7, c_8 \geq 0$.
  • Quadratic inequalities: e.g., $4c_1c_7 \geq c_9^2$ and $4c_2c_8 \geq c_{10}^2$.
  • Higher-order non-linear constraints involving sums and differences of coefficients (e.g., Eq. 15 and Eq. 19), which encode an infinite set of linear inequalities.
• Geometric Structure: The bounds define a spectrahedral geometry. Specifically, in certain subspaces, the allowed region forms an "elliptope" (a spectrahedron defined by a positive semidefinite Gram matrix). The paper explicitly maps the vertices and boundaries of this geometry, showing that the determinant-zero locus corresponds to Cayley's four-nodal cubic surface.
• Amplitude Hierarchies: The derived bounds imply hierarchies for mixed-helicity amplitudes, even in the non-forward limit ($t \neq 0$). For instance, $|M_{++}^{++'}|^2 \lesssim M_{++}^{++} M_{++'}^{++'}$.
• Two-Sided Bounds: The analysis yields two-sided bounds for various low-energy amplitudes, circumscribed by functions of other helicity amplitudes, providing tighter constraints than one-sided bounds alone.
• UV Completion Mapping: The authors analyze how specific UV completions populate this geometry:
  • Kinetic-Mixing Portals: These generally lie on the surface of the IR elliptope (rank-2 contributions). Specific limits (e.g., millicharged particles or decoupled dark sectors) lie on the edges or vertices of the geometry.
  • Dark-Axion Portals: Tree-level completions involving a massive pseudoscalar saturate specific bounds and lie on the vertices of the elliptope.
  • Hybrid Models: If both kinetic mixing and axion portals exist, or if the UV sector is richer, the contributions are generically of rank 3, placing the theory inside the elliptope rather than on its boundary.

Significance
The paper claims to provide the first complete positivity analysis for the photon–massless-dark-photon EFT in a currently relevant setting. The results constitute the strongest model-independent theoretical constraints available for this EFT. By establishing a "spectrahedral geometry" for the Wilson coefficients, the work connects low-energy observables to fundamental principles of causality and unitarity. Furthermore, it offers sharp IR diagnostics for viable dark-sector UV physics, demonstrating how different UV completions (kinetic mixing vs. axion portals) occupy distinct, identifiable regions within the positivity geometry. This framework extends the "EFT-hedron" program to mixed gauge sectors, organizing EFT data through convex hulls and spectrahedral lifts.