A Non-Renormalization Theorem for Local Functionals in Ghost-Free Vector Field Theories Coupled to Dynamical Geometry

TL;DR

The work proves a non-renormalization theorem for ghost-free generalized Proca theories coupled to gravity, shown within a controlled EFT framework and a precise decoupling limit. By classifying admissible local counterterms and analyzing a decoupled high-energy sector, the authors demonstrate that quantum corrections cannot renormalize the classical derivative self-interactions responsible for the constraint structure; any induced operators carry extra derivatives and are suppressed by strong-coupling scales $\,\Lambda_3$ and $\,\Lambda_4$. The results hold to all loop orders in the EFT and extend flat-space non-renormalization properties to curved spacetimes, thereby preserving the classical degrees of freedom and the perturbative stability of the theory in the presence of dynamical geometry. This structural stability supports the use of ghost-free Proca theories as consistent infrared modifications of gravity within their EFT domain and clarifies the role of the decoupling limit in bounding radiative corrections. Overall, the paper provides a rigorous foundation for the radiative robustness of vector-tensor systems with derivative self-interactions in curved backgrounds.

Abstract

We establish a non-renormalization theorem for a class of ghost-free local functionals describing massive vector field theories coupled to dynamical geometry. Under the assumptions of locality, Lorentz invariance, and validity of the effective field theory expansion below a fixed cutoff, we show that quantum corrections do not generate local operators that renormalize the classical derivative self-interactions responsible for the constraint structure of the theory. The proof combines an operator-level analysis of the space of allowed local counterterms with a systematic decoupling-limit argument, which isolates the leading contributions to the effective action at each order in the derivative expansion. As a consequence, all radiatively induced local functionals necessarily involve additional derivatives per field and are suppressed by the intrinsic strong-coupling scales of the theory. In particular, the classical interactions defining ghost-free vector field theories are stable under renormalization, and any additional degrees of freedom arising from quantum corrections appear only above the effective field theory cutoff. This result extends known non-renormalization properties of flat-space vector theories to the case of dynamical geometry and provides a structural explanation for their perturbative stability to all loop orders.

Figures (11)

• Figure 1: The two mixed one-loop 1PI diagrams inducing quantum corrections to the Proca two-point function, both accompanied by a symmetry factor of 2. The vertices required for these diagrams arise from the perturbative expansion of the Lagrangian in the minimal model at cubic and quartic order that involve mixed vector-tensor terms, i.e. from $\mathcal{L}^{(3)}_{h\,\delta A^2}\subset\mathcal{L}^{(3)}$ and $\mathcal{L}^{(4)}_{h^2\delta A^2}\subset \mathcal{L}^{(4)}$.
• Figure 2: The two mixed one-loop 1PI diagrams inducing quantum corrections to the graviton two-point function, both accompanied by a symmetry factor of 2. The vertices required for these diagrams arise from the perturbative expansion of the Lagrangian in the minimal model at cubic and quartic order that involve mixed vector-tensor terms, i.e. from $\mathcal{L}^{(3)}_{h\,\delta A^2}\subset\mathcal{L}^{(3)}$ and $\mathcal{L}^{(4)}_{h^2\delta A^2}\subset \mathcal{L}^{(4)}$.
• Figure 3: The two mixed one-loop 1PI diagrams inducing kinetic mixing at the quantum level, both accompanied by a symmetry factor of 2. The vertices required for these diagrams arise from the perturbative expansion of the Lagrangian in the minimal model at cubic and quartic order that involve mixed vector-tensor and pure Proca terms, i.e. from $\mathcal{L}^{(3)}_{h\,\delta A^2},\,\mathcal{L}^{(3)}_{\delta A^3}\subset\mathcal{L}^{(3)}$ and $\mathcal{L}^{(4)}_{h\delta A^3}\subset \mathcal{L}^{(4)}$.
• Figure 4: The two mixed-sector one-loop 1PI bubble diagrams inducing quantum corrections to the Proca two-point function, when constructed from the decoupling limit. The vertices required for these diagrams arise from the scalar-vector-tensor interactions in $\mathcal{L}^{(3)}_{\rm{DL}}$. Dashed lines denote external legs or propagators of the scalar field $\phi$ that couples to the graviton and the gauge-invariant vector field $F_{\delta A}$, whose propagators and external legs are depicted by wavy lines.
• Figure 5: The tadpole diagram in the decoupling limit, giving quantum corrections to the Proca two-point function. The left diagram corresponds to the one-loop contribution in unitary gauge, whereas the right one represents the surviving configuration in the decoupling limit, with dashed lines indicating external scalar legs.

Theorems & Definitions (20)

• Definition 2.1: Ghost-free vector field theories on curved spacetimes
• Proposition 2.1
• Remark 2.1: Radiative stability from structural properties
• Definition 2.2: Perturbative expansion about a background configuration
• Proposition 2.2: Euler--Lagrange equations for the background fields
• Lemma 2.1: Flat background solutions
• Remark 2.2: stationary point of the action
• Proposition 2.3: Quadratic action and second variation
• Corollary 2.1: Perturbative degrees of freedom
• Definition 3.1: Decoupling limit