Asymptotic Safety in Generalized Proca Theories

Generalized Proca Theories are the most general higher-derivative extensions of a massive vector field that retain second-order equations of motion. They are phenomenologically interesting as models of dynamical dark energy that, unlike scalar-tensor theories, can naturally accommodate cosmological anisotropies. A key open question is whether such theories can be fundamental. As a first step in this direction, we investigate whether they admit an ultraviolet completion within a quantum field theory framework, working with a truncation comprising up to four powers of the Proca field and up to two derivatives. We find a triplet of non-Gaussian ultraviolet fixed points, that lie very close to one another. Only one of them features a non-tachyonic Proca mass and could thus serve as a consistent ultraviolet completion for Generalized Proca Theories. We name it the Proca fixed point. We discuss its stability and contrast its features with those of the standard Reuter fixed point of the asymptotic safety scenario for quantum gravity and matter. In particular, we show that the Gaussian and Reuter fixed points lie on singular hypersurfaces of the flow of Generalized Proca Theories, yet can act as quasi-fixed points in certain regimes.

💡 Research Summary

The paper investigates whether Generalized Proca Theories (GPTs)—the most general higher‑derivative extensions of a massive vector field that retain second‑order equations of motion—can be UV‑complete within the framework of asymptotic safety. The authors focus on a truncation that includes all operators up to four powers of the Proca field and up to two derivatives, which reduces the infinite theory space to six independent couplings: the cosmological constant Λ, Newton’s constant G, the Proca mass coupling G₂, and three higher‑order self‑interaction couplings G₄,₁, G₄, and G₄,₂.

Using the functional renormalization group (FRG) in the Wetterich form, they employ a background‑field split (g_{μν}= \bar g_{μν}+h_{μν}, A_μ=\bar A_μ+\hat A_μ) and choose a constant background Proca field together with an Einstein‑manifold metric. The gravitational sector is gauge‑fixed in the harmonic (Feynman) gauge (α=β=1), while the Proca sector requires no gauge fixing because the mass term explicitly breaks U(1) symmetry, making the longitudinal mode physical. A “type I” regulator depending only on the background Laplacian and the sharp (limit) shape function R_k(p²/k²)=(k²−p²)θ(k²−p²) is used to regularize all fluctuation fields, including the longitudinal mode.

The flow equation is evaluated by expanding the second functional derivative of the effective average action, performing a Parker‑Fulling expansion of the propagator, and applying heat‑kernel techniques to handle the resulting operator structures. The resulting beta functions are expressed in terms of dimensionless couplings λ=Λ/k², g=G k², g₂=G₂ k², g₄,₁, g₄, and g₄,₂. Numerical analysis of these beta functions reveals three non‑Gaussian ultraviolet fixed points that lie extremely close to each other in coupling space; the authors refer to this set as the “Proca triplet.”

Crucially, the three fixed points differ in the sign of the dimensionless Proca mass coupling g₂. Only the fixed point with g₂ > 0 (i.e., a non‑tachyonic Proca mass) is physically admissible; the authors name it the “Proca fixed point.” This fixed point possesses five relevant directions (positive critical exponents), which is larger than the typical three or four relevant directions found for the standard Reuter fixed point in pure gravity or gravity‑matter systems. Consequently, the predictive power of the Proca fixed point is reduced, and its stability is weaker: some critical exponents form complex conjugate pairs, leading to oscillatory RG trajectories, and the flow exhibits a stronger dependence on the regulator choice.

The paper also examines the role of singular hypersurfaces in theory space where the beta functions diverge. Both the Gaussian fixed point (GFP) and the Reuter fixed point lie on such hypersurfaces when the Proca mass vanishes, implying that they act as quasi‑fixed points only in limited regimes. The presence of a physical longitudinal mode thus fundamentally alters the RG structure compared to standard Einstein‑Maxwell or scalar‑tensor models.

In summary, the authors demonstrate that within their truncation GPTs admit a non‑Gaussian UV fixed point, providing the first evidence that these vector‑field extensions can be asymptotically safe. However, the fixed point’s reduced stability, larger number of relevant directions, and sensitivity to truncation indicate that further work—such as extending the operator basis, exploring alternative regulators, and studying more general backgrounds—is required before a definitive claim of UV completeness can be made. The study opens a new avenue for investigating vector‑mediated dark‑energy models within a quantum‑gravity‑compatible framework.