Cosmology from asymptotically safe Proca theories

Abstract: Effective field theories for cosmology offer a powerful framework to investigate the dynamics of space--time and address longstanding open puzzles. In this work, we initiate a programme to analyse the ultraviolet completion of vector--tensor quantum field theories within the asymptotic safety paradigm, focusing on generalised Proca theories with a vector condensate. This enables us to assess whether a consistent fundamental UV completion exists and to constrain the set of viable infrared scenarios. Using the non--perturbative functional renormalisation group, we identify several fixed points, including Proca--type candidates, and, among them, a particularly remarkable one with four relevant directions: two associated with gravity and two induced by matter. This provides evidence for the non--perturbative renormalisability of vector--tensor theories. We further outline how the resulting UV critical surface constrains late--time cosmology.

• The paper identifies an interacting Proca-type fixed point with exactly four relevant directions through non-perturbative fRG analysis.
• The paper employs a systematic truncation and gauge-fixing scan to robustly map the UV fixed-point landscape in generalized Proca theories.
• The paper demonstrates that UV completion in these theories imposes predictive constraints on cosmological observables and IR parameters.

Cosmology from Asymptotically Safe Proca Theories: Technical Summary and Analysis

Motivation and Framework

The paper addresses the possibility of ultraviolet (UV) completion for cosmologically relevant vector–tensor (V–T) theories, specifically generalized Proca theories (GPTs), within the paradigm of asymptotic safety (AS). The inadequacy of General Relativity (GR) at fundamental scales (e.g., quantum gravity, dark sector dynamics, cosmological tensions) motivates extensions coupling gravity with non-trivial matter sectors. The central question: Which effective field theories (EFTs) for gravity and matter admit consistent, interacting UV fixed points (FPs)?

GPTs generalize Proca models by introducing broken $U(1)$ symmetry and derivative self-interactions, yielding viable cosmological models with healthy propagating degrees of freedom. The authors employ the functional renormalization group (fRG) to search for FPs in the coupled gravity–Proca system, focusing on the scale dependence of the effective action, $\Gamma_k$, which interpolates between microscopic and IR actions. This approach evaluates how UV-complete cosmological scenarios constrain possible IR (late-time universe) realizations.

Construction of the Effective Action and RG Flow

GPTs are parametrized using invariants $X = (Z_A/2)A_\mu A^\mu$ and $\Theta = (Z_A/4)F_{\mu\nu}F^{\mu\nu}$, enabling the construction of the quantum effective action with an infinite tower of operators. The gravitational sector is incorporated via an expansion in the metric and the gauge-fixed approach, while derivative self-interactions and condensate dynamics for $A_\mu$ are introduced.

Truncation is applied for computational tractability:

• Taylor expansion in $X - X_0$ (where $X_0$ is $\langle X \rangle$)
• Canonically relevant and marginal operators retained, higher-order terms omitted
• Curvature-square ($R^2$, $C^2$) and $\Gamma_k$0 terms excluded in this analysis
• Wavefunction renormalization and mass parameters for the graviton ($\Gamma_k$1, $\Gamma_k$2) and Proca field ($\Gamma_k$3, $\Gamma_k$4) included

The RG flow of couplings is determined by evaluating $\Gamma_k$5-functions using projections onto appropriate tensor structures of the correlation functions, computed in the fluctuation approach. Gauge-fixing flexibility ($\Gamma_k$6 parameter) is introduced to interpolate between longitudinal and transverse vector dynamics, allowing a thorough scan of FP landscapes.

Fixed Point Analysis and Numerical Results

The FP structure is systematically mapped in the gravity–Proca theory space. Five primary FP branches are identified:

• Minimally Coupled FP: Vanishing matter self-interactions, five relevant directions, critical exponents invariant under gauge-fixing parameter, trajectories flow to minimally coupled Abelian gauge theory in IR
• Interacting Proca-type FP: Nonvanishing Proca mass and quartic self-coupling, four relevant directions (two gravity, two Proca sector), distinct from minimally coupled FP, no IR trajectory to Abelian gauge theory
• Gemini FP Pairs (Gemini$\Gamma_k$7, Gemini$\Gamma_k$8): Each pair consists of two branches related by sign flip in $\Gamma_k$9, finite cubic Proca coupling, three/five relevant directions, existence constrained to $X = (Z_A/2)A_\mu A^\mu$0, critical exponents indicate strong operator mixing
• Interacting Proca$X = (Z_A/2)A_\mu A^\mu$1 FP: Only FP with nonzero matter self-couplings in strict Proca limit ($X = (Z_A/2)A_\mu A^\mu$2), four relevant directions, critical exponents stable under truncation extensions

Across all FPs, the critical exponents and FP coordinates are tabulated for representative $X = (Z_A/2)A_\mu A^\mu$3 values. Notably, the interacting Proca-type and Proca$X = (Z_A/2)A_\mu A^\mu$4 FPs have relatively few relevant directions, implying non-perturbative renormalizability and potentially higher predictivity.

Strong Numerical Claims:

• Existence of an interacting Proca-type FP with exactly four relevant directions (two gravity, two matter) in the analyzed truncation
• Critical exponents and FP coordinates stable (robust) under gauge-fixing scan and truncation extensions
• Cubic and quartic Proca couplings are found to be irrelevant in minimally coupled theory but relevant in interacting branches

Theoretical and Practical Implications

This work provides non-perturbative evidence that GPTs, as cosmological EFT candidates, can be UV-completed in the AS framework. The restricted number of relevant directions at the interacting Proca-type FPs implies a constrained space of IR parameters, resulting in correlations among cosmological observables (e.g., expansion history, perturbation growth) not present in bottom-up EFT approaches.

The UV-completion acts as a selection principle: If cosmological data favor an IR EFT lying outside the AS landscape, this signals the need for new physics beyond the considered GPTs. UV-consistent models offer a minimal pathway to a predictive cosmological extension of the Standard Model.

Contradictory/Stark Claims:

• Marginal operators ($X = (Z_A/2)A_\mu A^\mu$5, $X = (Z_A/2)A_\mu A^\mu$6, etc.) omitted here were found in other works to be irrelevant; their inclusion is not expected to alter the main results substantially.
• The relevance assignment for $X = (Z_A/2)A_\mu A^\mu$7 operator differs from earlier background-field studies, suggesting caution in operator-level mapping, as strong mixing is observed.

Outlook for AI and Cosmological Theory

Future extensions should:

• Expand the truncation to include neglected marginal and higher-order operators
• Allow for full momentum-dependent vertices in correlation function flows
• Evolve RG trajectories from UV FPs to IR and translate into cosmological observables for data-driven parameter constraints
• Quantify UV-imposed constraints in cosmological parameter fitting, potentially reducing the degeneracy in searches for new physics

From an AI research perspective, the application of advanced symbolic computation and diagrammatic flow analysis techniques (e.g., FORM/DoFun/VertEXpand toolchains) is crucial for carrying out high-dimensional non-perturbative RG computations. The technical approach outlined provides a template for similar analyses in modified gravity, dark sector, and particle cosmology.

Conclusion

The paper demonstrates, through non-perturbative fRG analysis, that GPTs coupled to gravity admit asymptotically safe UV completions in several branches, notably exhibiting an interacting Proca-type FP with four relevant directions. This provides a pathway for UV selection in cosmological EFTs, yielding predictive constraints among IR observables. The results contribute towards a theoretical foundation for narrowing cosmological models through UV consistency, and highlight the importance of systematic RG analyses for guiding the search for physics beyond the cosmological Standard Model.