Strict renormalizability as a paradigm for fundamental physics

Abstract: An important theoretical achievement of the last century was the realization that strict renormalizability can be a powerful criterion to select Lagrangians in the framework of perturbative quantum field theory. The Standard Model Lagrangian (without gravity) is strictly renormalizable from a perturbative point of view. On the other hand, the inclusion of gravity seems not to respect this criterion, since general relativity is perturbatively non-renormalizable. The aim of this work is to provide concrete evidence that strict renormalizability is still a valid criterion even when applied to gravity. First, we show that adding quadratic curvature terms to the Einstein-Hilbert action gives rise to a strictly renormalizable theory known as quadratic gravity. Second, we argue that this unique theory represents the most conservative approach to quantum gravity and, at the same time, is highly predictive, as it can explain new physics beyond general relativity already in the sub-Planckian regime. In particular, it provides one of the best fits to the CMB anisotropies via Starobinsky inflation and makes sharp cosmological predictions that can be tested in the near future. Finally, we comment on the (super-)Planckian regime and conclude with a historical note.

Strict Renormalizability as a Paradigm for Fundamental Physics

The incorporation of gravity into quantum field theory (QFT) has long posed a challenge due to the perturbative non-renormalizability of general relativity (GR). Luca Buoninfante's paper makes compelling arguments for adopting strict renormalizability as a viable criterion for constructing quantum gravity. The study systematically explores various aspects of strict renormalizability and its implications for both theoretical and observational physics.

Quadratic Gravity and Strict Renormalizability

Quadratic gravity emerges as a noteworthy candidate for a strictly renormalizable theory of gravity. By augmenting the Einstein-Hilbert action with quadratic curvature terms, such as $R^2$ and the Weyl term $C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma}$, the theory attains strict renormalizability, circumventing the issues of non-renormalizable divergences typical in GR. In this framework, the quadratic terms elevate the superficial degree of divergence to a constant four, irrespective of loop orders, thus requiring only a finite set of counterterms. This characteristic makes quadratic gravity a predictive theory amenable to empirical validation and falsification.

Implications for Early Universe Cosmology

The addition of the $R^2$ term introduces a scalar degree of freedom, commonly referred to as the scalaron, whose dynamics can potentially drive inflationary phases in the early universe. Quadratic gravity, particularly in the form of Starobinsky's model, provides one of the best fits to the Cosmic Microwave Background (CMB) anisotropies, as reported by recent data. The scalaron's mass, derived from CMB data, is approximately $3.0 \times 10^{13}$ GeV, leading to a large $c_0$ parameter on the order of $6.5 \times 10^{9}$, emphasizing sub-Planckian scales where novel physics manifest.

Runaway Ghosts and Causal Structure

A central concern in theories with higher derivative terms is the potential instability induced by ghost degrees of freedom. Quadratic gravity accommodates a massive spin-two ghost, raising questions about its physical viability. However, Buoninfante's paper revisits various quantization prescriptions—such as negative norm states, anti-Feynman prescription for ghost particles, and fakeons—to mitigate ghost-related issues and uphold unitarity. The presence of ghosts could suggest acausal effects at very high energies, implying that causality in quadratic gravity may be an emergent phenomenon relevant only at macroscopic levels.

Cosmological Predictions and Future Prospects

The paper forecasts that the tensor-to-scalar ratio—a key observable in inflationary cosmology—provides sharp predictions from quadratic gravity. Current and forthcoming precision measurements aim to constrain this ratio ($r$), potentially confirming the wide-ranging implications of the strict renormalizability framework. If validated, quadratic gravity may significantly influence the paradigm of quantum gravity as well as theoretical physics at large.

Challenges and Open Questions

Despite its theoretical consistency, several questions remain about quadratic gravity, particularly concerning the predictions of scattering amplitudes at ultra-Planckian energies where perturbative methods may falter. The robustness of asymptotically free behaviors, and how amplitudes evolve in non-perturbative settings, could require innovative approaches or resummation techniques. The paper advocates for more research to ascertain whether quadratic gravity can offer a comprehensive UV-complete framework.

Historical Insights and Future Developments

Buoninfante's work encourages revisiting historical insights on QFT and renormalizability, drawing parallels with the adoption of QED and non-Abelian gauge theories as strictly renormalizable models in the past. Should quadratic gravity's predictions hold, the theory could represent another instance where traditional QFT principles triumph over speculative alternatives. The paper calls for persistent engagement with strict renormalizability, suggesting that current and future empirical tests could affirm its role as a foundational principle in theoretical physics.

In conclusion, the exploration of strict renormalizability as applied to quantum gravity offers an enticing avenue for bridging GR with QFT. By carefully assessing theoretical constructs and empirical constraints, researchers are poised to refine the understanding of fundamental physics, potentially validating or challenging longstanding assumptions in quantum gravity.