De Sitter-Invariant Renormalization Scheme

• De Sitter-Invariant Renormalization Scheme is a method to remove ultraviolet and infrared divergences in quantum field theory on de Sitter space while preserving the SO(1,4) symmetry.
• It systematically controls secular growth by absorbing loop-induced time dependence into coupling redefinitions via invariant counterterms and a Weierstrass transform.
• This approach refines predictions in inflationary cosmology by ensuring that physical observables are free from spurious, time-dependent artifacts.

A de Sitter-invariant renormalization scheme is a procedure for removing ultraviolet (UV) and infrared (IR) divergences in quantum field theory (QFT) defined on de Sitter space that preserves the symmetry group of the de Sitter background at each step of the calculation. Such schemes are designed to avoid spurious violations of de Sitter invariance in observables, prevent the introduction of nonphysical secular (time-dependent) growth in correlation functions, and ensure that the counterterms and finite renormalizations respect the underlying spacetime isometries.

1. De Sitter Invariance and the Structure of IR Divergences

In QFT on four-dimensional de Sitter space, fields with masses $m \ll H$ (where $H$ is the Hubble parameter) exhibit enhanced IR fluctuations. In conventional treatments, this leads to the appearance of secular terms in correlation functions—logarithmic growth in time which appears to "break" de Sitter symmetry and threatens the consistency of cosmological perturbation theory.

However, as demonstrated in (Palma et al., 28 Jul 2025), the essential origin of this secular growth is not intrinsic to the physics but an artifact of non-invariant regularization and renormalization schemes, particularly those employing comoving cutoffs or non-covariant subtractions. If one instead uses a de Sitter-invariant scheme—defined by manifest invariance under the de Sitter isometry group $SO(1,4)$ and implementing Wilson's integration axioms—the structure of divergences is controlled by the number of interaction vertices ($V$) in a diagram, not the number of loops ($L$). In this framework, each vertex in an $n$-point function contributes a logarithm in conformal time, and all time dependence beyond tree level can be absorbed into a multiplicative renormalization of couplings, yielding no additional secular growth from loops.

Specifically, the generic structural formula for superhorizon $n$-point functions is:

$$D_T(\vec{k}_1,\ldots,\vec{k}_n;\tau) \propto \delta^{(3)}\left(\sum_{i=1}^n \vec{k}_i\right) \sum_s A_s^{(T)}(\vec{k}_1,\ldots,\vec{k}_n) \prod_{a=1}^V \ln[-\tau f_a^{(T)}(\vec{k}_1,\ldots,\vec{k}_n)]$$

where $V$ is the number of interaction vertices, and $A_s^{(T)}$ and $f_a^{(T)}$ are analytic functions of the external momenta determined by the tree topology $T$ (Palma et al., 28 Jul 2025).

2. Systematic IR Regulation and Renormalization

Many of the most severe infrared divergences in de Sitter arise from "daisy" diagrams—loops in which the internal momentum begins and ends at the same vertex. These diagrams generate terms proportional to the coincident two-point function $G(0;\tau)$, which is either a divergence or a constant depending on regularization.

Within de Sitter-invariant schemes, these divergences are systematically removed order by order via a Weierstrass transform of the couplings. Explicitly, for a scalar field with non-derivative interactions, the bare coupling $\lambda_n$ is replaced by a renormalized coupling $\bar{\lambda}_n$:

$$\lambda_n = \sum_{L=0}^{\infty} \frac{(-1)^L}{L!} \left(\frac{\sigma_{\text{tot}}^2}{2}\right)^L \bar{\lambda}_{n+2L}$$

where $\sigma_{\text{tot}}^2$ is the properly regularized coincident variance (which can be set to zero in a scale-invariant scheme, or assigned a physical cutoff-dependent value). This summation absorbs the concatenated daisy loops into the definition of the coupling, ensuring that physical $n$-point functions are finite and free of secular artifacts (Palma et al., 28 Jul 2025).

When considering diagrams with loops involving external momentum flow, de Sitter-invariant regulation ensures that the loop IR divergence does not introduce new time dependence. Instead, these divergences become proportional to tree-level terms or can be absorbed by nonlocal counterterms that are themselves de Sitter-invariant.

3. Absence of Genuine Secular Growth and Role of Initial State

A detailed distinction must be made between massless scalars with shift symmetries (e.g., Goldstone modes) and massive or general light scalars without such symmetries. For massless shift-symmetric fields, all possible physical observables are constructed from derivatives, and secular growth is absent since the shift invariance prevents the growth of physical correlation functions.

For light, non-derivatively-coupled scalars, although correlation functions can diverge strongly as $m \to 0$, these divergences remain invariant under the de Sitter group. When renormalization and observable construction proceed in an invariant way, no physical secular time dependence appears in connected correlators: any superficial time dependence can either be absorbed into the definition of coupling constants or is due to the specific choice of UV/IR regulator and not to intrinsic physics.

Therefore, neither massless nor light scalar fields in de Sitter, when treated with a symmetry-preserving renormalization scheme, exhibit genuine secular growth in the superhorizon limit (Palma et al., 28 Jul 2025).

4. Critique and Modification of the Stochastic Formalism

The stochastic inflation formalism, built upon the Fokker–Planck equation for the coarse-grained field’s probability distribution, is often invoked to resum secular IR effects. The derivation typically employs a moving, time-dependent comoving cutoff, which breaks de Sitter invariance:

$$\frac{d\rho}{dt} = \frac{H^3}{8\pi^2} \frac{\partial^2 \rho}{\partial \phi^2} + \frac{1}{3H} \frac{\partial}{\partial\phi} (\rho V')$$

However, as (Palma et al., 28 Jul 2025) explains, the entire secular structure in this framework is a direct consequence of this breaking: the time dependence is not intrinsic to the de Sitter QFT, but rather is induced by the moving cutoff in the ultraviolet or by non-invariant filters that artificially distinguish modes by their comoving scale. When the filter is replaced by a fixed (de Sitter-invariant) physical cutoff, all secular time dependence in observables disappears except for the controlled, vertex-induced logarithms prescribed by the QFT expansion.

Thus, a de Sitter-invariant renormalization scheme reveals that the standard stochastic resummation must be revised; in particular, the effective noise (diffusion coefficient) and drift terms must be computed using invariant QFT methods and will generally differ from those derived via non-invariant smoothing techniques.

5. Implications for Effective Field Theory and Cosmology

Adoption of a fully de Sitter-invariant renormalization scheme (one satisfying Wilson's axioms for integration and respecting the spacetime symmetries at each step) has several notable implications:

• All UV and IR divergences in loops can be removed in a systematic, symmetry-preserving way using local and, where necessary, nonlocal counterterms that are themselves de Sitter-invariant.
• The only secular time dependence appearing in $n$-point functions is that associated with the number of interaction vertices ($V$): no new secular growth emerges from increasing loop order ($L$). All enhanced IR contributions are thus organized and resummed via coupling redefinitions, not by time-dependent corrections to observables.
• For shift-symmetric fields, all physical correlation functions are manifestly constant (modulo slow rolling from explicit symmetry breaking) even if the two-point function itself diverges.
• The stochastic formalism must be modified to properly reflect the de Sitter-invariant dynamics; its standard implementation is not the correct resummation once symmetry-preserving renormalization is used.

A plausible implication is that physical predictions for inflationary observables (e.g., non-Gaussianities, higher $n$-point functions) must be revisited whenever the standard QFT-in-de Sitter expansion is replaced by a de Sitter-invariant renormalization. Secular effects previously attributed to quantum loop growth may be artifacts of non-invariant regularization, and the true quantum corrections may be milder, especially for light fields.

6. Summary Table: Comparison of Renormalization Schemes

Scheme Type	IR Divergences Removed	Secular Growth in Correlators	De Sitter Invariance Preserved
Comoving Cutoff/Stochastic	Not fully (spurious spikes)	Appears at all loop orders	No (cutoff breaks invariance)
De Sitter-invariant	Systematically, all orders	Controlled, only by $V$	Yes

In conclusion, the main insight from (Palma et al., 28 Jul 2025) is that a de Sitter-invariant renormalization scheme, built on symmetry-preserving regularization and reparametrization of couplings, systematically removes IR divergences and eliminates nonphysical secular growth from loops, thus aligning quantum field theory in de Sitter with the underlying spacetime symmetries and altering the interpretation of IR effects during inflation.