Existence theorem on the UV limit of Wilsonian RG flows of Feynman measures

Technical Summary: Existence Theorem on the UV Limit of Wilsonian RG Flows of Feynman Measures

Problem Statement
In the nonperturbative formulation of Euclidean signature Quantum Field Theory (QFT), the vacuum state is characterized by a Wilsonian Renormalization Group (RG) flow of Feynman measures. A central difficulty in constructive QFT is the definition of the interacting measure $\mu = e^{-V} \cdot \gamma$, where $\gamma$ is a Gaussian measure on distributional fields and $V$ is an interaction potential. Since $\gamma$ lives on distributional fields while $V$ typically requires pointwise multiplication of fields (which is ill-defined for distributions), the product measure cannot be naively defined.

Wilsonian regularization addresses this by considering a family of measures $(\mu_\eta)$ on regularized (smooth) fields linked by coarse-graining operators. A critical open question addressed in this paper is whether a "nonterminating" Wilsonian RG flow—defined as a family of measures extending to arbitrary regularization strengths—admits a well-defined limit measure $\mu$ on the space of unregularized distributional fields. Furthermore, the paper investigates whether the relative interaction potentials between two such flows (e.g., an interacting flow and a free Gaussian reference flow) admit a UV limit, and under what conditions these limits preserve properties like lower bounds on the potential.

Methodology
The authors employ rigorous functional analysis and measure theory on spaces of distributions, specifically within the framework of Euclidean field theories on flat spacetime ($\mathbb{R}^N$).

1. Mathematical Framework: The study utilizes the space of tempered distributions $S'$ and Schwartz functions $S$. Coarse-graining operators are defined as convolution operators $C_\eta$ with kernels $\eta$ belonging to specific subsets of $S$ (e.g., regulators with Schwartz frequency tails, non-vanishing tails, or strictly bandlimited tails).
2. Factorization Properties: The core of the proofs relies on strong factorization properties of convolution operators in $S$. Specifically, the authors utilize theorems stating that any Schwartz function can be factored into a convolution of two other Schwartz functions, and that compact sets in $S$ can be factored similarly. These properties allow the construction of a "parent" function from the flow of regularized functions.
3. Bochner–Minlos Theorem: The existence of the UV limit measure is established by constructing a continuous, positive-definite function $Z$ on $S$ (the Fourier transform of the measure) from the family of regularized Fourier transforms $(Z_\eta)$. The Bochner–Minlos theorem then guarantees the existence of a unique sigma-additive measure on $S'$ corresponding to $Z$.
4. Radon–Nikodym and Absolute Continuity: To address relative interaction potentials, the authors analyze the absolute continuity of measures. They use the Radon–Nikodym theorem to relate the density functions of interacting and reference flows, proving that if a relative density exists at one scale, it exists for all scales and admits a UV limit.
5. Lower Semicontinuous Envelopes: In the case studies, the authors utilize the concept of lower semicontinuous envelopes and "greedy extensions" to define potentials on the full distribution space, analyzing conditions under which these extensions yield non-zero measures.

Key Contributions and Results

1. Existence of UV Limit Measure (Theorem 14, Corollary 15):
   The paper proves that any nonterminating Wilsonian RG flow of Feynman measures $(\mu_\eta)$ admits a unique UV limit measure $\mu$ on the space of distributional fields $S'$. The regularized measures in the flow are exactly the marginal measures of $\mu$ obtained by pushing forward via the coarse-graining operators: $\mu_\eta = (C_\eta)_* \mu$. This establishes a factorization property where the flow originates from a single ultimate measure.

2. Existence of UV Limit Relative Potential (Corollary 21):
   If two Wilsonian RG flows, $\mu_\eta$ and $\gamma_\eta$, are related by a relative interaction potential (density) $f_\eta = e^{-V_\eta}$ at a specific coarse-graining scale, then this relationship holds for all scales. Crucially, there exists a UV limit potential $V$ (and density $f$) such that the limit measures satisfy $\mu = f \cdot \gamma$.

3. Preservation of Lower Bounds (Theorem 25):
   If the regularized relative interaction potential $V_\eta$ is bounded from below at a specific scale, the UV limit potential $V$ is also bounded from below by the same bound (in the sense of essential supremum with respect to the reference measure).

4. Rigidity of Reference Gaussian Measures (Remark 23):
   The authors demonstrate that if a Wilsonian RG flow is described as a free Gaussian measure modified by a running potential, the parameters of the reference Gaussian measure (mass and field renormalization) cannot "run" (i.e., they must remain constant) modulo the smoothing by the regulator. This contrasts with informal RG approaches where parameters often run.

5. Case Studies and Renormalizability (Section 5):

   • Bounded Potentials: Models with interaction potentials bounded both from below and above (e.g., "basin-shaped" potentials or sine-Gordon models) are shown to be nonperturbatively renormalizable in arbitrary dimensions. Their UV limit measures are well-defined and non-zero.
   • $\phi^4$ Theory: The paper provides a rigorous analysis of the $\phi^4$ model. It confirms that for dimensions $N > 1$, the naive extension of the $\phi^4$ potential to distributional fields results in a potential that is $+\infty$ almost everywhere, leading to a zero measure. This aligns with the known triviality or non-existence issues in higher dimensions.
   • Non-Wilsonian Flows: The authors clarify that many constructive approaches in the literature (e.g., lattice limits or specific counterterm constructions for $\phi^4$ in $d=3,4$) produce sequences of measures that converge to a UV limit but do not themselves form a Wilsonian RG flow (they do not satisfy the marginality condition). However, the resulting limit measure does induce a Wilsonian RG flow.

Significance
The paper provides a rigorous existence theorem for the UV limit of Wilsonian RG flows, establishing that such flows are not merely families of regularized approximations but are fundamentally derived from a single, well-defined measure on distributional fields. This result validates the structural consistency of the Wilsonian approach in Euclidean QFT.

The authors emphasize that their results allow for a clear distinction between models that are nonperturbatively renormalizable (those admitting a UV limit with a bounded potential) and those that are not. Specifically, the theorems suggest that in 4-dimensional spacetime, interactions like $\phi^4$ are disfavored within the strict Wilsonian framework (as they fail to produce a non-zero measure with a bounded potential), whereas bounded potentials (like basin-shaped or competing Higgs models) are favored.

The work also clarifies the relationship between constructive QFT methods (which often construct a limit measure via non-Wilsonian sequences) and the Wilsonian RG formalism, showing that while the construction path may differ, the resulting UV limit measure naturally generates a consistent Wilsonian flow. This bridges the gap between existence proofs of measures and the structural requirements of the renormalization group.