We consider a functional relation between a given Wilsonian renormalization group (RG) flow, which has to be related to a specific coarse-graining procedure, and an infinite family of (UV cutoff) scale-dependent field redefinitions. Within this framework, one can define a family of Wilsonian proper-time (PT) exact RG equations associated to an arbitrary regulator function. New applications of these RG flow schemes to the Ising Universality class in three dimensions in the derivative expansion are shortly illustrated.
We consider a functional relation between a given Wilsonian renormalization group (RG) flow, which has to be related to a specific coarse-graining procedure, and an infinite family of (UV cutoff) scale-dependent field redefinitions. Within this framework, one can define a family of Wilsonian proper-time (PT) exact RG equations associated to an arbitrary regulator function. New applications of these RG flow schemes to the Ising Universality class in three dimensions in the derivative expansion are shortly illustrated.
The functional renormalization group (FRG) is a powerful tool that has facilitated the investigation of various strongly interacting theories, encompassing systems characterized by both bosonic and fermionic variables. Specifically, this project focuses on the analysis of theories with O(N) symmetry with bosonic scalar fields. These theories are of particular interest due to their broad applicability in real physical systems. For example, Z2 symmetry (the O(1) model) describes the well-known Ising universality class, which in turn models the liquid-gas transition. On the other hand, the O(2) model belongs to the universality class of the XY model, used to describe the transition from fluid to superfluid in 4 He, while the O(3) model, known as the Heisenberg model, describes the ferromagnetic transition in isotropic materials. Finally the O(0) model is related to the self-avoiding walk (SAW). To characterize these theories, various methods have been implemented within the framework of the renormalization group. Perturbation theory has been applied since the early days of the subject, specifically in the form of the ε-expansion, which has reached high loop order in recent years. Alternatively, a non-perturbative approach can be chosen, which is known as the non-perturbative renormalization group (NPRG). Within the NPRG, there are several approximation schemes, with the derivative expansion being particularly noteworthy. The ultimate goal of these efforts is to calculate the critical exponents, which define the given universality class. The standard non-perturbative method involves solving an equation that depends on a regulator or cut-off, which represents one of the main challenges of this approach. Although the regulators are designed so that the theory does not depend on them at the scales of interest, the results for the critical exponents can vary depending on the regulator used once approximations are made. In this project, we propose an approach that eliminates the need for an explicit mass cut-off significantly simplifying the calculation of critical exponents while mitigating the regulator dependence of the results. In particular, we show that it is possible to use dimensional regularization (DR) beyond the ε-expansion in the context of RG calculations of critical properties. Based on this we propose a new functional RG scheme called Functional Dimensional Regularization (FDR) and apply it to the O(N) model in three dimension, finding excellent agreement with state-of-the-art computations
Coarse graining from within: Wilson-Fisher universality on S3
Wilsonian renormalization is usually formulated in momentum space, but on curved backgrounds momentum shells have no invariant meaning. We replace them by an intrinsic spectral cutoff, ordering modes by the covariant Laplacian and setting the cutoff resolution by the system size in renormalization group (RG) units. For a scalar field on S3, this yields a covariant, momentum-free RG flow whose trace is an exact sum over spherical harmonics. The standard flat-space flow is recovered when the sphere is large compared with the coarse-graining scale. As a nontrivial test, the compact spectral flow realizes Wilson-Fisher universality without momentum shells: the interacting fixed point survives at finite resolution, has one relevant direction, and approaches its flat-space counterpart smoothly, with critical exponents only weakly affected by the compact spectrum.
We introduce and develop Functional Dimensional Regularization (FDR), a novel functional renormalization group (RG) framework that extends the principles of dimensional regularization beyond the perturbative ε-expansion. The key insight is to define the FDR beta functions, valid in any continuous dimension d and characterized by threshold functions, as a sum over all critical dimensions of the corresponding scheme-independent beta functions computed in DR. We demonstrate that this formalism exhibits all the hallmarks of a fully-fledged functional RG. Within the local potential approximation, we perform a general fixed point analysis, recovering the ε-expansion for all multi-critical models and establishing a direct connection to functional scaling solutions, spike plots, and eigen-perturbation spectra. Extending the framework to the second order of the derivative expansion, we compute the critical exponents for the Ising universality class in d=3, providing estimates for the anomalous dimension η and RG spectrum that converge rapidly and show favorable agreement with state-of-the-art results. We present a detailed comparison with non-perturbative and proper-time RG approaches, highlighting structural similarities and key differences. In particular, we introduce a ``distillation'' procedure that allows one to systematically derive the FDR flow from any other one-loop-exact RG scheme. Our results establish FDR as a self-contained and competitive RG framework that combines the technical simplicity of dimensional regularization with the versatility of the functional approach.
The use of nonperturbative Wilsonian functional renormalization group (FRG) equations within an unavoidable approximation scheme produces physical results that depend on the choice of the coarse-graining regulator. This dependence of physical observables on the regulator can be minimized according to the principle of minimal sensitivity (PMS). For the 1PI formulation based on the Wetterich-Morris equation, we propose to investigate the optimal choice within a space of compactly supported polynomial regulators parametrized by a set of variables. We test this approach using the derivative expansion of the effective average action up to fourth order, applying it to the scalar field theory representing the three-dimensional Ising universality class and minimizing the dependence of the anomalous dimension on the regulator. At fourth order, we find appreciable corrections to the critical exponents that improve the predictions, compared to previous one parameter analyses. We also observe families of compact regulators that rapidly converge towards an optimal functional form as a small number of parameters is progressively introduced.
Renormalization group (RG) flows for systems subject to an external field can be derived either by fixing the field or by fixing the conjugate order parameter. We show how to implement these boundary conditions within the functional renormalization group approach. We derive and compare the corresponding flow equations for ϕ4 theory and show that some scaling variables and scaling exponents associated with the linearized flow of the relevant three-point vertex in the vicinity of the Ising fixed point depend on the boundary conditions. We also discuss the effect of elasticity on Ising criticality. In this case two of the fixed points (namely the renormalized Ising fixed point and the spherical fixed point) obtained for constant strain are shifted to physically inaccessible regimes of parameter space when we perform the RG at constant stress.
This walkthrough derives, from substrate primitives, the structural backbone of Wilsonian renormalization applied to the canonical Event Density (ED) PDE — and shows what the RG actually says about ED as a coarse‑grained theory. In 1971, Kenneth Wilson introduced the renormalization group: instead of solving a system at one scale, track how its couplings flow as you integrate out short‑distance structure. Fixed points govern universal behavior; relevant and irrelevant operators determine which features survive coarse‑graining. The framework reshaped statistical physics and quantum field theory. Applied to the canonical ED PDE, three structural results emerge: (1) Form‑closure of the operator basis. Under Wilsonian coarse‑graining (momentum‑shell integration + rescaling), the ED operator basis is *closed*: no new operators appear at tree level or one loop. The ED PDE’s structure isa theorem, not an ansatz. (2) ED is not an RG fixed point at physical couplings. The beta functions are non‑trivial. With dynamical exponent z = 2, M₀ is marginal, P₀ and ζ are relevant, τ is irrelevant. The couplings flow; ED is regime‑dependent, not scale‑invariant. (3) Three distinct flow regimes. The RG flow produces a UV regime of free two‑channel diffusion, an intermediate ED‑activeregime where the full operator structure matters, and an IR regime of linear gapped diffusion. ED‑specific predictions live only in the intermediate window. The derivation chain: canonical two‑channel ED PDE (δ coupled to participation mode v) with finite operator basis → momentum‑shell split and rescaling → form‑closure at tree and one loop → beta‑function computation at z = 2 → fixed‑point catalog (Gaussian fixed point, two fixed lines, IR sink at infinity) → flow invariants → two crossover scales (ℓ_v for participation‑slaving onset; ℓ_ξ for gap‑dominated decay) → ED window ℓ_v < ℓ < ℓ_ξ → three effective PDEs in UV, intermediate, and IR regimes. The structural payoff: ED’s cross‑scale invariance is *operator‑structure* invariance within a bounded window, not coupling‑invariance across all scales. The “one PDE, three domains” claim means the same operator basis governs behavior across the intermediate regime, even though the couplings run. The RG walkthrough establishes ED’s self‑consistency: the canonical ED PDE is a closed, well‑behaved object under Wilsonian coarse‑graining, with a defined applicability window and controlled limits. This is not a derivation of a new equation — it is a structural validation that ED’s cross‑scale invariance survives RG analysis exactly as advertised.