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arXiv:2609.25238

Functional Dimensional Regularization

Piero Beretta, Alessandro Codello

Published Sep 21, 2026• 0 citations
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Abstract

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 $\varepsilon$-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 $\varepsilon$-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.

DOI
10.48550/arxiv.2609.25238

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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\varepsilonε-expansion. The key insight is to define the FDR beta functions, valid in any continuous dimension ddd 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\varepsilonε-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=3d=3d=3, providing estimates for the anomalous dimension ηηη 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.
We show that it is possible to use dimensional regularization (DR) beyond the usual ε\varepsilonε-expansion in the context of renormalization group (RG) calculations in Critical Phenomena. Based on this fact, we propose a new functional RG scheme - Functional Dimensional Regularization (FDR) - and apply it to a scalar theory in three dimensions. We compute the critical exponents of the Ising universality class directly in d=3d=3d=3 under various typical approximations. The method that emerges combines the agility typical of DR with the generality proper of functional RG. Moreover, at a given order of approximation, FDR seems to provide faster convergence and better estimates than other functional RGs.
Continuous phase transitions provide a natural setting for understanding universality. Near criticality, the correlation length diverges, microscopic details become increasingly irrelevant, and physical observables obey scaling laws characterized by universal critical exponents. Within the Functional Renormalization Group (FRG), this scale dependence is described through the Effective Average Action, which progressively incorporates fluctuations as the momentum scale is lowered. A first part of this thesis is developed within the Local Potential Approximation (LPA), where we study multicritical fixed points of scalar field theories. Their universal properties are compared with results obtained at higher orders in the derivative expansion. Particular attention is devoted to the relation between global fixed-point potentials and their local representations through finite polynomial truncations. We show that the convergence of such expansions is constrained by the analytic structure of the global solution, and that expanding around a suitable nonzero stationary point significantly improves the reconstruction of multicritical branches. Beyond the structure of individual fixed points, a central part of this thesis concerns RG irreversibility. In two dimensions, Zamolodchikov’s c-theorem describes the flow from the ultraviolet to the infrared through a monotonically decreasing c-function. Within the FRG framework, we develop a variational procedure to reconstruct RG trajectories between fixed points and integrate the corresponding c-function along them. The resulting quantity displays monotonic behaviour and becomes increasingly stable as the truncation order is raised, providing a manifestation of RG irreversibility within the approximation considered. Finally, close to the upper critical dimensions, the linearized FRG flow around the Gaussian fixed point is naturally organized in terms of Hermite polynomials. Using the epsilon-expansion, we derive a curvature-based candidate c-function and obtain an analytic explanation for the alternating curvature of successive multicritical fixed points and for their progressive accumulation around the Gaussian fixed point.
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
We study the 3d Ising universality class using the functional renormalization group. With the help of background fields and a derivative expansion up to fourth order we compute the leading index, the subleading symmetric and antisymmetric corrections to scaling, the anomalous dimension, the scaling solution, and the eigenperturbations at criticality. We also study the cross correlations of scaling exponents, their dependence on dimensionality, and the numerical convergence of the derivative expansion. Collecting all available data from functional renormalization group studies to date, we estimate that systematic errors are in good agreement with findings from Monte Carlo simulations, ϵϵϵ-expansion techniques, and resummed perturbation theory.
tware architecture is developed that separates the physical model, numerical algorithms, and post-processing analysis into independent computational layers. The framework incorporates multi-start fixed-point searches, numerical Jacobian construction, eigenvalue and eigenvector analysis, residual-based convergence diagnostics, and automated extraction of universal critical exponents. The architecture is designed to accommodate higher-order truncations, alternative regulator choices, and continuation methods without requiring substantial modifications to the numerical infrastructure. The present implementation investigates scalar field models using quartic and higher-order polynomial truncations with a self-consistent treatment of the anomalous dimension within an LPA′-type framework. Validation procedures include Gaussian fixed-point recovery, truncation reduction tests, numerical convergence studies, and stability analysis. The resulting fixed-point structure and critical exponents are benchmarked against established Functional Renormalization Group calculations, providing a quantitative assessment of the computational implementation and its current level of agreement. Beyond the immediate numerical results, this work establishes a reproducible computational platform for systematic investigations of regulator dependence, truncation convergence, and dimensional continuation. Within the broader SDEL program, the framework serves as a foundation for exploring whether scale-dependent emergent dynamics can be formulated consistently alongside established renormalization-group methods. The paper therefore contributes both a validated computational methodology and a structured pathway for investigating future theoretical extensions while maintaining clear separation between established FRG techniques and proposed SDEL-specific developments.
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.