Functional Dimensional Regularization
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 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.
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BibTeX
@misc{beretta2026functional,
title = {{Functional Dimensional Regularization}},
author = {Piero Beretta and Alessandro Codello},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25238},
eprint = {2609.25238},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25238},
}
RIS
TY - UNPB
TI - Functional Dimensional Regularization
AU - Piero Beretta
AU - Alessandro Codello
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.25238
UR - https://arxiv.org/abs/2609.25238
AB - 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.
ER -