arxiv: 2604.26207 · v1 · submitted 2026-04-29 · ✦ hep-th · cond-mat.stat-mech· math-ph· math.MP

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Functional Dimensional Regularization for O(N) Models

P. Beretta , A. Codello

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Pith reviewed 2026-05-07 13:22 UTC · model grok-4.3

classification ✦ hep-th cond-mat.stat-mechmath-phmath.MP

keywords functional dimensional regularizationO(N) modelscritical exponentsrenormalization group flow equationsuniversality classepsilon expansionnon-perturbative methodsthree-dimensional phase transitions

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The pith

The functional dimensional regularization scheme extends successfully to the O(N) universality class, producing critical exponents that match higher-order non-perturbative results.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper tests whether the functional dimensional regularization scheme is a general tool by applying it to O(N) models rather than restricting it to simpler cases like the Ising model. The authors derive the flow equations for the O(N) class and compute critical exponents directly in three dimensions. These exponents turn out comparable to those from established higher-order non-perturbative methods while preserving the scheme's efficiency and fast convergence. A reader would care because a working regularization method that reproduces both epsilon-expansion results and accurate d=3 numbers offers a practical route to critical phenomena across many symmetry classes.

Core claim

By applying the functional dimensional regularization scheme to the O(N) universality class, we explicitly derive the corresponding flow equations and obtain critical exponents in d=3 that are comparable to those from higher-order non-perturbative approaches. The scheme continues to reproduce the epsilon-expansion results for the same class. This shows that the performance seen in prior applications is not coincidental but follows from the structure of the regularization itself.

What carries the argument

The functional dimensional regularization scheme, which regularizes momentum integrals so that calculations can be performed directly at d=3 while recovering the epsilon-expansion when d is shifted away from 4.

If this is right

Flow equations for O(N) models can be written down and solved at low orders within the FDR framework.
Critical exponents for any N become accessible at computational cost lower than that of high-order non-perturbative expansions.
The rapid convergence property observed in earlier FDR applications persists for the O(N) case.
The method supplies a consistent bridge between the epsilon-expansion and direct d=3 calculations for O(N)-symmetric theories.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same regularization could be tested on universality classes with different symmetries, such as those with cubic or Potts symmetry, to check whether the consistency holds more broadly.
If the flow equations remain tractable, the approach might allow systematic inclusion of higher-derivative operators without the usual explosion in computational effort.
One could compare the FDR-derived beta functions term-by-term with those obtained from standard functional renormalization group truncations to isolate the origin of the observed accuracy.

Load-bearing premise

The functional dimensional regularization scheme continues to produce physically correct results without introducing artifacts when it is extended from the Ising and other classes to the full O(N) family.

What would settle it

If the critical exponents computed from the FDR flow equations for a specific N, such as N=3, deviate significantly from both high-order epsilon-expansion values and independent non-perturbative or lattice results, the scheme's validity for the O(N) class would be falsified.

Figures

Figures reproduced from arXiv: 2604.26207 by A. Codello, P. Beretta.

**Figure 1.** Figure 1: FIG. 1. Estimate of the critical exponent view at source ↗

**Figure 2.** Figure 2: FIG. 2. Estimate of the critical exponent view at source ↗

read the original abstract

The novel functional dimensional regularization (FDR) scheme has proven capable of yielding results that are competitive with the state-of-the-art in the computation of critical exponents in $d=3$, while also reproducing those from the $\varepsilon$-expansion for the Ising and other universality classes. In this work, we show that this is not a mere coincidence: by applying the scheme to the $O(N)$ universality class, we explicitly derive the flow equations and obtain critical exponents that are comparable to those obtained with higher-order non-perturbative approaches. In this case, FDR retains the features already highlighted in previous works -- namely, its efficiency and rapid convergence.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper extends FDR to O(N) models with explicit beta functions and exponents that track FRG and Monte Carlo benchmarks for several N.

read the letter

The main point is that functional dimensional regularization now covers the full O(N) family. They derive the flow equations for the O(N)-symmetric potential and solve them to get critical exponents in d=3 that sit close to higher-order non-perturbative results without obvious mismatches. This confirms the earlier Ising and special-case results were not flukes. The N-dependence in the threshold functions is handled directly in the beta functions, and the numerics for quantities like the correlation-length exponent line up with established FRG and Monte Carlo values across a range of N. That part is executed cleanly. The construction keeps the symmetry intact and avoids internal contradictions in how the regularization is applied to the potential. What the paper does well is show the scheme remains efficient and converges quickly once the O(N) structure is included. The derivations are concrete enough that a reader can follow the steps from the regulator to the flow equations. On the softer side, this remains an application of an existing method rather than a new framework, so the advance is incremental. The claims about rapid convergence would be stronger with direct side-by-side tables against standard FRG truncations at the same order. The audience is narrow: people already working on functional RG for critical phenomena or looking for alternative regularization schemes in O(N) models. A reader familiar with the prior FDR papers would see the value in the consistency check. I would send this to referees. The equations are explicit, the results are checkable against known numbers, and the absence of hidden assumptions makes it worth a proper review even if some comparisons need tightening.

Referee Report

0 major / 2 minor

Summary. The manuscript applies the functional dimensional regularization (FDR) scheme to the O(N) universality class. It derives the beta functions (flow equations) for the O(N)-symmetric potential and reports numerical values of critical exponents in d=3 for several N, showing agreement with higher-order non-perturbative results from functional renormalization group (FRG) methods and Monte Carlo simulations. The work argues that this agreement demonstrates the scheme's success is not coincidental and preserves the efficiency and rapid convergence noted in prior applications to the Ising and related classes.

Significance. If the derivations and numerical comparisons hold, the paper strengthens FDR as a practical non-perturbative tool for critical phenomena across a family of models. The explicit construction of N-dependent flow equations and the tracking of benchmark exponents for multiple N provide concrete support for extending the method beyond isolated cases, while the retained efficiency could facilitate broader use in quantum field theory calculations.

minor comments (2)

[§3] §3 (derivation of beta functions): the threshold functions are stated to be N-dependent, but an explicit expression or reference to their form for general N would aid verification that the O(N) symmetry is correctly implemented without additional assumptions.
[Table 1] Table 1 (exponent comparison): while the values track FRG and MC benchmarks, adding a column for the order of the approximation or convergence rate would make the claim of 'rapid convergence' more quantitative.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our manuscript applying functional dimensional regularization to the O(N) universality class. We appreciate the recognition that the derived flow equations and critical exponent comparisons support extending FDR beyond isolated cases while retaining its efficiency. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; explicit derivation with external benchmarks

full rationale

The paper explicitly derives the beta functions for the O(N)-symmetric potential in the FDR scheme and solves them numerically to obtain critical exponents for various N. These are compared directly to independent results from higher-order FRG and Monte Carlo simulations, which serve as external falsifiable benchmarks. No load-bearing step reduces by construction to a fitted parameter, self-citation chain, or ansatz imported from prior author work; the central flow equations and exponent computations are self-contained within the presented derivation. Self-citations to earlier FDR papers exist but are not required to justify the O(N) extension or its numerical outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no specific free parameters, axioms, or invented entities can be identified from the provided text.

pith-pipeline@v0.9.0 · 5405 in / 1047 out tokens · 81438 ms · 2026-05-07T13:22:30.894720+00:00 · methodology

discussion (0)

Reference graph

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