A theoretical and numerical study of the three-dimensional Ising universality class reports a smaller mismatch in one tracked conformal constraint at the higher of two tested derivative-expansion orders. The comparison was made near a regulator setting chosen by a whole-field criterion, meaning the calculation searched for the setting that minimized conformal-constraint violation across the studied range of field values. In that diagnostic, the reported violation at O(∂⁴) was of order one-quarter of the O(∂²) result.
The work is a preprint identified as arXiv:2608.25103v1 and dated 25 August 2026. It studies ϕ4 theory with the functional renormalization group as a calculation of the three-dimensional Ising universality class.
A calculation built around internal checks
The calculation numerically integrated derivative-expansion flow equations from the ultraviolet scale to the infrared scale. That procedure was used to determine fixed-point potentials, anomalous dimensions and universal critical exponents. The model was treated at next-to-next-to-leading order, written as O(∂⁴), and included a source for composite operators.
After locating a fixed point, the study obtained critical exponents by linearizing the renormalization-group flow around it. It focused on perturbations associated with ν and ω, two critical-exponent sectors examined in the calculation, and varied the multiplicative regulator factor α to see how the results changed with the regulator profile.
At the studied truncation, six conformal constraints were independent and distinct from constraints imposed by scale invariance. The checks came from conformal Ward identities applied to vertex functions and from selected momentum structures. The O(∂⁴) analysis addressed four constraints that had not previously been accessible and followed one particular constraint across the O(∂²) and O(∂⁴) approximations.
The regulator choice shaped the comparison
For ν and ω, conformal-constraint violations were lower near regulator values where the critical exponents showed minimal sensitivity to the regulator. The minima in the constraints coincided with extrema in the calculated observables within numerical accuracy. The study refers to the minimal-sensitivity test as PMS and to the constraint-based choice as PMC.
With the exponential regulator, the PMS entries were ν = 0.63061(66) and ω = 0.8263(55). The listed PMC estimates ranged from 0.63061 to 0.63193 for ν and from 0.81665 to 0.82625 for ω. The parenthetical figures are table-defined estimates based on differences between the O(∂⁴) and O(∂²) results divided by four, rather than statistical confidence intervals.
The same pattern appeared with the Wetterich regulator. Its PMS entries were ν = 0.63028(51) and ω = 0.8265(54), while the listed PMC estimates ranged from 0.63028 to 0.63139 for ν and from 0.82335 to 0.82646 for ω. The reported results showed quantitative and qualitative agreement between the exponential and Wetterich regulators, with lower conformal violations linked to lower sensitivity to the regulator profile.
What the pattern does not settle
The whole-field criterion was important because a pointwise choice did not always describe the calculation’s behavior across the field range. For some constraints, choosing α from the lower violation at ρ = 0 did not match the global behavior: most violations at ρ = 0 were negligible, a global minimum appeared at high α, and the behavior at non-zero ρ differed.
The reported alignment between smaller conformal violations and low-sensitivity regions, together with the improvement in the tracked constraint at the higher tested order, is presented as evidence relevant to convergence toward conformal consistency. The paper explicitly states that it provides no formal proof that the derivative expansion converges. The evidence is limited to the scalar three-dimensional Ising/ϕ4 calculation, the O(∂²) and O(∂⁴) truncations, and the exponential and Wetterich regulator families tested here.
G.D.P. acknowledges support from PEDECIBA and grant FCE-3-2024-1-180709 from Uruguay’s Agencia Nacional de Investigación e Innovación.
Paper data and sources
Original title: Convergence of the conformal Ward identity in the derivative expansion approximation
Authors: Jorge Ibañez, Matthieu Tissier, Gonzalo De Polsi
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-25
DOI: Not available
Original paper · Full text