Local subtractions and renormalization-group monotonicity: exact defect and sphere flows
Abstract
We ask when local subtractions of the sphere free energy yield a function that decreases monotonically along a renormalization-group flow. The subtractions are polynomials in D=R d/dR that remove the local counterterms and leave fixed-point values unchanged, in a fixed renormalization prescription. We prove that if the subtracted free energy C approaches its infrared value from below, with ordered endpoints and a power-law approach to both fixed points, no such subtraction is monotone, and that further factors with real zeros cannot remove a nonmonotonicity already present. For a three-dimensional flow to a gapped phase, C−CIR=4π2(M2−3M0)/R+…, where M0 and M2 are spectral moments of the flat-space stress-tensor two-point function, so every subtraction fails when M2<3M0, as for a free conformally coupled massive scalar. In two dimensions the leading term is always negative, so no unitary flow to a gapped phase whose subtracted free energy starts at cUV/3 admits a monotone subtraction; the massive Dirac fermion on S2 is an exact example. In the conformally coupled massive O(N) model, M2−3M0 is negative at leading order in N for every finite nonnegative quartic coupling. At criticality it vanishes at that order and its first correction is negative. At weak positive coupling it is negative for every curvature coupling and every finite N. For the Gross--Neveu flow from an interacting fixed point, it is negative already at leading order. In contrast, we prove monotonicity for Robin boundary flows on every sphere Sp with p≥2 and for massive Dirac fields on every odd sphere Sd with d≥3. For two solvable double-trace families we find exact thresholds: a monotone subtraction exists if and only if Δ̂≥1/2 on a spherical surface defect, and Δ≥(3−√(37/13))/2 on S3 at leading large N.
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BibTeX
@misc{scardino2026local,
title = {{Local subtractions and renormalization-group monotonicity: exact defect and sphere flows}},
author = {Francesco Scardino},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.02447},
eprint = {2608.02447},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.02447},
}
RIS
TY - UNPB
TI - Local subtractions and renormalization-group monotonicity: exact defect and sphere flows
AU - Francesco Scardino
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2608.02447
UR - https://arxiv.org/abs/2608.02447
AB - We ask when local subtractions of the sphere free energy yield a function that decreases monotonically along a renormalization-group flow. The subtractions are polynomials in D=R d/dR that remove the local counterterms and leave fixed-point values unchanged, in a fixed renormalization prescription. We prove that if the subtracted free energy C approaches its infrared value from below, with ordered endpoints and a power-law approach to both fixed points, no such subtraction is monotone, and that further factors with real zeros cannot remove a nonmonotonicity already present. For a three-dimensional flow to a gapped phase, C−C_IR=4π^2(M_2−3M_0)/R+…, where M_0 and M_2 are spectral moments of the flat-space stress-tensor two-point function, so every subtraction fails when M_2<3M_0, as for a free conformally coupled massive scalar. In two dimensions the leading term is always negative, so no unitary flow to a gapped phase whose subtracted free energy starts at c_UV/3 admits a monotone subtraction; the massive Dirac fermion on S^2 is an exact example. In the conformally coupled massive O(N) model, M_2−3M_0 is negative at leading order in N for every finite nonnegative quartic coupling. At criticality it vanishes at that order and its first correction is negative. At weak positive coupling it is negative for every curvature coupling and every finite N. For the Gross--Neveu flow from an interacting fixed point, it is negative already at leading order. In contrast, we prove monotonicity for Robin boundary flows on every sphere S^p with p≥2 and for massive Dirac fields on every odd sphere S^d with d≥3. For two solvable double-trace families we find exact thresholds: a monotone subtraction exists if and only if Δ̂≥1/2 on a spherical surface defect, and Δ≥(3−√(37/13))/2 on S^3 at leading large N.
ER -