Could four-dimensional ϕ4 theory exist beyond the usual triviality arguments?
Abstract
Do lattice triviality, suspected Borel non-summability and the perturbative Landau pole exclude an interacting four-dimensional theory satisfying the Osterwalder--Schrader axioms and reproducing every renormalized coefficient of positive-coupling O(N) ϕ4 theory? An example involving one real function separates vanishing cutoff limits along its allowed bare trajectories from a nonzero result obtained by renormalizing the coefficients first; an exponentially small correction preserves its expansion and removes the pole in its running coupling. For the massless two-point expansion of the singlet composite ΦaΦa at leading order in 1/N, ordinary Borel summation fails even for Gaussian smearing. The arithmetic mean of two Borel integrals along opposite imaginary rays defines a real distribution on every four-dimensional Schwartz momentum or position test, with finite-order remainder bounds uniform on bounded test sets. Two alternative covariances with nonnegative mass spectral weights remove the Euclidean pole and construct generalized free fields. After a fixed contact subtraction, their Euclidean extensions preserve every coefficient of the collective correlations obtained by Wick pairing. The interacting finite-N family remains open. We also state a conditional interacting route through the physical scalar correlations of a larger theory (the parent) with heavy particles. A concrete N=1 gauge--Higgs--fermion candidate has a tuned one-loop trajectory on which all interaction couplings tend to zero in the ultraviolet, and a stable tree vacuum with additional masses comparable to the heavy scale. Constructing the parent and its corrected ultraviolet trajectory, and proving coefficient decoupling and the exact remainder bounds as the quartic coupling tends to zero and the family's heavy masses grow, remain open.
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BibTeX
@misc{ragin2026could,
title = {{Could four-dimensional $ϕ^4$ theory exist beyond the usual triviality arguments?}},
author = {Anton Ragin},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12672},
eprint = {2610.12672},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12672},
}
RIS
TY - UNPB
TI - Could four-dimensional ϕ^4 theory exist beyond the usual triviality arguments?
AU - Anton Ragin
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.12672
UR - https://arxiv.org/abs/2610.12672
AB - Do lattice triviality, suspected Borel non-summability and the perturbative Landau pole exclude an interacting four-dimensional theory satisfying the Osterwalder--Schrader axioms and reproducing every renormalized coefficient of positive-coupling O(N) ϕ^4 theory? An example involving one real function separates vanishing cutoff limits along its allowed bare trajectories from a nonzero result obtained by renormalizing the coefficients first; an exponentially small correction preserves its expansion and removes the pole in its running coupling. For the massless two-point expansion of the singlet composite Φ_aΦ_a at leading order in 1/N, ordinary Borel summation fails even for Gaussian smearing. The arithmetic mean of two Borel integrals along opposite imaginary rays defines a real distribution on every four-dimensional Schwartz momentum or position test, with finite-order remainder bounds uniform on bounded test sets. Two alternative covariances with nonnegative mass spectral weights remove the Euclidean pole and construct generalized free fields. After a fixed contact subtraction, their Euclidean extensions preserve every coefficient of the collective correlations obtained by Wick pairing. The interacting finite-N family remains open. We also state a conditional interacting route through the physical scalar correlations of a larger theory (the parent) with heavy particles. A concrete N=1 gauge--Higgs--fermion candidate has a tuned one-loop trajectory on which all interaction couplings tend to zero in the ultraviolet, and a stable tree vacuum with additional masses comparable to the heavy scale. Constructing the parent and its corrected ultraviolet trajectory, and proving coefficient decoupling and the exact remainder bounds as the quartic coupling tends to zero and the family's heavy masses grow, remain open.
ER -