One-loop renormalization group flow of the translation-invariant noncommutative Yukawa theory
Abstract
We study the one-loop renormalization group flow of the translation-invariant noncommutative pseudoscalar Yukawa theory on four-dimensional Euclidean Moyal space. The two inequivalent orderings of the Moyal-star Yukawa vertex give rise to two independent couplings g1 and g2, whose beta functions form a coupled nonlinear system; every one-loop divergence is absorbed by a counterterm already present in the action, establishing one-loop renormalizability of the theory. We solve the one-loop system analytically: it decouples in the variables u=g12+g22 and v=g12−g22, the quartic coupling is obtained by a Riccati reduction on the symmetric surface g1=g2, and the generic asymmetric flow is reduced to a single quadrature by the exact invariant ℐ=(g1g2)3/(g12−g22)4. Two dynamical consequences follow. The ratio r=v/u has a UV-attractive symmetric surface r=0 and IR-attractive asymmetric directions r=±1, so that the flow restores the symmetry between the two Moyal orderings towards the ultraviolet and amplifies any initial ordering asymmetry towards the infrared, as a fractional power of the logarithm. In consequence the Yukawa-induced coefficient of the non-planar 1/(θ2p2) structure is dynamically suppressed towards p→ 0; the singularity itself is not removed, and remains the task of the IR-improving term. We also solve the dimensionful sector (M2, m, a2 and b) in closed form on the symmetric critical trajectory, and compare throughout with the commutative theory.
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BibTeX
@misc{bouchachia2026oneloop,
title = {{One-loop renormalization group flow of the translation-invariant noncommutative Yukawa theory}},
author = {Karim Bouchachia and Smain Kouadik},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.29213},
eprint = {2608.29213},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29213},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - One-loop renormalization group flow of the translation-invariant noncommutative Yukawa theory
AU - Karim Bouchachia
AU - Smain Kouadik
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2608.29213
UR - https://arxiv.org/abs/2608.29213
AB - We study the one-loop renormalization group flow of the translation-invariant noncommutative pseudoscalar Yukawa theory on four-dimensional Euclidean Moyal space. The two inequivalent orderings of the Moyal-star Yukawa vertex give rise to two independent couplings g_1 and g_2, whose beta functions form a coupled nonlinear system; every one-loop divergence is absorbed by a counterterm already present in the action, establishing one-loop renormalizability of the theory. We solve the one-loop system analytically: it decouples in the variables u=g_1^2+g_2^2 and v=g_1^2−g_2^2, the quartic coupling is obtained by a Riccati reduction on the symmetric surface g_1=g_2, and the generic asymmetric flow is reduced to a single quadrature by the exact invariant ℐ=(g_1g_2)^3/(g_1^2−g_2^2)^4. Two dynamical consequences follow. The ratio r=v/u has a UV-attractive symmetric surface r=0 and IR-attractive asymmetric directions r=±1, so that the flow restores the symmetry between the two Moyal orderings towards the ultraviolet and amplifies any initial ordering asymmetry towards the infrared, as a fractional power of the logarithm. In consequence the Yukawa-induced coefficient of the non-planar 1/(θ^2p^2) structure is dynamically suppressed towards p→ 0; the singularity itself is not removed, and remains the task of the IR-improving term. We also solve the dimensionful sector (M^2, m, a^2 and b) in closed form on the symmetric critical trajectory, and compare throughout with the commutative theory.
ER -