On the Origins of the Strong CP Problem
Abstract
The conventional strong CP problem arises from the tension between an unrestricted physical parameter θ̅ and the experimental constraint lvertθ̅rvert≲10−10. In the standard formulation, a global topological classification of gauge-field configuration space leads to integer-valued sectors, θ-vacua, and θ as a physical vacuum angle. Alternatively, a physical flavor-singlet pseudoscalar phase may be introduced through the quark mass matrix. The axial anomaly relates these descriptions through the invariant combination θ̅=θ+argdet M. We ask the logically prior question of whether known physical principles or observables require the introduction of an independent physical flavor-singlet CP-violating parameter in QCD. We distinguish consequences of local gauge invariance and causal locality from those requiring additional global assumptions or the introduction of a physical flavor-singlet CP-violating parameter. For constant θ, the topological charge density is a total four-divergence with no local variational response to compactly supported field variations, and a nonzero θ coupling is not required by perturbative renormalization. No established QCD phenomenology requires a physical θ̅ parameter. Without such a parameter, the topological susceptibility, axial anomaly, 't~Hooft vertex, Leutwyler--Smilga and Witten--Veneziano relations, standard lattice QCD results, semiclassical instanton solutions, and established phenomenological successes of QCD all remain. Nothing prevents the introduction of θ̅ as an independent physical parameter, which is consistent with these same QCD results. In this sense, establishing that the strong CP problem is an \emph{unavoidable} feature of QCD requires showing why a physical θ̅ parameter is \emph{required}, rather than merely allowed.
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BibTeX
@misc{williams2026origins,
title = {{On the Origins of the Strong CP Problem}},
author = {Anthony G. Williams},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2607.10272},
eprint = {2607.10272},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.10272},
}
RIS
TY - UNPB
TI - On the Origins of the Strong CP Problem
AU - Anthony G. Williams
PY - 2026
DA - 2026/08/29
PB - arXiv
DO - 10.48550/arxiv.2607.10272
UR - https://arxiv.org/abs/2607.10272
AB - The conventional strong CP problem arises from the tension between an unrestricted physical parameter θ̅ and the experimental constraint lvertθ̅rvert≲10^−10. In the standard formulation, a global topological classification of gauge-field configuration space leads to integer-valued sectors, θ-vacua, and θ as a physical vacuum angle. Alternatively, a physical flavor-singlet pseudoscalar phase may be introduced through the quark mass matrix. The axial anomaly relates these descriptions through the invariant combination θ̅=θ+argdet M. We ask the logically prior question of whether known physical principles or observables require the introduction of an independent physical flavor-singlet CP-violating parameter in QCD. We distinguish consequences of local gauge invariance and causal locality from those requiring additional global assumptions or the introduction of a physical flavor-singlet CP-violating parameter. For constant θ, the topological charge density is a total four-divergence with no local variational response to compactly supported field variations, and a nonzero θ coupling is not required by perturbative renormalization. No established QCD phenomenology requires a physical θ̅ parameter. Without such a parameter, the topological susceptibility, axial anomaly, 't~Hooft vertex, Leutwyler--Smilga and Witten--Veneziano relations, standard lattice QCD results, semiclassical instanton solutions, and established phenomenological successes of QCD all remain. Nothing prevents the introduction of θ̅ as an independent physical parameter, which is consistent with these same QCD results. In this sense, establishing that the strong CP problem is an \emph{unavoidable} feature of QCD requires showing why a physical θ̅ parameter is \emph{required}, rather than merely allowed.
ER -