The strong CP problem revisited and solved by the gauge group topology

TL;DR

This paper proposes a novel solution to the strong CP problem by treating the theta angle as a gauge group topology operator, whose value is dynamically determined by the fermionic mass term and the infinite volume limit.

Contribution

It introduces a new perspective on the theta angle as a gauge group topology operator, providing a stable, radiatively corrected mechanism to solve the strong CP problem.

Findings

The theta angle is identified as a spectrum point of a gauge topology operator.

The fermionic mass term determines the phase, fixing the theta angle dynamically.

The mechanism is validated in the massive Schwinger model and extended to QCD with chiral symmetry-preserving boundary conditions.

Abstract

We exploit the non-perturbative result that the $\theta$ angle which defines the vacuum structure is not a $c$-number free parameter, as suggested by the instanton semi-classical approximation, but instead one of the points of the spectrum of the central operator $\tilde{\theta}$ which describes the gauge group topology. Hence, the value of such an angle should not be \textit{a priori} fixed, but rather be determined, as any quantum operator, by the infinite volume limit of the functional integral, where the fermionic mass term uniquely fixes the phase, with $< \tilde{\theta} > = \theta_M$, the mass angle. Such an equality is stable under radiative corrections performed before the infinite volume limit of the functional integral. The mechanisms is carefully controlled in the massive Schwinger model with attention to the infrared problems, to the volume effects induced by boundary terms and with a careful discussion of the infinite volume limit of the functional integral. The extension to the QCD case relies on the choice of modified APS boundary conditions which do not break chiral symmetry, allowing for the crucial role of the fermionic mass term in uniquely determining the phase, with a solution of the strong $CP$ problem.