Physical remnant of electroweak theta angles

TL;DR

The paper shows that, in addition to the QCD theta angle, the Standard Model contains a physical electroweak theta angle that becomes observable only on non-simply connected manifolds. By analyzing chiral rotations and the associated anomalies, it identifies two invariant combinations, $\bar{\theta}_3$ and $\bar{\theta}_{21}$, which remain unchanged under arbitrary chiral rotations of SM fermions; $\bar{\theta}_3$ corresponds to the familiar $\bar{\theta}_\text{QCD}$ and $\bar{\theta}_{21}$ coincides with the effective QED theta angle after electroweak symmetry breaking. The third section demonstrates that, after EWSB, the EW theta structure yields a QED-like term with coefficient $\tfrac{1}{2}\theta_2+\theta_1$, i.e., $\bar{\theta}_\text{QED}$, while the $W$ and $Z$-related contributions decouple as total divergences, implying a physical observable in nontrivial topology. The work furthermore notes a coincidental emergence of the same coefficient combination from both chiral-rotation shifts and Lagrangian decomposition, suggesting potential constraints on SM fermion representations via second-order Dynkin indices.

Abstract

In addition to the well-known quantum chromodynamical theta angle, we show that the Standard Model has another theta angle which is invariant under arbitrary chiral rotations of quarks and leptons. The new theta angle coincides with the quantum electrodynamical theta angle which may be observable in a nontrivial spacetime topology.