Witten Effect in $3$-Form Description of $θ$-vacua

TL;DR

The paper investigates how θ-vacua, encoded through a massless 3-form description tied to topological susceptibility, influence magnetic monopoles via the Witten effect. It formulates θ-vacua in terms of a 3-form field, promotes the topological density to an integration constant, and uses a numerical relaxation of a ’t Hooft-Polyakov monopole in backgrounds with nonzero topological density to demonstrate the induced electric charge. Key findings show that the monopole acquires an electric charge proportional to the background and that the monopole exhibits electric-field–induced polarization; these results depend on the Higgs-to-vector mass ratio and align with the constrained instanton picture underpinning TSV. The work provides a bridge between quantum θ-physics and classical equations of motion, offering insights into nonperturbative vacuum structure and potential implications for electroweak θ-vacua and related monopole phenomena.

Abstract

The $θ$-vacua of a gauge theory admit an equivalent formulation as vacua of a massless Chern-Simons $3$-form, which originate from the topological susceptibility of the vacuum. This formulation provides a framework in which the physical manifestations of the $θ$-angle, which are quantum in origin, can be captured at the level of effective classical equations of motion. Within this framework, we derive the Witten effect, demonstrating that in the background of a massless $3$-form, the magnetic monopole indeed acquires an electric charge proportional to $θ$. This result, in particular, provides evidence that instantons, even when constrained by the Higgs effect, maintain a non-zero topological susceptibility of the vacuum. In addition to the Witten effect, we numerically demonstrate that a magnetic monopole exhibits polarizability when placed in a constant background electric field.

Figures (4)

• Figure 1: Polarizability of the monopole with respect to $m_h/m_v$. For $m_h \lesssim m_v$, the Polarizability decays approximately with $(m_h/m_v)^{-1}$, and for $m_h \gtrsim m_v$, it stays approximately constant.
• Figure 2: Witten electric charge with respect to $G\tilde{G}$ for different values of $m_h/m_v$. The electric charge is directly proportional to $G\tilde{G}$ as can be seen from the linear fit (gray lines).
• Figure 3: Witten charge with respect to $m_h/m_v$ for different $G\tilde{G}$. The electric charge approaches a constant value for large $m_h$ with a $(m_h/m_v)^{-1}$ behavior (gray lines).
• Figure 4: A background electric field (for example, coming from an electric dipole) can polarize the vacuum through virtual dyon-antidyon pairs of electric charge $\pm \theta g$ and magnetic charge $\pm g^{-1}$.