Magnetic Helicity, Magnetic Monopoles, and Higgs Winding

TL;DR

This work identifies a fundamental correction to the conventional link between magnetic helicity change and $F \wedge F$ in the presence of magnetic monopoles. It derives a general formula, $\Delta \mathcal{H} = \int_{M'} F \wedge F + \frac{4\pi}{e} \sum_i \int_{\Sigma_i} F$, by introducing a punctured spacetime $M'$ and Dirac-string contributions on worldsheets $\Sigma_i$, ensuring a gauge-consistent, well-defined helicity change. The authors extend the analysis to non-Abelian theories, showing that in the Georgi–Glashow model the IR helicity relates to UV invariants as $\mathcal{H}_{IR} = \frac{16\pi^2}{g^2} (N_{CS} + N_H)$ when heavy gauge modes decouple, and derive analogous results for electroweak-like theories where $e^2 \mathcal{H}_{EM} = 16\pi^2 (N_{CS} + N_H)$ under similar decoupling. The findings have implications for the evolution of primordial magnetic fields and for baryon-number dynamics via the chiral anomaly, highlighting that Higgs winding and Dirac-string effects can modify prior estimates of baryogenesis scenarios and suggesting new avenues for numerical exploration.

Abstract

Changes in magnetic helicity are often discussed across a variety of fields, from condensed matter physics to early universe cosmology. It is frequently stated that the helicity change is given by the integral of the gauge field strength tensor and its dual over spacetime, $\int F \wedge F$. However, this is incorrect when magnetic monopoles once exist in the spacetime. In this paper, we show the correct formula of the helicity change in such a case for the Maxwell theory with the magnetic monopoles. We also discuss what happens when we embed the Maxwell theory with magnetic monopoles into non-Abelian gauge theories. We show that a similar formula holds for the 't Hooft--Polyakov monopole. In particular, we find the winding numbers and the zeroes of the Higgs field in the non-Abelian gauge theory play a crucial role in the helicity change. The same discussion is also applicable to the electroweak theory, and we discuss the implication of our findings to the baryon number change via the chiral anomaly in the early universe.

Figures (5)

• Figure 1: An example of the configuration where the helicity is not well-defined. Some current supports the constant magnetic field $\vec{B}$ inside the torus. The red blob represents a magnetic monopole. The local patches $U_N$ and $U_S$ are shown in orange and green, respectively. The vector potential is defined in the direction of the orange and green surfaces from the center of the monopole. The two vector potentials are related by the large gauge transformation.
• Figure 2: (Left) A schematic representation of the punctured manifold $M'$ with neighbourhoods of 't Hooft loops. 't Hooft loops $L_i$ and $L_j$ are shown as red loops. The scooped-out regions around the loops are shown as pink regions. The arrow near $L_i$ indicates the direction of the 't Hooft loop. (Right) A schematic picture when we take a time slice of the punctured manifold $M'$. The time slice is shown in the left figure as a gray slanted rectangle. The 't Hooft line is a pair of monopole and anti-monopole punctures on the time slice, which are shown as red dots with $+$ and $-$ signs, respectively. The intersection of the time slice with $L_i$ is shown as an orange spherical surface. The scooped regions are shown as pink balls surrounding the monopole and anti-monopole punctures.
• Figure 3: (Left) A schematic representation of the punctured manifold $M"$ with infinitesimally small $B_i$. In addition to $M'$, described in Fig. \ref{['fig:thooft_loop_cut_off']}, we indicate $B_i$ as blue regions. (Right) A schematic picture when we take a time slice of the punctured manifold $M"$. The time slice is shown in the left figure as a gray slanted rectangle. In addition to the monopole and anti-monopole punctures as in Fig. \ref{['fig:thooft_loop_cut_off']}, $B_i$, which is the three-dimensional cylinder-like region connecting the monopole and anti-monopole punctures, is shown as a blue region. Note that, in this figure, $B_i$ looks like a straight cylinder, but just as Dirac strings, it can be any shape conguent to a cylinder, as long as it connects the monopole and anti-monopole punctures and infinitisimally thin.
• Figure 4: A toroidal magnetic flux with a monopole moving along the $z$-axis. The toroidal magnetic flux is centered at the origin $O$, with radius $R_0$. The monopole is represented by the red dot at the bottom, moving upwards along the $z$-axis.
• Figure 5: A schamatic illustration of the helicity change in the Higgs phase. At the initial state, two linked vortex loops are present, and the helicity is nonzero, as shown in the leftmost figure. At some point in time, a monopole--anti-monopole pair is nucleated, as shown in the middle figure. The monopole and anti-monopole are represented by the red dots and a possible choise of the Dirac string between them is shown as the blue dashed line. They move along the vortex loop as shown by the pink arrows in the middle figure. They eventually annihilate, leaving only one vortex loop in the final state, as shown in the rightmost figure. The helicity is zero in the final state.