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Electrons embedded in liquid $^3$He form mesoscopic bubbles with radii largecompared to the interatomic distance between $^3$He atoms, voids of$N_{bubble}\approx 200$ $^3$He atoms, generating a negative ion with a largeeffective mass that scatters thermal excitations. We develop scattering theoryof Bogoliubov quasiparticles by negative ions embedded in $^3$He-A thatincorporates the broken symmetries of $^3$He-A, particularly time-reversal andmirror symmetry in a plane containing the chiral axis $\hat{\bf l}$. Multiplescattering by the ion potential, combined with Andreev scattering by the chiralorder parameter, leads to a spectrum of Weyl Fermions bound to the ion thatsupport a mass current circulating the electron bubble - the mesoscopicrealization of chiral edge currents in superfluid $^3$He-A films. A consequenceis that electron bubbles embedded in $^3$He-A acquire angular momentum, ${\bfL}\approx -(N_{bubble}/2)\hbar\,\hat{\bf l}$, inherited from the chiral groundstate. We extend the scattering theory to calculate the forces on a movingelectron bubble, both the Stokes drag and a transverse force, ${\bf F}_{W} =\frac{e}{c}{\bf v}\times{\bf B}_{W}$, defined by an effective magnetic field,${\bf B}_{W}\propto\hat{\bf l}$, generated by the scattering of thermalquasiparticles off the spectrum of Weyl Fermions bound to the moving ion. Thetransverse force is responsible for the anomalous Hall effect for electronbubbles driven by an electric field reported by the RIKEN group. Our resultsfor the scattering cross section, drag and transverse forces on moving ions arecompared with experiments, and shown to provide a quantitative understanding ofthe temperature dependence of the mobility and anomalous Hall angle forelectron bubbles in normal and superfluid $^3$He-A. We also discuss our resultsin relation to earlier theoretical work on negative ions in superfluid $^3$He.

Electron bubbles and Weyl Fermions in chiral superfluid $^3$He-A