Fermion Zero Modes and Fermion number $1/2$ of the Electroweak Monopole

Fermion bound states in the background of the electroweak SU(2) monopole are investigated for various values of gauge coupling constant $g$, the Higgs self-coupling constant $\lambda$, and the Yukawa coupling constant $y_q$. Numerical solutions to the set of coupled differential equations for various selected points in the parameter space reveal only a zero mode, for which we also present an analytic argument. We show that the monopole profile functions and the zero mode wave function become more localized with increasing $g$ and $\lambda$, while the right-handed component of the latter decreases with $g$. However, as expected, this component increases with $y_q$. We find that the zero mode in the limit $g\to 0$ differs from the zero mode held by the Higgs alone, highlighting the nonlinear and nonperturbative character of the system. Finally, we prove the spectral mirror symmetry of the fermion, whence, together with the existence of the zero mode, we infer the fermion number $1/2$ of the electroweak monopole.