Zero mode of a fermion chirally coupled to a soliton.

Abstract

We investigate the constraints under which a chirally coupled fermion may have a zero-energy normalizable eigenstate. In the (1+1)-dimensional case, the fermionic null mode is exactly solvable for a class of solitons with an arbitrary topological quantum number having magnitude greater than \textonehalf{}. In 3+1 dimensions, a hedgehog soliton $\ensuremath{\theta}(r)$ is coupled to a fermionic field. It is assumed that $\ensuremath{\theta}(0)=n\ensuremath{\pi}$, where $n$ is an integer and $\ensuremath{\theta}(\ensuremath{\infty})=0$. It is proved that a necessary (but not sufficient) condition for a zero-energy nodeless fermionic ground state is that $n$ be an odd integer.