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 1/2. In 3+1 dimensions, a hedgehog soliton θ(r) is coupled to a fermionic field. It is assumed that θ(0)=nπ, where n is an integer and θ(∞)=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.