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The scattering of Dirac fermions on the sine-Gordon kink is studied bothanalytically and numerically. To achieve invariance with respect to a discretesymmetry, the sine-Gordon model is treated as a nonlinear $\sigma$-model with acircular target space that interacts with fermionic isodublets through theYukawa interaction. It is shown that the diagonal and antidiagonal parts of thefermionic wave function interact independently with the external field of thesine-Gordon kink. The wave functions of the fermionic scattering states areexpressed in terms of the Heun functions. General expressions for thetransmission and reflection coefficients are derived, and their dependences onthe fermion momentum and mass are studied numerically. The existence conditionis found for two fermionic zero modes, and their analytical expressions areobtained. It is shown that the zero modes do not lead to fragmentation of thefermionic charge, but can lead to polarization of the fermionic vacuum. Thescattering of the diagonal and antidiagonal fermionic states is found to besignificantly different; this difference is shown to be due to the differentdependences of the energy levels of these bound states on the fermion mass, andis in accordance with Levinson's theorem.

Scattering of fermionic isodoublets on the sine-Gordon kink