  Zero-mass lines result in appearance of linear dispersion modes for Dirac
fermions. These modes play an important role in various physical systems.
However, a Dirac fermion may not precisely follow a single zero-mass line, due
to either tunneling between different lines or centrifugal forces. Being
shifted from a zero-mass line the Dirac fermion acquires mass which can
substantially influence its expected "massless" behavior. In the paper we
calculate the energy gap caused by the tunneling between two zero-mass lines
and show that its opening leads to the delocalization of linear dispersion
modes. The adiabatic bending of a zero-mass line gives rise to geometric
phases. These are the Berry phase, locally associated with a curvature, and a
new phase resulting from the mass square asymmetry in the vicinity of a
zero-mass line.
