  Fermion-number fractionalization without breaking of time-reversal symmetry
was recently demonstrated for a field theory in $(2+1)$-dimensional space and
time that describes the couplings between massive Dirac fermions, a
complex-valued Higgs field carrying an axial gauge charge of 2, and a U(1)
axial gauge field. Charge fractionalization occurs whenever the Higgs field
either supports vortices by itself, or when these vortices are accompanied by
half-vortices in the axial gauge field. The fractional charge is computed by
three different techniques. A formula for the fractional charge is given as a
function of a parameter in the Dirac Hamiltonian that breaks the spectral
energy-reflection symmetry. In the presence of a charge $\pm1$ vortex in the
Higgs field only, the fractional charge varies continuously and thus can take
irrational values. The simultaneous presence of a half-vortex in the axial
gauge field and a charge $\pm1$ vortex in the Higgs field re-rationalizes the
fractional charge to the value 1/2.
