Gapless excitations of axially symmetric vortices in systems with tensorial order parameter

We extend the results of previous work on the vortex order parameter in systems similar to the Ginzburg-Landau description of superfluid ${}^3$He in the bulk B phase. Specifically, we consider vortices preserving an axial $U(1)$ symmetry. We determine the conditions required by the ${\beta }_i$ parameters to allow for an energetically favorable development of the off-diagonal antisymmetric and symmetric-traceless elements satisfying the axial symmetry from the trace-only ansatz of the order parameter. The number and type of gapless moduli appearing on the classical low energy theory of axial vortices is determined. The time-dependent part of the Ginzburg-Landau free energy is then considered to determine the number of quantized modes emerging from the gapless moduli fields.