  We study the axial quasi-normal modes of neutron stars in scalar-tensor
theories with massive scalar field including a self-interacting term in the
potential. Various realistic equations of state including nuclear, hyperonic
and hybrid matter are employed. Although the effect of spontaneous
scalarization of neutron stars can be very large, binary pulsar observations
and gravitational wave detections significantly constrain the massless
scalar-tensor theories. If we consider a properly chosen nonzero mass for the
scalar field, though, the scalar-tensor parameters cannot be restricted by the
observations, resulting in large deviation from pure general relativity. With
this motivation in mind, we extend the universal relations for axial
quasi-normal modes known in general relativity to neutron stars in massive
scalar-tensor theories with self-interaction using a wide range of realistic
EOS. We confirm the universality of the scaled frequency and damping time in
terms of the compactness and scaled moment of inertia for neutron stars with
and without massive scalarization.
