  We investigate the internal structure and the mass-radius relation of neutron
stars in a recently proposed scalar-tensor theory dubbed asymmetron in which a
massive scalar field undergoes spontaneous scalarization inside neutron stars.
We focus on the case where the Compton wavelength is shorter than 10 km, which
has not been investigated in the literature. By solving the modified Einstein
equations, either purely numerically or by partially using a semianalytic
method, we find that not only the weakening of gravity by spontaneous
scalarization but also the scalar force affect the internal structure
significantly in the massive case. We also find that the maximum mass of
neutron stars is larger for certain parameter sets than that in general
relativity and reaches 2 solar mass even if the effect of strange hadrons is
taken into account. There is even a range of parameters where the maximum mass
of neutron stars largely exceeds the threshold that violates the causality
bound in general relativity.
