  We probe the Ioffe-Regel limits of glasses with repulsions near the
zero-temperature jamming transition by measuring the dynamical structure
factors. At zero temperature, the transverse Ioffe-Regel frequency vanishes at
the jamming transition with a diverging length, but the longitudinal one does
not, which excludes the existence of a diverging length associated with the
longitudinal excitations. At low temperatures, the transverse and longitudinal
Ioffe-Regel frequencies approach zero at the jamming-like transition and glass
transition, respectively. As a consequence, glasses between the glass
transition and jamming-like transition, which are hard sphere glasses in the
low temperature limit, can only carry well-defined longitudinal phonons and
have an opposite pressure dependence of the ratio of the shear modulus to the
bulk modulus from glasses beyond the jamming-like transition.
