  In the presence of attraction, the jamming transition of packings of
frictionless particles corresponds to the rigidity percolation. When the range
of attraction is long, the distribution of the size of rigid clusters, $P(s)$,
is continuous and shows a power-law decay. For systems with short-range
attractions, however, $P(s)$ appears discontinuous. There is a power-law decay
for small cluster sizes, followed by a low probability gap and a peak near the
system size. We find that this appearing ``discontinuity'' does not mean that
the transition is discontinuous. In fact, it signifies the coexistence of two
distinct length scales, associated with the largest cluster and smaller ones,
respectively. The comparison between the largest and second largest clusters
indicates that their growth rates with system size are rather different.
However, both cluster sizes tend to diverge in the large system size limit,
suggesting that the jamming transition of systems with short-range attractions
is still continuous. In the framework of the two-scale scenario, we also derive
a generalized hyperscaling relation. With robust evidence, our work challenges
the former single-scale view of the rigidity percolation.
