  In this work, we compare the $\gamma$-ray spectra available in literature
from 11 middle aged supernova remnants (SNRs) interacting with molecular clouds
(MCs). It is found that 5 remnants prefer a smoothly broken power law proton
spectrum with similar power law index but different break energy. The rest of
the SNRs need updated data to test whether a spectral break is preferred in the
proton spectrum. Then we compare the $\gamma$-ray spectra from all 11 SNRs with
the prediction from widely accepted escaping scenario and direct interaction
scenario. We show that current $\gamma$-ray data is inconsistent with the
escaping model statistically, as it predicts a diversity of $\gamma$-ray
spectra which is not detected in the observation. We also find that ambient CRs
can be very important for the $\gamma$-ray emission in the MCs external to W28
and W44, which requires further investigation. In the direct interaction
scenario, we focus on re-acceleration of pre-existing ambient CRs. The model
can produce the overall profile of $\gamma$-ray data with different
acceleration time, but it suggests a transition of seed particles in the
evolution of SNR. Whether such transition indeed exists has to be tested by
future observation. In the end, we propose that radiative SNR without MC
interaction can also produce a significant amount of $\gamma$-ray emission. One
good candidate is S147. With accumulated Fermi data and CTA in future we expect
to detect more remnants like S147.
