Constraining Cosmic-Ray Acceleration and Escape in Middle-Aged Supernova Remnants with GeV-TeV Gamma-Ray Observations

In this work, we perform a systematic, time-dependent study of the gamma-ray emission from four representative middle-aged SNRs (W51C, IC 443, W44, W28), incorporating both CRs within the remnant shells and escaped CRs interacting with surrounding molecular clouds. We compare our results with GeV--TeV gamma-ray observations from Fermi-LAT, H.E.S.S., MAGIC, and LHAASO, including a dedicated analysis of the Fermi-LAT data for regions A and B associated with W28. We find that the observed spectra favor steeper CR injection spectra with indices of α about4.2 -- 4.3, maximum proton energies of about 100 -- 300 TeV, diffusion coefficients below the Galactic average, and CR acceleration efficiencies from a few to tens of percent. In particular, the VHE emission detected by LHAASO from W51C is more naturally explained by escaped CRs interacting with a nearby molecular cloud. We also investigate the contribution of escaped CRs to the VHE emission from IC 443, W44, and W28. We further demonstrate that escaped CRs can substantially enhance the TeV neutrino flux from middle-aged SNRs, improving their prospects as potential neutrino sources. These results provide new constraints on CR acceleration and escape in middle-aged SNRs and highlight the important role of escaped CRs in shaping their high-energy gamma-ray and neutrino emission.