The middle-aged supernova remnant (SNR) W44 has recently attracted attention because of its relevance regarding the origin of Galactic cosmic-rays. The gamma-ray missions AGILE and Fermi have established, for the first time for a SNR, the spectral continuum below 200 MeV which can be attributed to neutral pion emission. Confirming the hadronic origin of the gamma-ray emission near 100 MeV is then of the greatest importance. Our paper is focused on a global re-assessment of all available data and models of particle acceleration in W44, with the goal of determining on a firm ground the hadronic and leptonic contributions to the overall spectrum. We also present new gamma-ray and CO NANTEN2 data on W44, and compare them with recently published AGILE and Fermi data. Our analysis strengthens previous studies and observations of the W44 complex environment and provides new information for a more detailed modeling. In particular, we determine that the average gas density of the regions emitting 100 MeV - 10 GeV gamma-rays is relatively high (n= 250 - 300 cm^-3). The hadronic interpretation of the gamma-ray spectrum of W44 is viable, and supported by strong evidence. It implies a relatively large value for the average magnetic field (B > 10^2 microG) in the SNR surroundings, sign of field amplification by shock-driven turbulence. Our new analysis establishes that the spectral index of the proton energy distribution function is p1 = 2.2 +/- 0.1 at low energies and p2 = 3.2 +/- 0.1 at high energies. We critically discuss hadronic versus leptonic-only models of emission taking into account simultaneously radio and gamma-ray data. We find that the leptonic models are disfavored by the combination of radio and gamma-ray data. Having determined the hadronic nature of the gamma-ray emission on firm ground, a number of theoretical challenges remains to be addressed.
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