Late decaying axino as CDM and its lifetime bound

The axino with mass in the GeV region can be cold dark matter(CDM) in the galactic halo. However, if R-parity is broken, for example by the bilinear terms ${\mu }_{\alpha }$, then axino($\tilde{a}$) can decay to $\nu +\gamma$. In this case, the most stringent bound on the axino lifetime comes from the diffuse photon background and we obtain that the axino lifetime should be greater than $3.9\times {10}^{24}{\mathrm{\Omega }}_{\tilde{a}}h$ s which amounts to a very small bilinear R-parity violation, i.e. ${\mu }_{\alpha }<1$ keV. This invalidates the atmospheric neutrino mass generation through bilinear R-violating terms within the context of axino CDM.