Very-high-energy gamma rays from cosmic rays escaping from Galactic black hole binaries

We solve the cosmic-ray diffusion around a Galactic black hole binary (microquasars) by considering the finite size of the escape region and the continuous cosmic-ray injection. We find that the energy spectrum of escaping cosmic rays in the gamma-ray emission region is described by a broken power law spectrum with one or two spectral breaks even though the total spectrum of escaping cosmic rays is a single power law spectrum. Using the solution for the diffusion equation, we construct a unified picture that explains spatially extended very-high-energy gamma rays from five microquasars observed by HAWC and LHAASO. The comparison of our unified model and observed data suggest that all five microquasars have the same energy spectrum of the escaping CRs, $dN/dE \propto E^{-2}$, the same diffusion coefficient, and the same emission region. The hard energy spectrum without the high-energy cutoff supports the idea that the origin of Galactic CRs beyond PeV energies is Galactic black hole binaries.