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We consider an axion cloud around a black hole with background magnetic fields. We calculate the decay rate of the axion cloud due to the axion-photon conversion associated with the axion-photon coupling. For simplicity, we consider the situation where the axion configuration is dominated by a solution for the eigenvalue equation equivalent to that for the Hydrogen atom, and the coupling term can be evaluated by a successive perturbation method. For the monopole background, we find the decay rate of the axion cloud is given by ∼q2κ2(GM)5μ8, where μ, M, G, κ and q are the axion mass, black hole mass, gravitational constant, coupling constant of the axion-photon coupling and monopole charge, respectively. For the uniform background magnetic field, we obtain the decay rate of the axion cloud ∼B20κ2(GM)7μ6, where B0 is the magnetic field strength. Applying our formula to the central black hole in our galaxy, we find that the value of the decay rate for the case of the uniform magnetic field is comparable to the growth rate of the superradiant instability with κ∼10−12GeV−1, B0∼103G and μ∼10−18eV. The ratio is 105 times larger for the monopole magnetic field with the same values of the parameters.
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