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Non-analyticity of the groud state energy of the Hamiltonian for Hydrogen atom in non-relativistic QED

We derive the ground state energy up to the fourth order in the fine structure constant $α$ for the translation invariant Pauli-Fierz Hamiltonian for a spinless electron coupled to the quantized radiation field. As a consequence, we obtain the non-analyticity of the ground state energy of the Pauli-Fierz operator for a single particle in the Coulomb field of a nucleus.