Infrared Structure of the Sub-Ohmic Spin-Boson Model

We analyze the infrared structure of the sub-Ohmic spin-boson model through its retarded Green function and identify a spectral criterion governing long-time dynamics. The reduced two-level system exhibits an intrinsic analytic infrared exponent n_IR = 2, while coupling to a bath with spectral density proportional to omega to the power s generates a non-analytic retarded self-energy that scales as minus i omega to the power s. The infrared behavior is determined by the competition between these analytic and spectral contributions and is controlled by the inequality gamma < n_IR, where gamma denotes the leading infrared exponent of the self-energy. In the sub-Ohmic regime 0 < s < 1, the non-analytic term becomes asymptotically dominant, yielding algebraic long-time behavior proportional to t to the power s minus 1, in contrast with the exponential relaxation obtained from local analytic approximations. These results provide a compact organizing criterion for identifying regimes in which local derivative expansions do not control the infrared dynamics of dissipative two-level systems.