Fixed point annihilation for a spin in a fluctuating field

A quantum spin impurity coupled to a critical free field (the Bose-Kondo model) can be represented as a 0+1D field theory with long-range-in-time interactions that decay as $|t-t'|^{-(2-\delta)}$. This theory is a simpler analogue of nonlinear sigma models with topological Wess-Zumino-Witten terms in higher dimensions. In this note we show that the RG flows for the impurity problem exhibit an annihilation between two nontrivial RG fixed points at a critical value $\delta_c$ of the interaction exponent. The calculation is controlled at large spin $S$. This clarifies the phase diagram of the Bose-Kondo model and shows that it serves as a toy model for phenomena involving fixed-point annihilation and "quasiuniversality" in higher dimensions.