Critical lines in the pure and disordered $O(N)$ model

We consider replicated $O(N)$ symmetry in two dimensions within the exact framework of scale invariant scattering theory and determine the lines of renormalization group fixed points in the limit of zero replicas corresponding to quenched disorder. A global pattern emerges in which the different critical lines are located within the same parameter space. Within the subspace corresponding to the pure case (no disorder) we show how the critical lines for non-intersecting loops ($-2\leq N\leq 2$) are connected to the zero temperature critical line ($N>2$) via the BKT line at $N=2$. Disorder introduces two more parameters, one of which vanishes in the pure limit and is maximal for the solutions corresponding to Nishimori-like and zero temperature critical lines. Emergent superuniversality (i.e. $N$-independence) of some critical exponents in the disordered case and disorder driven renormalization group flows are also discussed.