Critical Probability Distributions of the order parameter at two loops II: $O(n)$ universality class

We show how to compute the probability distributions of the order parameter of the $O(n)$ model at two-loop order of perturbation theory generalizing the methods developed for computing the same in case of the Ising model \cite{Sahu:2025bkp}. We show that even for the $O(n)$ model, there exists not one but a family of these probability distribution functions indexed by $\zeta$ which is the ratio of system size $L$ to the bulk correlation length $\xi_{\infty}$. We also compare these PDFs to the Monte-Carlo simulations and the existing FRG results for the $O(2)$ and $O(3)$ models.