Entanglement length scale separates threading from branching of unknotted and non-concatenated ring polymers in melts

We look at the structure and dynamics of unknotted and non-concatenated ring polymers in melt conditions and investigate the interplay between double folding and threadings. Double folding is postulated in current theories which describe the ring as a tree-like double folded object, and evidence of this are supported from simulations on a single ring level. In contrast threadings, penetration of a ring through the surface of another ring, are neglected by theories but recent works show that they are present and may be fundamental for the dynamics of such systems. In this work, we reconcile this dichotomy using Monte-Carlo simulations of the ring melts with different bending rigidity. We find that rings are indeed double folded (more strongly for stiffer rings) on and above the entanglement length scale, while threadings are localized on smaller scales. In the stiffer melts we show the first evidence of the presence of an underlying tree-like structure with self avoiding walk statistics, while more flexible chains do not show this feature. As for the threadings, despite they create only a small opening in the double folded structure, the threading loops can be numerous and their length can exceed substantially the entanglement scale. We link the threading constraints to the divergence of the relaxation time when a fraction of rings are pinned. Current theories do not predict such divergence and predict faster than measured diffusion of rings, pointing at the relevance of the threading constraints in unpinned systems as well. Finally, revision of the theories with explicit threading constraints might elucidate the validity of the conjectured existence of topological glass.