Braiding statistics of loop excitations in three dimensions

While it is well known that three dimensional quantum many-body systems can support non-trivial braiding statistics between particle-like and loop-like excitations, or between two loop-like excitations, we argue that a more fundamental quantity is the statistical phase associated with braiding one loop $\alpha$ around another loop $\beta$, while both are linked to a third loop $\gamma$. We study this three-loop braiding in the context of $(Z_N)^K$ gauge theories which are obtained by gauging a gapped, short-range entangled lattice boson model with $(Z_N)^K$ symmetry. We find that different short-range entangled bosonic states with the same $(Z_N)^K$ symmetry (i.e. different symmetry-protected topological phases) can be distinguished by their three-loop braiding statistics.