  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
$(\mathbb{Z}_N)^K$ gauge theories which are obtained by gauging a gapped,
short-range entangled lattice boson model with $(\mathbb{Z}_N)^K$ symmetry. We
find that different short-range entangled bosonic states with the same
$(\mathbb{Z}_N)^K$ symmetry (i.e. different symmetry-protected topological
phases) can be distinguished by their three-loop braiding statistics.
