  We study Abelian braiding statistics of loop excitations in three-dimensional
(3D) gauge theories with fermionic particles and the closely related problem of
classifying 3D fermionic symmetry-protected topological (FSPT) phases with
unitary symmetries. It is known that the two problems are related by turning
FSPT phases into gauge theories through gauging the global symmetry of the
former. We show that there exist certain types of Abelian loop braiding
statistics that are allowed only in the the presence of fermionic particles,
which correspond to 3D "intrinsic" FSPT phases, i.e., those that do not stem
from bosonic SPT phases. While such intrinsic FSPT phases are ubiquitous in 2D
systems and in 3D systems with anti-unitary symmetries, their existence in 3D
systems with unitary symmetries was not confirmed previously due to the fact
that strong interaction is necessary to realize them. We show that the simplest
unitary symmetry to support 3D intrinsic FSPT phases is
$\mathbb{Z}_2\times\mathbb{Z}_4$. To establish the results, we first derive a
complete set of physical constraints on Abelian loop braiding statistics.
Solving the constraints, we obtain all possible Abelian loop braiding
statistics in 3D gauge theories, including those that correspond to intrinsic
FSPT phases. Then, we construct exactly soluble state-sum models to realize the
loop braiding statistics. These state-sum models generalize the well-known
Crane-Yetter and Dijkgraaf-Witten models.
