  We evaluate the hadronic contribution to the $g-2$ of the muon by deriving
the low-energy limit of quantum chromodynamics (QCD) and computing in this way
the hadronic vacuum polarization. The low-energy limit is a non-local
Nambu--Jona-Lasinio (NJL) model that has all the parameters fixed from QCD, and
the only experimental input used is the confinement scale that is known from
measurements of hadronic physics. Our estimations provide a novel analytical
alternative to the current lattice computations and we find that our result is
close to the similar computation performed from experimental data. We also
comment on how this analytical approach technique, in general, may provide
prospective estimates for hadronic computations from dark sectors and its
implication in BSM model-building in future.
