  The Sa spiral NGC 4138 is known to host two counter-rotating stellar disks,
with the ionized gas co-rotating with one of them. We measured the kinematics
and properties of the two counter-rotating stellar populations to constrain
their formation scenario. A spectroscopic decomposition of the observed
major-axis spectrum was performed to disentangle the relative contribution of
the two counter-rotating stellar and one ionized-gas components. The
line-strength indices of the two counter-rotating stellar components were
measured and modeled with single stellar population models that account for the
alpha/Fe overabundance. The counter-rotating stellar population is younger,
marginally more metal poor, and more alpha-enhanced than the main stellar
component. The younger stellar component is also associated with a star-forming
ring. The different properties of the counter-rotating stellar components of
NGC~4138 rule out the idea that they formed due to bar dissolution. Our
findings support the results of numerical simulations in which the
counter-rotating component assembled from gas accreted on retrograde orbits
from the environment or from the retrograde merging with a gas-rich dwarf
galaxy.
