  A search for the lepton-flavor violating decay of the Higgs boson and
potential additional Higgs bosons with a mass in the range 110-160 GeV to an
e$^{\pm}\mu^{\mp}$ pair is presented. The search is performed with a
proton-proton collision dataset at a center-of-mass energy of 13 TeV collected
by the CMS experiment at the LHC, corresponding to an integrated luminosity of
138 fb$^{-1}$. No excess is observed for the Higgs boson. The observed
(expected) upper limit on the e$^{\pm}\mu^{\mp}$ branching fraction for it is
determined to be 4.4 (4.7) $\times$ 10$^{-5}$ at 95% confidence level, the most
stringent limit set thus far from direct searches. The largest excess of events
over the expected background in the full mass range of the search is observed
at an e$^{\pm}\mu^{\mp}$ invariant mass of approximately 146 GeV with a local
(global) significance of 3.8 (2.8) standard deviations.
