  We explore the Higgs mass spectrum in a class of Two Higgs Doublet Models
(THDMs) in which a scalar SU(2)_L doublet interacts only with quarks, while the
second one interacts only with leptons. The spectrum includes two CP-even Higgs
bosons, either of which can account for the SM-like Higgs boson, and the
spectra involving light Higgs bosons receive strong impacts from the LEP
results and the current collider analyses. We find that a consistent spectrum
can involve a CP-odd Higgs boson as light as about 10 GeV, while the lightest
CP-even Higgs boson cannot be lighter than about 55 GeV when m_A ~ 28 GeV.
These analyses can rather bound the low tan beta region which can also
accommodate an observed excess in dimuon events at m_mumu ~ 28 GeV. A
lepton-specific class of THDMs (LS-THDM) can predict such an excess through A
-> mu mu decays, while the solutions can be constrained by the A -> tau tau
mode. After constraining the solutions with the consistent ranges of sigma(pp
-> bbA -> bb tau tau), a largest excess at about 1.5 sigma at 8 TeV center of
mass (COM) energy and 2 sigma at 13 TeV COM is observed for tan beta ~ 12 and
m_A ~ 28 GeV in the sigma(pp -> bbA -> bb mu mu) events.
