The SU(3) simplest little Higgs model in its original framework without the
so-called mu term inevitably involves a massless pseudoscalar boson eta, which
is problematic for b-physics and cosmological axion limit. With the mu term
introduced by hand, the eta boson acquires mass m_eta ~ mu, which can be
lighter than half the Higgs boson mass in a large portion of the parameter
space. In addition, the introduced mu term generates sizable coupling of
H-eta-eta. The Higgs boson can dominantly decay into a pair of eta's especially
when mH below the WW threshold. Another new decay channel of H->Z+eta can be
dominant or compatible with H -> WW for mH above the Z+eta threshold. We show
that the LEP bound on the Higgs boson mass is loosened to some extent due to
this new H->eta eta decay channel as well as the reduced coupling of H-Z-Z. The
Higgs boson mass bound falls to about 110 GeV for f=3-4 TeV. Since the eta
boson decays mainly into a bb pair, H-> eta eta -> 4b and H-> Z eta -> Z bb
open up other interesting search channels in the pursuit of the Higgs boson in
the future experiments. We discuss on these issues.Comment: major modification considering the simplest little Higgs model with
the mu ter
