  The Higgs portal interaction to a singlet sector of the standard model (SM)
gauge group is widely-studied. In this Letter, we show that a quantum effect is
important if the Higgs field mixes with another singlet scalar field whose
decay rate is larger than the mass difference between the two mass eigenstates.
This effect may be interpreted as the quantum Zeno effect. In either the
quantum mechanics or the quantum field theory, we show that the resulting
propagating mode is not the eigenstate of the mass matrix, but it is
approximately the eigenstate of the interaction. As a consequence, the
decoupling of the mixing effect happens at the infinity limit of the decay
width of the exotic scalar even if the na\"{i}ve mixing parameter is not small.
With a finite decay width of the exotic scalar, we derive the effective mass of
the propagating mode in the SM sector, its decay rate, and the couplings at the
1-loop level. It turns out that the mixed mass eigenstates can mimic the
discovered 125 GeV Higgs boson. This fuzzy Higgs boson can be obtained in a
simple perturbative renormalizable model. It is consistent with the 125 GeV SM
Higgs boson when the mass difference is smaller than ${\cal O}(0.1)$GeV (${\cal
O}(1)$GeV) for ${\cal O}(1)$ (${\cal O}(0.01)$) mixing. We argue the possible
natural scenario for the tiny mass splitting and the possibility that the upper
bound of the mass difference is larger for a strongly-coupled singlet sector.
To probe the fuzzy Higgs boson scenario, it is difficult to directly produce
the singlet sector particles. Nevertheless, the future Higgs factories may
probe this scenario by precisely measuring the Higgs boson invisible decay rate
and the deviation of the Higgs coupling. Applications of the mechanism are also
mentioned.
