  Unraveling general properties of Green's functions of quantum dissipative
systems is of both experimental relevance and theoretical interest. Here, we
study the spin-boson model as a prototype. By utilizing the Majorana- Fermion
representation together with the polaron transformation, we establish a
theoretical approach to analyze Green's functions of the spin-boson model. In
contrast to conventional perturbation theories either in the tunneling energy
or in the system-bath coupling strength, the proposed scheme gives reliable
results over wide regimes of the coupling strength, bias, as well as
temperature. To demonstrate the utility of the approach, we consider the
susceptibility as well as the symmetrized spin correlation function (SSCF)
which can be expressed in terms of Green's functions. Thorough investigations
are made on systems embedded in Ohmic or sub-Ohmic bosonic baths. We found the
so-obtained SSCF is the same as that of the non-interacting blip approximation
(NIBA) in unbiased systems while it is applicable for a wider range of
temperature in the biased systems compared with the NIBA. We also show that a
previous perturbation result is recovered as a weak coupling limit of the
so-obtained SSCF. Furthermore, by studying the quantum criticality of the
susceptibility, we confirm the validity of the quantum-toclassical mapping in
the whole sub-Ohmic regime.
