  The influence of helicity on the stability of scaling regimes, on the
effective diffusivity, and on the anomalous scaling of structure functions of a
passive scalar advected by a Gaussian solenoidal velocity field with finite
correlation time is investigated by the field theoretic renormalization group
and operator product expansion within two-loop approximation. The influence of
helicity on the scaling regimes is discussed and shown in the plane of
exponents $\epsilon-\eta$, where $\epsilon$ characterizes the energy spectrum
of the velocity field in the inertial range $E\propto k^{1-2\epsilon}$, and
$\eta$ is related to the correlation time at the wave number $k$ which is
scaled as $k^{-2+\eta}$. The restrictions given by nonzero helicity on the
regions with stable fixed points which correspond to the scaling regimes are
analyzed in detail. The dependence of the effective diffusivity on the helicity
parameter is discussed. The anomalous exponents of the structure functions of
the passive scalar field which define their anomalous scaling are calculated
and it is shown that although the separate composite operators which define
them strongly depend on the helicity parameter the resulting two-loop
contributions to the critical dimensions of the structure functions are
independent of helicity. Details of calculations are shown.
