Based on a sample corresponding to

4.3 \times 10^{6}

produced

τ

-pair events, we have studied hadronic dynamics in the decay

τ^{-} \to ν_{τ} π^{-} π^{0} π^{0}

in data recorded by the CLEO II detector operating at the CESR

e^{+} e^{-}

collider. The decay is dominated by the process

τ^{-} \to ν_{τ} a_{1}^{-} (1260),

with the

a_{1}^{-}

meson decaying to three pions predominantly via the lowest dimensional (mainly S-wave)

ρ^{-} π^{0}

Born amplitude. From model-dependent fits to the Dalitz plot and angular observables in bins of

3 π

mass, we find significant additional contributions from amplitudes for

a_{1}

decay to

σ π,

f_{0} (1370) π,

and

f_{2} (1270) π,

as well as higher dimensional

a_{1} \to ρ π

and

ρ^{'} π

amplitudes. Notably, the squared

σ π

amplitude accounts for approximately 15% of the total

τ^{-} \to ν_{τ} π^{-} π^{0} π^{0}

rate in the models considered. The data are well described using couplings to these amplitudes that are independent of the

3 π

mass. These amplitudes also provide a good description for the

τ^{-} \to ν_{τ} π^{-} π^{+} π^{-}

Dalitz plot distributions. We have searched for additional contributions from

τ^{-} \to ν_{τ} π^{'} - (1300) .

We place

90 %

confidence level upper limits on the branching fraction for this channel of between

1.0 \times 10^{- 4}

and

1.9 \times 10^{- 4},

depending on the

π^{'}

decay mode considered. The

π^{-} π^{0} π^{0}

mass spectrum is parametrized by a Breit-Wigner form with a mass-dependent width which is specified according to the results of the Dalitz plot fits plus an unknown coupling to an

a_{1} \to K^{*} K

amplitude. From a

χ^{2}

fit using this parametrization, we extract the pole mass and width of the

a_{1},

as well as the magnitude of the

K^{*} K

coupling. We have also investigated the impact of a possible contribution from the

a_{1}^{'} (1700)

meson on this spectrum. Finally, exploiting the parity-violating angular asymmetry in

a_{1} \to 3 π

decay, we determine the signed value of the

τ

neutrino helicity to be

h_{ν_{τ}} = - 1.02 \pm 0.13 (stat) \pm 0.03 (s y s t + m o d e l),

confirming the left-handedness of the

τ

neutrino.

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