  Interest has been directed recently towards low energy implications of a
non-trivial conformal sector of an effective field theory with an IR fixed
point (\Lambda), manifest in terms of ``unparticles'' with bizarre properties.
We re-examine the implications of the limits on decay lifetimes of solar
neutrinos for unparticle interactions. We study in detail the fundamental
parameter space (\Lambda, M) and derive bounds on the energy scale M
characterizing the new physics. We work strictly within the framework where
conformal invariance holds down to low energies. We first assume that couplings
of the unparticle sector to the Higgs field are suppressed and derive bounds
with \Lambda in the TeV region from neutrino decay into scalar unparticles.
These bounds are significant for values of the anomalous dimension of the
unparticle operator 1.0 < d < 1.2. For a region of the parameter space, we show
that the bounds are comparable to those arising from production rates at high
energy colliders. We then relax our assumption, by considering a more natural
framework which does not require a priori restrictions on couplings of
Higgs-unparticle operators, and derive bounds with \Lambda in meV region from
neutrino decay into vector unparticles. Such low scales for the IR fixed point
are relevant in gauge theories with many flavors.
