  Charge transport in doped quantum paralectrics (QPs) presents a number of
puzzles, including a pronounced $T^2$ regime in the resistivity. We analyze
charge transport in a QP within a model of electrons coupled to a soft
transverse optical (TO) mode via a two-phonon mechanism. For $T$ above the
soft-mode frequency but below some characteristic scale ($E_0$), the
resistivity scales with the occupation number of phonons squared, i.e., as
$T^2$. The $T^2$ scattering rate does not depend on the carrier number density
and is not affected by a crossover between degenerate and non-degenerate
regimes, in agreement with the experiment. Temperatures higher than $E_0$
correspond to a non-quasiparticle regime, which we analyze by mapping the Dyson
equation onto a problem of supersymmetric quantum mechanics. The combination of
scattering by two TO phonons and by a longitudinal optical mode explains the
data quite well.
