  Paraexcitons, the lowest energy exciton states in Cu$_{2}$O, have been
considered a good system for realizing exciton Bose-Einstein condensation
(BEC). The fact that their BEC has not been attained so far is attributed to a
collision-induced loss, whose nature remains unclear. To understand collisional
properties of cold paraexcitons governing their BEC, we perform here a
microscopic consideration of the s-wave paraexciton-paraexciton scattering. We
show its two-channel character with incoming paraexcitons coupled to a
biexciton, which is a Feshbach resonance producing a paraexciton loss and a
diminution of their background scattering length. The former elucidates the
mechanism of the long-observed paraexciton loss, which turns out to be
inefficient at temperatures near one Kelvin and below, whereas the latter makes
the paraexciton scattering length in strain-induced traps negative under stress
exceeding a critical value. Our rough estimates give this value of order of one
kilobar, hence already moderate stress creates a serious obstacle to attaining
a stable paraexciton BEC. Thus our results indicate that BEC of trapped
paraexcitons might be achieved at a subkelvin temperature, but only under low
stress.
