  We present an analytical low-energy theory of piezoelectric electron-phonon
interactions in undoped Weyl semimetals, taking into account also Coulomb
interactions. We show that piezoelectric interactions generate a long-range
attractive potential between Weyl fermions. This potential comes with a
characteristic angular anisotropy. From the one-loop renormalization group
approach and a mean-field analysis, we predict that superconducting phases with
either conventional s-wave singlet pairing or nodal-line triplet pairing could
be realized for sufficiently strong piezoelectric coupling. For small
couplings, we show that the quasi-particle decay rate exhibits a linear
temperature dependence where the prefactor vanishes only in a logarithmic
manner as the quasi-particle energy approaches the Weyl point. For practical
estimates, we consider the Weyl semimetal TaAs.
