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.
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