Absence of emergent supersymmetry at superconducting quantum critical points in Dirac and Weyl semimetals

Supersymmetry plays a crucial role in superstring theory and high-energy physics, but has never been observed in experiments. Recently, an effective space-time supersymmetry (SUSY) was argued to emerge in the low-energy region by tuning Dirac or Weyl semimetal to approach a superconducting (SC) quantum critical point (QCP), at which the Dirac or Weyl fermion and the bosonic order parameter are both massless. Here, we study under what circumstances can SUSY be realized at a quantum critical point. We demonstrate that the Yukawa-type coupling between the massless fermion and massless boson can dynamically generate an infinite number of non-supersymmetric terms in the effective field theory of the boson. Owing to these terms, no space-time supersymmetry emerges at the SC QCPs. The results provide important constraint on the exploration of emergent SUSY in condensed matter systems.