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 was argued to emerge in the low-energy region by tuning Dirac or Weyl semimetal to approach a superconducting quantum critical point, at which the Dirac or Weyl fermion and the bosonic order parameter are both massless. Here, we study under what circumstances can space-time supersymmetry 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 superconducting quantum critical points. The results provide important constraint on the exploration of emergent space-time supersymmetry in condensed matter systems.