  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.
