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By using the Landau-Ginzburg-Wilson paradigm, we show that, near a quantumcritical point (QCP), Cooper pairs at zero temperature would obey a nonlinearrelativistic equation, where the imaginary time emerges as a novel dimension.This relativistic equation is applicable to certain superconductors at zerotemperature for which the Faber-Pippard coherence length formula holds at andabove the upper critical dimension. Here, we further show that the relativisticequation leads to a testable prediction in the vicinity of the QCP $T_c=0$,with $T_c$ being the transition temperature. That is, for 2D overdoped (clean)superconducting films, when the parameter $T_c/(c_0v_F)$ is lower than acharacteristic scale, the Lorentz symmetry of relativistic equation arouses ananomalous scaling $\xi_0 \propto T_c^{-1.34}$, where $\xi_0$ denotes thezero-temperature coherence length, $v_F$ denotes the Fermi velocity, and $c_0$denotes the Faber-Pippard coefficient. However, when the parameter$T_c/(c_0v_F)$ is large enough, the Lorentz symmetry may be broken so that theFaber-Pippard scaling $\xi_0 \propto T_c^{-1}$ is restored.

Superconducting quantum criticality and the anomalous scaling: A nonlinear relativistic equation