  By introducing the imaginary time, Gor'kov's Ginzburg-Landau equation at zero
temperature can be extended to an exact relativistic form without any
phenomenological parameter, which is intended to describe the zero-temperature
overdoped cuprate. By using such a relativistic equation, we have shown that
the two-class scaling observed in the overdoped side of single-crystal
$La_{2-x}Sr_xCuO_4$ (LSCO) films [Nature 536, 309-311 (2016)] can be derived
exactly. In this paper, we further test the validity of the relativistic
Ginzburg-Landau equation. By applying the perturbation method into this
equation, we theoretically predict that near the superconductor-metal
transition point in the overdoped side of LSCO films, the zero-temperature
correlation length $\xi(0)$ and the transition temperature $T_c$ should yield a
novel scaling $\xi(0)\propto T_c^{-\sigma}$ with a critical exponent $\sigma
\approx 1.31 $ (up to the two-loop approximation). Here, we propose a
diffraction experiment between $X$-rays and zero-temperature LSCO films to
measure the critical exponent $\sigma$.
