  Recently, Bozovic et al. reported that [Nature 536, 309-311 (2016)], in the
overdoped side of the single-crystal $La_{2-x}Sr_xCuO_4$ (LSCO) films, the
transition temperature $T_c$ and zero-temperature superfluid phase stiffness
$\rho_s(0)$ will obey a two-class scaling law: $T_c=\gamma \cdot
\sqrt{\rho_s(0)}$ for $T_c \leq T_Q$ and $T_c \propto \rho_s(0)$ for $T_c \geq
T_M$, where $\gamma=(4.2 \pm 0.5) K^{1/2} $, $T_Q \approx 15 K$, and $T_M
\approx 12 K$. They further pointed out that the parabolic scaling observed in
the highly overdoped side indicates a quantum phase transition from a
superconductor to a normal metal. In this paper, we propose a quantum partition
function (QPF) for zero-temperature Cooper pairs, by which one can effectively
distinguish between mean-field and quantum critical behaviors. We theoretically
show that the two-class scaling law can be exactly derived by using the QPF,
and the theoretical values of $\gamma$, $T_Q$, and $T_M$ are well in accordance
with experimental measure values. Our analyses indicate that the linear scaling
$T_c \propto \rho_s(0)$ is a mean-field behavior, while the parabolic scaling
$T_c=\gamma \cdot \sqrt{\rho_s(0)}$ is a quantum critical behavior.
