  We computed the contribution of the Compton and Bremsstrahlung processes with
a hidden light $U(1)_D$ neutral boson $\gamma_D$ to the white dwarf G117-B15A
anomalous cooling rate, as well as the white dwarf luminosity functions (WDLF).
We demonstrated that for a light mass of hidden photon ($m_{\gamma_D} \ll$ a
few keV), compatible results are obtained for the recent Sloan Digital Sky
Survey and the SuperCOSMOS Sky Survey observation, but the stringent limits
would be imposed on the kinetic mixing $\epsilon$. We performed $\chi^2$-tests
to acquire a quantitative assessment on the WDLF data in the context of our
model, computed under the assumption of different kinetic mixing $\epsilon$,
the age of the oldest computed stars $T_D$, and a constant star formation rate
$\psi$. Then taken together, the WDLF analysis of 2$\sigma$ confidence interval
$\epsilon = \left( 0.37^{+0.35}_{-0.37}\right) \times 10^{-14}$ is barely
consistent with the cooling rate analysis at 2$\sigma$ regime $\epsilon =
\left( 0.97^{+0.35}_{-0.37} \right) \times 10^{-14}$. The two approaches used
here agree with each other in yielding an anomalous cooling rate of white dwarf
in this luminosity range.
