  We report the differential yields at mid-rapidity of the Breit-Wheeler
process ($\gamma\gamma\rightarrow e^{+}e^{-}$) in peripheral Au+Au collisions
at $\sqrt{s_{_{\rm{NN}}}} = $ 54.4 GeV and 200 GeV with the STAR experiment at
RHIC, as a function of energy $\sqrt{s_{_{\rm{NN}}}}$, $e^{+}e^{-}$ transverse
momentum $p_{\rm T}$, $p_{\rm T}^{2}$, invariant mass $M_{ee}$ and azimuthal
angle. In the invariant mass range of 0.4 $<$ $M_{ee}$ $<$ 2.6 GeV/$c^{2}$ at
low transverse momentum ($p_{\rm T}$ $ < $0.15 GeV/$c$), the yields increase
while the pair $\sqrt{\langle p_{\rm T}^{2} \rangle}$ decreases with increasing
$\sqrt{s_{_{\rm{NN}}}}$, a feature is correctly predicted by the QED
calculation. The energy dependencies of the measured quantities are sensitive
to the nuclear form factor, infrared divergence and photon polarization. The
data are compiled and used to extract the charge radius of the Au nucleus.
