  Materials tuned to a quantum critical point display universal scaling
properties as a function of temperature $T$ and frequency $\omega$. A
long-standing puzzle regarding cuprate superconductors has been the observed
power-law dependence of optical conductivity with an exponent smaller than one,
in contrast to $T$-linear dependence of the resistivity and $\omega$-linear
dependence of the optical scattering rate. Here, we present and analyze
resistivity and optical conductivity of La$_{2-x}$Sr$_x$CuO$_4$ with $x=0.24$.
We demonstrate $\hbar\omega/k_{\mathrm{B}} T$ scaling of the optical data over
a wide range of frequency and temperature, $T$-linear resistivity, and optical
effective mass proportional to $\sim \ln T$ corroborating previous specific
heat experiments. We show that a $T,\omega$-linear scaling Ansatz for the
inelastic scattering rate leads to a unified theoretical description of the
experimental data, including the power-law of the optical conductivity. This
theoretical framework provides new opportunities for describing the unique
properties of quantum critical matter.
