  The anomalous strange metal phase found in high-$T_c$ cuprates does not
follow the conventional condensed-matter principles enshrined in the Fermi
liquid and presents a great challenge for theory. Highly precise experimental
determination of the electronic self-energy can provide a test bed for
theoretical models of strange metals, and angle-resolved photoemission can
provide this as a function of frequency, momentum, temperature and doping. Here
we show that constant energy cuts through the nodal spectral function in
(Pb,Bi)$_{2}$Sr$_{2-x}$La$_x$CuO$_{6+\delta}$ have a non-Lorentzian lineshape,
meaning the nodal self-energy is $k$ dependent. We show that the experimental
data are captured remarkably well by a power law with a $k$-dependent scaling
exponent smoothly evolving with doping, a description that emerges naturally
from AdS/CFT-based semi-holography. This puts a spotlight on holographic
methods for the quantitative modelling of strongly interacting quantum
materials like the cuprate strange metals.
