  Recent photoemission experiments \cite{dessau} reveal that the excitations
along the nodal region in the strange metal of the cuprates, rather than
corresponding to poles in the single-particle Green function, exhibit power-law
scaling as a function of frequency and temperature. Because such power-law
scaling is indicative of a scale-invariant sector, as a first step, we
perturbatively evaluate the electron self-energy due to interactions with
scale-invariant unparticles. We focus on a $G_0W$ type diagram with an
interaction $W$ mediated by a bosonic scalar unparticle. We find that, in the
high-temperature limit, the imaginary part of the self-energy
$\mathrm{Im}\Sigma$ is linear in temperature. In the low-temperature limit,
$\mathrm{Im}\Sigma$ exhibits the same power law in both temperature and
frequency, with an exponent that depends on the scaling dimension of an
unparticle operator. Such behavior is qualitatively consistent with the
experimental observations. We then expand the unparticle propagator into
coherent and incoherent contributions, and study how the incoherent part
violates the density of states (DOS) and density-density correlation function
sum rules (f-sum rule). Such violations can, in principle, be observed
experimentally. Our work indicates that the physical mechanism for the origin
of the power-law scaling is the incoherent background, which is generated from
the Mott-scale physics.
