  Recent photoemission spectroscopy measurements [arXiv:1509.01611] on cuprate
superconductors have inferred that over a wide range of doping, the imaginary
part of the electron self-energy scales as
$\Sigma^{\prime\prime}\sim(\omega^2+\pi^2T^2)^a$ with $a=1$ in the overdoped
Fermi-liquid state and $a<0.5$ in the optimal to underdoped regime. We show
that this non-Fermi-liquid scaling behavior can naturally be explained by the
presence of a scale-invariant state of matter known as unparticles. We evaluate
analytically the electron self-energy due to interactions with fermionic
unparticles. We find that, in agreement with experiments, the imaginary part of
the self-energy scales with respect to temperature and energy as
$\Sigma^{\prime\prime}\sim T^{2+2\alpha}$ and $\omega^{2+2\alpha}$, where
$\alpha$ is the anomalous dimension of the unparticle propagator. In addition,
the calculated occupancy and susceptibility of fermionic unparticles, unlike
those of normal fermions, have significant spectral weights even at high
energies. This unconventional behavior is attributed to the branch cut in the
unparticle propagator which broadens the unparticle spectral function over a
wide energy range and non-trivially alters the scattering phase space by
enhancing (suppressing) the intrinsic susceptibility at low energies for
negative (positive) $\alpha$. Our work presents new evidence suggesting that
unparticles might be important low-energy degrees of freedom in strongly
coupled systems such as the cuprate superconductors.
