  Recent photoemission spectroscopy measurements (T. J. Reber et al.,
arXiv:1509.01611) of cuprate superconductors have inferred that the self-energy
exhibits critical scaling over an extended doping regime, thereby calling into
question the conventional wisdom that critical scaling exists only at isolated
points. In particular, this new state of matter, dubbed a power-law liquid, has
a self-energy whose imaginary part scales as
$\Sigma^{\prime\prime}\sim(\omega^{2}+\pi^{2}T^{2})^{\alpha}$, with $\alpha=1$
in the overdoped Fermi-liquid state and $\alpha\leq0.5$ in the optimal to
underdoped regime. Previously, we showed that this self-energy can arise from
interactions between electrons and unparticles, a scale-invariant sector that
naturally emerges from strong correlations. Here, taking the self-energy as a
given, we first reconstruct the real part of the self-energy. We find that the
resultant quasiparticle weight vanishes for any doping level less than optimal,
implying an absence of particle-like excitations in the underdoped regime.
Consequently, the Fermi velocity vanishes and the effective mass diverges for
$\alpha\leq\frac{1}{2}$, in agreement with earlier experimental observations.
We then use the self-energy to reconstruct the spectral function and compute
the superconducting $T_c$ within the BCS formalism. We find that the $T_c$ has
a dome-like structure, implying that broad scale invariance manifested in the
form of a power-law liquid is the likely cause of the superconducting dome in
the cuprates.
