We derive the neutral fermion excitation spectrum in the resonating-valence-bond state based on a fermion-boson field theory. It is formally shown that charge-
e bosons are created upon doping while the neutral fermion excitation remains gapless, predicting a linear low-temperature specific heat $C\ensuremath{\propto}\ensuremath{\gamma}T$ in the superconducting state. Bose condensation in doped samples leads to the observed superconductivity with true "off-diagonal long-range order."
We consider an isotropic Fermi liquid in two dimensions near the
n=2 Pomeranchuk instability in the charge channel. The order parameter is a quadrupolar stress tensor with two bosonic shear modes with polarizations longitudinal and transverse to the quadrupolar momentum tensor. Longitudinal and transverse bosonic modes are characterized by dynamical exponents ${z}_{\ensuremath{\parallel}}=3$ and ${z}_{\ensuremath{\perp}}=2$, respectively. Previous studies have found that such a system exhibits multiscale quantum criticality with two different energy scales $\ensuremath{\omega}\ensuremath{\sim}{\ensuremath{\xi}}^{\ensuremath{-}{z}_{\ensuremath{\parallel},\ensuremath{\perp}}}$, where $\ensuremath{\xi}$ is the correlation length. We study the impact of the multiple energy scales on the electron Green's function. The interaction with the critical ${z}_{\ensuremath{\parallel}}=3$ mode is known to give rise to a local self-energy that develops a non-Fermi-liquid form, $\ensuremath{\Sigma}(\ensuremath{\omega})\ensuremath{\sim}{\ensuremath{\omega}}^{2/3}$ for frequencies larger than the energy scale $\ensuremath{\omega}\ensuremath{\sim}{\ensuremath{\xi}}^{\ensuremath{-}3}$. We find that the exchange of transverse ${z}_{\ensuremath{\perp}}=2$ fluctuations leads to logarithmically singular renormalizations of the quasiparticle residue
Z and the vertex $\ensuremath{\Gamma}$. We derive and solve renormalization-group equations for the flow of
Z and $\ensuremath{\Gamma}$, and show that the system develops an anomalous dimension at the nematic quantum critical point (QCP). As a result, the spectral function at a fixed $\ensuremath{\omega}$ and varying
k has a non-Lorentzian form. Away from the QCP, we find that the flow of
Z is cut at the energy scale ${\ensuremath{\omega}}_{\text{FL}}\ensuremath{\propto}{\ensuremath{\xi}}^{\ensuremath{-}1}$, associated with the
z=1 dynamics of electrons. The ${z}_{\ensuremath{\perp}}=2$ energy scale, $\ensuremath{\omega}\ensuremath{\sim}{\ensuremath{\xi}}^{\ensuremath{-}2}$, affects the flow of
Z only if one includes into the theory self-interaction of transverse fluctuations.