  We find "{\it chiral symmetry breaking}" at finite energies in U(1) spin
liquid, corresponding to critical particle-hole composite states with twice of
the Fermi momentum (2$k_{F}$). We investigate this Fermi surface problem based
on the Nambu-Eliashberg theory, where the off diagonal pairing self-energy is
introduced to catch the Aslamasov-Larkin vertex correction. This approach is
quite parallel with the case of superconductivity, where such Aslamasov-Larkin
quantum corrections in the particle-particle channel are well known to be
responsible for superconducting instability, formulated as the Nambu-Eliashberg
theory in an elegant way. We obtain the pairing self-energy, which vanishes at
zero energy but displays the same power law dependence for frequency as the
normal Eliashberg self-energy. As a result, even the pairing self-energy
correction does not modify the Eliashberg dynamics without the Nambu spinor
representation, where thermodynamics is described by the typical $z = 3$
scaling free energy. We discuss physical implication of the anomalous
self-energy identical to the conventional Eliashberg normal self-energy,
focusing on thermodynamics.
