  We investigate the electronic properties of the boson mode in a three-point
fermion loop. In this framwork, the single-particle excitation and the
many-body local (in imaginary time and momentum space) field effects are
investigated in IR or UV limits with the density fluctuation induced by
external potential (or bosonic frequency). While in the limit of vanishing
effect of external potential, which is equivalent to UV limit of the fermionic
frequency, the conserving approximation can be applied together with the
Luttinger-Ward analysis, in which case the anomalous contribution to the
fermion self-energy or the expectation value of many-body interaction term,
which is $g^{2}\langle \Delta^{\dag}\Delta\rangle$ ($g$ is the irreducible
particle-particle vertex and $\Delta$ is the boson field operator), vanishes,
and results in a Hartree-Fock type momentum- and frequency-independent fermion
self-energy. The correlator $\langle\Delta^{\dag}\Delta\rangle$ is positive
which can be obtained through the local moment sum rule of dynamical
susceptibility.
