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We investigate the electronic properties of the boson mode in a three-pointfermion loop. In this framwork, the single-particle excitation and themany-body local (in imaginary time and momentum space) field effects areinvestigated in IR or UV limits with the density fluctuation induced byexternal potential (or bosonic frequency). While in the limit of vanishingeffect of external potential, which is equivalent to UV limit of the fermionicfrequency, the conserving approximation can be applied together with theLuttinger-Ward analysis, in which case the anomalous contribution to thefermion self-energy or the expectation value of many-body interaction term,which is $g^{2}\langle \Delta^{\dag}\Delta\rangle$ ($g$ is the irreducibleparticle-particle vertex and $\Delta$ is the boson field operator), vanishes,and results in a Hartree-Fock type momentum- and frequency-independent fermionself-energy. The correlator $\langle\Delta^{\dag}\Delta\rangle$ is positivewhich can be obtained through the local moment sum rule of dynamicalsusceptibility.

Electronic properties of the boson mode in a three-point fermion loop and the emergent SYK physics