  Fermi surface of heavy electron systems plays a fundamental role in
understanding their variety of puzzling phenomena, for example, quantum
criticality, strange metal behavior, unconventional superconductivity and even
enigmatic phases with yet unknown order parameters. The spectroscopy
measurement of typical heavy fermion superconductor CeCoIn$_{5}$ has
demonstrated multi-Fermi surface structure, which has not been in detail
studied theoretically in a model system like the Kondo-Heisenberg model. In
this work, we make a step toward such an issue with revisiting the
Kondo-Heisenberg model. It is surprising to find that the usual self-consistent
calculation cannot reproduced the fermionology of the experimental observation
of the system due to the unfounded sign binding between the hopping of the
conduction electrons and the mean-field valence-bond order. To overcome such
inconsistency, we assume that the sign binding should be relaxed and the
mean-field valence-bond order can be considered as a free/fit parameter so as
to meet with real-life experiments. Given the fermionology, the calculated
effective mass enhancement, entropy, superfluid density and Knight shift are
all in qualitative agreement with the experimental results of CeCoIn$_{5}$,
which confirms our assumption. Our result supports a $d_{x^{2}-y^{2}}$-wave
pairing structure in heavy fermion material CeCoIn$_{5}$. In addition, we have
also provided the scanning tunneling microscopy (STM) spectra of the system,
which is able to be tested by the present STM experiments.
