  To understand better the quantum structure of field theory and standard model
in particle physics, it is necessary to investigate carefully the divergence
structure in quantum field theories (QFTs) and work out a consistent framework
to avoid infinities. The divergence has got us into trouble since developing
quantum electrodynamics in 1930s, its treatment via the renormalization scheme
is satisfied not by all physicists, like Dirac and Feynman who have made
serious criticisms. The renormalization group analysis reveals that QFTs can in
general be defined fundamentally with the meaningful energy scale that has some
physical significance, which motivates us to develop a new symmetry-preserving
and infinity-free regularization scheme called loop regularization (LORE). A
simple regularization prescription in LORE is realized based on a manifest
postulation that a loop divergence with a power counting dimension larger than
or equal to the space-time dimension must vanish. The LORE method is achieved
without modifying original theory and leads the divergent Feynman loop
integrals well-defined to maintain the divergence structure and meanwhile
preserve basic symmetries of original theory. The crucial point in LORE is the
presence of two intrinsic energy scales which play the roles of ultraviolet
cut-off $M_c$ and infrared cut-off $\mu_s$ to avoid infinities. The key concept
in LORE is the introduction of irreducible loop integrals (ILIs) on which the
regularization prescription acts, which leads to a set of gauge invariance
consistency conditions between the regularized tensor-type and scalar-type
ILIs. The evaluation of ILIs with ultraviolet-divergence-preserving (UVDP)
parametrization naturally leads to Bjorken-Drell's analogy between Feynman
diagrams and electric circuits. The LORE method has been shown to be applicable
to both underlying and effective QFTs.
