  Through defining irreducible loop integrals (ILIs), a set of consistency
conditions for the regularized (quadratically and logarithmically) divergent
ILIs are obtained to maintain the generalized Ward identities of gauge
invariance in non-Abelian gauge theories. Overlapping UV divergences are
explicitly shown to be factorizable in the ILIs and be harmless via suitable
subtractions. A new regularization and renormalization method is presented in
the initial space-time dimension of the theory. The procedure respects
unitarity and causality. Of interest, the method leads to an infinity free
renormalization and meanwhile maintains the symmetry principles of the original
theory except the intrinsic mass scale caused conformal scaling symmetry
breaking and the anomaly induced symmetry breaking. Quantum field theories
(QFTs) regularized through the new method are well defined and governed by a
physically meaningful characteristic energy scale (CES) $M_c$ and a physically
interesting sliding energy scale (SES) $\mu_s$ which can run from $\mu_s \sim
M_c$ to a dynamically generated mass gap $\mu_s=\mu_c$ or to $\mu_s =0$ in the
absence of mass gap and infrared (IR) problem. It is strongly indicated that
the conformal scaling symmetry and its breaking mechanism play an important
role for understanding the mass gap and quark confinement.
