Symmetry Principle Preserving and Infinity Free Regularization and renormalization of quantum field theories and the mass gap

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.