The paper by G. 't Hooft discusses the behavior of renormalized field theories under scale transformations using the continuous dimension method. This method allows for a finite perturbation expansion without introducing additional regulator diagrams, preserving symmetries like local gauge invariance. The key findings include: 1. **Scaling Properties of Renormalized Parameters**: The scaling behavior of renormalized parameters \(\lambda_R\) and \(m_R\) is described by differential equations, which are closely related to the Callan-Symanzik equation. These equations are derived from the residues of poles at \(n=4\) and do not require diagram calculations. 2. **Identities for Coefficients**: Equations are derived that relate the coefficients \(a_v\) and \(b_v\) in the series expansion of the bare parameters \(\lambda_B\) and \(m_B\) to the renormalized parameters \(\lambda_R\) and \(m_R\). These identities are crucial for understanding the scaling properties of the theory. 3. **Generalization to Multiple Parameters**: The scaling behavior of multiple parameters in a renormalizable theory is generalized, showing that the scaling properties of physical parameters are determined by their residues at \(n=4\). 4. **Scaling Behavior of Specific Theories**: The scaling behavior of theories like the \(\lambda \phi^4\) theory and quantum electrodynamics is analyzed, demonstrating that the small-distance behavior of these theories is not described by the usual perturbation expansion but can be accurately determined in the infrared region. 5. **Renormalization and Perturbation Expansion**: The paper shows that if the bare coupling constant \(\gamma_B\) in \(\varphi^4\) theory or \(g_B^2\) in pure Yang-Mills theory is given a specific \(n\) dependence, the theory is finite at \(n \to 4\). This result is significant for understanding the finiteness of the full theory. 6. **Mass Renormalization**: The paper also derives the mass renormalization in the same way, showing that only one-loop infinities contribute to the mass renormalization, while only the one and two-loop infinities determine the coupling constant renormalization. The paper concludes by emphasizing the importance of these equations in understanding the scaling behavior of dimensionally regularized field theories and the exact determination of the singular behavior of bare parameters at \(n \to 4\).The paper by G. 't Hooft discusses the behavior of renormalized field theories under scale transformations using the continuous dimension method. This method allows for a finite perturbation expansion without introducing additional regulator diagrams, preserving symmetries like local gauge invariance. The key findings include: 1. **Scaling Properties of Renormalized Parameters**: The scaling behavior of renormalized parameters \(\lambda_R\) and \(m_R\) is described by differential equations, which are closely related to the Callan-Symanzik equation. These equations are derived from the residues of poles at \(n=4\) and do not require diagram calculations. 2. **Identities for Coefficients**: Equations are derived that relate the coefficients \(a_v\) and \(b_v\) in the series expansion of the bare parameters \(\lambda_B\) and \(m_B\) to the renormalized parameters \(\lambda_R\) and \(m_R\). These identities are crucial for understanding the scaling properties of the theory. 3. **Generalization to Multiple Parameters**: The scaling behavior of multiple parameters in a renormalizable theory is generalized, showing that the scaling properties of physical parameters are determined by their residues at \(n=4\). 4. **Scaling Behavior of Specific Theories**: The scaling behavior of theories like the \(\lambda \phi^4\) theory and quantum electrodynamics is analyzed, demonstrating that the small-distance behavior of these theories is not described by the usual perturbation expansion but can be accurately determined in the infrared region. 5. **Renormalization and Perturbation Expansion**: The paper shows that if the bare coupling constant \(\gamma_B\) in \(\varphi^4\) theory or \(g_B^2\) in pure Yang-Mills theory is given a specific \(n\) dependence, the theory is finite at \(n \to 4\). This result is significant for understanding the finiteness of the full theory. 6. **Mass Renormalization**: The paper also derives the mass renormalization in the same way, showing that only one-loop infinities contribute to the mass renormalization, while only the one and two-loop infinities determine the coupling constant renormalization. The paper concludes by emphasizing the importance of these equations in understanding the scaling behavior of dimensionally regularized field theories and the exact determination of the singular behavior of bare parameters at \(n \to 4\).