Renormalization is a fundamental concept in quantum field theory (QFT) and plays a central role in the renormalization group (RG) approach. The RG is a powerful tool for studying the behavior of physical systems across different scales. It allows us to understand how the properties of a theory, such as coupling constants and correlation functions, change as we zoom in or out on different length or energy scales.
In the context of QFT, renormalization refers to the process of removing divergences that arise in perturbative calculations. When performing calculations in QFT, such as computing scattering amplitudes or correlation functions, we encounter infinite quantities due to the presence of ultraviolet (UV) divergences. These divergences arise when we attempt to perform calculations at arbitrarily high energies or small distances.
Renormalization involves systematically accounting for these divergences by introducing counterterms, which are additional terms in the Lagrangian or the parameters of the theory. These counterterms are chosen in such a way that they cancel out the divergences, resulting in finite and well-defined predictions for observables.
The RG approach, on the other hand, focuses on understanding the behavior of a theory under scale transformations. It allows us to analyze how the theory's parameters change as we move between different energy scales or length scales. The RG flow describes the evolution of these parameters, and it can reveal the fixed points and the relevant behavior of the theory.
Renormalization group methods provide a framework to study the RG flow and understand how the theory behaves in the UV (high energy) and IR (low energy) limits. By examining the behavior of the couplings and correlation functions under RG transformations, we can gain insights into the universality and critical behavior of physical systems.
In summary, renormalization is the process of removing divergences in QFT calculations, while the RG approach studies the behavior of a theory under scale transformations. Renormalization plays a crucial role in the RG framework by ensuring that the theory's parameters are well-defined and that meaningful predictions can be made across a wide range of energy scales.