In quantum field theory (QFT), certain integrals can lead to divergent expressions, meaning that the integrals do not have a finite value. These divergences arise due to the infinite nature of the theory and can be categorized into different types. Two common types of divergent expressions in QFT are ultraviolet (UV) divergences and infrared (IR) divergences.
Ultraviolet (UV) Divergences: UV divergences arise when integrating over momentum modes that are arbitrarily large, corresponding to high-energy or short-distance behavior. These divergences indicate that the theory is sensitive to physics at arbitrarily short distances, which is not physically meaningful. UV divergences often appear in loop diagrams involving virtual particles. They can be regularized by introducing a cutoff scale or using renormalization techniques to remove the infinities and obtain finite, meaningful results.
Infrared (IR) Divergences: IR divergences, on the other hand, arise when integrating over momentum modes that are arbitrarily small, corresponding to low-energy or long-distance behavior. These divergences arise due to the long-range nature of certain interactions or massless particles in the theory. IR divergences can appear in certain processes involving massless particles, such as the emission of soft photons or gluons. In practice, IR divergences are often dealt with by including additional soft-gluon or soft-photon resummation techniques to obtain well-defined and physically meaningful results.
It is worth noting that divergent expressions in QFT are not inherent flaws in the theory but rather reflect the challenges of dealing with infinite quantities. Renormalization techniques, such as regularization and subtraction procedures, are employed to handle these divergences and extract finite and physically meaningful results from quantum field theory calculations.