Problem of the Landau Poles in Quantum Field Theory: from N. N. Bogolyubov to the Present Day

Abstract

A review of problems associated with the unphysical Landau pole in propagators of quantum particles is given. Approaches to eliminating this pole within the framework of electrodynamics and effective theories of strongly interacting particles are investigated. The asymptotic behavior at large momenta in the scalar theory ϕ⁴ in the two-particle (bubble) approximation is investigated. To formulate a calculational model in the two-particle approximation, we use an iterative scheme for solving the Schwinger–Dyson equation in the formalism of a bilocal field source. The main problem is to develop a recipe for numerical analysis of the solutions of the obtained nonlinear equation for the amplitude at small interaction distances (large values of the momentum) for different values of the constant. The nontrivial behavior of the amplitude in the deeply inelastic region of momenta is determined. The positions of the unphysical poles (the Landau poles) in the expression for the amplitude in the deeply inelastic region of momenta are identified.