Momentum space of constant curvature and the “rainbow” approximation in quantum field theory

Abstract

The behavior of the mass operator is studied in “rainbow” graph approximation in the momentum space of constant curvature with the group of motions SO(4,1). The infrared divergences occuring there are eliminated by a multiplicative renormalization. When x⩽4ι ⁻² (whereι is the “fundamental length”), the resulting asymptotic (x ≫ m² _c) expressions for the mass operator Σ_R (x) and its imaginary part are analytic in the coupling constant at zero, while in the domain x≫4ι ⁻² a logarithmic branching occurs, and the function grows linearly. The assumption that there are “superheavy particles” in nature (with m ²_c ≫hι ⁻²) in the asymptotic domain x≫4ι ₋₂ leads to a violation of the positive definiteness for the imaginary part of the mass operator.