Interpretation of finite-part consistency conditions in η-regularised one-loop ILIs

Investigate the origin and interpretation of the finite-part consistency conditions γ[θ0] − γ[η0] = 1/4 and γ[κ0] − γ[η0] = 5/12 for logarithmically divergent one-loop irreducible loop integrals, and the analogous relations γ[θ2] − γ[η2] = 1/4 and γ[κ2] − γ[η2] = 5/12 for quadratically divergent integrals, where γ[η] = −∫0^∞ η′(x) ln x dx and η, θ, κ denote the regulator functions used for scalar and tensor ILIs in η regularisation; determine why these specific constants arise and what principle fixes them within the framework enforcing Wu’s gauge-invariance consistency relations.

Background

In enforcing gauge invariance via Wu’s consistency relations for a general non-abelian gauge theory with Dirac fermions, the authors express the one-loop vacuum polarisation and higher-point functions in terms of η-regularised irreducible loop integrals (ILIs). For logarithmically divergent ILIs (α = 0), cancellation of divergences yields finite-part constraints γ[θ0] − γ[η0] = 1/4 and γ[κ0] − γ[η0] = 5/12. For quadratically divergent ILIs (α = −1), after eliminating the quadratic divergences using enhanced regulators, analogous finite-part constraints emerge: γ[θ2] − γ[η2] = 1/4 and γ[κ2] − γ[η2] = 5/12.

Here γ[η] is the finite integral −∫0^∞ η′(x) ln x dx, and η, θ, κ are the regulator functions used, respectively, for scalar, rank-2 tensor, and rank-4 tensor ILIs. Although the authors exhibit choices of rescaled enhanced regulators that satisfy these relations, they explicitly state that they currently lack a deeper understanding of why these particular constants appear or what principle determines them.