Logarithmic coupling functionals and polyhomogeneous asymptotics

Develop a systematic treatment of genuinely logarithmic scalar–vector–tensor coupling functionals within a polyhomogeneous Bondi–Sachs framework, determining how logarithmic couplings modify asymptotic expansions, mass and angular-momentum aspects, flux finiteness, and the conformal-frame resolution of non-minimal couplings.

Background

The analysis derives asymptotic-flatness conditions under regular Taylor expansions and separately considers power-law singularities. It then discusses logarithmic couplings such as X ln(X/μ⁴) and (Φ−ϕ₀) ln(Φ−ϕ₀), which fall outside the power-law class used in the main analysis.

For marginal operators, logarithmic behavior generates logarithmic tails and requires a polyhomogeneous expansion involving powers of 1/r and ln r. The paper argues that the conformally coupled sector is more difficult because a logarithmic derivative of G₄ prevents the total-derivative mechanism that resolves the constant-derivative case. A complete treatment is left unresolved.