Closed-form expression for A(ν_sys, ν_env) in super-Hubble purity scaling

Determine a closed-form expression for the coefficient A(ν_sys, ν_env) that is defined in Eq. (A_coeff_def) of Appendix App:PT_mass and governs the super-Hubble scaling of the perturbative purity for a spectator system scalar σ coupled via the cubic interaction V_c = g φ^2 σ to an environment scalar φ in de Sitter space. Specifically, derive an analytic formula for A(ν_sys, ν_env) across its parameter range (notably when Re(ν_env) < 3/4), where A(ν_sys, ν_env) controls the asymptotic behavior 1 − γ_k(η) ∝ A(ν_sys, ν_env) (−kη)^{−2ν_sys} in the regime −kη ≪ 1.

Background

In the paper’s analysis of decoherence for spectator scalars during de Sitter expansion, the authors derive perturbative expressions for the mode-by-mode purity γ_k(η) in the super-Hubble regime when the system and environment interact through the cubic coupling V_c = g φ^2 σ. They show that for Re(ν_env) < 3/4, the late-time scaling of the purity is controlled by a coefficient A(ν_sys, ν_env), leading to 1 − γ_k(η) ∝ A(ν_sys, ν_env) (−kη)^{−2ν_sys}.

Although the authors provide special-case evaluations (e.g., ν_sys = 3/2) and asymptotic estimates at large imaginary ν_env (heavy environment), they explicitly state that a general closed-form expression for A(ν_sys, ν_env) could not be obtained. Establishing such a formula would clarify the dependence of the decoherence rate on the masses and couplings of the system and environment and enable more precise analytic control of the super-Hubble behavior.