Numerical and Physical Interpretation of the Sunset–Tadpole Ratio in the TT Sector of History-Dependent Gravity

Abstract

We present a numerical analysis and physical interpretation of the sunset-to-tadpole ratio R_TS = T_tad / T_sun arising in the transverse-traceless (TT) sector of the functional renormalization group (FRG) treatment of history-dependent gravity. Building on the TT vertex structure established in previous work, we perform a numerical validation of the two loop contributions and their ratio within the implemented B.27-C TT kernel. At fixed relative external momenta p/k, we find numerical agreement with the expected k^4 scaling of the individual contributions, T_sun = k^4 T_sun(p/k, m_reg^2 / k^2) and T_tad = k^4 T_tad(p/k, m_reg^2 / k^2), over the range k = 0.25...4, with relative scatter below 0.002%. Consequently, R_TS is numerically stable under variations of the FRG scale k at fixed p/k, with relative standard deviation σ_rel(R_TS) ≲ 6×10^-4 % for the Litim regulator. We obtain R_TS^Litim(p/k=0.05) = 26.264 and R_TS^Litim(p/k=0.15) = 24.765. A comparison with the exponential regulator reveals a regulator dependence of ΔR_TS ≃ -9.7%...-13.8% within the present truncation. We conclude that R_TS is a numerically robust but regulator-dependent diagnostic quantity characterizing the relative weight of the two surviving TT loop topologies, rather than a regulator-independent prediction within the present truncation.