High-Post-Newtonian-Order Dynamics Induced by Tail-of-Tail Interactions: The Non-Geodesic Terms

We compute the tail-of-tail contribution to the conservative dynamics of eccentric, non-spinning compact binaries to relative 1PN order and to $\mathcal O(e_t^{12})$. Using the $1$PN quasi-Keplerian dynamics in harmonic coordinates, we derive the Delaunay-averaged Hamiltonian at $5.5$PN and $6.5$PN order and match it to the effective-one-body description, allowing us to determine the corresponding contributions to the non-geodesic EOB $Q$ potential through the $\mathcal{O}(p_r^{12})$, including the dependence on the symmetric mass ratio up to $\mathcal{O}(\nu^2)$. The terms linear in the mass ratio reproduce the available first-order self-force results, while the quadratic terms provide qualitatively new eccentric second-order self-force predictions arising from the tail-of-tail terms. We independently rederive the averaged Hamiltonian using a Fourier--Bessel decomposition of the hereditary interaction. Applying the first law at fixed orbital frequencies, we recover the known first-order self-force redshift through $\mathcal O(e^{12})$ and obtain the complete tail-of-tail contribution to the second-order inverse redshift at $5.5$PN and $6.5$PN through $\mathcal O(e^{10})$.