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Conservative second-order gravitational self-force on circular orbits and the effective one-body formalism

We consider Detweiler's redshift variable $z$ for a nonspinning mass $m_1$ in circular motion (with orbital frequency $Ω$) around a nonspinning mass $m_2$. We show how the combination of effective-one-body (EOB) theory with the first law of binary dynamics allows one to derive a simple, exact expression for the functional dependence of $z$ on the (gauge-invariant) EOB gravitational potential $u=(m_1+m_2)/R$. We then use the recently obtained high-post-Newtonian(PN)-order knowledge of the main EOB radial potential $A(u ; ν)$ [where $ν= m_1 m_2/(m_1+m_2)^2$] to decompose the second-self-force-order contribution to the function $z(m_2 Ω, m_1/m_2)$ into a known part (which goes beyond the 4PN level in including the 5PN logarithmic term, and the 5.5PN contribution), and an unknown one [depending on the yet unknown, 5PN, 6PN, $\ldots$, contributions to the $O(ν^2)$ contribution to the EOB radial potential $A(u ; ν)$]. We indicate the expected singular behaviors, near the lightring, of the second-self-force-order contributions to both the redshift and the EOB $A$ potential. Our results should help both in extracting information of direct dynamical significance from ongoing second-self-force-order computations, and in parametrizing their global strong-field behaviors. We also advocate computing second-self-force-order conservative quantities by iterating the time-symmetric Green-function in the background spacetime.