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Using a recently presented numerical code for calculating the Lorenz-gauge gravitational self-force (GSF), we compute the O (m) conservative correction to the precession rate of the small-eccentricity orbits of a particle of mass m moving around a Schwarzschild black hole of mass M. Specifically, we study the gauge-invariant function (x), where is defined as the O (m) part of the dimensionless ratio (^{}ₑ/^{}_) ^2 between the squares of the radial and azimuthal frequencies of the orbit, and where x=Gc^-3 (M+m) ^{}_^2/3 is a gauge-invariant measure of the dimensionless gravitational potential (mass over radius) associated with the mean circular orbit. Our GSF computation of the function (x) in the interval 0<x1/6 determines, for the first time, the strong-field behavior of a combination of two of the basic functions entering the effective one-body (EOB) description of the conservative dynamics of binary systems. We show that our results agree well in the weak-field regime (small x) with the 3rd post-Newtonian (PN) expansion of the EOB results, and that this agreement is improved when taking into account the analytic values of some of the logarithmic-running terms occurring at higher PN orders. Furthermore, we demonstrate that GSF data give access to higher-order PN terms of (x) and can be used to set useful new constraints on the values of yet-undetermined EOB parameters. Most significantly, we observe that an excellent global representation of (x) can be obtained using a simple ``2-point'' Pad\'e approximant which combines 3PN knowledge at x=0 with GSF information at a single strong-field point (say, x=1/6).
Barack et al. (Wed,) studied this question.