A new analytical method for self-force regularization II. Testing the efficiency for circular orbits

In a previous paper, based on the black hole perturbation approach, we formulated a new analytical method for regularizing the self-force acting on a particle of small mass $\mu$ orbiting a Schwarzschild black hole of mass $M$, where $\mu \ll M$. In our method, we divide the self-force into the $\tilde S$-partand$\tilde R$-part\mathrm{.}Allthesingularbehaviorsarecontainedinthe$\tilde S$-part,andhencethe$\tilde R$-part is guaranteed to be regular. In this paper, focusing on the case of a scalar-charged particle for simplicity, we investigate the precision of both the regularized $\tilde{S}$-part and the $\tilde{R}$-part required for the construction of sufficiently accurate waveforms for almost circular inspiral orbits. For the regularized $\tilde S$-part, we calculate it for circular orbits to 18 post-Newtonian (PN) order and investigate the convergence of the post-Newtonian expansion. We also study the convergence of the remaining $\tilde{R}$-part in the spherical harmonic expansion. We find that a sufficiently accurate Green function can be obtained by keeping the terms up to $\mathrm{\ell }=13$.