Self-Force on a Scalar Charge in Circular Orbit around a Schwarzschild Black Hole

In an accompanying paper, we have formulated two types of regulariz_ationmethods to calculate the scalar self-force on a particle of charge $q$ movingaround a black hole of mass $M$, one of which is called the ``power expansionregularization''. In this paper, we analytically evaluate the self-force (whichwe also call the reaction force) to the third post-Newtonian (3PN) order on thescalar particle in circular orbit around a Schwarzschild black hole by usingthe power expansion regularization. It is found that the $r$-component of theself-force arises at the 3PN order, whereas the $t$- and $\phi$-components,which are due to the radiation reaction, appear at the 2PN and 1.5PN orders,respectively.Comment: 16 pages, no figure, Minor typos correcte