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We present a calculation of the scalar field self-force (SSF) acting on ascalar-charge particle in a strong-field orbit around a Kerr black hole. Ourcalculation specializes to circular and equatorial geodesic orbits. Theanalysis is an implementation of the standard mode-sum regularization scheme:We first calculate the multipole modes of the scalar-field perturbation usingnumerical integration in the frequency domain, and then apply a certainregularization procedure to each of the modes. The dissipative piece of the SSFis found to be consistent with the flux of energy and angular momentum carriedby the scalar waves through the event horizon and out to infinity. Theconservative (radial) component of the SSF is found to be attractive (inwardpointing) for $r_0>r_{\rm c}(a)$ and repulsive (outward pointing) for$r_0<r_{\rm c}(a)$, where $a$ is the Kerr spin parameter, $r_0$ is theBoyer-Lindquist orbital radius, and $r_{\rm c}$ is a critical $a$-dependentradius at which the conservative SSF vanishes. When the motion is retrogradethe conservative SSF is repulsive for all $r_0$ (as in the Schwarzschild case).The dominant conservative effect of the SSF in Schwarzschild spacetime is knownto be of 3rd post-Newtonian (PN) order (with a logarithmic running). Ournumerical results suggest that the leading-order PN correction due to the blackhole's spin arises from spin-orbit coupling at 3PN, which dominates the overallSSF effect at large $r_0$. In PN language, the change-of-sign of the radial SSFis attributed to an interplay between the spin-orbit term ($\propto-ar_0^{-4.5}$) and the "Schwarzschild" term ($\propto r_0^{-5}\log r_0$).

Self force on a scalar charge in Kerr spacetime: circular equatorial orbits