= Solution
For $k_R=0$, the detuning is $\omega_{m0}=m(\Omega_\varphi-\Omega_{\mathrm p})$. The <Gaussian resonance broadening> factor selects the <corotation resonance>, where the star's azimuthal <frequency> matches the <pattern speed>. The derivative driving the flux is just $m\partial f_0/\partial J_\varphi$. In fact this contribution alone has the form
$$
\Delta F_0\big|_{k_R=0}=\partial_{J_\varphi}\left[\frac{\sqrt{8\pi^5}\,m^2}{\Gamma}\mathcal R_\Gamma(\omega_{m0})\int_0^\infty|u_{m0}|^2\partial_{J_\varphi}f_0\,dJ_R\right].
$$
A single <Fourier series> harmonic changes the <action variables> in the direction $(m,k_R)$, because $\dot{\mathbf J}=-i\mathbf k\delta\phi_{\mathbf k}e^{i\mathbf k\cdot\boldsymbol\theta}$ plus its conjugate. For this harmonic, $\Delta J_R=0$: stars change <angular momentum> and shift their <epicyclic guiding center> without a leading-order increase of random radial motion. This is <stellar radial migration> driven by the <corotation resonance>. The resonance condition explains the absence of leading heating: $\Delta E\simeq\Omega_\varphi\Delta J_\varphi=\Omega_{\mathrm p}\Delta J_\varphi$ at the <corotation resonance>.
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