Dipole transition matrix elements for systems with power-law potentials

We study the behavior of dipole matrix elements for systems bound by power-law potentials of the form $V\left(r\right)\mathrm{}\sim r^{\alpha }$, which are useful in the descriptions of quarkonium systems. The experimental feature for which further understanding is sought is the apparent suppression of the transition $\Upsilon \mathrm{}\left(3S\right)\mathrm{}\to {\chi }_b\gamma$. We find that this matrix element actually vanishes in a power-law potential $r^{\alpha }$ for a certain power ${\alpha }_0\approx -0.4\mathrm{}$. The suppression of transitions between states with different numbers of nodes in their radial wave functions is a universal property of most physically interesting power-law potentials. We derive results in the limit of large orbital angular momenta $l$, checking that they agree with the known answers for the Coulomb and spherical oscillator potentials. For states with $n_r$ nodes in their radial wave functions, we find that the matrix elements $〈n_r,l\left|r\right|\mathrm{}{n}_r,l+1\mathrm{}〉$ behave as $l^{\frac{2}{(2+\alpha )}}$ for small $n_r$ and large $l$. Transitions with $\Delta n_r=\pm 1$ behave with respect to those with $\Delta n_r=0\mathrm{}$ as $\frac{\mathrm{const}}{\sqrt{l}}$, with constants calculated for each $n_r$. Moreover, we find that $\frac{〈n_r=0,l\left|r\right|\mathrm{}{n}_r=2,l-1\mathrm{}〉}{〈n_r=0,l\left|r\right|\mathrm{}{n}_r=0,l+1\mathrm{}〉}\to \frac{\Phi \left(\alpha \right)}{l}$ as $l\to \infty$, where $\Phi \left(\alpha \right)$ is calculated explicitly.