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(r)∼rα, 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 Υ(3S)→χbγ. We find that this matrix element actually vanishes in a power-law potential rα for a certain power α0∼-0.4. 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 nr nodes in their radial wave functions, we find that the matrix elements left-angle nr,l|r|nr,l+1 right-angle behave as l2/(2+α) for small nr and large l. Transitions with Δnr=±1 behave with respect to those with Δnr=0 as const/ √l, with constants calculated for each nr. Moreover, we find that left-angle nr=0,l|r|nr=2,l-1 right-angle/left-angle nr=0, l|r|nr=0,l+1 right-angle →Φ(α)/l as l→∞, where Φ(α) is calculated explicitly