A tale of Bethe logarithms: leptonic widths of $\chi_{cJ}$ and Lamb shift

The rare annihilation decays of $P$-wave spin-triplet quarkonia into lepton pair have to proceed via two-photon intermediate state, which are plagued with the infrared divergence symptom. We recognize that the physical root of the IR divergence and its remedy is the same as the Lamb shift in QED. In this work we provide a complete solution to this IR problem by including the effect of the higher Fock component of the $\chi_{cJ}$ state, {\it viz.}, the $c\bar{c}(^3S_1^{(1)})$ pair accompanied with a very long wavelength photon. Adding the contributions from the leading and next-to-leading order Fock components together, we arrive at the IR finite and factorization scale independent predictions for leptonic widths of $\chi_{cJ}(nP)$. The Bethe logarithms associated with these exclusive reactions are found to have rather different traits from those associated with Lamb shift. We present numerical predictions by employing several influential quark potential models. The predicted leptonic widths of $\chi_{cJ}(nP)$ are sizable and the observation prospect for $e^+e^-\to \chi_{cJ}(1P,2P)$ at BESIII looks bright. The future observation of $e^+e^-\to X(3872)$ will shed important light on the $c\bar{c}$ content of the $X(3872)$ meson.