Threshold for the breakdown of NRQCD predictions in P-wave Dalitz decays

Determine quantitatively the exact photon-energy threshold below which theoretical predictions for the differential Dalitz-decay distributions of P-wave heavy quarkonia \(\chi_{QJ}\to l^{+}l^{-}\gamma\) become unreliable because of soft-photon effects and the breakdown of NRQCD factorization.

Background

The paper finds that the differential decay distributions for the P-wave states $\chi_{cJ}$ and $\chi_{bJ}$ develop a soft-photon infrared singularity as the dilepton invariant mass approaches the parent-meson mass, corresponding to the photon energy becoming small. The authors explain that reliable total-width predictions require the contribution of the Fock state $|Q\bar{Q}(^3S_1^{[1]})\gamma\rangle$, while the practical analysis imposes a lower photon-energy cut to avoid the experimentally undetectable soft-photon region.

For differential distributions, however, the precise boundary between a theoretically reliable region and the region where NRQCD factorization becomes unreliable is not established. Identifying this threshold would clarify the domain of applicability of the NLO NRQCD predictions and guide consistent choices of photon-energy cuts in future experimental comparisons.