Next-to-leading-order QCD corrections to a vector bottomonium radiative decay into a charmonium

TL;DR

This paper calculates NLO QCD corrections to bottomonium radiative decays into charmonium states, improving theoretical predictions and revealing discrepancies with experimental data, thus advancing understanding of heavy quarkonium decay processes.

Contribution

First computation of two-loop NLO corrections for bottomonium to charmonium radiative decays using NRQCD, with improved numerical methods and analysis of scale dependence.

Findings

NLO corrections are moderate for $ ext{η}_c$ and $ ext{χ}_{c2}$, marginal for $ ext{χ}_{c0}$ and $ ext{χ}_{c1}$.

NLO corrections significantly reduce scale dependence in decay widths and branching fractions.

Predicted decay widths and branching ratios show notable discrepancies with experimental measurements.

Abstract

Within the framework of nonrelativistic QCD (NRQCD) factorization, we calculate the next-to-leading-order (NLO) perturbative corrections to the radiative decay $\Upsilon\to \eta_c(\chi_{cJ})+\gamma$. Both the helicity amplitudes and the helicity decay widths are obtained. It is the first computation for the processes involving both bottomonium and charmonium at two-loop accuracy. By employing the Cheng-Wu theorem, we are able to convert most of complex-valued master integrals (MIs) into real-valued MIs, which makes the numerical integration much efficient. Our results indicate the $\mathcal{O}(\alpha_s)$ corrections are moderate for $\eta_c$ and $\chi_{c2}$ production, and are quite marginal for $\chi_{c0}$ and $\chi_{c1}$ production. It is impressive to note the NLO corrections considerably reduce the renormalization scale dependence in both the decay widths and the branching fractions for $\chi_{cJ}$, and slightly improve that for $\eta_c$. With the NRQCD matrix elements evaluated via the Buchm\"uller-Tye potential model, we find the decay width for $\eta_c$ production is one-order-of-magnitude larger than $\chi_{cJ}$ production, which may provide a good opportunity to search for $\Upsilon\to \eta_c+\gamma$ in experiment. In addition, the decay width for $\chi_{c1}$ production is several times larger than those for $\chi_{c0,2}$. Finally, we find the NLO NRQCD prediction for the branching fraction of $\Upsilon\to \chi_{c1}+\gamma$ is only half of the lower bound of the experimental data measured recently by {\tt Belle}. Moreover, there exists serious contradiction between theory and experiment for $\Upsilon\to \eta_c+\gamma$. The discrepancies between theory and experiment deserve further research efforts.