Two-loop QCD corrections to $η_b \to J/ψ+ γ$

Abstract

We present a next-to-leading-order (NLO) analysis of the rare radiative decay $η_b \to J/ψγ$, a flavor-changing transition between bottomonium and charmonium, within the framework of non-relativistic QCD (NRQCD). We systematically compute the complete $\mathcal{O}(α_s)$ corrections, which include the one-loop QCD corrections to the QED-initiated amplitudes and the two-loop corrections to the QCD-initiated ones. The branching ratio is enhanced from $2.06^{+2.82}_{-1.32}\times10^{-7}$ at LO to $7.53^{+5.67}_{-1.16}\times10^{-7}$ at NLO, representing an increase by a factor of about 3.65. The theoretical uncertainties caused by renormalization scale and $m_{b/c}$ masses are also discussed. Furthermore, the renormalization scale dependence is reduced at NLO.

Figures (4)

• Figure 1: The typical QED diagrams.
• Figure 2: The typical QCD diagrams.
• Figure 3: Real and imaginary part of $f^{(0/1)}(r)$ and $g^{(0/1)}(r)$ from $r=0.01-0.8$, where the two-loop SDC $f^{(1)}(r)$ is given only in several mass ratio, i.e., $r = \frac{(1.5\pm0.1)^2}{(4.7\pm0.1)^2}$.
• Figure 4: The branching fraction for $\eta_b\to J/\psi\gamma$ versus renormalization scale $\mu$ with $m_b = 4.7\pm0.1\ \rm GeV$, $m_c = 1.5\pm0.1\ \rm GeV$ and $\Gamma[\eta_b] = 10\ \rm MeV$.