Two loop QCD corrections to $e^+ e^- \to J/ψ+ η_c$ in asymptotic expansion

TL;DR

This work addresses the NNLO QCD corrections to the short-distance coefficients for the process $e^+e^-\to J/\psi+\eta_c$ within NRQCD by deriving a comprehensive asymptotic expansion in $r=16m_c^2/s$ up to $r^{15}$. The NNLO terms $f^{(1)}(r)$ and $f^{(2)}(r)$ are obtained using differential-equation methods for master integrals and then expressed as series in $r$ and $\ln r$, with explicit OS mass renormalization and MS coupling; mass conversions to the $\overline{\text{MS}}$ scheme are implemented to provide scheme-dependent predictions. The asymptotic expressions converge well for $r<0.8$, enabling reliable cross-section predictions across a wide energy range, though accuracy worsens near threshold; the OS and MS predictions are broadly consistent and align with existing BABAR/Belle data. This work also clarifies the impact of renormalization-scale and charm-mass uncertainties and furnishes a practical framework for phenomenology and potential resummations in heavy-quarkonium production.

Abstract

Within the framework of NRQCD, the short-distance coefficients (SDCs) for the process $e^+e^-\to J/ψ+η_c$ have been obtained up to NNLO in asymptotic expansions over $r={16m_c^2}/{s}$ up to $r^{15}$. Although these asymptotic expressions are deviated from the full results near the threshold $r= 1$, they provide excellent approximations to the full results for $r<0.8$, with deviations less than $3\%$. Therefore, these asymptotic expressions offer reliable applications for phenomenological predictions across a wide range of center-of-mass energies $\sqrt{s}$. Utilizing these asymptotic expressions, we present phenomenological predictions for the cross sections in both the on-shell mass scheme and the $\overline{\rm MS}$ mass scheme, with the uncertainty arising from the renormalization scale $μ_R$ included. The $μ_R$ uncertainty for predictions from the $\overline{\rm MS}$ mass scheme is slightly larger than that from the on-shell mass scheme, which is partly attributed to the helicity flip in the process $e^+e^-\to J/ψ+η_c$. We observe that both mass schemes yield quite similar predictions, and our theoretical results are consistent with the available experimental data.

Figures (4)

• Figure 1: Some typical Feynman diagrams. (a) is the LO amplitude, (b)-(e) illustrate the NLO corrections, and (f)-(o) depict the NNLO contributions. The final diagram (o) illustrates the “light-by-light” part.
• Figure 2: Comparison of the asymptotic expressions with the exact results for real part of $f^{(2)}(r)$.
• Figure 3: Comparison of the asymptotic expressions with the exact results for the imaginary part $f^{(2)}(r)$.
• Figure 4: The cross section as a function of $\sqrt{s}$ at various levels of accuracy in $\alpha_s$. The left panel corresponds to predictions from the OS mass scheme, while right panel corresponds to predictions from the $\overline{\rm MS}$ mass scheme. For the OS mass scheme prediction, We take $m_c=1.5$ GeV. The shaded bands represent the uncertainty arising from varying $\mu_R$ from $3$ GeV to $\sqrt{s}$.