The next-to-next-to-leading-order QCD corrections to $e^{+}{e}^{-}\to \hat{I}{\cdot }_c/\ddot{I}{‡}_{cJ}+{\hat{I}}^3$ at B factories

Abstract

We investigate the processes $e^{+}{e}^{-}\to \hat{I}{\cdot }_c+{\hat{I}}^3$ and $e^{+}{e}^{-}\to \ddot{I}{‡}_{cJ}+{\hat{I}}^3$ at B factories within the NRQCD factorization framework, computing the corresponding helicity amplitudes through $\mathcal{O}(\hat{I}{\pm }_s^2)$. The short-distance coefficients are obtained as series expansions in $r=\frac{4m_c^2}{s}$ around $r=0,1/3,2/3,1$, using the method of differential equations. By combining the expansions from all four points, we construct composite asymptotic expressions that reproduce the exact results accurately over the full range $0\le r\le 1$, with relative errors below $0.1\%$ over most of the domain and remaining under $1\%$ elsewhere. Analytic expressions for the leading and next-to-leading logarithmic terms are extracted in the limit $r\to 0$. Using these results, we compute the unpolarized cross sections and observe that the perturbative corrections are small for $\ddot{I}{‡}_{c0}+{\hat{I}}^3$, moderate for $\ddot{I}{‡}_{c1}+{\hat{I}}^3$, and substantial for $\hat{I}{\cdot }_c+{\hat{I}}^3$ and $\ddot{I}{‡}_{c2}+{\hat{I}}^3$. Theoretical prediction for $\ddot{I}{‡}_{c1}+{\hat{I}}^3$ is consistent with the {\tt Belle} measurement within $2\ddot{I}ƒ$, showing good agreement between theory and experiment. We also predict the angular distribution parameters $\hat{I}{\pm }_{\hat{I}}^H¸$, which are insensitive to NRQCD matrix elements and exhibit small theoretical uncertainties. These parameters further display good stability across different perturbative orders. With the high luminosity anticipated at {\tt Belle 2}, future experimental measurements will thus provide a clear test of NRQCD factorization.

35 pages, 7 figures, 3 tables, 1 attached file