Next-to-Next-to-Leading-Order Radiative Corrections to $e^{+}{e}^{-}\to {\chi }_{cJ}+\gamma$ at $B$ factory

Abstract

Within the nonrelativistic QCD (NRQCD) factorization framework, we have computed the $\mathcal{O}({\alpha }_s^2)$ corrections to the exclusive production of $P$-wave spin-triplet charmonia ${\chi }_{cJ}(J=0,1,2)$ accompanied by a hard photon at $B$ factories. For the first time, we have explicitly verified the validity of NRQCD factorization for exclusive $P$-wave quarkonium production at two-loop order. Unlike the ${\chi }_{cJ}$ electromagnetic decay processes, the $\mathcal{O}({\alpha }_s^2)$ corrections are found to be smaller than the $\mathcal{O}({\alpha }_s)$ corrections in all three channels $e^{+}{e}^{-}\to {\chi }_{c0,1,2}+\gamma$. In particular, the $\mathcal{O}({\alpha }_s^2)$ corrections appear moderate for ${\chi }_{c1}$ case, and insignificant for ${\chi }_{c0}$. In addition, the next-to-next-to-leading order (NNLO) predictions for the production rates of ${\chi }_{c0,1}+\gamma$ are insensitive to the renormalization and factorization scales. All of these features may indicate that perturbative expansion in these two channels exhibits a decent convergence behavior. By contrast, both the $\mathcal{O}({\alpha }_s)$ and $\mathcal{O}({\alpha }_s^2)$ corrections to the ${\chi }_{c2}+\gamma$ production cross section are sizable, which even reduces the Born order cross section by one order of magnitude after including the NNLO perturbative correction. Taking the values of the long-distance NRQCD matrix elements from nonrelativistic potential model, our prediction to ${\chi }_{c1}+\gamma$ production rate is consistent with the recent {\tt Belle} measurement. The NNLO predictions for the ${\chi }_{c0,2}+\gamma$ production rates are much smaller than that for ${\chi }_{c1}+\gamma$, which seems to naturally explain why the $e^{+}{e}^{-}\to {\chi }_{c0,2}+\gamma$ channels have escaped experimental detection to date.

typos corrected, some references added

References in corpus (3)

• Quark mass relations to four-loop order
• QCD corrections to ${\mathbf{e}}^{+}{\mathbf{e}}^{-}\to \mathbf{J}/\mathbf{\psi gg}$ at B Factories
• Bottom and Charm Mass determinations from global fits to $Q\bar{Q}$ bound states at N${}^3$LO

Cited by in corpus (8)

• $Z$ boson radiative decays to a $P$-wave quarkonium at NNLO and LL accuracy
• $Z$ boson radiative decays to a $S$-wave quarkonium at NNLO and NLL accuracy
• $P$-wave quarkonium wavefunctions at the origin in the $\overline{\mathrm{MS}}$ scheme
• Study on ${\eta }_{c2}({\eta }_{b2})$ electromagnetic decay into double photons
• Next-to-leading-order QCD corrections to a vector bottomonium radiative decay into a charmonium
• $\mathcal{O}({\alpha }_s{v}^2)$ corrections to hadronic decay of vector quarkonia
• $X(3860)$ production in association with $J/\psi$ via $e^{+}{e}^{-}$ annihilation at Belle
• Next-to-leading order analysis of $J/\psi +\gamma$ production in photon-photon collisions at CEPC