Exclusive production of P-wave charmonia through two virtual photons in electron-positron annihilation

We present a relativistic study of the exclusive double-charmonium processes $e^{-}{e}^{+}\to {\gamma }^{\ast }{\gamma }^{\ast }\to {\chi }_{c1}+{\eta }_c$ and $e^{-}{e}^{+}\to {\gamma }^{\ast }{\gamma }^{\ast }\to h_c+h_c$ at $\sqrt{s}=10.6$ GeV in the framework of the $4\times 4$ Bethe-Salpeter equation. Since these channels are forbidden in single-photon annihilation, they proceed purely through two-photon quark-rearrangement mechanisms and thus provide a clean probe of relativistic quarkonium dynamics. Using parameters fixed from mass spectroscopy in our earlier work, we obtain leading-order predictions for the total cross sections and their energy dependence. The predicted cross sections are of order ${10}^{-3}$ fb and exhibit a hierarchy that originates from the radial structure of the meson wave functions, which determines the overlap integrals contributing to the production amplitude. While the absolute magnitudes of the cross sections differ from NRQCD estimates, the qualitative ordering of the channels is consistent with NRQCD expectations. We find a smooth power-law falloff of the cross section for both ${\chi }_{c1}+{\eta }_c$ and $h_c+h_c$. These results provide benchmark predictions for future searches at high-luminosity $e^{-}{e}^{+}$ colliders.