Production of axial-vector mesons at $e^{+}{e}^{-}$ collisions with double-tagging as a way to constrain the axial meson LbL contribution to muon g-2 and/or hyperfine splitting of muonic hydrogen

We calculate cross sections for production of axial-vector $f_1(1285)$ mesons for double-tagged measurements of the $e^{+}{e}^{-}\to e^{+}{e}^{-}{f}_1(1285)$ reaction. Different ${\gamma }^{\ast }{\gamma }^{\ast }\to f_1(1285)$ vertices from the literature are used. Both integrated cross section as well as differential distributions are calculated. Predictions for a potential measurement at Belle II are presented. Quite different results are obtained for the different vertices proposed in the literature. Future measurements at $e^{+}{e}^{-}$ colliders could test and/or constrain the ${\gamma }^{\ast }{\gamma }^{\ast }\to f_1(a_1,f_1^{\mathrm{\prime }})$ vertices and associated form factors, known to be important ingredients for calculating contributions to anomalous magnetic moment of muon and hyperfine splitting of levels of muonic atoms.