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We calculate cross sections for production of axial-vector $f_1(1285)$ mesonsfor double-tagged measurements of the $e^+ e^- \to e^+ e^- f_1(1285)$ reaction.Different $\gamma^* \gamma^* \to f_1(1285)$ vertices from the literature areused. Both integrated cross section as well as differential distributions arecalculated. Predictions for a potential measurement at Belle II are presented.Quite different results are obtained for the different vertices proposed in theliterature. Future measurements at $e^+ e^-$ colliders could test and/orconstrain the $\gamma^* \gamma^* \to f_1 (a_1, f_1')$ vertices and associatedform factors, known to be important ingredients for calculating contributionsto anomalous magnetic moment of muon and hyperfine splitting of levels ofmuonic atoms.

physical review. d/physical review. d.

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