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In this work, we perform a detailed study of transition form factors foraxial-vector meson production via the two-photon fusion process $\gamma^*\gamma^* \to 1^{++}$, with space-like virtual photons in the initial state anda $P$-wave axial-vector quarkonium in the final state. In this analysis, weemploy the formalism of light-front quarkonium wave functions obtained from asolution of the Schr\"odinger equation for a selection of interquark potentialsfor $Q \bar Q$ interaction. We found the helicity structure and covariantdecomposition of the matrix elements that can be generically applied for any $q\bar q$ axial-vector meson $\gamma^* \gamma^* \to 1^{++}$ transition, while ournumerical results are given for the phenomenologically relevant charmonium$\chi_{c1}$ state. We present the helicity form factors as functions of bothphoton virtualities. We also obtain, that $Q F_{\rm LT}(Q^2,0)/F_{\rmTT}(Q^2,0) = {\rm const.}$

Light-front approach to axial-vector quarkonium $\gamma^* \gamma^*$ form factors