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The $γ^* γ^*\toη_c$ transition form factor

We study the $γ^* γ^*\toη_c$ transition form factor, $F_{η_cγγ}(Q_1^2,Q_2^2),$ with the local-duality (LD) version of QCD sum rules. We analyse the extraction of this quantity from two different correlators, $<PVV>$ and $<AVV>,$ with $P,$ $A,$ and $V$ being the pseudoscalar, axial-vector, and vector currents, respectively. The QCD factorization theorem for $F_{η_cγγ}(Q_1^2,Q_2^2)$ allows us to fix the effective continuum thresholds for the $<PVV>$ and $<AVV>$ correlators at large values of $Q^2=Q_2^2$ and some fixed value of $β\equiv Q_1^2/Q_2^2$. We give arguments that, in the region $Q^2\ge10$--$15 GeV^2$, the effective threshold should be close to its asymptotic value such that the LD sum rule provides reliable predictions for $F_{η_cγγ}(Q_1^2,Q_2^2).$ We show that, for the experimentally relevant kinematics of one real and one virtual photon, the result of the LD sum rule for $F_{η_cγ}(Q^2)\equiv F_{η_cγγ}(0,Q^2)$ may be well approximated by the simple monopole formula $F_{η_cγ}(Q^2)={2e_c^2N_cf_P}(M_V^2+Q^2)^{-1},$ where $f_P$ is the $η_c$ decay constant, $e^2_c$ is the $c$-quark charge, and the parameter $M_V$ lies in the mass range of the lowest $\bar cc$ vector states.