The $\gamma^* \gamma^*\to\eta_c$ transition form factor

TL;DR

This paper uses QCD sum rules to analyze the $ o ext{transition form factor}$ for $ ext{eta}_c$ meson, providing predictions for its behavior at high momentum transfer and a simple monopole approximation for experimental conditions.

Contribution

It introduces a method to extract the $ ext{eta}_c$ transition form factor using local-duality sum rules and fixed effective thresholds, offering reliable predictions at high $Q^2$ and a simple monopole model for experiments.

Findings

Effective thresholds approach asymptotic values at $Q^2 ext{≥} 10$--$15$ GeV$^2$.

LD sum rule predictions align with monopole approximation for one real and one virtual photon.

Parameter $M_V$ is within the mass range of lowest $ar cc$ vector states.

Abstract

We study the $\gamma^* \gamma^*\to\eta_c$ transition form factor, $F_{\eta_c\gamma\gamma}(Q_1^2,Q_2^2),$ 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 $F_{\eta_c\gamma\gamma}(Q_1^2,Q_2^2)$ 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 $\beta\equiv Q_1^2/Q_2^2$. 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 $F_{\eta_c\gamma\gamma}(Q_1^2,Q_2^2).$ We show that, for the experimentally relevant kinematics of one real and one virtual photon, the result of the LD sum rule for $F_{\eta_c\gamma}(Q^2)\equiv F_{\eta_c\gamma\gamma}(0,Q^2)$ may be well approximated by the simple monopole formula $F_{\eta_c\gamma}(Q^2)={2e_c^2N_cf_P}(M_V^2+Q^2)^{-1},$ where $f_P$ is the $\eta_c$ 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.