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### Description

We derive the light-front wave function (LFWF)

representation of the $\gamma^* \gamma^* \to \eta_c(1S)\,,\eta_c(2S)$

transition form factor $F(Q_1^2, Q_2^2)$ for two virtual photons

in the initial state. For the LFWF, we use different models

obtained from the solution of the Schr\"odinger equation for

a variety of $c \bar c$ potentials. We compare our results to

the BaBar experimental data for the $\eta_c(1S)$

transition form factor, for one real and one virtual photon.

We observe that the onset of the asymptotic behaviour is strongly delayed

and discuss applicability of the collinear and/or massless limit.

We present some examples of two-dimensional distributions for

$F (Q_1^2,Q_2^2)$.

A factorization breaking measure is proposed and factorization breaking

effects are quantified and shown to be almost model independent.

Factorization is shown to be strongly broken, and a scaling of

the form factor as a function of $\bar Q^2 = (Q_1^2 + Q_2^2)/2$ is obtained.

In addition, we present a thorough analysis of $\eta_c(1S,2S)$

quarkonia hadroproduction in $k_\perp$-factorisation in the framework of

the light-front potential approach for the quarkonium wave function.

The off-shell matrix elements for the $g^* g^* \eta_c(1S,2S)$ vertices

are derived. We discuss the importance of taking into account the gluon

virtualities.

We present the transverse momentum distributions of $\eta_c$ for

several models of the unintegrated gluon distributions.

Our calculations are performed for four distinct parameterisations

for the $c\bar c$ interaction potential consistent with the meson spectra.

We compare our results for $\eta_c(1S)$ to measurements by the

LHCb collaboration and present predictions for $\eta_c(2S)$ production.

References:

1.) I. Babiarz, V.P. Goncalves, R. Pasechnik, W. Sch\"afer and

A. Szczurek, ''The ${\gamma^* \gamma^* \to \eta_c (1S,2S)}$ transition

form factors for spacelike photons'',

Phys.Rev. {\bf D100} 054018 (2019)

2.) I.~Babiarz, R.~Pasechnik, W.~Sch\"afer and A.~Szczurek,

``Prompt hadroproduction of $\eta_c(1S,2S)$ in the $k_T$-factorization approach,''

JHEP {\bf 2002}, 037 (2020) [arXiv