Eur. Phys. J. C (2025) 85: 1474
https://doi.org/10.1140/epjc/s10052-025-15226-2

Regular Article - Theoretical Physics

Transition form-factor for ${\eta }_Q$ at NNLO in the strong coupling ${\alpha }_s$ and with all-order $v^2$ resummation

¹ Institute of Nuclear Physics, Polish Academy of Sciences, ul. Radzikowskiego 152, 31-342, Kraków, Poland
² CNRS, IJCLab, Université Paris-Saclay, 91405, Orsay, France
³ College of Mathematics and Natural Sciences, University of Rzeszów, ul. Pigonia 1, 35-310, Rzeszów, Poland

^a This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 29 September 2025
Accepted: 16 December 2025
Published online: 28 December 2025

Abstract

In this work, we discuss both relativistic and perturbative QCD corrections to the transition form-factor ${\mathcal{F}}_{{\eta }_Q}(t_1,t_2)$ for the process ${\gamma }^{\ast }(q_1){\gamma }^{\ast }(q_2)\leftrightarrow {\eta }_Q(P)$ with dependencies on the normalised photon virtualities $t_1=q_1^2/m_Q^2$ and $t_2=q_2^2/m_Q^2$, where $m_Q$ is the heavy quark mass. We resum a class of relativistic corrections to all orders in the relativistic parameter ${⟨v^2⟩}_{{\eta }_Q}$. In addition, we include perturbative QCD corrections up to next-to-next-to-leading order (NNLO) in the strong coupling constant ${\alpha }_s$. This involves the computation of two-loop amplitudes with two off-shell photons. We explore three different phenomenological applications of our transition form-factors for the charmonium case. We first study the ratio $|{\mathcal{F}}_{{\eta }_c}(t_1,0)|/|{\mathcal{F}}_{{\eta }_c}(0,0)|$ for the single space-like photon case and compare our results with the existing ${\eta }_c$ measurements from the BaBar collaboration. Secondly, we consider observables for the case of double space-like photons and discuss the impact of the different corrections. The NNLO corrections for this case are presented here for the first time in the literature. Finally, we revisit the decay width $\mathrm{\Gamma }[{\eta }_c\to \gamma \gamma ]$ and compare it with the existing PDG value.