Heavy-ion double-charge-exchange and its relation to neutrinoless double-beta decay

We introduce the formalism to describe heavy-ion Double-Charge-Exchange (DCE) processes in the eikonal approximation. We focus on the low-momentum-transfer limit -- corresponding to the differential cross section at $\theta=0^\circ$ -- and, for the first time, we show that it is possible to factorize the DCE cross-section in terms of reaction and nuclear parts. Whereas in the $\theta\neq0^\circ$ case the nuclear part is a convolution of the beam and target nuclear matrix elements (NMEs), for $\theta=0^\circ$ we demonstrate - for the first time- that the transition matrix elements can be written as the sum of Double-Gamow-Teller (DGT) and Double Fermi (DF) type parts, and that they can both be further factorized in terms of target and projectile NMEs. By making use of the Interacting Boson Model (IBM) formalism, we also show that the DGT and total parts of the neutrinoless double-beta decay NMEs are in linear correlation with DCE-DGT NMEs. This confirms the hypothesis of a linear correlation between them, as introduced in [Phys.\ Rev.\ Lett.\ {\bf 120}, 142502 (2018)]. The possibility of a Two-Step Factorization (TSF) of the very forward differential DCE-cross-section and the emergence of DGT and DF types for the DCE nuclear matrix elements, combined with a linear correlation between DCE-DGT and 0$\nu\beta\beta$ NMEs, opens the possibility of placing an upper limit on neutrinoless double-beta decay NMEs in terms of the DCE experimental data at $\theta=0^\circ$.