$\Upsilon$ electroproduction at HERA, EIC, and LHeC within the nonrelativistic QCD framework

Based on the nonrelativistic QCD framework, we study the $\Upsilon$ production in semi-inclusive deep inelastic electron-proton scattering (SIDIS) at HERA, EIC, and LHeC, with the main aim of assessing the viability of observing $\Upsilon$ electroproduction at the three colliders. The color-octet (CO) contributions are found to have a crucial effect on both the integrated and differential cross sections, serving to further establish the significance of the CO mechanism. By setting the kinematic cuts to $p_{t,\Upsilon}^{\star 2}>1~\textrm{GeV}^{2}$, $W>50$ GeV, $2<Q^2<100~\textrm{GeV}^2$, and $0.3<z<0.9$, only a few electroproduced $\Upsilon$ events can be generated at HERA, partially accounting for its lack of measured $\Upsilon$ electroproduction. However, under the same cut conditions, the EIC and LHeC can accumulate about $8.7\times10^{2}$ and $3.7\times10^{4}$ reconstructed $\Upsilon$ events in one operation year, respectively, which manifestly indicates the prospect of detecting the $\Upsilon$-related SIDIS processes at the two forthcoming $ep(eA)$ colliders.