Forward ${\chi }_c$ production in $pp$ and $pA$ collisions within the CGC+NRQCD framework

We study the forward ${\chi }_{cJ}$ production in the low $p_{\perp }$ region in proton-proton $(pp)$ and proton-nucleus $(pA)$ collisions at the LHC within the framework of non-relativistic QCD (NRQCD) factorization and the Color Glass Condensate (CGC) effective field theory. Incorporating the small-$x$ gluonic evolution via the running-coupling Balitsky-Kovchegov equation with various initial conditions, we systematically analyze the cross section ratio $R_{{\chi }_c}={\sigma }_{{\chi }_{c2}}\mathrm{/}{\sigma }_{{\chi }_{c1}}$, and find that the theoretical predictions weakly depend on these inputs. Our predictions align well with the LHCb measurements for $p\mathrm{P}\mathrm{b}$ collisions, however, overshoot those for $pp$ collisions. We further investigate the feed down fractions $F_{{\chi }_{c1,2}\to J\mathrm{/}\psi }={\sigma }_{{\chi }_{c1,2}\to J\mathrm{/}\psi }\mathrm{/}{\sigma }_{J\mathrm{/}\psi }$. We notice that with the experimental measurement of prompt $J\mathrm{/}\psi$ cross section as inputs in the denominator the CGC+NRQCD predictions broadly match the data when choosing the standard McLerran-Venugopalan initial conditions. In contrast, implementing purely theoretical CGC+NRQCD predictions for both $J\mathrm{/}\psi$ and ${\chi }_c$ production yield systematic discrepancies across all tested parameters. This highlights a persistent challenge in achieving a simultaneous, unified description of ${\chi }_c$ and prompt $J\mathrm{/}\psi$ production within the current framework.