Systematic study of bottomonium production in proton-proton collisions at LHC energies

We present a comprehensive study of $\mathrm{{\rm Y}}(nS)$ ($n$ = 1, 2, 3) production in proton-proton ($pp$) collisions at various LHC energies and rapidity ranges within the framework of leading order non-relativistic quantum chromodynamics (NRQCD) factorization. The transverse momentum ($p_{\mathrm{T}}$)-dependent production cross-sections are calculated, incorporating both direct and feed-down contributions. Specifically, feed-down from $\mathrm{{\rm Y}}(2S)$, $\mathrm{{\rm Y}}(3S)$, ${\chi }_{bJ}(1P)$, and ${\chi }_{bJ}(2P)$ states to $\mathrm{{\rm Y}}(1S)$ is included, while $\mathrm{{\rm Y}}(2S)$ receives contributions from $\mathrm{{\rm Y}}(3S)$ and ${\chi }_{bJ}(2P)$. No significant feed-down is considered for $\mathrm{{\rm Y}}(3S)$. The computed cross-sections and cross-section ratios among different $\mathrm{{\rm Y}}$ states are compared with experimental measurements from ALICE, ATLAS, CMS and LHCb. It is found that the experimental cross-sections and cross-section ratios are well described within the theoretical uncertainties arising due to the choices of the factorization and renormalization scales for $p_{\mathrm{T}}$ $>$ 4 GeV and $p_{\mathrm{T}}$ $>$ 0 GeV, respectively. Furthermore, the cross-section ratios exhibit a clear saturation behavior beyond $p_{\mathrm{T}}$ $\approx$ 40 GeV.