$\Upsilon(nS)$ Production within Jets at the LHC
Heavy quarkonium production inside jets offers a sensitive probe of QCD dynamics and bound-state formation mechanisms. While recent studies demonstrate that charmonium-in-jet observables effectively discriminate among competing nonrelativistic QCD (NRQCD) long-distance matrix element (LDME) sets, whether this discriminating power persists in the bottomonium sector remains an open question. Here, we present the first phenomenological study of $\Upsilon(1S)$, $\Upsilon(2S)$, and $\Upsilon(3S)$ production inside jets using the fragmenting jet function (FJF) framework at next-to-leading order (NLO), incorporating DGLAP evolution, threshold resummation, and feeddown contributions from higher bottomonium states. In sharp contrast to charmonium, we find that bottomonium-in-jet momentum-fraction ($z_H$) distributions exhibit a universal shape that is remarkably insensitive to the choice of LDME sets. We show that this universality stems from the strong dominance of the S-wave spin-triplet color-octet ($^3S_1^{[8]}$) production mechanism reinforced by $\chi_b$ feeddown transitions. Our predictions capture both the characteristic large-$z_H$ peak and the spectral broadening with increasing jet transverse momentum observed in recent CMS measurements. These results establish a clear physical distinction between charmonium and bottomonium fragmentation inside jets, providing a theoretical benchmark for future high-precision measurements at the LHC.