$Υ(nS)$ Production within Jets at the LHC

Abstract: 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 $Υ(1S)$, $Υ(2S)$, and $Υ(3S)$ 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 ($<sup>3S_1<sup>{[8]}$) production mechanism reinforced by $χ_b$ 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.

• The paper presents the first phenomenological study of Υ(1S), Υ(2S), and Υ(3S) production inside LHC jets using NLO NRQCD fragmentation functions, FJF factorization, DGLAP evolution, threshold resummation, and comprehensive feeddown.
• The predicted zH spectra are nearly universal across four substantially different LDME sets because color-octet 3S1 fragmentation dominates, while χb feeddown reinforces this behavior and reduces the observables’ ability to distinguish NRQCD models.
• The resummed framework reproduces CMS peak structures and their momentum-dependent broadening, but overpredicts low-zH production, highlighting the need for finite-mass corrections, higher-order calculations, and complete scale uncertainties.

Overview

This paper presents the first phenomenological study of $\Upsilon(1S)$, $\Upsilon(2S)$, and $\Upsilon(3S)$ production inside jets at the LHC within the fragmenting jet function (FJF) framework (2608.18922). The calculation combines next-to-leading-order (NLO) NRQCD fragmentation functions with timelike DGLAP evolution and threshold resummation of the short-distance coefficients, and incorporates feeddown from higher bottomonium states. The central finding is a sharp contrast with the charmonium sector: while quarkonium-in-jet momentum-fraction ($z_H$) distributions discriminate effectively among competing nonrelativistic QCD (NRQCD) long-distance matrix element (LDME) sets for charmonium, the bottomonium $z_H$ spectra are nearly universal in shape across four substantially different LDME sets. This universality is traced to the dominance of the S-wave spin-triplet color-octet channel ${}^3S_1^{[8]}$, reinforced by $\chi_b$ feeddown that is itself dominated by the same mechanism.

Theoretical framework

The analysis uses the semi-inclusive FJF formalism in Soft-Collinear Effective Theory (SCET), factorizing the cross section for $pp\to(\mathrm{jet}\,H)+X$ into perturbatively calculable parton production cross sections convolved with FJFs $\mathcal{G}_i^H(z,z_H,p_TR,\mu)$, where $z = p_T/p_T^i$ and $\Upsilon(2S)$0. Three scales enter: the hard scale $\Upsilon(2S)$1, the jet scale $\Upsilon(2S)$2, and the NRQCD matching scale $\Upsilon(2S)$3. The FJFs are matched onto NRQCD FFs through perturbative jet functions, retaining the leading color-singlet and color-octet channels $\Upsilon(2S)$4, $\Upsilon(2S)$5, $\Upsilon(2S)$6, and $\Upsilon(2S)$7.

The FFs are evolved from $\Upsilon(2S)$8 to $\Upsilon(2S)$9 and then to $\Upsilon(3S)$0 via timelike DGLAP evolution at leading-logarithmic accuracy, resumming collinear logarithms associated with the hierarchy $\Upsilon(3S)$1. In addition, threshold logarithms arising as $\Upsilon(3S)$2 are resummed in Mellin space for the gluon-induced contributions to the $\Upsilon(3S)$3 and $\Upsilon(3S)$4 channels and the $\Upsilon(3S)$5 channels relevant for $\Upsilon(3S)$6 feeddown. An appendix comparison shows that resummation shifts predictions by roughly 20–30% over intermediate $\Upsilon(3S)$7, up to about 50% in some bins, and regularizes the fixed-order threshold enhancement near $\Upsilon(3S)$8; the effect is modest but controlled rather than signaling perturbative instability.

A limitation worth noting: the FJF is derived in the massless-parton limit, so finite heavy-quark mass corrections are absent — an approximation whose consequences become visible in the comparison with data discussed below.

LDME classification and feeddown

The paper classifies available bottomonium LDME sets into four categories based on the relative signs of $\Upsilon(3S)$9 and $z_H$0 (which determine constructive versus destructive interference, since the $z_H$1 short-distance coefficient is negative over most of the relevant $z_H$2 range) and the size of $z_H$3: Brambilla et al. (Category 1, same-sign octet channels, small $z_H$4), Gong et al. (Category 2, negative $z_H$5, large $z_H$6), Feng et al. (Category 3, opposite signs), and Han et al. (Category 4, vanishing $z_H$7, maximal $z_H$8). The Gong set, originally extracted at $z_H$9, is evolved to the common scale $z_H$0 using the fixed-order NRQCD renormalization-group equation, which mixes $z_H$1 with $z_H$2 (and analogously $z_H$3 with $z_H$4 for $z_H$5). This mixing is itself part of the explanation for why the $z_H$6 contribution remains more prominent in bottomonium than in charmonium, where large cancellations occur.

Feeddown is treated comprehensively: $z_H$7 radiative transitions and $z_H$8, $z_H$9 cascades, with branching fractions from the Particle Data Group and, for ${}^3S_1^{[8]}$0, from theoretical predictions. Because the daughter state inherits only a fraction of the parent momentum, feeddown shifts weight toward smaller ${}^3S_1^{[8]}$1 and broadens the spectrum, most strongly for ${}^3S_1^{[8]}$2. A practical caveat is that no Category 2 LDME set exists for the excited states, and the Gong-based ${}^3S_1^{[8]}$3 feeddown is omitted because those states were not included in that fit.

Numerical results and comparison with CMS

The central phenomenological result concerns inclusive production in the CMS fiducial region (${}^3S_1^{[8]}$4 TeV, anti-${}^3S_1^{[8]}$5 jets with ${}^3S_1^{[8]}$6, ${}^3S_1^{[8]}$7, muon-level cuts applied at particle level). Although direct-production channel compositions differ markedly among LDME sets — Brambilla shows strong ${}^3S_1^{[8]}$8–${}^3S_1^{[8]}$9 cancellation, Gong has a negative $\chi_b$0 yielding a negative direct contribution near $\chi_b$1, Feng exhibits constructive interference, and Han is purely $\chi_b$2-dominated — the inclusive spectra converge to a universal shape. Two mechanisms produce this convergence: the $\chi_b$3 gluon fragmentation coefficient carries a threshold-enhanced $\chi_b$4 structure that dominates the large-$\chi_b$5 peak, and the $\chi_b$6 feeddown, which is comparable to or larger than the direct component over much of the $\chi_b$7 range for $\chi_b$8, is itself dominated by $\chi_b$9 because the $pp\to(\mathrm{jet}\,H)+X$0 contribution is strongly suppressed after summing over the multiplet.

The predictions reproduce the main features of the CMS measurements for all three states: the characteristic large-$pp\to(\mathrm{jet}\,H)+X$1 peak and its systematic broadening toward smaller $pp\to(\mathrm{jet}\,H)+X$2 with increasing jet transverse momentum, driven by DGLAP evolution. Notably, the bottomonium threshold structure survives over the CMS kinematic range, whereas for charmonium it is washed out already at moderate $pp\to(\mathrm{jet}\,H)+X$3 — a mass-dependent difference in evolution strength that the authors identify as physically meaningful. However, a clear discrepancy remains: the predicted low-$pp\to(\mathrm{jet}\,H)+X$4 enhancement is systematically larger than observed by CMS, even after fiducial acceptance corrections. The authors state plainly that the origin is not understood within the current framework, listing finite-mass corrections to the FJF and missing higher-order perturbative corrections as candidate sources. Uncertainty bands include only LDME uncertainties; scale variations are deliberately excluded to isolate LDME sensitivity, which means the quoted agreement does not constitute a full uncertainty assessment.

Additional results strengthen the physical picture. At fixed quarkonium transverse momentum ($pp\to(\mathrm{jet}\,H)+X$5 GeV), all LDME sets again yield similar shapes but differ visibly in absolute normalization, with the ordering varying across the three states (Brambilla largest for $pp\to(\mathrm{jet}\,H)+X$6, Feng largest near the peak for $pp\to(\mathrm{jet}\,H)+X$7). Jet-radius dependence is pronounced: reducing $pp\to(\mathrm{jet}\,H)+X$8 from 0.8 to 0.4 enhances the threshold region and hardens the spectrum, since less radiation is clustered into the jet. This indicates that quarkonium-in-jet observables probe not only fragmentation dynamics but also the jet definition, motivating multi-radius measurements. Predictions for LHCb kinematics ($pp\to(\mathrm{jet}\,H)+X$9, $\mathcal{G}_i^H(z,z_H,p_TR,\mu)$0) show an even sharper large-$\mathcal{G}_i^H(z,z_H,p_TR,\mu)$1 peak due to reduced fragmentation evolution at lower $\mathcal{G}_i^H(z,z_H,p_TR,\mu)$2.

Limitations and open questions

Several limitations qualify the conclusions. The massless-FJF approximation is the most concrete: the unexplained excess at low $\mathcal{G}_i^H(z,z_H,p_TR,\mu)$3 relative to CMS may plausibly originate there, but this remains unresolved. The DGLAP resummation is performed only at LL accuracy, and higher-order semi-inclusive jet function corrections to the evolution are known to exist but are not included. NNLO corrections to the $\mathcal{G}_i^H(z,z_H,p_TR,\mu)$4 channel have recently become available but are not incorporated, so NLO accuracy is maintained uniformly across channels at the cost of omitting these improvements. Scale-variation uncertainties are omitted from the data comparison entirely. Finally, whether the universality observed here persists in TMD-sensitive observables — such as the quarkonium transverse momentum relative to the jet axis, which CMS also measures — is left open, as is the quantitative impact of finite-mass effects at lower transverse momenta.

Conclusion

This work establishes that bottomonium-in-jet $\mathcal{G}_i^H(z,z_H,p_TR,\mu)$5 distributions behave qualitatively differently from their charmonium counterparts: despite substantial differences among LDME sets, the inclusive $\mathcal{G}_i^H(z,z_H,p_TR,\mu)$6 spectra exhibit a universal shape governed by $\mathcal{G}_i^H(z,z_H,p_TR,\mu)$7 fragmentation, reinforced by $\mathcal{G}_i^H(z,z_H,p_TR,\mu)$8 feeddown dominated by the same channel. Consequently, the discriminating power that quarkonium-in-jet observables possess in the charmonium sector largely disappears in the bottomonium sector, where instead the observables serve as a consistency test of the $\mathcal{G}_i^H(z,z_H,p_TR,\mu)$9-dominated fragmentation picture. The resummed NLO FJF framework describes the CMS-measured peak structure and its $z = p_T/p_T^i$0-dependent broadening, while leaving the low-$z = p_T/p_T^i$1 excess as a concrete open problem tied to finite-mass and higher-order effects.