Triply Heavy $Ω$ Baryons with JETHAD: A High-Energy Viewpoint

Abstract: We investigate the leading-power fragmentation of triply heavy $Ω$ baryons in high-energy hadronic collisions. Extending our previous work on the $Ω{3c}$ sector, we release the full OMG3Q1.0 family of collinear fragmentation functions by completing the description of the charm channel and delivering the novel $Ω{3b}$ functions. These hadron-structure-oriented functions are constructed from improved proxy-model calculations for heavy-quark and gluon fragmentation, matched to a flavor-aware DGLAP evolution based on the HF-NRevo scheme. For phenomenological applications, we employ the (sym)JETHAD multimodular interface to compute and analyze NLL/NLO$^+$ semi-inclusive $Ω_{3Q}$ plus jet distributions at the HL-LHC and FCC. This work consolidates the link between hadron structure, rare baryon production, and resummed QCD at the energy frontier.

• The paper presents a diquark-inspired fragmentation approach for triply heavy Ω baryon production using full NLO calculations for both charm and bottom channels.
• It employs a hybrid high-energy factorization method that merges collinear PDFs/FFs with BFKL resummation to achieve stable predictions.
• Phenomenological results predict observable cross sections at HL-LHC and FCC with narrow uncertainty bands, highlighting robust theoretical control.

Triply Heavy $\Omega$ Baryons at High Energy: Structure, Fragmentation, and Collider Phenomenology

Introduction and Motivation

The study "Triply Heavy $\Omega$ Baryons with JETHAD: A High-Energy Viewpoint" (2604.01871) provides a comprehensive analysis of triply heavy baryons, focusing primarily on the $\Omega_{3c}$ ($ccc$) and $\Omega_{3b}$ ($bbb$) states, in the context of high-energy hadronic collisions. The work addresses the construction of variable-flavor number scheme (VFNS) fragmentation functions (FFs), their DGLAP evolution, and the impact of high-energy logarithmic resummation on phenomenological predictions at the HL-LHC and FCC. The theoretical framework combines a diquark-inspired proxy model for FFs, a detailed factorization approach, and a state-of-the-art computational infrastructure (JETHAD).

Fragmentation Functions for Triply Heavy Baryons

Theoretical Construction

Triply heavy baryon FFs, denoted as {\tt OMG3Q1.0}, are formulated within a diquark proxy model that captures the dominant nonperturbative dynamics of baryon formation. The model utilizes a two-step mechanism, where a constituent heavy quark fragments into a color-antitriplet diquark, which then hadronizes nonperturbatively into the final-state baryon. This theoretical construction allows for a consistent treatment of both quark- and gluon-initiated fragmentation channels:

• Quark channel: Leading-order and next-to-leading order SDCs are evaluated, combining perturbative kinematics with a phenomenological hadronization function (Peterson-type).
• Gluon channel: Initiated at higher perturbative order, the gluon fragmentation involves the formation of three heavy quarks via a $gg \to QQQ$ topology and is implemented at NLO.

The analytic structure of these FFs is governed by the convolution of partonic SDCs with the hadronization function. The hadronization parameter $\tau_P$ is fixed following scaling arguments for multi-heavy systems.

Figure 1: Representative leading-order diagrams for the diquarklike proxy model describing the initial-scale collinear fragmentation of a constituent heavy quark (left) and a gluon (right) into a triply heavy $\Omega_{3Q}$ baryon.

DGLAP Evolution and Heavy-Flavor Thresholds

The initial-scale FFs are evolved to collider-relevant scales using the {\tt HF-NRevo} approach. This implements DGLAP evolution with explicit accounting for threshold effects and variable matching scales for the initiation of charm and gluon channels. The resulting {\tt OMG3Q1.0} grids are tabulated in {\tt LHAPDF6} format, optimized for direct phenomenological application.

Figure 2: Factorization-scale dependence of the {\tt OMG3Q1.0} NLO FFs for $\Omega_{3c}$ (left) and $\Omega$0 (right) at fixed $\Omega$1.

A key numerical result is the dominance of the constituent-heavy-quark FF over the gluon (and dynamically generated nonconstituent quark) channels throughout the relevant $\Omega$2 range, with gluon contributions suppressed by two or more orders of magnitude.

High-Energy Hybrid Factorization and Resummation

Formalism

For single and multi-differential cross section calculations involving semi-inclusive $\Omega$3 plus jet final states, the paper employs a hybrid factorization scheme. This formalism consistently merges collinear (DGLAP) and high-energy (BFKL) resummation:

• Impact Factors: The observable is factorized into proton PDFs, evolved fragmentation functions for $\Omega$4 production, and jet-energy/kinematic constraints.
• BFKL Kernel: The inclusion of high-energy LL and NLL energy logarithms is performed via the BFKL evolution kernel, with selected NLL$\Omega$5 terms.

  Figure 3: Schematic of the hybrid collinear/BFKL factorization for semi-inclusive $\Omega$6 plus jet production.

The factorization scale is set to the natural kinematic value $\Omega$7, enabling control of MHOUs through scale variation.

Phenomenological Predictions: HL-LHC and FCC Sensitivity

Rapidity-Differential Distributions

The $\Omega$8 distributions for $\Omega$9jet and $\Omega_{3c}$0jet production are computed for HL-LHC ($\Omega_{3c}$1 TeV) and FCC ($\Omega_{3c}$2 TeV), for $\Omega_{3c}$3.

Figure 4: Rapidity-differential distribution of semi-inclusive $\Omega_{3c}$4 plus jet at HL-LHC (left) and FCC (right); shaded bands represent MHOU uncertainties.

Figure 5: Rapidity-differential distribution for semi-inclusive 4b plus jet production, analogous cuts.

Strong enhancement—an order of magnitude or more—is observed in the cross section when moving from HL-LHC to FCC kinematics. The $\Omega_{3c}$5 cross section dominates over the tetraquark channel by 2–3 orders of magnitude, especially at large rapidity separations. The behavior is monotonic in $\Omega_{3c}$6, with high-energy resummation stabilizing theoretical predictions and reducing scale uncertainties compared to fixed-order.

Transverse-Momentum-Differential Observables

Differential distributions in $\Omega_{3c}$7 for $\Omega_{3c}$8 exhibit smooth monotonic decrease and persistently stable theoretical uncertainties across the explored phase space:

Figure 6: Transverse-momentum-differential spectra for $\Omega_{3c}$9jet at HL-LHC and FCC; top and bottom panels correspond to $ccc$0 and $ccc$1.

Resummation effects become more pronounced at high $ccc$2 as deviations from DGLAP fixed-order approximations grow, particularly at FCC energies.

Implications and Outlook

The comprehensive analysis shows that triply heavy $ccc$3 baryons—most notably $ccc$4—are robust probes for both the nonperturbative structure of cold QCD matter and the dynamics of the high-energy regime at future hadron colliders. The simultaneous inclusion of heavy-quark and gluon fragmentation channels allows for the examination of potential intrinsic heavy-flavor content and the assessment of competing color and threshold dynamics. The demonstrated "natural stability" of high-energy resummation when connected to VFNS-based FFs is a significant technical result, with the size of theoretical uncertainties under control even when varying MHOU-relevant scales.

Practically, the $ccc$5 channel will be pivotal for future experimental searches and could serve as an anchor for validating FF determinations against global fits, especially as heavy-flavor tagging and vertexing improve at the FCC. The framework and FF sets ({\tt OMG3Q1.0}) are provided in public repositories for future collider phenomenology studies.

Conclusion

This work establishes the theoretical and phenomenological framework for the production of triply heavy baryons in high-energy collisions. By constructing and evolving diquark-inspired, hadron-structure-sensitive fragmentation functions and embedding them in a hybrid high-energy/collinear resummed factorization scheme, the paper achieves precision predictions across LHC and FCC relevant kinematic domains. The strong sensitivity of semi-inclusive $ccc$6 observables to both perturbative and nonperturbative effects, together with the demonstrated numerical stability and the practical availability of FF grids, position these channels as benchmark probes for future experimental and theoretical QCD programs.