Fragmentation of fully heavy tetraquarks: The TQ4Q1.1 functions as a case study

TL;DR

This work develops the TQ4Q1.1 collinear fragmentation functions for fully heavy tetraquarks in the $0^{++}$, $1^{+-}$, and $2^{++}$ channels for charm and bottom flavors, using NRQCD short-distance coefficients and color-composite LDMEs within a heavy-flavor VFNS. It evolves these inputs with a two-step DGLAP procedure that accounts for heavy-quark and gluon thresholds via the HF-NRevo scheme and propagates nonperturbative LDME uncertainties into the FFs. The authors provide ${ m NLL/NLO^+}$ predictions for tetraquark–jet production at HL-LHC and FCC within the HyF framework, including rapidity-interval distributions and novel angular-multiplicity observables, plus realistic event-yield estimates. They identify the axial-vector channel as a particularly clean and stable probe of high-energy QCD dynamics and demonstrate broad feasibility for discovering or constraining fully heavy tetraquarks at future colliders, thereby connecting exotic hadron spectroscopy with precision QCD and high-energy phenomenology.

Abstract

We extend the study of exotic matter formation via the TQ4Q1.1 set of collinear, variable-flavor-number-scheme fragmentation functions for fully charmed or bottomed tetraquarks in three quantum configurations: scalar ($J^{PC} = 0^{++}$), axial vector ($J^{PC} = 1^{+-}$), and tensor ($J^{PC} = 2^{++}$). We adopt single-parton fragmentation at leading power and implement a nonrelativistic Quantum Chromodynamics (NRQCD) factorization scheme tailored to tetraquark Fock-state configurations. Short-distance inputs at the initial scale are modeled using updated calculations for both gluon- and heavy-quark-initiated channels. A threshold-consistent Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution is then applied via the novel Heavy-flavor nonrelativistic-evolution (HF-NRevo) hybrid scheme. We provide the first systematic treatment of uncertainties from nonperturbative color-composite long-distance matrix elements (LDMEs), as well as from perturbative hard-scattering (H-MHOUs) and fragmentation-scale inputs (F-MHOUs), assessed separately and in combination. To support phenomenology, we compute NLL/NLO$^+$ cross sections for tetraquark-jet systems at the HL-LHC and FCC within the hybrid collinear and high-energy factorization (HyF) as implemented in (sym)JETHAD, incorporating angular multiplicities as key observables sensitive to high-energy QCD dynamics. We also provide expected event yields based on realistic luminosity scenarios, offering a concrete benchmark for experimental searches. This work connects the investigation of exotic hadrons with state-of-the-art precision QCD.

Figures (11)

• Figure 1: LO diagrams illustrating the collinear fragmentation of a gluon (left) and a heavy quark (right) into a fully heavy $S$-wave tetraquark in a color-singlet configuration. Parton interactions on the left-hand side of each diagram depict the perturbative short-distance coefficients (SDCs), while red blobs on the right-hand side portray the nonperturbative long-distance matrix elements (LDMEs) describing the final hadronization step.
• Figure 2: $z$-shape of the TQ4Q1.1 FFs for scalar tetraquarks $T_{4c}(0^{++})$ (left) and $T_{4b}(0^{++})$ (right), evaluated at various energy scales. Upper (lower) plots refer to the heavy-quark (gluon) initiated channels. Filled bands in the main panels represent the combined effect of uncertainties from perturbative F-MHOUs and nonperturbative LDME variations. The first lower panel shows the impact of perturbative F-MHOUs (replica envelope normalized to the central one), while the second displays LDME variations as ratios to the central value.
• Figure 3: $z$-shape of the TQ4Q1.1 FFs for axial-vector tetraquarks $T_{4c}(1^{+-})$ (left) and $T_{4b}(1^{+-})$ (right), evaluated at various energy scales. Upper (lower) plots refer to the heavy-quark (gluon) initiated channels. Filled bands in the main panels represent the combined effect of uncertainties from perturbative F-MHOUs and nonperturbative LDME variations. The first lower panel shows the impact of perturbative F-MHOUs (replica envelope normalized to the central one), while the second displays LDME variations as ratios to the central value.
• Figure 4: $z$-shape of the TQ4Q1.1 FFs for tensor tetraquarks $T_{4c}(2^{++})$ (left) and $T_{4b}(2^{++})$ (right), evaluated at various energy scales. Upper (lower) plots refer to the heavy-quark (gluon) initiated channels. Filled bands in the main panels represent the combined effect of uncertainties from perturbative F-MHOUs and nonperturbative LDME variations. The first lower panel shows the impact of perturbative F-MHOUs (replica envelope normalized to the central one), while the second displays LDME variations as ratios to the central value.
• Figure 5: Energy shape of the TQ4Q1.1 FFs for scalar (upper), axial-vector (central), and tensor tetraquarks (lower), evaluated at $z = \langle z \rangle \simeq 0.5$. Left (right) panels are for fully charmed (bottomed) states. Ratios between the new TQ4Q1.1 functions (this work) and the previous TQ4Q1.0 sets Celiberto:2024mab are shown in the ancillary panels for scalar states.