The $b\bar{b}c\bar{c}$ Fully Heavy Tetraquark at the LHC and the FCC: Multimodal Fragmentation and Observability
We investigate the production of the all-heavy mixed-flavor $b\bar b c\bar c$ tetraquark through a multimodal fragmentation setup combining direct-multiquark and scalar-diquark mechanisms. Standard Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution is applied while preserving threshold consistency through the Heavy-Flavor NonRelativistic evolution (HF-NRevo) scheme. The resulting fragmentation functions (FFs) are then embedded into hybrid high-energy and collinear factorization at next-to-leading order for semi-inclusive tetraquark-plus-jet production. We compare the two mechanisms through rapidity-separation distributions at the 14 TeV High-Luminosity LHC (HL-LHC) and at the 100 TeV Future Circular Collider (FCC), finding a clear phenomenological preference for the direct-multiquark channel over the scalar-diquark mode. Integrated cross sections and expected event yields provide favorable production-level prospects for future searches at the HL-LHC, with substantially enhanced statistics at FCC energies. These results establish a quantitative connection between microscopic formation mechanisms and the prospective observability of the $b\bar b c\bar c$ tetraquark at present and future hadron colliders.