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

We extend the study of exotic matter formation via the set of collinear, variable-flavor-number-scheme fragmentation functions for fully charmed or bottomed tetraquarks in three quantum configurations; scalar ( <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mrow> <a:msup> <a:mrow> <a:mi>J</a:mi> </a:mrow> <a:mrow> <a:mi>P</a:mi> <a:mi>C</a:mi> </a:mrow> </a:msup> <a:mo>=</a:mo> <a:msup> <a:mrow> <a:mn>0</a:mn> </a:mrow> <a:mrow> <a:mo>+</a:mo> <a:mo>+</a:mo> </a:mrow> </a:msup> </a:mrow> </a:math> ), axial vector ( <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:msup> <c:mi>J</c:mi> <c:mrow> <c:mi>P</c:mi> <c:mi>C</c:mi> </c:mrow> </c:msup> <c:mo>=</c:mo> <c:msup> <c:mn>1</c:mn> <c:mrow> <c:mo>+</c:mo> <c:mo>−</c:mo> </c:mrow> </c:msup> </c:math> ), and tensor ( <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:msup> <e:mi>J</e:mi> <e:mrow> <e:mi>P</e:mi> <e:mi>C</e:mi> </e:mrow> </e:msup> <e:mo>=</e:mo> <e:msup> <e:mn>2</e:mn> <e:mrow> <e:mo>+</e:mo> <e:mo>+</e:mo> </e:mrow> </e:msup> </e:math> ). 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 () 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 <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mi>NLL</g:mi> <g:mo>/</g:mo> <g:msup> <g:mrow> <g:mi>NLO</g:mi> </g:mrow> <g:mo>+</g:mo> </g:msup> </g:math> cross sections for tetraquark-jet systems at the HL-LHC and FCC within the hybrid collinear and high-energy factorization (HyF) as implemented in , 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.