# Hadronization
**Hadronization** is the non-perturbative process in quantum chromodynamics (QCD) by which quarks and gluons (partons) produced in high-energy collisions combine into the color-neutral hadrons observed in detectors. The related term *fragmentation* is more specific, denoting the breakup of a partonic system into a set of primary hadrons, while hadronization is more generic and can also include decays of short-lived resonances[1](https://pmc.ncbi.nlm.nih.gov/articles/PMC6394252/). Because the relevant momentum transfers are small, the strong coupling is large and perturbation theory cannot describe the transition; instead, it is modeled by phenomenological schemes such as string fragmentation, cluster decay, and [quark coalescence](https://www.edgepedia.org/quark-coalescence), implemented in [Monte Carlo](https://www.edgepedia.org/monte-carlo) event generators[2](https://link.springer.com/article/10.1140/epjc/s10052-024-13641-5) • [3](https://research.monash.edu/en/publications/general-purpose-event-generators-for-lhc-physics/).
| Key fact | Detail |
|---|---|
| Physical setting | Non-perturbative QCD: the parton-to-hadron transition occurs at momentum scales where the strong coupling is too large for perturbative calculation[2](https://link.springer.com/article/10.1140/epjc/s10052-024-13641-5) |
| String model | PYTHIA implements the Lund string fragmentation model, with a constant string tension \( \kappa \sim 1 \) GeV/fm motivated by the linear confinement potential seen on the lattice[2](https://link.springer.com/article/10.1140/epjc/s10052-024-13641-5) |
| String breaking | Breaks follow a Schwinger-type tunneling probability \( \exp(-\pi m_{\perp q}^{2}/\kappa) \); producing charm quarks by string breaking is suppressed, \( \exp(-\pi m_{\mathrm{c}}^{2}/\kappa) \lesssim 10^{-11} \)[2](https://link.springer.com/article/10.1140/epjc/s10052-024-13641-5) |
| Cluster model | HERWIG uses the cluster model based on color pre-confinement: gluons split non-perturbatively into \( \mathrm{q}\bar{\mathrm{q}} \) pairs that form clusters, which fission and decay into hadrons[2](https://link.springer.com/article/10.1140/epjc/s10052-024-13641-5) |
| Heavy-ion regime | In the quark-gluon plasma formed in nucleus-nucleus collisions, hadronization proceeds near the crossover temperature \( T_{\mathrm{c}} \sim 155 \) MeV from lattice QCD, and quark coalescence (2→1 for mesons, 3→1 for baryons) becomes a dominant mechanism[4](https://cds.cern.ch/record/2838475/files/2211.04384.pdf?version=1) • [2](https://link.springer.com/article/10.1140/epjc/s10052-024-13641-5) |
| Coalescence signatures | Enhancement of baryon-to-meson ratios and quark-number scaling of elliptic flow, \( v_{2}/n_{\mathrm{q}} \) versus \( p_{\mathrm{T}}/n_{\mathrm{q}} \) with \( n_{\mathrm{q}} = 2 \) for mesons and 3 for baryons[2](https://link.springer.com/article/10.1140/epjc/s10052-024-13641-5) |
| Universality broken | LHC measurements show heavy-flavor fragmentation fractions in pp collisions differ significantly from those in e⁺e⁻ and ep collisions, so hadronization cannot be modeled with universal fragmentation functions alone[2](https://link.springer.com/article/10.1140/epjc/s10052-024-13641-5) |
## Why hadronization is non-perturbative
QCD calculations based on factorization separate what can be computed perturbatively from what cannot. In the factorization approach, heavy-flavor hadron production cross sections are built from parton distribution functions, a perturbative partonic cross section in powers of \( \alpha_{\mathrm{s}} \), and fragmentation functions, which parameterize the probability that a quark gives rise to a hadron carrying a fraction \( z_{\mathrm{Q}} \) of the parent momentum[2](https://link.springer.com/article/10.1140/epjc/s10052-024-13641-5). [Fragmentation functions](https://www.edgepedia.org/fragmentation-functions) for light charged (\( \pi^{\pm} \), \( K^{\pm} \), and \( p/\bar{p} \)) and neutral (\( K_{S}^{0} \) and \( \Lambda/\bar{\Lambda} \)) hadrons are extracted through global fits to data, particularly accurate e⁺e⁻ reactions at LEP, and then applied to HERA, Tevatron, and RHIC data through factorization universality[5](https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.82.2489). The dynamical process behind these fitted functions must be modeled phenomenologically.
## String fragmentation
**The Lund string model** pictures the collapsed color confinement field between a color charge and an anticolor charge as a relativistic string with constant tension \( \kappa \sim 1 \) GeV/fm, a choice motivated by the linear potential observed at non-perturbative scales on the lattice and encapsulated in the QCD Cornell potential[2](https://link.springer.com/article/10.1140/epjc/s10052-024-13641-5). As color-connected partons move apart, the string stretches until breaking becomes energetically favorable; at each break a light-flavored quark-antiquark or diquark-antidiquark pair is produced, splitting the string into smaller systems until on-shell hadrons remain[2](https://link.springer.com/article/10.1140/epjc/s10052-024-13641-5).
The break probability follows a Schwinger-type tunneling formula,
\[ \exp\left(\frac{-\pi m_{q}^{2}}{\kappa}\right) \exp\left(\frac{-\pi p_{\perp q}^{2}}{\kappa}\right) = \exp\left(\frac{-\pi m_{\perp q}^{2}}{\kappa}\right), \]
where \( m_{\perp}^{2} = m^{2} + p_{\perp}^{2} \) is the transverse mass[2](https://link.springer.com/article/10.1140/epjc/s10052-024-13641-5). The Gaussian form strongly suppresses breaks with high transverse momentum and heavier quarks: charm and heavier flavors cannot be produced by string breaking, since \( \exp(-\pi m_{\mathrm{c}}^{2}/\kappa) \lesssim 10^{-11} \), so heavy quarks arise only from perturbative processes[2](https://link.springer.com/article/10.1140/epjc/s10052-024-13641-5). The longitudinal energy sharing is governed by a left-right symmetric fragmentation function \( f(z) = N \, z^{-1}(1-z)^{a} \exp(-b m_{\perp}^{2}/z) \), whose parameters \( a \) and \( b \) are tuned to data[2](https://link.springer.com/article/10.1140/epjc/s10052-024-13641-5). For a heavy-quark endpoint, the string drags the quark along a hyperbola rather than a straight line, reducing the area available for breaks; the Bowler modification accounts for this with an additional factor \( 1/z^{r_{Q} b m_{Q}^{2}} \) in the fragmentation function[2](https://link.springer.com/article/10.1140/epjc/s10052-024-13641-5).
The Lund model also carries a space-time structure: linear confinement makes the energy-momentum and space-time pictures of a simple \( \mathrm{q}\bar{\mathrm{q}} \) system proportional, and a full space-time picture of hadronization has been developed for generic multiparton systems as part of the Pythia event generator[1](https://pmc.ncbi.nlm.nih.gov/articles/PMC6394252/). Extensions such as rope formation, shoving, and an environment-dependent string tension introduce string interactions that have been proposed as a microscopic source of the collective behavior observed even in small collision systems[1](https://pmc.ncbi.nlm.nih.gov/articles/PMC6394252/) • [6](https://doi.org/10.3390/universe10010046).
## Cluster fragmentation
**The cluster model**, used in HERWIG, rests on color pre-confinement. After the parton shower, gluons split non-perturbatively, \( g \rightarrow \mathrm{q}\bar{\mathrm{q}} \), and each color-singlet \( \mathrm{q}\bar{\mathrm{q}} \) pair forms a cluster[2](https://link.springer.com/article/10.1140/epjc/s10052-024-13641-5). Clusters heavier than a threshold undergo recursive fission, in which a light \( \mathrm{q}\bar{\mathrm{q}} \) pair is created from the vacuum and the daughter-cluster masses are sampled as \( M_{1,2} = m_{1,2} + (M - m_{1,2} - m_{q})\,\mathcal{R}^{1/P} \) with a random number \( \mathcal{R} \) and a splitting parameter \( P \); no diquark formation is permitted in fission[2](https://link.springer.com/article/10.1140/epjc/s10052-024-13641-5). Each remaining cluster then decays isotropically into two hadrons, with both quark-antiquark and diquark-antidiquark formation allowed, so mesonic clusters can yield either two mesons or a baryon-antibaryon pair[2](https://link.springer.com/article/10.1140/epjc/s10052-024-13641-5).
## Coalescence in heavy-ion collisions
In nucleus-nucleus collisions a color-deconfined quark-gluon plasma ([QGP](https://www.edgepedia.org/quark-gluon-plasma)) forms, and hadronization happens in a hot, dense medium full of thermal light partons with which a produced quark can recombine[2](https://link.springer.com/article/10.1140/epjc/s10052-024-13641-5). [Lattice QCD](https://www.edgepedia.org/lattice-qcd) places the confined-to-deconfined transition at a crossover with \( T_{\mathrm{c}} \sim 155 \) MeV, so hadronization occurs near this temperature[4](https://cds.cern.ch/record/2838475/files/2211.04384.pdf?version=1). The recombination process, modeled as a 2→1 coalescence for mesons or 3→1 for baryons, is crucial for reproducing shifts in baryon-to-meson ratios and the collective motion of final-state hadrons[2](https://link.springer.com/article/10.1140/epjc/s10052-024-13641-5). Coalescence models for hadron formation from the QGP were developed and compared against abundant experimental evidence from the [Relativistic Heavy Ion Collider](https://www.edgepedia.org/relativistic-heavy-ion-collider) (RHIC)[7](https://www.annualreviews.org/content/journals/10.1146/annurev.nucl.58.110707.171134).
Coalescence differs from fragmentation in its momentum structure: in fragmentation the final hadron always carries less energy than the parent quark, whereas coalescing partons combine their transverse momenta, pushing the average hadron \( p_{\mathrm{T}} \) to larger values[2](https://link.springer.com/article/10.1140/epjc/s10052-024-13641-5). A distinctive prediction is quark-number scaling of elliptic flow, in which \( v_{2}(p_{\mathrm{T}}) \) collapses onto a universal curve \( (1/n_{\mathrm{q}})\,v_{2}(p_{\mathrm{T}}/n_{\mathrm{q}}) \), with moderate breaking expected at low \( p_{\mathrm{T}} \) from the finite hadron wave-function width, resonance feed-down, and radial flow[2](https://link.springer.com/article/10.1140/epjc/s10052-024-13641-5). In a radially expanding medium, recombination also tends to equilibrate particle velocities toward the medium expansion velocity, and recombination is an essential ingredient for reproducing measured D-meson \( R_{\mathrm{AA}} \) and \( v_{2} \)[2](https://link.springer.com/article/10.1140/epjc/s10052-024-13641-5).
## What LHC data changed
The standard expectation had been that only the large parton reservoir and high energy density of nucleus-nucleus collisions would make coalescence relevant, so baryon-to-meson enhancement was not expected in pp collisions, which were thought to resemble e⁺e⁻ collisions where in-vacuum fragmentation dominates[2](https://link.springer.com/article/10.1140/epjc/s10052-024-13641-5). LHC heavy-flavor baryon measurements contradicted this: fragmentation fractions of charm and beauty quarks into baryons differ significantly from those measured in e⁺e⁻ and ep collisions[2](https://link.springer.com/article/10.1140/epjc/s10052-024-13641-5). In coalescence-plus-fragmentation approaches such as the Catania model, D-meson production in pp collisions remains dominated by fragmentation while coalescence dominates baryon production, explaining the large measured \( \Lambda_{\mathrm{c}}^{+}/\mathrm{D}^{0} \) ratio; in AA collisions the dense quark reservoir makes recombination dominant for both heavy mesons and baryons[2](https://link.springer.com/article/10.1140/epjc/s10052-024-13641-5). The same baryon enhancement can also be reproduced within PYTHIA through color reconnections in topologies that emerge with increasing numbers of multi-parton interactions[2](https://link.springer.com/article/10.1140/epjc/s10052-024-13641-5).
## Hadronization in event generators
General-purpose Monte Carlo generators for LHC physics, including Ariadne, Herwig++, Pythia 8, and Sherpa, translate parton-level predictions into hadron-level events using the string and cluster models for hadron formation, together with models of the underlying event and hadron decays[3](https://research.monash.edu/en/publications/general-purpose-event-generators-for-lhc-physics/). Model parameters are tuned to experimental data, and systematic variations of the hadronization model are among the uncertainty variations that are not well suited to event reweighting, motivating machine-learning approaches to reweighting between samples for High-Luminosity LHC computing[8](https://link.springer.com/article/10.1007/s41781-021-00055-1).
## References
1. [The space–time structure of hadronization in the Lund model, Eur. Phys. J. C (via PMC)](https://pmc.ncbi.nlm.nih.gov/articles/PMC6394252/)
2. [Towards the understanding of heavy quarks hadronization: from leptonic to heavy-ion collisions, Eur. Phys. J. C (2024)](https://link.springer.com/article/10.1140/epjc/s10052-024-13641-5)
3. [General-purpose event generators for LHC physics, Physics Reports 504, 145 (2011)](https://research.monash.edu/en/publications/general-purpose-event-generators-for-lhc-physics/)
4. [CERN-EP-2022-227 (QCD transition at Tc ∼ 155 MeV)](https://cds.cern.ch/record/2838475/files/2211.04384.pdf?version=1)
5. [Hadronization of partons, Rev. Mod. Phys. 82, 2489 (2010)](https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.82.2489)
6. [String Interactions as a Source of Collective Behaviour, Universe 10, 46 (2024)](https://doi.org/10.3390/universe10010046)
7. [Coalescence Models for Hadron Formation from Quark-Gluon Plasma, Annu. Rev. Nucl. Part. Sci. 58, 177 (2008)](https://www.annualreviews.org/content/journals/10.1146/annurev.nucl.58.110707.171134)
8. [Challenges in Monte Carlo Event Generator Software for High-Luminosity LHC, EPJ Research Infrastructures (2021)](https://link.springer.com/article/10.1007/s41781-021-00055-1)
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