AMPT: A Multi-Phase Transport Model

• The model provides a multi-phase framework covering fluctuating initial conditions, a parton cascade via ZPC, and dual-mode hadronization.
• AMPT employs both default string fragmentation and string melting with quark coalescence to study particle production and flow.
• It delivers practical insights into collective flow, particle correlations, and heavy-flavor production across RHIC and LHC energies.

A Multi-Phase Transport Model (AMPT) is a self-contained phenomenological model, and more recently a self-contained kinetic theory-based description, for relativistic nuclear collisions. It follows the space-time evolution of an event from the fluctuating initial condition through a parton cascade, hadronization, and a hadron cascade. In practical use, AMPT exists in a default version, in which hadrons emerge from Lund string fragmentation, and a string melting version, in which excited strings are converted into quarks and antiquarks and hadronize through quark coalescence. Since the public release of the source code in 2004 and the detailed 2005 description of its physics, the model has been continually extended to address collective flow, particle production, heavy flavor, small-system correlations, femtoscopy, strangeness enhancement, and light nuclei production across RHIC and LHC energies (Lin, 2014, Lin et al., 2021).

1. Core architecture

AMPT is organized into four primary components: a fluctuating initial condition, a parton cascade, hadronization, and a hadron cascade. In the early and still widely used implementations, the initial condition is provided by HIJING, which supplies spatial and momentum distributions of minijet partons and soft excited strings. The partonic stage is evolved by Zhang’s Parton Cascade (ZPC), which describes two-body elastic parton scatterings with cross sections derived from perturbative QCD and regulated by a screening mass. The late hadronic stage is handled by the ART hadronic transport model (Zhang et al., 2018, Lin et al., 2021).

The distinction between the two main AMPT modes lies in hadronization. In the default version, partons recombine with parent strings and hadrons are produced by Lund string fragmentation. In the string melting version, all excited strings are first converted into quarks and antiquarks, and hadronization proceeds through quark coalescence. The string melting formulation is central in applications where partonic collectivity, quark coalescence, or hadron-species systematics are the focus (Zhang et al., 2018, Lin et al., 2021).

The partonic scattering strength is often parameterized through the parton-parton cross section,

$\sigma_{pp} = \frac{9\pi \alpha_s^2}{2\mu^2},$

where $\alpha_s$ is the strong coupling constant and $\mu$ is the screening mass. In applied studies, varying $\sigma_{pp}$ is used to probe the sensitivity of flow and correlation observables to the strength of partonic interactions (Magdy, 2021).

Component	Implementation	Function
Initial condition	HIJING	Minijet partons and soft strings
Parton cascade	ZPC	Elastic parton scatterings
Hadronization	Lund fragmentation or quark coalescence	Parton-to-hadron conversion
Hadron cascade	ART	Hadronic rescattering and decays

2. Versions, parameterizations, and code evolution

The public AMPT releases were v1.11 and v2.11 in 2004, followed by v1.21 and v2.21 in 2008. Subsequent development introduced multiple “test” versions with additional processes and user hooks, including deuteron and anti-deuteron interactions, user-defined hadron insertion at the start of the hadron cascade, access to participant and spectator nucleon information, event triggering based on an initial minijet threshold, probe embedding with back-to-back $q/\bar q$ pairs, extraction of parton collision histories, variable nuclear shadowing strength, and support for deformed nuclei such as uranium-238 (Lin, 2014).

A central tunable ingredient of AMPT is the Lund string fragmentation function,

$$f(z) \propto z^{-1} (1-z)^a \exp\left(-\frac{b m_T^2}{z}\right),$$

with free parameters $a$ and $b$. These parameters control yields and transverse-momentum spectra and have been retuned in several applications. A study of $\phi$ meson production found that strange hadrons, especially $\phi$ mesons, are more sensitive to the Lund parameters than pions, that the same parameter set can describe $\alpha_s$0 yields at $\alpha_s$1 and $\alpha_s$2 GeV, and that a different set is needed at $\alpha_s$3 GeV. This suggests that at low energy the underlying mechanism for particle production is different compared to top RHIC energies (2002.01201).

Modernization of the initial condition has been a major line of development. Updated AMPT implementations replaced the original free-proton PDFs with CTEQ6.1M and implemented impact parameter-dependent nuclear shadowing through EPS09sNLO or EPS09s. Because the older parameter choices no longer reproduced $\alpha_s$4 and $\alpha_s$5 cross sections with the new PDFs, the two-component initial-condition parameters $\alpha_s$6 and $\alpha_s$7 were refitted. For $\alpha_s$8 collisions at LHC energies, a nuclear scaling of the minijet cutoff,

$\alpha_s$9

was introduced, motivated by the color glass condensate, to avoid overestimating particle yields in central collisions (Zhang et al., 2019).

Heavy-flavor production required additional changes. The updated heavy-flavor sector removed the transverse-momentum cutoff on initial heavy-quark production and included the corresponding heavy-flavor cross section in the total minijet cross section. Together with the modern nPDF implementation, these changes yielded a much better description of open-charm yields and $\mu$0 spectra in $\mu$1 and $\mu$2 collisions over a wide energy range (Zheng et al., 2019).

Hadronization itself has also been revised. In the older string-melting coalescence algorithm, meson, baryon, and antibaryon numbers were separately conserved in each event. A newer quark-coalescence prescription, associated in the literature with He and Lin, removed this forced separate conservation and retained only net baryon number conservation. A related refinement in later AMPT developments allows baryon-versus-meson formation to depend on spatial proximity through a parameter $\mu$3:

$\mu$4

These changes improved the description of baryon yields and correlation observables (Zhang et al., 2018, Lin et al., 2021).

3. Collective flow, fluctuations, and mode coupling

AMPT has been extensively used to connect event-by-event initial-state geometry to final-state anisotropic flow. In this framework, the azimuthal distribution is expanded as

$\mu$5

and, for lower harmonics, the response is often expressed as

$\mu$6

Studies in Au+Au collisions at $\mu$7 GeV showed that relative eccentricity fluctuations in AMPT account for the observed elliptic-flow fluctuations, in agreement with STAR measurements. The same work found that the Elliptic-Power function is a promising candidate for the event-by-event probability density function of both $\mu$8 and $\mu$9, and that non-zero correlations between different symmetry planes and harmonics are present in both the initial and final state (Zhou et al., 2015).

Higher-order flow harmonics have been analyzed in terms of linear and nonlinear contributions. In Au+Au collisions at $\sigma_{pp}$0 GeV, AMPT calculations showed that $\sigma_{pp}$1, $\sigma_{pp}$2, and related linear and nonlinear components are sensitive to the parton-scattering cross section, while the nonlinear response coefficients and symmetry-plane correlations are essentially insensitive to $\sigma_{pp}$3. This separation suggests that flow magnitudes encode final-state transport strength more strongly than the normalized mode-coupling observables, which remain dominated by initial geometry and harmonic mixing (Magdy, 2021).

Event Shape Engineering (ESE) has been used within AMPT to test the independence of linear and nonlinear higher-order flow modes. In Au+Au collisions at $\sigma_{pp}$4 GeV, using the string-melting scenario and $\sigma_{pp}$5-selected event classes, AMPT showed that the nonlinear contributions to $\sigma_{pp}$6 and $\sigma_{pp}$7 increase with $\sigma_{pp}$8, whereas the linear contributions remain flat as a function of $\sigma_{pp}$9. The study interpreted this pattern as weak or negligible coupling between linear and nonlinear modes and proposed ESE as a practical tool for data sets in which low multiplicity or limited statistics preclude more demanding Pearson-coefficient analyses (2002.04583).

AMPT has also been used to compare flow-fluctuation estimators. In identified-particle elliptic flow at $q/\bar q$0 GeV, the ratio of $q/\bar q$1 measured with a first-order spectator plane to that measured with a second-order participant plane was proposed as a direct fluctuation measure. The spectator-plane result was found to agree quantitatively with $q/\bar q$2, while the fluctuation magnitude showed weak particle-species dependence and weak transverse-momentum dependence (Magdy et al., 2020).

A broader correlator program extended this logic to multi-particle observables. In Au+Au collisions at $q/\bar q$3 GeV, SC and ASC were found to be sensitive to both initial and final state effects, whereas NSC and NASC were largely insensitive to the AMPT viscosity setting and therefore primarily sensitive to initial-state fluctuations and correlations. In that study, HIJING served as a non-flow baseline, and subevent methods suppressed residual short- and long-range non-flow contributions (Magdy, 2022).

4. Hadronization, chemical composition, and composite production

The hadronization stage is one of the most distinctive parts of AMPT phenomenology. In small systems at LHC energies, the string-melting version with new quark coalescence was required to reproduce the pronounced near-side depression in same-sign baryon-baryon and antibaryon-antibaryon angular correlations observed by ALICE. Default AMPT and other Monte Carlo models failed to reproduce this effect, while the improved coalescence algorithm yielded baryon-baryon and antibaryon-antibaryon correlations that are nearly identical, as expected at LHC energies, and recovered the away-side enhancement as well (Zhang et al., 2018).

The hadronization prescription also affects bulk chemistry and strangeness. In RHIC Au+Au collisions, $q/\bar q$4 mesons were shown to be substantially more sensitive to the Lund fragmentation parameters than non-strange pions; at low energy, the preferred parameter set differed from that at 39 and 200 GeV, indicating a change in the particle-production mechanism (2002.01201). For multistrange baryons, a further extension introduced species-dependent hyperon enhancement factors $q/\bar q$5 into the coalescence criterion,

$q/\bar q$6

for hyperon formation. This extended AMPT model gave a reasonable description of the multiplicity dependence of strangeness enhancement in Pb+Pb, Au+Au, high-multiplicity $q/\bar q$7, and $q/\bar q$8+Pb collisions, with fitted coalescence factors depending primarily on system size (Shao et al., 2020).

AMPT output has also been coupled to a naive coalescence afterburner for light nuclei. In that framework, final-state nucleons from string-melting AMPT are grouped into deuterons, tritons, and helium-3 if they satisfy phase-space coalescence conditions. The associated coalescence parameter is written as

$q/\bar q$9

with the usual symmetric-system approximation at midrapidity. This AMPT-plus-afterburner setup reproduced qualitative trends, and often quantitative trends at higher $$f(z) \propto z^{-1} (1-z)^a \exp\left(-\frac{b m_T^2}{z}\right),$$0, in STAR and ALICE data for light nuclei yields and coalescence parameters (Bailung et al., 2023).

A major interpretive issue concerns the mass splitting of azimuthal anisotropies. In AMPT, the overall $$f(z) \propto z^{-1} (1-z)^a \exp\left(-\frac{b m_T^2}{z}\right),$$1 is mainly generated by an anisotropic escape mechanism rather than hydrodynamic flow, yet mass splitting at low transverse momentum is still observed. Detailed studies showed that the splitting is small immediately after hadronization, especially when resonance decays are included, and that most of the mass splitting develops during the hadronic rescattering stage even though the contribution of that stage to the total charged-hadron $$f(z) \propto z^{-1} (1-z)^a \exp\left(-\frac{b m_T^2}{z}\right),$$2 is small. The same qualitative pattern was found for heavy-ion and small-system collisions and for both $$f(z) \propto z^{-1} (1-z)^a \exp\left(-\frac{b m_T^2}{z}\right),$$3 and $$f(z) \propto z^{-1} (1-z)^a \exp\left(-\frac{b m_T^2}{z}\right),$$4. The implication is explicit: mass splitting of low-$$f(z) \propto z^{-1} (1-z)^a \exp\left(-\frac{b m_T^2}{z}\right),$$5 $$f(z) \propto z^{-1} (1-z)^a \exp\left(-\frac{b m_T^2}{z}\right),$$6 is not a unique signature of hydrodynamic collective flow (Li et al., 2016, Li et al., 2016).

5. Correlation observables, small systems, and femtoscopy

AMPT has been used to study correlation observables well beyond anisotropic flow. In Au+Au collisions at $$f(z) \propto z^{-1} (1-z)^a \exp\left(-\frac{b m_T^2}{z}\right),$$7–$$f(z) \propto z^{-1} (1-z)^a \exp\left(-\frac{b m_T^2}{z}\right),$$8 GeV, neighboring-bin multiplicity correlations as a function of pseudorapidity exhibited a systematic energy dependence: for $$f(z) \propto z^{-1} (1-z)^a \exp\left(-\frac{b m_T^2}{z}\right),$$9 GeV the final-state short-range correlation has a trough at central pseudorapidity, whereas for $a$0 GeV it has a peak at central pseudorapidity, with a flat behavior at 19.6 GeV. Comparisons with and without the ART phase showed that hadronic scattering raises the overall correlation strength but does not alter the trough-versus-peak structure; the different shapes were attributed mainly to partonic evolution and the following hadronization scheme (Wang et al., 2015).

In high-multiplicity $a$1 collisions at $a$2 TeV, the string-melting AMPT model has been extended to include sub-nucleon structure through a constituent-quark “3 quarks” scenario. In that setup, the model source function and correlation function for identical pions were analyzed through the Koonin-Pratt equation,

$a$3

with Gaussian and Cauchy source parameterizations. The study found that the measured ALICE one-dimensional source radii and the common $a$4 scaling behavior can be described by the AMPT model with sub-nucleon structure (Wang et al., 27 Jan 2025).

A related line of work examined two-pion HBT interferometry for partially coherent sources in heavy-ion collisions. There, longitudinal and transverse coherent emission lengths were introduced into pion generation coordinates before building the correlation function, and the resulting HBT radii and chaoticity parameter $a$5 were closer to STAR and ALICE data than the corresponding results for fully chaotic sources. The fitted Gaussian correlation form,

$a$6

provided a compact characterization of the extracted radii and coherence effects (Wang et al., 2023).

AMPT has also been adapted to track jetlike broadening in the partonic phase. In a transport study of Au+Au collisions at $a$7 GeV, the angular correlation between a high-$a$8 probe parton and the medium partons it scattered with was found to broaden with the number of parton-parton collisions suffered by the probe and with the probe azimuth from in-plane to out-of-plane. Because the analysis used the recorded collision history of each probe parton, it provided a purely transport-model reference for collisional broadening and pathlength-dependent jet-medium interaction studies (Edmonds et al., 2016).

6. Limitations, interpretive issues, and future directions

AMPT has accumulated a substantial record of phenomenological success, but several limitations have been documented explicitly. Earlier versions violated electric-charge conservation for two distinct reasons: the hadron cascade included charged kaons but not neutral kaons, so the code converted $a$9 and $b$0 before the cascade and then partially reconverted them afterward; and some hadronic reactions used isospin-averaged cross sections with randomly selected final-state charges, which could admit charge-forbidden channels. These issues were identified as targets for correction in future releases (Lin, 2014).

Model-data discrepancies have also been noted. At FAIR-scale beam energies, neither the default nor the string-melting version exactly matched the NA49 elliptic-flow measurements in Pb+Pb collisions at $b$1 GeV, with the largest disagreements at low $b$2 and in mid-central to peripheral events. In multistrange baryon production, standard AMPT underpredicted yields such as $b$3 by a factor of two or more before the introduction of additional coalescence factors. In jetlike-correlation studies, the model provided a purely collisional baseline because radiative energy loss was not included (Sarkar et al., 2018, Shao et al., 2020, Edmonds et al., 2016).

These limitations are closely tied to ongoing development priorities. Documented future directions include updated nuclear parton distributions and shadowing, dynamical parton recombination based on local parton density or local energy density, inelastic parton reactions, explicit gluonic roles in coalescence, improved initial conditions with more realistic sub-nucleonic structure, extensions of the parton cascade for high-$b$4 physics, more complete hadronic afterburners, unified treatment across beam energies, and coupling AMPT initial conditions and hadronic transport to hydrodynamic evolution in hybrid models. Continued public release of improved code and documentation has been presented as part of the model’s long-term role as both a stand-alone simulator and a flexible framework for hybrid descriptions of relativistic nuclear collisions (Lin et al., 2021, Lin, 2014).