GiBUU Transport-Theoretical Framework

• GiBUU is a comprehensive transport framework that simulates non-equilibrium hadronic, leptonic, and nuclear reactions using semiclassical and off-shell dynamics.
• It incorporates both non-relativistic and relativistic mean-field potentials, validated by empirical nuclear ground state properties and sophisticated collision modeling.
• The framework accurately captures final-state interactions and multiparticle reactions, making it essential for studies in neutrino oscillations, heavy-ion collisions, and related phenomena.

GiBUU (Giessen Boltzmann-Uehling-Uhlenbeck) is a comprehensive transport-theoretical framework developed to describe the non-equilibrium dynamics of hadronic, leptonic, and nuclear reactions across a broad range of energies and projectiles. Built upon the traditions of nonequilibrium quantum many-body theory, GiBUU models the full time-evolution of complex nuclear reactions, from the initial interaction through to the detailed final-state configurations, by propagating particle phase-space distributions under mean-field potentials and dynamically evolving collision integrals. Key features include extensions to off-shell dynamics, rigorous treatment of final-state interactions, and application to diverse phenomena such as pion production, heavy-ion collisions, dilepton emission, and lepton-induced reactions on nuclei.

1. Foundations: Transport Theory and the BUU Equation

At its core, GiBUU employs semiclassical transport theory on a quantum-statistical basis to evolve the one-particle phase-space distribution function $f(x, p, t)$ for each particle species. The central kinetic equation is the Boltzmann-Uehling-Uhlenbeck (BUU) equation:

$$\frac{\partial f(x, p)}{\partial t} + \{H(x, p), f(x, p)\} = C(x, p)$$

where the drift term involves the Poisson bracket $$\{H, f\} = (\partial H/\partial p) \cdot (\partial f/\partial x) - (\partial H/\partial x) \cdot (\partial f/\partial p)$$, $H(x, p)$ is the single-particle Hamiltonian including mean-field potentials, and $C(x,p)$ is the collision term encompassing gain and loss from elastic/inelastic scatterings, decays, and absorption channels (Buss et al., 2011).

GiBUU supports both non-relativistic (Skyrme-like) and relativistic mean-field (RMF, e.g., non-linear Walecka model) potentials for nucleons and hadrons. These are tuned to reproduce empirical properties such as optical potentials and saturation density.

The framework generalizes to off-shell transport for broad hadronic resonances. It evolves the generalized Wigner function $F(x, p) = 2\pi g f(x, p) A(x, p)$ incorporating the particle spectral function $A(x, p)$, and solves a modified kinetic equation with explicit off-shell corrections ("Z-factors") and an off-shell potential (OSP) approach for mass evolution (Buss et al., 2011).

2. Implementation: Potentials, Initialization, and Collision Terms

Potentials

• Non-relativistic (Skyrme-like):

$$U_N(x, p) = A (\rho(x)/\rho_0) + B (\rho(x)/\rho_0)^\gamma +$$ momentum-dependent terms.

• Relativistic Mean-Field (RMF):

Lagrangian includes $\sigma$, $\omega$, and $\rho$ exchange with scalar ($S$) and vector ($V^\mu$) potentials.

Ground State Initialization

Nuclear ground states are initialized using empirical (Woods-Saxon or oscillator) density distributions. Local nucleon momenta follow the (relativistic) Thomas-Fermi distribution,

$$f_{n,p}(r,p) = \Theta(p_F(r) - |p|), \quad p_F(r) = [3\pi^2 \rho(r)]^{1/3}.$$

Recent advances employ self-consistent energy-density functionals, ensuring constant Fermi energies and stable ground states across the nucleus (Gallmeister et al., 2016).

Collision Term

The collision operator $C(x, p)$ is decomposed by process order:

Term	Physical process	Notation/Comments
$C^{(1)}$	One-body decays (e.g., resonance decay)	Includes width, Pauli/Bose factors
$C^{(2)}$	Two-body collisions (NN, $\pi$N, etc.)	In-medium masses/cross sections
$C^{(3)}$	Three-body (multi-hadron) collisions	Geometrical/statistical treatment

For in-medium reactions, kinematic variables (e.g., invariant mass $s$) and flux factors are corrected for mean-field potentials and effective masses. Pauli blocking and Bose enhancement are implemented explicitly. Off-shell effects are incorporated where relevant (e.g., broad resonances).

3. Off-Shell Transport and Spectral Functions

GiBUU notably incorporates off-shell transport for broad resonances and short-lived hadrons, allowing particles to propagate with masses deviating from their vacuum ("on-shell") values. The off-shell evolution employs the spectral function $A(x, p)$, and particle propagation includes additional equations (generalized Hamiltonian with Z-factors) and an off-shell potential ansatz to ensure that as particles leave the dense medium, their invariant masses return to physical pole values (Buss et al., 2011).

This is essential for correctly describing the production and propagation of, for example, the $\Delta$ resonance or the $\rho$ meson, as seen in the modeling of dilepton production and pion dynamics in nuclear reactions (Weil et al., 2012).

4. Modeling Final-State Interactions (FSI)

The propagation of secondary hadrons produced in the initial reaction vertex is handled via dynamic coupled-channel transport using the same BUU equation framework. Secondary processes include:

• Elastic and inelastic hadron–hadron scattering
• Charge-exchange reactions (e.g., $\pi^+ n \to \pi^0 p$)
• Absorption (e.g., $\pi N \to \Delta$, then $\Delta N \to NN$)
• Multi-particle emission, fragmentation, and clustering

FSI redistribute energy and momentum among final-state particles, profoundly modulating experimental observables such as kinetic energy and multiplicity spectra of outgoing mesons and nucleons (Buss et al., 2011, Lalakulich et al., 2011, Lalakulich et al., 2013). Accurate FSI treatment is critical for extraction of neutrino oscillation parameters, as FSI can convert resonance-induced events into CCQE-like topologies or result in pionless final states.

5. Channel Structure and Unified Multiparticle Reactions

GiBUU treats all relevant reaction channels coherently, including:

• Quasi-elastic (QE), resonance (RES) excitation, nonresonant background (BG), and deep inelastic scattering (DIS):

$$\sigma_{\text{tot}} = \sigma_{\text{QE}} + \sigma_{\text{RES}} + \sigma_{\text{BG}} + \sigma_{\text{DIS}}$$

The model smoothly interpolates between these contributions in the "Shallow Inelastic Scattering" (SIS) regime via switching functions of the invariant mass $W$ (Lalakulich et al., 2011, Lalakulich et al., 2013).

• 2p2h (two-particle-two-hole) interactions:

Based on fits to empirical electron-scattering structure functions $W_1^e(Q^2, \omega)$ (Gallmeister et al., 2016), with baryonic isospin scaling for neutrino reactions.

• Resonance broadening and in-medium modifications:

E.g., for the $\Delta$ resonance:

$$\Gamma_\Delta^{\text{medium}} = \Gamma_\Delta^{\text{PB}} - 2\,\text{Im}(\Sigma_\Delta)$$

where $\Gamma_\Delta^{\text{PB}}$ is Pauli-blocked width and $\Sigma_\Delta$ the self-energy (Yan et al., 28 Jul 2025).

• Density-dependent cross-section modifications:

$$\sigma_{NN \to N\Delta}(\rho_N) = \sigma_{NN \to N\Delta}(0) \exp\left(-\alpha \frac{\rho_N}{\rho_0}\right)$$

with a medium-suppression parameter $\alpha$ (Yan et al., 28 Jul 2025).

6. Applications, Benchmarking, and Model Validation

GiBUU's comprehensive treatment enables its application across a wide reaction scope:

Application Domain	Example Systems/Observables	Role of FSI and Transport
Pion-induced reactions	$\pi$-nucleon/nucleus, DCX, absorption	Energy redistribution, absorption
Proton/antiproton-induced	Knockout, fragmentation, annihilation	Strong in-medium absorptive FSI
Heavy-ion collisions	Meson yields, flow, multifragmentation	Three-body collisions, mean fields
Photon, electron, neutrino	Inclusive QE, RES, DIS, CC/NC pion production	XX
Dilepton production	$NN \to NR$, $\rho$ decay, HADES data	Off-shell vector mesons

Model validation is achieved through cross-comparisons with electron, photon, pion-induced, and heavy-ion data, ensuring that the nuclear ground state initialization and FSI physics are robust across different probes and energy regimes (Lalakulich et al., 2013, Lalakulich et al., 2011, Weil et al., 2012). These external constraints are essential for credible neutrino-nucleus predictions.

7. Monte Carlo Simulation and Systematics in Modern Neutrino Experiments

The GiBUU framework underpins Monte Carlo event generators for realistic neutrino-nucleus interaction modeling, producing event-by-event samples that are compatible with experimental detector simulations (Soplin et al., 2023). Weighted events (with possibly negative weights from interference terms) are transformed to unweighted samples via acceptance-rejection, maintaining fidelity to the original cross-section modeling.

Integration with standard tools (e.g., GENIE) is realized through common data formats and handling of detector geometries and fluxes. For propagation of systematic uncertainties, infrastructure is in development to generate variants of model libraries reflecting uncertainties in nuclear density, resonance widths, medium modifications, and scaling parameters (Soplin et al., 2023). This systematic suite enables robust prediction and uncertainty quantification in multi-experiment contexts.

Recent studies reveal that while individual datasets (e.g., MINERvA or MicroBooNE) can be accommodated by specific choices of in-medium modifications, achieving a unified description is challenging, a reflection of the rich complexity of nuclear dynamics and open issues in FSI modeling (Yan et al., 28 Jul 2025).

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In summary, GiBUU constitutes a unified, microscopically-constrained transport-theoretical platform, solving semiclassical kinetic equations with off-shell dynamics in order to simulate both elementary and composite nuclear reactions. Its flexible framework unifies the modeling of initial hard scatterings and the intricate cascades of final-state interactions, providing a benchmark for interpreting experimental data and exploring open problems in nuclear dynamics, resonance behavior, and nuclear medium effects, especially in the context of contemporary neutrino oscillation and nuclear physics research.