Relativistic Distorted Wave Impulse Approximation (RDWIA)

• RDWIA is a relativistic quantum framework that models lepton and hadron scattering using the impulse approximation and Dirac equation for nucleon states.
• It incorporates complex optical potentials to account for distortion effects and final-state interactions, ensuring accurate description of exclusive and inclusive reactions.
• RDWIA has been benchmarked against (e,e'p), (p,2p), and neutrino scattering data, enhancing the extraction of nuclear structure observables and reaction dynamics.

The Relativistic Distorted Wave Impulse Approximation (RDWIA) is a quantum-mechanical framework for modeling nuclear reactions involving lepton or hadron scattering from nuclei, where both the initial and final nucleon wave functions are consistently treated in a relativistic formalism. It underpins precision descriptions of exclusive, semi-inclusive, and inclusive processes such as $(e, e'p)$, $(p,2p)$, and neutrino-induced quasielastic scattering, playing a critical role in extracting nuclear structure observables, determining nucleon spectral functions, and interpreting results from modern high-energy nuclear experiments.

1. Fundamental Principles and Mathematical Structure

The RDWIA approach is founded on the impulse approximation: the external probe (electron, neutrino, or hadron) is assumed to interact with a single nucleon inside the nucleus, while the remaining nucleons act as spectators. Distortion effects—arising from the interaction of the incoming and outgoing particles with the nuclear medium—are incorporated by solving the Dirac equation for both initial (bound) and final (continuum) nucleon states in the presence of complex scalar and vector (S-V) potentials (optical potentials).

The central object is the transition amplitude between an initial bound state $\psi_i({\bf r})$ and a final state $\psi_f({\bf r})$, linked by the nuclear current operator $J^\mu({\bf r})$ and modulated by the momentum transfer ${\bf q}$: $$M \propto \int d^3r\, \psi_f^\dagger({\bf r})\, J^\mu({\bf r})\, \psi_i({\bf r})\, e^{i{\bf q}\cdot{\bf r}}$$ The final-state nucleon is described by a distorted outgoing wavefunction, $\psi_f$, that includes effects of final-state interactions (FSI) encoded in the optical potential. This formulation is used across processes including $(e,e'p)$, $(p,2p)$, and neutrino-nucleus scattering (Alvarez-Rodriguez et al., 2010, Meucci et al., 2011, Butkevich et al., 2017).

The cross section is ultimately constructed from bilinear products of such amplitudes, summed and integrated as required, and is often written as: $$\frac{d\sigma}{d\Omega} \propto \sum_f \left| \langle \psi_f| J^\mu({\bf q}) | \psi_i \rangle \right|^2 \delta(E_f - E_i - \omega)$$ where $\omega$ is the energy transfer, and the integration over undetected final states connects exclusive, semi-inclusive, and inclusive observables.

2. Relativistic Treatment of Bound and Scattering States

In RDWIA, the bound-state wave functions are generated by solving the Dirac equation with real scalar and vector mean-field potentials, typically derived from relativistic mean field (RMF) theory or fitted to reproduce nuclear shell structure. This provides realistic relativistic nucleon momentum distributions, including high-momentum tails essential for short-range correlation physics (Kaki, 2017, Alvarez-Rodriguez et al., 2010).

For the outgoing nucleon, the Dirac equation is solved with a complex energy- and A-dependent relativistic optical potential, incorporating both absorption (imaginary part, for inelastic FSI) and elastic distortion (real part). The form is: $$\left[ \boldsymbol{\alpha} \cdot \mathbf{p} + \beta(m + S({\bf r})) + V({\bf r}) \right] \psi_f({\bf r}) = E \psi_f({\bf r})$$ The optical potential parameters are commonly constrained by elastic proton-nucleus scattering data to ensure physical FSI modeling (Alvarez-Rodriguez et al., 2010, Nikolakopoulos et al., 2022, Franco-Patino et al., 2023).

Distorted waves correct for attenuation, phase modifications, and spin-orbit effects, providing quantitatively accurate observables. For low nucleon energies ($T_N \lesssim 100$ MeV), the limitations of semi-classical cascade models become apparent, as full RDWIA with optical potentials captures quantum interference phenomena absent from cascade-based approaches (Nikolakopoulos et al., 2022).

3. Role and Impact of Final-State Interactions

Final-state interactions represent a dominant source of suppression and redistribution in exclusive and semi-inclusive cross sections. RDWIA treats FSI via the inclusion of an optical model whose imaginary part (absorption) removes flux corresponding to channels not explicitly considered (e.g., multi-nucleon knockout or non-elastic channels). The real part modulates the outgoing nucleon's phase and trajectory.

In exclusive channels, such as $(e,e'p)$ or $(p,2p)$, inclusion of the full complex optical potential is imperative to accurately describe the observed spectroscopic factors and momentum distributions (Atkinson et al., 2018, Yoshida et al., 2021, Kolar et al., 2023). For inclusive processes, in which flux lost to inelastic channels should still be counted, only the real part of the optical potential (rROP) is used, or, alternatively, relativistic Green's function (RGF) methods are employed to redistribute lost strength (Meucci et al., 2011, Nikolakopoulos et al., 2022).

Failure to correctly model FSI leads to systematic biases in cross-section normalization, lepton energy distributions, and hadronic kinematic observables, affecting neutrino energy reconstructions and oscillation parameter extractions in neutrino experiments (Butkevich et al., 2018, Franco-Patino et al., 2023).

4. Extensions: Two-Body Currents, Multi-Nucleon Correlations, and Spectroscopic Factors

While RDWIA excels in describing one-nucleon knockout, two-body meson-exchange currents (MEC) and short-range correlations (SRC) induce significant multinucleon emission, especially at high missing momentum and in transverse response channels. Recent RDWIA implementations incorporate such effects phenomenologically by adding calculated 2p–2h response functions or by replacing the traditional one-body operator with effective two-body operators, as necessitated by rigorous relativistic treatments (Bolton et al., 2010).

Spectroscopic factors—quantifying the probability of removing a nucleon from a given shell—are naturally extracted within RDWIA frameworks employing nonlocal dispersive optical models (DOM), which self-consistently constrain both bound and scattering wave functions. Notably, discrepancies between spectroscopic factors extracted from $(e,e'p)$ and $(p,2p)$ reactions using DOM+DWIA analyses highlight the remaining challenge of accurately modeling in-medium NN interactions and possible limitations of current factorization approaches (Yoshida et al., 2021, Atkinson et al., 2018).

5. Implementation in Large-Scale Monte Carlo and Benchmarking against Data

The RDWIA formalism has been integrated into neutrino event generators such as NEUT, allowing accurate simulation of CCQE and semi-inclusive reactions with detailed nucleon kinematics and improved modeling of transverse kinematic imbalance variables (McKean et al., 15 Feb 2025). The generator combines RDWIA for the primary lepton-nucleon vertex with an intranuclear cascade to simulate post-emission nucleon rescattering.

RDWIA-based models, when benchmarked against T2K, MINERvA, and MicroBooNE differential data, demonstrate improved $\chi^2$ and a more accurate reproduction of proton kinematics and cross-section distributions compared to factorized or Fermi-gas-based approaches. Table below summarizes representative advantages:

Feature	Factorized Models	RDWIA
Kinematics	Simplified (FG/PA)	Fully quantum, relativistic
FSI	Cascade/semi-class.	Dirac optical potential (full FSI)
Spectral Info	Smeared, nonlocality often neglected	Self-consistent (e.g., DOM)
Multi-nucleon	Typically phenomenological or omitted	Explicit 2p–2h, improved operators

This improved accuracy is particularly significant in reproducing data sensitive to FSI and to nucleon-nucleon correlations, and in regions where scaling violations or dynamical correlations are important (e.g., forward lepton scattering, high missing momentum) (Franco-Patino et al., 2022, Nikolakopoulos et al., 2023, Bolton et al., 2010).

6. Sensitivities and Limitations

RDWIA predictions are sensitive to the choice and parameterization of the optical potential, treatment of nonlocality (Perey and Darwin factors), and effective nucleon current operator. For $(p,pN)$ and $(e,e'p)$ reactions, deep shell nuclear structure and spectroscopic strengths depend on the interplay of correlations and mean-field inputs. Despite the sophistication of RDWIA, persistent challenges remain in:

• Explicitly modeling SRC and three-nucleon force effects at high missing momenta (Iqbal et al., 2019).
• Extending the approach to consistently include multi-nucleon knockout in an unfactorized, fully relativistic context.
• Reconciling differences in extracted spectroscopic factors between $(e,e'p)$ and $(p,2p)$ using unified optical potential models (Yoshida et al., 2021).
• Quantifying uncertainties due to input nuclear structure and in-medium effective NN interactions.

A continued focus is placed on developing relativistic, nonlocal, and dispersive self-energies to accurately describe both bound and continuum nucleon states, and on integrating more microscopic two-body and many-body current operators into the RDWIA formalism.

7. Current and Future Directions

RDWIA serves as both a benchmark and a high-fidelity tool in contemporary nuclear and neutrino physics. Recent advancements include:

• Incorporation of DOM-based bound and scattering wave functions to address nonlocality and depletion effects (Atkinson et al., 2018).
• Integration with event generators (e.g., NEUT) that facilitate systematic tuning to experimental data and exploration of different nuclear targets and energies (McKean et al., 15 Feb 2025).
• Expansion to describe semi-inclusive and multi-nucleon emission measured in large LArTPCs at accelerator-based neutrino facilities (Franco-Patino et al., 2023, Nikolakopoulos et al., 2022).
• Systematic benchmarking of FSI modeling by comparing optical potential and cascade-based treatments, especially at low kinetic energies (Nikolakopoulos et al., 2022).
• Improved extraction of in-medium modifications to nucleon structure via double and super-ratio analyses of polarization transfer observables (Kolar et al., 2023).

Ongoing refinement aims to address remaining discrepancies in cross-section normalization, shape, and nucleon emission kinematics, and to reduce systematic uncertainties in the extraction of nuclear response functions and neutrino oscillation parameters. This encompasses direct implementation of lattice-QCD-consistent nucleon form factors and explicit coupling to advanced many-body nuclear structure models.

RDWIA thus remains a central theoretical and computational framework in precision lepton- and hadron-nucleus scattering, critical for the interpretation of both current and next-generation nuclear physics and neutrino oscillation experiments.