Relativistic Partial-Wave Analysis

• Relativistic partial-wave calculation is a framework that employs covariant methods to analyze scattering and bound-state problems while upholding Lorentz invariance.
• It decomposes wavefunctions into angular momentum eigenstates using techniques like Clebsch–Gordan coefficients to simplify complex interactions.
• The methodology combines analytic and numerical strategies to extract spectra, phase shifts, and observables in particle, nuclear, and atomic physics.

Relativistic partial-wave calculation is a comprehensive framework used to analyze scattering and bound-state problems in quantum field theory and relativistic quantum mechanics, generalizing the standard angular-momentum separation techniques to systems and amplitudes governed by Lorentz invariance. This formalism underlies modern approaches to two- and three-body problems in particle and nuclear physics, atomic and hadronic transitions, effective field theories, and celestial amplitudes, where the dynamical equations admit covariant partial-wave expansion, precise orthogonality relations, and numerical implementation in high-energy regimes.

1. Covariant Formulation and Wave Equations

Relativistic partial-wave schemes start from covariant two- and three-body equations. For two particles, the system is described by reduced phase-space coordinates, relative four-momentum, and Lorentz-invariant distance, with central vector or scalar potentials introduced by minimal coupling or mass shifts, respectively. Explicit forms for scalar–scalar, scalar–fermion, and fermion–fermion cases employ Klein-Gordon (in Feshbach–Villars form) and Dirac Hamiltonians, yielding matrix equations of size $4\times 4$ (scalar–scalar), $8\times 8$ (scalar–fermion), or $16\times 16$ (fermion–fermion) in the presence of interactions (Giachetti et al., 2018).

For three-particle systems, the mass-Casimir operator formulation encodes Poincaré invariance, with interactions added to the free mass operator respecting spin and cluster properties. Dynamical equations, such as the Bethe–Salpeter–Faddeev or relativistic Faddeev equations, are formulated in terms of Jacobi momenta and constituent spins, with kernels depending only on these covariant variables. Special care is taken for separable and rank-six kernels in nucleon-nucleon scattering (Bondarenko et al., 2015, Bondarenko et al., 2011).

2. Partial-Wave Expansion and Basis Construction

Rotational symmetry and Lorentz invariance allow decomposition of wavefunctions and amplitudes into irreducible angular momentum eigenstates. In two-body cases, spherical harmonics $Y_{\ell m}$ or spinor spherical harmonics (for Dirac states) are coupled with orbital and spin degrees of freedom to form total angular momentum states. Central potentials reduce the problem to coupled ordinary differential equations for partial-wave radial components (e.g., $u_\ell(r)$), while interaction terms induce mixing between different partial waves (Giachetti et al., 2018, Yerokhin et al., 2011).

For multi-particle systems, expansions are performed in momentum-space bases combining orbital angular momentum, constituent spins, and isospin (SU(2) or higher flavor symmetry) via Clebsch–Gordan and Wigner rotation matrices. In the general Poincaré-invariant case, amplitudes are constructed by gluing Clebsch–Gordan coefficients for initial and final states and projecting onto orthonormal irreducible representations (Shu et al., 2021). For celestial amplitudes, relativistic partial waves are built as eigenfunctions of quadratic and quartic Poincaré Casimir operators with explicit orthogonality and completeness relations (Law et al., 2020).

3. Integral Equations, Matrix Elements, and Kernel Evaluation

Solving the dynamical equations requires deriving system-specific integral equations for the partial-wave-projected amplitudes. In two-body systems, radial equations admit analytic solutions for special cases (e.g., Klein-Gordon spectra for scalar–scalar Coulomb), but generally necessitate numerical integration and matching conditions for bound- and scattering-state boundary conditions (Giachetti et al., 2018, Hoffmann, 2021).

Three-body problems involve coupled integral equations in two or three variables (momentum and energy), with kernels incorporating one-particle exchange (OPE) and short-distance three-body forces. Symmetric and asymmetric formalisms allow flexible parameterization and numerical recoupling; explicit expressions for OPE kernels, short-distance $K_3$ matrices, and ladder amplitudes are given for generic spinless systems and nucleon clusters (Briceño et al., 2024, Jackura et al., 18 Jul 2025). Analytical methods are complemented by computational algebra for recoupling coefficients, particularly for systems with nontrivial isospin and angular-momentum structure (Witala et al., 2011, Jackura et al., 2020).

4. Orthogonality, Inner Products, and Normalization

A defining feature of relativistic partial-wave formalism is the explicit construction of hermitian inner products and measures under which Casimir operators are self-adjoint. In amplitude-based approaches, partial waves are orthogonal under on-shell Lorentz-invariant phase-space integrals, and completeness relations ensure expansion of arbitrary amplitudes (Law et al., 2020, Shu et al., 2021). For celestial amplitudes, the orthogonality of Jacobi-polynomial partial waves in each Mandelstam channel is established using ODE-derived measures on real kinematic variables. Inner product formulas allow direct extraction of partial-wave coefficients from amplitudes and facilitate the calculation of unitarity cuts and anomalous-dimension matrices in EFT (Shu et al., 2021).

Normalization of bound and scattering states uses standard integrals over the radial coordinate and angular variables, with regularity at the origin and decay at infinity enforced by boundary conditions on the radial wave components (Giachetti et al., 2018). In fermionic and multi-channel cases, correlated normalization across coupled waves maintains probabilistic consistency (Bondarenko et al., 2011).

5. Spectral Extraction, Observable Calculations, and Physical Applications

Observed spectra and cross sections follow from solving partial-wave integral equations and constructing scattering amplitudes via the complex phase shift $\delta_\ell$ or S-matrix parametrizations. For bound states, discrete energy levels are extracted by enforcing boundary conditions or requiring vanishing determinants of matching matrices. Scattering states are identified by asymptotic behavior and phase-shift relations (e.g., via Bessel–Neumann matching) (Giachetti et al., 2018, Hoffmann, 2021).

Partial-wave analysis is critical in atomic (relativistic double photoionization, hyperfine splitting), nuclear (relativistic NN, three-nucleon form factors), and hadronic sectors (quarkonium decays, three-pion resonance bands). Radiative electromagnetic decay rates, charge form factors, and Dalitz-plot intensities are computed by projecting transition amplitudes onto relevant partial waves and integrating over physical phase space (Pei et al., 2022, Bondarenko et al., 2020, Jackura et al., 18 Jul 2025). Relativistic corrections, higher multipoles, retardation effects, and multi-channel mixing (e.g., $S$–$D$ coupling in NN interactions, $S$–$P$–$D$–$F$ mixing in quarkonium) are quantitatively significant in high-$Z$, high-energy, or multi-body regimes.

6. Numerical Strategies and Systematic Uncertainties

Numerical methods are tailored to the stiffness and singularity structure of relativistic radial and momentum-space equations. Techniques include double-shooting with Padé improvement, Gaussian or spline discretization, matrix inversion, iterative solution schemes (Lanczos, power method), and complex-contour rotation (Giachetti et al., 2018, Jackura et al., 2020, Bondarenko et al., 2015). Treatment of bound-state poles (via explicit $i\epsilon$ prescription or semi-analytic subtraction) and systematic extrapolation (polynomial fitting in grid size) are essential to control finite-$N$ errors and ensure compliance with unitarity (Jackura et al., 2020). High-dimensional integrals in charge form factor calculations are addressed by Wick rotation, Monte-Carlo sampling, and analytic residue evaluation at pole crossings (Bondarenko et al., 2020).

7. Extensions, Generalizations, and Research Directions

Relativistic partial-wave analysis extends uniformly across two- and three-body quantum systems with arbitrary spin, isospin, particle content, and interaction structure. In effective-field-theory contexts, partial-wave expansion is crucial for operator basis construction, loop amplitude projection, and renormalization; spinor-helicity methods generalize the formalism to massless amplitudes and celestial variables (Shu et al., 2021, Law et al., 2020). In nuclear and hadronic physics, ongoing work explores symmetrization of spectator amplitudes, recoupling algebra, and manifestly covariant treatment of multi-channel and multi-resonance problems (Briceño et al., 2024, Jackura et al., 18 Jul 2025). Applications to lattice QCD, precision atomic theory, and photoproduction continue to emphasize the role and versatility of covariant partial-wave frameworks in confronting modern data and theoretical developments.

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Relevant arXiv references:

• (Giachetti et al., 2018) Giachetti & Sorace, "Two body relativistic wave equations"
• (Witala et al., 2011) Witała et al., "Three-nucleon force in relativistic three-nucleon Faddeev calculations"
• (Yerokhin et al., 2011) Yerokhin & Surzhykov, "Relativistic theory of the double photoionization of helium-like atoms"
• (Briceño et al., 2024) Jackura et al., "Partial-wave projection of relativistic three-body amplitudes"
• (Jackura et al., 2020) Jackura et al., "Solving relativistic three-body integral equations in the presence of bound states"
• (Hoffmann, 2021) Hoffmann, "Finite perturbation theory for the relativistic Coulomb problem"
• (Law et al., 2020) Pasterski et al., "Relativistic partial waves for celestial amplitudes"
• (Shu et al., 2021) Cohen et al., "Constructing the general partial waves and renormalization in EFT"
• (Pei et al., 2022) Li et al., "Partial wave effects in the heavy quarkonium radiative electromagnetic decays"
• (Bondarenko et al., 2020) Bondarenko et al., "Relativistic rank-one separable kernel for helium-3 charge form factor"
• (Bondarenko et al., 2011) Bondarenko et al., "Relativistic complex separable potential for describing the neutron-proton system in $^3S_1$–$^3D_1$ partial-wave state"
• (Bondarenko et al., 2015) Bondarenko et al., "Relativistic three-nucleon calculations within the Bethe-Salpeter approach"
• (Jackura et al., 18 Jul 2025) Jackura et al., "Symmetrizing relativistic three-body partial wave amplitudes"