Equations to code
This page maps each equation of Godfrey and Isgur (1985) to the code that implements it. Names in code font without a link are internal functions; they are listed so that you can find the implementation, not as supported API. Paths are relative to the repository root.
Spectrum: Section II and Appendix A
| paper | content | implementation |
|---|---|---|
| Eq. (1) | kinetic term in relativistic_hamiltonian (src/hamiltonian.jl) for the mesh; ho_p2_matrix and the matrix square root in src/harmonic_oscillator_basis.jl for the oscillator basis | |
| Eqs. (2)–(3) | color-singlet | static_coulomb_G, static_confinement_S (src/running_coupling.jl); parameters in ConfinementPotential |
| Eq. (4) | contact and tensor hyperfine terms | through Eq. (A15), below |
| Eqs. (6)–(7) | vector and Thomas spin-orbit terms | through Eqs. (A15)–(A16), below |
| Eqs. (12)–(13), Fig. 2 | running coupling | alpha_s_q, alpha_s_r; constants ALPHA_COEFFS, ALPHA_GAMMAS (src/constants.jl) |
| Eq. (14) and text | staged diagonalization | fixed_spectrum (stage 1), add_intra_meson_mixing (stage 2), add_isoscalar_annihilation (stage 3) |
| Eqs. (A7)–(A8) | Gaussian smearing of a potential | direct convolution smear_3d_radial (src/smearing_appendix_a.jl), used only by the diagnostic AppendixASmearing3D method |
| Eq. (A9) | smearing width | contact_smearing_sigma |
| Eqs. (A12)–(A14) | closed-form smeared | smeared_coulomb_G_closed, smeared_confinement_S_closed (src/smearing_appendix_a.jl) |
| after Eq. (A14) | momentum factors | appendix_a_momentum_sandwich_matrix (src/hamiltonian.jl) for the central term; RelativisticFactors holds the |
| Eq. (A15) | contact, vector spin-orbit and tensor operators | smeared_contact_kernel (src/contact_hyperfine.jl); fine_structure_radial_kernels and the shared angular assembly in src/spin_fine_structure.jl |
| Eq. (A16) | scalar (Thomas) spin-orbit operator | same as (A15), scalar_11 and scalar_22 kernels |
| Eq. (A17) | ho_operator_matrix, ho_p2_matrix, ho_r2_matrix; the variational OscillatorSolver |
Every spin-dependent operator is built once and used both to assemble the Hamiltonian and to report its contribution, so the reported pieces always add up to the mass. The finite-difference and oscillator solvers share the angular and mass algebra; they differ only in how radial matrices are represented.
Details that matter
The contact density is the Laplacian of the smeared Coulomb potential,
The tensor operator uses the Pauli
, so the Appendix A spin bracket carries a factor . Eq. (A16) contains only the confinement derivatives of the
and pairs; the Coulomb derivative enters through (A15) alone. The contact term acts in every partial wave. Smearing gives it finite range, and Eqs. (23)–(26) of the paper keep it in P waves.
Mixing: Section II D and Table III
| paper | content | implementation |
|---|---|---|
| text near Eq. (14) | tensor | tensor_mixing_components, TensorMixing |
| text near Eq. (14) | antisymmetric spin-orbit | spin_orbit_mixing_components, AntisymmetricSpinOrbit |
| Eq. (16) | general annihilation matrix element | isoscalar_general_annihilation_solution (src/pseudoscalar_annihilation.jl), isoscalar_annihilation_block |
| Eq. (17) | smeared wavefunction at the origin | _sL_smearing_factor (src/pseudoscalar_annihilation.jl), from the momentum-space wave |
| Eq. (18a) | pseudoscalar P1 amplitude | PaperP1Annihilation |
| Eq. (18b) | pseudoscalar P2 amplitude, solved as a mass-dependent fixed point | PaperP2Annihilation |
| Table III | amplitudes P1/P2 constants | AnnihilationAmplitudes, [annihilation] in the parameter file |
Diagonalization of every mixing block goes through one routine, diagonalize_mixing_block, which returns a MixingResult.
Decays: Section IV and Appendices B–D
| paper | content | implementation |
|---|---|---|
| Appendix B, Eqs. (B1)–(B36) | flavor, spin and spatial wavefunction conventions | QuarkModelTransitions/src/flavor_algebra.jl, spin_algebra.jl |
| Eq. (19) | pseudoscalar emission | PseudoscalarEmission (pseudoscalar_emission.jl) |
| Appendix C, Eqs. (C1)–(C3) | helicity amplitudes, partial waves, width | partial_wave_projection (transition_amplitudes.jl), decay_width |
| Table IV | single-Gaussian reduced amplitudes | the Table V reference backend (strong_decays.jl), used by GIPaper |
| Appendix D | E1, M1 and M2 photon transitions | PhotonEmission, photon_transition_class (photon_emission.jl, radiative_decays.jl) |
| Eqs. (D7)–(D9) | leptonic widths | LeptonicCurrent, leptonic_decay_factor (annihilation_widths.jl) |
| Table VII (a)–(c) | leptonic, two-photon and gluonic annihilation | LeptonicCurrent, TwoPhotonAnnihilation, GluonicAnnihilation |
| Table VII (d) | charge radii | charge_radius_squared |
Central-potential variants
CentralPotentialMethod selects how the spin-independent potential is built. Only one of them is the paper's; the rest exist to measure how much each ingredient matters:
| method | construction |
|---|---|
AppendixAMomentumSandwich | closed-form |
AppendixAClosedForm | closed-form |
PointwiseCentral | unsmeared |
Coulomb1DSmearing, AppendixASmearing3D, AppendixADerivativeG | diagnostic smearing approximations (mesh only) |
central_potential_path reports which one a parameter set uses.