Same-J Mixing-Angle Audit (13 GI85 captions and 6 GK91 targets)
Generated by checks/audit_mixing_angles.jl on 2026-09-27. Numerics: OscillatorSolver — nbasis = 24:8:80 adaptive, ΔE ≤ 0.1 MeV, beta in [0.25, 2.35] GeV (22 bracket points, refined to 0.002 GeV), 6 levels/channel.
Computed-vs-paper comparison of every same-J (^1L_L, ^3L_L) mixing angle quoted in the text of Godfrey & Isgur, Phys. Rev. D 32, 189 (1985): strange (Fig. 4 caption), charmed (Fig. 7 caption), and b-flavored (Fig. 9 caption) sectors. Model angles come from the public spectrum API (compute_spectrum with explicit OscillatorSolver(); StateMixing records with mechanism antisymmetric_spin_orbit), quark masses u/d 0.220, s 0.419, c 1.628, b 4.977 GeV from data/parameters.provisional.toml.
Closed as an implementation investigation. The missing all-L contact term is fixed. The large GI85 caption-angle mismatch is a documented historical reference discrepancy: the corrected calculation agrees with the later GI-model results. The origin of the 1985 values is unknown; no parameters are adjusted to match them. Targets and provenance: ledger, machine-readable comparison. Historical mechanism studies are available in Git history.
Convention mapping (derived once, applied uniformly)
The paper prints the same rotation matrix in all three captions:
[ Q_low ] [ cos(theta_nL) sin(theta_nL) ] [ n ^1L_L ]
[ Q_high ] = [ -sin(theta_nL) cos(theta_nL) ] [ n ^3L_L ]so the lower-mass eigenstate is Q_low = cos(theta) |n ^1L_L> + sin(theta) |n ^3L_L> with cos(theta) >= 0 implied by the quoted range.
The production spectrum diagonalizes one complete block containing every requested radial singlet and triplet with this L,J. For each quoted nL, we take the lower of the two physical states assigned to its singlet/triplet precursors, project its shared eigenvector onto those two exact basis rows, fix the singlet component positive, and report theta = atan2(v_triplet,nL, v_singlet,nL). Thus the angular convention is identical to the paper while radial-state mixing remains present, so
theta_paper = theta_model (identity angular mapping; the model value is a projection of the complete radial block).
Two residual sign conventions enter only through the off-diagonal element c and are pinned once, globally:
Constituent ordering.
Meson(mq, f1, f2)is thef1 f2barmeson; the antisymmetric spin-orbit operator carries(1/m1^2 - 1/m2^2)/2and flips sign under quark <-> antiquark exchange. Flavors are ordered exactly as the paper's sector labels:u sbar,c ubar,c sbar,b ubar,b sbar,b cbar.Basis phase. The angular matrix element is
<^1L_L| L.(S1-S2) |^3L_L> = +sqrt(L(L+1))withS1the quark (f1) spin (spin_orbit_mixing_components,src/spin_fine_structure.jl).
Consistency check (an outcome, not a fit): with this fixed mapping the model reproduces the paper's sector-wide sign pattern from the mass ordering alone — strange (u quark, heavy antiquark) angles positive, charmed (heavy quark, light antiquark) angles negative.
Computed vs paper
Both columns use the paper convention above. Delta = computed - paper.
| Sector | nL | Computed theta (deg) | Paper theta (deg) | Delta (deg) |
|---|---|---|---|---|
| u sbar | 1P | +4.3 | +34 | -29.7 |
| u sbar | 1D | +12.8 | +33 | -20.2 |
| u sbar | 2P | +14.2 | +15 | -0.8 |
| u sbar | 1F | +12.6 | +32 | -19.4 |
| u sbar | 2D | +20.2 | +25 | -4.8 |
| u sbar | 1G | +10.8 | +33 | -22.2 |
| c ubar | 1P | -25.8 | -41 | +15.2 |
| c ubar | 1D | -38.1 | -39 | +0.9 |
| c sbar | 1P | -39.6 | -44 | +4.4 |
| c sbar | 1D | -38.4 | -39 | +0.6 |
| b ubar | 1P | -28.0 | -43 | +15.0 |
| b sbar | 1P | -41.1 | -45 | +3.9 |
| b cbar | 1P | +69.2 | -53 | +122.2 |
Later GI-model reference: Godfrey–Kokoski (1991), Table I
Same model parameters and lower-state convention. The printed K angle is -5 degrees for s ubar; constituent exchange gives +5 degrees for our u sbar. P_s is the singlet probability within the same-n projection. The 1-degree printed step is precision, not an assigned uncertainty or a pass threshold.
| Sector | GI85 angle | GK91 angle | Computed angle | Computed − GK91 | GK91 P_s | Computed P_s |
|---|---|---|---|---|---|---|
| us | +34 | +5 | +4.3 | -0.7 | 0.992 | 0.994 |
| cu | -41 | -26 | -25.8 | +0.2 | 0.808 | 0.811 |
| cs | -44 | -38 | -39.6 | -1.6 | 0.621 | 0.594 |
| bu | -43 | -31 | -28.0 | +3.0 | 0.735 | 0.780 |
| bs | -45 | -40 | -41.1 | -1.1 | 0.587 | 0.567 |
| bc | -53 | +68 | +69.2 | +1.2 | 0.140 | 0.127 |
No later target is substituted for a GI85 row. The seven caption rows without a GK91 counterpart remain visible; missing later targets are not counted as agreement.
Reading the deviations
The five largest direct-angle differences are:
b cbar 1P: model +69.2 deg, paper -53 deg (Delta +122.2 deg); same-n projection leaves 0.00% norm in other radial rows.u sbar 1P: model +4.3 deg, paper +34 deg (Delta -29.7 deg); same-n projection leaves 0.02% norm in other radial rows.u sbar 1G: model +10.8 deg, paper +33 deg (Delta -22.2 deg); same-n projection leaves 0.00% norm in other radial rows.u sbar 1D: model +12.8 deg, paper +33 deg (Delta -20.2 deg); same-n projection leaves 0.03% norm in other radial rows.u sbar 1F: model +12.6 deg, paper +32 deg (Delta -19.4 deg); same-n projection leaves 0.00% norm in other radial rows.
Near-degenerate singlet/triplet diagonals remain sensitive to small matrix changes; when the physical low/high labeling is reversed, the complementary angle in the diagnostics is the relevant comparison. The explicit outside- projection percentage distinguishes that convention issue from genuine cross-radial composition in the complete block.
The antisymmetric element is itself cancellation-sensitive: its vector color-magnetic and scalar/Thomas pieces usually have opposite signs. The survival percentage below is |vector + Thomas| / (|vector| + |Thomas|). A small value indicates sensitivity to the relative strength of the two Appendix-A spin-orbit kernels; it does not establish an implementation error.
Diagnostics (raw block data)
Corrected (pre-mixing) diagonal masses E_s = E(^1L_L), E_t = E(^3L_L), same-n off-diagonal c = c_vector + c_Thomas, the two assigned physical masses, projected angle, complementary angle (low/high labels exchanged), and norm carried by other radial rows. All angles already use the paper convention (identity mapping).
| Sector | nL | E_s (GeV) | E_t (GeV) | E_s - E_t (MeV) | vector c (MeV) | Thomas c (MeV) | total c (MeV) | survives (%) | low (GeV) | high (GeV) | theta (deg) | complement (deg) | other radial norm (%) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| u sbar | 1P | 1.3518 | 1.3657 | -13.90 | +7.88 | -8.93 | -1.05 | 6.3 | 1.3516 | 1.3656 | +4.32 | -85.68 | 0.02 |
| u sbar | 1D | 1.7905 | 1.8038 | -13.30 | +4.11 | -7.31 | -3.20 | 28.0 | 1.7896 | 1.8044 | +12.85 | -77.15 | 0.03 |
| u sbar | 2P | 1.8972 | 1.9279 | -30.73 | +2.30 | -10.62 | -8.31 | 64.3 | 1.8953 | 1.9302 | +14.21 | -75.79 | 0.03 |
| u sbar | 1F | 2.1306 | 2.1428 | -12.19 | +2.44 | -5.31 | -2.87 | 37.0 | 2.1300 | 2.1434 | +12.60 | -77.40 | 0.00 |
| u sbar | 2D | 2.2383 | 2.2544 | -16.13 | +1.93 | -8.78 | -6.85 | 63.9 | 2.2359 | 2.2571 | +20.17 | -69.83 | 0.03 |
| u sbar | 1G | 2.4224 | 2.4329 | -10.54 | +1.60 | -3.70 | -2.09 | 39.4 | 2.4220 | 2.4333 | +10.82 | -79.18 | 0.00 |
| c ubar | 1P | 2.4575 | 2.4634 | -5.95 | -25.35 | +29.10 | +3.75 | 6.9 | 2.4557 | 2.4652 | -25.79 | +64.21 | 0.00 |
| c ubar | 1D | 2.8272 | 2.8341 | -6.96 | -14.64 | +28.89 | +14.25 | 32.7 | 2.8160 | 2.8453 | -38.14 | +51.86 | 0.00 |
| c sbar | 1P | 2.5503 | 2.5516 | -1.33 | -18.00 | +21.46 | +3.46 | 8.8 | 2.5474 | 2.5545 | -39.56 | +50.44 | 0.00 |
| c sbar | 1D | 2.9101 | 2.9159 | -5.78 | -10.74 | +23.15 | +12.41 | 36.6 | 2.9003 | 2.9258 | -38.45 | +51.55 | 0.00 |
| b ubar | 1P | 5.7788 | 5.7811 | -2.33 | -31.68 | +33.40 | +1.72 | 2.6 | 5.7779 | 5.7820 | -27.97 | +62.03 | 0.00 |
| b sbar | 1P | 5.8582 | 5.8587 | -0.45 | -24.95 | +26.60 | +1.65 | 3.2 | 5.8568 | 5.8601 | -41.12 | +48.88 | 0.00 |
| b cbar | 1P | 6.7470 | 6.7412 | +5.87 | -9.42 | +6.81 | -2.61 | 16.1 | 6.7402 | 6.7480 | +69.16 | -20.84 | 0.00 |
Paper sources
u sbar: Fig. 4 caption.c ubar: Fig. 7 caption.c sbar: Fig. 7 caption.b ubar: Fig. 9 caption.b sbar: Fig. 9 caption.b cbar: Fig. 9 caption.