Table VII Audit — Annihilation Amplitudes

All kinematic M and M_P values use the pinned PDG 2026 experimental registry. M̃ and S_L remain wavefunction integrals. NaN denotes an unavailable experimental assignment; no GI/model mass fallback is used. Width-reference measurements below are legacy comparisons, not updated PDG 2026 width inputs.

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.

Generated by julia GIPaper/checks/audit_table_vii.jl. All 61 canonical rows in the four subtables are accounted for: 57 are computed from the model wavefunctions and four hypothetical toponium rows are explicitly unavailable. The only fitted quantity anywhere is the charge-radius smearing exponent f = 0.2 (the paper's own π⁺ fit); every other amplitude is parameter-free.

Both slices run on the Eq. (17)-style smeared momentum integral over the jₗ-transformed radial wave. Phase convention throughout: each radial wave's outermost antinode is positive, so the wavefunction-at-origin — and every amplitude below — flips sign once per radial node. The overall sign of a single width is unobservable (a rephasing of |M>); matching the paper's alternating signs under one consistent convention demonstrates the node structure is right.

Gluonic decays (part c)

amp = sqrt(prefactor) · S_L(Ψ),   amp² = Γ
  Γ(¹S₀→2g) = 8π α_s²/(3 m_Q²) |S₀|²      Γ(³S₁→3g) = 40(π²−9)/(81 m_Q²) α_s³ |S₀|²
  Γ(³P₂→2g) = 32π α_s²/(45 m_Q²) |S₁|²    Γ(³P₀→2g) = 8π α_s²/(3 m_Q²) |S₁|²

S_L(Ψ) is the Eq. (17) smeared wavefunction-at-origin (wavefunction_origin_smearing), α_s = α_s(M) at the meson mass, m_Q the constituent quark mass. Waves come from the paper-order native finite-HO-basis full fixed-channel diagonalization of H_central + V_spin: the smeared contact for the S-waves (singlet/triplet split) and the literal A15-A16 spin-orbit+tensor operator for the ³P_J chi rows — the same treatment used for every other subtable below (W6). Amplitudes in MeV^(1/2).

16 gluonic rows scored; median |model|/|paper| = 1.05; signs agree on 16/16. The two hypothetical t-tbar rows (eta_t, zeta) are not modelled (no top constituent mass in the 1985 set).

decay	M (GeV)	α_s(M)	S_L	model MeV^½	paper MeV^½	ratio	sign
eta_c -> 2g	2.98	0.260	+0.3385	+4.945	+4.700	1.05	✓
psi -> 3g	3.10	0.256	+0.2645	+0.435	+0.420	1.04	✓
eta'_c -> 2g	3.64	0.237	-0.2136	-2.850	-2.700	1.06	✓
psi' -> 3g	3.69	0.236	-0.1964	-0.286	-0.280	1.02	✓
eta_b -> 2g	9.40	0.183	+0.8656	+2.915	+2.500	1.17	✓
Upsilon -> 3g	9.46	0.183	+0.6853	+0.223	+0.210	1.06	✓
eta'_b -> 2g	10.00	0.181	-0.5791	-1.927	-1.700	1.13	✓
Upsilon' -> 3g	10.02	0.181	-0.4956	-0.159	-0.150	1.06	✓
Upsilon'' -> 3g	10.36	0.180	+0.4324	+0.137	+0.130	1.05	✓
Upsilon''' -> 3g	10.58	0.179	-0.3981	-0.125	-0.110	1.14	✓
chi_2c -> 2g	3.56	0.240	+0.1272	+0.886	+0.880	1.01	✓
chi_0c -> 2g	3.42	0.244	+0.1851	+2.544	+2.500	1.02	✓
chi_2b -> 2g	9.91	0.181	+0.1778	+0.306	+0.350	0.87	✓
chi_0b -> 2g	9.86	0.181	+0.2639	+0.881	+0.820	1.07	✓
chi'_2b -> 2g	10.27	0.180	-0.1914	-0.327	-0.370	0.88	✓
chi'_0b -> 2g	10.23	0.180	-0.2640	-0.875	-0.820	1.07	✓

Leptonic decays (part a)

f = coefficient · factor
  P_P   = M⁻¹ M̃^(-1/2) K[√(m₁m₂/E₁E₂)]     (¹S₀ singlet wave)
  V_V   = M⁻² M̃^(+1/2) K[√(m₁m₂/E₁E₂)]     (³S₁ triplet wave)
  V'_V  = M⁻² M̃^(+1/2) K[(m/E)(1 − m/E)]   (³D₁ central wave)
  P'_A1 = M⁻² M̃^(+1/2) K[m·p/E²]           (³P₁ central wave)
  K[w]  = (2π)^(-3/2) ∫d³p (4π)^(-1/2) Φ_L(p) w(p)

coefficient is the printed quark-charge/flavor factor of the formula column (e.g. 2√3 for f_π, (16/3)^(1/2) for f_ψ); M is the experimental PDG 2026 mass from the shared registry; M̃ is the model mock mass <E₁+E₂> over the same wave (mock_meson_mass). The tabulated amplitude is dimensionless (the weak rows are f_P/M_P: the experiment column shows f_π/M_π = 0.95).

The formula column is for unmixed/ideally-mixed states; the paper's numerical column additionally folds in Table III isoscalar mixing, which is a small effect for the near-ideally-mixed ω/φ — part of the residual here. The hypothetical t-tbar ζ row is not modelled.

24 leptonic rows scored; median |model|/|paper| = 1.03; signs agree on 24/24.

decay	factor	M (GeV)	M̃ (GeV)	value	f model	f paper	ratio	sign
pi -> mu nu	P_P	0.140	1.650	+0.3939	+1.3644	+1.3000	1.05	✓
K -> mu nu	P_P	0.494	1.665	+0.1374	+0.4760	+0.4700	1.01	✓
D -> mu nu	P_P	1.870	2.553	+0.0389	+0.1347	+0.1300	1.04	✓
F -> mu nu	P_P	1.968	2.724	+0.0509	+0.1764	+0.1700	1.04	✓
B -> mu nu	P_P	5.279	5.756	+0.0096	+0.0333	+0.0330	1.01	✓
1^1S_0(bc) -> mu nu	P_P	6.274	7.143	+0.0272	+0.0941	+0.0910	1.03	✓
tau -> A1 nu	Pp_A1	1.230	1.290	+0.0351	+0.0993	+0.1100	0.90	✓
tau -> K*(892) nu	V_V	0.892	1.261	+0.0880	+0.3047	+0.3000	1.02	✓
rho -> e+ e-	V_V	0.775	1.090	+0.0809	+0.1982	+0.2000	0.99	✓
omega -> e+ e-	V_V	0.783	1.090	+0.0794	+0.0648	+0.0700	0.93	✓
phi -> e+ e-	V_V	1.019	1.435	+0.0978	-0.1129	-0.1100	1.03	✓
rhoS -> e+ e-	V_V	1.465	1.361	-0.0148	-0.0362	-0.0370	0.98	✓
rhoD -> e+ e-	Vp_V	1.720	1.393	+0.0133	+0.0154	+0.0190	0.81	✓
omegaS -> e+ e-	V_V	1.410	1.361	-0.0160	-0.0130	-0.0120	1.09	✓
omegaD -> e+ e-	Vp_V	1.670	1.393	+0.0141	+0.0054	+0.0060	0.91	✓
phiS -> e+ e-	V_V	1.680	1.645	-0.0246	+0.0284	+0.0270	1.05	✓
phiD -> e+ e-	Vp_V	NaN	1.655	+NaN	+NaN	-0.0120	—	✗
psi -> e+ e-	V_V	3.097	3.730	+0.0533	+0.1230	+0.1200	1.03	✓
psi' -> e+ e-	V_V	3.686	3.833	-0.0283	-0.0653	-0.0630	1.04	✓
psi'' -> e+ e-	Vp_V	3.774	3.804	+0.0064	+0.0070	+0.0110	0.64	✓
psi''' -> e+ e-	V_V	4.040	3.936	+0.0207	+0.0479	+0.0460	1.04	✓
Upsilon -> e+ e-	V_V	9.460	10.394	+0.0247	-0.0285	-0.0260	1.10	✓
Upsilon' -> e+ e-	V_V	10.023	10.404	-0.0159	+0.0184	+0.0170	1.08	✓
Upsilon'' -> e+ e-	V_V	10.355	10.460	+0.0130	-0.0151	-0.0140	1.08	✓
Upsilon''' -> e+ e-	V_V	10.579	10.520	-0.0115	+0.0133	+0.0120	1.11	✓
1^3D_1(bb) -> e+ e-	Vp_V	NaN	10.361	+NaN	+NaN	-0.0008	—	✗

Two-photon decays (part b)

A(P→γγ)   = √6   q_eff (α/m) (M/M̃)^(3/2) (1/2π)     ∫d³p φ_P(p) [m/E]
A(³P₂→γγ) = −√(4/5) q_eff (α/m) (M/M̃)^(3/2) (2/π)^(1/2) ∫dp p² Φ(p) [m·p/E²]

q_eff = Σ aᵢ eᵢ² is the state's effective squared charge (flavor amplitude times quark charges: (e_u²−e_d²)/√2 for π/A2, 4/9 for cc̄, 1/9 for bb̄/ss̄); α the fine-structure constant; M the meson mass, M̃ the mock mass. Amplitude² is Γ; units follow the paper (π in eV^½, the rest in keV^½).

The clean-flavor rows are below; the strongly-mixed isoscalar pseudoscalars (eta, eta', eta_r, eta'_r) follow in their own table (they use the final P1-mixed spectrum). f/f' use ideal tensor mixing (f₂ nonstrange, f₂' = ss̄); the hypothetical t-tbar eta_t is not modelled.

8 clean-flavor two-photon rows; median |model|/|paper| = 1.06; signs agree on 8/8.

decay	kind	M (GeV)	q_eff	model	paper	unit	ratio	sign
pi -> gamma gamma	P	0.135	+0.236	+2.507	+2.600	eV^½	0.96	✓
pi' -> gamma gamma	P	1.300	+0.236	-1.056	-1.000	keV^½	1.06	✓
eta_c -> gamma gamma	P	2.984	+0.444	+2.766	+2.600	keV^½	1.06	✓
eta'_c -> gamma gamma	P	3.638	+0.444	-2.381	-2.200	keV^½	1.08	✓
eta_b -> gamma gamma	P	9.399	+0.111	+0.732	+0.620	keV^½	1.18	✓
A2 -> gamma gamma	P2	1.318	+0.236	-1.197	-1.200	keV^½	1.00	✓
f -> gamma gamma	P2	1.275	+0.393	-1.898	-1.900	keV^½	1.00	✓
f' -> gamma gamma	P2	1.517	+0.111	-0.434	-0.250	keV^½	1.73	✓

Isoscalar-mixed pseudoscalars (η, η', η_r, η'_r) — a coherent sum over the final state's signed [1nn, 1ss, 2nn, 2ss] native components from compute_isoscalar_spectrum / physical_components (Sec. V A), using the physical mock-meson M_P. Ideal mixing cannot be used here: it inverts the η<η' ordering. Median |model|/|paper| = 0.78; signs agree on 3/3.

decay	M_P (GeV)	M model (GeV)	1nn	1ss	2nn	2ss	model keV^½	paper keV^½	ratio	sign
eta -> gamma gamma	0.548	0.531477	+0.624840	-0.771776	+0.110556	+0.041400	+0.436	+0.500	0.87	✓
eta' -> gamma gamma	0.958	0.999850	+0.567712	+0.552460	+0.591002	+0.152339	+1.020	+1.300	0.78	✓
eta_r -> gamma gamma	1.294	1.495649	-0.251628	-0.160238	+0.585749	-0.753595	-1.826	-2.700	0.68	✓
eta'_r -> gamma gamma	NaN	1.799422	-0.473245	-0.271060	+0.543500	+0.638102	+NaN	-2.200	—	✗

The η<η' γγ ordering — backwards under ideal mixing — is reproduced. Mixed-state signs use positive overlap with the assigned unmixed precursor; the remaining magnitude spread reflects the model's sensitivity to the P1 mixing amplitudes and the light-pseudoscalar masses. The hypothetical t-tbar eta_t is not modelled.

Charge radii (part d)

r_E² = Σᵢ eᵢ [ <rᵢ²> + (3/4mᵢ²) ∫d³p |φ(p)|² (mᵢ/Eᵢ)^{2f} ]

rᵢ = (mⱼ/M)·r is quark i's position from the meson center of mass, <r²> the position-space expectation over the ¹S₀ wave, and the second term the relativistic smearing of the quark position (f = 0.2, the paper's one fit, fixed on the π⁺). charge_radius_squared; values as signed r_E² (fm²) and the radius sign·√|r_E²| (fm).

2 charge-radius predictions scored (the π⁺ is the f fit anchor); median r_E²(model)/r_E²(paper) = 1.02; signs agree on 3/3.

meson	r_E² model fm²	r_E² paper fm²	radius model fm	radius paper fm	ratio	sign
pi+ (fit)	+0.4235	+0.4356	+0.651	+0.660	0.97	✓
K+	+0.3537	+0.3481	+0.595	+0.590	1.02	✓
K0	-0.1029	-0.0900	-0.321	-0.300	1.14	✓

Hypothetical top-flavor rows

These four printed Table VII predictions cannot be recomputed because Table II contains no top constituent mass and the active GI parameter set therefore has no toponium wavefunction. Substituting the current quark pole mass would define a new model rather than reproduce the 1985 calculation.

decay	paper value	unit	local result	reason
zeta -> e+ e-	+0.01	dimensionless	—	no Table II top constituent mass
eta_t -> gamma gamma	+1.5	keV^½	—	no toponium wavefunction
eta_t -> 2g	+1.1	MeV^½	—	no toponium wavefunction
zeta -> 3g	+0.10	MeV^½	—	no toponium wavefunction

Decay widths (Eqs. D7-D9)

Γ(P→ℓν)   = G² f_P² m_ℓ² / (8π M_P) · (M_P² − m_ℓ²)²        (D7)
Γ(V→ℓ⁺ℓ⁻) = (4π/3) α² M_V f_V²                              (D8)
Γ(τ→A₁ν)  = G² f_A1² m_τ³ M_A1²/(16π) [1−M_A1²/m_τ²][1+2M_A1²/m_τ²] (D9)

Table VII lists the decay constants f, not the widths; Eqs. (D7)-(D9) (leptonic_pseudoscalar_width, dilepton_vector_width, axial_tau_width) turn a constant into a width. The formulas are validated against experiment by feeding the measured constant:

check	fed	model width	measured	note
D7 π→μν	f_π/M_π = 0.936 (exp)	2.626e-17 GeV	2.528e-17 GeV	+4% is the Cabibbo cos²θ_C the reduced G² omits
D8 ψ→ee	f_ψ = 0.0896 (exp)	5.550e-06 GeV	5.55e-06 GeV	round-trip

Applied to the model's own constants (physical masses; the paper's reduced G²/α² carry no CKM or QCD radiative factor), the D8 dilepton widths carry the Table VII f_V residual squared:

decay	f_model	width (D8)	measured	ratio	note
rho -> e+ e-	+0.1982	6.791e-06 GeV	7.040e-06 GeV	0.96
psi -> e+ e-	+0.1230	1.045e-05 GeV	5.550e-06 GeV	1.88	QCD (1−16α_s/3π) not in D8
psi' -> e+ e-	+0.0653	3.511e-06 GeV	2.340e-06 GeV	1.50
Upsilon -> e+ e-	+0.0285	1.715e-06 GeV	1.340e-06 GeV	1.28

The paper tabulates decay constants; D8 widths are derived here using the same PDG mass as the corresponding constant. Radiative corrections are not included.

Reading

The comparison tests GI wavefunctions and operators at modern experimental kinematics. Differences may reflect both numerical reproduction and changed mass inputs. Model eigenvalues belong to the separate spectrum comparison. Experimental state correspondences and unavailable inputs are listed in the shared mass registry. No mass is selected to improve agreement.