Mixing angles: a paper erratum
The captions of Figs. 4, 7 and 9 of the 1985 paper quote thirteen singlet–triplet mixing angles for strange, charmed and
The comparison
In the paper's convention (lower state
| system | GIModel | GI 1985 caption | Godfrey–Kokoski 1991 |
|---|---|---|---|
| +4.3° | +34° | +5° | |
| −25.8° | −41° | −26° | |
| −39.6° | −44° | −38° | |
| −28.0° | −43° | −31° | |
| −41.1° | −45° | −40° | |
| +69.2° | −53° | +68° |
The GIModel column is from the recorded report, computed with the paper's oscillator method (OscillatorSolver()). The finite-difference solver agrees within 0.1° (see Heavy-light mesons and mixing). Most higher states (
Why the implementation is not at fault
1. A real bug was found and fixed first. The first version of the code applied the smeared contact interaction only to S waves. Appendix A and Eqs. (23)–(26) of the paper keep it in every partial wave, because smearing gives it a finite range. After the fix, the singlet–triplet gaps of all eight comparable two-state systems matched the paper and the median angle difference fell from 33° to 15°. What remained was concentrated in the antisymmetric spin-orbit term.
2. The same authors' later calculations agree with GIModel. The Godfrey–Isgur model with the 1985 parameters was used again in later papers. Their angles agree with GIModel within a few degrees, including the sign of the
S. Godfrey and R. Kokoski, Phys. Rev. D 43, 1679 (1991), Table I:
, , , , , . An early draft of this paper (July 1986) already contains these angles, together with the same contact, spin-orbit and tensor expectation values that GIModel gives, and presents them as the 1985 calculation without comment. R. Blundell, S. Godfrey and B. Phelps, Phys. Rev. D 53, 3712 (1996):
consistent with the Godfrey–Kokoski value. S. Godfrey, Phys. Rev. D 70, 054017 (2004):
. S. Godfrey and K. Moats, Phys. Rev. D 93, 034035 (2016):
and . S. Godfrey, K. Moats and E. S. Swanson, Phys. Rev. D 94, 054025 (2016):
and .
3. The published masses are consistent with the model. A P-wave multiplet of an unequal-mass meson is described by four spin-dependent strengths: symmetric spin-orbit, tensor, contact, and antisymmetric spin-orbit. Extracting them from the paper's own printed masses and angles, the first three agree with GIModel in all eleven systems. Only the antisymmetric strength implied by the caption angles differs, by factors of 4 to 18. The masses barely depend on that strength, because the diagonal gap dominates; the angles depend on it directly.
The conclusion
The captions of Figs. 4, 7 and 9 appear to contain angles that the 1985 calculation itself did not produce. Where they came from is unknown. GIModel is not tuned toward them: its parameters and operators are the paper's, and the reproduction reports keep both the 1985 captions and the later values as comparison targets, never replacing one with the other.
When you compare with the 1985 paper, compare mixing angles with the later Godfrey–Isgur values as well as with the captions.
Still open
The small