Conventions and units

This page collects the conventions used throughout the package. Signs of amplitudes and mixing angles depend on them, so check here before comparing a number with another calculation.

Units

quantity	unit
masses, energies, momenta	GeV
distances	(  fm)
decay widths from decay_width	MeV
M1 transition moments	nuclear magnetons
E1 and M2 photon amplitudes
two-photon amplitudes

Field names carry their unit when it is not obvious: mass_GeV, central_GeV, contact_shift_GeV, beta_GeV. The parameter file stores quark masses and the constant in MeV (m_c_MeV, c_MeV); the loader converts them to GeV.

State labels

A level is written   and labeled in code as "n^(2S+1)L_J", for example "1^3P_2":

• n ≥ 1 counts radial excitations, starting at 1 in every partial wave (so the lowest P wave is 1P, not 2P);

• the multiplicity is 1 for spin singlets and 3 for spin triplets;

• L is the letter S, P, D, F, G for  .

Parity and charge conjugation follow from the quantum numbers:    and, for self-conjugate mesons,   .

After mixing, a physical state keeps the label of the unmixed level it is assigned to, by ascending mass. The label identifies the state; its composition is given by physical_components.

Quark order

Meson(mq, f1, f2) is the meson: f1 is the quark and f2 the antiquark. Meson(mq, :c, :u) is (a ) and Meson(mq, :u, :s) is (a ).

The masses are symmetric, but one sign is not: the antisymmetric spin-orbit operator contains   and changes sign when quark and antiquark are exchanged. Mixing angles of and are therefore opposite.

Mixing angles

For unequal-mass mesons, the two   states mix. GIModel writes the lower physical state as

with the spin-orbit angular matrix element       , where is the spin of the quark (f1). This matches the convention in the captions of Figs. 4, 7 and 9 of the paper. The coefficients are read directly from physical_components or from state.mixings[1].components.

With this convention the model gives positive angles for (light quark, heavier antiquark) and negative angles for , , and (heavy quark, lighter antiquark), as in the paper.

Phases of wavefunctions

• Radial waves. Every radial wave is normalized with its outermost lobe positive. A radially excited wave therefore changes sign at the origin once per node. The large-radius convention matches the one stated after Eqs. (B35)–(B36) of the paper.

• Orbital states. Spherical harmonics follow the Condon–Shortley convention.

• Angular-momentum coupling. States couple orbital angular momentum first, then spin: .

• Spin and flavor wavefunctions. Singlet and triplet spin states follow Eqs. (B26)–(B29), and flavor states follow Eqs. (B1)–(B15), including the signs of charged and heavy-light mesons.

• Mixed states. Every mixing eigenstate, spectroscopic or annihilation, is chosen to have a positive overlap with the basis state it is assigned to, so a small admixture with a negative sign does not flip the whole state.

An overall phase of an external state changes the sign of an amplitude but never a width. Relative signs between interfering amplitudes are fixed by the conventions above.

Flavor states

The light quarks are degenerate, so the model does not distinguish from dynamically. The flavor label matters in two places:

• Charges. Photon emission and leptonic currents need explicit :u or :d; the averaged :q is rejected there.

• Isospin states. A neutral pion is  . Build it from :u and :d spectra as a PhysicalState with the two components (see Strong decays beyond the paper).

For isoscalar annihilation mixing, the nonstrange channel is Meson(mq, :q, :q) and stands for  .

Pseudoscalar emission

The operator of Eq. (19) uses the phase conventions of Appendix C of the paper: helicity amplitudes with the emitted meson along , converted to partial waves as     . For identical daughters, exchange symmetry   removes forbidden waves and the allowed ones carry a factor . The width is

in GeV, returned in MeV by decay_width.

All mass-mixing eigenvectors use positive overlap with their assigned unmixed state, ordered by unmixed mass. If that overlap is zero, the largest component sets the phase. This convention is shared by spin-orbit, tensor, and annihilation mixing. diagonalize_mixing_block(...; phase_anchor=1) explicitly requests a common first-row anchor when comparison with another convention requires it.