The model

This page describes the Hamiltonian that GIModel solves. It follows Sec. II and Appendix A of Godfrey and Isgur (1985); equation numbers refer to that article. The goal is to show which physical idea sits behind each column of a computed spectrum.

The idea in one paragraph

A meson is treated as a constituent quark and antiquark bound by a flavor-independent interaction. At long distance the interaction is linear confinement; at short distance it is one-gluon exchange with a running coupling. What makes the model relativized is that the kinetic energy is fully relativistic, and every potential is smeared over a distance of order the inverse quark mass and multiplied by momentum-dependent factors. These modifications stand in for relativistic effects without solving a Bethe–Salpeter equation, and they let one set of parameters describe mesons from the pion to the .

The Hamiltonian

In the meson rest frame, Eq. (1) reads

where and are the constituent masses and is the relative momentum. A tilde marks a relativized operator. The three potentials are:

term	physics	shown in the spectrum as
	spin-independent: Coulomb plus linear confinement	part of central
	color-magnetic contact and tensor interactions	contact, and the tensor part of fine str
	spin–orbit: color-magnetic plus Thomas precession	the spin-orbit part of fine str

The nonrelativistic starting point

Without relativization, the spin-independent potential of a color-singlet pair is

is one-gluon exchange with color factor and is the confining scalar potential with string tension and offset .

The running coupling

The strong coupling is parametrized in momentum space by three Gaussians (Eq. (12) and the Fig. 2 caption), which in position space become

The coupling vanishes as   (asymptotic freedom) and saturates at   at large distance. These six numbers are fixed by the paper and live in the code as constants; alpha_s_r and alpha_s_q evaluate the profile.

Smearing

The quarks are not treated as point particles. Each potential is convolved with a Gaussian of width (Eqs. (A7)–(A9)), where

with   GeV and  . Heavy quarks are more localized and are smeared less. contact_smearing_sigma returns .

Because is a sum of error functions, the smeared Coulomb potential has a closed form (Eqs. (A12)–(A14)):

Smearing the linear potential also gives a closed form,

Momentum-dependent factors

Relativistic kinematics also modify the strength of each interaction. Appendix A replaces the nonrelativistic factors by functions of  , placed symmetrically on both sides of the potential so that the operator stays Hermitian. For the Coulomb part of the central potential,

For each spin-dependent term , the factor is

with a separate exponent , , and for the contact, tensor, vector spin-orbit and scalar spin-orbit terms. With   the nonrelativistic becomes . The pairwise terms below use the appropriate mass pair (, or ) in both the smearing width and the factor.

Spin-dependent terms

All spin-dependent potentials come from derivatives of the smeared and , with and the quark and antiquark spins.

Contact hyperfine. The Laplacian of gives a smeared delta function:

Since   is for spin singlets and for triplets, this term splits from and from . Because the delta function is smeared, it acts in every partial wave, not only in S waves.

Tensor.

It is nonzero only in spin triplets with   on the diagonal, and it mixes    with    (for example with ).

Spin–orbit. The one-gluon exchange (vector) part and the Thomas precession from the scalar confining potential are

The two usually have opposite signs, so the net spin-orbit splitting is a partial cancellation.

When  , the spin-orbit operator has a part proportional to    that does not conserve total spin . It mixes the spin singlet with the spin triplet at  . This is why the physical and , or and , are superpositions of and . For equal masses (, ) the operator vanishes, and so does the mixing.

Annihilation mixing

For self-conjugate isoscalar mesons, the pair can annihilate into gluons and reappear as another flavor. This mixes , , and heavier states with the same quantum numbers. The paper models it with a matrix element proportional to the smeared wavefunctions at the origin (Eqs. (16)–(18)):

• Pseudoscalars (, ) mix strongly. Eq. (18) gives two versions, P1 and P2, each with a nonperturbative and a perturbative term.

• Vectors (, ) and tensors (, ) mix weakly, through the general Eq. (16) with amplitudes and from Table III.

Isoscalar flavor mixing shows how to compute this.

The three stages of the calculation

Godfrey and Isgur solve the model in stages (text near Eq. (14) and Appendix A). GIModel follows the same order:

1. Fixed sectors. For each , assemble the kinetic, central, contact, diagonal spin-orbit and diagonal tensor terms into one matrix and diagonalize it. This gives masses and radial wavefunctions that already include the spin-dependent forces.

2. Spectroscopic mixing. Using those eigenstates as a basis, add the off-diagonal tensor (  ) and antisymmetric spin-orbit ( ) matrix elements, and diagonalize each block.

3. Annihilation mixing. For isoscalars, add the flavor-changing annihilation matrix elements and diagonalize again.

In code, stage 1 is fixed_spectrum, stage 2 is add_intra_meson_mixing, and stage 3 is add_isoscalar_annihilation. compute_spectrum runs stages 1 and 2; compute_isoscalar_spectrum runs all three.

Numerical method

The paper solves stage 1 by expanding the radial wavefunction in harmonic-oscillator functions. It uses one oscillator scale per sector, chosen variationally, and enlarges the basis until the energies converge. Matrix elements of functions of are computed in momentum space, matrix elements of functions of in position space, and products are joined by inserting a complete set of states (Eq. (A17)):

GIModel implements this as OscillatorSolver. It also provides FiniteDifferenceSolver, which solves the same Hamiltonian on a radial mesh. The paper does not use this second method; it serves as an independent check. Solvers and convergence compares them.