The Schrödinger-Dirac equations became an established theory because of their ability to account for atomic spectra in the early stages of development. Similarly, any viable hadron theory must be able to account at least approximately for the meson spectra. Quantum Chromodynamics (QCD) (see e.g. [1] ), the main stream strong interaction theory, has failed to do this, after decades of work and lattice computations. Therefore, the low energy, nonperturbative end of QCD has to be abandoned.

On the other hand, the scalar strong interaction hadron theory (SSI) can approximately but adequately account for the masses of the ground state pseudoscalar mesons 0^- (singlet) and vector mesons 1^- (triplet) [2] [3] [4] . However, predictions of the spectra of the excited states of these mesons using the same linearized equations turned out to contradict data [5] ; the spacing between the energy levels according to (7) below turned out to be too small. The nonlinear strong interaction potential was called in to mitigate this difficulty phenomenologically ( [4] Section 5.5-7).

This difficulty is incompatible with a viable SSI. The purpose of this paper is to resolve it and provide predictions in rough agreement with data without the above phenomenology.

In SSI, the interaction potential between the quark and the antiquark in a meson is given in ( [2] 7.2, [4] 3.2.8),

Here, is the interquark distance vector and also denotes the “hidden” relative space, , the d_m’s integration constants of the fourth order differential equation ( [2] 6.9, [4] 3.1.11), g_s the strong interaction coupling constant, y₀ (singlet) and (triplet) the rest frame meson wave functions in. In the case of zero orbital momentum, l = 0, these wave functions are determined by ( [4] 3.2.5b, 3.4.1, 3.4.2a, 3.4.3)

derived from ( [4] 3.2.10b, 3.2.11a). Here m_p,r are quark masses of flavors p, r, J = 0 refers to singlet and J = 1 to triplet and (2) becomes ( [4] 3.2.17)

In early 1990’s, when the above work was in progress, potential models suggested a confinement potential of the Coulomb plus linear type ( [6] , [7] §14.3.2). The nonlinear Φ_cJ leads to linear confinement at large r ( [2] 7.11b [4] 3.2.19). The Coulomb term d_m/r was kept and the quadratic term d_m2r² dropped ( [4] 3.2.20). In this case, the linearized (4) with Φ_cJ®0 is of the same form as that for the hydrogen atom and the ground state solutions are given by ( [4] 4.3.1-3)

where the second subscript refers to radial quantum number n_r = 0. These wave functions, being plane waves in the laboratory frame, vanish when the normalization volume Ω à ¥ ( [4] 4.7.2) so the above assumption Φ_cJ ® 0 holds. The meson mass E_J0 is given by a slightly extended ( [4] 4.4.1)

This result with n_r = 0 and the empirical ( [4] 5.2.2) together with six pseudoscalar meson masses determine the five quark masses, d_m, and d_m0 ( [2] 10.2, [3] Table 1, [4] 5.2.3, Table 5.1). The results are summarized in Table 1 below.

Table 1 and (7) predict many other ground state pseudoscalar and vector meson masses with good approximation ( [3] , [4] 5.2.3, Tables 5.3-5).

However, for radially excited pseudoscalar and vector mesons, n_r ³ 1 and (7) shows that the spacings between successive radially excited meson masses, analogous to those between excited states in a hydrogen atom, are too small and are decreasing rapidly with increasing n_r, contrary to data ( [4] Tables 5.6-7). In an attempt to remove this discrepancy, an assumption ( [4] 4.7.5) was made in which the normalization volume Ω à Ω_c is made finite in (6) for the excited states. The wave functions in the nonlinear Φ_cJ in (5) now no longer vanish. Since Φ_cJ is a positive quantity, it will increase the meson masses.

Not being able to treat this complex nonlinear problem, Φ_m(r) in (1) was replaced by unknown parameters ( [4] 5.5.2) which are determined by using data points, the masses of chosen excited mesons. In this way, the results in ( [4] Tables 5.6-7) were obtained, after having spent many data points.

Obviously, the above treatment failed to account for the spectra of excited mesons.

In reviewing the above treatment, it is seen that there is no compelling justification to drop d_m2r² in (1), (4), as was done below (5), except that the linearized (4) with Φ_cJ = 0 turns out to have no converging solution. Therefore, put d_m = 0 and keep the quadratic confining d_m2r² in (4). The solution analogous to (6) is of harmonic oscillator type and reads

where Ω denotes the nomalization box in ( [4] 4.2.8) for the ground state (8b).

The series in (8a) terminates when

Here, the other root leads to divergent wave function at r = 0 and is dropped. Now, (7) is changed to

which differs from (7) only in the last term. The above ground state results for n_r = 0 mentioned in Table 1 and the two lines below it can now be taken over if the last two columns in Table 1 are replaced by

Comparison of (7) to (10) shows that the latter gives much larger spacings between the excited states. Application of (10) to the radially excited states in ( [4] Table 5.6, 5.7) are given in Table 2 and Table 3 below.

The predicted masses E_th are 7% - 18% smaller than the measured ones E_exp in Table 2 and 4% - 18% in Table 3. These differences may tentatively be attributed to the neglected nonlinear potential Φ_cJ(r) of (5) in (4) in order to arrive at (10). Actually, (4) and (5) have been solved numerically using an iterative procedure. ( [2] Section 10, [3] Section 7), also explained in ( [4] Section 5.6), for the first and second radially excited singlet and triplet mesons in Section 2 (d_m2 = 0). The wave functions as well as the associated nonlinear potential Φ_cJ(r) are plotted in ( [2] Figure 1 and Figure 2, [3] Figure 2 and Figure 3). These computations depend upon the choices of unknown parameters, the amplitudes of the wave function in Φ_c in (2) or the finite sizes of the normalization box Ω_c of ( [4] 4.7.5a) or, equivalently, N_CJn of ( [3] 4d) which is a volume integral over y_J(r)². Therefore, the contribution of Φ_cJ(r) to E_th can presently not be uniquely determined. More strictly, an dependence is implicitly introduced in (4.7.4a) so that the general wave function y_J(X,x) of ( [4] 3.1.5) is no longer separable in the laboratory coordinates and relative coordinates, rendering the problem not manageable.

Qualitatively, follow Section 5.5 of [4] and let Φ_cJ(r) in (4) be replaced by constants. Correct predictions are achieved if shown in the parentheses ( ) in Table 2 and Table 3. If the nonlinear potential contribution is small relative to the quadratic confining potential, << the last term in (10). Then (8a) and hence also Φ_cJ(r) of (5) still hold approximately; becomes independent of flavor or quark masses and be a constant in each column of Table 2 and Table 3. The values DE²/4 in the parentheses are however not constant in each column. This due to that they are not small but comparable to the last term in (10) = 3d_h, 5d_h, 7d_h... = 0.21, 0.35, 0.49… Therefore, is flavor-dependent and this dependence, as well as that for y_J, increase with decreasing normalization volume Ω à Ω_c, which generally decrease with increasing mass. These are seen from the numbers in the parenthese (..) in Table 2 and Table 3 in which DE²/4 become large for heavier mesons. Thus, the nonlinear potential Φ_cJ does contribute to the meson masses, in this case by 9-18% mentioned above.

The wave function is given by ( [4] 3.4.2a), , where θ and j are angles and l ³ 1. in (2) now depend upon these angles and (5) and hence also (4) no longer hold. For r ® 0, the wave function (8a) µ r^s which vanishes for s = l ³ 1 and is therefore independent of the angles. For large r, the r² term in (1) dominates over of (2) which by itself is proportional to r ( [4] 3.2.19). In these both limits. the equivalent of ( [4] 3.4.2b) analogous to (4) reads

As was mentioned in ( [4] §5.7.1), the two solutions for small and large r determined from (12) are independent of the angle θ and are to be connected by an unknown solution dependent upon both r and θ in the intermediate r region. This nonseparable problem starting from ( [4] 3.2.10b) is, as in [4] §5.7.1, beyond the reach of the present work.

Not being to treat this problem adequately, follow the procedures that led to Table 2 and Table 3 in order to obtain some estimates. Linearizing (12) by putting Φ_c0 = 0, (10) with n = 0 and s = l is obtained. It gives the masses of the mesons in Table 5.8 of [4] listed in Table 4 below.

These results are not surprisingly coarser than those in Table 2 and Table 3 because the neglected nonlinear Φ_c0 in (12) actually contains angle dependence in the intermediate r region; more equations are required. This turns out actually to be the case; the predicted masses E_th are now 14% - 38% smaller than the measured E_exp, as compared to 9% - 18% for the radially excited mesons. Again replace Φ_c0 in (12) by constants and put it equal to shown in the parentheses (..) in Table 4. The values in them are now much greater than the last term in (10) = 3d_h, 5d_h, 7d_h... = 0.21, 0.35, 0.49…, showing that the nonlinear potential F_c0 contributes much more to the masses than does the quadratic confining potential.

Like the behavior mentioned at the end of Section 4, ΔE²/4 here also increase with the meson masses and, in addition, also with increasing orbital angular momentum l.

Note that the predicted values in the second columns in Table 2 and Table 3 are the same for the same quark content. This is due to that (10) cannot distinguish between s = J = 1, l = 0 in Table 3 from J = 0, s = l = 1 in Table 4. The differences are due to that the neglected F_c1 and F_c0 are different; the latter also depends upon the angles in the intermediate r region. The associated radial and orbital quantum numbers n_r and l, meaningful at r ® 0 and ¥, are coupled in that region and may lead to a new pair of quantum numbers.

For orbitally excited triplet mesons he classification ( [4] §5.7.2) remains the same.

At very low energies, classical mechanics fails and has to be replaced by quantum mechanics, which can however go back to classical mechanics when the energy is sufficiently high. This is not true in the reverse direction. Similarly, QCD fails at low energies and has to be replaced by an appropriate low energy theory, here the SSI, which analogously can go over to QCD in a high energy region [8] , Chapter 14 of [4] . Again, this is not true in the reverse direction.

The role of the nonlinear potential (2) and (5) needs be investigated in an attempt to remove the discrepancies given in the parentheses (…) in Tables 2-4.

The book [4] remains the same up to equation (3.2.19); but (3.2.20), d_m2 = 0, needs be changed to d_m = 0 which will lead to changes in rest of the book. Thus, the decay rate calculations in Chapters 6-8 need be revised.

Hoh, F.C. (2017) On Excited Meson Spectra in the Scalar Strong Interaction Hadron Theory. Journal of Modern Physics, 8, 1127-1133. https://doi.org/10.4236/jmp.2017.88073