In the context of non-relativistic quark potential models, our results may perhaps seem reasonable. In potential models, the pion is the most tightly bound meson, and as such the pion wavefunction should be the least sensitive of meson wavefunctions to any long range confining potential, since the pion wavefunction is smaller at large distances than any other meson wavefunction. Exactly how sensitive the wavefuntion is to the presumed confinement potential will of course be model dependent. Another relevant point is that potential model calculations of meson scattering reveal that the confinement part of the potential contributes only a small fraction of the pion scattering amplitude [80]. This is not true for other mesons like the rho [80].

Comparison with non-relativistic potential models begs the question: "Why do non-relativistic potential models work?". Manohar and Georgi has suggested that the answer might be that quarks are non-relativistic [81]. The idea is that the quarks gain "constituent masses" of order 300 MeV through spontaneous chiral symmetry breaking, and that the effective coupling of these "constituent quarks" is small. The hydrogen atom analogy suggests that the "constituent quark" velocity should be equal to the effective coupling alpha_s = g^2/4 pi , so an effective coupling alpha_s « 1 implies slowly moving, non-relativistic quarks. Glashow has estimated this effective coupling by examining the ratio of colour and electromagnetic hyperfine splittings in baryons, he found [82] alpha_s approx 0.28. If the effective coupling is really this small, then quarks are non-relativistic, and most gluon corrections would be small, which may explain some of the success of our model.

It is also interesting to compare the results of our model with those of the vector meson dominance model [59]. This model determines the coupling constants of the chiral Lagrangian by assuming that the only contribution comes from vector and axial-vector meson exchange. The fact that this model arrives at similar values for the coupling constants (see table 6.3) as our model which involves calculating quark loops may be a little surprising, but it is very reminiscent of the fairly old concept of quark-hadron duality [83,84]. The idea of duality is that a "suitably" averaged physical hadronic cross section should be approximately equal to a cross section calculated using perturbative quarks and gluons. Duality becomes an exact correspondence in the limit that the quarks are weakly bound^2.

A more refined version of duality is given by QCD sum rules [84,85]. At the simplest level, the use of QCD sum rules involves calculating hadronic process in terms of quark and gluon loops, and including non- perturbative corrections (eg. insertions of vacuum condensates of quark and gluon fields). QCD sum rules are not successful in describing radial excitations or mesons with large spin [84], but they are successful in describing low-lying mesons with spin \le 2. Sum rule ideas suggest that this success results from these mesons being relatively insensitive to confinement effects [84]. We can see that there is a vague relation between our model and the sum rule approach in that we attempt to include dynamical quark mass, which is a non-perturbative effect that is related to the quark condensate (eq. 2.7).

For future work it would be interesting to see what the model predicts for non-leptonic weak decays of pseudoscalar mesons (eg. K --> pi pi). It is also may be possible to apply our approach to the case of radial excitations of the GB's. It is well known that there are potentially many solutions to the Schwinger-Dyson equation for the self-energy [78d]. These different solutions are thought to belong to different vacua of the theory. In the ladder approximation, it can be shown that the solution that corresponds to the stable vacuum (corresponding to the smallest action) should be monotonically decreasing function of Euclidean momentum, and positive [78e]. The other solutions for the self-energy cross through zero at least once; these solutions should correspond to meta-stable vacua. If this is so, then we would expect that fluctuations of the order parameter in these meta-stable vacua correspond to GB's of these vacua. Such "meta-GB's" would be heavier than the standard GB's simply because the meta-stable vacua have higher energy densities than the stable vacua. Thus a tower of meta-stable vacua would imply a tower of "meta-GB's". An interesting observation to make is that the "wavefunctions" (that is the corresponding S(p)'s) have different numbers of nodes, as would be expected of radial excitations.

The conclusion that we might draw from this work is that so far the minimal model which represents GB's as fluctuations of a nonlocal order parameter is in good agreement with experiment, but there are more comparisons with experiment to be made.

Footnotes for Chapter 8:

^1 The form of S(p) for small momentum p (corresponding to large distances) may contain some information about confinement, but as we saw in section 6.2 the loop integrals are not sensitive to small momenta.

^2 For example, this is the case when the quark mass is very large, and the running coupling at the scale of the quark mass is small.