Nonlocal Models of Goldstone Bosons ... Conclusions
8.1 - Summary and Speculations
We have found that our minimal model (including current quark masses
and minimal nonlocal coupling to external electroweak gauge fields)
predicts coupling constants for the effective chiral Lagrangian that are in
surprisingly good agreement with the phenomenological chiral Lagrangian
coupling constants extracted from experiment. We have also calculated the
decay constant of a hypothetical techni-Higgs particle, but there is no way
to test this result at the present time. The dynamical quark mass is used as
a nonlocal order parameter for spontaneous chiral symmetry breaking, and
it acts as a natural regulator in the quark loop integrations, rendering all
but one of the parameters of the effective chiral Lagrangian finite.
However, we have no a priori argument to show that our model is an
approximation to quantum chromodynamics (QCD). We have virtually
ignored gluons, except for their role in chiral symmetry breaking. Most
importantly we have not made any attempt to account for confinement^1.
The obvious question to ask is: "Why does the model work?". To attempt
to answer this question we turn to a comparison with other models of
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 . This is not true for
other mesons like the rho .
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 .
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  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 . 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
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 , 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 . 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
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.