Far from resonance peaks, the role played by V-mesons can be reproduced
by other dynamical mechanisms. This is the case for charge radii and
their related slope parameters, both defined around . In VMD,
the previously discussed expressions for
lead to simple
formulae and reasonable values for the pseudoscalar charge radii
These values satisfy the
VMD-relation but are only
in rough agreement with data (eqs.(14) and (15)).
The situation can easily be improved, as discussed in ref. [15],
and extended to the transition F.F. and their slope parameters (b
and
). In all
these cases (the
case is somehow exceptional, as we already
discussed), the V-meson mass fixes correctly the scale with
near
.
But constituent-quark model ideas can also be useful in this
field. A simple quark-loop gives finite predictions for charge radii,
which are quite close to those from VMD provided the quark mass is
fixed to , near the constituent quark-mass value
(see, ref. [16] for details and related work). SU(3)-breaking
can similarly be introduced through the
mass difference.
Further reasonable predictions are also known for the slope parameters
[16] thus supporting the general
idea of
- or QCD-duality, i.e., that VMD (or mesonic)
contributions and (fermionic) quark-loop effects are somehow dual or
equivalent and, in particular, cannot be added [18].
ChPT is, however, the more valid and QCD inspired approach to account
for the pseudoscalar charge radii. The charge radius is
unambiguously predicted to be
near experiment. For
charged pions and kaons one obtains reasonable values and the
VMD-like relation
.
Improvements and further details can be found in ref. [15] in
these proceedings.