It is not incorrect to state that G
PT is more general just because it
admits a considerably wider range of values of certain low energy constants and
of the current quark masses than the standard scheme. On the other hand, the
standard 
PT claims that the values of the 
constants 
,...,
are well under control, both determinig them from data [2]
and estimating
them via resonance saturation [20,21]. Similar claims are
often made about the
values of the light quark masses [22]. The standard values of the low
energy parameters can indeed be justified within the set of assumptions
underlying the standard 
PT. However, beyond this framework, the same
experimental data, the same sum rules, etc..., can often yield rather different
results. We start by discussing the constants 
from the point of view of
G
PT.
5a - 
and 
These two constants appear in 
, which is the part
common to both 
and 
. Their measurements
via the electromagnetic radius of the pion and the radiative decay 
[2] should not be altered
in G
PT. Furthermore, 
and 
are, respectively, related to two and three point functions of vector and axial
vector currents dominated by vector and axial vector meson poles
[21,23]. This makes
the estimates of 
and of 
rather stable.
5b - 
, 
and 
These remaining three constants of 
are more difficult to
measure. Although they are not directly related to explicit symmetry breaking
effects, they enter the observables (
form factors [24], 
D-waves [2]) together with r-dependent loop corrections. Consequently,
for lower values of r, the standard determination of 
, 
and
is slightly modified [25], by not more than a factor of two.
The estimates of these constants via resonance saturation is also more involved
since, in addition to vector and axial vector mesons, 
, 
and 
receive a contribution from scalar exchanges, which are not known so well
[20]. The
reason of this complication is the fact that, unlike 
and 
, 
,
and 
are related to four point functions of the vector and axial
currents.
5c - 
and 
Let us concentrate on the 
constant 
(
violates the
Zweig rule) and, for simplicity, let us stick to the leading large 
behaviour, denoting the leading part of 
by 
. One has (Eq.
(32)),
where we have used (30). The standard determination of 
[2]
(or, equivalently, of 
including
the chiral logarithms) would replace 
by 
, leading to the value 
. Within G
PT, this last step could be misleading,
provided r is well below 
. Using instead the leading order
formula (26a) and neglecting the Zweig rule violating parameter 
, one
obtains
Hence, the value of 
one extracts from Eq. (48) crucially depends
on the quark mass ratio r: For 
one gets the ``standard
value'' 
, whereas for r decreasing down to
, 
increases up to infinity. The estimates of 
via
resonances merely concern scalar exchanges, whose description is considerably
more ambiguous [26] than in the case of vector and axial vector mesons
[21].
5d - 
, 
and 
The values of these 
constants are at the heart of our
discussion of symmetry breaking effects. 
violates the Zweig rule in the
channel and it will not be discussed here. The combination 
can be related to the deviation 
from the
Gell-Mann--Okubo mass formula [2] (see Eqs. (10) and (32)). The standard
evaluation of 
based on this relation suffers from a similar bias
as in the case of 
: The expression for 
involves the
factor 
in the denominator. As a result, the value
of 
is even more sensitive to r than 
, and it can
actually be considerably larger than the standard value [2] 
. A separate measurement of 
is usually based on
the relation with the isospin breaking quark mass ratio R (34), which is
known from different sources [2,9,14]. In G
PT, the relation
between 
and R
can easily break down, for the same reason which could invalidate the
elliptic relation
(36): The importance of unduly neglected 
contributions. As a
consequence, the standard value [2] of 
can be underestimated by as much as two orders of magnitude. Writing
the renormalization group invariant mass parameter 
can be estimated from
Eqs. (26a,b). In the standard scheme (
), 
is expected to be at a
GeV scale, whereas in G
PT, 
can be as small as 
, or even
smaller.
The estimate of 
from resonance contributions can be obtained from
the two point function (3) which satisfies the superconvergent sum rule
[26]
Saturating this sum rule by nothing but the pion and a single 
state of
mass 
,
one obtains 
GeV, leading to a value for 
which is
of the standard order of magnitude 
. Unfortunately, the above
argument does not by itself support the standard picture, and can be
turned around: If
, the pion contribution (
) can hardly dominate the
pseudoscalar component of the sum rule (51). In order to balance the scalar
contribution, one has to include an excited 
state of mass
, 
MeV. It is then easy to see that the mass
parameter 
in Eq. (50) can take any value between zero and 
[26].
Quite generally, the introduction of J=0 resonances, compatible with chiral
symmetry and with the short distance properties of QCD correlation functions,
into the effective
lagrangian [20] is more tricky and more ambiguous
than in the case of J=1 states [21]. The authors of Ref. [20]
have, for
instance, decided to disregard the contribution of the 
nonet.
Doing so, they have a priori eliminated the low 
alternative.
The standard 
PT rewrites the expansion in quark masses as an expansion in
powers of 
. In the vicinity of the
antiferromagnetic critical point, 
, the coefficients of the latter
expansion, viz. 
...
, blow up, and the expansion has to be redefined.
G
PT is precisely such a redefinition. At the leading order, it is
characterized by a single undetermined parameter not present in the standard
scheme, the quark mass ratio 
. One may say that G
PT
parametrizes the deviations from standard predictions of 
PT in terms of
the deviation of r from its standard value 
[2].
5e - Running light quark masses
In G
PT, the decrease of the quark mass ratio r from 
to
is likely to be interpreted as an increase in the value of the
running quark mass 
by a factor of 
. The order of
magnitude of the mass differences 
and 
should
remain essentially unchanged. In this connection, it should be stressed that
all existing estimates of 
use in one way or the other the
assumption 
. This is obvious for those approaches which deduce the
value of 
from the estimates of 
[8,9]. It is however even true in the case of direct quantitative
determinations of 
from QCD sum rules [17]. The latter
express the square of 
as a weighted integral of the
spectral function (43). Nothing is known experimentally about 
beyond the one pion intermediate state contribution. The only existing attempt
to fill this gap makes use of the low 
behaviour (44) of
as given by the
standard 
PT in order to normalize the whole 
contribution
represented by a broad 
Breit-Wigner peak. In this way, the
result 
is obtained [17].
If, instead,
the 
contribution is normalized using the G
PT result (45), the
above value of 
is increased by a factor 
. (Let us
note in passing that within G
PT, the possibility of having 
appears even less likely than in the standard case [14].)
In general, G
PT admits and expects a larger absolute strength of the
divergence of the 
axial current away from the pion pole than the
standard scheme could possibly tolerate. The physical origin of this increase
would be ascribed to the importance of the ``
contribution'',
which can and should be checked experimentally: The 
component of the spectral function (43) can be measured in high statistics
decay experiments [18], and in this way, the
determination of 
could be put on a solid experimental
basis.
Concluding this section, it is worth emphasizing that nothing in the preceeding
discussion indicates that the standard determination of the 
's and of the
light quark masses is internally inconsistent. The standard 
PT together
with the standard value of the low energy parameters is a perfectly
self-consistent scheme. However, it is not the only possible consistent
scheme and it does not contain its proper justification.