To estimate the values of the 
coefficients
we study the 
behaviour of the vector-vector correlation
function where one can compare the theoretical predictions with
experimental results.
The two-point vector correlation function is defined as
where 
is the flavoured vector quark current:
with
the Gell-Mann matrices normalized as 
.
Lorentz covariance and 
invariance imply for the
the following structure:
where 
, with 
euclidean.
is zero at all orders in the chiral limit.
For 
the following expression holds at NPLL order [1]:
where the running of 
and 
at NPLL order can be
extracted from eq. (14) with 
terms set to zero.
The real part of the invariant 
function is related to
its imaginary part through a standard dispersion relation
For a review on QCD spectral Sum rules and the calculation of QCD two-point
Green's functions see [5].
The imaginary part is given in terms of the experimentally known total
hadronic ratio of the 
annihilation in the isovector channel
with
We have performed a comparison between the QR model parametrization
(20), valid in the energy region 
,
and the prediction obtained from a
modelization of the experimental data on 
[6]
in the channel with the 
meson quantum numbers (I=1, J=1).
For a determination of the function 
in the
high 
region (i.e. beyond the cutoff 
) see [7].
We adopted the following parametrization of the experimental hadronic isovector ratio:
This is a generalization of the one proposed in ref. [8], where
corrections due to the finite width of the rho meson have not been included.
KeV is the 
width and
is the total widht of the neutral 
[9].
We used the leading logarithmic approximation for 
:
Expression (24) includes a dependence of
the 
channel upon the 
width and the contribution from the
continuum starting at a threshold 
[8].
For the running of 
we used a value of 260 MeV
for 
, according to the average experimental value
MeV [9] and with 
flavours.
The results are practically insensitive to the
running corrections and our leading log approximation turns
out to be adequate.
To extract information on 
coefficients of
the NTL logarithmic corrections
we made a best fit of the first derivative of the 2-point function coming
from the parametrization (24) of the experimental data:
where the derivative of the VV function in the QR model is given by:
We have used
MeV for the IR cutoff and 
GeV for
the UV cutoff, determined by a global fit in ref. [2].
In fig.(1) we show the 
behaviour of the derivative
of the experimental 2-point function, the curve from the best fit,
which has been done in the region: 
GeV, and
the derivative of the ENJL prediction with quark-bubbles resummation.
Figure 1:
The derivative of the experimental vector invariant function 
(solid line)
the curve from the best fit in the region 
GeV (dashed line)
and the ENJL prediction (dot-dashed line).
The best values of the two free coefficients are
The 
of the fit has been defined as 
and the 
are defined assuming a
of uncertainty on the experimental data.
A 
has been obtained.
The ENJL prediction differs by roughly a 
from the experimental curve
at 0.8 GeV. Most of this discrepancy can be accounted for with the
corrections that we have calculated.
The invariant function 
obtained from the best fit automatically
match the ENJL function at 
, because we have normalized the
corrections to vanish at 
:
The
function obtained with the values (28) and with the
matching of eq. (29) is plotted in fig.(2) and compared
with the ENJL prediction (i.e. including the resummation
of linear chains of quark bubbles and including only logarithmic corrections).
Figure 2:
from the QR model (dashed line), obtained from the
fitted derivative of fig. (1) by imposing the matching
with the ENJL function at 
, versus 
from the ENJL model (solid line).
The difference between the two curves reaches a 
at 0.7 GeV.
