The 
cross section at lowest order is obtained from
in eq.(25) and turns out to be
where s is the square of the total CM energy and 
nb. This lowest order cross section is shown
(dotted line) in Fig.(4).
Figure 4:
Cross-section for 
as a function of
the total CM energy : lowest order result (dots), including
counter-term contributions at the next order (dashes),
full ChPT prediction (with loop and counterterm
corrections) at next-to leading order (dot-dashes) and
the ``all-order'' result (full line).
Next order corrections in ChPT include the effects
of loops and counterterms as in eq.(28). For the latter, the
resonance saturation assumption implies 
, thus increasing the lowest order amplitude
up to a 
at 
GeV as also shown (dashed line) in
Fig.(4).
The associated loop corrections are given by 
in eq.(29).
These loop corrections are considerably smaller
than those coming from the corresponding counterterms (around a 15% in the
amplitude) and slightly increase the 
cross-section,
as shown (dot-dashed line) in Fig.(4). This curve represents the
full ChPT
prediction at next-to-leading order for 
and is
expected to reproduce future data in the low energy region.
Around the
resonance masses the cross-section is quite different as indicated by the
value at the 
-peak [20] 
nb,
shown with error bars in Fig.(4).
Attempts
to improve the situation in ChPT would imply the evaluation of higher order
loop corrections and the corresponding counterterms. In general and for
values of 
, next order loop corrections in ChPT are
found to be around 10-
of the preceding order amplitude,
smaller than those coming from other uncertainties in our model
and the values of its parameters.
By contrast, corrections coming from counterterms have been shown to be
larger than the ones from the loops
and the evaluation of higher-order ones, under our assumption of
resonance saturation, is trivial.
As before, the introduction of the whole vector-meson form-factor
(instead of its truncated series
) represents an ``all-order'' estimate of our
resonance dominated counterterms. Taking into account the physical finite
widths of the 
and 
mesons, this amounts to write
to first order in the loop corrections.
The corresponding prediction is also shown (solid line) in
Fig.(4). The agreement
at the 
-peak (161 nb vs 
nb from experiment
[20]) is essentially a consequence of having used 
, and 
MeV and 
[13], quite close to 
MeV and
as measured in [20] from the 
cross-section at the 
peak. Near this peak the
-contribution is obviously dominant, but at lower energies, loop
effects, counterterms and the 
resonance curve represent a substantial
fraction of the total cross-section.
The 
cross-section can be analyzed along the same
lines. Eq.(48) becomes
and the various contributions arising from this expression have been plotted in Fig.(5), with the same notation employed for Fig.(4).
Figure 5: Cross-section for 
as a function of
the CM energy, distinguishing different contributions near and
around the omega peak from
lowest order(dots), counterterms (dashes) and chiral loops (dotdashes).
The interference effects among the various terms are now important
even on the 
-peak.
Experimental data for 
in this energy region are
known [20] but affected by large error bars.
Turning now to the 
transition, 
, allowing
for the final state photon to be also off-mass-shell (
) as in
, one gets
Eq.(48) leads to the
cross-section plotted in Fig.(6) for
and 
, 
and 0 (dashed, solid and
dotted lines). One obtains an 
-peak cross-section 
, 
and 
nb,
respectively.
Figure 6: Cross-section for 
and
for the quoted values of 
and 
. VMD (solid line)
requires 
. The dashed line corresponds
to the alternative model with 
.
The corresponding experimental value 
nb
[20,17]
favours the first two possibilities but new experiments could
contribute to clarify the situation discriminating among the different ratios
. This is not the case for the
cross-section (also shown in
Fig.(6)) where the
predictions for different values of 
are quite similar due to the
dominance of small 
values which reduces the sensitivity on
as seen in eq.(48).
Moreover, for this particular process and well below the resonance region
one could expect non-negligible contributions coming from the scattering
channel 
with two spacelike photons [19].
In summary, 
, 
and
cross-sections at low energy seem particularly interesting to test
the saturation of the ChPT counterterms by resonances, which in
this case are expected to be the well-known 
and
vector-mesons.
