Let us discuss the properties of the partial wave amplitudes, 
, and what
we need
to know to calculate them [13,14,3,4]. Firstly, these amplitudes are
analytic functions of
s. They have a left hand cut starting at s=0 from the pion exchange Born term and
then other cuts running to the left from 
generated
by 
, 
and other exchanges of mass M. Of course, only the
nearby part of the left hand cut from s=0 to 
is really known
from one pion exchange [3]. For 
,
form-factor damping in the
vertex (where 
or 
)
as well as other exchanges affect the left hand cut
discontinuity.
The partial wave amplitudes also have a right hand cut generated by
final state interactions. At low energy, the only possible strong interaction
is 
. Then the final state pions will scatter strongly back to 
.
Indeed, it
is in this way that the cross-section for 
can readily become non-zero :
, where the first process in the chain can occur by the
Born term.
Fortunately, such effects are exactly calculable, thanks to two body unitarity.
Above inelastic threshold, which effectively means above 
threshold near 1
GeV, 
can also go to 
, which in turn can scatter back to 
. Though
unitarity still constrains these contributions, one needs information on
and 
scattering, as well as 
, to know how. This means
it is more
difficult to implement the constraints of unitarity when many channels enter.
However below roughly 1 GeV, elastic unitarity enforces
Watson's theorem [15] that makes the
phase of the partial waves for 
for each I and J, 
, equal to the
phase of
the corresponding 
amplitude, 
. This is exceedingly useful,
since knowledge
of the phase of an amplitude largely determines the behaviour of its modulus
--- amplitudes being analytic functions. A simple example of this is the phase
rising rapidly from
to 
. The modulus then has to peak at a position and width
wholly correlated with the phase variation. This relationship is exemplified
by the
well-known Breit-Wigner formula. The general relation between the phase and
the modulus of the amplitude is embodied in the Omnès representation [16]. Thus
knowing the phase of a partial wave amplitude, 
, from
threshold to infinity, we can define a function
(the Omnès function)
where in the region of elastic unitarity 
, the 
phase shift, independent of the two photon helicity
.
This function, 
, has the phase 
by construction.
The way convergent dispersion relations work, 
at low energies
fortunately does not require detailed knowledge of the phase 
above 1 GeV.
We will return to this later.
How this information can be used to compute 
scattering is explained
in detail
in Ref. 6. Here we sketch the methodology.
We can write an appropriately subtracted dispersion relation for each
partial wave amplitude, specified by I, J and 
. For the S-waves
(J=0, 
), these are twice subtracted [17] : two subtractions
to suppress
the dependence at low energies on what the distant left and right hand cut
discontinuites are. That is, there should be only a weak
dependence on both higher mass cross-channel exchanges and the
phases of the 
amplitudes above 1 GeV. The lack of knowledge of these
terms is parametrized by two subtraction constants that are fixed by two
crucial low energy constraints. Firstly, Low's theorem that states each partial wave amplitude
equals its corresponding Born term at s=0 --- that follows from QED gauge
invariance. Secondly, from chiral dynamics we have a prediction
in the low energy region for the
amplitude minus its Born term. Thus, for instance the neutral channel
S-wave has a zero at 
---
in one loop
Chiral Perturbation Theory (
PT ) [18,19] this is at 
.
In general,
all we know is that these near threshold corrections
are 
. These low energy limits fix the two
subtraction constants, 
, in the I=0 and 2 
S-wave amplitudes. Thus
The functions 
have the complete left hand cut
and no right hand cut. They are given by
where 
is the one pion exchange Born term and
,
denote the contributions
to the left-hand cut generated by exchanges with 
and 
quantum numbers, respectively.
Then the combinations in the charged and neutral channels are :
According to Low's theorem :
,
so in turn this means :
Recalling 
(Eq. (1)), the S-wave
Born amplitude
is, for example, [1,3]
with a cut for 
.
The fixing of subtraction constants by appeal to chiral dynamics only affects the S-wave amplitudes. The higher partial waves satisfy essentially once subtracted dispersion relations on dividing out their known threshold behaviour, so that only Low's QED theorem is needed for these waves. Thus [3]
where the 
are the appropriate partial waves of the left hand cut
components of Eq. (8).
Using Eqs. (3, 5-10), the cross-sections can
then be deduced from these partial
wave amplitudes.
It is useful for our later discussion, though not necessary for our low energy calculations, to note that these partial wave amplitudes , for all J, can be expressed quite generally [3] as :
where the function 
has the left hand cut, Eq. (8),
and 
the right hand cut, Eq. (5), and 
is a
real polynomial. While 
is the full left hand cut function, it is often
convenient to model this function by some 
(for instance, the
Born term) and rewrite Eq. (11) as
and 
(with 
labels suppressed)
may now be more complicated functions along the
left hand cut, but they will continue to be smooth along the right hand cut away from their
singularities. In physical terms, the first term models the simple well
understood cross-channel exchanges and then 
incorporates the
rest as direct channel contributions. Near 
threshold,
is just the pion exchange Born term and there are
essentially no direct channel effects other than those of final state
interactions automatically included in Eqs. (11,12). There
to a good approximation. It is this fact that allows the low energy cross-section to be accurately predicted as we discuss in the next section.
One can go to slightly higher 
masses, 500 MeV or so, by taking
in Eq. (12) with 
still zero. However, above
that energy, direct channel effects ( or equivalently heavier mass
cross-channel exchanges ) become increasingly important and definite
predictions possible near threshold must give way to fitting
data [3,4,11].
This is necessary to determine the form of 
and
hence the direct channel couplings.
Figure 5: Integrated cross-section for
as a function of the 
invariant mass, 
. The data are
from Crystal Ball [3]
scaled to the full angular range by a factor of 1.25.
The line marked 
is the prediction of one loop Chiral Perturbation Theory [18,19].
The curve marked 
" is the dispersive calculation
using Weinberg phases [20],
while that
labelled 
" are the predictions
from experimental 
phases
as described in [6] --- both with just 
-exchange
for the left hand cut.
