In the context of the experiments at the 
--factory
an accurate knowledge of the radiative Bhabha scattering
cross section is mandatory
because the process constitutes the main background for
physics experiments with
tagging facilities [12,13].
Theoretical predictions are required for
various experimental configurations which include an extremely very forward
angular set--up, as can be seen from the following list:
, 
, with
few mrad, 
MeV and
MeV, both totally inclusive on the positron and
the photon and with acceptance cuts on them
(this includes: single tagging rates
for small angle scattered electron (SAST),
double tagging rates for both electron and positron at small angle (SADT),
coincidence rate of electron and photon in the forward direction (SAEPC));
, with 
of order
few MeV;
;
),
but accompanied by
a photon of energy 
at a minimum angle
from any of the final state fermions (LLDT).
Although analytic results for differential and total cross sections of
are available in the literature
(refs. [15]--[17] is an indicative list of such calculations),
a detailed analysis of all single and double tagging configurations
demands an exact evaluation of the matrix element in
the very forward direction, where the momentum transfer is of order 
and the 
scattering angles 
,
with 
at DA
NE.
Unfortunately, in these extreme kinematical conditions the
squared matrix element of ref. [18], with the finite mass corrections
obviously included, is inapplicable, namely it leads to an unphysical
negative result for hard photon
emission collinear to the very forward electron direction. Indeed, as discussed
in [19], the CALKUL collaboration's result [18]
is valid only in the limit 
whereas in the very forward direction the minimum four--momentum
exchange 
is given by:
up to 
. Moreover,
as a consequence of the approximation used in ref. [18],
``off--diagonal'' terms of the form
,
associated to initial--final state interference, are missing. However,
for very small electron scattering angles they become of the same order of
``diagonal'' terms
or 
describing initial and final state radiation respectively.
This is due to the fact that in the very forward region initial
and final radiation cones overlap and interfere and therefore
the contribution of ``off--diagonal'' mass
terms become more and more important when the electron scattering angle goes to
zero.
For the above reasons, we computed by using SCHOONSCHIP [20]
the complete squared matrix element
of the (gauge invariant) t--channel diagrams associated
with electron radiation. Positron radiation and electron and positron
interference can be safely neglected being suppressed by the
stringent constraints on the electron energy.
Then the full expression for the
squared matrix element, including all
, 
and 
terms,
reads as follows [19]:
where
with 
, 
. The other invariants
are defined as follows:
where 
and 
are the four--momenta of the incoming and
outgoing electron (positron). The squared
matrix element (24) coincides
with the result obtained independently in [21].
Based on the analytic result (24), the
semi--analytical program
PHIPHI [22] has been developed in order to obtain
phenomenological predictions of interest at DA
NE. Some results are
shown in Fig. 3 -- 5 together with comparisons with those
existing in the literature.
In Fig. 3 the total cross section of small angle single
tagging at DA
NE obtained by integrating the electron
energy over the range 
= 190 MeV is shown
as a function of the
maximum electron energy 
. The solid and dashed lines
correspond to the integration of
the spectrum 
of ref. [16]
in two different domains of the electron scattering angle,
namely 
mrad (dashed line) and
mrad (solid line). Our results
are represented by the open circles
and squares showing a fully satisfactory agreement
with the integration of the analytical result of [16]. For the energy
and angle intervals of interest at DA
NE, i.e.
MeV and
MeV with 
mrad the total cross
section is about 35, 50 mb respectively.
Fig. 4 shows a comparison for the
total cross section of SAST between the recent results
of ref. [23] and ours obtained with
mrad , 
mrad and
mrad and the electron energy within
= 190 MeV,
as a function of the maximum electron
energy 
. More in detail, in ref. [23] an approximate formula
for the total cross section of SAST in the forward region,
with a radiative energy loss of at least 
and momentum
transfer in the range 
and 
, is given as:
with
and the function
takes the form:
As can be seen by Fig. 4, the approximate
result derived in [23] provides a good estimate of the SAST
cross section with an accuracy of order 5--10 % when the kinematical
configuration proposed
in ref. [13] (
mrad,
250 MeV 
440 MeV) is assumed. In the limit of small
electron scattering angles (
mrad) the relative deviation
decreases as 
increases, consistently with the soft, collinear and
no--recoil approximation adopted in ref. [23].
The total cross section for photons emitted in the forward
region is shown in Fig. 5. The results are obtained by integrating the photon
energy 
over the range
as a function of the
minimum photon energy 
.
The different curves correspond to 
in the regions 
mrad
(dashed line), 
mrad (dotted line),
mrad (dash--dotted line),
(solid line). The physical
content of Fig. 5 is that the radiation is essentially emitted
within a cone of half--opening angle of 2 mrad, as expected from the
collinear nature of the bremsstrahlung mechanism.
Further numerical results for the total cross section of small angle electron--photon double tagging, of small--large and large--large electron--positron double tagging can be found in [19]. Our results, obtained by the code PHIPHI, has been checked very recently by the Monte Carlo generator BBBREM [24] and found in very good agreement [25]. In particular, for the unconstrained kinematics, we also agree with the single photon spectrum and the total cross section derived in refs. [15,16].
We would like to point out, in conclusion, that
for the SLDT case, where for kinematical reasons
no mass divergences appear, the ultrarelativistic matrix element
of ref. [18] can
be safely used. In the LLDT case mass divergences do appear but the condition
is fulfilled so that complete CALKUL result
do apply. In all the other cases
(SAST, SADT, SAEPC and LS) one has both mass divergences and very small angles,
so that the full matrix element of eq. (24), including quartic and
sextic mass correction terms, must be used.
