INFN - Laboratori Nazionali di Frascati, May 1995
From: The Second DAFNE Physics Handbook
Chapter 10: Two Photon Initiated Processes
Eds. L. Maiani, G. Pancheri, N. Paver
Supported by the INFN, by the EC under the HCM contract number CHRX-CT920026 and by the authors home institutions.
G.Pancheri INFN,Laboratori Nazionali di Frascati,I00044, Frascati (Rome), Italy
cm
,
is only logarithmically divergent as 
.
In this note we present a simple expression which can be of use to approximate the differential and integrated cross-section for the process
in the forward region. This process, the so-called radiative Bhabha scattering, is used at electron-positron colliders as a luminosity monitor [1] and the relevant cross-section is well established in the literature [2,3,4,5,6]. Our aim in this note is to derive an expression which is a good approximation to the exact results, and from which one can easily estimate the expected total rates.
As discussed in the previous section in this handbook, process (1)
constitutes an important background to 
or 
[7].
In order to disentangle
this process from the hadronic background due to the annihilation
channel, it is being considered to provide the planned detectors
with tagging facilities in the forward region.
In the forward region, only small values of the momentum transfer
are considered and the no-recoil approximation can be applied.
In this approximation, and to lowest order in 
, the cross-section for
process (1), can be written as
where the elastic electron-positron cross-section is given by
and 
is the probability
that photons with momentum k are emitted.
The relativistic invariants
for the elastic scattering process, are defined as
and the probability 
is given by[8,9]
with
for radiation emitted by particles of 4-momentum 
. The
factor 
according to the charge of the emitting
particle or antiparticle : the (-) sign corresponds to
photon emission from an electron (positron) in the final(initial)
state, and the (+) sign to emission from an electron (positron)
in the initial (final) state.
Upon integration over the photon directions, one obtains the energy spectrum as
with E the beam energy and
the electron's scattering
angle. This photon spectrum
correctly describes collinear and almost collinear photons,
but only in the soft photon approximation. To include hard
collinear emission, one can make the substitution
The function 
is obtained performing the angular integration
on the photon direction
with the null 4-vector, 
. Since
is a relativistic invariant[8], it can
be conveniently expressed in terms of the Mandelstam variables, i.e.
with
In the large energy, fixed t limit, the terms with s and u dependence cancel out[10], since
and the radiative spectrum 
takes the simple form
i.e.
The function 
is plotted in fig.1 together with two
curves which approximate its behaviour for small and large values of 
.
One notices that,
as t goes to zero, 
also goes to zero,
i.e.
In order to obtain the total rate, one integrates eq.(2) over the momentum transfer t, between the two limits obtained from the exact kinematics of process (1) for small angle scattering. In the forward region the maximum value for the momentum transfer -t is approximated as
where 
is the maximum angular opening of the electron detector
in the forward region.
For the experimental set up at DA
NE,
the largest scattering angle allowed by the small angle tagging system,
SAT, is
expected to be 
.
Figure 1: The radiative spectrum 
is plotted as a full line for the exact
expression from eq.(12), and for the two
approximations respectively valid at small (dots) and large -t/
(dashes).
For the smallest momentum transfer (see the Appendix ) one obtains
The total cross-section in the forward region,
with a radiative energy loss of at least 
, is now given by
with
While the exact integration
of eqs.(14) and (15) contains non leading 
terms, the main
contribution to the integral comes from the small
t region, where
the differential cross-section for process (1) takes the
form
We have used the small t, large s limit for the elastic
electron-positron cross-section 
and have neglected
the second (higher order term) in the expression for 
.
This expression shows how
the radiative
spectrum regularizes the 
singularity.
We notice that this phenomenon is
a coherence effect, obtained from
an exact cancellation between all the terms of eq.(9), including
the constant ones. Because of this cancellation, the singular, small t
behaviour
is softened and the cross-section exhibits only a
logarithmic singularity.
A particularly simple expression, for the total radiative Bhabha cross section, can be obtained in the soft photon approximation, i.e.
with 
.
Using the kinematic limits, with 
and 
as described,
the integrated cross-section takes the form
with 
. Retaining only the leading logarithmic
contribution, one then gets
with
and 
. The LLO expression just obtained coincides with
the one from ref. [2,3], in the same limit. For radiative Bhabha
scattering, from all four charged particles, these authors obtain the
differential expression
whose integrated rate coincides with eq.(18) in the small x limit. The total rate of electrons hitting a small angle detector along one of the beam directions, is obtained (i) by halving the above cross-sections, since only photons from either electrons or positrons are detected in single tagging [11] and (ii) by multiplying with the expected luminosity.
We find that
the expected rate of electrons which have lost an energy larger that 70 MeV
is of the order of 30 MHz, corresponding to 0.03 electrons per nanosecond,
an acceptable rate, which should not interfere with the tagging of electrons
from the radiative 
process
of interest at DA
NE.
A more precise and exact calculation for both the integrated and the differential cross sections can be found in the next section of this Handbook.