The processes of electron-positron annihilation into photon pairs and of pair creation by photons are of interest both theoretically and experimentally. Electron-positron annihilation takes places with high rates at colliders such as LEP, SLC and TRISTAN. It is therefore important to account for this reaction correctly in order to carry out reliable analyses of experimental data. Moreover this is also important for the development of new, higher energy electron-positron colliders and for the planning of new experiments because this process is a large source of background. Theoretically the process is well understood ever since it was first calculated by P.A.M. Dirac in 1930. We shall see later that the study of annihilation into photons is also a first step towards the understanding of a similar process, the annihilation of quark-antiquark pairs into gluons, which has acquired importance in elementary particle physics in recent years.

When you enter the annihilation process in the CompHEP session
you will see that only one Feynman diagram is generated
(Fig. 4). This should come as some surprise since
you probably know that by crossing you should expect each of the two Compton
diagrams to give rise to one Feynman diagram for the annihilation process
(see, for instance, [2] Section 7.8, [1] Section 6.15).
The reason for this is simple: the two diagrams which you expect have
only different 4-momenta and polarization vectors attached to the
final state photons, *topologically* however they are identical.
The single diagram displayed by CompHEP is therefore equivalent to
two diagrams with labelled external lines. When you let
CompHEP carry out the squaring of the amplitude and then inspect
the squared diagrams, you will see that there are two squared diagrams
and not three. One of these represents the two topologically identical
diagrams with different photon momenta and polarizations, the second
one is the expected interference term.

Now carry out the analytical calculations of the squared matrix element
applying the familiar procedure of invoking option
*Symbolic calculations*. We can explore the annihilation process by
performing a numerical experiment using the option
*Numerical calculator*.

Let us begin by choosing a CMS energy of 64 GeV, characteristic for the
electron-positron collider TRISTAN. The total cross section will be
displayed as
. Assuming the nominal luminosity of TRISTAN,
, such a cross section means the
production of one photon pair every 40 seconds. This is a high rate,
more than an order of magnitude greater than for any other annihilation
process at this energy. Therefore the two-photon annihilation
could turn out to be a very intensive source of background for other
physical processes. However let us look at the photon angular
distribution using option *Angular dependence*. You will see the
curve reproduced here in Fig. 5. The main feature of this
distribution is the forward/backward symmetry which is always observed
when there are two identical particles in the final state. Furthermore
we note a very strong peaking at
and .
Experimentally this implies that most of the photons escape in the beam pipe.

To get a more quantitative picture about the narrow peaks at the edges
let us store the results in the form of a table using option
*Save result in a file* and then view the table with
the help of function key `F5`. The differential cross section equals
at
and drops
by nine orders of magnitude between
and
, and stays flat between
and . Next let us cut out a typical pair of cones around
the beam pipe, where particles remain undetected (see section
), using option *Set angular range*:

f-angle between in - e1 and out - A cos(f): min = -0.996200 max = 0.996200

This corresponds to an angular cut of . As a result the total cross section is reduced to . This means that about 80% of the photons escape undetected in the beam pipe or, conversely, that the number of photons actually seen in the detector is significantly less than the total number of photons produced by annihilation. Nevertheless this rate is still large and such events can be a significant background to other reactions.

Let us calculate now the energy dependence of the total cross section for
three cases: for the whole angular range and for the angular intervals
(
)
and
(
).
To get the corresponding plots use the menu *Parameter dependence*
and select the options *Total Cross Section* and *Energy*.
The package will prompt you for the
energy range you want to examine, for the scale (logarithmic or
linear) and for the number of points (21 by default). Let us choose
the following values:

Parameter : Energy min 10 max 100 Scale norm Number of points 21

As soon as these parameters are entered, the numerical calculation
is done automatically. When the calculation is completed you can
inspect the results using options *Show plot* and
*Save result in a file*. The latter option produces tables in the
form of files with the names *tab_N.txt*, where
. Here in Fig. 6 we have
combined the three curves in one plot.
From these curves we infer that the
total cross section has the smooth behaviour characteristic of
collisions of structureless
particles. The drop with energy is independent of the angular range.
Presently we shall see that it is close to a law.

To get the analytical answer for the squared matrix element you must repeat
all steps which we have made in the case of Compton scattering for both of
the squared amplitudes and then add the results to get the final expression:

You can check that the squared matrix elements have the same functional
form in the two cases of Compton scattering and of the annihilation
process. To get one formula from the other one has to exchange the two
Mandelstam variables, and and take account of some additional
factors. One of these is connected with the Fermi statistics of
electrons and positrons. As there is one fermion in the final state of
Compton scattering and there are no fermions in the case of annihilation,
those two matrix elements have opposite signs (see section ).
There is furthermore a factor of which arises in the annihilation
channel because of the presence of two identical particles, the two photons,
in the final state (see section ).
Such very strong similarity between the matrix elements of the processes
under consideration is based on the fact that the same set of Feynman
diagrams contribute. This is a consequence of a fundamental property
called *crossing symmetry*.^{3}In the case of processes we can say that crossing symmetry
relates the process to the process
, where and denote the antiparticles of and .
These processes are represented by the same set of Feynman diagrams.
However you have to remember that in CompHEP Feynman diagrams are
constructed
without permutation over final identical particles (this operation is
performed in later steps of the calculation).

The differential cross section of annihilation is given by

(see the normalization in Section ).

It remains to integrate the differential cross section at fixed energy
over the kinematical range of ,

(see section ) and hence get the following formula for the total cross section:

At high energies,
, this simplifies to the asymptotic
formula

and you see that the total cross section decreases with increasing energy almost as fast as (cf. Fig. 6).

Finally consider the third reaction which is related by crossing symmetry to Compton scattering and to electron-positron annihilation, i.e the creation of electron-positron pairs in collisions, . This is the process reversed in time with respect to the annihilation reaction. There are two Feynman diagrams representing the scattering amplitude. According to crossing symmetry the squared matrix element has the same dependence on the kinematical variables as the squared matrix element of the annihilation process but it is twice as large because of the absence of the combinatorial factor of that was needed in the case of annihilation to account for the two identical particles in the final state. We leave it to the reader to verify this statement by doing the next problem.

**Problem.**Using the CompHEP REDUCE output obtain analytical answer for the squared matrix element of process and compare it with that of the annihilation channel.

- ... symmetry.
^{3} - for a detailed study of crossing symmetry consult, for instance, [2].