




Pomeron in Elastic Scattering











Fazal-e-Aleem and Sohail Afzal Tahir

Centre for High Energy Physics
University of the Punjab
Lahore-54590, Pakistan








Contributed to
Physics in Collisions, Frascati, Italy
June 1998









Pomeron in Elastic Scattering





Fazal-e-Aleem and Sohail Afzal Tahir

Centre for High Energy Physics
University of the Punjab
Lahore-54590, Pakistan
E-mail: faleem@hotmail.com





ABSTRACT

Pomeron has been an object of intense study these days in elastic and deep inelastic
scattering. We still do not know where it is a soft or hard Pomeron. In our study we
analyze the Pomeron in elastic scattering taking the entire data into consideration.





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Strong interactions continue to be a challenge in theoretical high energy physics. Problem is
even more difficult to handle in the non-perturbative or soft region [1]. Some attempts have been
made to study the same in the purturbative region using QCD [2]. In order to have a deeper
understanding of this problem pomeron is being employed within the framework of Regge theory
and to relate it to QCD. The pomeron is related, in a hadronic scattering process, to the rightmost
singularity of the partial scattering amplitude in the complex angular momentum plane. The
scattering amplitude is associated to the presence of an exchanged object with vacuum quantum
numbers, which leads to a power law, increase of the total cross section with centre of mass energy.

Experimental Measurements for Elastic Scattering

Current status of the measurements of various parameters for elastic scattering is as follows:
The total and differential cross section, T and d/dt, elastic cross section, el; the local slope
parameter, B and ratio of the real and imaginary parts of the scattering amplitude, have been
measured by several authors [3-24] at CERN-ISR, CERN-SPS, and FERMILAB. Experiments
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currently active in pp elastic scattering are - CDF, E710 [25] and E811 at FERMILAB.
Measurements from the Cosmic ray data corresponding to LHC energy [26] have also been reported
recently for pp elastic scattering. There is general consistency of the experimental data measured at
different colliders except the CDF results at FERMILAB. Their results are T = 80.03  2.24 mb, B =
16.98  0.25 (GeV/c)2, el = 19.70  0.85 mb at s = 1.8 TeV. This is significantly different from
E710 results of T = 72.2  2.7 mb, B = 16.99  0.47 (GeV/c)2, el = 16.6  1.6 mb. Recently E-811
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collaboration has measured pp elastic scattering in the small momentum region. The data is being
analysed currently and results at 1.8 TeV are expected shortly. These results will be of quite
significance in the light of discrepancy of CDF and E710 data.

Pomeron in S-Matrix theory

The pomeron was first introduced in the early sixties in the framework of the complex
angular-momentum theory, to describe high-energy soft processes. After expanding the scattering
amplitude in partial waves, it is assumed that the partial wave amplitude a(j,t) is dominated by an
isolated, simple, moving pole located in the complex angular-momentum plane j at some value (t),
a(j,t) = (j) / ((j - (t)),
using a well-defined mathematical prescription known as the Sommerfeld-Watson transform, it is
found that the asymptotic behaviour of the elastic-scattering amplitude A(s,t) in the limit s  m2, t
0, s is given by
A(s,t) = () () (s/s0) ,

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where (), known as signature factor, can be written as
() = {1 + exp (i) }/ sin , =  1 .
In the above equation, (t) is known as the residue function, (t) is called Regge trajectory and
s0 is a scale parameter.
A single pole exchange (i.e. a simple moving pole in the complex angular momentum plane)
is the simplest ingredient normally used as a basic part of the dynamics and it leads to predict a
power behaviour of the scattering amplitudes, More realistic models may involve more complicated
j-plane structures, either generated by unitarity (Regge cuts) and/or coming from a more involved
input (like a double pole) [27]. Perturbative QCD calculations indicate that the pomeron may be a
complicated j-plane structure. Phenomenology of hadronic reactions, on the other hand, also points
independently, towards a complicated j-plane structure whereby integrated hadronic cross-sections
grow as Ins or In2s (a kind of growth traditionally attributed to cuts in the complex angular-
momentum plane).
.There are many confusing ideas about the pomeron . One of the most persistent is that there
are two pomerons - a soft and hard. Soft (used to explain diffraction phenomena) and hard (used to
explain small x phenomena in DIS and calculated from perturbative quantum chromodynamics,
pQCD). There are different competing approaches for the explanation of soft and hard Pomeron.
More specifically, Pomeron can be visualized as a complicated entity which in different dynamical
situations may have different manifestations but whose origin is always the same, diffraction. The
twofold interpretation of single object or phenomenon. It however has a clear origin: the
conventional pomeron studied in hadronic physics is a soft phenomenon, out side the range of
applicability of perturbative quantum chromodynamics. What has come to be known, as the hard
pomeron is something that can be calculated from perturbative quantum charomodynamics.. It is
likely that there is only one object and the diversity of its manifestations reflects merely the diversity
of the reactions in which it can occur and of the physical and kinematics situations in which it is
investigated. More specifically, we visualize the pomeron as a very complicated entity, which will, in
fact, be a function of different sets of variables depending on the reaction one is looking at. In
different dynamical situations it may have different manifestations with the same origin. In our paper
we will take up the role of Pomeron in elastic scattering. The recent small-x data collected at HERA
were interpreted by some authors as a manifestation of the hard pomeron, and thus as an argument in
favour of the existence of two pomerons [28]. However, much better data, more conclusive analyses
and even better understanding will perhaps help us in having a clearer picture of the Pomeron in the
near future
We now know that elastic and total cross section data which poses much problem for the
physicists specially in the diffractive region is well described by Regge theory. In addition to

Pomeron, Regge trajectories A2, , , ... are exchanged at lower energies (10< s < 53 GeV) [29].
However, as the energy increases only the pomeron with an intercept 1.07 can account for the total
cross section data up to Tev energies. In literature there are various values for the soft pomeron


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which lye in the range of 1.07  0.02. This is mainly dependent on the choice of range of
experimental data and error bins at very high energies. It is also true for the differential cross section
in the diffractive region. For the large momentum transfers Pomeron alone is not sufficient and
contribution of other trajectories or Odderon (specially in the vicinity of dip region) is desired. At the
same time it appears that contribution of the cuts is also essential.
The focus will also be on the recent progress made in understanding of the role of soft
interactions in the hard processes, in particular in deep inelastic lepton scattering. This will include
the new data on low-x deep inelastic scattering and diffraction from the experiments H1 and ZEUS
at HERA with particular emphasis on theoretical developments and perspectives. We will also take
in to account the expected results with higher luminosity measurements expected from HERA, Run
II at Fermilab Tevatron collider, COMPASS, polarized RHIC and LHC in years 2000 and beyond.
In general the process is deeply related to non-perturbative phenomena and the only possibility to
understand as much as possible of it is to consider processes involving the scattering of highly virtual
photons or onia-particles.





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