Although the data are well described by single pomeron exchange, an ever-increasing cross-section will ultimately violate what physicists call unitarity ­ the probability of something happening becomes greater than 100% ­ clearly impossible. This is avoided by the exchange of more pomerons as the energy increases. These contribute with alternating signs and with increasing strength, and ultimately damp down the energy dependence.

A fundamental theorem due to the French physicist Marcel Froissart says that the total cross-section cannot increase faster than log2s and multiple pomeron exchanges can ensure this. The Froissart bound is asymptotic and is not relevant at present energies. At present energies the effect of multiple pomeron exchanges is rather small. Even at Fermilab Tevatron collision energies of 1.8 TeV, they contribute only about 10% to the total cross-section and less at lower energies. The effect of adding a small two-pomeron exchange contribution is that the experimental result for e is an effective value and a little less than that of the single pomeron.

Total cross-sections only provide information on the pomeron trajectory at t = 0. They tell us nothing about the t-dependence of the trajectory. This can only be obtained from studying the angular distribution of the protons ­ the differential cross-section ­ when they scatter off each other.

It appears that the pomeron trajectory is linear, just like the meson trajectories, but with different numbers in the formula: a(t) = 1 + e0 + a't. The data indicate that the slope parameter a' = 0.25, a number first obtained in 1973 from an analysis of the ISR data.

At small values of t the differential cross-section can be represented by an exponential: e-bt . The slope parameter, b, is energy-dependent, and Regge theory can predict this energy dependence in terms of the trajectory. The effect is that b increases as the energy increases, so that the forward peak of the differential cross-section becomes steeper, a phenomenon that is known as "shrinkage". For a' = 0.25 the predicted change in b from ISR to Tevatron energies is 3.5 GeV­2. This is in excellent agreement with the data and is something of a triumph for Regge theory.

The pomeron provides a simple, economical and accurate description of high-energy scattering, but what is it? When we introduced the meson trajectories, we linked them to physical particles for positive t values and then extrapolated to the scattering region where t is negative. Is the pomeron trajectory similarly linked to physical particles? Can one not simply reverse the process for the pomeron and extrapolate from the scattering region to the physical region? The extrapolation is a longer one, but assuming that the linear trajectory is valid for all values of t, it predicts a physical particle of spin-2 at a mass of about 2 GeV. This cannot be one of the known mesons: they are already members of the meson trajectories and it has the wrong mass and spin to be associated with them. This means that this "pomeron particle", if it exists, must be something other than a quark­antiquark state. The favourite interpretation is that it is a "glueball", composed of two gluons.

Quantum chromodynamics

It is natural to try to relate the pomeron to our understanding of quantum chromodynamics, the field theory of quarks and gluons, and to describe it in terms of gluon exchange. The simplest picture, that the principal effect is just the exchange of two gluons, was proposed independently in 1975 by American physicist Francis Low and Israeli physicist Shmuel Nussinov.

Subsequent development of this idea, principally at Cambridge and Heidelberg, has shown that it is compatible with all soft-pomeron phenomenology except for one aspect: two-gluon exchange gives a constant, not a rising, cross-section. The exchanged gluons must interact with each other to produce an increase.

How can one measure the quark and gluon content of the pomeron? In 1985, Swedish physicist Gunnar Ingelman and American physicist Peter Schlein suggested that the content of the pomeron could be established in ways analogous to those used to finding the quark­gluon content of the proton by probing it in high-momentum transfer reactions.

The first results for the pomeron were obtained by the UA8 experiment at the CERN proton­antiproton collider, establishing the validity of the concept and technique, but this experiment could not distinguish between quarks and gluons. This has been achieved by the H1 and ZEUS experiments at the HERA electron­proton collider at DESY. A clear separation can be made and gluons are found to constitute about 90% of the pomeron.

However, another question immediately arises: are the UA8 and HERA experiments probing the soft pomeron or something entirely different? To study quark­gluon content requires small distances and high-momentum transfer, while the natural realm of the soft pomeron is at large distances and low-momentum transfer.

Gluon "ladder"
Gluon "ladder"

Perturbative pomeron

A candidate for another pomeron, the "perturbative" or "hard" pomeron, emerged as early as 1975 in the work of Russian physicist Lev Lipatov and his colleagues. The hard pomeron is essentially a gluon "ladder" ­ the exchange of two gluons with gluon rungs joining them (figure 4). Because of this interaction the hard pomeron does give energy dependence, predicting e of about 0.5.

Unsolved mystery
Unsolved mystery

This idea was given tremendous impetus by results from the H1 and ZEUS experiments at HERA when it was discovered that the greater the momentum transfer, the greater the energy-dependence of the cross-section. The change from the low-momentum transfer, long-range interaction to the large-momentum transfer, short-range interaction is shown in figure 5. There is at present no consensus on the real reason for this dramatic change, and there are several models on the market that fit the data rather well. British physicists Sandy Donnachie and Peter Landshoff have taken the two-pomeron scenario seriously, and the curves in figure 5 are the result of their fit to the data.

The phenomenological pomeron has enjoyed remarkable success in describing a range of data, and that part of the story will not alter. However, the existence of the hard pomeron is still speculative, and its nature and theoretical explanation remain questions for the future. Data on photon­photon interactions at high energies from the LEP experiments will provide a new and powerful probe of this concept.

It is already clear from the measurements of the real photon­photon total cross-section by the L3 and OPAL experiments at CERN's LEP electron­positron collider that more than the soft pomeron is required. Proton­proton data from Brookhaven's RHIC collider, which is scheduled to come into operation this year, will complement existing proton­antiproton data and test our implicit assumption that there is no difference between proton­proton and proton­antiproton scattering at high energies. In the longer term, CERN's LHC will provide the long lever arm necessary to investigate the effect of multiple pomeron exchange and possibly to seek the presence, at some level, of the hard pomeron in purely soft interactions.