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Behaviour of Total and Elastic Cross-Sections at Very High Energies

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Behaviour of Total and Elastic Cross-Sections at Very High Energies CERN-TH.7437/94 9 OCR Output CERN—TH-7437-94 BEHAVIOUR OF TOTAL AND ELASTIC CROSS-SECTIONS AT VERY HIGH ENERGIES Tai '1`sun WU Gordon McKay Laboratory, Harvard University, Cambridge, MA 02138, USA and Theoretical Physics Division, CERN CH - 1211 Geneva 23 CERN-TH.7437/94 September 1994 BEHAVIOR OF TOTAL AND ELASTIC CROSS SECTIONS AT VERY HIGH ENERGIES TAI TSUN WU Gordon McKay Laboratory Harvard University Cambridge, MA 02138, USA, and Theoretical Physics Division, CERM Geneva, Switzerland 1. Introduction In this talk, I would like to present an overview of the original theoretical prediction of the increasing total cross section on the basis of gauge quantum field theory, phenomeno logical predictions from this theoretical understanding, comparison with later experimental data, and expected future data from HER.A. The discussion will be limited to elastic scattering, since it is the simplest scattering process. Through the optical theorem, the total cross section—which gives the overall strength of the scattering process———is proportional to the imaginary part of the elastic scattering amplitude in the forward direction. Therefore, the theoretical understanding of elastic scattering is often of primary importance. Some of these developments can be found in an excellent book by Herbert Fried, the organizer of this workshop: Herbert M. Fried, Fhnctional Methods and Eikonal Models, Editions Frontieres (1990). A few years earlier, Hung Cheng and I also wrote a book on this subject: Hung Cheng and Tai Tsun Wu, Expanding Protons: Scattering at High Energies, The MIT Press (1987). A recent topic of current interest, of course not covered in either of these books which are a few years old, is the data from the electron-proton colliding accelerator HQERA at DESY, Hamburg, Germany. This is discussed in some detail in Section 5. 2. The Beginning: 1967-1970 In 1967, Hung Cheng and I learned that the United States Atomic Energy Commis sion (predecessor of the Department of Energy) had decided to fund the Fermi National Accelerator Laboratory, Batavia, Illinois. This was for the construction and operation of a 200-GeV proton accelerator, later upgraded to 400 GeV. When a 200-GeV proton hits a proton at rest, the energy of each proton in the center-of-mass system is about 10 GeV. If we say that a particle is extremely relativistic if its energy is at least ten times its rest mass, then for the first time it would be possible to observe the interaction of two extremely relativistic protons. Because of this exciting news, we asked ourselves what we could say about such scat tering processes at very high energies. It seemed reasonable to expect some simple features in such a limit, perhaps analogous to optics as a limiting case of electromagnetic theory where the wavelength is small. How could we get hold of these simple features? The approach we decided to take was to study the high-energy behavior of quantum field theory. Cheng was an expert on Regge-pole theory, and I had some experience in getting Regge poles and cuts from ¢>“ field theory. However, we both felt that, in order to .. ]_ -·OCR Output be able to make theoretical predictions for the scattering processes at very high energies, we needed take the best quantum field theory and study its high-energy behavior without preconceived prejudices as to the outcome. At that time, the only definitive piece of information was the Froissart—Martin bound. In the sixties, it was easy to decide which quantum field theory was the best: the only well-understood one was quantum electrodynamics. Since our interest was in using QED as a clue for high-energy hadron scattering, we made two modifications to QED: We per mitted a nonzero photon mass, and we allowed the coupling constant to be larger than the fine-structure constant. Neither changed the calculation in any appreciable way. Perhaps influenced by the optical limit of Maxwell’s equations, we decided to investigate first the high-energy behavior of Compton scattering, i.e., the scattering of a vector gauge particle by a fermion. To the lowest order, the total Compton scattering cross section approaches zero as the energy increases without bound. This is also true to the next, or fourth, order. Therefore, Cheng and I concentrated on calculating the sixth—order cross sections. This was one of the longest calculations that I have engaged in, and it took us over a year to complete. The result of this sixth-order asymptotic calculation gave a total cross section that no longer goes to zero at infinite energy? Thus at high energies, the sixth-order matrix element is much larger than the second- and fourth—order ones. We found this result to be very satisfying: for example, unlike the second- and fourth-order results, this sixth—order total cross section at least bears some resemblance to that of the optical scattering from a sphere. Of course this calculation also gives the elastic differential cross section. The high energy behavior of this cross section is somewhat complicated and is most compactly ex pressed in terms of what we call "impact factors." The same impact factor for the vector gauge particle also appears in the eighth-order scattering of two such vector particles. Once the sixth—order Compton scattering cross section at high energies was understood, we gradually became more and more efficient in extracting the high-energy behavior of various processes. Our calculations became much more exciting when we found extra factors of ln s in the higher-order diagrams, giving the first indication that the total cross section may increase with energy rather than approaching a constant value. Figure 1 shows the orders of magnitude of some of these higher-order terms. Last year at the Blois conference, Cheng explained this development in detail and very well} I therefore refer the reader to his presentation. About sixteen months after obtaining the sixth-order result for Compton scattering, Hung Cheng and I were able to predict that at very high energies the total cross sections increase without bound for all hadronic scattering. In our 1970 paper in Physical Review Letters,5 the abstract says: “At infinite energy, we predict that: (1) aw, approaches ininity; (2) the ratio of the real part to the imaginary part of the forward elastic amplitude approaches zero; (3) ac,/am, approaches 1/2; (4) the width of the diffraction peak approaches zero; its product with aw, is a constant. We give theoretical evidences based on massive quantum electrodynamics as well as experimental evidence in support of these predictions, and a physical picture for high-energy scattering? By the time this volume comes out next year, it will be the twenty-fifth anniversary of this first prediction of increasing total cross sections. I am happy to say that these predictions have fared well after all these years. - 2 OCR Output Order: Feynman Diagrams 2 so + many others 4 so lo 1 O gs in s 14 2 14 g s( Zn s) where Fig. 1. The orders of magnitude of the matrix elements for Compton scattering in an Abelian gauge theory when s is large but t is fixed. [s is the square of the ¢6I1t81`—0f-1*11888 energy and —t is the square of the momentum transfer.] - 3 OCR Output As mentioned above, the original motivation for our investigation was provided by the approval of the Fermi National Accelerator Laboratory by the U. S. Atomic Energy Commission. However, before 1970, we did not expect such a drastic prediction of the increasing total cross section. For the purpose of obtaining experimental verification for this prediction, the Intersecting Storage Rings (ISR) at CERN, Geneva, Switzerland, was the most promising accelerator. We therefore discussed with the experimentalists, especially Giorgio Bellettini, about measuring the total proton-proton cross section at ISR. Naturally he raised the good question as to how much increase was to be expected in this total cross section. On the basis of the so-called tower diagrams in quantum field theory, we had learned that the effective interaction strength increases with energy in the form of 81+c (1) (1¤.q)·= To answer the question of Bellettini, we needed to be able to find an approximate determi nation of the value of c, since the dependence on c' is only of secondary importance. In order to determine the value of c, we asked ourselves whether there were any ex perimental data that we could use. At that time, the best experimental evidence for the increasing cross section was the measurement of the ratio of the real to the imaginary parts of the proton—proton forward elastic cross section: it showed a distinct tendency to cross over to positive values.“ However, it was diflicult to get an estimate for Bellettini of the increase in total cross section from these data. Since the values of c and c' are independent of the hadronic scattering process under consideration, we could use any hadronic data. Before the Fermilab accelerator and the CERN ISR, the highest-energy accelerator in the world was that at Serpukhov, Russia. That accelerator produced 70 GeV protons, and there were secondary 1r and K beams of both charges, up to nearly the same energy. Thus there were data for pp, pp, 1r+p, 1r"p, K + p, and K “p elastic scattering from Serpukhov.7 Of these six total cross sections, five are decreasing and only one is increasing in the Serpukhov energy range, the increasing one being that of K +p. Moreover, the K +p total cross section is below that for K 'p.
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