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, , Cambridge, MA 02138, USA and Theoretical Physics Division, CERN CH - 1211 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 , 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, , 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 . 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 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. While the almost universal belief at that time was that these two total cross sections would approach the same constant value at very high energies due to the Pomeranchuk theorem, we took the opposite view that this K "p total cross section as measured at Serpukhov can be used to estimate the increase in all total cross sections, including the pp cross section to be measured at the CERN ISR and the Fermilab accelerator. Figure 2 shows this original estimateg of the increase in pp total cross section obtained by this method and given to Bellettini for the planning of the experiment at the CERN ISR. The result of this rough estimate was that the increase of the total pp cross section from its minimum value to the ISR energy of 53 GeV is almost 3 mb. The year after this phenomenological estimate was obtained, the total cross section at this energy was measured by both the Pisa-Stony Brook Collaboration,1° of which Bellettini is a prominent member, and the CERN-Rome Collaboration,11 and it was found that this estimate was too low by about 35%. In view of the crudeness of the assumptions involved in this estimate, the result of Fig. 2 was considered to be a theoretical triumph.

.. 4 OCR Output so

45

ANTI Pnorou

moron /'POMERON commeurrou ’ /»TO Tomi. cnoss sacrnou

3% noo wooo ioooo S(GeVl

Fig. 2. An early rough prediction of rising pp and pp total cross sections, carried out before the first experimental observation of the rise in 1973.

The natural next step was to use these ISR data·as1°” an input in addition to the Serpukhov data.? However, there was still no way to determine the value of c'. For this reason, two sets of phenomenological parameters were obtained at this second stage, one set with c' = 0 and the other with c' = 1. For these two sets, the corresponding values of c were of course different:

For c' = 0, c = 0.08; For c' = 1, `c = 0.2. (2)

Once these parameters were obtained, it was possible to predict not only the pp and the pp total cross sections, but also those for ‘ll'+p, 7l’_p, K +p, and K `p. The result of this early phenomenology is shown in Fig. 3 together with the experimental data available in 1973. In this energy range, there is virtually no difference between the two sets of parameters. Various other quantities can also be obtained easily besides the total cross sections. One example, the ratio of the real and imaginary parts of the elastic amplitude in the forward direction, is shown in Fig. 4 together with the experimental data available at that time. Diffraction scattering was also studied. It should perhaps be mentioned that the value of c = 0.08 as given in Eq. (2) for c' = 0 was later used extensively by many authors on increasing total cross sections. 15 Because of the important contributions of Pomeranchuk" to the theoretical study of total cross sections, the word “pomeranchukon," later shortened to “pomeron," is used to describe the exchanged object that is responsible for elastic scattering. The theoretical prediction and later experimental verification of rising total cross sections mean that the pomeron is a more complicated and hence more interesting object than originally visualized. This pomeron is of course responsible for diffraction scattering. *°OCR Output

.. 5 grown. (mb)

45 45 lN\ ANTI PROTON

40 40 moron

H P 25 25

ITP

_ K P

20 20

1<‘ P

|5 I5

IO IOO S(GeV)2 ` IOOO |0,000

Fig. 3. Comparison of the phenomenology with the parameters of Eq. (2) with the experimental dm from Serpukhov,7 Fermi1ab,13 and 1sR.*°·¤·¤

It should be emphasized that there is only one pomeron. In particular, the so-called soft pomeron and the hard pomeron are merely aspects of the same object. For example, with reference to Eq. (2) which gives two different parametrizations of this pomeron, the value of 0.08 (1.08 when 1 is added) is sometimes considered to pertain to the soft pomeron, while the value of 0.2 (similarly 1.2 when 1 is added) to the hard pomeron.

- 6 .. OCR Output O2 pnoron moron rrnsru: scarremnc

,21f/ Of l- pp xi

`*—1¤1¤ -¤·l 1i

-02-l:i 10*

$(GeV)

Fig. 4. Theoretical values of oz, the ratio of real to imaginary parts of the pp and pp forward scattering amplitude, together with the experimental data available up to 1973.

4. Theoretical Predictions and Experimental Verifications: 1977-1993 After obtaining the results of Figs. 3 and 4 from the early phenomenological analyses in 1973 and 1974, we were not able to make any further progress on this topic. The major problems were clear: (a) The value of c' remained to be determined; and (b) we needed to take better account of the exchange of other objects, such as the p Regge pole. This problem remained dormant until I was fortunate to meet Claude Bourrely and Jacques Soffer of Marseille, France. They were great experts on a number of subjects, including spin physics and phenomenology in general. Since they found the problem of increasing total cross sections both interesting and challenging, we decided to collaborate on working out explicit theoretical predictions to be compared with future experimental data. As of now, I have been collaborating with them for nearly twenty years. In order to solve problem (a) above, Bourrely, Soffer and I carried out a systematic study of the experimental data available up to 1979. This lead unambiguously to the conclusion that c' is positive: 17 c' = 0.756. (2) This result has the following important implication: The pomeron is not a Regge pole (for which c' must be zero), but is instead a Regge cut above 1. A few years later, this systematic study was repeated"; using additional data and this value of c' was revised to 0.748 (see below)—a reduction of an insignificant amount——meaning that the determination of c' is quite stable.

- 7 OCR Output The present model (sometimes referred to as the impact picture) for proton-proton and antiproton-proton elastic differential cross section is given by the following simple formulas, when Coulomb contributions are includedz ls

¤(8. ¢) = ¤"’(8. i) i ¤°(¢). (4) where the upper sign is for pp while the lower one ts for pp, aN (s, t) is the hadronic amplitude and a° (t) is the pp Coulomb amplitude. The Coulomb amplitude is that of West and Yennie,*° and the hadronic amplitude at high energies is given by

a”(8, r) = is / J,,(b,/‘-¢)(1 - e·°·»<··’·>)bd1», (5) where _ 9¤(8. 6) = S¤($)F(b‘). (6) when the Regge background is neglected. The energy dependence is given by the crossing symmetric version of Eq. (1), SC ,uC ' .—°(8) _ (ln s)°' + (ln u) (7) In Eq. (7), u is the third Mandelstam variable and both s and ·u are expressed in (GeV) The t-dependence of a" (s, t) is controlled by F(b”) whose Fourier transform is taken to be

F(¢) = f lG(¢)ll(¤+”” ¢)/(G—¢)l»2 (8) where G'(t) is given by Gm = [(1 - ¢/miie —¢/m%>1 ·‘ (9)

The values of the parameters are

c = 0.167, c' = 0.748, mi = 0.586 GeV, mz = 1.704 GeV, 0. = 1.953 GeV, f = 7.115 GeV (10)

These values have not changed at all during the last ten years. In fact, they have changed only slightly over the previous five years. Since these formulas are rather simple, anyone with a computer can calculate the theoretical predictions and compare with experimental results. The inclusion of p and w exchanges makes the formulas only slightly more complicated. The most extensive tabula tion of the results from these formulas was published several years ago.Such2° comparisons between theoretical predictions and later experimental results have been carried out on many occasions, and the reader is referred to the literature. Instead, I would like to men tion only a couple of examples. First, as already discussed in Sec. 2, the third prediction of the 1970 papers was that the ratio as,/aw, of the integrated elastic cross section and the total cross section approaches 1/2 as the center-of-mass energy approaches infinity. At the time of this prediction, this ratio for pp scattering was less than 0.2 and decreasing. This prediction was not verified experimentally until over a decade later by the UA4 Collaboration.21

... 8 OCR Output 10“

UA4 x/S = 541 sev

102

10* 0.05 0.1 0.15

It! (GeV/¢)

Fix- 5‘ Comparison of theoretical prediction and new experimental measurement by UA4at24 ,/E = 541 GeV in the small-t region.

As a second example, I would like to relate the story of the ratio of the real and imag inary parts of the elastic scattering amplitude in the forward direction. The experimental data on this quantity“ provided the first indication that the total cross section may not be approaching a constant“ at infinite energy. Later experimental data up to the early eighties gave good confirmation of the theoretical calculations. However, in 1987, the UA4 Collabo ration published an experimental value for this ratio of O.24:b0.04,” much above the predic tion of 0.13. This discrepancy led to a flurry of theoretical activities claiming new physics. Not believing this new physics, Bourrely, Soifer and I analyzed the UA4 datau in detail and concluded that their value of 0.24:l:0.04 was questionable.” It was nevertheless gratify ing when this same Collaboration repeated their experiment with more extensive data and found that, instead of 0.2:l:0.04, this ratio is 0.135:h0.015.2‘ A comparison of the theoretical prediction and the new data is shown in Fig. 5; a more detailed comparison is in Ref. 25. Some further comparisons between our theoretical predictions and the recent data from the CERN Collider and the Fermilab Collider will be presented by André Martin. 2°

5. Photon-Proton Scattering at HERA Energies So far, I have described only pp and pp scattering, sincethese have been the most reliable source of experimental information on the rising total cross sections, first predicted in 1970.5 With the operation of HERA, there will soon be a second source of reliable information about rising total cross sections. Although the theoretical development was initially based on U(1) gauge theory, the phenomenology has been applied almost exclusively to hadronic scattering. In the seventies, we were not sufliciently far-sighted to realize the possibility of constructing electron-proton

- 9 OCR Output pp

<¤> (b)

YP

(cl

Fig. 6. Multi·tower diagrams for (a) pp and pp, (b) 1rp, and (c) yp scattering. colliding accelerators to study the scattering of and protons at very high energies. This remarkable feat has now been accomplished at the Deutsches Electronen-Synchrotron in Hamburg, Germany, under the leadership of Bjorn Wiik and Gus Voss. This new ac celerator is called HERA, with an initial proton energy of about 800 GeV and an electron energy of about 30 GeV, leading to a center-of-mass energy of about 300 GeV. This makes it possible to study very high-energy yp scattering. The theory of rising total cross sections is most easily obtained by studying the multi tower diagrams. These are shown in Fig. 6a for pp and pp and in Fig. 6b for 'Il‘+p and 1r'p. Clearly the corresponding multi—tower diagrams for yp are those in Fig. 6c. The similarity of the diagrams in Fig. 6b and Fig. 6c are striking, the major differences being: i) the electromagnetic coupling e appears twice in Fig. 6c, and ii) the photon is its own anti-particle. Because of these two differences, it is natural to expect that

¤i¤¤(yz>) = (¤¤¤st)¤[¤»·(¤*p) +

- .. OCR Output 180

7 P

160

140

120

5 ‘!

100 *—— * 10 102 x/s (GeV)

Fig. 7. yp total cross section as a function of the center-of-mass energy.

Collaborations are also shown together with the previous data at lower energies.The2° HERA data are surely going to improve rapidly, and we are eager to see a much more accurate determination of the yp cross sections. In fact, while the points shown in Fig. 7 are the most recent ones published by the H1 and ZEUS Collaborations, they are based on their 1992 data. They have not yet finished analyzing the 1993 data, but this is to be expected any time now. These more accurate data will provide a stringent test of the accuracy of Eq. (11). Thus, we have encountered the expanding proton unambiguously in two different situ ations, first the rising total cross sections of pp and pp, and secondly that of yp. To us this is especially satisfying: we began with the study of Compton scattering at high energies twentyfrive years ago, and now HERA is providing data on Compton scattering at high energies. The cycle is complete.

Acknowledgments I am grateful to the organizers of this Workshop, especially to Herbert Fried, for such an interesting and pleasant meeting. I also wish to thank the Theoretical Physics Division of CERN for its kind hospitality. This work was supported in part by the the U.S. Department of Energy under Grant No. DE-FG02-84ER40158 with Harvard University.

- OCR Output References M. Froissart, Phys. Rev. 123 (1961) 1053; A. Martin, Nuovo (Jim. 42 (1966) 930; 44 (1966) 1219. H. Cheng and T. T. Wu, Phys. Rev. Letts. 22 (1969) 666; Phys. Rev. 182 (1969) 1868, 1899. T. T. Wu, Phys. Rev. 104 (1956) 1201. H. Cheng, Proc. of the Vth Blois Workshop, eds. H. M. Fried, K. Kang, and C.-I. Tan (World Scientific, Singapore, 1994), p. 236. H. Cheng and T. T. Wu, Phys. Rev. Letts. 24 (1970) 1456. K. J. Foley et al., Phys. Rev. Letts. 19 (1967) 857. S. P. Denisov et al., Phys. Letts. B36 (1971) 415. I. Ya. Pomeranchuk, Sov. Phys. JETP 7 (1958) 499. H. Cheng, J. K. Walker, and T. T. Wu, Paper 524, 16th Int. Conf. on High Energy Physics, 1972 (unpublished). 10. S. R. Amendolia et al., Phys. Letts. B44 (1973) 119. 11. U. Amaldi et al., Phys. Letts. B44 (1973) 112. 12. H. Cheng, J. K. Walker, and T. T. Wu, Phys. Letts. B44 (1973) 97. 13. G. Charlton et al., Phys. Rev. Letts. 29 (1972) 515; F. T. Dao et al., Phys. Rev. Letts. 29 (1972) 1627; J. W. Chapman et al., Phys. Rev. Letts. 29 (1972) 1686. 14. Aachen-CERN-Haivard-Geneva-Torino Collaboration, paper presented by C. Rubbia at 16th Int. Conf. on High Energy Physics, 1972 (unpublished). 15. See, for example, A. Donnachie and P. V. Landshoff, Nucl. Phys. B231 (1983) 189. 16. H. Cheng, J. K. Walker, and T. T. Wu, Phys. Rev. D9 (1974) 749. 17. C. Bourrely, J. Soffer, and T. T. Wu, Phys. Rev. D19 (1979) 3249. C. Bourrely, J. Soffer, and T. T. Wu, Nucl. Phys. B247 (1984) 15. 19. G. B West and D. R. Yennie, Phys. Rev. 172 (1968) 1413. 20. C. Bourrely, J. Soifer, and T. T. Wu, Z. Phys. C'—Particles E4 Fields 37 (1988) 369. 21. UA4 Collaboration, R. Battiston et al., Phys. Letts. B115 (1982) 333; B117 (1982) 126; and B127 (1983) 472. 22. UA4 Collaboration, D. Bernard et al., Phys. Letts. B198 (1987) 583. 23. C. Bourrely, J. Soffer, and T. T. Wu, Mod. Phys. Letts. A6 (1991) 2973. 24. UA4.2 Collaboration, C. Augier et al., Phys. Letts. B316 (1993) 448. 25. C. Bourrely, J. Soffer, and T. T. Wu, Phys. Letts. B315 (1993) 195. 26. A. Martin, this Workshop. 27. C. Bourrely, J. Soffer, and T. T. Wu, CERN Preprint TH-7387 (1994), Phys. Letts. B (to be published). 28. H1 Collaboration, T. Ahmed et al., Phys. Letts. B299 (1994) 374; ZEUS Collaboration, M. Derrick et al., DESY Preprint 94-032 (1994). 29. D. O. Caldwell et al., Phys. Rev. Letts. 40 (1978) 1222.

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