I. Ya Pomeranchuk and Physics at the Turn of the Century: Proceedings of the International Conference Moscow, Russia, 24 - 28 January 2003

and Physics at the Turn of the Century editors

A, Berkov N. Narozhny L. Okun

I. *Ja Pomeranchuk and Physics at the Turn of the Century

-r

••»

*SA4:

s**ftl

•>*VvJj*i : oo. However experimental data show that at s ~ 100 GeV 2 total cross sections of hadronic interactions have a weak energy dependence and they slowly increase with energy at higher energies. In the Regge pole model this can be related to the pole, which has an intercept ap(0) w 1. This pole is usually called pomeron or the vacuum pole, because it has the quantum numbers of the vacuum -

32

positive signature, parity and G (or C) parity and isospin 1=0. A value of the intercept of the pomeron is of crucial importance for the Regge theory. If ap(0) = 1, as it was assumed initially, then all the total interaction cross sections tend to a constant at very high energies. This theory has however some intrinsic difficulties and must satisfy to many constraints in order to be consistent with unitarity. Besides experimental data indicate that e r ^ rise with energy. Thus at present the supercritical pomeron theory with a_p(0) > 1 is widely used. In a model with only Regge poles taken into account an assumption that ap(0) > 1 would lead to a power like increase of total cross sections, thus violating the Proissart bound. However in this case other singularities in the j - plane,- moving branch points 19 ' 20 should be taken into account and their contributions allow one to restore unitarity and to obtain the high energy behavior of scattering amplitudes which satisfy to the Proissart bound.

Regge poles in

QCD

An astonishing linearity of trajectories for Regge poles corresponding to the known qq-st&tes indicates to essentially nonperturbative, string-like dynamics of these objects in QCD. A nonperturbative method, which can be used in QCD for description of large distance dynamics, is the 1/Nexpansion (or topological expansion) 5 - 6 ' 7 . In this approach the quantities 1/NC 5 or 1/AT// 6 ( Nif is the number of light flavors) are considered as small parameters and amplitudes and Green functions are expanded in terms of these quantities. In QCD Nc = 3 and Nif ~ 3 and the expansion parameter does not look small enough. However we shall see below that in most cases the expansion parameter is l/N* ~ 0.1. In the formal limit Nc —> oo (Nif/Nc —> 0) QCD has many interesting properties and has been intensively studied theoretically. However this approximation is rather far from reality, as resonances in this limit are infinitely narrow ( r ~ l/Nc). The case when the ratio Nij /Nc ~ 1 is fixed and the expansion in 1/-/V;/ (or l/iVc) is carried out 6 seems more realistic. This approach is called sometimes by the topological expansion, because the given term of this expansion corresponds to an infinite set of Feynman diagrams with definite topology. The first term of the expansion corresponds to the planar diagrams of the type shown in Fig.3 for the binary reaction. These diagrams always have as border lines the valence quarks of the colliding hadrons. At high

33

energies they should correspond to exchanges by secondary Regge poles OLR(P, A2,U>, ..) "made of light quarks. The s-channel cutting of the planar diagram of Fig.3a) is shown in Fig.3b). Here and in the following we do not show internal lines of gluons and quark loops.

c c a)

b)

Figure 3. a ) Planar diagrams for reaction ab —> cd, b) same for reaction ab —> X. Full lines denote quarks, wavy lines - gluons.

Let us note that the planar diagrams correspond to annihilation of a valence quark and an antiquark, belonging to colliding hadrons. The topological classification of diagrams in QCD leads to many relations between parameters of the reggeon theory, hadronic masses, widths of resonances and total cross sections (for a review see 12 ) . All these relations are in a good agreement with experiment. A contribution of the planar diagrams to the total cross section decreases with energy as l / s ( 1 _ a R ( 0 » « l/y/s. A calculation of Regge-trajectories in QCD is a difficult problem even for planar diagrams. It was considered in the paper 21 using the method of Wilson-loop path integral 22 . It was shown that under a reasonable assumption about large distances dynamics: minimal area law for Wilson loop at large distances (W) ~ exp(—aSmin), it is possible to calculate spectrum of q^-states. The resulting spectrum for light quarks with a good

34

accuracy is described by a very simple formula M2 - — = L + 2nr + a

(6)

Z7TCT

where L and nr are orbital and radial quantum numbers. This spectrum corresponds to an infinite set of linear Regge trajectories similar to the one of dual and string models. In ref.21 spin effects have not been taken into account. Realistic calculations of masses of hadrons which take into account spin effects, perturbative interactions at small distances and quark loops have been carried out recently 23 . The resulting Regge trajectories are in a good agreement with experimental ones shown in Fig.2. Glueballs and the pomeron

in

QCD

The pomeron in 1/A^-expansion of QCD corresponds to a class of cylindertype diagrams shown in Fig.4. Such diagrams for elastic scattering or reactions without quantum numbers exchange in the t-channel, shown in Fig.4a), where the valence quarks of colliding hadrons are conserved in the process of interaction. The s-channel cuttings of these diagrams correspond to the multiparticle production configurations, shown in Fig.4b). These configurations correspond to production of two chains of particles and each chain has the same structure as the one shown for the planar diagram of Fig.3b). From the t-channel point of view the cylinder diagrams are due to exchange by gluons in the t-channel. From the topological classification point of view the cylinder diagrams correspond to a sphere with two boundaries, given by the valence quarks lines of colliding hadrons. It is very important to calculate the pomeron trajectory in QCD. Perturbative calculations of the pomeron in QCD have been carried out by L. Lipatov and collaborators 24 (BFKL pomeron) many years ago. The pomeron is related to a sum of ladder type diagrams with exchange by reggeized gluons. Reggeization of gluons (as well as quarks) is an important property of QCD (at least in perturbation theory). In the leading approximation of the perturbation theory an expression for the intercept of the pomeron is 24 : A = aP(0)-l = ^ ^ a

s

(7)

7T

In this approximation A RS 0.5. NLO corrections have been calculated recently 25 and strongly decrease LO results for A. The value of A depends on a choice of renormalization scheme and renormalization scale. The choice

35

a) Figure 4.

b)

a ) Cylinder type diagrams and b ) cutting of these diagrams in the s-channel.

of a physical (BLM) scheme leads to a stable results for A, which practically does not depend on virtuality of a process and A « 0 . 1 7 2 6 . An influence of the non-perturbative effects on the pomeron trajectory and its relation to the spectrum of glueballs was considered in papers 27>28J using the method of Wilson-loop path integral discussed above for the case of gg-Regge poles. While experimental situation for glueballs is not yet settled, lattice simulations yield an overall consistent picture of lowest (< 4 GeV) mass spectrum. In the approximation when the spin effects and quark loops are neglected the spectrum of two-gluon glueballs is determined by the expression for the Wilson loop at large distances given above with the only difference that the string tension {afUnd) for the qq system is changed to aadj- Thus the mass spectrum for glueballs is given by eq.(6) with the change a -» cradjThe value of aadj can be found from the string tension a fund of qq system, multiplying it by | , as it follows from Casimir scaling observed on the lattices. Taking experimental Regge slope for mesons a' = 0.89 GeV~2 one obtains Ofund = 0.18 GeV2 and oadj « 0.40 GeV2. The spin splittings for glueball masses were calculated in paper 28 assuming that spin effects can be treated as small perturbations. The calculated spectrum of glueballs 28 is in a perfect agreement with the results of lattice calculations . The calculated glueball Regge trajectory is close to qq Regge trajectories

36

/ , / ' in the region of small t. In this region a mixing between gluonic and qq Regge trajectories is important. The mixing effects and account of small distances by QCD perturbation theory allow to obtain phenomenologically acceptable intercept of the pomeron trajectory and lead to an interesting pattern of vacuum trajectories in the positive t-region 28 . I would like to emphasize that in the approach discussed above the pomeron singularity contains both non-perturbative effects and perturbative dynamics. Thus the resulting "physical" pole is a state due to both "soft" and "hard" interactions. Thus the pomeron in QCD has a very rich and interesting structure. Regge poles are not the only singularities in the complex angular momentum plane. Exchange by several reggeons in the i-channel leads to moving branch points (or Regge cuts) in the j-plane 19>20. Contributions of these singularities to scattering amplitudes T n (s,0) ~ s 1 + n A are especially important at high energies for A > 0. The whole series of n-pomeron exchanges should be summed. An account of these multipomeron exchanges in the i-channel leads to unitarization of scattering amplitudes. The Gribov reggeon diagrams technique 29 allows one to calculate contributions of Regge cuts to scattering amplitudes. Prom the point of view of 1/N-expansion n-pomeron exchange contributions correspond to more complicated topologies with "handles" . So each topological class of surfaces is characterized by a given number of boundaries (rib) and handles (rih)- Topological expansion allows one to give a complete classification of diagrams and to determine their dependence on the parameter 1/N. The diagrams of a given topological class have the following dependence on 1/N Tnb,nh ~ ( l / A r b + 2 n "

(8)

Thus the contribution of planar diagrams (rib = 1, n>h = 0) to the scattering amplitude is ~ 1/N, the cylinder diagram (rib = 2, n/j = 0) contribution is ~ (1/N)2 and the diagram of two-pomeron exchange (rib = 2, rih = 1) is ~ (1/N)4. Eq.(8) is valid for 4-point functions (amplitudes). Note that the ratio of the cylinder to the planar diagram is ~ 1/iV , however for the amplitudes with vacuum quantum numbers in the t-channel the cylinder-type diagrams will dominate at high energies, because its relative contribution increases as sap(°)-aR(°) a s energy increases. This is one of the manifestations of the dynamical character of the 1/N-expansion. In many cases the type of the process (and the number of boundaries) is fixed by the quantum numbers in the t-channel. In this case the expansion is

37

in the number of handles and, according to Eq. (8), the expansion parameter is (1/N)2 as it was mentioned above.

3.

High-energy hadronic interactions

The topological or 1/N-expansion discussed in the previous section gives a useful classification of all QCD diagrams. It becomes especially predictive if a definite space-time picture for these diagrams is given. The process of hadronic interactions at high energies can be related to production of new objects,- color tubes or strings 8>9'10. Planar diagrams of Fig.3 can be interpreted as annihilation of valence quarks of colliding hadrons and formation of color tubes, which decay later to two (Fig.3a) or several (Fig.3b) final hadrons 11>12. It is possible to show that planar diagrams lead naturally to Regge asymptotic behavior for binary processes and to derive the rules for fragmentation functions, which describe transformation of the color tube to hadrons 30>12. In the same way one can relate the cylinder-type diagrams to a process of color octet exchange in the t-channel, which leads to formation and subsequent decay of two tubes (strings) 12 ' 14 .Fragmentation of each string has the same properties as in the planar case. The string model provides a simple picture of interaction for diagrams of 1/N-expansion. The detailed models of strings breaking have been developed 10 and are widely used now in Monte Carlo simulations of multiparticle production. In models based on reggeon theory and the space-time picture of 1/TV-expansion in QCD 12 ' 14 it is usually assumed that the pomeron corresponding to the cylinder-type diagrams is a simple Regge pole with ap(0) > 1. The value of ap(0) is determined from analysis of experimental data. For a "supercritical" pomeron higher terms of the topological expansion associated with exchange by several pomerons in the t-channel are also important. This is due to the fact that, though the exchange by npomerons is ~ l/(N2)n it is enhanced by the factor (s/so)nA. The s-channel discontinuities of these diagrams are related to processes of production of 2k (k < n) chains of particles. An important ingredient of the reggeon approach to high-energy hadronic interactions are the AGK-cutting rules 31 . These rules define schannel discontinuities of the n-pomeron exchange contributions and thus determine a multiparticle content of arbitrary reggeon diagrams. In the method based on 1/A^-expansion these rules give a possibility to determine

38

the cross sections for 2k-chains (strings) production (with any number of uncut pomerons) if the contributions of all n-pomeron exchanges to the forward elastic scattering amplitude are known. They can be calculated using the reggeon diagram technique 29 . In most of calculations the diagrams of the eikonal type has been taken into account 14 - 12 . For example in the " quasieikonal" approximation (for account of more complicated diagrams with interactions between pomerons see below) the cross sections of 2k-chains production o^ have the form 32 -1

i

1 - exp(-z) £

-

k > 1,

(9)

»=o where aP = 8irjP exp(A£), z = ^Jpi exp(A£), £ = In ^ . The quantity C = 1.5 takes into account modification of the eikonal approximation due to intermediate inelastic diffractive states. The simplest parameterization of the pomeron exchange 2>(£, 0 = IP exp[ap(0)£ + (R2 + a'Pt)t]

(10)

has been used in Eq.(9). This model will be used below for description of different aspects of highenergy hadronic interactions. The parameters of the pomeron exchange 7 P , R2, A and a'P were determined from the fit to the experimental data on the total interaction cross sections and differential cross section of elastic pp, pp-scattering at high energies 33>34. The most important parameter A = 0.12 4- 0.14 in the "quasieikonal" approximation. It should be noted that the value of A becomes larger than in the "quasieikonal" approximation (A w 0.2) if the interactions between pomerons is taken into account 35 . For super-high energies, when £ ^> 1 er(tot)(£) has the Froissart type behavior

„(0 , ^

e

,

(n)

The slope of the diffraction cone increases asymptotically also as £ 2 . Inclusive cross sections and multiplicity distributions can be obtained in this approach by summing over hadronic production for all processes of 2k-chains formation. ,

oo

^ = £^(0/*^) ayc

0

(12)

39

*N(0 = y£12. In QGSM all these functions can be determined theoretically and are expressed in terms of intercepts of known Regge poles 3 0 ' 1 2 . Thus, contrary to other models, where fragmentation functions are determined from experimental data, in QGSM practically all parameters are fixed theoretically. The inclusive spectra in this model automatically have the correct triple-Regge limit for x —> 1, double-Regge limit for x -» 0 and satisfy to all conservation laws.

Comparison

with

experiment

In this section predictions of the QGSM model are compared with experimental data. After all the parameters of the Pomeranchuk-pole have been determined from a fit to experimental data on the total cross sections of pp, pp-interactions and the slope of the diffraction cone in elastic ppscattering, the predictions of the QGSM for different characteristics of multiparticle production at high energies practically do not contain new free parameters. For a sum of all n-pomeron exchange diagrams of the eikonal-type (without interaction between pomerons) there is a cancellation of their contributions to the single particle inclusive spectra in the central rapidity region for n > 2 3 1 . So only the pole diagram contributes and inclusive spectra increase with energy as fa ~ (s/so) A Rapidity (and pseudorapidity) distributions of charged particles in PP(PP)-interactions at different energies are well described in QGSM 11 0j

s

_> oo

(15)

In reggeon theory the Pomeranchuk theorem is automatically satisfied if the leading Regge singularity (pomeron) has vacuum quantum numbers. In this case differences of cross sections for particles and antiparticles decrease as powers of s for s -» oo ,

M«) = *THs) - ^\a)

= ]>>(0)(VS)(1-a-(0))

(16)

43

where the sum is over poles with negative signature and C-parity (w,p). Existing data on Acr(s) are in perfect agreement with the prediction of reggeon approach and do not indicate any violation of the Pomeranchuk theorem . A singularity in the j-plane with negative signature and C-parity and an intercept close to unity ("odderon") 39 , which could lead to a violation of the Pomeranchuk theorem, appears in the QCD perturbation theory 40 . It is a bound state of three reggeized gluons. In the approach, which takes into account nonperturbative effects 28 , described above, the lowest state made of three gluons connected by strings with quantum numbers 3 has rather large mass (M « 4 GeV) and the leading Regge trajectory with negative signature and C-parity has a very low intercept a^g < — 1. Contrary to the pomeron case in the small t region its mixing with qq trajectories (u,<j>) is weak and there is no "odderon" in this approach. Thus an experimental search for the "odderon" in the small t region is very important for understanding the dynamics of j-plane singularities in QCD. Searches for the "odderon" in the processes jp -> ir°N*,^p -> f°N* have been carried out at HERA. No signals have been found and limits on "odderon" contributions to cross sections of these processes have been established.

5. Conclusions Results presented in this review demonstrate that the reggeon theory is useful and universal approach for investigation of interactions of hadrons and nuclei at high energies. The pomeron is the main object of this approach. The challenging problem for high-energy hadronic physics is to establish the dynamical nature of the pomeron. The analysis of this problem in QCD with inclusion of both nonperturbative and perturbative effects shows that the pomeron has a very rich dynamical structure. Connection between reggeon theory, 1/N-expansion in QCD and string model leads to many relations between parameters of this theory and allows one to understand many characteristic features of strong interactions. The Quark-Gluon Strings and DPM models developed in the framework of this approach give a good description of many characteristics of high-energy hadronic interactions and give predictions for energies of future accelerators. The multipomeron exchanges are very important at high energies. With an account of these unitarization effects it is possible to understand and

44 quantitatively describe high-energy hadronic interactions and small-x deep inelastic scattering. Interactions between pomerons are essential in hadronnucleus and especially in nucleus-nucleus collisions.

References 1. E.L. Feinberg and I.Ya. Pomeranchuk, Doklady Akad. Nauk SSSR 94, 439 (1903); Suppl. Nuovo Cim. Ill, 052 (1956). 2. I.Ya. Pomeranchuk, ZhETF 34, 725 (1978). 3. V.N. Gribov and I.Ya. Pomeranchuk, Phys. Rev. Lett. 8, 343 (1962); 8, 312 (1962); Nucl. Phys. 38, 516 (1962); Phys. Rev. Lett. 9, 239 (1962). 4. S.C. Frautschi, M. Gell-Mann, F. Zacharisen, Phys. Rev. 126, 2204 (1962). 5. G. t'Hooft, Nucl Phys. B72, 461 (1974). 6. G. Veneziano, Phys. Lett. 52B, 220 (1974). 7. G. Veneziano, Nucl. Phys. B117, 519 (1976). 8. A. Casher, J. Kogut, L. Susskind, Phys. Rev. D10, 732 (1974). 9. X. Artru, G. Mennesier, Nucl. Phys. B70, 93 (1974). 10. B. Andersson, G. Gustafson and C. Peterson, Phys. Lett. 71B, 337 (1977); B. Andersson, G. Gustafson and C. Peterson, Z. Phys. C I , 105 (1979). 11. A.B. Kaidalov, JETP. Lett. 32, 474 (1980); A.B. Kaidalov, Phys. Lett. 116B, 459 (1982); A.B. Kaidalov and K.A. Ter-Martirosyan, Phys. Lett. 117B, 247 (1982); A.B. Kaidalov and K.A. Ter-Martirosyan, Sov. J. Nucl. Phys. 39, 979 (1984); 40, 135 (1984). 12. A.B. Kaidalov, in Proceedings of International Conference "QCD at 200 TeV", ed. L.Cifarelli and Yu. Dokshitzer, Plenum Press (1992), p.l.; A.B. Kaidalov, Surveys in High Energy Physics, 13, 265 (1999). 13. A. Capella et a l , Z. Phys. C 3 , 329 (1980); A. Capella, and J. I r a n Thanh Van, Phys. Lett. 114B, 450 (1982); Z. Phys. CIO 249 (1981). 14. A. Capella, U. Sukhatme, C.-I. Tan and J. Tran Thanh Van, Phys. Rep. 236, 225 (1994). 15. T. Regge, Nuovo Cim. 14, 951 (1959). 16. V.N. Gribov, ZhETF 41, 667 (1961). 17. G.F. Chew and S.C. Frautschi, Phys. Rev. Lett.7, 394 (1961). R. Blankenbecler and M. Goldberger, Phys. Rev. 126 766 (1962). 18. Froissart M. Phys. Rev. 123 1053 (1959). 19. S. Mandelstam, Nuovo Cim. 30 1148 (1963) 20. V.N. Gribov, I.Ya. Pomeranchuk and K.A. Ter-Martirosyan , Phys. Lett. 9, 269 (1964); Yad. Fiz. 2, 361 (1965) 21. A. Dubin, A. Kaidalov, Yu. Simonov, Phys. Lett. 323, 41 (1994) ; Yad. Fiz. 56, 213 (1993) . 22. Yu.A. Simonov, Nucl. Phys. B307 512 (1988); Yad. Fiz. 54, 192 (1991). 23. Yu.A. Simonov, hep-ph/0210309. 24. E.A. Kuraev, L.N. Lipatov, V.S. Fadin, Sov. Phys. JETP 44,

45

25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40.

41. 42. 43.

433 (1976); 45, 199 (1977); Ya.Ya. Balitsky, L.N.Lipatov, Sov. J. Nucl. Phys. 28, 822 (1979). V. S. Fadin, L. N. Lipatov, Phys. Lett. 429B 127 (1998). G. Camici, M. Ciafaloni, Phys. Lett. 430B 349 (1998). S.J.Brodsky et al, JETP Lett. 70, 155 (1999). Yu.A. Simonov, Phys. Lett. 249B, 514 (1990). A.B. Kaidalov, Yu.A. Simonov, Phys. lett. 477B, 163 (2000); A.B. Kaidalov, Yu.A. Simonov, Yad. Fiz. 63 8, 1507 (2000). V.N. Gribov, JETP 53, 654 (1967). A.B. Kaidalov, Sov. J. Nucl. Phys. 45, 902 (1987). V.A. Abramovski, O.V. Kancheli, V.N. Gribov, Yad. Fiz. 18, 595 (1973). K.A. Ter-Martirosyan, Phys. Lett. 44B 377 (1973). A.B. Kaidalov, K.A. Ter-Martirosyan, Yu.M. Shabelski, Sov. J. Nucl. Phys. 43, 822 (1986). K.A. Ter-Martirosyan, Sov. J. Nucl. Phys. 44. 817 (1986). A.B. Kaidalov, L.A. Ponomarev, K.A. Ter-Martirosyan, Sov. J. Nucl. Phys. 44, 468 (1986). A.B. Kaidalov and O.I. Piskunova, Sov. J. Nucl. Phys. 4 1 , 816 (1985); Zeit. Phys. C30, 145 (1985). N.S. Amelin et al., Sov. J. Nucl. Phys. 5 1 , 327 (1990); 52, 172 (1990). A.B. Kaidalov, Phys. Reports 50, 157 (1979). K. Kang, B. Nicolescu, Phys. Rev. D l l , 2461 (1975). D. Joynson et al. Nuovo Cim. 30A, 345 (1975). L.N. Lipatov, Phys. Lett. 309B, 394 (1993); P. Gauron, L.N. Lipatov and B. Nicolescu, Phys. Lett. 304B, 334 (1993); L.D. Faddeev and G.P. Korchemsky , Phys. Lett. 342B, 311 (1994); R.A. Janik and J. Wosiek, Phys. Rev. Lett. 82 1092 (1999). A. Capella, E. Ferreiro, A.B. Kaidalov and C.A. Salgado, Nucl. Phys. B593, 336 (2001), Phys. Rev. D63, 054010 (2001). B.B. Back et al (PHOBOS Collaboration), Phys. Rev. Lett. B 7 102303 (2001). A. Capella, A. Kaidalov, J. I r a n Thanh Van, Heavy Ion Phys. 9, 169 (1999).

T H E U N E X P E C T E D ROLE OF FINAL STATE I N T E R A C T I O N S IN D E E P INELASTIC SCATTERING

P. HOYER* Department of Physical Sciences and Helsinki Institute of Physics POB 64, FIN-00014 University of Helsinki, Finland E-mail: [email protected]

1. Introduction Deep Inelastic Scattering, e + N —> e + X (Fig. 1) is one of our most precise tools for investigating the substructure of hadrons and nuclei. Soon after the first observation of scaling at SLAC, it was shown1 that the DIS structure functions can be interpreted as the probability density of constituents in the target. Thus, in terms of the Fock state wave functions ipn of a target nucleon N, DIS would measure the parton density fcfj. 0

Target wf is probed at equal LC time (6)

+

An a; -ordered picture of the scattering is shown in Fig. 2a. The rescattering of the struck quark on the gluon field of the target influences the DIS cross section insofar as it occurs within the Ioffe coherence length (4). Even though the rescattering occurs over a finite distance in time and space the LC time is infinitesimal, x+ ~ \jv as given in (6). Hence one may regard the rescattering effects as part of an "augmented" LC wave function. Such a wave function contains physics specific to the DIS probe, and is thus distinct from the usual wave function of an isolated hadron. 2.2. Dipole

frame

I shall refer to the target rest frame where the photon moves along the positive z-axis as the Dipole frame: q = (v, 0j_, -Y^/v2 + Q2). This frame is related to the DIS frame by a 180° rotation around a transverse axis. Such a rotation is a dynamical transformation for a theory quantized at x+ = 0 . Hence there is no simple relation between the DIS and dipole frames in the a

T h e virtual photon is here regarded as an external particle with negative squared mass q2 = —Q2 and thus, exceptionally, q+ < 0 in (5).

50

(a)

(b)

Figure 2. LC time ( x + = t + z) ordered dynamics of deep inelastic scattering, (a) The DIS frame (5), where the virtual photon momentum qz ~ — v. The photon hits a target quark, which Coulomb rescatters before exiting the target A. The increase in t is compensated by a decrease of z such that the photon probes the target at an instant of x+. (b) The Dipole frame (7), where qz ~ -\-v. The photon splits into a qq pair at LC time x+ ~ XjIm^XB before the target. The DIS cross section is given by the scattering of the qq color dipole in the target.

case of non-perturbative target structure (the two frames can obviously be connected for a perturbative target model). The kinematics is inverted wrt. that of (5) and (6), q+

q

s

J_

~2v Q2

-^7

a.

~2v ' 1

0 (7)

Hence in the dipole frame the scattering occurs over a finite LC time x+ ~ Li. Since q+ is positive there can be a 7* -> qq transition before the interaction in the target. The qq forms a color dipole whose target cross section is determined by its transverse size. In the aligned jet model which corresponds to the parton model and thus to lowest order in QCD - the quark takes nearly all the longitudinal momentum, pi" ~ q+ ~ 1v, while p j ~ t^qCDjra^XBThe transverse separation of the quark pair is then rj_(qq) ~ 1/^-QCD ~ 1 fm. Such a transversally large qq dipole has a non-perturbative cross section, which is the parameter in the dipole frame that corresponds to the parton distribution of the DIS frame. The qq pair multiple scatters in the target as shown in Fig. 2b. At low XB also the antiquark momentum is large and one expects pomeron exchange, i.e., diffractive DIS, as well as shadowing of the target 6 . The dipole frame is natural for modelling these features of the data, where

51

T(q)

|

T(q)

|

p-krk2

T(p)

(a)

r(i)

i

i

(b) Figure 3. Two types of final state interactions, (a) Scattering of the antiquark (p2 line), which in the aligned jet kinematics is part of the target dynamics, (b) Scattering of the current quark (pi line). For each LC time-ordered diagram, the potentially on-shell intermediate states are marked by dashed lines.

phases and interference effects play a key role 7 . If it were possible to "turn off" rescattering in the DIS frame by choosing LC gauge A+ = 0, the pomeron and interference effects of the dipole frame would have to be built into the target wave function itself (which, as I noted above, has no dynamical phases). Thus there would be a preformed pomeron in the target, and shadowing would imply that quarks with low XB would be suppressed in nuclei (relative to nucleons). I shall next discuss why rescattering effects cannot be avoided even in the DIS frame. This makes the physics of the DIS and dipole frames much more similar. In particular, diffraction and shadowing arise from the interference of rescattering amplitudes in both frames.

52

3. Interactions between spectators We normally regard the target as "frozen" during the DIS process. I first recall why this is legitimate in Feynman gauge, and then discuss why there are relevant spectator interactions in LC gauge. 3.1. Spectator

interactions

in Feynman

gauge

+

The generic, x -ordered Feynman diagram shown in Fig. 3a involves a gluon exchange between two target spectator lines (in the aligned jet kinematics only the struck quark carries asymptotically large momentum, p^ ~ 2v). Three intermediate states, marked by dashed lines, can kinematically be onshell and thus contribute to the discontinuity of this forward diagram. Since Pi is much bigger than all target momenta the three discontinuities occur at essentially the same value of pf. Hence they form a higher order pole, i.e., their contributions cancel to leading order in the Bj limit. The same argument applies to any diagram involving spectator-spectator exchanges between the virtual photon vertices. Exchanges between a spectator and the struck quark (Fig. 3b) are, on the other hand, expected to contribute at leading twist and to build the path-ordered exponential in (2). Elastic scattering of the struck quark implies k+ ~ 1/z/, which is of the same order as p+. There are then two distinct large momenta in Fig. 3b, pj~ and (Pi + fo)-, a n d the three discontinuities do not cancel. The above argument holds in Feynman gauge, where all contributions to the discontinuity arise from on-shell configurations. 3.2. Spectator

interactions

in LC gauge

The singularity of the LC gauge propagator (3) at k+ = 0 blurs the distinction between spectator-spectator and spectator-struck quark interactions (Figs. 3a and 3b, respectively). Since the k+ = 0 poles are gauge artifacts the sum of their residues must add up to zero. However, this cancellation requires that diagrams of type 3a and 3b are added. The path ordered exponential in (2) reduces to unity in A+ = 0 gauge. In perturbation theory this occurs through a cancellation1" between the —g^v (Feynman) and (n^k" + k^n'/)/k+ (LC gauge artifact) parts of the propagator (3), for interactions of the struck quark such as in Fig. 3b. b

Depending on the prescription used at k+ = 0 this cancellation can occur separately for each Feynman diagram, or only in their combined contribution to the DIS cross section, as explained in Ref.8.

53

h.

y*(g) T(P)

T(p') (a)

(b)

(c)

Figure 4. A scalar abelian model for deep inelastic scattering with one-, two, and threegluon exchanges. At each order only one representative diagram is shown. In the XB —> 0 limit and at the orders considered no other final states contribute to the total DIS cross section.

Once the k+ = 0 poles of the LC gauge propagator are "used up" in diagrams like Fig. 3b to cancel the Feynman gauge contribution, the corresponding spurious poles in diagrams like Fig. 3a give a non-vanishing contribution to the leading twist DIS cross section. In fact, their sum must equal the contribution from the path ordered exponential in Feynman gauge. Thus gauge independence is achieved, and the simplification of the exponential (alias rescattering of the struck quark) is accompanied by a complication in the spectator system.

4. A perturbative model Most of the features discussed above were verified in an perturbative model calculation 4 . The one-, two- and three-gluon exchange amplitudes were evaluated in the Bj limit for XB —>• 0 in both Feynman and LC gauge, using various prescriptions for the k+ = 0 poles, and in covariant as well as a:+-ordered perturbation theory. In this way the gauge dependence of the individual diagrams could be seen to be compatible with the gauge independence of the physical cross section. In particular, the fact that the k+ = 0 poles are absent in the complete sum of diagrams was verified. In impact parameter space the n-gluon exchange amplitudes were (for n = 1,2,3) found to have the form An oc V(-y*)Wn, where V is the virtual photon wave function and W a rescattering factor of the qq pair. This corresponds to the structure expected in the dipole frame at small XB, where the quark pair is created by the photon and then rescatters in the target. The perturbative calculation can of course be equivalently done in the DIS frame.

54

Summing the gluon exchanges to all orders gives

5>i=

2 sin 'g W(f±,R±)/2

M{p2

92W(f±,R±)/2

,rj_,R±)

(8)

where W(rx,R±)

=-log

( — R ^ —

(9)

with rj_ the transverse size of the produced quark pair, R± the distance between the pair and the target (see Fig. 4b), and g the gauge coupling. Hence the higher order corrections reduce the magnitude of the Born amplitude A\, and thus also of the DIS cross section. This "shadowing" effect arises from destructive interference between the rescattering amplitudes An.

T(q)

A2=

!

^r "E> T(p')

Figure 5. The two-gluon exchange amplitude A2 is purely imaginary at low i g . The intermediate state indicated by the dashed line is thus on-shell and the full amplitude is given by the product of the two subamplitudes on either side of the cut. (Only one of the contributing Feynman diagrams is shown.)

The rescattering gives the amplitudes complex phases. While the Born (Ai) amplitude is real, the two-gluon exchange amplitude A? is purely imaginary in the XB —> 0 limit, as mandated by analyticity and crossing. Hence the intermediate state between the gluon exchanges in Fig. 4b is on-shell. This allows to write the full amplitude as a product of two single gluon exchange subamplitudes (Fig. 5). The subamplitudes are on-shell and hence gauge invariant: The feature of on-shell intermediate states, and hence rescattering, is present in Feynman as well as in LC gauge. The A2 amplitude is also the lowest order contribution to diffractive DIS, characterized by color singlet exchange. This leading twist part of the total DIS cross section thus explicitly arises from rescattering of the struck quark in the target.

55

. k+q

P-k spectator' system proton

(b)

(a)

Figure 6. (a) The struck quark can be asymmetrically distributed in azimuth wrt the virtual photon direction if it suffers final state interactions and the target is transversely polarized, (b) A model diagram with a final state interaction giving rise to a transverse spin asymmetry at leading twist. (The figures are from Ref. 9 .)

5. Single spin asymmetry The realization that rescattering gives rise to complex phases in DIS amplitudes led to interesting developments for the transverse spin asymmetry in DIS 9 ' 10 . The data 11 indicated a sizeable target spin asymmetry, in apparent conflict with a theorem 12 that such effects are higher twist. The asymmetry for a transversely polarized target is given by the imaginary part of the interference between target helicity flip and non-flip, SSA ~ Im(T\T*_x), and is thus sensitive to phase differences. The gluon rescattering contribution in the model Feynman diagram of Fig. 6b provides such a phase, resulting in an SSA at leading twist 9 . The rescattering phases are contained in the path ordered exponential of the parton distribution (2). The presence of the exponential violates the assumptions that were made 12 in deriving the theorem, and indeed allows a leading twist asymmetry 10 . This effect is unrelated to the transverse spin carried by the struck quark, making a measurement of the latter more challenging. 6. Discussion Our discussion has centered on why the effects of the path ordered integral Pexp ig I Jo

dw A+(w

)

(10)

56

in the expression (2) of the parton distribution cannot be eliminated. The exponential reduces to unity in light cone gauge (A+ = 0), but this merely shifts its effects to the spectator system. The exponential arises from the rescattering of the struck quark in the target. This makes the scattering amplitudes complex, giving rise to diffraction and shadowing phenomena in DIS. Such physical effects cannot be eliminated by a gauge choice. Some misunderstandings occur due to imprecise use of the term "final state interactions". In the present context we should differentiate between interactions at three time scales (in the target rest frame): (1) The hadronization time of the struck quark. Due to Lorentz dilation this is proportional to the quark energy: t = x° oc v. The total DIS cross section is independent of interactions occurring at the hadronization time scale. (2) The Ioffe time tj = v/Q2 = l/(2mivXB), which is finite in the Bj limit. Interactions occurring within the Ioffe time affect the DIS cross section. The struck quark rescattering which gives rise to the path ordered exponential occurs at this time scale. Note that the corresponding LC time x+ = t + z oc \jv is infinitesimal in the DIS frame, due to a cancellation between the time t and the distance z when the photon moves in the negative z-direction. (3) The exact LC time x+ of the virtual photon interaction. This is relevant for QCD quantized at a given x+, and in x + -ordered perturbation theory. Interactions occurring before the photon interaction build the light cone wave function of the target. Struck quark rescattering occuring after this time gives rise to dynamic phases which are specific to DIS, and are not present in the wave function of an isolated hadron. The struck quark rescattering that occurs at the Ioffe time is within the time (or x+) resolution of the virtual photon. It is therefore possible to regard those interactions as part of an "augmented" target wave function, specific to the hard inclusive probe. Since the interactions occur at infinitesimal x+ oc \jv and the exchanged gluons likewise carry k+ oc 1/v, the augmented wave function apparently differs from the standard one through "zero modes". Such modes 13 are possible for theories quantized on a lightlike surface, which allows causal (light-like) connections. It is sobering to realize that many questions are still open concerning such a basic QCD process as deep inelastic scattering, more than 30 years after it first was discovered.

57 Acknowledgments I would like to t h a n k the organizers of this meeting for their kind invitation. T h e work I have presented is based on work done with Stan Brodsky, Nils Marchal, Stephane Peigne and Francesco Sannino. I am grateful to them and many others for helpful discussions.

References 1. S. D. Drell and T.-M. Yan, Phys. Rev. Lett. 24, 181 (1970); S. D. Drell, D. J. Levy and T.-M. Yan, Phys. Rev. D l , 1035 (1970). 2. J. C. Collins and D. E. Soper, Nucl. Phys. B194, 445 (1982); J. C. Collins, D. E. Soper and G. Sterman, Nucl. Phys. B261, 104 (1985), Nucl. Phys. B308, 833 (1988), Phys. Lett. B438, 184 (1998) and review in Perturbative Quantum Chromodynamics, (A.H. Mueller, ed.,World Scientific Publ., 1989, pp. 1-91); G. T. Bodwin, Phys. Rev. D 3 1 , 2616 (1985), Erratum Phys. Rev. D34, 3932 (1986). 3. S. J. Brodsky, M. Diehl and D. S. Hwang, Nucl. Phys. B596, 99 (2001) [hepph/0009254]; M. Diehl, T. Feldmann, R. Jakob and P. Kroll, Nucl. Phys. B596, 33 (2001) [hep-ph/0009255]. 4. S. J. Brodsky, P. Hoyer, N. Marchal, S. Peigne and F. Sannino, Phys. Rev. D65, 114025 (2002) [hep-ph/0104291]. 5. J. C. Collins, arXiv:hep-ph/0106126. 6. V. N. Gribov, Sov. Phys. JETP 29, 483 (1969) and 30, 709 (1970). 7. G. Piller and W. Weise, Phys. Rept. 330, 1 (2000) [arXiv:hep-ph/9908230]. 8. A. V. Belitsky, X. Ji and F. Yuan, Nucl. Phys. B656, 165 (2003) [arXiv:hepph/0208038]. 9. S. J. Brodsky, D. S. Hwang and I. Schmidt, Phys. Lett. B530, 99 (2002) [arXiv:hep-ph/0201296]. 10. J. C. Collins, Phys. Lett. B536, 43 (2002) [arXiv:hep-ph/0204004]. 11. A. Airapetian et al. [HERMES Collaboration], Phys. Rev. Lett. 84, 4047 (2000) [arXiv:hep-ex/9910062]. 12. J. C. Collins, Nucl. Phys. B396, 161 (1993) [arXiv:hep-ph/9208213]. 13. S. J. Brodsky, H. C. Pauli and S. S. Pinsky, Phys. Rept. 301, 299 (1998) [arXiv:hep-ph/9705477].

OVERVIEW OF RESULTS FROM THE STAR E X P E R I M E N T AT RHIC*

K. F I L I M O N O V Nuclear Science Division, Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720, USA E-mail: [email protected]

The Relativistic Heavy-Ion Collider (RHIC) provides A u + A u collisions at energies up to ^/s N N =200 GeV. STAR experiment was designed and constructed to investigate the behavior of strongly interacting matter at high energy density. An overview of some of the recent results from the STAR collaboration is given.

1. Introduction One of the fundamental predictions of the Quantum Chromodynamics (QCD) is the existence of a deconfined state of quarks and gluons at the energy densities above 1 GeV/fm 3 1 . This strongly interacting medium, the Quark Gluon Plasma (QGP), may be created in the laboratory by the collision of heavy nuclei at high energy. The current experimental program at the Relativistic Heavy-Ion Collider (RHIC) is aimed at detecting the new state of matter and studying its properties. The Relativistic Heavy-Ion Collider is located at Brookhaven National Laboratory (New York, USA). It is capable of colliding gold ions from v /s N N =20 to 200 GeV per nucleon pair, protons up to ^/s=500 GeV, and asymmetric systems like deuterons with heavy nuclei. RHIC is also the first polarized proton collider at high energies, opening new opportunities "This work was supported by the Division of Nuclear Physics and the Division of High Energy Physics of the Office of Science of the U.S. Department of Energy, the United States National Science Foundation, the Bundesministerium fur Bildung und Forschung of Germany, the Institut National de la Physique Nucleaire et de la Physique des Particules of France, the United Kingdom Engineering and Physical Sciences Research Council, Fundagao de Amparo a Pesquisa do Estado de Sao Paulo, Brazil, the Russian Ministry of Science and Technology and the Ministry of Education of China and the National Science Foundation of China.

58

59

to study the spin structure of the proton, Diffractive processes in high electromagnetic fields can also be studied in ultra-peripheral heavy-ion collisions. 2. S T A R D e t e c t o r The Solenoidal Tracker at RHIC (STAR) is one of two large detector systems constructed at RHIC 2 . The layout of the STAR experiment is shown in Figure 1. The main detector in STAR is the world's largest Time Projection Chamber (TPC) 3 measuring trajectories of charged particles at mid-rapidity (\r]\ < 1.4) with full azimuthal coverage. A solenoidal magSiftn

i iBtet

-\\x. r:''

'

-

* •"*•*
> ' 1 • ' ' 1 • Au+Au 200 GeV central o Au+Au 130 GeV central n Pb+Pb 17 GeV central (NA44)

CD 0.8 0.6 A H Q. 0.4 V

%

A Au+Au 200 GeV peripheral A p+p 200 GeV

0.2 0

I

0.2

0.4

mass

0.6

0.8

1

.

.

.

1.2

[GeV/c ]

Figure 4. Above: Extracted (p±) versus centrality for n~ (triangles), K~ (squares), and p (circles) from STAR. Below: Extracted (p±) versus particle mass for five collision systems.

64

consistent with transverse radial flow, which appears to be stronger at RHIC than SPS. A blast wave fit to the central n~, K~, and p spectra provides a transverse radial flow velocity of (PT) — 0.55C at 130 GeV, and ~0.60c at 200 GeV, with c the speed of light. The fit kinetic freezeout temperature parameters show at most a mild decrease going from SPS (Tfo~110 MeV) to RHIC (T fo ~100 MeV) energies. 4. Hard Physics High PT hadrons are produced in the initial collisions of incoming partons with large momentum transfer. Hard scattered partons fragment into a high energy cluster (jet) of hadrons. In elementary e + e~ and pp collisions, jet cross sections and single particle spectra at high transverse momentum are well described over a broad range of collision energies by perturbative QCD (pQCD). Partons propagating through a dense system may interact with the surrounding medium radiating soft gluons at a rate proportional to the energy density of the medium. The measurements of radiative energy loss (jet quenching) in dense matter may provide a direct probe of the energy density 12 . 4.1. High PT: Suppression

of Inclusive

Spectra

Hadrons from jet fragmentation may carry a large fraction of jet momentum (leading hadrons). In the absence of nuclear medium effects, the rate of hard processes should scale with the number of binary nucleon-nucleon collisions. The yield of leading hadrons measured in Au+Au collisions at v /s^'=130 GeV has been shown to be significantly suppressed 13 , indicating substantial in-medium interactions. The high statistics 200 GeV data extended the measurements of hadron spectra to pr =12 GeV/c 14 . Figure 5 (above) shows inclusive invariant pr distributions of charged hadrons within |JJ| = 0) for both centralizes is consistent with that measured in p + p collisions. This indicates that the same mechanism (hard parton scattering and fragmentation) is responsible for high transverse momentum particle production in p+p and Au+Au collisions. The away-side (back-to-back) correlations in peripheral Au+Au collisions may be described by an incoherent superposition of jetlike correlations measured in p+p and elliptic flow. However, back-to-back jet production is strongly suppressed in the most central Au+Au collisions. This indicates a substantial interaction as the hard-scattered partons or their fragmentation products traverse the medium. The ratio of the measured Au+Au correlation excess relative to the p + p correlation is: IAA(A<j>uA)[D^A"-B(l C*

a

+ 2vl cos(2A4>))]

d(A^)DPP

•

W

The ratio can be plotted as a function of the number of participating nucleons (Npart). IAA is measured for both the small-angle (\A 2.24 radians) regions. The ratio should be unity if the hard-scattering component of Au+Au collisions is simply a superposition of p+p collisions unaffected by the nuclear medium. These ratios are given in Figure 8 for the trigger particle momentum ranges indicated. For the most peripheral bin (smallest Npari), both the small-angle and backto-back correlation strengths are suppressed, which may be an indication of initial state nuclear effects such as shadowing of parton distributions or scattering by multiple nucleons, or may be indicative of energy loss in a dilute medium. As Npart increases, the small-angle correlation strength increases, with a more pronounced increase for the trigger particles with lower pr threshold. The back-to-back correlation strength, above background from elliptic flow, decreases with increasing Npart and is consistent with zero for the most central collisions.

70

4

^

:

J

A

lx

0.5

0 0

50 100 150 200 250 300 350 400

Npart

i

1 ] •

•

-

A

i

i

1 1 i

i

i

•

; •

1

i—i

:S L*

' i.

—

i—i


3 GeV/c, and disappearance of back-to-back jets, are all consistent with a picture in which observed hadrons at pr > 3 — 4 GeV/c are fragments of hard scattered partons, and partons or their fragments are strongly scattered or absorbed in the nuclear medium. The observed hadrons therefore result preferentially from hard-scattered partons generated on the periphery of the reaction zone and heading outwards. These observations appear con-

71

sistent with large energy loss in a system t h a t is opaque to t h e propagation of high-momentum partons or their fragmentation products. T h e upcoming results from d-Au collisions at y / sj^7=200 GeV will help in determining whether the observed effects are due to the final state interactions.

References 1. J. W. Harris and B. Muller, Ann. Rev. Nucl. Part. Sci. 46, 71 (1996). 2. K.H. Ackermann et al. [STAR Collaboration], Nucl. Instr. Methods. A 499, 624 (2003). 3. M. Anderson et al, Nucl. Instr. Methods. A 499, 659 (2003). 4. C. Adler et al. [STAR Collaboration], Phys. Rev. Lett. 86, 4778 (2001), Erratum-ibid. 90, 119903 (2003). 5. C. Adler et al. [STAR Collaboration], Phys. Rev. Lett. 89, 092301 (2002). 6. G. Van Buren [STAR Collaboration], Nucl. Phys. A 715, 129c (2003). 7. T. S. Ullrich, Nucl. Phys. A 715, 399c (2003). 8. P. Braun-Munzinger, D. Magestro, K. Redlich and J. Stachel, Phys. Lett. B 518, 41 (2001). 9. O. Barannikova and F. Wang [STAR Collaboration], Nucl. Phys. A 715, 458c (2003). 10. C. Adler et al. [STAR Collaboration], Phys. Rev. Lett. 87, 262302 (2001). 11. C. Adler et al. [STAR Collaboration], arXiv:nucl-ex/0206008. 12. R. Baier, D. Schiff and B. G. Zakharov, Ann. Rev. Nucl. Part. Sci. 50, 37 (2000). 13. C. Adler et al. [STAR Collaboration], Phys. Rev. Lett. 89, 202301 (2002). 14. J. L. Klay [STAR Collaboration], Nucl. Phys. A 715, 733c (2003). 15. X. N. Wang, nucl-th/0305010; private communication. 16. I. Vitev and M. Gyulassy, Phys. Rev. Lett. 89, 252301 (2002). 17. D. Antreasyan et al. Phys. Rev. D 19, 764 (1979); P. B. Straub et al., Phys. Rev. Lett. 68, 452 (1992). 18. I. Vitev and M. Gyulassy, Phys. Rev. C 65, 041902 (2002). 19. X. N. Wang, Phys. Rev. C 63, 054902 (2001). 20. M. Gyulassy, I. Vitev and X. N. Wang, Phys. Rev. Lett. 86, 2537 (2001). 21. S. Voloshin and Y. Zhang, Z. Phys. C 70, 665 (1996). 22. K. H. Ackermann et al. [STAR Collaboration], Phys. Rev. Lett. 86, 402 (2001). 23. C. Adler et al. [STAR Collaboration], Phys. Rev. Lett. 90, 032301 (2003). 24. K. Filimonov [STAR Collaboration], Nucl. Phys. A 715, 737c (2003). 25. C. Adler et al. [STAR Collaboration], Phys. Rev. Lett. 90, 082302 (2003).

I. YA. POMERANCHUK AND SYNCHROTRON RADIATION G. N. KULIPANOV, A. N. SKRINSKY Budker Institute of Nuclear Physics, 630090 Novosibirsk, Russia. In this paper, it is considered the development of I.Ya. Pomeranchuk's ideas given in his pioneering works concerning radiation of relativistic electrons in the magnetic field. Realization of these ideas has led to the appearance of a new research direction related to creation of the synchrotron radiation sources. The use of synchrotron radiation opened up now the new possibilities of research both for various scientific fields and applications 1.

History

I.Ya.Pomeranchuk was a very high level physicist, a very good theoretician, and his level is obvious when you consider his contribution in the field of elementary particle physics, in fundamental nuclear physics or in theory of nuclear reactors. The same is true for such a limited area as synchrotron (or magnetostrahlung) radiation. Pomeranchuk's interest in this kind of radiation (appeared at the end of 1930s) was of astrophysical origin: what happens when charged cosmic ray particles travel in the magnetic field of the Earth? The answer was [1] - they lose their energy emitting electromagnetic radiation. And he obtained very simple and fundamental formula 9 AF — —1• r 2 LU -'

~ „

R

2

class "transv

• v2 • T I

^path

which gives energy losses of (relativistic) charged particle upon passing of Lpass in transversal magnetic field B, ransv . For given energy, the energy losses are proportional to the M" 4 , hence much more important for electrons than, say, for protons. Now it is well known: in many astrophysical cases such radiation plays dominant role, and observation of it gives very valuable information on magnetic field structure in variety of astronomical objects. But in this article we deal with another face of the Pomeranchuk radiation radiation of high energy particles in accelerators and storage rings. Together with D.D.Ivanenko [2] and than with L.A.Artsimovich [3], I.Ya.Pomeranchuk was the first to analyze this phenomenon. It is called now synchrotron radiation and in already for 40 years plays crucial role for beams behavior (it produces synchrotron radiation cooling, and gives possibility to reach, in specific cases, ultimately low beam emittance or 72

73

ultimately high luminosity of colliders, and also limits the energy of cyclic electron accelerators, as was predicted in [3]). In the article [3] the basic characteristics of synchrotron radiation are studied and obtained for the fist time: • • • •

angular distribution (1/y). radiation spectrum (^.~R/y3). the radiation of non-interacting system of electrons (and limits for such approach). radiation influence on electron trajectory at betatron: (energy losses - but oscillation damping, beam compression, etc.).

On the other hand, its use for ultra-violet, vacuum ultra-violet and X-ray experiments gave rise for 10 orders of magnitude in mean brilliance - the main characteristics of radiation sources for crystallography, diffractometry and spectroscopy, for topography and holography, for time resolved experiments, etc. Brightness defines the exposure time for great variety of experiments and technological applications [4]. The Pomeranchuk pioneer synchrotron radiation studies described above limit his important contribution to this field. Below, we present very brief description of the status and convincing prospects of the development of SR sources in the world and specifically in Russia. 2.

Synchrotron radiation sources in the world

At present time there are about 30 functioning SR sources in the world and 15 ones under construction (see Fig. 1). The basic characteristic that determines practical merit of any SR source is the spectral brightness (B^). It is equal to the number of photons (Nph) emitted within a given spectral range (AAA) per unit time (At) and from unit area of a source (AS) to a unit solid angle (AQ):

AX/XAtASAQ.

74

Syn c h r o t r o n s

,1 J

KSRS (Moscow) TSC (Zelerograd) OB.SV (Pubna) SSRC (Novosibirsk) South Korea PLS (Poha-3 v ,

I U in t h e wo IRussia

Germany ANKA (Karlsrufte* BBSSt (fcsrtint 08LTA (Dortmund) EISA-II (senru MASYLftB (Mambu Denmark ASTMD (AartwS) *

Spain # - LIS 'i France ESRF («3r*nob!e) LURE (Orsat) j4 , CftMD (Battn louge) «Duje) CHESS (Cornell) NSLS (Brookhs* SRC (Stoyg¥*m)l SSRL (Stanford) SURF (Gathei5b*irg)°* Brazil LrttS (Campmss;

Ital* 0AFNE (Frarrjh) ELrtTBA (TnMtr) UK DIAMOND (Oli(o«) SPS (Daresbury) Sweden MAK (Uind) Switzerland SIS (Valligeft)

s cWm» , BSRr »«•>'!«)

HSRC ( H l r o s h i m t f t ^ • NAHIMANA ( C h * f > NEW SUBARU !HrtfjeiO PHOTON FACTOR 1, one might find a larger probability if the energy is split in several fragments, and those fragments collide to produce several black holes of smaller masses. We discuss here such process in the case where the fragments are gravitons, and the number n of produced black holes is large, n > 1, so that the typical energy u ~ E/n of each

91

graviton is much smaller than E. On the other hand, it is assumed here that the typical invariant mass in pairwise collisions of the gravitons is still larger than the Planck mass, so that one could apply the geometric formula for creation of "small" black holes in those collisions. The latter condition allows to only consider the range of n up to n ~ \[GE. For the estimate of the effect of the energy fragmentation into gravitons we start with considering a single soft graviton bremsstrahlung in a collision involving ultrarelativistic particles. The term in the amplitude, corresponding to emission of a soft graviton with momentum k by massless particle "a" with energy Ea 3> w, can be simply found in the physical gauge in the c m . frame, where the components of the graviton tensor amplitude h^v are only spatial, traceless, and transversal to the graviton momentum k:

Aa=V^G^fi^.

(6)

u> 1 - cos 0 Here i and j stand for the spatial indices, and p1 is a unit vector in the direction of the momentum of the particle a, and 8 is the angle between that direction and the graviton momentum. Also it is assumed that the graviton tensor amplitude is canonically normalized, i.e. g^ — 77M„ 4- ylQirG h^. One can further notice that for the physical components of the graviton tensor amplitude, the product plp>hij is in fact proportional to sin2 6. Thus unlike in a bremsstrahlung of massless vector bosons (e.g. photons) there is no forward peak in the emission of gravitons for an ultrarelativistic particle, i.e. in the massless limit. The total amplitude of a soft graviton emission is given by the sum of the amplitudes of emission, as in eq.(6), by all the energetic particles. In particular this generally leads to that, unlike in the familiar case of bremsstrahlung of vector particles, the direction of emission of soft graviton is not associated with the direction of any particular incoming or outgoing particle. This is most explicitly illustrated by the graviton bremsstrahlung in a collision of two massless particles at energy E = y/s forming a static (in the cm.) massive object (as in the discussed production of a single large black hole). The total amplitude of emission can be written in the c m . frame as

A = Vm^G §-plpkhik ( 2w

1

\ 1 —cos0

+

* ), 1 + COS0/

(7)

92

which results in totally isotropic probability of emission of the graviton dw =

2Gs 7T

dudSl UJ 4-7T

(8)

where dVt is the differential of the solid angle. It is important for what follows that the effective strength of the source of soft gravitons is determined by the large energy E of the projectile, rather than by the soft graviton energy u>. Proceeding to considering the fragmentation of the energy of the initial particles, we first discuss a condition that would ensure that the produced fragments do not subsequently fall into a common large black hole. Most conservatively, i.e. in a manner most favorable to the idea of geometric cross section for black hole production, it is assumed here that all objects moving at transverse distances shorter than the gravitational radius of the largest possible common black hole, r 0 = 2 G y/s, are likely to fall into the large black hole. Thus only the fragments, that move at transverse distances larger than rn will be considered as avoiding that fall. One can readily verify that this condition limits the transverse momenta of the "fall safe" fragments as k± < 1/ro. The longitudinal distance at which a fragment with energy w is emitted can be estimated as I ~ to/k2^, and this distance is also larger than ro, once the above condition for k± is satisfied.

Figure 1. A representative of the type of graphs considered here for multiple black hole production. The circles stand for black holes and the lines denote gravitons. (The initial incoming particles are also drawn as gravitons for simplicity.)

93

Let us estimate now the amplitude for production of n black holes due to collisions of soft virtual (in fact almost real) gravitons, under the assumption of the geometric cross section. A generic graph for this process is shown in Fig.l. According to the previous discussion of soft graviton emission the factor in the amplitude describing production of black hole in the collision of i-th and j-th gravitons can be estimated as / 2

where f(q ) is the coupling of two gravitons to a state of black hole with mass Mjj = q2. In evaluating the amplitude we treat the logarithmic integrals as being of order one, which is sufficient for estimating the lower bound on the amplitude. In this approximation the result of the integration over ki (with the restriction k± < I/TQ for both ki and kj) can be estimated as / d4ki/(k2 k2) ~ 0(1), and w; ujj ~ q2 = M\. Then the cross section for producing n smaller black holes can be estimated as

d

°n ~ ( ^ ft f ^ £ l/(^)l2 ^ r P(«2) K •

do)

where the index a enumerates the produced black holes, and p(M^) is the density of states of a black hole at mass M # . The factor (n!)~ 2 in eq.(10) arises from the number (2n) of identical (virtual) gravitons, and we neglect weaker in n factors, i.e. behaving as powers of n or as cn with c being a numerical constant. With the constraint q± < 1/ro the integration over the momentum qa of the black hole can be estimated (again, up to a logarithmic factor) as J d3q/q° ~ l/r^ « l/(G 2 s). Furthermore, the geometric cross section, that is assumed here for the purpose of this calculation, implies that \f(q2)\2p(q2)~G2(q2)2,

(11)

which according to eq.(10) results in the estimate of the cross section as d

-n-7^f[G2Sdq2a. V

';

(12)

a=l

For production of n black holes, each with mass ranging up to (the lower bound on) the cross section can thus be estimated as

E/n,

G2 s2 \ " •

(13)

94

In obtaining this estimate for the cross section the graphs with graviton self-interactions were neglected. The contribution of emission of gravitons by gravitons would enhance the amplitude, and eq.(13) can still be used as a lower bound. A more serious problem arises from loop graphs with rescattering of gravitons. These graphs generally would modify the amplitude by order one. However a reliable estimate of the effect runs into the general problem of calculating loop graphs in quantum gravity, which is not readily solvable at present. This undoubtedly makes the status of the estimate (13) less certain, although it does not look any less certain than that of the geometric cross section. Clearly, the estimate (13) implies that a t G s > 1 the total cross section should grow exponentially atot ~ ex.p(^/G~~s), and should be dominated by production of 0(y/G s) small black holes, each having mass of order the Planck mass G - 1 / 2 . I believe that this behavior illustrates the intrinsic inconsistency of the assumption of a geometric cross section for black hole production. Namely, assuming the geometric formula for production of a single black hole, one arrives at the conclusion that the channel with a single black hole should make only an exponentially small fraction of the total cross section. Thus in any unitary picture, where the total cross section does not grow exponentially with energy, the partial cross section with production of one large black hole should be exponentially small, in contradiction with the assumption of geometric cross section.

3. Statistical and path integral considerations As is well known a large black hole can be considered as a thermodynamic object with the density (number) of states N described by its entropy SH, which for a non-rotating (and non-charged) black hole is given by SH = 47rG Mjj, and M = expS/f.The total probability of production of the black hole states can then be written as P(few ->H) = '%2 \A{few - • H)\2 ~ N \A{few ->• H)\2 .

(14)

m On the other hand, by reciprocity the amplitude A(few —> H) is related 9 to the amplitude of decay of each state of the black hole into the considered state of "few" particles: \A(few -> H)\2 = \A(H -> few)\2. The probability of such decay can be estimated from the black hole evaporation

95 law with the temperature T# = l/(47rr/f): P(H -> few) ~ \A(H ->• few)\2 ~ exp

- >

—-

= exp

~TH)

'

(15) where E{ are the energies of individual particles. Thus, using the reciprocity, the probability in eq.(14) can be evaluated as P(few

-> ff) ~ exp (s„

- j £ \

= exp (-4TTGM 2 H )

,

(16)

which describes the exponential suppression of production of a large black hole, i.e. with G Mfj 3> 1. This agreement should come as no surprise, since the expression in (16) contains the free energy Fn = MH — TJJ SH, in agreement with the general thermodynamic expression for the probability as being given by exp(—F/T). It should be noted, that the estimate (15) of the decay probability from the evaporation of the black hole is not entirely without a caveat. Namely the standard consideration of evaporation 10 , leading to the Gibbs factor exp(—Ei/T) per each particle, neglects the back reaction of the radiated particles on the black hole. In the process of decay into few particles the black hole disappears, and the effects of back reaction should be quite important. One might expect however, that these effects do not drastically change the exponent, estimated from the evaporation formula. Indeed, if the number of ("few") particles n is a large number n ^> 1, the emission of each of these particles does not significantly affect the mass of the remaining black hole. Thus one might expect that the back reaction gives corrections to pre-exponent decreasing for large n. Extrapolating this behavior down to small n and eventually down to n = 2 may significantly change the pre-exponent in eq.(15), but the back reaction effects are unlikely to compete with the large exponential factor. A modification of the Gibbs factor, effectively eliminating the discussed exponential suppression was advocated 11 by approximating an instantaneous decay of a black hole into two (or few) particles by a sequential emission of small energy fragments. However the applicability of such approximation in the discussed problem at least requires a further consideration. Furthermore, the agreement of the result from the presented here estimate with that from the path integral consideration, outlined below, can be argued as a reasoning for no substantial modification of the Gibbs factor.

96 The result in eq.(16) can be formulated as a quantitative assessment of the effect of "the rapid growth of the density of black hole states at large mass" 3 . As can be seen, the entropy of the black hole, as large as it is, is still not sufficient for the number of states Af = exp(Sn) to overcome the Gibbs factor exp(—M/f/Ttf). In other words, the free energy FH = MH - TH SH of the black hole is positive. An equivalent to the statistical consideration line of reasoning can be developed also using the path integral approach to calculating the scattering amplitudes. The process under discussion is of the type few -» H, where the initial state contains few particles (including the case of just two particles colliding), and H stands for a black hole with mass MH ^> Mpi. The specification "few" for the number of particles implies here that the number of particles n is not considered as a large parameter. The transition amplitude for the discussed process is given by the path integral A(few

- • JJ) = /

exp(il[g,

]) VVg

(17)

J few(t— — oo)

over all the field trajectories starting with incoming few particles in the distant past and ending as an outgoing black hole at t = +00, and where I[g, ] is the action functional depending on the metric g and all the rest fields, generically denoted as cf>. The probability then is given, up to nonexponential flux factors, by

P(few -¥ H) = J2 A*A ,

(18)

H

where the sum runs over the states of the black hole. For a large black hole a semiclassical calculation is justified. In such calculation a classical black hole exists starting from an instance of time to ("the moment of creation") to t = +00, so that the amplitude A contains the factor exp(il[g]\^) with the classical action of the black hole, described by the metric g, and calculated from the time to to t = +00. As usual, the integration in eq.(17) over an overall shift of to gives rise to the energy conservation 6 function in the transition amplitude, thus for the purpose of evaluating the magnitude of A the value of t 0 can be fixed arbitrarily, e.g. at to = 0. Furthermore, in order to dampen the oscillatory integrand in the path integral in eq.(17) at large t = T the integration over time in the action should be shifted to the lower complex half-plane of t: T = T\- iT2. Then in the product A^A the oscillatory part, corresponding to the integration over the real axis of

97

t, cancels, and the result for the product is determined only by integration along the imaginary axis: exp (2 IE[9]\oT2)

= exp ( - IE\g]\%)

,

(19)

where IE[{J] is the Euclidean space action for a black hole. Thus one arrives at the conclusion that the probability in eq.(18) contains an exponential factor, determined by IE[3]'P{few -> ff) ~ exp{-IE\g)).

(20)

It should be pointed out that the saddle-point expression (20) describes the entire sum over the states of the black hole. This follows from the fact that one classical configuration for the black hole with given 'global' parameters: mass MH, angular momentum J, and electric charge Q, corresponds to all quantum states with these values of the parameters. The classical action in eq.(20) is well known from the Gibbons-Hawking calculation 12 . For a non-charged black hole a the Gibbons-Hawking result is expressed in terms of MH and the angular momentum J as /#[