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

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Figure 8. Ratio of Au+Au and p + p (Equation 3) for small-angle (squares, \A 2.24 radians) azirauthal regions versus number of participating nucleons for trigger particle intervals 4 < p£%g < 6 GeV/c (solid) and 3 < Py* 9 < 4 GeV/c (hollow). The horizontal bars indicate the dominant systematic error (highly correlated among points) due to the uncertainty in V2-

5. Summary At RHIC, a system with low net-baryon density at mid-rapidity is produced. 2/3 of the baryons come from baryon-antibaryon pair production, while 1/3 of mid-rapidity net baryons come from the initial nuclei. The system undergoes chemical freeze-out at Tch ~ 175 MeV and /x& « 25 — 50 MeV. The single particle pr spectra suggest that the system undergoes further elastic re-scattering until final freeze-out at (fir) ~ 0.50 —0.6c and a kinetic freeze-out temperature T/ 0 w 100 MeV. The striking phenomena that have been observed at high pr in nuclear collisions at RHIC: strong suppression of the inclusive hadron yield in central collisions, large elliptic flow which saturates at pr > 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 <• k , (Xsufcufea) SPRINGS (Hrm*)ii ; tivsoR (rttwakr) ' WSXfrok(O)

Singapore HELIOS 2 (Eingapoyr) Taiwan SftfcC (Hsmehy)

Australia Australian Synchrotron (Melbourne) *

Figure 1. The map of location of functioning SR sources and ones under construction. Figure 2 demonstrates a well known now diagram devoted to the history, current status and prospects of increasing the brightness of X-ray sources. Since the time of invention of first X-ray tubes, i.e. for 60 years, their brightness has been improved evolutionary by a factor of about 100. The use of electron synchrotrons and then electron-positron storage rings as X-ray synchrotron radiation sources has enabled the world SR community to perform, since the 1970s, a purposeful work on revolutionary increase the brightness of SR sources. Changing over from synchrotrons to storage rings increased brightness by about 102 - 103 times. Further rise was due to the usage of multipole wigglers for generation of X-ray. These devices establish a sign-alternative magnetic field on a rather long section of the orbit, thereby permitting to concentrate the radiation to a single beam, with an n-fold increase of its intensity (n = 101 - 102 is the number of wiggler poles). Dedicated storage rings - second generation SR sources have offered the possibility to reduce the emittance of electron (positron) beams and, hence, to decrease the area of an emission source and to rise the brightness approximately by one more order of magnitude. Third generation of the SR sources having even still the lower emittance and higher energy, make it possible to employ undulators [5,6,7] as sources of X-ray radiation. Due to the interference of radiation from all undulator poles these devices generate quasimonochromatic radiation with monochromaticity A%/k « 1/n and spectral brightness n2 times higher as compared with the radiation from bending magnets. Taking into account the fact that undulators with n =103 are now in use and in the nearest future super long undulators with n =104 are expected to be employed, we may surely predict an

75 increase of the brightness of X-ray sources by a factor of 103 in the next decade, as is shown in Fig.2.

1»*»8 1«*17 1»»16 1»*1S

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coceeoo — I —

i»*e1950

1960

1970

~ r

LEADING EDGE OF COMPUTING SPEED {Millions or orwatlofw/sac) j —

1980 1990 Calendar Y*«r

1«+0 1»-1

—I

3*»

l»«1

»16

2020

Figure 2. This diagram illustrates the brightness of the SR sources that are being used, designed and constructed over the world in compare with the world mean of the maximal computer speed.

76 Electron Gun Linear Aecaterator

Synchrotron Radiation Enperifflsfttal Hwsftes*-/,

1 ton-long Beamllrte

Long

Beamline jSj Figure 3. The highest energy and SR brightness third generation SR source SPring (Japan).

£

HSYNCHMTRON RADIATION

t

UO-TECHNOLOC]

MATERIAL SCIENCES HTSC

ECOLOGY \

PRODUCTION CONTROL CATALYSIS

NAN07ECHNOLOGY

MICROELECTRONICS AND MICROMECHANICS

Figure 4. Current research and technological applications of synchrotron radiation.

77

3.

Synchrotron radiation in Russia

In the fifties and the sixties the experiments on study of the properties of synchrotron radiation at the first synchrotrons of the Physical Institute RAN (Moscow) attracted attention to the possible research and technological applications of SR beams. But interest in SR on the part of physicists with various specialities, biologisys and chemists was rose especially sharply after construction of electron storageringsof high energy (of the order of 1 GeV and higher). The first experiments with using of SR from the storage ring were started at Novosibirsk in 1973 at VEPP-3. The number of research teams at the Budker INP grew fast and in 1981 officially was organized the Siberian Synchrotron Radiation Center (SSRC) on the base of laboratories and electron-positron storage rings - SR sources VEPP-2M (used from 1984 to 2001), VEPP-3 (from 1973 up to now) and VEPP-4M (from 1983 up to now) of the Budker Institute of Nuclear Physics (Novosibirsk). It remains the main site of synchrotron radiation andfreeelectron laser (FEL) investigations in Russia. SSRC is (he research center, which is open and free of tax for the research teamsfromRussia andfromabroad.

VEFP-4M

* r ^ O 1 s* '• Ginxon (4J0 MHz) "*****" " # 2, LINAC (50 MeV) 3. Convenor S. Synchrotron f (350 MeV) "*

ROKK-1M ^J

DetertorKEBK.

Figure 5. The scheme of the VEPP-3/VEPP-4M accelerators complex. SR - the bunkers for synchrotron radiation experiments.

78

The research program of SSRC included the following directions: research studies and development of new technologies with the use of synchrotron radiationfromthe VEPP-3 and VEPP-4M storage rings; • construction of experimental equipment for works with SR as the beamlines, experimental stations, X-ray optics, monochromators, and detectors; • development and construction of the accelerators - dedicated sources of synchrotron radiation for another Russian and foreign centers; • development and construction of wigglers and undulators; • development of thefreeelectron laser and Siberian Center of Photochemistry. • education and professional training of students and post-graduates. The second Russian SR Center is Kurchatov SR Source (KSRS, Moscow) - a specialized accelerator facility intended for generation and application of SR beams. The KSRS facility includes two electron storage rings, Siberia-1 for 450 MeV (in operation from 1982) and Siberia-2 for 2.5 GeV (in operation from 2001). Both SR sources for KSRS were completely developed and manufactured at Budker INP. The third Russian SR Center is being constructed at JINR, Dubna. The "Dubna Electron Synchrotron" (DELSY-Project) is aimed at the construction of the SR source of "a nearly third generation". DELSY source is based on the equipment supplied by the High Energy Physics (NIKHEP), Amsterdam, the Netherlands. Parameters of the DELSY SR beams will be substantially improved with the installation of one or two high field wigglers, produced in the Budker INP. The Russian SR Technological Center will be organized at Zelenograd (Moscow Region). A dedicated storage ring - SR Source "TSC" (Technological Storage Complex) for Zelenograd was designed at BINP by the beginning of 1990s. All the systems have been designed and manufactured at Budker INP in the period 1992-1996. However, by 1996 financial support of the project ceased completely. The financing of the project is recently renewed and in 2002 the "TSC" foreinjector was assembled and commissioned. •

4.

Insertion devices- advanced instruments for generation of the intense photon beams

The history of application of the periodic magnetic systems for generation of synchrotron radiation had been started in 1947 from the work by V.Ginzburg [5]. The interest to this systems considerably increased after publication in 1972 the articles by V.Bayer, et al., [6] and D.Alferov, et al., [7]. This articles were a basis for the posterior theoretical and practical development of the wigglers and

79 undulators - modern devices for the generation of the intense white or monochromatic polarized photon beams. Table 1. Synchrotron radiation sources in Russia.

SR source

Location

VEPP-2M

Novosibirsk

VEPP-3

Novosibirsk

Energy, GeV

0.7 2.0

Max. current, mA 600 e50x50 e+e-

Status

150

used up to in since

from 1974 2000 operation 1973

VEPP-4M

Novosibirsk

6.0

40

in operation since 1983

Siberia-1

Moscow

0.45

360

in operation since 1982

Siberia-2

Moscow

2.5

100

TSC

Zelenograd

1.6

(300)*

DELSY

Dubna

1.2

(200)*

4.1. Superconductive

wigglers -powerful

in operation since 2000 Assembling construction and assembling * preliminary value

incoherent SR generators.

Since 1979, Budker INP has developed and manufactured the superconductive wigglers with high induction of magnetic field. Table 2 presents the main parameters of all superconductive wigglers developed at Budker INP. The unique combination of the photon beam properties from the modem SR sources (high brightness and total power, tunable monochromatization, tunable polarization, short-pulse structure of the SR beam, spatial and temporal coherence) determined a wide spectrum of their research and technological applications. The use of synchrotron radiation opened up now the new possibilities of research both for various scientific fields and applications. The main of this fields itemized in the Fig.4.

80 Table 2.

Superconductive wigglers designed and manufactured at the Budker INP.

Purpose VEPP-3 (BINP) Spiral undulator (BINP) VEPP-2 (BINP) Siberia-1 (Moscow) PLS (Korea) LSUCAMD (USA) Spring-8 (Japan) WLS (BESSYII, Germany) PSF-WLS (BESSYII, Germany) HMI(BESSYII, Germany) ELETTR A (Italy)

Year of start

Wiggler Storage ring e" Numb, of Total Vert, Pole poles Energy, beam magn. length aper., curr., field, (main anc GeV length, mm mm corr.) T A mm Max

SR jower ,kW

197 9

3.5

20

45

900

8

2

198 4

0.47

16

12

250

13

0.65

8

5

120

600

15

0.65

5.8

1+2

350

22

0.45

0.1

7.68

1+2

170

800

26

2

0.1

3.6

199 8

7.55

1+2

172

972

32

1.5

0.3

5.3

200 0

10.3

1+2

200

104 2

20

8

0.1

100

200 0

7.5

1+2

172

972

32

1.9

0.5

13

200 1

7.5

1+2

172

972

32

1.9

0.5

13

200 2

7.0

13+4

72

136 0

14

1.9

0.5

56

200 2

3.5

45+4

30

168 0

11

2

0.1

6

198 4 198 5 199 5

0.1

2.8

Application of high-field wigglers gives the possibility of shifting the radiation spectrum of the already-constructed sources to the region of shorter waves and of significant increase of the radiation power. That expands greatly the possibilities of the existing SR sources and makes it possible to carry out new experiments. Each of the wigglers has its own features and was designed specially for the specific storage ring. For instance, a unique feature of the superconductive wiggler for BESSY-II is the high level of stability and uniformity of magnetic field.

81 This wiggler will be applied as a standard for calibration experiments with the use of the PTB laboratory (Germany). Value of magnetic field in the wiggler is measured with an NMR detector. Basing on these measurements, the feedback system corrects current in the wiggler windings and restores the required value of magnetic field.

Figure 6. 10 T superconducting wiggler for "SPring-8" (2000).

4.2. Elliptical wigglers and undulators for generation of radiation with tunable polarization Budker INP has developed and manufactured a different kinds of wigglers and undulators for Russian and foreign synchrotrons (see a sample in Fig.7), including unique devices with fast switching of polarization sign and universal ones (planar, elliptic or spiral radiation for your choice). The elliptical wiggler and undulator designed for obtaining of ellipticaly polarized synchrotron radiation with rapid change of polarization sign was manufactured and tested at Budker INP in 1996 and 1998. In 1997 elliptical wiggler was installed at NSLS-II (BNL, USA) and undulator was installed in 1999 at SR source APS (Argonne National Laboratory, USA). The time required for switching of polarization sign is 5 ms. The four universal electromagnetic undulators ordered by Duke University (Northern Carolina, USA) have the separate power supply of the windings allows application of these undulators as planar, elliptic or spiral ones.

82

Figure 7. General view of one of two halfs of elliptical electromagnetic undulator for advanced SR source SLS (Switzerland). Period - 21.2 cm, maximal field - 0.6 T vertical and 0.12 T horizontal. Started up in 2001. 5.

A source of bright beams of slow positrons

Production of intense low-energy positrons with synchrotron radiation was proposed by authors of this paper in 1988 (G.Kulipanov, A.Skrinsky, [8]) In 1996 Budker INP and the Japan SR center SPring-8 signed the agreement on scientific cooperation, to realize the project for generation of bright beams of slow positrons with SR application (A.Ando, HLKamitsubo, G.Kulipanov, et al., [9]). A record field 10.3 T wiggler (presented above, Fig. 6) developed at Budker INP for the storage ring SPring-8 (Japan) with energy 8 GeV is a source of the very hard synchrotron radiation (a photon energy exceeds 1 MeV) and the key component for the Russian Japanese source of slow positrons (Fig. 8). In 1999, the wiggler magnetic system in a cryostat was manufactured and after successful tests delivered to SPring-8. In February 2000, it was successfully put into operation separatelyfromthe storage ring. In 2002, the wiggler was installed at SPring-8 and experiments with a beam at low currents were started. Another key component of the Project is a moderator (a decelerator of positrons) is in the process of its development. Proposed intensity of the slow positron source on the 100 kW SRbeam from the wiggler at SPring-8 can reach a level of- 1-51010 e7fi.

83

Figure 8. Scheme of generation of slow positrons using SC wiggler.

Free electron lasers Free electron lasers (FEL) are based on the phenomenon of amplification of the photon flux passing through undulator placed at the relativistic electron beam. First FEL was developed under the direction of J.Madey (Duke University, USA) [10]. It was successfully started up in 1976. In the next, 1977 year at the Budker INP had been developed a conception of the new type of free electron laser - Optical Clystron (A.Skrinsky, N.Vinokurov, [11]). It made possible useful increase of amplification with the use of a special buncher magnet placed in the center of undulator. Using this scheme, the ultra-violet generation with record wavelength of 0.24 microns had been successfully obtained in the Budker INP at the storage ring VEPP-3 in 1989. The maximum power of optical clystron installed at the storage ring is fundamentally limited and progress of the powerful FEL is possible only with using of linac or similar type of single-pass accelerators (A.Skrinsky, N.Vinokurov, 1978, [12]). At present, scientists work intensely on construction of high-power (of an average power > 1 kW) infrared FELs. Creation of industrial photochemical technologies requires an average power -10 kW and higher at rnonochromaticity not worse than several hundredths of percent. The Budker INP full-scale FEL under construction bases on the microtronrecuperator and will have a re-tuning range from 2 to 50 microns, generating 30 picoseconds pulses with energy of 5 mJ and average power up to 100 kW (Fig.9).

84

Figure 9. Scheme of the free electron laser on the base of microtron-recuperator: 1 - the electron gun, 2 - the bending magnets, 3 - the RF cavities, 4 - the injection magnets, 5 - the extraction magnets, 6 and 7 - the quadrupole magnets in the common and separate tracks, 8 - the magnetic system of the FEL, 9 - the worked-out electron beam dump. The first stage of the facility includes the full-scale RF system of the microtronrecuperator and only one track. Maximal energy of microtron-recuperator in this scheme is 14 MeV. Some important parameters of the first stage FEL are as follows: a re-tuning range from 100 to 300 microns, generating 20 - 100 picoseconds pulses, pulse power 1-7 MW, and average power up to 7 kW (Fig. 10).

I

Cavities

jj

Bending magnets

l l l l l l l l l l l l l Undulators | £ j | |

Buncher

|

Quadrupoles Q] Mirrors

Q Solenoids ft

Outcoupler

Figure 10. Scheme of the one-track accelerator-recuperator and submillimeter FEL On April 4, 2003 at the first stage accelerator-recuperator at 12 MeV energy obtained first FEL lasing at 100-micron wavelength.

7.

The future synchrotron radiation sources

7.1. Conceptual design oftheSR

source MARS

In 1997, a new concept of the SR-source of the fourth generation MARS (Multi-pass Accelerator Recuperator Source) was formulated in the Budker INP by N.Vinokurov, G.Kulipanov and A.Skrinsky [13] and presented in their report at the Conference "SRI97" in Japan [14]. The MARS SR Source was assumed to be realized not on the base of the storage ring but on the base of the microtron-recuperator. In this case, the one flight regime of radiation in the magnetic field is realized for each electron bunch and various

85 diffusion phenomena have no time to increase beam emittance and an energy spread. In such an accelerator, the brightness of undulator radiation can surpass by two-three orders the brightness of the SR sources of the third generation in the range of wavelengths ~ 0.14nm. After five years of work on the project and discussions at a number of international conferences the present-day version of the project was finally formed (Fig.8) with the following main parameters: maximum energy of electrons ~ 8 GeV, mean current up to 510 mA, horizontal emittance is less than 0.01 nm*rad, relative spread of energy in a beam ~ 0.001 %. 4 long undulalors N ~ \0* +A-IA

undulalors magnets

40 short undulalors JV~10!-10

accelerating sections A£ = 1.8GeK

E = 5 MeV | J T = 5-l5kW

[_\ E = 5 MeV 7 = l-3m/( sn-10' 'rn-rad

Figure 11. Conceptual design of the SR source MARS (2002). The parameters of SR beams from MARS are near the answers the basic requirements for the fourth generation SR source: • full spatial coherence; • as high as possible temporal coherence (AE / E < 10-4 without additional monochromatization); • the averaged brightness of the sources has to exceed 1023 -1024 photons s-1 mrrr^ •

mrad-2 (0.1 % bandwidth); the increase of brightness must not be accompanied by an increase of the full photon flux for minimization of the problem with optics and sample degradation;

86

•

high peak brightness of more than 1026 photons s-1 mm-2 mrad-2 (0.1 % bandwidh)-l is important for some experiments.

Table 3. Comparison of MARS with various types of the coherent X-ray sources.

MARS

ESRF

LCLS FEL

Storage ring

at linac

Wavelength, ran

0.1

0.15

0.1

Electron energy, GeV

6

14

8

Average current, A

.2

3x10^

5xlO' 3

Peak current, A

3.4 x 103

1

Relative energy spread

2xl04

1 x 10"5

Parameter

Emittance, nm E.

5 x 10'2

4 2.5 x 10"2

Acceleratorrecuperator

3 x 10"3

Undulator period, cm

4.2

3

1.5

Undulator length, m

5

120

150

Coherent flux,

6xl0'2

2xl014

7xl013

io- 2

IO 3

\04

1020

4 x 1022

3 x 1023

lxlO 3 3

3xl0 26

15

10

1

1

1

1

ntintnn/«

Bandwidth Average brightness, ph/s/mm2/mrad2/0.1 %BW Peak brightness, —//— Transverse size of

0x 350

source (standard

Oy

8

Radiation transverse divergence (standard deviation), urad

°V

13

°,

3

87

Further work on the Project MARS requires experimental studies and tests of the basic operation principles of the accelerator-recuperator. This work started in 2002 at the first stage of the accelerator-recuperator for the free electron laser of the Siberian Center of Photochemistry, which will be the prototype of the first stage of the X-ray radiation source MARS. References 1. LYaPomeranchuk., JETP9,915 (1939); Journal ofPhysics of USSR 2,65 (1940). 2. D.D. Ivanenko, I.Ya.Pomeranchuk., DAN USSR 44,343. (1944); Phys. Rev., 65, 343 (1944). 3. L.A.Artsimovich, I.Ya.Pomeranchuk., Journal of Physics of USSR IX 267 (1945); JETP 16,379(1946). 4. G.N.Kulipanov, A.N.Skrinsky., Soviet Physics - Uspekhi 20,559 (1977). 5. V.Ginzburg., Izv. AN USSR (Phys.ser.) 11,163 (1947). 6. V.N.Baier, V.M.Katkov, V.M.Strakhovenko., JETP 63,2121 (1972). 7. D.F.Alferov, Yu.A.Bashmakov, E.G.Bessonov., JETP42,1921 (1972). 8. G.N.Kulipanov, A.N.Skrinsky. Synchrotron radiation and its application. Memoirs on I.Ya.Pomeranchuk., "Nauka", Moscow, p. 246,1988. 9. A.Ando, H.Kamitsubo, G.Kulipanov, et al., Jof Synchrotron Radiation 3, 201 (1996). 10. L.Ellias, D.Madey., Phys. Rev. Lett. 36,717 (1976). 11. A.N.Skrinsky, N.A. Vinokurov., Budker INP Preprint No 77-59, Novosibirsk, 1977. 12. A.N.Skrinsky, N.A.Vinokurov., Budker INP Preprint No 78-88, Novosibirsk, 1978. 13. G.N.Kulipanov, A.N.Skrinsky, N.A.Vinokurov., Budker INP Preprint No 97-103, Novosibirsk, 1997. 14. G.N.Kulipanov, A.N.Skrinsky, N.A.Vinokurov., J. ofSynchrotron Radiation 5, 176 (1998).

BLACK HOLES FROM PARTICLE COLLISIONS AT T R A N S - P L A N C K I A N ENERGIES?*

M. B . V O L O S H I N William Institute

I. Fine Theoretical of Theoretical

Physics Institute, Minneapolis, 55426, and and Experimental Physics, Moscow, 117259, E-mail: [email protected]

USA Russia

The recently popular topic of creation of black holes of a multidimensional gravity at future accelerators is discussed. I present arguments, based on general scattering theory, on thermodynamics, and on the path integral method, that suggest that the production of a large black hole in a high energy particle collision is quite unlikely. This pessimistic conclusion however does not imply a suppression of processes with production of multiple small black holes, each with a mass of the order of the appropriate Planck scale, provided that such black holes make sense.

1. Introduction The question "If two particles collide at an energy far above the Planck mass (e.g. of order of the solar mass!), would they produce a macroscopic black hole?" has been around for at least as long as I can remember. However there was not much of widespread theoretical enthusiasm in answering this question, owing to its obvious 'eternally academic' nature, since there exists no way of arranging a single collision of particles at energies even close to the Planck scale of the standard gravity. The situation has drastically changed recently with the advent of models with multidimensional gravity 1 ' 2 , in which our 4-dimensional space time is embedded in a D = 4 + Ndimensional space-time, where the extra N dimensions are compactified in a manifold with volume VN = LN (so that L is the typical linear size of the compact manifold). In such models the standard Einstein-Hilbert action for the gravity in D dimensions reads as I{D) = (M*)2+N

j R(D) ^^)dixdNy

•This work is supported in part by the DOE grant DE-FG02-94ER40823.

88

(1)

89 where the coordinates in our 4-dim space are denoted as x, and in the extra dimensions as y, and M* is the D dimensional analog of the Planck scale, i.e. the mass scale in the gravity action. The standard 4-dim gravity then is an effective theory of the modes that are constant in the extra coordinates y, and the 4-dim action is found by a straightforward integration:

7(4) = LN {M*f+N J % ) 7 = ^ d 4 z ,

(2)

which relates the standard Planck scale Mpi to M* and L: MPI = M* (LM*)N/2

.

(3)

The latter relation implies that the mass scale M* for the D dimensional gravity can be much lower than the standard Mpi, provided that the size L is sufficiently large. The values of L that one is allowed to assume are constrained from above by the known limits on behavior of gravity at short distances, since at distances, shorter than L, the observed 4-dim gravity would be modified, and would display a true D dimensional behavior at distances -C L. The current measurements of the Newton's gravity law are taken down to distances just below one millimeter. Thus assuming for the sake of argument a value of L near the current bound, L ~ 1 mm, and also assuming that the number of extra dimensions N equals two, one reproduces the standard Planck mass with a reasonably small M*: M* ~ 1 TeV. The possible low value of the D dimensional equivalent of the Planck scale puts the trans-Planckian energies within the range in principle attainable at future colliders, and thus makes the question about black hole production in particle collisions well relevant to a possible phenomenology of high energy interactions. It was in fact claimed in the literature 3,4 that the production of multidimensional black holes, growing with energy, should be the dominant process at high energies. These claims are based on the idea, that goes back to an (unpublished) paper by Banks and Fischler5: "General Relativity predicts that when the impact parameter . . . is smaller than a critical value Rs, a black hole is formed. . . . Rs is of order the Schwarzchild radius of the corresponding black hole . . . ". The word "predicts" in this quote is put in italics by me in order to point out that neither a reference to this 'prediction' is given in [5] nor was I able to trace such reference. Clearly, if this idea is correct, the cross section for production of black holes should grow with the black hole mass and with energy in the collision. Indeed, the Schwarzchild radius rjj of a black hole in D = 4 + N dimensions is related 6

90 to the black hole mass MH as

(M„\^

1 rH(X

W {JFJ

(4)

'

and the cross section is expressed by the geometric formula a ~ irrH . Since the maximal mass of the black hole that can be created in a collision with the total c m . energy E = y/s scales as E, the most probable process would be when a large fraction of the total energy goes into a single black hole, and the cross section for such process would grow as a power of the energy: 1

/ E \

°BH ~ M^ [W)

^

•

(5)

The purpose of this talk, based on the papers 7,8 , is to present arguments that the claimed behavior of black hole production in particle collisions at trans-Planckian energies is quite unlikely, and that production of large black holes with mass MH 3> M* should in fact be exponentially suppressed in a power of the ratio (Mg/M*). On the other hand it may well be that the total cross section of inelastic gravitational scattering at such energies indeed has to grow as a power of energy (unless the gravity is modified) and be dominated by creation of multiple objects with masses of the order of the Planck scale. One line of reasoning to be presented is based on the general scattering theory applied to the gravity theory, and the other one is based on considering a semiclassical black hole as a thermodynamic object. The latter line of reasoning will be also supplemented by (an equivalent) path integral consideration. Both these types of argument point to a suppression of production of large black holes. In what follows, for definiteness, the standard 4-dimensional theory will be assumed, so that N = 0 and Mpi = M*, with only few remarks being made further on multidimensional and also lower-dimensional cases. 2. Energy fragmentation and multi-black-hole production The geometric formula a ~ 7r rH implies that the cross section for the black hole production grows quadratically in the mass MH, since TH = 2G MH with G being the Newton's constant. Clearly, at a far trans-Planckian energy E — y/s, such that G E2 3> 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 /#[
= 2irGMH

h + -=L=j

,

(21)

where G is the Newton's constant, and j = J/(GMH) is the angular momentum in units of its maximal possible value. From this expression and eq.(20) one concludes that the probability of production of a large semiclassical black hole is necessarily exponentially suppressed. Moreover, the total probability of production of black holes with mass MH and with different angular momenta is dominated by the contribution of slowly rotating black holes. Indeed, the summation over the partial waves with different J is Gaussian and the exponential factor is determined by small J: P (few -> H{MH))

=

^(2J+l)P(/ew->i7(MH,J))~exp(-47rGM|r)

.

(22)

j

In other words, the production of rapidly rotating black holes with j ~ 0(1), which is argued in Ref.[3] to be a typical process, is in fact even more "Clearly, in a few particle process it is impossible to produce a black hole with a macroscopic charge.

98 heavily suppressed by the semiclassical exponent, while the typical angular momenta contributing to the sum (22) are (J) ~y/GMH. There are at least few points, which merit discussion in connection with derivation of the formula in eq.(20). The first one relates to specifying the imaginary time Ti for the limit of integration in eq.(19). According to Ref.[ 12] the black hole metric is periodic on the Euclidean section of space-time with the period determined by (the inverse of) the Hawking temperature TR- The full path integral should contain summation over the number of wrappings of the classical 'trajectory' over the period, with the action in eq.(21) being that for one period. Although it is trivial to perform the summation of the geometric series for the sum over the number of periods, within the saddle point method it would be inconsistent to keep the terms with higher exponential suppression. In any event, the corrections from higher exponents are small inasmuch as the black hole is semiclassical, i.e. GM£»1. The second point is that the formula in eq.(20) could be derived by applying the optical theorem to the expression in eq.(18). Indeed, in this approach one could write the probability as the imaginary part of the forward scattering amplitude, with a black hole in the intermediate state: P{few -> H) ~ Im A(few -> H -> few) .

(23)

The latter amplitude can be evaluated from the path integral, with the semiclassical trajectory containing a black hole in the intermediate state. The imaginary part is contributed by the configuration where the black hole is "on shell", i.e. it exists over an infinite (Minkowski) time. Correspondingly, the action for such black hole (with appropriate complex shifts of the integration contour in t at the infinities) is exactly the one calculated by Gibbons and Hawking 12 . However a possible problem with application of the optical theorem, and generally of unitarity, to processes involving black holes is that the unitarity itself is put into question 13 in such setting. On the other hand, even if the unitarity condition requires a modification in the presence of black holes, it would be quite unlikely that the optical theorem is broken by large exponential factors. Finally, the third point that merits discussion is the relation between the production of a black hole by 'few' initial particles (small n), and 'many' initial particles (large n). The first case involves an essentially quantum initial state, while in the latter the initial state can also be considered classically, so that the whole process of formation of a black hole becomes classical. I believe that it is the simplistic extrapolation of a classical

99 collapse down to the case of few initial particles, which is the source of the existing in the literature claims 3 ' 4 that the cross section for the process few —» H has essentially no suppression. Within the present state of the art the interpolation between these two extremes can be viewed (although not fully described) from two starting points: starting from a quantum description at small n and increasing n, and starting from classically large n and decreasing the number of initial particles. The consideration presented in this section clearly implies the former approach, i.e. starting from small n. The number of initial particles enters the calculation of the amplitude as a pre-exponential factor n! determined by permutations of the particles in the initial state. Clearly, at sufficiently large n the factorial overcomes the exponential suppression, and the process becomes unsuppressed, which is in qualitative agreement with the classical collapse. However a quantitative description of such interpolation runs into the problem that at large n there are generally n2 pair wise interactions between the particles, and the initial state cannot be treated perturbatively. The situation almost literally repeats that encountered in the studies of the B + L violation in the standard electroweak theory and in related model processes14: the B + L violating processes are not suppressed at high temperature, or high fermion density, exceeding the sphaleron barrier, and this behavior can be understood at the microscopic level as dominated by (semi)classical multiparticle processes, while in few particle collisions the exponential suppression persists, even if the available energy is well above the barrier and no tunneling through the sphaleron barrier is required (the most recent discussion of this subject can be found in Ref. [15]).

4. Classical limit and discussion It is instructive to rewrite the exponential suppression factor from eq.(21) in the cross section with restored explicit dependence on the Planck's constant h 2-KGM% CJ{MH, J) ~ exp

h

1

{

+

v/l-J2/(G2M£)

(24)

This dependence implies that in the classical limit, h—> 0, this cross section should vanish. Also the characteristic range of the impact parameter b contributing to the cross section can be found from the previously mentioned estimate of the dominating values of the angular momentum,

100

J ~ VGhMH, as b=-^-~VGh. (25) v ; Mh In other words, the characteristic impact parameter is of order the Planck length and does not depend on the energy of colliding particles, and it also vanishes in the limit 7i-» 0. The conclusion about vanishing in the classical limit cross section and the effective impact parameter for production of a large black hole can be confronted with results of purely classical analyses. Such comparison however turns out to be quite paradoxical. Namely, Eardley and Giddings 16 , using an approach going back to unpublished remarks by Penrose, considered formation of the so-called marginally trapped surface in a classical collision of two Aicheburg-Sexl17 shock waves, which model the metrics of ultrarelativistic particles. A marginally trapped surface in D dimensions, by definition, is a closed D — 2 dimensional surface the outer null normals of which have zero convergence. It is believed that existence of such surface S in a metric implies also existence of a horizon, which either coincides with S or encompasses <S. Eardley and Giddings argued that a finite marginally trapped surface is formed in collisions within a finite range of the impact parameter in any dimensions and have also explicitly found S for D = 4. According to their results the marginally trapped surface is formed up to the maximal value of the impact parameter in the collision bmax = 1.61 Gy/s, so that the classical cross section gets the lower bound of 8.1 G s2. If the surface area of S is interpreted as the minimal value of the area of the horizon of a single formed black hole, their result corresponds to that black holes with masses up to at least 0.45 y/s are produced in the collisions. A similar apparent contradiction (in my interpretation) also arises in a somewhat simpler setting of a (2+1) dimensional gravity with a negative cosmological term. This model of an anti de Sitter (AdS) gravity is known to possess black hole solutions (so-called BTZ black holes). It has been shown some time ago that in this model there are classical solutions leading from two colliding ultrarelativistic (classical) particles to a BTZ black hole both at zero impact parameter 18 as well as at a nonzero one 19 . On the other hand the (2+1) dimensional AdS gravity is holographically equivalent20 to the Liouville theory on closed 2-dimensional surface. The probability of a black hole creation in collision of two particles is then related to a four-point Green function in the Liouville theory. For creation of a large

101

black hole this Green function can be evaluated semiclassically, and the result 21 also contains an exponential suppression, where the negative power of the exponent becomes infinite in the limit h -> 0. Although in the final version of the paper the author 21 argues that the exponential suppression is compensated by the large number of the black hole states, I believe that such conclusion is invalid, and the found exponential suppression refers to the sum over all the states of a large black hole. Indeed, as previously mentioned, a single WKB trajectory in the path integral describes the sum over all the degenerate quantum states. The resolution of the described contradiction between the classical analyses of appearance of a horizon, corresponding to a large black hole at the instance of the two particle collision, and an infinite suppression of large black hole production in the h —> 0 limit of the semiclassical expressions, might be in that the inevitable strong quantum fluctuations of the metric at the instance of the collision give rise to fragmentation of the horizon into a configuration, corresponding to many small black holes, in defiance of generally adopted classical properties of a horizon evolution. Then the classical analyses of the cross section for producing some horizon can be reconciled with the statistical/semiclassical analyses, which do not require a suppression of production of multiple small black holes. The conclusion that the total cross section of gravitational inelastic scattering should grow (up to possible extra logarithmic factors) as a power of energy, specifically as E2 in the standard 4-dim theory, looks to be inevitable, much in the same way as a similar growth of the cross section was inevitable in the old four-fermion theory of weak interactions 22 . The discussed issue relates to the structure of the dominant final states, i.e. one large black hole versus multiple objects, each with a mass of the order of Planck scale. Clearly, if multidimensional models are of any relevance to the reality, the signature of these two types of final states should be quite different. Indeed, a black hole decays through evaporation with temperature TH ~ rjj1. The heavier is the black hole the lower is its temperature and thus the softer are the individual decay products with larger multiplicity. If a large fraction of the energy in the collision goes into a single black hole, the typical energy of each of the final fragments should be (cf. eq.(4)) e ~ M* (M*/E)1/(-N+1\ and the multiplicity of the fragments should scale as (E/M*)(-N+2^^N+lS>. On the contrary, in the argued here picture, where individual objects in the final state have mass of order M*, one obviously should expect the typical energy of the final fragments to be e ~ M* and the multiplicity scaling as E/M*.

102 In conclusion, I believe t h a t , irrespectively of the relevance of multidimensional models of gravity, the 'eternally academic' question of particle collisions at trans-Planckian energies is interesting on its own, and t h a t with sufficient effort, it can be answered within our present theoretical means.

References 1. N. Arkani-Hamed, S. Dimopoulos and G. Dvali, Phys.Lett. B429, 263 (1998). 2. I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos and G. Dvali, Phys.Lett. B436, 257 (1998). 3. S.B. Giddings and S. Thomas, Phys.Rev. D65, 056010 (2002). 4. S. Dimopoulos and G. Landsberg, Phys.Rev.Lett 87, 161602 (2001). 5. T. Banks and W. Fischler, Report RU-99-23, UTTG-03-99, Jun 1999. [hepth/9906038] unpublished. 6. R.C. Myers and M.J. Perry, Ann. Phys. (N.Y.) 172, 304 (1986). 7. M.B. Voloshin, Phys.Lett. B518, 137 (2001). 8. M.B. Voloshin, Phys.Lett. B524, 376 (2002). 9. G. 't Hooft, Nucl.Phys. B (Proc. Suppl.) 43, 1 (1995). 10. S.W. Hawking, Commun.Math.Phys. 43, 199 (1975). 11. S.N. Solodukhin, Phys.Lett. B533, 153 (2002). 12. G.W. Gibbons and S.W. Hawking, Phys.Rev. D15, 2752 (1977). 13. S.W. Hawking, Phys.Rev. D14, 2460 (1976); Commun.Math.Phys., 87 395 (1982); Phys.Lett, B209, 39 (1988). 14. An account of this development with a list of referencies can be found in: M.B. Voloshin, Non-Perturbative Methods, Proc. XXVII Int. Conf. on High Energy Phys., Glasgow, 1994, Ed. P.J. Bussey and I.G. Knowles, IOP Pub., Bristol, 1995; p.121. 15. F. Bezrukov et.al., Suppression of Baryon Number Violation in Electroweak Collisions: Numerical Results, Moscow INR report INR-TH-2003-6, May 2003; [hep-ph/0305300]. 16. D.M. Eardley and S.B. Giddings, Phys.Rev. D66, 044011 (2002). 17. P.C. Aichelburg and R.U. Sexl, Gen.Rel.Grav. 2, 503 (1971). 18. H-J. Matschull, Class.Quant.Grav. 16, 1069 (1999). 19. S. Hoist and H-J. Matschull, Class.Quant.Grav. 16, 3095 (1999). 20. K. Krasnov, Class.Quant.Grav. 19, 3977 (2002). 21. K. Krasnov, Class.Quant.Grav. 19, 3999 (2002). 22. A.D. Dolgov, V.I. Zakharov and L.B. Okun, Sov. J.Nucl.Phys. 15, 451 (1972).

COSMOLOGY AT T H E T U R N OF CENTURIES*

A. D. D O L G O V INFN,

sezione ITEP,

di Ferrara, Bol.

Via Paradiso, 12 - 44100 Ferrara, and Cheremushkinskaya, 25, Moscow, Russia E-mail: [email protected]

Italy

A brief review of the present-day development of cosmology is presented for mixed physical audience. The universe history is briefly described. Unsolved problems are discussed, in particular, the mystery of the cosmological constant and dark energy, the problems of dark matter, and baryogenesis. A brief discussion of the cosmological role of neutrinos is also presented.

1. Introduction Probably the most important scientific breakthrough of the last quarter of the previous century was related to tremendous progress in cosmology. From a poor relative of distinguished family of fundamental science characterized by the words prescribed to L. Landau: "always in error but never in doubt", cosmology turned, or is turning, into an exact science and possibly the most interesting one. It is full of mysteries, unsolved problems, puzzling phenomena and strongly indicates that there exists new physics beyond the standard model. The stunning development of cosmology is due to a combination of two factors: 1) development of new theoretical methods, application of particle physics and quantum field theory to the early and contemporary universe; 2) new observational technique and devices, much more precise then even recently used ones and exploration of new windows to the universe (electromagnetic radiation in all wave length ranges, gravitational waves, neutrinos, high energy cosmic rays). Taken together they have led to Great Cosmological Revolution which keeps on going, turning into "permanent * The hospitality of the Research Center for the Early Universe of the University of Tokyo, where the preparation of this contribution for publication was completed, is gratefully acknowledged.

103

104

revolution" almost on Leo Trotsky own terms but, fortunately, not so bloody. Cosmological parameters are now quite accurately known with even brighter perspectives for the nearest future. One impressive example is the baryon asymmetry of the universe or, better to say, the ratio of the cosmological number density of baryons to the number density of photons in cosmic microwave background radiation (CMBR). When the first works on the baryon asymmetry were written this ratio was known as P = ns/n-y = 1 0 _ 9 ± 1 . Now it is 0 = (6 ± 1) • l f r 1 0 . The progress is striking. 2. Cosmological Parameters An important indication that cosmology is entering the club of exact sciences is the precision in determination of the values of basic cosmological parameters. The progress of the recent years is achieved, to a large extent, thanks to the measurements of the angular fluctuations of the CMBR 1 and detailed study of large scale structure (LSS) of the universe2. The universe expansion law, V = Hr, is usually expressed in terms of the dimensionless Hubble parameter h = H/100 km/sec/Mpc. According to the present day measurements, it is h = (0.72 — 0.73) ± 0.05. Previously for a long time it's value was between 1 and 0.5 with unclear systematic arrows. With this accurate determination of h the critical value of the cosmological energy density becomes pc = °

p o

' = 10" 2 9 g/cm 3 (Zi/0.73)2 = 5.62 kev/cm 3 (/i/0.73) 2

(1)

07T

The contribution of any form of matter into total cosmological energy density is expressed in terms of the parameter ftj = Pj/pc', sometimes a more accurately determined quantity is fljh2. The total energy density in the universe is quite close to the critical one, fltot = 1.02 ± 0.02 in agreement with inflationary theory 3 . Visible matter (either shining or absorbing light) gives a minor contribution into total mass, flvis « 0.005 (see e.g. ref. 4 ). The total fraction of baryonic matter is about an order of magnitude larger, £}& = 0.044 ± 0.004. The answer to the question where are the unseen 90% of baryons is not completely clear by now. A much larger contribution to O comes from some unknown form of matter, the so called dark matter, while a better word could be "invisible", £lDM = 0.27 ± 0.04. Though unknown what, it is still normal matter, presumably, though not surely, with a normal non-relativistic equation of

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state with zero pressure p = 0 and positive energy (mass) density, p > 0. Dark matter is believed to posses the usual gravitational interactions. Much more puzzling, even mysterious, is the dominant contribution to the cosmological energy density, the so called dark energy, with QDE = 0.73 ± 0.04. It has negative pressure density, p = wp

(2)

with w < —0.8, so it could be vacuum energy (or, in other words, cosmological constant) for which w = — 1. Its impact on the cosmological expansion is anti-gravitating, i.e. it leads to accelerated expansion 5 . This statement follows from the Friedman/Einstein equation for the second time derivative of the cosmological scale factor a(t): a 4nG „ , - = z-(p + 3p) 3 a 6 and acceleration of the expansion would be positive if p < — p/3, of course under the standard assumption that p is positive. For negative p anti-gravity could be easily created but such theories would possess quite unpleasant pathological features if special care is not taken. In addition to the direct observation of the accelerated expansion by high red-shift supernovae, there is an indirect argument in favor of nonzero vacuum-like energy. Namely with fi = 1 and h = 0.73 the age of the universe would be about 9 Gyr, while nuclear-chronology and stellar evolution theory give the number 12-14 Gyr. With non-zero vacuum energy, Qvac = 0.7 the universe needs considerably more time to reach the presentday state and its age would well agree with the data. Theory of large scale structure formation also supports a non-zero value of the vacuum energy. One more comment is worth making. Different forms of matter/energy density have similar values at the present time: Oj = 0.044, fim = 0.27, QDE = 0.73, though their physical origin could be quite different and unrelated at least at the level of our present understanding. They may naturally differ by many orders of magnitude. Moreover, densities of nonrelativistic matter and dark energy evolve in different ways in the course of the cosmological expansion (see sec. 3). This makes the problem even more profound. At the moment this cosmic coincidence (or cosmic conspiracy) problem is not understood at all. 3. Cosmological Constant/Dark Energy Cosmological constant (or lambda-term) was introduced to equations of general relativity by Einstein in 1918 when he found out that the

106

equations did not admit stationary solution in cosmological situation with homogeneous and isotropic matter source T^: -^W

_

o 9»"R

~ ^-9nv = SKGNT^

(4)

Later, after the Friedman's cosmological solution and the Hubble's observation of the universe expansion, Einstein strongly rejected the idea of cosmological constant and considered it as the "biggest blunder" of his life. For a long time after that the majority of the astronomical/cosmological establishment even refused to hear about it, though were were notable exceptions like e.g. Lemaitre, De Sitter, Eddington. The attitude of majority could be characterized by the words written by G. Gamow in his autobiography book "My World Line": "Lambda rises its nasty head again", after astronomical data indicated an accumulations of quasars near red-shift z = 2. In a sense Gamow was right because these data were explained without cosmological constant but it seems impossible to avoid non-zero A today. According to the contemporary point of view, A-term can be interpreted as the energy-momentum tensor of vacuum and should be positioned in the r.h.s. of the Einstein equation: Rpu ~ \ 9»vR

= &7TGN

(T$>

+ Pvacg^)

(5)

Quantum field theory immediately leads to a very serious trouble 6 . The energy of the ground state is generally non-vanishing and, moreover, for a single species it is infinitely divergent: Pvac =9s

77^7 V V + ™2 = oo4

(6)

Here m is the mass of the field, gs is the number of spin states of the field and it is assumed that the field in question is a bosonic one. One cannot live in the world with infinitely big vacuum energy, so Zeldovich assumed that bosonic vacuum energy should be compensated by vacuum energy of fermionic fields. Indeed, vacuum energy of fermions is shifted down below zero and is given by exactly the same integral as (6) but with the opposite sign. (This is related to the condition that bosons are quantized with commutators, while fermions are quantized with anti-commutators.) So, if there is a symmetry between bosons and fermions such that for each bosonic state there exists a fermionic state with the same mass and vice versa, then the energy of vacuum fluctuations of bosons and fermions would be exactly compensated, giving zero net result. This assertion

107 6

was made before the the pioneering works on super-symmetry 7 were published. However, since super-symmetry is broken this compensation is not complete and non-compensated amount of vacuum energy in any softlybroken super-symmetry must be non-vanishing, pvacV ~ m susy — 108 GeV 4 . This is 55 orders of magnitude larger than the cosmological energy density pc « 1 0 - 4 7 GeV 4 , eq. (1). Supergravity, i.e. locally realized supersymmetry, allows vanishing vacuum energy even in the broken phase but at the expense of fantastic fine-tuning by more than 120 orders of magnitude. Somewhat smaller contribution but still significant (mismatch by 55 orders of magnitude), and in a sense more troubling, comes from the vacuum condensates in quantum chromodynamics (QCD). It is practically an experimental fact that vacuum in QCD is not empty but filled by quark 8 and gluon condensates 9 with the energy density about 10~ 2 — 10~ 4 GeV 4 . What else "lives" in vacuum and exactly "eats up" all these contributions (with accuracy 1 0 - 4 5 or maybe even 10 - 1 2 0 ) is the biggest mystery in modern physics. The reviews of vacuum energy problem and possible solutions (none is satisfactory at the moment) can be found in refs. 10 . Another side of the problem emerges from the observed acceleration of the universe: it means that either the vacuum energy-momentum tensor is non-vanishing, TM„ = pvac9\iv 7^ 0, or there exists something unknown with negative and large by absolute value pressure, p = wp with w < —1/3. According to the covariant conservation of energy-momentum: p=-3H(p

+ p),

(7)

the energy density of matter with the equation of state (2) evolves in the course of expansion as p ~ a~3(1~w\ In particular, for vacuum energy p = const. A puzzling feature is why the energy densities of non-relativistic matter which evolves as pm ~ a~3 and dark (vacuum) energy are almost coinciding today, though their cosmological evolution were completely different. This may be explained by the so called quintessence 11 , that is by a new scalar field with vanishingly small mass and negligible interaction with other matter fields, except for gravity. Equation of motion of this field might have a tracking solution whose energy density closely follows the dominant one, though a fine-tuning is necessary to realize such a solution. On the other hand, even if the problem of coincidence of pvac and pm today, may be solved by the tracking solution, this new field does not explain why the vacuum energy is so close to zero. A model which in principle could solve both these problems is based on dynamical adjustment of vacuum energy by a new massless scalar 12 ,

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vector 13 , or tensor 14 field coupled to gravity in such a way that this new field is unstable in De Sitter space-time with respect to formation of vacuum condensate whose energy would compensate the source (i.e. the original vacuum energy) in accordance with the general principle of Le Chatellier. Such models have a generic property that the vacuum energy is compensated, however the compensation is not complete but only down to the terms of the order of the critical energy density at the running time moment t. So at each moment there is a non-compensated remnants (dark energy) with Q, ~ 1 with equation of state which may differ very much from the usual ones. Phenomenologically such models describe decaying vacuum energy and contain many of the later suggested "quintessentuial" ideas. The only, but important, shortcoming is that a realistic cosmological model based on this mechanism has not yet been found. An important quantity which may lit some light on the physical nature of dark energy is the value of the parameter w which describes the equation of state of dark energy (2). The present-day data agrees with w = — 1, i.e. dark energy could be the vacuum energy. However, other values of w are not excluded, moreover, it is possible that w is not a constant but a function of time and even that equation of state does not exist, so p cannot be expressed locally through p. In a normal field theory the energy dominance condition, \p\ < p is usually fulfilled. However, dealing with such a strange entity as the dark energy we cannot exclude that w < — 1. In this quite unusual case the energy density rises(!) in the course of expansion, p > 0 (see eq. (7)). An example of field theory leading to such a regime can be found in ref.14: it is a theory of free massless vector (or tensor) field with the Lagrangian density L = V^-l/Vfi''//2. Such a theory in a curved background is unstable with respect to formation of vacuum condensate of the time component Vt with the energy and pressure densities: pv

= V2/2 + 3H2Vt2/2, 2

2 2

pv = Vt /2 - 3H V /2

(8) - HV

2

- 2HVtVt

(9)

There is no equation of state p = p(p) in this theory but anyhow pv + pv may be negative and a new type of cosmological singularity: H ~ (to - t ) ~ 3 / 2

(10) 14

can be reached in a final time resulting in "tearing apart" the universe . A different model of negative (p + p) based on a scalar field with a wrong sign of the kinetic term and a discussion of a similar singularity can be found in

109

ref.15. The pathological properties of such theories probably indicate that a negative p + p is impossible, though it is not yet rigorously forbidden. Understanding the problem of vacuum energy compensation may have a noticeable impact on cosmology (and quite probably on fundamentals of quantum field theory). The mechanism that ensures this compensation may change the standard picture of the cosmological evolution. However, at the moment there is no strong demands for such changes. On the other hand, some discrepancies which may exist in big bang nucleosynthesis (see below sec. 5.4) or possibly in the formation of large scale structure could be cured by the dark energy. A related strong challenge is to understand what is the dark energy and a very important information about it can be obtained through a more accurate measuring of w in the future.

4. Dark Matter Second most important unsolved problem in cosmology (which may also have a strong impact on particle physics and field theory) is the problem of dark matter. The latter, most probably, consists from normal but yet undiscovered particles or fields. There are many (maybe, too many) possibilities discussed in the literature and at the moment we do not know which one of these hypothetical objects plays the role of dominant matter constituent of our world, or maybe there are even several of them. For a recent review see e.g. the lectures 16 Though the dark matter is not observed directly its existence seems to be firmly established. First, the total fraction of non-relativistic matter in the universe is ~ 0.3, while the amount of baryonic contribution is about 0.05 (see sec. 2). The latter is determined by two independent methods: by measurements of angular fluctuations of CMBR and by observations of abundances of light elements produced at BBN. Another argument in favor of cosmological domination of non-baryonic matter is based on the theory of large scale structure formation. Structures in baryo-dominated universe can only be formed after hydrogen recombination which took place at T « 3000 K, i.e. at red-shift zrec = 103 when the matter became electrically neutral. Before that period a large light pressure experienced by electrons (and as a result, by protons) prevented them from gravitational clusterization. After recombination, initially small density perturbations, dp/p rose as the cosmological scale factor and hence could increase only by the factor zrec = 103. On the other hand, the measured angular fluctuations of CMBR temperature

110

are quite small, ST/T <(a f e w ) x l 0 - 5 . Hence it follows under the standard assumption of adiabatic density perturbations (this assumption is now confirmed by CMBR data) that the latter should be also small at recombination, Sp/p < 10~ 4 . Hence they should remain small at the present epoch, even after amplification by 3 orders of magnitude. So one could conclude that there should be some other form of matter which does not interact with light and which started to form structures long before recombination. On the other hand, much larger density perturbations at small scales are not formally excluded by observations and they might allow an efficient structure formation in purely baryonic universe. Still, the combined data are very much against structure formation without dark matter. In particular, inflation predicts flat spectrum of perturbations with spectral index n = 1. From observations follows n = 0.93 ± 0.03. If this spectral behavior remains true at small scales then larger density perturbations, mentioned above, would not be present. Most probably cosmological dark matter consists of cold relics from Big Bang. It is the so called cold dark matter (CDM). Two most popular candidates for the latter are the lightest supersymmetric particle (which could be as heavy as a few hundreds GeV) and axions (which would be extremely light, about 10~ 5 eV). However, such simple forms of dark matter meet serious problems in description of details of large scale structure. There are several troubling features. In particular, CDM cosmology predicts dark halos with steep cusps 17 , while observations indicate that halos have a constant density cores (see e.g. ref. 18 ). Theory also predicts too many, by factor 5, galactic satellites 19 . Comparison 21 of galactic cluster abundances at high [z > 0.5) and low redshifts is compatible with the theory of cluster evolution for a very low cosmological matter density, ftm « 0.17, which is much smaller than the value, ~ 0.3, obtained by other methods. And at last, the galaxies in CDM simulations have considerably smaller angular momenta, than those observed (see the paper 20 and references therein). All these inconsistencies could be either prescribed to shortcomings of numerical simulations and, in particular, to neglecting some essential physical processes, or, more probably, to a real crisis in cold dark matter cosmology. Several new forms of dark matter were suggested to overcome the CDM problems: warm dark matter (WDM), self-interacting dark matter and, in particular, long discussed mirror or shadow matter (which is a special case of self-interacting dark matter), decaying cold dark matter, etc. Models with non-canonical forms of the spectrum of primordial

111

fluctuations and models with mixed adiabatic and isocurvature fluctuations were also considered. (For a recent review and a list of references see the papers 22 , here is neither space not time to discuss these issues in detail.) It seems that a simple canonical model with one form of dark matter and flat (inflationary) spectrum of perturbations does not satisfactory describe details of cosmic large scale structure and some deviations from the standard scenario seem to be necessary. It makes the situation more complicated and more interesting. It is worth mentioning that models with several forms of dark matter, e.g. mixed CDM+WDM, make the problem of cosmic conspiracy even more profound because, in addition to similar magnitudes of Ub, fiTO, and SIDE one has to explain close contributions to ft of several new forms of DM.

5. Main Events in Cosmological Evolution Here we will briefly discuss the history of the universe and physical phenomena necessary for creation of the present state of the world where we can live.

5.1. Before

beginning

Nothing is known about the universe prior for a certain temporal moment. So we cannot extend our history infinitely backward in time. It may be so because theory of quantum gravity is yet missing and there is no way to describe the state of the universe when the characteristic curvature and energy density had Planckian magnitudes. Possibly even time and space in our classical understanding did not exist "at that time". There are some attempts to go beyond Planck scales or, better to say, before Big Bang exploring e.g. string cosmology23. However, it is difficult to say if this or any other continuation through big-bang was indeed realized. Another possibility of periodically oscillating universe between big bang and big crunch was suggested ages ago (a list of the early papers and discussion can be found in the books 24 ) and was recently revitalized 25 as an alternative to inflation. Such models have an evident difficulty: in contraction phase already evolved large inhomogeneities become even larger and catastrophic formation of black holes during contraction phase may endanger the scenario. A recent discussion of behavior of perturbations in oscillating or bouncing universe can be found in the papers 26 .

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5.2.

Inflation

The earliest period of the universe evolution whose existence seems to be on firm grounds is the inflationary epoch. Today one could even say that inflation is an experimental fact. After the original suggestion by Guth 3 and first realistic models 27 there appeared zillions of papers on the subject; for recent reviews one can address ref.28 During relatively short inflationary period, when the universes exponentially expanded, a(t) ~ exp(Hjt), the proper initial conditions for creation of the present-day universe have been secured. During this period the cosmological energy density was, by assumption, dominated by a slow varying scalar field, inflaton, with the vacuum-like energymomentum tensor, TM„ « g^vpj. As we have already mentioned above, such energy-momentum tensor creates cosmological gravitational repulsion and hence it explains the origin of the observed today expansion. If inflation was sufficiently long, HiAti > 70, then the universe would be flat, VI = 1 ± 10~ 4 in accordance with observations (in fact the necessary duration of inflation depends upon the temperature of the universe heating and might be somewhat smaller than 70 Hubble times). The universe would be almost homogeneous and isotropic on large scales. This explains why CMBR coming from different directions has almost the same temperature - one should keep in mind that without inflation the regions on the sky separated by more than one degree would never communicate to each other. Simultaneously with the "smoothing down" the universe, inflaton created small density perturbations but at astronomically large scales which much later became seeds of large scale structure formation. In non-inflationary cosmology no reasonable mechanism of creation of density perturbations was known and the problem of their generation at astronomically large scale remained a great mystery. A unique explanation of these previously unexplainable features would be enough to consider inflation as an established fact. Moreover, there are some quantitative predictions of inflationary scenario supported by the astronomical data. First is of course a prediction of flat universe, fi = 1. Second is a prediction of flat (Harrison-Zeldovich29) spectrum of perturbations with spectral index n = 1. In fact inflationary spectrum usually slightly deviates from the flat one, depending upon the concrete model of inflation (for a review see e.g. ref. 30 ). According to the recent WMAP data (fifth paper in ref. 1 ), the index deviates from unity, n = 0.93 ± 0.03. It could be a worrying sign but maybe the deviation is not

113

significant taking into account possible statistical and systematical errors. On the other hand, as argued in ref. 31 , the graceful exit from inflation demands 0.92 < n < 0.97 and the WMAP data can be considered as confirmation of simple inflationary scenarios. However, even if the spectral index happened to be outside these bounds, it would not mean that inflation is excluded. There could be e.g. some other forms of perturbations, in particular, isocurvature ones (see e.g. the review32 for possible mechanisms and recent papers 33 and references therein) which would have a different spectrum and which may explain a noticeable deviation of n from unity. So as a whole inflation is a great success. It is observationally confirmed and theoretically beautiful. Now we have to fix some details: what is the inflaton, in which potential it evolves, are there gravitational waves generated at inflationary stage, etc. This is of course highly non-trivial, especially because there could be some other sources of perturbations, except for the inflaton. The measurements of CMBR angular spectrum and polarization gives some hopes to progress in this direction. Existence of inflation means in particular that there should be new physics beyond the standard model (SM) because there is no space for the inflaton in the frameworks of SM. 5.3.

Baryogenesis

Predominance of matter over antimatter was one of the biggest cosmological puzzles of the first two thirds of the XX century. Its solution was outlined by Sakharov 34 in 1967 and is now commonly accepted. The mechanism is based on three conditions: 1) Non-conservation of baryonic charge; 2) Breaking of C and CP symmetries; 3) Deviation from thermal equilibrium in primeval plasma. All these three conditions are known to be true either from experiment or because they are well justified theoretically. Moreover, the existence of the charge asymmetric universe itself is a strong indication to non-conservation of baryons, otherwise inflation would not be possible. The time and temperature interval where baryogenesis took place strongly depends on concrete model and may vary in a very wide range from GUT or even almost Planck scales down to a fraction of GeV. For the reviews and more recent quotations see refs. 32 ' 35 . There are many models of baryogenesis considered in the literature. Possibly an incomplete list includes: (1) Baryogenesis in out-of-equilibrium heavy particle decays 34 . (2) Electroweak baryogenesis36.

114

(3) (4) (5) (6) (7)

Baryogenesis by supersymmetric baryonic charge condensate 37 . Spontaneous baryogenesis38. Baryo-through-lepto-genesis39. Baryogenesis in black hole evaporation 40 . Baryogenesis by topological defects (domain walls, cosmic strings, magnetic monopoles) 41 .

The problem is to understand what of the above is indeed realized and how it can be verified. The second part is non-trivial because usually a model of baryogenesis has to explain only one number the ratio of baryonic charge number density to the number density of photons in CMBR, /? = 6 • 10~ 10 , and there could be many models giving the same baryon asymmetry. It seems established that electro-weak baryogenesis, which might proceed in the frameworks of the standard model, is not efficient enough to produce the observed asymmetry. All other models demand new physics. So the baryon asymmetry of the universe is another cosmological fact that indicates to existence of physics beyond the standard model. There are plenty of models of baryogenesis which simultaneously with excess of matter over antimatter in our neighborhood predict that there could be astronomically large domains of antimatter and maybe even not too far from us. Such models are discussed in ref.42. In this connection expected BESS, Pamela, or AMS experiments searching for cosmic antinuclei 4 He or heavier ones would be of primary importance (for a review see e.g. ref. 43 ). 5.4. Big bang

nucleosynthesis

Big bang nucleosynthesis (BBN) took place in a relatively old and cold universe in the temperature interval from a few MeV down to 60-70 keV and time in the range from roughly 1 sec up to 300 sec. Physical processes involved are well known: they include low energy weak interactions and low energy nuclear physics. In contrast to phenomena considered in the previous subsections when we were in terra incognita and had to use our imagination based on reasonable theoretical models, now we are on firm grounds of well established physics. The only unknown parameter that enters theoretical calculations of the light element abundances is the ratio of baryon-to-photon number densities, P = UB/TIJ, which a few years ago was found from BBN itself. Now can be independently determined from CMBR. A reasonable concordance of the two values of (3 shows that the theory is in a good shape and that there was

115

no influx of photons into the cosmic plasma between neutrino decoupling at T « 1 MeV and hydrogen recombination at red-shift z « 103 or T « 0.26 eV. The accurate Bose-Einstein shape of the CMBR spectrum indicates the same but in a slightly shorter temperature interval, roughly speaking, starting from the red-shift 107 (see e.g. the reviews 44,45 ). During those several minutes the following light elements have been produced: deuterium (~ 3 • 10 - 5 , relative to hydrogen by number), helium3 (about the same number as 2 H), helium-4 (23-24% by mass), and a little of 7 Li (a few x 10~ 10 ). Accuracy of calculations is mostly determined by the uncertainties in the nuclear interaction rates and is quite good. According to the analysis of ref.46, the theoretical accuracy is at the level < 0.1% for i He, better than 10% for 2H and is about 20-30% for 7Li. Anyhow, in all the cases theoretical uncertainty is much smaller than the observational precision. The latter suffers from two serious problems: systematic errors and poorly understood evolutionary effects. They are reviewed e.g. in refs. 47 . Despite existing uncertainties, theory reasonably well agrees with observations. However, there are indications to possible discrepancies. The results of different groups measuring deuterium at high redshifts, which may be a primordial one, are in noticeable disagreement with each other. Moreover, the measured abundances of 4 He and 2 H seem to correspond to somewhat different values of (3. It is not clear if these discrepancies are serious and indicate some unusual physics (degeneracy of cosmic neutrinos, non-negligible role of dark energy at BBN, some new particles present at BBN, etc) or after a while all measurements will come to an agreement between themselves and with the value of (3 inferred from CMBR. Hopefully it will not take more than a decade.

5.5. Large scale structure

formation

During an initial period of the universe life-time (but after inflation) the universe was very smooth, practically homogeneous and isotropic. It remained such between the epoch of inflation and the end of radiation domination (RD) era. This period lasted roughly speaking about 100,000 years or until red-shift zeq « 104. Small density perturbations generated at inflation were almost frozen during RD period and were practically unnoticeable. However, their importance at matter dominated (MD) regime is difficult to overestimate. At MD-stage these small density perturbations became unstable with respect to gravitational attraction, they started to rise forming seeds from which astronomical large scale structures such as

116

galaxies, their clusters and superclusters evolved. Development of such huge objects, their temporary evolution, and power spectrum primarily depend upon the form of primordial density perturbations and properties of dark matter. In particular, detailed study of the large scale structure at different scales2 can help to establish essential properties of dark matter particles prior to their possible discovery in direct experiments. To some extent this subject is discussed in sec. 4. An interpretation of the astronomical data and conclusion about properties of DM-particles depend upon the hypothesis about the perturbation spectrum. Usually the spectrum of primordial density perturbations is taken in one parameter power law form with an arbitrary spectral index n. Such shape is justified for the flat spectrum (with n = 1) which is scale free and on dimensional grounds this is the only possible form of the spectrum. In the case that a dimensional parameter exists, any function of this parameter, and not only a power law, is a priori allowed. Though, as is mentioned above, the nearly flat spectrum is supported by inflation and quite well agrees with the data, noticeable deviations from this type of spectrum are possible. Such deviations would be very interesting for physics of the early universe and, in particular, for possible mechanisms of creation of density fluctuations but simultaneously it would make it more difficult to obtain conclusions about DM-particles from the LSS data. 5.6. Future of the

universe

In Friedman cosmology with the normal matter content the ultimate fate of the universe is completely determined by the spatial curvature: for flat (zero curvature) and open (negative curvature) geometry the universe will expand forever, while closed universe (positive curvature) will stop expanding and will recollapse to big crunch. One can see that from the Friedman equation: \aj

orript

az

where the sign of the constant k determines the sign of curvature. The energy density p of any normal matter with positive pressure drops in the course of expansion faster than I/a3, the limiting case corresponds to nonrelativistic matter with p = 0. Hence the curvature term will dominate at large a(t) and determine the universe destiny. If the parameter w connecting pressure and energy densities, eq. (2), may be negative, then for w < —1/3 the pressure density which evolves as p ~ a~3(1+w\ would decrease slower than k/a2 so initially inessential

117

curvature term would always remain such and expansion would never stop in any geometry. Thus in a distant future only gravitationally binded objects will continue to exist, while distant galaxies will disappear from the sky. It would not create a big difference for a naked eye if stellar luminosity remains the same during some billion years. Unfortunately this is not so because the Sun and other stars will exhaust their nuclear fuel and the world will fall into (almost) complete darkness and cold which may be slightly lit up and heated by possible proton decay. For the proton lifetime TP ~ 10 33 years the solar luminosity generated by the decay will be 20 orders of magnitude smaller than the present-day solar luminosity. In this epoch the Earth will be much stronger "illuminated" and heated by her own decaying protons. Nevertheless, life will hardly be possible in such uncomfortable conditions. The only chance for survival could be catalysis of proton decay by e.g. magnetic monopoles48 if they exist. The consumption of the whole solar mass will allow to maintain life on the Earth for about 1020 years. Using to this end other stars in the galaxy may extend the life-time up to 1030 — 1032 years, if other civilizations do not interfere. All the story may end much faster if p < — p and the universe will evolve to catastrophic expansion singularity (10) in 10-100 Gyr. In this case not only astronomically large objects but even atoms and elementary particles may be destroyed. However this conclusion is model dependent and an inhomogeneous component of the dark energy field Vt(t,x) may stop the catastrophe. 6. Neutrinos in Cosmology The universe is filled with neutrinos - 55 neutrinos and the same number of antineutrinos per cm 3 for each neutrino flavor - and though they are very light and weakly interacting their sheer number makes them cosmologically important. Knowing the number density of cosmic neutrinos one can immediately deduce an upper limit on their mass 49 : Y,m^a

< 9 4 e V f i m / i 2 = 15 eV

(12)

a

where Vtm = 0.3 and h2 = 0.5 have been substituted and the sum is taken over all neutrino species, a = e, /z, r. Since neutrinos were relativistic during a large part of the universe history, their long free streaming path would suppress structure formation in neutrino dominated universe at the scales below M = 10 1 7 Mo(m„/eV)~ 2 where MQ is the solar mass. Hence massive neutrinos

118

cannot be the dominant dark matter particles and their mass density should be below fi„ < 0.1. Correspondingly J2amv* < 5eV (further discussion and references to this and other subjects discussed below can be found in the review 45 ). More detailed studies of the large scale structure together with WMAP data on angular fluctuations of CMBR allowed to improve this limit down to J^Q m„ o < 0.7eV. Already today astronomy is able to constraint m„ with better accuracy then direct experiment. Future Planck mission and more data on LSS will possibly push this limit down to ~ 0.1 eV or will measure the neutrino mass. This would be a unique example when the mass of an elementary particle is determined by telescopes. Apart from LSS, neutrinos can be traced through BBN and CMBR. BBN permits to constrain the number of neutrino families50. Keeping in mind the existing accuracy in extracting primordial abundances of 2 H and 4 He from observations (see sec. 5.4) one can impose the upper limit on the number of additional neutrino families, ANV < 0.3 - 0.5 with justified expectations to strengthen this limit down to about 0.1 in the near future. At the present time the CMBR data are not competitive with the upper limit obtained from BBN. Still the analysis 51 of the CMBR data indicates, independently from BBN, that cosmological relic neutrinos indeed exist and the number of families is confined between: 1 < Nv < 6. One may hope for a drastic improvement of this result if the expected sensitivity of the Planck mission at per cent level is achieved, so that it may register even non-equilibrium corrections to the energy spectrum of relic neutrinos at the level of 3%. These corrections are predicted to result from non-equilibrium e+e~- annihilation in the primeval plasma and deviations from ideal gas approximation at the epoch when the universe was about 1 sec old (see discussion and references in the review 45 ). Consideration of BBN permits also to derive bounds on mixing between active and possible sterile neutrinos, on the magnitude of cosmological charge asymmetry, on magnetic moment of neutrinos, etc.

7. Conclusion Great progress in cosmology at the end of the previous century helped to understand the universe much better and simultaneously led to discovery of many new puzzles, problems, and even mysteries. Today cosmology tells us that fundamental physics is not completed and there surely exist new phenomena outside well established known physics. We need to discover much more to understand the observed features of the universe.

119

To start, the greatest mystery of (almost) complete cancellation of vacuum energy is not (or cannot be) resolved in the frameworks of known physics. The next, possibly not so striking, question about the nature of dark energy also remains unanswered. As for dark matter, it may possibly be explained by some extension of the realm of the existing particles by addition either stable lightest supersymmetric particle or axion. There are some more good candidates for dark matter particles or objects but it is not yet established which one of them makes dark matter. Moreover, the CDM crisis (see sec. 4) possibly demand either several forms of DM, thus deepening the cosmic conspiracy puzzle, or particles with rather strange properties unexpected in simple, theoretically motivated generalizations of the standard model. Baryogenesis might in principle operate based on the existing minimal standard model of particle physics but the latter is not sufficiently productive to create the observed baryon asymmetry of the universe. Hence new particles or fields are necessary. Hopefully joint efforts of experimentalists and theoreticians will help to resolve some or, in further perspective, all these problems but new mysteries and discoveries are waiting for us on the way and this is what makes cosmology so interesting now. To conclude, cosmology had very productive period during last quarter of the previous century, it is in the process of exciting development today, and bright future with many new discoveries is coming.

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123 51. P. Crotty, J. Lesgourgues and S. Pastor, astro-ph/0302337; S. Hannestad, astro-ph/0303076; E. Pierpaoli, astro-ph/0302465; V. Barger, J.P. Kneller, H.-S. Lee, D. Marfatia and G. Steigman, hepph/0305075.

I N T E G R A B I L I T Y OF T H E P O M E R O N I N T E R A C T I O N S IN THE MULTI-COLOUR QCD

L. N. L I P A T O V * Petersburg Nuclear Physics Institute, Orlova Roscha, Gatchina, 188300, St. Petersburg, Russia E-mail: [email protected]

It is demonstrated, that the hamiltonian describing possible interactions of the Reggeized gluons in the leading logarithmic approximation (LLA) of the multicolour QCD has the properties of conformal invariance, holomorphic factorization and duality. It coincides with the hamiltonian of the integrable Heisenberg model with the spins being the Mobius group generators. With the use of the Baxter-Sklyanin representation we calculate intercepts of the colourless states constructed from three and four reggeized gluons and anomalous dimensions of the corresponding high twist operators. The relation between the integrability of the high energy dynamics in the multi-colour QCD and an extended N = 4 supersymmetry is discussed taking into account the next-to-leading corrections to the BFKL and DGLAP equations and the C F T / A d S correspondence.

1. Introduction The scattering amplitude A(s, t) in the Regge kinematics of high energies 2E = yfs and fixed momentum transfers q — yf—i in the leading logarithmic 2

approximation (LLA) a s l n s ~ 1, as = f^ —> 0 (g is the QCD coupling constant) is obtained by calculating and summing the largest contributions ~ (as ln(s)) n in all orders of the perturbation theory. In the case of gluon quantum numbers in the crossing channel t it has the Regge-like form 1 A(s,t)~sjW,

j(t) = l + w(t),

where j{t) is the gluon Regge trajectory. The function tu(t) in LLA is given below ^) = -A^ln^+0(54), *Work partially supported by grant INTAS 2000-366.

124

125

where A is a factious gluon mass regularizing the infrared divergency and Nc is the rank of the gauge group (for QCD iVc = 3 ) . In the total crosssection this divergency is cancelled because the probability of the soft gluon emission contains the same contribution but with an opposite sign 1 . The above behaviour of A(s, t) means, that the gluon belongs to a family of particles with odd spins j and masses \ft lying on the Regge trajectory j = j(t) with a negative signature. The gluon Regge trajectory was calculated in two first orders of the parturbation theory 2 . It is possible, that in a non-perturbative approach j(t) is a linear function near t = 0. In LLA the most essential process at high energies s = (PA + PB)2 — 1 [PA + PB1)2 1S t n e production of an arbitrary number of gluons with momenta kr (r — 1,2, ...,n, fco = PA1, kn+\ = PB1) in the multi-Regge kinematics when their pair energies ^fsk (sk — (kT-i + kr)2) are large and and momentum transfers qm = PA~ J2T=o ^r a r e nxec ^- I n this kinematics the production amplitude has the multi-Regge form 1 A

— o„r c i

^2->2+n — ^*J- AA'

Mti)

i!

-+2 9 1

L

rdi

c2ci

Mt2)

zl

—>2 1 2

i

rd2

<"(*»+i)

n+i

C 3 C 2 - " -£2 1 n+1

L

rcn+i

BB> '

where tm = —~t2n- The gluon colours are enumerated by indices A,A',ci,di.... The vertices TC^A, and r ^ + l C r describe respectively the scattering of the external particles due to an exchange of the reggeized gluon and the production of a gluon from the reggeized gluon. They contain apart from the structure constants fd,a,b of the gauge group SU(NC) also the factors depending on the helicities of the gluons. The helicity of the scattered gluon is conserved: TC^A, = —ifCAA'S\A\A,. The production vertex for the gluon with a fixed helicity contains the factor rA,"+1. Here we introduced the complex components qr = q], +iq2, q* and kr, k* of twodimensional momenta ~~q*r and k r, respectively. Thus, in the cross-section for the gluon production the square of T has an additional propagator |fcr|~ in the transverse subspace. The high energy behaviour of total cross-sections a ~ sA is related to the i-channel exchange of the Pomeron - the Reggeon with vacuum quantum numbers and the positive signature. The intercept A of its Regge trajectory is assumed to be small. In leading and next-to-leading approximations the Pomeron is a composite state of two reggeized gluons. Its wave function satisfies the equation of Balitsky, Fadin, Kuraev and Lipatov (BFKL) 1 . This equation is used for the description of structure functions for the deepinelastic lepton-hadron scattering together with the DGLAP equation 3 . In the impact parameter space ~ft the BFKL equation has the Schrodinger-like

126

form

5

Ef(pi,pt)

= H12f(pi,pt),

(i)

where in LL A the eigenvalue E of the ground state is related to the intercept A of the Pomeron as follows

A=-0E

(2)

and the Hamiltonian is given below in an operator form #12 = In b i | 2 + In H

2

*ffi (in \p12\2) p\p2 + 4 7 . (3) V J |Pi| \P2\ Here two first terms are contributions of the Regge trajectories of two virtual gluons, the third term appears from the Fourie transformation of a product of two effective vertices r|f* Cr ~ qrq*+i/^r for the gluon production in the multi-Regge kinematics and 7 = — \P(1) is the Euler constant (see 4 ) . The infrared divergency at A —> 0 is cancelled between the terms corresponding to the virtual and real contributions. We introduced the complex components of the gluon coordinates pk = Xk + iyk, p\ (pu = Pi — P2) and the canonically conjugated momenta pk = id/(dpk), Pi = id/(dp*k). In LLA the BFKL equation is invariant under the Mobius group transformations 5 + 2 Re

Pk ->•

apk + b —;,

(4)

cpk + a where a,b,c,d are arbitrary complex parameters. Its solutions belong to the principal series of unitary representations of the Mobius group. For this series the conformal weights m = 1/2 + iv + n/2, fh = 1/2 + iv - n/2 (5) are expressed in terms of the anomalous dimension 7 = 1 + 2%v (with real v) and the integer conformal spin n of the local operators 0TO-^(/5$). The conformal weights are related to the eigenvalues M2fm,m=m{m-l)fm,fh, 2

M*2fmtm

= m(fh - l)/ m ,m

(6)

2

of the Casimir operators M and M* of the Mobius group (for the Pomeron the number of reggeons n = 2) M

2

= [ J 2 \fc=l

M

n /

=Y,2MrMs=-T,P2rs9rds, r<s

M*2 r<s

= (M2Y,

(7)

127

Here M% are the group generators Mzk=pkdk,

M+ = -p2kdk

M-=dk,

(8)

and dk = d/(dpk). The eigenfunctions of the Casimir operators M2,M*2 and the hamiltonian H12 can be considered as three-point functions of a twodimensional conformal field theory and presented in the Polyakov anzatz as follows 5

(

\ m /

-£*-)

*

\ m

(-^V) .

P10P20/

0)

\P10P20J

Putting this anzatz in the BFKL equation one can calculate the energy 1 Em>fh = 4 R e f * ( i + iv + M ) + *) - 4 * ( 1 ) . (10) Note, that Em^ can be written in the form compatible with the holomorphic separability of if 12 4 Em,m =£m+efh, The minimum olEm^

£m = *(m) + * ( 1 - m) - 4*(1).

(11)

is obtained for v = n = 0 and equals m i n S m , S i = - 8 1n2.

(12)

Therefore the total cross-section in LLA grows rather rapidly at~sA,

A=^ATcln2

(13)

and exceeds the Froissart limit <jt < In2 s. For the photon virtualities in the deep-inelastic ep scattering measured in DESY the effective coupling constant g is such, that A « 0.5. Next-to-leading corrections give a possibility to find the region of applicapability of the BFKL equation 2 . In particular one can obtain for the Pomeron intercept A « 0.2 6 , which coincides with a phenomenological estimate of the soft Pomeron intercept obtained many years ago in the papers of K. Ter-Martirosyan, A. Kaidalov and others. Moreover, the total cross-section for the annihilation of virtual photons into hadrons is calculated in this approximation in an agreement with experimental data obtained in CERN. In the framework of the optimal renormalization scheme one can verify that the Mobius invariance obtained earlier in LLA 5 is valid approximately even after taking into account next-to-leading terms in the BFKL kernel 6 . Nevertheless the next-to-leading corrections to the

128

intercept of the Pomeron do not lead to the unitarization of scattering amplitudes and an disagreement with the Froissart bound for the total cross-section remains. A self-consistent approach to the construction of the unitary S -matrix at high energies in the perturbative QCD should take into account the use of the effective action for the regeized gluon interactions 7 . But we consider below a more simple method related to solutions of the Bartels, Kwiecinski and Prascalowich (BKP) equations for the composite states of n reggeized gluons 8 .

2. Reggeon states in the multi-colour QCD The BFKL Pomeron is a composite state of two reggeized gluons. But this property is valid only in the leading and next-to-leading approximations. Generally we should take into account also multi-gluon components of the Pomeron wave function. The Regge trajectories and multi-gluon couplings for the field theory of reggeon interactions can be obtained from QCD with the use of the effective action constructed in ref. 7 . In a generalized LLA one can neglect the non-conservation of a number of reggeized gluons in the i-channel taking into account only their pair interaction given (up to a colour factor) by the BFKL Hamiltonian H^i- In such way we obtain the BKP equation 8 r

Em,m4>m,m = Hl/jm,m

, H =

^ l
Hrl

pa'T'a _ ' •

T

< •N

(14)

c )

Here T" implies the gauge group generator acting on the colour index of the gluon r. We took into account also that due to the Mobius invariance of H the wave function and energies are enumerated by the conformal weights m and fh for the corresponding representations of the Mobius group. The intercepts A mi m entering in the asymptotic contribution a ~ s A to at from the Feynman diagrams with n reggeized gluons in the i-channel are expressed through Em>m as follows 92NC &m,m —

0 _22

8TT

•^/m,fn

\*-")

Really the wave function VVn.m is proportional to a discontinuity of the ^-channel partial wave fj(t) on its j-plane cut and the maximal eigenvalue A is equal to its position in the w-plane (u — j — 1). Due to the Mobius invariance the singularities of fj(t) are of the form A/W — A and do not move with t. The eigenfunction ipm,m(~pi> pt> •••> Pni P^) describing the composite

129

state ^m^h(p^)

°f

n

reggeized gluons depends on their impact parameters

In a particular case of the Odderon, being a composite state of three reggeized gluons with the charge parity C — — 1 and signature Pj = — 1, the colour factor is unique and coincides with the completely symmetric tensor dabc 9 . Therefore taking into account, that in this state each pair of gluons is projected into the adjoint representation, the equation for the wave function / m ,m m front of datc is simplified as follows 8 Em,m

fm,m

= « ( # 1 2 + # 1 3 + # 2 3 ) fm,m-

(16)

The eigenvalue of this equation is related to the high-energy behaviour of the difference of the total cross-sections app and aPp for interactions of particles p and anti-particles p with a target 9 c?pp - oVp ~ s A m ' ™ .

(17)

Due to the Bose symmetry the wave function is completely symmetric /m,m(p1*, P%> P$> P#) = fm,m(P$,

~P~t, pti P&) = / m , m ( P f \ P~t, Pi*-, ~p1>)- (18)

Note, that the other solution proportional to a completely anti-symmetric tensor fabc has the anti-symmetric wave function and describes the state with the Pomeron quantum numbers C = Pj = 1. To simplify the structure of the equation for colourless composite states for a general case of n reggeized gluons we consider the multi-colour limit Nc —> oo 4 . According to t'Hooft only planar diagrams are essential in the multi-colour QCD. In our case it means, that providing, that we describe the colour structure of the gluon r by an hermitian matrix AaTf of the rank iVc with its Green function presented by a pair of quark and antiquark lines, only cylinder-type diagrams in the ^-channel survive. It is enough to consider the irreducible contribution for which the wave function has the following colour structure i>m,m{P~i>->Pli'
^2 fm,m(P~U>->PiZ'>Pft) {ii,...,i„}

tr

(Tail

...Tai«)

, (19)

where the summation is performed over all permutations {ii...in} of gluons 1,2,...,n. At large Nc each term in the sum satisfies the Schrodinger equation and therefore for the function fm,m{J>ii 7>$> •••> P~n> 7$) symmetric under the cyclic transmutations fm,m{p~t,

~P$, -P~n\ P&) = fmMP~Z>

P~l> -P^l'i

P&)

(20)

130

we obtain the simplified equation similar to the Odderon case n

1

r=l

For iVc —• oo only neighbouring gluons interact each with another and the factor 1/2 is related to the fact, that each pair of these gluons is in an adjoint representation. Here it was implied, that Hn>n+i = i?i,„. It is remarkable, that the Hamiltonian H in the multicolour QCD has the property of the holomorphic separability 10 : H = ±(h + h*), [M*]=0,

(22)

where the holomorphic and anti-holomorphic Hamiltonians n

n

* = $>k.fc+i,/»* = X X * + i k=i

(23)

k=i

are expressed in terms of the corresponding BFKL hamiltonians

10

hk,k+i = log(pfc Pk+i) + Pk1 log(pfc,fc+i) Pk +Pklllog(pk,k+i)Pk+i

: +27. (24)

Owing to the holomorphic separability of H, the wave function fm,m(pt, •••, Pn', ~pi$) has the property of the holomorphic factorization 10 : / m , m ( p t , - - - , P ^ ; P ^ ) = Ylcr,l

/m(/>l, - > Pn, Po) f'm(Pl>->Pn'>Po)>

(25)

r,l

where r and I enumerate degenerate solutions of the Schrodinger equations in the holomorphic and anti-holomorphic sub-spaces: e

m Jm

=

il Jm ; ^rh Jm

==

"

Jm > ^m^rh

=

7y{^rn > £-m) •

\^l

Similarly to the case of two-dimensional conformal field theories, the coefficients cr,j are fixed by the single-valuedness condition for the wave function / TO ,m(pi\ P~i, —,Prl> P&) m the two-dimensional ~p*-space. Note, that in these conformal models the holomorphic factorization of the Green functions is a consequence of the invariance of the operator algebra under the infinitely dimensional Virasoro group n .

131

3. Integrability of the reggeon calculus in the multi-colour QCD It is easily verified, that for the pair hamiltonian one can write another representation 10 hk,k+i = Pk,k+i log(pfc Pk+i) Pkjc+i + 2 log(P*,*+i) + 2 7 •

(27)

As a result, there are two different normalization conditions for the wave function r

"

i;=/nJ

n

=/n. r

d pr

I I *V,r+l / r=l

r=l

J

d pr

r=l

II Pr/

(28)

r=l

compatible with the hermicity properties of H. Indeed, the transposed Hamiltonian h* is related to h by two similarity transformations 10

ht= prh p x

r+i

n n r =n Pr,r+i ^ n ^. r=l

r=l

r=l

(29)

r=l

Therefore fr commutes [h, A}=0 with the differential operator

(30)

10

A = Pl2p23-PnlPlP2-Pn-

Furthermore motion

12

(31)

, there is a family {qr} of mutually commuting integrals of [qr,qs] = 0 , [qT,h] = 0.

(32)

They are given below Qr

=

/ _,

Pi\i2 Pi2h---PirhPhPi2---Pir-

(33)

U<J2<-.-
In particular g n is equal to A and 52 is proportional to M2. The generating function for these integrals of motion coincides with the transfer matrix T(u) for the XXX model 12 13 T(u) = tr(Li(u)L 2 (u)...L f l (u)) = ^ u n - r g r >

(34)

r=0

where the L-operators are Lk(u)=(U

+ P kPk 2

\

-PkPk

Pk

u-pkpk

) - « G ! ) + ( - » ) i"'1'"" <35>

132

The transfer matrix is the trace of the monodromy matrix t(u) : T{u) = tr (t(u)), t{u) = Li(u)L 2 (u)...L n (u). It can be verified that t(u) satisfies the Yang-Baxter equation

K\ («) Ki (v) ''III («-«) = i£? («-«) 4 («) 4 («),

(36) 12 13

(37)

where Z(iu) is the i-operator for the well-known Heisenberg spin model

rtfaM = «>V1Kl+iS%6?l.

(38)

The commutativity of T(u) and T(v) [T(u),T(v)}=0

(39)

is a consequence of the Yang-Baxter equation. If one will parametrize t(u) in the form t ( u ) =

(*(«)+J3(«) i-(«) ), V j+(«) jo(«)-j3(w)y

(4Q)

this equation is reduced to the following Lorentz-covariant relations for the currents j^(u): ftM,

JVOO] = [ i » , > ( « ) l = ^f~

{f{u)f{v)

- j"(v)j'(u)).

(41)

Here ty.vpa is the antisymmetric tensor (£1230 = 1) in the fourdimensional Minkovski space and the metric tensor rfv has the signature (1, —1, —1, —1). This form follows from the invariance of the Yang-Baxter equations under the Lorentz transformations. The generators for the spacial rotations coincide with that of the Mobius transformations M. The commutation relations for the Lorentz algebra are given below: [Ms,Mt]=iestuMu,

[Ms,Nt]=iestuNu,

[Ns,Nf]

where N are the Lorentz boost generators. The commutativity of the transfer matrix T(u) hamiltonian h 12 13

= iestuMu

, (42)

with the local

[T(u),h} = 0

(43)

is a consequence of the relation: \Lk{u) Lk+1{u),hk,k+1] for the pair Hamiltonian hk,k+i-

= -i {Lk(u) - Lk+1{u))

(44)

133

In turn, this relation follows from the Mobius invariance of hk,k+i and from the identity: (Mk,k+i)

k,k+li

=

,Nk,k+l

(45)

4Nk,k+i,

where Mktk+1 = Mk+ Mk+l,

Nkik+1 =Mk-

Mk+1

(46)

are the Lorentz group generators for the two gluon state. To verify the above identity one should take into account, that the pair hamiltonian hk,k+i depends only on the Casimir operator (Mkik+l) is diagonal

and therefore it

hk,k+i \mk,k+i) = (*(m fc , fc+ i) + *(1 - mk
(47)

in the conformal weight representation: [Mk,k+i)

\rnk,k+i) = mk
- 1) \mk,k+i) •

(48)

Further, using the commutation relations between Mkik+i and Nk
= 0, one can verify that the operator

Nk:k+i has non-vanishing matrix elements only between the states Imt^+i) and \mktk+i ± 1 ) . As a result, the above identity for Nk>k+i turns out to be is a consequence of the known recurrence relations for the \P-functions: *(m) = * ( m - 1) + l / ( m - 1), $(1 - m) = *(2 - m) + l / ( m - 1). (49) The pair Hamiltonian /ifc.jt+i can be expressed in terms of a small-u asymptotics for the fundamental L-operator of the integrable Heisenberg model with spins being the generators of the Mobius group 13 14 + ...).

(50)

The fundamental L-operator acts on functions f(pk,pk+i) defined by the relation

and Pktk+i is

Lktk+i(u)

- Pk
Pk,k+i f{Pk,Pk+i) = f(pk+i,pk). The operator Lk,k+i satisfies the linear Yang-Baxter equation Lk(u)Lk+l(v)

Lk
- v) Lk+i(v) Lk(u).

(51) 13

(52)

134

This equation can be solved in a way similar to that for hk,k+i and the proportionality constant is fixed from the triangle Yang-Baxter equation Li3(u) L23(v) L12(u - v) = Ll2(u - v) L23(v) L13(u).

(53)

To find a representation of the operators obeying the Yang-Baxter commutation relations, the algebraic Bethe anzatz can be used 13 . To begin with, in the above parametrization of the monodromy matrix t(u) in terms of the currents j M (u), one should construct the pseudovacuum state |0) satisfying the equations j+(«)|0> = 0.

(54)

However, these equations have a non-trivial solution only if the above Loperators are regularized as follows 'Si„.\ _ ( u + PkPk -iS Pk \ i : , " o , : x .u~ . PkPk+ / : , ,*<x5 -PkPk+2ipkS

(55)

J*(«)=

by introducing a small conformal weight 6 —> 0 for reggeized gluons Another possibility is to use the dual space corresponding to 8 = —1 14 . For the above regularization the pseudovacuum state is

t56)

i * ) = n "** It is also an eigenstate of the transfer matrix: T(u)\5) = 2j0(u)\S)

= ((u-i6)n

+ (u + iS)n) \5).

(57)

Furthermore, all excited states are obtained by applying to \S) the product of the operators j _ (v) \viv2...vk)

= j-(vi)

j-{v2)...j-(vk)

\S).

(58)

They are eigenfunctions of the transfer matrix T(u) with the eigenvalues:

f (U) = ( « + isr n ^ ^ + (u - isr n ^ ^ , r~l

r

r=l

providing that the spectral parameters vi,v2,...,Vk solutions of the set of the Bethe equations 13 i6\n vs + iS J for s = l,2...k.

xx

jTV.-vr-i vs - vr + i

r

(59)

are chosen to be

135

Due to the above relations the Baxter function denned as follows k

Q(u) = JJ(u-i; r )

(61)

f(u) Q(u) = (u - iS)n Q(u + i) + (u + iS)n Q(u - i).

(62)

satisfies the equation

13 14

Here T(u) is an eigenvalue of the transfer matrix T{u). The eigenfunctions of h and qk can be expressed in terms of a solution Q^ (u) of this equation using the Sklyanin anzatz 15 \viv2...vk)

= QW(S 1 ) Q<*)(u 2 )...QW(u„_ 1 )|<J>,

(63)

where the integral operators ur are zeros of the current j _ (u) n-l

j - (u) = c Y[ (u - uT) •

(64)

r=l

Thus, the problem of finding the wave functions and intercepts of composite states of reggeized gluons is reduced to the solution of the Baxter equation 13 14 . We shall consider the Baxter-Sklyanin approach below. 4. Duality property of Reggeon interactions The integrals of motion qT and the Hamiltonian h are invariant under the cyclic permutation of gluon indices i -> i + 1 (i = l,2...n), corresponding to the Bose symmetry of the Reggeon wave function at Nc —)• oo. It is remarkable that above operators are invariant also under the more general canonical transformation 16 : Pi-i,i -> Pi ->• Pi,i+i,

(65)

combined with reversing the order of the operator multiplication. This duality symmetry is realized as an unitary transformation only for a vanishing total momentum: n

r=l

The wave function ipm,m of the composite state with ~~$ = 0 can be written in terms of the eigenfunction fmm of the integrals of motion qk and ql for k — 1,2...n as follows -y-^/m.mCPl'.pt.-.iOn;^)(67)

136

Taking into account the hermicity of the total Hamiltonian n

n

H+ = JJ \Pk,k+i\-2H

n

10 12

:

n

Yl | P w | 2 = H\Pk\2H]l\pk\-2,

(68)

fc=i fc=i fc=i fc=i the solution i/>^~ m of the complex-conjugated Schrodinger equation for ~p* = 0 can be expressed in terms of ipm,m as follows : n V^,m(Pl£,P2$,-) = J J | W , f c + i r 2 ( ^ m , m ( ^ , P 2 t , - ) * •

(69)

Because i/Vn.m is also an eigenfunction of the integrals of motion A = qn and .A* with their eigenvalues Am and AJ^ = A„ 10 : Alpm,m — -Vi Vm.m i

iPm,m = ^fhWm,m

A

>

-A = P\2---Pn\

Pi •••Pn i ( ' 0 )

one can verify that the duality symmetry takes a form of the following integral equation for i[>mtm 1 6 ;

V V m ( p l l - , ^ ) _ f "TT1 (Pp'k-i,k A e * * ^ ^ |\

I on

~ /

11

oT

11 I

- ^ -r>

2 Vm,m(.Pl2»-.Pni;-

(71)

In the case of the Odderon the conformal invariance fixes the solution of the Schrodinger equation 10

(

\ m/3 / * * * \ m/3 Pl2 P23 P31 \ Pl2P2z£zi\ n2

n2 n2

[ n*2 n*2 n*2

PlO P20 P30 /

,

t

^

fm,m(*)

\PlO P20 P30 J

(72)

up to an arbitrary function / m , m (~2^) of one complex variable x being the anharmonic ratio of four coordinates x = ^ ^ . (73) PlO PZ2

Owing to the Bose symmetry of the Odderon wave function, fm,fh(~&) is transformed simple under the substitutions x —> 1 — x, x —>• l/x 16 . The Odderon wave function tpm,m(Pi]) at i f = 0 can be written as /

\ m—1

/

*

\

M

(-^V )

\Pl2P3lJ

\Pl2p3lJ

1>mM) = (- -)

m—1

*».*(*) , - = — , (74) P32

where

XmM ]

^

=

y 2,\x-zf

{•*=*))

l ^ r ^ y J •(75)

137

In fact this function is proportional to XmM^)

fi-m,i-m(~^)'-

~ (1(1 - ^ ) ) 2 ( m " 1 ) / 3 (**(1 - X * ) ) 2 ^ - 1 ' / 3 / 1 _ m , 1 - S i ( ^ ) , (76)

which is a certain realization of the linear dependence between two representations (m,fh) and (1 — m, 1 — m). The corresponding reality property for the Mobius group representations can be presented in a form of the integral relation XmM^)

=f ^ (

X

- Z)2m~2

(** - Z*)2™~2 X l - r M - A ( ^ )

(77)

for an appropriate choice of phases for the functions Xm,fh a n d Xi-m,i-mThe duality equation for Xm,ih(~^) can be written in the pseudodifferential form 16 : | z ( l - z)\2 {idf~m

(id*)2'™

^-m.i-inCt)

= |Am,m| ( < / > l - m , l - m ( ^ ) ) * ,

(78) where ip^l-rnC?)

= (Z(l - Z)f~m

(**(1 - Z*))1-*

Xm,mC*)

•

(79)

The normalization condition for the wave function

llWmll2 = / , .,, J

X

, | 2 \
(80)

\x(l — X)\

is compatible with the duality symmetry. For the holomorphic factors ip(m\x) the duality equations have the simple form 16 :

am^m\x)

= A V 1 _ m ) ( z ) , ai-^-^ix)

= Xl~m^m\x),

(81)

where am = x(l-x)p1+m

.

(82)

If we consider p as a coordinate and x —1/2 as a momentum, the duality equation for the most important case m = 1/2 can be reduced to the Schrodinger equation with the potential V(p) = y/\p~3^2 16 . For each eigenvalue A there are three independent solutions (p\m' (x, A) of the third-order ordinary differential equation corresponding to the diagonalization of the operator A 10 -ix(l

- x) (x(l - x)d2 + (2 - m) ((1 - 2x)d - l + m))dip = \p.

(83)

138

In the region x -> 0 they can be chosen as follows 17 oo

m)

4

(x,X)=J2

^

(A) x" > ^

W

=1

-

(84)

oo

m

(x, A) = Y, ^

W ^ + V^ & A) ln X > flim) = 0 '

( 85 )

fc=0 oo

^W^^cgjA)***'",

4m)(A) = l.

(86)

fc=0

Due to the above differential equation the coefficients ak, ck and c^ satisfy certain recurrence relations. From the single-valuedness condition near ~# = 0, we obtain for the total wave function the following representation:

cl(vimHx,X)lpl™\x\X^+-X)

. (87)

The complex coefficients ci,C2 and the eigenvalues A are fixed from the conditions of the single-valuedness of / m ,m(pi\ P~t, ~p$\ ~P&) at p^ = pf (i = 1,2) and the Bose symmetry 1 7 . With the use of the duality equation we have 16 M = |A|.

(88)

Another relation Im—

= Im{m-l+fh-1).

(89)

can be derived 16 if we shall take into account, that the complex conjugated representations y>m<m and (p x _ m i l _„ of the Mobius group are related by the above discussed linear transformation. One can verify from the numerical results of ref. 17 that both relations for c\ and c^ are fulfilled. If we introduce for general n the time-dependent pair hamiltonian hk,k+i (t) in the form hk,k+i{t) =exp(iT(u)t)

hktk+i exp(-iT(u)t)

,

(90)

in the total holomorphic hamiltonian h we can substitute hk,k+i -> hktk+i(t)

(91)

139

due to the commutativity of h and T(u), On the other hand, as a result of the rapid oscillations at t —> oo each pair Hamiltonian hk,k+i(t) is diagonalized in the representation, where the transfer matrix T{u) is diagonal: hk,k+i(oo) = /ib,jfe+i ((72,93,...&,).

(92)

In the case of the Odderon hk,k+i (oo) is a function of the total conformal momentum M2 and of the integral of motion q3 = A which can be written as follows: A = y [M22 , M23] = £ [M23, M*2] = ll [M23, Mf3] .

(93)

In a general case of n reggeized gluons, one can use the Clebsch-Gordan approach, based on the construction of common eigenfunctions of the total momentum 52 with the eigenvalue m(m — 1) and a set < Mk \ of the commuting sub-momenta, to find all operators M | fc+1 in the corresponding representation. However to diagonalize h we should perform an unitary transformation to the representation, where T(u) is diagonal. The kernel of this integral transformation satisfies some recurrence relations (see 1 6 ). 5. Odderon Hamiltonian and integrals of motion One can present the holomorphic Hamiltonian for n reggeized gluons in the form explicitly invariant under the Mobius transformations 16

" = t (108 ( ^^ £?[ \

Plk+l

\Pk+l,0 Pk+l,k+2

dk)+log(Pk^°Pl^ J

\Pk-l,0 Pk-l,k-2

O-2W) J

J (94)

by introducing the coordinate po of the composite state. Let us consider in more details the Odderon case. Using for its wave function the conformal anzatz fm(Pl,P2,P3-,P0) = ( — 1 ¥>m(x), X = Pl2P30 , (95) VP20P30 / P10P32 one can obtain the following Hamiltonian for the function
log (d + ^ j

+ log (x2(d + j - ^ - ) ) +

+ log ((1 - x)2 (d - | ) ) + log (d-~)-

(96)

140

It is convenient to introduce the logarithmic derivative P = xd as a new momentum. With the use of the relations of the type log(9) = - log(x) + 9{-xd),

log(a;2a) = log(d) + 21og(ar) - ^ ,

and expanding the functions in a series over x, one can obtain the Odderon hamiltonian in the normal order 16 : h

°°

- = - log(z) + V(l -P)

+ V-(--P) + <M™ - P) - 3^(1) + £

xk

fk(P), (97)

yhere

™--h\{rtt + -Fr^±TTit=0

Here tW

(-!)*-« T(m + t) ((t -k)(m

+ t)+m

fc/2)

jfer(m-ife + t + i)r(t + i)r(jb-t + i) '

l

j

Because h and P< = iA commute each with another, h is a function of P . In particular, for large B this function should have the form: ,

oo

_=log(B) + 3 7 + ^ ^ .

(100)

r=l

"

The first two terms of this asymptotic expansion were calculated in 10 . The series is constructed in inverse powers of P 2 , because h should be invariant under all modular transformations, including the inversion x —> 1/x under which B changes its sign. The same functional relation should be valid for the eigenvalues e/2 and /i = i A of these operators. For large /i it is convenient to consider the corresponding eigenvalue equations in the P representation, where x is the shift operator z = exp(-—),

(101)

after extracting from eigenfunctions of B and h the common factor ¥>m(P) = r ( - P ) r ( l - P) T(m - P) exp(iTrP) $ m ( P ) . The function $ m ( P ) can be expanded in series over l//i oo

n=0

(102)

<98

141

where the coefficients $J^ (P) turn out to be the polynomials of order An satisfying the recurrence relation: *nm(P)

=

p

ST(k -l)(k-l-m)

((k - m) ^ \ k

- 1) + (k - 2) ^Zi(k

- 1 - m)) -

Jfe=l

-

m

-^(fc-l)(A;-l-m)((/c-m)$r1^-l) + (fc-2)$^(fc-l-m)), fe=i

(104) valid due to the duality equations written below after the substitution x fi —> x for a definite choice of the phase of $ m ( P ) $ i _ m ( P + 1 - m) - - P (P - 1) (P - m) * i _ m ( P - m) = $ m ( P ) , M $ m ( P + m) - - P (P - 1) (P + m - 1) $ m ( P + m - 1) = * i _ m ( P ) . (105) Indeed, we obtain after changing the argument P —• P — m in the second equation and adding it with the first one $m(P)-$m(P-l) = - (P - 1) (P - 1 - m) ((P - m)$m(P

- 1) + (P - 2) $ ! _ m ( P - m - 1)),

which leads to the above recurrence relation. Note that the summation constants $£j(0) in this recurrence relation have the anti-symmetry property *m(0) = - * r _ r o ( 0 ) ,

(106)

which guarantees the fulfilment of the relation

*s,M = *r_m(o)

(lor)

being a consequence of the duality relation. On the other hand, taking into account that the last relation is correct and rm = $™_TO(0) — 3>m(0) i s an anti-symmetric function, we can chose $ ^ ( 0 ) = —rm/2 because adding a symmetric contribution will redefine the initial condition 3>^(P) = 1.

142

The most general solution of the duality equation is our function $ TO (P) multiplied by an arbitrary symmetric constant. With the use of the recurrence relation in particular we have $ m ( l ) = $m(0). The Odderon energy can be expressed in terms of $m(P)

as follows

| = logOu) + 3 7 + -gp log $ m ( P ) p

+

*™( ) f^^kfk(P-k)*l(P-k)f[(P-r)(P-r

+ l)(P-r-m

r=1

+ l) (108)

and this expression does not depend on P due to the commutativity of h and B. Because for P —> 1 we have

and $ m ( l ) = $ m (0), one can obtain the more simple expression for energy

It is possible to express e through the values of the function <J>TO(P) in other integer points s

^=log(/,) + 3 7 +
f^^fk(s-k)*m{s-k)l[(s-r)(s-r

+ l)(s-r-rn

+ l).

This representation is equivalent to previous one due to the reccurence relations for <5?m,i-m(-P) following from the duality equation. One can fix 3>(P) at some P in an accordance with the duality relation without a loss of the generacy $ m ( l + m) = * m ( m ) = * m ( l ) = $ m (0) - 1.

(110)

143

For other integer arguments of $TO we have the recurrence relations following from the eigenfunction equation for the integral of motion ( 1

^(2)=(l

+

-

m )

( 2 M

-

m )

),^(

S +

l)^

1 + fc^> (2s - m)) *m(.) - '('-ms-mns-m-1)

*ra(2 +

_

m ) = f 1 + (l±f0(2 ± f0

(

f e -L 771)5

\

1+ — (2s + m) 1 $ m ( s + m) _(s + m)(s + m - l ) V ( s - l ) m _ /z2 The solution of these equations is a polynomial in /x _1 and m 2 ( * - l ) - l 2k k=0

1=0

We can calculate also subsequently the derivatives in these points defining $^(l)=e(m).

(Ill)

Namely, one obtains

W j u " - * - ' ) . , , ) ^ (112) and ^ ( s + 1) = ( l +S-^~^(2s - m ) ) ^ ( s ) s(s-l)2(s-m)2(s-m-n M

2

, , ,. 1 $ ^ ( s - 1) + - (s(s - m) (2s - m))' $ m ( s ) M

- ^ 2 (s(s - l) 2 (s - m)2(s -m- 1))' $ m ( s - 1). (113) M

The parameter e(m) is fixed from the condition, that the duality equation for <£m(s) is valid at P —> oo. This requirement can be formulated in a more simple way if one returns to the initial definition of the eigenfunction *m(P) =

(pm(P)exp(iTrP)

r(-p)r(i - p)T(m - p)np

144

and presents m{P) will be generally different. Because the Odderon wave function constructed as a bilinear combination of these solutions in the holomorphic and anti-holomorphic subspaces should have a definite total energy, the quantization of /i should arise as a result of the coincidence of the holomorphic energies for different solutions similar to the case of the Baxter-Sklyanin approach considered below. By solving the recurrence relation for $ ^ ( P ) and putting the result in the above expression for the energy, we obtain the following asymptotic expansion for e/2 16 :

| = >°sM + * + ( i ! + liO" " V>? ~ > ( 4185 V 2050048

2151 _U2]2 49280^ ' '

+

" W)

JS

\ _1 / 965925 \ J_ '") ^ + ^37044224 + '") n6 + '" ' (114)

This expansion can be used with a certain accuracy even for the smallest eigenvalue \x = 0.20526, corresponding to the ground-state energy e = 0.49434 17 . For the first excited state with the same conformal weight m = 1/2, where e = 5.16930 and \i = 2.34392 [9], the energy can be calculated from the above asymptotic series with a good precision. The analytic approach, developed in this section, should be compared with the method based on the Baxter equation 17 . One can derive from above formulas also the representation for the Odderon hamiltonian in the two-dimensional space It: IE = h + h* = 12 7 + ln(|:r|4 |d| 2 ) + lnfll - x\A \df) + (x - l ) m (x* - 1)™ (ln(|d| 2 ) + ln(|x| 4 |5| 2 )) (x - l ) " " 1 (x* - 1)"™+ {-x)m

{-x'T

(ln(|l - x\A \d\2) + ln(|9| 2 ) ) {-x)~m

(-i*)"" .

(115)

The logarithms in this expression can be presented as integral operators with the use of the relation

(116)

145

This representation can be used to find the eigenvalue of the Hamiltonian for the eigenfunction of the integrals of motion B and B* with their vanishing eigenvalues p = p* = 0:

The corresponding wave function fm,m(p~t, ~p~%, pt; ~p&) is invariant under the cyclic permutation of coordinates pt ~* P~$ -> pt -> pt but it is symmetric under the permutations ~pt -H- ~p$ only for even value of the conformal spin n = fh — m, where the norm HvJm.mlli. is divergent due to the singularities at x = 0, 1, co. It is the reason, why the solution exists only for the case fh-m

= 2k + 1, fc = 0 , ± l , ± 2 , . . . .

(118)

Owing to the Bose symmetry of the wave function, this state corresponds to the /-coupling and has the positive charge-parity C. It could be responsible for the small-re behaviour of the structure function gi{x). Using the above representation for H, we obtain {•#),

(119)

v

where E m ^ is the corresponding eigenvalue for the Pomeron Hamiltonian

EU = & + 4

(120)

and ^

= V(l-m)+V(m)-2
(121)

p

The minimal value of E m ^ is obtained at fh — m = ±1 and corresponds to u> = 0. For the case of odd n = fh — m, the norm \
:

37T \x{\

= Re

^(m)

+

^

_

m) +

^m^

+

^ j _m^_

4^(1^

_

— X)\

(122) Note, however, that the norm \ip\2 is divergent. It is possible, that the divergency disappears for a more general solution with a non-vanishing value of A. Using the duality transformation it is possible to obtain from above function
146

6. Baxter-Sklyanin representation Thus, the problem of finding solutions of the Schrodinger equation for the reggeized gluon interaction is reduced to the search of a representation for the monodromy matrix satisfying the Yang-Baxter bilinear relations 12 . It is convenient to work in the conjugated space 14 , where the monodromy matrix is parametrized as follows,

F(U) = Z B ( « ) . . . Z 1 ( „ ) = ( ^ W ) )

(123)

where Lk (u) is given below

Lk(u)=(U

+ Pk Pk0 2

V PkPkO

u

~Pkpl° V

(124)

- PkPkO )

The pseudo-vacuum state annihilated by the operators C(u) and C* (u) has the form 14 ¥0\pt,pt,...,pt;p^)

= f[~^.

(125)

To construct the n-reggeon states with physical values of conformal weights m, rh in the framework of the Bethe Ansatz one can use the BaxterSklyanin approach 13 - 15 . To begin with, one should introduce the Baxter function satisfying the equation (see 14 21 22 ) A
A(A; jJ) = ^ H )

f c

Hk A""* , Mo = 2, /ii = 0, /x2 = m(m - 1 ) . (127)

Here we assume in accordance with ref. 2 1 , that the eigenvalues fit — ik Qk of integrals of motion are real. The eigenfunctions of the holomorphic Schrodinger equation can be expressed through the Baxter function Q(A) using the Sklyanin Ansatz 15.

f(pi,P2,-,pn;Po)

= Q(Ai; m,ju)Q(A 2 ; m,/z)...Q(A„_i; m,/2)* ( 0 ) , (128)

147

where Ar are the operator zeroes of the matrix element B{u) of the monodromy matrix: n —1

n

B(u) = - P JJ (« - Ar), P = Y,Pkr=l

(129)

k=l

In ref. 21 we performed the unitary transformation of the wave function for the composite state of n reggeized gluons from the coordinate representation to the Baxter-Sklyanin representation in which the operators Ar are diagonal 21 (see also 2 2 ) . As a consequence of the single-valuedness condition for its kernel the arguments of the Baxter functions Q(A) and Q(A*) in the holomorphic and anti-holomorphic sub-spaces are quantized (see 21 2 2 ) : iV \ = a + i—

N ,

\*=<j-i-

,

(130)

where a and N are real and integer numbers, respectively. In ref.21 a general method of solving the Baxter equation for the n-Reggeon composite state was proposed and the wave functions and intercepts of the composite states of three and four reggeons were constructed. It turns out 21 , that there is a set of independent Baxter functions Q^ (t = 0,1, ...,n — 1) having multiple poles simultaneously in the upper and lower half-A planes in the points A = ik (k = 0, ± 1 , ±2,...). Using all these functions one can construct the normalizable total Baxter function Qm,m,p ( A J without poles at a = 0 21

Qm,rh,n (t) = Y,c^Q(t)

(A; m ' $ ®{l) (A*; " i > / ? )

(131)

by adjusting for this purpose the coefficients Ct,iThe total energy -Em,m can be expressed in terms of the Baxter function (see ref. 21 )

£=

\, 1 ^^^ ln [( A -^" 1 ( A *-^" 1 l A | 2 "^---P)](132)

Since the function QTOj ^ p ( A J is a bilinear combination of the Baxter functions QW(X) and Q^(X*) (t,l = 1,2, ...,n), the holomorphic energies for all solutions Q^> should be the same. This leads to a quantization of the integrals of motion qk 21 •

148

Let us rewrite the Baxter equation for n reggeon composite state in a real form introducing the new variable x = -iX, U(x,p) Q{x,p) = (x + 1)" Q{x + l,ft) + {x- \)n Q(x - 1,/Z),

(133)

where n

Sl(x,fl) = Y/(-l)k»kXn-k

(134)

k=0

and fio = 2 , Mi = 0 , H2 = m(m - 1), assuming that the eigenvalues of the integrals of motion \xu (k > 2) are real numbers. To solve the Baxter equation we introduce a set of the auxiliary functions for r = 1,2, ...,n — 1

fr(x,p) = ]T

HP) (x-iy-1

O/(M)

(=0 L (x-iy

9i{P)

(135)

where the coefficients 5;,...,aj satisfy the recurrent relations obtained by inserting fr instead of Q(x) in the Baxter equation, but with other initial conditions a 0 = 1, &o = — = 90 = 0.

(136)

Note, that all functions fr(x,jl) are expressed in terms of a subset of pole residues ai,...,zi for fn_i(x,p). There are n 'minimal' independent solutions Q^(x,p) (t = 0,1,2,...,n— 1) of the Baxter equation having border poles at positive integer x and (n — 1 — £)-order poles at negative integer x 2 1 t

n-l-t

r=l

(137) where the meromorphic functions fr(x,ft) were defined above and \isr = (—l)r/xr. Such form of the solution is related with the invariance of the Baxter equation under the substitution x —» — x, /7 —• \is. The coefficients C?\p) , C^ n ~ 1 _ t ) (/?) and /3'(/Z) are obtained imposing the validity of the Baxter equation at x —> oo, lim xk Q{i){x,p)

=0

, fc = l , 2 , . . . , n - 2 .

149

This leads to a system of n — 2 linear equations for the coefficients Cy . We normalize Q^(x,(l) by choosing c\t\t) = C^-tt\ji) = \ .

(138)

It is important to notice that three subsequent solutions Q^ for r = 1,2,..., n — 2 are linearly related as follows > > (/I) + 7T cot(7ra)] Q ( r ) ( i , M) = Q ( r + 1 ) (x, M) + a ( r ) (/2) Q ^ ^ ( i , P). (139) Indeed, the left and right-hand sides satisfy the Baxter equation everywhere including x —> oo and have the same singularities. Therefore due to the uniqueness of the 'minimal' solutions the quantity 7r cot(7ra;) Q(r\x, ft) can be expressed as a linear combination of Q^r~1^(x,jl), Q^(x,p) and Q( r + 1 )(x,/i). Furthermore, the coefficient in front of Q^T+l\x,fl) is chosen to be 1 taking into account our normalization of Q^(x,jT). The Baxter function in the total x-space is a bilinear combination of holomorphic and anti-holomorphic functions Q^r\ Therefore the holomorphic energy expressed in terms of the residues a^ = \,bo,a\,b\ of the poles closest to zero e

= h. + n = bo _ fhizL

(140)

should be the same for all solutions e^0^ = e^1) = ... = e^nh It leads to a quantization of the integrals of motion fik and the energy E 21 . The total energy of the composite state of n-Reggeons is the sum of the holomorphic and anti-holomorphic energies Em,m = Cm(P) + efh(fiS*) • It is valid for the wave function 4>m,m satisfying the Schrodinger equation in the Baxter-Sklyanin representation in the limit A, A* —> i 21 . We can obtain the analogous expression Em,m = em(Ms) + efnift*) by taking instead another limit A, A* —> —i. These two expressions for energies were derived from the Schrodinger equation with the hermitian hamiltonian 21 . Therefore they should coincide for its eigenfunction ^m,ihThis is possible only if the following property is fulfilled for the quantized values of /x: £m{P) + £ m ( £ S * ) = em(A«S) + £ « ( # * ) ,

150

It gives an additional constraint on the spectrum of the integrals of motion. One of the possible solutions of this constraint is that jl should be real or pure imaginary. Note, however, that providing, that the wave function Q(x) does not contain all possible bilinear combinations of the Baxter functions Q(r> and Q^s>*, the quantization conditions can be not so restrictive. 7. Intercepts of reggeons and anomalous dimensions of quasi-partonic operators The <32-dependence of the inclusive probabilities ni(x,\nQ2) to have a parton i with the momentum fraction x inside a hadron with the large momentum |"jf | —• oo can be found from the DGLAP evolution equation 3 . The eigenvalues of its integral kernels describing the inclusive parton transitions i —> k coincide with the matrix elements 7** (a) of the anomalous dimension matrix for the twist-2 operators 0 J ' with the Lorentz spins J = 2,3 For example, in the case of the pure Yang-Mills theory with the gauge group SU(NC) we have only one multiplicatively renormalized operator. Note, that in N = 4 supersymmetric gauge theory 20 there is one supermultiplet of such operators 19 . Its anomalous dimension is singular at the non-physical point u> = j — 1 —>• 0. In this limit one can calculate the anomalous dimension in all orders of perturbation theory 5

-*"(D ( ^ Y

7. = ^ •KUJ

\

+

...

(141)

ITU J

from the eigenvalue for the kernel of the BFKL equation in LLA 1 at n = 0: (T N

LOBFKL =

P*(l)

- *(7)

- *(1 - 7)] •

(142)

7T

One can find from the BFKL equation also anomalous dimensions of higher twist operators by solving the eigenvalue equation near other singular points 7 = — k (k = 1,2,...). But a more important problem is the calculation of the anomalous dimensions for the so called quasipartonic operators (see ref. 27 ) constructed from several gluonic fields and responsible for the unitarization of structure functions at high energies. The simplest operator of such type is the product of the twist-2 gluon operators. In the limit Nc -> 00 this operator is multiplicatively renormalized 23 . Let us consider now the high energy asymptotics of irreducible Feynman diagrams in which each of n reggeized gluons at iVc —• 00 interacts only with two neighbours. In the Born approximation the corresponding Green

151

function is a product of free gluon Green functions n " = i m \Pr ~ Pr\ • For small coupling constants as the full dimension for the operator related to the composite state of n reggeized gluons is approximately equal to the position of the pole (m + fh)/2 « n/2 in the eigenvalue of the Schrodinger equation for n regeized gluons w(m, m; fa,..., fxn) (see 21 ) asNc 2

2

'

w

(143)

Here 7^"^ is the anomalous dimension. The first relation is in an agreement with the result of calculations the anomalous dimension of diagrams with the i-channel exchange of several BFKL Pomerons 23 . It can be also obtained from the equation for matrix elements of quasi-partonic operators written with a double-logarithmic accuracy 24 . In particular, for the Odderon one can show that c^ = 0 according to an unpublished result of M. Ryskin and A. Shuvaev. Their result was confirmed by solving the Baxter equation and finding a pole singularity near m+m = 2 (instead of 3/2 as it could be expected from general formulas) 21 . For n = 4 in accordance with the above relation a pole singularity was found near m+m = 2 (see the discussion below). Moreover, similar to the case of the BFKL Pomeron the anomalous dimensions 73 and 74 were calculated for arbitrary a/to (see 2 1 ) , which is important for finding multi-Reggeon contributions to the deep-inelastic processes at small Bjorken's variable x. The quantity to/a was computed as a function of m = fh for the Odderon and for the fourReggeon state 21 . The holomorphic energies are expressed in terms of the ratio of residues for the single and double poles of the Baxter functions at a; = 1. These energies should be the same for the independent Baxter functions, which leads to the quantization of fi for given m 21 . One can find for m = fh = 1/2 the first roots numerically (cf. 21 ) /ii = 0.205257506 . . .

,

fi2 = 2.3439211...

,

fi3 = 8.32635 . . . (144)

with the corresponding energies E1 = 0.49434...

,

E2 = 5.16930...

,

E3 = 7.70234... .

(145)

The eigenvalues were computed as functions of m for 0 < m < 1 (see ). The energy decreases from E = E\ at m = 1/2 in a monotonic way. Only m = 0, 1 and | are physical values. For other m the curve describes the behaviour of the anomalous dimension for the corresponding 21

152

high-twist operator. The energy vanishes at m = 0,1, which follows from the expression given in ref.21. E(m, M = 0) =

. 7 . + ip(m) + i[)(l - m) - 2-0(1). sm(7rm)

Note, that E(m, fi ~ 0) describes an eigenvalue for which the function Q^ does not enter in the bilinear combination of the total wave function Qm> „ iM and therefore here the general method of quantization does not work. We obtain numerically at m —> 0 E(m) = 2.152 m - 2.754 m 2 + ..., m (m) = 0.375 A/TO - 0.0228 m+ .... (146) The state with m = 1 and m = 0 (or viceversa) is therefore the ground state of the Odderon corresponding to \n\ = 1. It has a vanishing energy for v —> 0 and is situated below the eigenstates with m = fh = 1/2. Note, that generally this solution is different from that found in ref. 18 because for it fi is non-zero. We continued the first eigenstate with n = 2 for m > 1 (see 2 1 ). The energy proceeds to decrease. The eigenstate with \n\ — 2 is absent on this trajectory because fi is pure imaginary in this interval and vanishes only at m = 1 and m — 2. Near m = 2 the energy tends to = oo while \x vanishes 2 E{m) = — — + 1 + 2 -

TO

, i/i = 2-m-

3 - (m - 2 ) 2 .

(147)

But for physical values m = l/2+iv+3/2, m = 1/2-M^ — 3/2 corresponding to \n\ = 3 and v -> 0 the total energy is finite E = 2. Thus, we obtain for the anomalous dimension at 7 = 2 — TO—>0 and

u » aNc TTU)

V 7TW /

V 7TW /

Let us consider now the Baxter equation for the Quarteton (four Reggeons state). A new integral of motion ^ = q^ appears here. The eigenvalues fi and g4 are assumed to be real, which is compatible with a single-valuedness of the wave function in the p-space. Following the general method presented in 21 we seek solutions of the Baxter equation for the Quarteton as a series of poles. The recurrence relations for the residues of the poles are obtained. Then the validity of the Baxter equation at infinity

153

is imposed, which gives further linear constraints on the pole residues. In such way three independent solutions Q^ of the Baxter equation was obtain. The energy of the four Reggeon state is expressed through the ratio of the residues of the poles of the Baxter functions at x = 1 (see 21 ) and due to our quantization rule is equal for the different solutions. The above equations for m = m = 1/2 numerically H = 0, q4= 0.1535892 , E = -1.34832 . H = 0.73833, qA = -0.3703 , E = 2.34105. One finds for the first eigenvalue with m = 0, rh = 1 corresponding to |n| = l // = 0 , q4 = 0.12167, E = -2.0799 . The ground state of the Quarteton with |n| = 1 has m = 0, m = 1. Its energy is lower than the energy E = —1.34832 of the state with m = fa = | . We have followed the first eigenvalue as a function of m for 0 < m < ^ (see 2 1 ) . Contrary to the Odderon case, the energy eigenvalue does not vanish for m = 0. The energy decreases with m for 0 < m < | and takes the value E = -2.0799 at m = 0. For m = 2 the lowest energy state of four Reggeons goes to — oo and 54 vanishes [73 is zero for all m in this state]. One obtains the following behaviour at m —> 2 E = —^— + 2 + 2 - m + ..., <74 = \{m - 2) 2 + .... m —I 4

(148)

Thanks to the m o 1 — m symmetry we have for m —>• — 1 4 £=

- + 3 + m + .... m+1 Now in order to compute the energy of the state with conformal spin n = 3 v —> 0 is used as a regulator. One obtains in this way, E = Ei (m) + JS7i (m) = 4 + 0(i/ 2 ) We can construct the anomalous dimension for 7 = 2 — m —> 0

7= 4

^+8 7TW

-)

+....

\ TTU J

The state with m — 3/2 (corresponding to n = 2, 1/ = 0) can be considered as a physical ground state for the Quarteton because for it the

154

eigenvalue of 94 is real. It has a large negative energy E = —5.863 lower then the energy E — —5.545 of the BFKL Pomeron constructed from two reggeized gluons. But for proving that this state is a physical ground state one should construct a bilinear combination of the corresponding Baxter functions to verify the normalizability of the corresponding solution. A large intercept for the state with the conformal spin 2 may lead to such unphysical results as negative cross sections. But it is known that the unitarization of scattering amplitudes is not solved within a framework where the number of Reggeons is fixed. 8. BFKL and D G L A P equations in N = 4 supersymmetric model It is possible to calculate next-to-leading corrections to the BFKL and DGLAP equations in supersymmetric gauge theories 25 26 . It is remarkable, that the eigenvalue of the BFKL kernel in the next-to-leading approximation is an analytic function of the conformal spin |n| and has the property of the hermitial separability 25 . It gives a possibility to continue it analytically to negative \n\ and to find the anomalous dimensions of the twist-2 operators in accordance with their direct calculation from the DGLAP equation 25 26 . The anomalous dimensions are obtained by an integer shift of the argument of an universal function QU) = ~S1{j)

(149)

+ 16Si(j)S 2 (j) + 8S3(j) -8S3(j) K(j) = j ( ^

+ 16 5 l i 2 (j)

+ S2(j) + S 2 (j))

5

*O-) = E £ , i=l

&,z(i) = £ ^ ( 0 -

(150)

5

*O-) = E ^ ,

(151)

t=l

(152)

i=l

In the leading logarithmic approximation the generalized DGLAP equations for the anomalous dimensions of the quasi-partonic operators 27 turns out to be integrable for the N = 4 supersymmetric theory 19 . Recently there was a great progress in the investigation of the N=4 SYM theory in a framework of the AdS/CFT correspondence 28 where the strong-coupling limit asNc -» 00 is described by a classical supergravity

155

in the anti-de Sitter space AdSs x S5. In particular, a very interesting prediction 29 was obtained for the large-j behaviour of the anomalous dimension for twist-2 operators 7(i)

= a(*)lnj,* = ^ 30

in the strong coupling regime (see also corrections): asNc\1/2

a

. . . „

+1+0

}™ = -{-^r)

(153) 31

and

for asymptotic

(Va.JV^-1/2N

{{-^r)

)•

^

Note, that in our normalization j(j) contains the extra factor —1/2 in comparison with that in Ref. 29 . On the other hand, all anomalous dimensions ji(j) and 7i(j) coincide at large j and our results for j(j) 25 26 allow one to find two first terms of the small-a s expansion of the coefficient a z-+0

7T

V

2

12

7 V 7T J

Note, that this result is obtained in the dimensional reduction scheme, but the coupling constant was taken in MS-scheme. To go from this expansion to the strong coupling regime we perform a resummation of the perturbative result using the method similar to the Pade approximation and taking into account, that for large iVc the perturbation series has a finite radius of convergency. Namely, we present a as a solution of the simple algebraic equation —

= - a + ^ — - - j a

.

(155)

Using this equation the following large-a s behaviour of a is obtained: <*sNc\1/2 , n„„ln , „ ffasNc^-1/2" a « - 1 . 1 6 3 2 I-=—M +0.67647 + 0 11-=—^]

(156)

in a rather good agreement with the above results based on the AdS/CFT correspondence. Note, that if we write for a the more general equation 2n

the coefficients CT for n > 3 can be chosen in such way to include all known information about a.

156

Further, for j —> 2 due to the energy-momentum conservation j(j) = (j - 2 ) 7 ' ( 2 ) , where the coefficient 7'(2) can be calculated from our results in two first orders of the perturbation theory 2 6 . On the other hand provided t h a t 7'(2) at large z = asNc/i: is proportional to z one can reproduce the intercept of the Pomeron j = 2 — O (1/2) obtained in the approach based on the A d S / C F T correspondence 3 2 3 3 . It should be taken into account 26 , t h a t in this limit 7 = 1/2 + iv 4- (j — l ) / 2 -» 1 for the principal series of unitary representations of the Moebius group appearing in the B F K L equation 2 . One can a t t e m p t o calculate the intercept of the Pomeron using also its perturbative expansion j — 1 = c\z + c^z2 with the coefficients ci,2 obtained in the last paper of Ref. 2 . After the P a d e resummation j — 1 = c\z/{\ — cxzjc-i) we obtain in the strong coupling regime j close to 2 in an approximate aggreement with the A d S / C F T estimate (see 3 2 33 ) . Note , however, t h a t in the upper orders of the p e r t u r b a t i o n theory the B F K L equation should be modified by including the contributions from multi-gluon components of the Pomeron wave function. I thank A.P. Bukhvostov, A.V. Kotikov, V.N. Velizhanin and H. de Vega for helpful discussions. References 1. L.N. Lipatov, Sov. J. Nucl. Phys. 23 (1976) 642; V.S. Fadin, E.A. Kuraev and L.N. Lipatov, Phys. Lett. B60 (1975) 50; Sov. Phys. J E T P 44 (1976) 443; 45 (1977) 199; Ya.Ya. Balitsky and L.N. Lipatov, Sov. J. Nucl. Phys. 28 (1978) 822. 2. V.S. Fadin, L.N. Lipatov, Phys. Lett. B429 (1998) 127; G. Camici and M. Ciafaloni, Phys. Lett. B430 (1998) 349.; A.V. Kotikov, L. N. Lipatov, Nucl. Phys. B582 (2000). 3. V.N. Gribov and L.N. Lipatov, Sov. J. Nucl. Phys. 18 (1972) 438, 675; L.N. Lipatov, Sov. J. Nucl. Phys. 20 (1975) 93; G. Altarelli and G. Parisi, Nucl. Phys. B126 (1977) 298; Yu.L. Dokshitzer, Sov. Phys. JETP 46 (1977) 641. 4. L. N. Lipatov, Pomeron in QCD, in "Perturbative QCD", ed. by A. N. Mueller, World Scientific, 1989; Small-x physics in the perturbative QCD, Physics Reports, 286 (1997) 132. 5. L.N. Lipatov, Sov. Phys. J E T P 63 (1968) 904. 6. S.J. Brodsky, V.S. Fadin, V.T. Kim, L.N. Lipatov, G.V. Pivovarov, JETP Letters 70 (1999) 155; 76 (2002) 306. 7. L.N. Lipatov, Nucl. Phys. B452 (1995) 369. 8. J.Bartels, Nucl. Phys. B175 (1980) 365; J. Kwiecinski and M. Prascalowicz, Phys. Lett. B 9 4 (1980) 413. 9. L. Lukaszuk, B. Nicolescu, Lett. Nuov. Cim. 8 (1973) 405; P. Gauron, L. Lipatov, B. Nicolescu, Phys. Lett. B 3 0 4 (1993) 334.

157 10. L.N. Lipatov, Phys. Lett. B251 (1990) 284; B309 (1993) 394. 11. A.A. Belavin, A.B. Zamolodchikov, A.M. Polyakov, Nucl. Phys. B241 (1984) 333. 12. L.N. Lipatov, hep-th/9311037, Padua preprint DFPD/93/TH/70, unpublished. 13. R.J. Baxter, Exactly Solved Models in Statistical Mechanics, (Academic Press, New York, 1982); V.O. Tarasov, L.A. Takhtajan and L.D. Faddeev, Theor. Math. Phys. 57 (1983) 163. 14. L.N. Lipatov, Sov. Phys. J E T P Lett. 59 (1994) 571; L.D. Faddeev and G.P. Korchemsky, Phys. Lett. B342 (1995) 311. 15. E. K. Sklyanin, Lect. Notes in Phys. 226, Springer-Verlag, Berlin, (1985). 16. L. N. Lipatov, Nucl. Phys. B548 (1999) 328. 17. R. Janik and J. Wosiek, Phys. Rev. Lett. 79 (1997) 2935; 82 (1999) 1092. 18. J. Bartels, L. N. Lipatov, G. P. Vacca, Phys.Lett. B477, 178 (2000). 19. L.N. Lipatov, Perspectives in Hadronics Physics, Proceedings of the ICTP conference (World Scientific, Singapore, 1997). 20. J.Maldacena, Adv. Theor. Math. Phys. 2 (1998) 231. 21. H. J. de Vega and L. N. Lipatov, Phys. Rev. D64, 114019 (2001); D66, 074013-1 (2002). 22. A. Derkachev, G. Korchemsky, A. Manashov, Nucl. Phys. B617 (2001) 375. 23. L. V. Gribov, E. M. Levin, M. G. Ryskin, Physics Reports, C100 (1983) 1; E. M. Levin, M. G. Ryskin , A. G. Shuvaev, Nucl. Phys. B387 (1992) 589; J. Bartels: Z. Phys. 60 (1993) 471; E. Laenen, E. M. Levin, A. G. Shuvaev, Nucl. Phys. B419 (1994) 39. 24. A. G. Shuvaev, hep-ph/9504341, unpublished. 25. A.V. Kotikov, L.N. Lipatov, hep-ph/0208220. 26. A.V. Kotikov, L.N. Lipatov, V.N. Velizhanin, Phys. Lett. B 557 (2003) 114. 27. A.P. Bukhvostov, G.V. Frolov, E.A. Kuraev, L.N. Lipatov, Nucl. Phys. 258 (1985) 601. 28. J. Maldacena, Adv. Theor. Math. Phys. 2, 231 (1998); Int. J. Theor. Phys. 38, 1113 (1998); S. S. Gubser, I. R. Klebanov and A.M. Polyakov, Phys. Lett. B 428, 105 (1998); E. Witten, Adv. Theor. Math. Phys. 2, 253 (1998). 29. S. S. Gubser, I. R. Klebanov and A.M. Polyakov, Nucl. Phys. B636, 99 (2002). 30. Yu. Makeenko, hep-th/0210256; M. Axenides, E.G. Floratos and A. Kehagies, hep-th/0210091. 31. S. Frolov and A. A. Tseytlin, JHEP 0206, 007 (2002). 32. J. Polchinski, M.J. Strassler, hep-th/0209211. 33. R. C. Brower, C.I. Tan, hep-th/0207144; R.A. Janik and R. Peschanski, Nucl. Phys. B625, 279 (2002).

QCD: CONFINEMENT, H A D R O N STRUCTURE A N D DIS*

Y U . A. S I M O N O V Institute

State Research Center of Theoretical and Experimental Moscow, 117218 Russia

Physics,

The main features of QCD, e.g. confinement, chiral symmetry breaking, Regge trajectories are naturally and economically explained in the framework of the Field Correlator Method (FCM). The same method correctly predicts the spectrum of hybrids and glueballs. When applied to DIS and high-energy scattering it leads to the important role of higher Fock components in the Fock tower of moving hadron, containing primarily gluonic excitations.

To the memory of my first teacher Isaak Yakovlevich Pomeranchuk: The Physicist, The Teacher, The Man. 1. Introduction The QCD is the (only) internally selfconsistent theory, defined by the only scale (parameter the string tension a or AQCD can be used for this purpose), which eventually explains fundamental structure of baryons and other hadrons and by this some 98% of all the mass in our world. The features of QCD are unique in their complexity: confinement, chiral symmetry breaking, string structure of hadrons generating Regge trajectories on one hand demonstrating nonperturbative (NP) interactions, and on another hand the applicability of perturbation theory for large Q and large momentum processes, where NP effects enter as corrections.The most popular theoretical approaches to these phenomena look fragmentary, i.e. magnetic monopoles are used for confinement, instantons for chiral *The partial support of the INTAS grants 00-110 and 00-366 is gratefully acknowledged.

158

159 symmetry breaking and pure perturbative expansions for high energy processes. The situation has improved with the introduction of the QCD sum rules x , where NP contributions are encoded in the form of local condensates. In this talk I shall describe a general method which allows to consider all NP effects on one ground - with the help of nonlocal Field Correlators (FC) 2 . It will be argued that the simplest of these correlators - the quadratic one is dominant and is responsible for confinement, chiral symmetry breaking, and the structure of meson and baryon spectra. When supplemented with the Background Perturbation Theory (BPT) the method allows to treat valence gluons in the confining background 3 . This yields the spectrum of hybrids and glueballs in good agreement with lattice simulations, and the new perturbation series for high energy processes, without IR renormalons and Landau ghost poles. As the new and unexpected element, it will be argued that hybrids play a more fundamental role in DIS and high-energy scattering - as the building blocks of the colliding hadron wave functions. The talk is organized as follows. In section 2 FC are introduced and confinement is related to their properties. In section 3 the Hamiltonian for valence components is written down and properties of meson spectrum are discussed. In section 4 this Hamiltonian is extended to the systems with valence gluons and spectrum of hybrids and glueballs is discussed. In section 5 the Fock towers are introduced for hadrons and the general matrix Hamiltonians is written down. In section 6 the role of higher (hybrid) Fock components in DIS is discussed and the gluon contribution to structure functions and proton spin is emphasized. The last section concludes the general picture of the QCD dynamics with the discussion of the selfconsistent calculation of the FC. 2.

The QCD vacuum structure. Stochastic vs coherent

The basic quantity which defines the vacuum structure in QCD is the field correlator (FC) £> (n) (a;i,...a; n ) =

(FfilUl(x1)^(xi,X2)F^l/2(x2)^(x2,x3)...Flln^(xn)^(xn,xi)), (1)

$(x,y)

= Pexpig

/

AM(z)dzM.

160

The set of FC (1) for n = 2,3,... gives a detailed characteristic of vacuum structure, including field condensates (for coinciding x\ = x2 = ...£„). On general grounds one can distinguish two opposite situations: 1) stochastic vacuum 2) coherent vacuum. In the first case FC form a hierarchy with the dominant lowest term £>(2)(a;i,a;2) = £>'2^(a;i - x2), while higher FC are fast decreasing with n. We shall call this situation the Gaussian Stochastic Approximation (GSA). In the second case all FC are comparable, and expansion of physical amplitudes as the series of FC is impractical. This is the case for the gas/liquid of classical solutions, e.g. of instantons, magnetic monopoles etc. The physical picture behind the situation of nonconverging FC series is that of the coherent lump(s), when all points in the lump are strongly correlated. To understand where belongs the QCD vacuum one can start with the Wilson loop in the representation D of the color group SU(3), WD(C)

= (trD exp(i 5 [ dznAlTM))

(2)

JC

The Stokes theorem and the cluster expansion identity allow to obtain the basic equation, which is used in most applications of the FCM (for more details see 2 ) W(C) =expJ2ij~T

[i>{n)(x1,...xn)d(x»lVl(x1)...der^Vtl(xn).

(3)

n

Here integration is performed over the minimal surface Smin inside the contour C defined in (2) and Dn is the so-called cumulant or the connected correlator, obtained from the FC Eq.(l) by subtracting all disconnected averages. From (3) one easily obtains that the Wilson loop has the arealaw asymptotics, W(C) ~ exp(—aSmin), for any finite number of terms rimax;n < nmax in the exponent (3). The string tension is expressed through D^n\ a = 1 J DW(Xl - x2)d2(Xl

- x2) + 0(D^n),n

> 4) = a2 +
(4)

Eq.(4) has several consequences: 1) confinement appears naturally for n = 2, i.e. in the GSA 2) the lack of confinement can be due to vanishing of all FC, or due to the special cancellation between the cumulants, as it happens for the instanton vacuum 4 , 3) for static quarks in the representation D of the color group SU(3), the string tension a2 is

161

proportional to the quadratic Casimir factor ( the Casimir scaling)

°2D) = T^A^^D

= \(S + ^ + ^+^ + 3,).

However for larger n, n > 4 the Casimir scaling is violated:
+ a2{C$f

+ a3C™ + ...

(6) D

It is remarkable that perturbative interaction of static quarks V^ \r) satisfies the Casimir scaling to the order 0(g6) considered so far 5 , a so the total potential V(D\r) = Vpert(r) + a^r + const is also Casimir scaling, if GSA works well. This picture was tested recently on the lattice 6 and confirmed the Casimir scaling with the accuracy around 1% in the range 0.1 < r < 1.1. fm. The full theoretical understanding of this fundamental fact is still lacking, both for the perturbative part and for the string tension. On the pedestrian level the Casimir scaling and the quadratic (Gaussian) correlator dominance implies that the vacuum is highly stochastic and any quasiclassical objects, like instantons, are strongly suppressed in the real QCD vacuum. The vacuum consists of small white dipoles of the size Tg made of neighboring field strength operators. The smallness of Tg might be an explanation for the Gaussian dominance since higher correlator terms in a are proportional to (FTg )"(Tg) - 1 , where F is the estimate of the average nonperturbative vacuum field, F ~ 500 (MeV) 2 . Lattice calculations of FC have been done repeatedly during last decades, using cooling technic 7 and with less accuracy without cooling 8 . (Recently another approach based on the so-called gluelump states was exploited on the lattice 9 and analytically 10 , which has a direct connection to FC). The basic result of 7 is that FC consists of perturbative part 0(l/xA) at small distances and nonperturbative part 0(exp(—x/T g )) at larger distances with Tg in the range Tg = 0.2 fm (quenched vacuum) and Tg = 0.3 fm ( 2 flavours). Calculations in 8 and 9 ' 10 , as well as sum rule estimates n yield a smaller value, Tg w 0.13 fm to 0.17 fm. This enables us in what follows to take the limit Tg -> 0 keeping a = const « 0.18 GeV 2 , and consider aTg as a small parameter of expansion, aTg
T h e diagrams of the order 0(gB) W.Wetzel (to be published)

violating the Casimir scaling have been found by

(5)

162

3. Hamiltonian for valence components There are two possible approaches to incorporating nonperturbative field correlators in the quark-antiquark (or 3q) dynamics. The first has to deal with the effective nonlocal quark Lagrangian containing field correlators 13 . From this one obtains first-order Dirac-type integro-differential equations for heavy-light mesons 12 ' 14 , light mesons and baryons 15 . These equations contain the effect of chiral symmetry breaking 12 ' 13 which is directly connected to confinement. The second approach is based on the effective Hamiltonian for any gauge-invariant quark-gluon system. In the limit Tg —> 0 this Hamiltonian is simple and local, and in most cases when spin interaction can be considered as a perturbation one obtains results for the spectra in an analytic form, which is transparent. For this reason we choose below the second, Hamiltonian approach 16>17. We start with the exact Fock-Feynman-Schwinger Representation for the qq Green's function (for a review see 1 8 ), taking for simplicity nonzero flavor case /•OO

rOO

G{qq'v) =

Jo

(trrin(mi

ds2(Dz)xy(Dz)xye~ K!-K2

dsi / Jo

- £i)W C T (C)r o u t (m 2 - D2))A

(7)

where K, = J*1 dTi(rrii + \(i^')2), Tint0ut = 1,7s,--- are meson vertices, and Wa(C) is the Wilson loop with spin insertions, taken along the contour C formed by paths (Dz)xy and (Dz)xy, Wa(C) = PFPAexp{ig

[ Jc

Apdzjx

x exp( 5 / ' a^F^dn -g [* a$F^dT2). (8) Jo Jo The last factor in (8) defines the spin interaction of quark and antiquark. The average (W^A in (7) can be computed exactly through field correlators (F(1)...F(TI))A, and keeping only the lowest one,(F(l)F(2)), which yields according to lattice calculation 6 accuracy around 1% 5 , one obtains (Wa{C))A~exp(-hf

ds^(l) J Smin

2

+ 'E

f

dsx„(2) +

J Smin

f a§dT* f ^^]<^(l)^(2)>)-

0)

163

The Gaussian correlator (Ffll/(1)F\(T(2)) = DM„,.\CT(1, 2) can be rewritten identically in terms of two scalar functions D(x) and D\{x) 2 , which have been computed on the lattice 7 to have the exponential form D{x),D\(x) ~ exp(—\x\/Tg with the gluon correlation length Tg w 0.2 fm. As the next step one introduces the einbein variables \ii and v; the first one to transform the proper times SJ,TJ into the actual (Euclidean) times U = z%\ One has 17 2/ii(t0 = ^ ,

dsi{D4z^)xy

/

= constjDfii(ti)(D3z^)xy.

(10)

The variable v enters in the Gaussian representation of the Nambu-Goto form for Smin and its stationary value v§ has the physical meaning of the energy density along the string. In case of several strings, as in the baryon case or the hybrid case, each piece of string has its own parameter i/'). To get rid of the path integration in (7) one can go over to the effective Hamiltonian using the identity Gq?(x,y)

= (x\exp(-HT)\y)

(11)

where T is the evolution parameter corresponding to the hypersurface chosen for the Hamiltonian: it is the hyperplane z± = const in the c m . case 1T. The final form of the c m . Hamiltonian (apart from the spin and perturbative terms to be discussed later) for the qq case is 17 ' 19

i=l

2fH

2J

+

2[/i!

(1 - C)2 + M2C2 + Jo W

- C)M/?)]

^r&+jf^

Here C = (Mi + JQ Pvd0)l{pi + M2 + f0 Pvd[3) and /ij and v((3) are to be found from the stationary point of the Hamiltonian dH0

_

o)

dH0

_

^ru=/.i -°' ^71—<°>-°-

(i3)

Note that Ho contains as input only mi,m,2 and a, where m, are current masses defined at the scale of 1 GeV. The further analysis is simplified by the observation that for L = 0 one finds v^ = or from (13) and Hi = \Jm2 + p 2 , hence Ho becomes the usual Relativistic Quark Model

164

(RQM) Hamiltonian H0(L

= 0) = J2 \Jm2i + P 2 + <"••

(14)

i=l

But Ho is not the whole story, one should take into account 3 additional terms: spin terms in (9) which produce two types of contributions: selfenergy correction 20

*~>-±*gr-

^ = - ' ( - ) , „
and spin-dependent interaction between quark and antiquark Hspin 21 which is entirely described by the field correlators D(x),Di(x), including also the one-gluon exchange part present in D\{x). Finally one should take into account gluon exchange contributions, which can be divided into the Coulomb part Hcoui — — f a " r ' r ', and Hrai including space-like gluon exchanges and perturbative self-energy corrections (we shall systematically omit these corrections since they are small for light quarks to be discussed below). In addition there are gluon contributions which are nondiagonal in number of gluons ng and quarks (till now only the sector ng = 0 was considered) and therefore mixing meson states with hybrids and glueballs 22 ; we call these terms HmiX and refer the reader to 22 and the cited there references for more discussion. Assembling all terms together one has the following total Hamiltonian in the limit of large JVC and small Tg (for more discussion see 2 3 ) : H = H0 + Hseif + Hspin + Hcoui + Hrad + HmiX.

(16)

We start with H0 = HR + Hstring • The eigenvalues M0 of HR can be given with 1% accuracy by 24 3 Ml x, SaL + 47rcr(n + - ) (17) where n is the radial quantum number, n = 0,1,2,... Remarkably M0 « 4^o, and for L = n = 0 one has /*o(0,0) = 0.35 GeV for a = 0.18 GeV 2 , and ^o is fast increasing with growing n and L. This fact partly explains that spin interactions become unimportant beyond L = 0,1,2 since they are proportional to d,T\dri ~ 4 * dt\dtv (see (9) and 2 3 ) . Thus constituent mass (which is actually "constituent energy") ^ 0 is "running". The validity of fio as a socially accepted "constituent mass" is confirmed by its numerical value given above, the spin splittings of light and heavy mesons 25 and

165

by baryon magnetic moments expressed directly through fio, and being in agreement with experimental values 26 . 4. Hamiltonian and bound states of valence gluons We now come to the gluon-containing systems, hybrids and glueballs. Referring the reader to the original papers 27 - 29 one can recapitulate the main results for the spectrum. In both cases the total Hamiltonian has the same form as in (16), however the contribution of corrections differs. For glueballs it was argued in 29 that H0 (12) has the same form, but with vrii = 0 and a -> aa^ = \o while Hseif — 0 due to gauge invariance. Thus one can retain in (16) only two main terms: H = H0 + Hspin while Hcoui was argued to be strongly decreased by loop corrections. The calculation in 29 was done for two-gluon and three-gluon glueball states and results are in surprisingly good agreement with lattice data for both systems (no fitting parameters have been used in 2 9 ) . We now coming to the next topic of this talk: hybrids and their role in hadron dynamics. We start with the hybrid Hamiltonian and spectrum. This topic in the framework of FCM was considered in 27 ' 28 The Hamiltonian H0 for hybrid looks like 23 . 27 - 28 H(h„b)

_ ml

m\

m+^+fig

PJ + Pl

Y ^ , _TA

+

HI

(18) Here p ? , p,j are Jacobi momenta of the 3-body system, Hseif is the same as for meson, while Hspin and Hcoui have different structure 28 . The main feature of the present approach based on the BPTh, is that valence gluon in the hybrid is situated at some arbitrary point on the string connecting quark and antiquark, and the gluon creates a kink on the string so that two pieces of the string move independently (however connected at the point of gluon). This differs strongly from the flux-tube model where hybrid is associated with the string excitation as a whole, but has a strong similarity to the treatment of gluons in the framework of the Lund model 30

Results for light and heavy exotic 1 *" hybrids are also given in 23 and are in agreement with lattice calculations. Typically an additional gluon in the exotic (L = 1) state "weights" 1.2-=-1.5 GeV for light to heavy quarks, while nonexotic gluon (L = 0) brings about 1 GeV to the mass of the total qqg system. Let us now consider the hybrid spectrum in more detail. First

166

of all we use for that 3-body problem the hyperspherical method, which works with accuracy of few percent 31 ' 32 . Then the whole spectrum is classified by the grand angular momentum K = 0,1,2,..., which is actually an arithmetic sum of all partial pair angular momenta in the system qqg. The lowest K = 0 states can be formed from the s-wave qq pair and s-wave valence gluon g, which gives the p + g and 7r + g systems, and vectors imply spin-one particle. In this way one obtains the classification K = 0, (TT + g) - 1+-, (p + g) - (2++, 1++, 0++) K = 1, (TT + (V x g)) - 1—, (p + (V x g)) - (2-+, 1-+, 0-+). The eigenvalues of the Hamiltonian (18) are easily obtained for the light quarks using the hypercentral (lowest K) approximation M(K = 0) = 1.872 GeV M(K = 1) = 2.45 GeV M(K = 2) = 2.90 GeV M{K = 3) = 3.27 GeV. Here the (negative) self-energy part of quarks Hseif is already added to masses. One has also in addition the color Coulomb part Hcoui and spindependent part HSpin, which contribute approximately (Hcoui) ~ —0.2 GeV and for spin-spin interaction approximately 0.08 GeV I — 1 I, where

1+1/ numbers inside brackets refer to 1 + 1 = 0,1,2 respectively from top to bottom. As a result, neglecting rather small contribution from Hrad and HmiX in (16), and the string correction, taking into account the moment of inertia of the string 1T, yielding around (-100 MeV), one has the approximate mass estimates for the lowest mass states of K multiplets: Mlow{K

= 0) s 1.42 GeV

Ml0W(K = 1 ) 3 1.9 GeV Mt0W{K = 2) S 2.45 GeV.

167

The mass value Miow{K = 1) agrees well with the lattice results for the mass of the 1 _ + state 33 . The recent unquenched calculation 34 yields the value which is somewhat lower. One should stress that the hybrid states, which start at the mass around 1.4 GeV, have a high multiplicity which grows exponentially with mass, as well as excited string states in bossonic string theory 35 . This fact has a very important consequence for high-energy processes, where the hybrid excitation is argued to be the dominant physical mechanism.

5. Hamiltonian and Fock states As was mentioned above the QCD Hamiltonian is introduced in correspondence with the chosen hypersurface, which defines internal coordinates {£&} lying inside the hypersurface, and the evolution parameter, perpendicular to it. Two extreme choices are frequently used, 1) the c m . coordinate system with the hypersurface X4 = const., which implies that all hadron constituents have the same (Euclidean) time coordinates £4 = const, i = 1, ...n, 2) the light-cone coordinate system, where the role of x\ and x^' is played by the x+,x+ components, x+ = x ° t p . To describe the structure of the Hamiltonian in general terms we first assume that the bound valence states exist for mesons, glueballs and baryons consisting of minimal number of constituents. To form the Fock tower of states starting with the given valence state, one can add gluons and qq pairs keeping the JPC assignment intact. At this point we make the basic simplifying approximation assuming that the number of colors 7VC is tending to infinity, so that one can do for any physical quantity an expansion in powers of 1/NC. Recent lattice data confirm a good convergence of this expansion for iVc = 3,4,6 and all quantities considered 36 (glueball mass, critical temperature, topological susceptibility etc.). Then the construction of the Fock tower is greatly simplified since any additional qq pair enters with the coefficient \/Nc and any additional white (e.g. glueball) component brings in the coefficient 1/N^. In view of this in the leading order of l/Nc the Fock tower is formed by only creating additional gluons in the system, i.e. by the hybrid excitation of the original (valence) system. Thus all Fock tower consists of the valence component and its hybrid equivalents and each line of this tower is characterized by the number n of added gluons. Then, the internal coordinates {£}„ describe coordinates and polarizations of n gluons in addition to those of valence constituents.

168

We turn now to the Hamiltonian H, assuming it to be either the total QCD Hamiltonian HQQD, or the effective Hamiltonian H(eH\ obtained from HQCD by integrating out short-range degrees of freedom. We shall denote the diagonal elements of H, describing the dynamics of the n-th hybrid excitation of s-th valence state (s = rn{ff},gg,2>g,b{f\f2fz} for mesons, 2-gluon and 3-gluon glueballs and baryons respectively with fa denoting flavour of quarks) as iJ„„. For nondiagonal elements we confine ourselves to the lowest order operators H„n+1 and # „ _ ! n describing creation or annihilation of one additional gluon, viz. H

9<,9 =9J «(x,0)o(x,0))g(x,0)d 3 x

Hg2g =9-fabcI ( V S - 5 „ o » a » a ^ x ,

(19)

(20)

and we disregard for simplicity the terms Hg3g. As it is clear from (19), (20), the first operator refers to the gluon creation from the quark line, while the second refers to the creation of 2 gluons from the gluon line. In what follows we shall be mostly interested in the first operator, which yields dominant contribution at large energies, and physically describes addition of one last cross-piece to the ladder of gluon exchanges between quark lines, while (20) corresponds in the same ladder to the as renormalization graphs, and to the graphs with creation of additional gluon line. The effective Hamiltonian in the one-hadron sector can be written as follows H = #(0) + V

(21)

where H^ is the diagonal matrix of operators, HW = {H&\Hi?,H$,-}

(22)

while V is the sum of operators (19) and (20), creating and annihilating one gluon. In (22) il„„ is the Hamiltonian operator for what we call the "n-hybrid", i.e. a bound state of the system, consisting of n gluons together with the particles of the valence component. In this way the n-hybrid for the valence p-meson is the system consisting of qq plus n gluons " sitting" on the string connecting q and q. Before applying the stationary perturbation theory in V to the Hamiltonian (21), one should have in mind that there are two types of excitations of the ground state valence Fock component: 1) Each of the

169

operators Hnn,n = 0,1,... has infinite amount of excited states, when radial or orbital motion of any degree of freedom is excited, 2) in addition one can add a gluon, which means exciting the string and this excitations due to the operator V transforms the n — th Fock component ipn into The wave equation for the Fock tower ^N{P, £} has the standard form HVN = (jf(°) + V)*N = EN
(23)

or in the integral form VN

= V{°) -G^V^N

(24)

where G^ is diagonal in Fock components,

o 0,(E) =

'

I^' G "™ <E) = ^ ] g f ^

<25)

and vtjy is the eigenfunction of H^°\ £(0)^(0)

=£W$(o)

(26)

and since H^ is diagonal, &N' has only one Fock component, &N' = ipn(P,{^}n), n = 0,1,2,..., and the eigenvalues E^' contain all possible excitation energies of the n-hybrid, with the number n of gluons in the system fixed,

EW=EP(P) = y/p* + Mlw.

(27)

Here {k} denotes the set of quantum numbers of the excited n-hybrid. From (24) one obtains in the standard way corrections to the eigenvalues and eigenfunctions. As a first step one should specify the unperturbed functions *&N , introducing the set of quantum numbers {k} defining the excited hybrid state for each n-hybrid Fock component ipn(P{€})', we shall denote therefore: *i?)=i{*}(P,{f}n),

n = 0,1,2,...

(28)

The set of functions 4>n{k} with all possible n and {k} is a complete set to be used in the expansion of the exact wave-function (Fock tower) \f jy:

*;v=Ec£W-wm{k}

(29)

170

Using the orthonormality condition J ^m{k}1Pri{p}dT

= SmnS{k}{p}

(30)

where dT implies integration over all internal coordinates and summing over all indices, one obtains from (15) an equation for cm{ky and EN, C

n{p}(EN

-

E

n{p})

= ^2 C™{k}Vn{p},m{k} m{fc}

(31)

where we have defined Vn{p},m{k} = J ^n{p}^^m{k}<^-

(32)

Consider now the Fock tower built on the valence component VV{K}> where v can be any integer. For V{K} = 0{0} this valence component corresponds to the unperturbed hadron with minimal number of valence particles. For higher values of V{K} the Fock component T/V{K} corresponds to the hybrid with v gluons which after taking into account the interaction is "dressed up" and acquires all other Fock components, so that the number N in (29) contains the "bare number" V{K} as its part iV = V{K},... (at least for small perturbation V). One can impose on \f jv the orthonormality conclusion

*+* M
/

(33)

m{k}

Expanding now in powers of V, one has C

m{k}

-<W<>{fc}{ K } +cm{k}

ENi„{K})

+ Cm{k}

+

-

= £ < % + E™ + £#> + ...

V^J

(35)

It is easy to see that EN' = 0, while for c^ one obtains from (31) the standard expression JV(l) _ n{P} ~

C

V n{p},u{^} ( 0 ) _ F(0) p

'

,„fi. ^0)

In what follows we shall be interested in the high Fock components, v + I, {k}, obtained by adding I gluons to the valence component v{n}. Using (31) and (34) one obtains W ("M) _ "u+l,{k} ~

V^ 2-*, {ki}...{ki}

Vv+l{k},v+l-l{ki} _E(0) E(0) ^V{K} ^u+l{k}

Vy+l-l{ki},u+l-2{k2} _E(0) ^ { K } -^1^+/ —l{fci}

E(0)

171

Vv+i{k,},v{K}

o(Vl+2)

(37)

Since V is proportional to g, one obtains in (34) the perturbation series in powers of as for cN and hence for *JV (29). One should note that as(Q2) is the background coupling constant, having the property of saturation for positive Q2 3 and the background perturbation series has no Landau ghost pole and is defined in all Euclidean region of Q2. The estimate of the mixing between meson and hybrid was done earlier in the framework of the potential model for the meson in 27 . In 24 the mixing between hybrid, meson and glueball states was calculated in the framework of the present formalism and we shortly summarize the results. One must estimate the matrix element (32) between meson and hybrid wave functions taking the operator V in the form of (19), where the operator of gluon emission at the point (x, 0) can be approximated as

x[exp(«k • x - ifit)elx)cx(k)

+ e<,A)c+(k) e x p ( - i k • x + ifit)]

(38)

Omitting for simplicity all polarization vectors and spin-coupling coefficients which are of the order of unity, one has the matrix element VMh = -jl== [ ^

(r)"^+(0, v)d3r

(39)

M where ^ ( r ) . V'/i( r i) r 2) a r e meson and hybrid wave functions respectively, and in (39) it is taken into account that the gluon is emitted (absorbed) from the quark position. Using realistic Gaussian approximation for the wave functions in (39) one obtains the estimate 24

VMh~g-

0.08 GeV.

(40)

A similar estimate is obtained in 22 for the hybrid-glueball mixing matrix element, while the meson-glueball mixing is second-order in (40). Hence the hybrid admixture coefficient (36) for the meson is VMh

n

=

,,.s

VMh =

(

} ^ ^ ^ ^ and for the ground state low-lying mesons when A M M / I ~ 1 GeV it is small, CMH ~ 0.1 — 0.15, yielding a 1-2% probability. However for higher

172

states in the region MM ^ 1 - 5 GeV, the mass difference i M j j / , of mesons and hybrids with the same quantum numbers can be around 200 MeV, and the mixing becomes extremely important, also for meson-glueball mixing, which can be written as

and VMH ~ VhG- It is clear that the iterative scheme described above can be useful only because hybrid excitation by one additional gluon "costs" around 1 GeV increase in mass, hence the coefficient c% (36) can be small. 6. Hybrid states and DIS As was stressed in the previous section, in the large Nc limit the higher Fock components which are excited by the external current (or incident hadron) are the multihybrid (or n-hybrid), states. It is convenient to consider these states in the light-cone formalism, following the line of derivation and most notations in 37 . Consider the n-hybrid with quark at the point z£ , antiquark at the point z^' and gluons at the points TT 'k=zl n. We also define pW = z^ — z^~l\ with *(°) = z(°) and z^n+^ = z^.' The action is A = K + aSmin

(43)

where the kinetic operator K and the minimal area Sm\n can be written as

K=^

+

^ + UT dz+[»Ma))2 + 2^) + M(ii6))2 + 2£6))

lHa

£ Jo

l^b

+ £/*((*}8,)2 + 2ii!))] i

r-T

crSmin = - /

"+1 n+1

o

rri

dz+J^

1 J

i=1

(44)

d(3i Jo

«iwr-

, ;»j +

(w'(i))2

(45) and we have defined

«i° = 4 i _ 1 ) (i - 0i) + A 4 ° . ™(i) = i '" 1 ) ( 1 - ft) + «/W r r z W - ^ * - 1 ) = p ( 0 .

fti(i)>

(46) (47)

173

As in

37

we introduce the total momentum P + , P+=Y,fii + Y, i=i

i=i

Vid/3 + fia + fJ.b-

(48)

J

°

At this point one can make a new important step and introduce the parton's quota Xi of the total momentum P+ to be associated with the Feynman variables Xi, i = 1,..., n, xa and Xb, which can be written as

_ iM + fZyiW +

xi —

fivi+iq-PW

(^4yj

"+ xa=x0

= (fia + f ^i(l - P)dp)/P+ Jo

xb = xn+i

(50)

= (m, + / vn+i0dj3)/P+. (51) Jo One can notice that Xi consists of three pieces: 1) the (+) -component of momentum of the valence gluon (m), 2) the (+) momentum of the preceeding piece (i) of string weighted with the factor (3 which takes into account that the string (i) is deformed by the motion of the gluon (i) while another end of the string is fixed 3) the (+) momentum of the string (i + 1) weighted with the factor (1 — (3) taking into account motion of the (i ^ 1) string due to the i-th gluon. Note that the parameter f3 in all strings (i), i = 1, ...n + 1 grows in one direction, e.g. from the left to the right. In this way the momentum of each piece of the string (i) is shared by two adjacent gluons: (i — 1) and (i), so that each parton quota Xi contains momentum of the parton (gluon or quark or antiquark) itself and of pieces of adjacent strings. These results imply several nontrivial consequences. First of all, one can see that the einbein factors fii, which played in the c m . system the role of constituent mass (energy) of gluon and quarks, and Vi played the role of energy density along the string, in the I.e. system they enter directly the Feynman parameters of gluons. In this way one can for the first time see the connection of the standard constituent quark (gluon) picture with the parton picture and calculate as in (50), (51) parton parameters through the (Lorentz boosted) constituent energies of quarks and gluons. Secondly in the I.e. wave-function of the n-hybrid the average values of m and i>i are equal for large n,fn = Di,i = 1,2, ...n, while for n — 1 one obtains fig = s/2jiq - V2p,q 3 8 .

174

Hence gluons carry more momentum on average than quarks in the nhybrid state. Therefore one expects that in DIS at large enough energy when the n-hybrid component of the hadron wave-function is excited, the contribution of gluons to the momentum sum rule and to the proton spin should be dominant. This expectation is consistent with the experimental data. Thus in the neutrino-isoscalar scattering the momentum sum rules for the quark part of F2{x, Q2) yield 39 0.44± 0.003, which implies that gluons carry more than 50% of the total momentum. For the proton spin one has the relation 40 i = | A £ ( / i ) + L,(/i) + J 9 (/i)

(52)

where the quark sigma-term experimentally is AE(jx = 1 GeV) = 0.2 ± 0 . 1 , and the most part of the difference between the l.h.s. and the r.h.s. is presumably due to the gluon spin contribution Jg(n). The detailed estimates of hybrid contributions to DIS and high-energy scattering will be published elsewhere 38 . 7. Conclusions It was explained above that the Field Correlator Method is a powerful tool for investigation of all nonperturbative effects in QCD. In particular it provides a natural mechanism of confinement, compatible with all lattice data, and explains the close connection of confinement and chiral symmetry breaking. The spectrum of mesons, glueballs and hybrids is calculated with a,as and current masses as the only fixed parameters used and this spectrum is in good agreement with lattice data and experiment. The latest development concerns the dominant role of hybrids in DIS and high-energy scattering and here the first qualitative results are consistent with experimental evidence. The author is grateful to the organizers of the Pomeranchuk International Conference for their excellent job, and to A.M.Badalian, K.G.Boreskov, A.B.Kaidalov and O.V.Kancheli for many stimulating discussions.

References 1. M.A.Shifman, A.I.Vainshtein and V.I.Zakharov, Nucl. Phys. B147 385, 448 (1979). 2. H.G. Dosch, Phys. Lett. B190, 177 (1987); H.G. Dosch and Yu.A. Simonov, Phys. Lett. B205, 339 (1988); Yu.A. Simonov, Nucl. Phys. B307, 512 (1988), fo a review see A.Di Giacomo,

175

3.

4. 5. 6. 7.

8. 9. 10. 11. 12.

13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.

24. 25.

26. 27.

H.G. Dosch, V.I. Shevchenko, and Yu.A. Simonov, Phys. Rep., in press; hepph/0007223. Yu.A. Simonov, Phys. At. Nucl. 58, 107 (1995); JETP Lett. 75, 525 (1993); Yu.A. Simonov, in: Lecture Notes in Physics v. 479, p. 139; ed. H. Latal and W. Schweiger, Springer, 1996. Yu.A.Simonov, Physics-Uspekhi, 39, 313 (1996). Yu.A.Simonov, JETP Lett. 71, 127 (2000); V.I.Shevchenko and Yu.A.Simonov Phys. Rev. Lett. 85, 1811 (2000). G.S.Bali, Phys. Rev. D62, 114503 (2000); S.Deldar, Phys. Rev. D62, 034509 (2000). A.Di Giacomo and H.Panagopoulos, Phys. Lett. B285, 133 (1992); A.Di Giacomo, E.Meggiolaro and H.Panagopoulos, Nucl. Phys. B483,371 (1997); A.Di Giacomo and E.Meggiolaro, hep-lat/0203012. G.S.Bali, N.Brambilla and A.Vairo, Phys. Lett. B42, 265 (1998). M.Foster and C.Michael, Phys. Rev. D59, 094509 (1999). Yu.A.Simonov, Nucl. Phys. B592, 350 (2001). M.Eidemueller, H.G.Dosch, M.Jamin, Nucl. Phys. Proc. Suppl. 86, 421 (2000). Yu.A. Simonov, Phys. At. Nucl. 60, 2069 (1997); hep-ph/9704301; Yu.A. Simonov and J.A. Tjon, Phys. Rev. D62, 014501 (2000); ibid 62, 094511 (2000). Yu.A.Simonov, Phys. Rev. D65, 094018 (2002); hep-ph/0201170. Yu.A. Simonov, Phys. At. Nucl. 63, 94 (2000). Yu.A. Simonov, Phys. At. Nucl. 62, 1932 (1999); hep-ph/9912383; Yu.A. Simonov, J.A. Tjon and J.Weda, Phys. Rev. D65, 094013 (2002). Yu.A. Simonov, Phys. Lett. B 226, 151 (1989), ibid. B 228, 413 (1989). A.Yu. Dubin, 'A.B. Kaidalov, and Yu.A. Simonov, Phys. Lett. B 323, 41 (1994); Phys. Atom. Nucl. 56, 1745 (1993). Yu.A. Simonov and J.A. Tjon, Ann. Phys. 300, 54 (2002). E.L. Gubankova and A.Yu. Dubin, Phys. Lett. B 334, 180 (1994). Yu.A. Simonov, Phys. Lett. B 515, 137 (2001). Yu.A. Simonov, Nucl. Phys. B 324, 67 (1989); A.M. Badalian and Yu.A. Simonov, Phys. At. Nucl. 59, 2164 (1996). Yu.A. Simonov, Phys. At. Nucl. 64, 1876 (2001). Yu.A. Simonov, QCD and Theory of Hadrons, in: "QCD: Perturbative or Nonperturbative" eds. L. Ferreira., P. Nogueira, J.I. Silva-Marcos, World Scientific, 2001, hep-ph/9911237. T.J. Allen, G. Goebel, M.G. Olsson, and S. Veseli, Phys. Rev. D64, 094011 (2001). A.M. Badalian, B.L.G. Bakker and V.L. Morgunov, Phys. At. 63, 1635 (2000); A.M. Badalian and B.L.G. Bakker, Phys. Rev. D64, 114010 (2001). B.O. Kerbikov and Yu.A. Simonov, Phys. Rev. D 6 2 , 093016 (2000). Yu.A. Simonov in: Proceeding of the Workshop on Physics and Detectors for DA&NE, Frascati, 1991; Yu.A. Simonov, Nucl. Phys. B (Proc. Suppl.) 23 B , 283 (1991);

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28. 29.

30. 31.

32. 33. 34. 35. 36. 37.

38. 39.

40.

Yu.A. Simonov in: Hadron-93 ed. T. Bressani, A. Felicielo, G. Preparata, P.G. Ratcliffe, Nuovo Cim. 107 A, 2629 (1994); Yu.S. Kalashnikova, Yu.B. Yufryakov, Phys. Lett. B 359, 175 (1995); Yu. Yufryakov, hep-ph/9510358. Yu.S. Kalashnikova and D.S. Kuzmenko, hep-ph/0203128. Yu.A. Simonov, Phys. Lett. B 249, 514 (1990); A.B. Kaidalov and Yu.A. Simonov, Phys. Lett. B 477, 163 (2000); Phys. At. Nucl. 63, 1428 (2000). B.Anderson, G.Gustafson and C.Peterson, Z.Phys. C l , 105 (1979); B.Anderson, G.Gustafson and B.SSderberg, Z.Phys. C20, 317 (1983). Yu.A.Simonov, Yad. Fiz. 3, 630 (1966); A.M.Badalian and Yu.A.Simonov, Yad. Fiz. 3, 1032 (1966); F.Calogero and Yu.A.Simonov, Phys. Rev. 183, 869 (1968); A.M.Badalian et.al., Nuovo Cim. A68, 577 (1970; M.Fabre de la Ripelle, Ann. Phys. (NY) 123, 185 (1979). Yu.A.Simonov, hep-ph/0212253. P.Lacock, C.Michael, P.Boyle and P.Rowland, Phys. Lett. B 4 0 1 , 308 (1997); C.Bernard et al., Phys. Rev. D56, 7039 (1997). C.Bernard et al., hep-lat/0301024. M.Green, J.Schwarz, E.Witten, Superstring theory, v.l, Cambridge Univ. Press, Cambridge, 1987. M.Teper, hep-ph/0203203 A.Yu.Dubin, A.B.Kaidalov, Yu.A.Simonov, Phys. Atom. Nucl. 58, 300 (1965), hep-ph/9408212; V.L.Morgunov, V.I.Shevchenko, Yu.A.Simonov, hep-ph/9704282. Yu.A.Simonov (in preparation). P.Berge et al., Z.Phys. C49, 187 (1991); G.G.Groot et al., Z.Phys. C l , 143 (1979); see also discussion in F.J.Yndurain, The theory Quark and Gluon Interactions, 3d Edition, Springer, Berlin-Heidelberg, 1999. B.W.Filippone and Xiangdong Ji, Adv. Nucl. Phys. 26 (2001) 1, hep-ph/0101224.

FINE T U N I N G IN LATTICE SU(2) G L U O D Y N A M I C S *

V. I. Z A K H A R O V Max-Planck Institut fur Physik Werner-Heisenberg Institut Fohringer Ring 6, 80805, Munich E-mail: [email protected]

We review recent data on monopoles and vortices in lattice SU(2) gauge theory. The non-Abelian action densities associated with the corresponding trajectories and surfaces turn to be ultraviolet divergent. On the other hand, the spatial densities of the monopoles and vortices scale in physical units, that is ( / m ) - 3 and (fm)~2, respectively. To check consistency of the emerging picture of nonperturbative fluctuations we consider constraints from the continuum theory on the ultraviolet behaviour of the monopoles and vortices. The constraints turn to be satisfied by the data in a highly non-trivial way. Namely, it is crucial that the monopoles populate not the whole of the four dimensional space but a twodimensional subspace of it.

1. Introduction By fine tuning one understands usually a particular problem arising in theory of charged scalar particles. Namely, expression for the scalar boson mass looks as m2H = 5M2rad - Ml ,

(1)

where SM^ad is the radiative correction while — MQ is a counter term. The problem is that the radiative corrections diverges quadratically in the ' T h i s mini-review is based on a talk presented to the Pomeranchuk memorial Conference held at the Moscow Institutes for Physics and Engineering (MIFI) in January 2003. the Conference was a moving event for me.I met Isaak Yakovlecvich Pomeranchuk while a student of MIFI (and he was a professor at the Institute). His devotion to physics and personality crucially influenced physicists around him, not even necessarily very close to him. The pain caused by his untimely death in 1966 is still alive for everyone of this circle. I am very thankful to the organizers of the Conference for their efforts. 177

178

ultraviolet (UV), d4k

2

UV ! / where a is the coupling and kuv is an ultraviolet cut off. If one uses the Planck mass for the cut off, Ay^ ~ (10 19 GeV)2, then to keep the mass of the charged (Higgs) boson in the 100 GeV region one should assume that the counter term is tuned finely to the value of the radiative correction and this tuning is readjusted with each order of perturbation theory. What is outlined above is the standard problem of the Standard Model but a priori one would bet that it has nothing to do with the vacuum state of the lattice SU{2) theory and with monopoles, which is a particular kind of the vacuum fluctuations. However, lattice measurements indicate strongly (see x and references therein) that the monopole mass is ultraviolet divergent, the same as the radiative correction in case of a point-like particle: ,w x const M{a)mon ~ , (2) a where a is the lattice spacing playing the role of an ultraviolet cut off (in coordinate space). Moreover, the mass in Eq (2) is directly related to the excess of the non-Abelian action associated with the monopoles. Observation (2) is the starting point of our discussion a . Let us emphasize that it is a pure phenomenological observation, with no theory involved. Moreover, it cannot be directly confronted with theoretical expectations since the monopoles themselves are defined not in terms of the original Yang-Mills fields but rather in terms of projected fields. Details of definitions can be found in review articles 3 . Here we give only a sketchy overview. To study monopoles, one starts with generating a representative set of vacuum configurations of the original Yang-Mills theory with the standard action. Then each configuration - and this is the central point - is projected into the closest configuration of 17(1) fields. The projection itself is in two steps. First, one uses gauge invariance to minimize, over the whole lattice the functional

R = £ [(4)2 + (4) 2 ] -

(3)

links

where A1 is the gauge potential and i(i = 1,2,3) are color indices. The meaning of minimizing R is that 'charged' fields are minimized. This a

To some extent, we follow the logic of 2 .

179

fixation of the gauge does not change physics, of course. However, at the next step, which is the projection itself, one sends A1^2 to zero generating in this way effective (7(1) fields, A3^. Finally, the monopoles are defined in terms of the projected potentials. In more detail, the monopole current, j™ on is related to violations of the Bianchi identities,

d^

= Con

,

(4)

where FM„ now is the field strength tensor constructed on the projected fields A3^ and F^ is dual to F^. A non-vanishing current (4) implies projected fields to be singular in the continuum limit. However, all the singularities are regularized by the lattice and the expression (4) is well defined 4 . An analysis of this type ends up with a net of monopole trajectories for each original non-Abelian field configuration. Theoretical task is then to interpret the data on the monopole clusters. Since singular non-Abelian fields would have infinite action, common wisdom tells us that such fields would drop off by themselves and, therefore, one would assume that the singularity (4) arises as an artifact of the projection. Observation (1) means that something is missing in this standard logic. Generically, we would call it "fine tuning" if singular field, defined on the UV scale a affect physics on the physical scale of order AQCD- (Relation to the fine tuning of the standard model will be clarified later.) The monopoles are known to produce a confining potential at large distances 3 . Thus, we confront the problem of fine tuning for vacuum fluctuations in SU(2) gauge theory. The outline of the review is as follows. In Sect. 2 we review well understood examples of fine tuning on the lattice. In Sect. 3 we summarize the data on the fine tuning in the lattice SU(2) gluodynamics. We will see that the data amount to a discovery of a novel type of fine tuning. In Sect. 4 we will argue that the singularities in the non-Abelian fields, observed on the lattice, do not contradict constraints from the continuum theory. Finally, Sect. 5 is conclusions.

180

2. Examples of fine tuning 2.1. Free

particle

Consider first a free particle with the classical action b : Sa = M(a)-L

,

(5)

where M(a) is a mass parameter and we reserved for its possible dependence on the lattice spacing, while L is a length of a trajectory of the particle, everything in the Euclidean space. Furthermore, define a propagator as a path integral: D(xi,xf;a)

=

^ Z exp( - Sct)

,

(6)

paths

where Xi j are the end points of trajectories. The summation in (6) can be performed explicitly. In the momentum space: D

(P'a)

=

^2Dfree(mlh)

,

(7)

where c is a constant depending on details of the ultraviolet regularization and mph is the propagating mass. The relation of mph to the bare mass M(a) introduced in (5) is as follows:

mlh = l(M(a) - ^ ) ,

(8)

where the constants in front of the ultraviolet factors (i.e., inverse powers of a) are in fact regularization dependent and hereafter we will have in mind hypercubic lattice. Eq (8) demonstrates that to keep the physical mass fixed, i.e. independent of a, one should tune the bare mass to a pure geometrical factor. One could proceed further and consider interaction as well. Thus, Eq (8) is only an example of relations which arise within the so called polymer approach to field theory. Here, we will use this approach only to derive a useful relation for the vacuum expectation value of the corresponding scalar field squared 6 ' 7 . Namely, the average value of the length of the particles trajectories in the vacuum is given by: (L)

b

= ^ l n Z ,

This subsection is mostly a text-book material, see, e.g., 5 .

(9)

181

where Z is the partition function. Moreover, one can replace: d dM

8 d ~* adm2ph "

[

j

The derivative with respect to mph, on the other hand, is related to the vacuum expectation of the |(/>|2 where 0 is a (complex) scalar field entering the standard formulation of field theory. Indeed, the standard Lagrangian contains a term mph\4>^'. Finally, (0|H2|0)

= §<£>•

(11)

Eq (11) relates quantities entering the standard and polymer representations of theory of a scalar field. For us, it is important that the lengths of the monopole trajectories are directly measurable.

2.2. Lattice

U(l)

Lattice f/(l) , see, e.g., 8 , is actually close to the case of free particle just considered. A new point is that M(a) is now calculable as energy of the magnetic field:

M{a)mon = -L / W r ~ ~

,

(12)

where one has to introduce an ultraviolet cut off, a since H ~ 1/r2 and the integral diverges at small distances. Note also that we kept explicit dependence on the electric charge e which is due to the Dirac quantization condition. Finally, and might be most noteworthy, Eq (12) does not contain contribution of the Dirac string. This is a privilege of the lattice regularization (for more details see, e.g., 9 ) . Note that upon substiting (12) into (8) we reproduce in fact (1). Now, if one tunes e 2 in such a way that mph = 0, where mph is defined in (8) the monopoles condense. This is confirmed by the lattice data. 2.3.

Percolation

Percolation is a common notion in papers on the monopoles (for a review of percolation theory see, e.g., 1 0 ). In most cases it is related to existence of an infinite cluster of monopoles, see n and references therein. Phenomenologically the percolating cluster is very important since the

182

confining potential for external heavy quarks is entirely due to this infinite cluster while finite clusters do not confine. Uncorrelated percolation is the simplest kind of percolation. In this case, one introduces a probability p,p < 1, for a link to be "open" and this probability does not depend on the neighbors. In our case, an open link would correspond to a link belonging to a monopole trajectory c . The probability to find a connected trajectory of length L is given by W{L)

= pL'a • NL

,

where L/a is the number of steps and Ni trajectories of the same length L. Moreover,

(13)

is the number of various

NL = 8 L / a .

(14)

Indeed, the monopoles occupy centers of cubes and at each step the trajectory can be continued to a neighboring cube. There are 8 such cubes for D = 4 d . Note that uncorrelated percolation is equivalent to a free field theory. Indeed we could interpret Eq (14) as a product of the suppression factor due to the action while Ni represents an entropy factor. Moreover, M(a)

= Inp/ a .

Note also that Eq. (14) neglects the effect of intersections. Such an approximation is valid for qualitative conclusions in any case. Moreover, in four dimension, D = 4 the effect of intersections is not large numerically 10 Clearly, there is a critical value of p,p = pc, when the probability to find a trajectory of any length is not suppressed. This is the point of phase transition to percolation. In the supercritical phase, p > pc there always exists a single infinite percolating cluster. Most amusing, if (p — pc)
,

(15)

where the critical exponent 0 < a < 1. c Monopoles are defined as end points of the Dirac strings and occupy centers of the lattice cubes. Alternatively, one can say that on the dual lattice monopoles occupy sites and the monopole trajectories are built up on the links on the dual lattice. In most cases, we do not mention that it is the dual lattice which is implied in fact. d For charged particles, which we are considering, the factor 8 in Eq (14) is to be replaced by the factor of 7. Indeed, if one and the same link is covered by a trajectory in the both directions, then the link does not belong to a trajectory at all. In the field theoretical language this cancellation corresponds to the fact particle and anti-particle have opposite charges. For simplicity of presentation we will keep Eq (14) without change

183

2.4.

Lattice Z?

Because of space considerations, we can be only very brief on the lattice Z2, details could be found, e.g., in 12 ' 13 . The plaquette action of the Z2 gauge theory is the trace of four matrices associated with the links. In turn, each link can be ascribed matrix ±1. In the limit of zero temperature, P —> 00 the ground state is all plaquettes equal to + 1 . At smaller (5 there appear excitations which are closed surfaces unifying all the negative plaquettes. The suppression due to the action is exp ( - Svort)

~ exp ( - 0—)

,

where A is the area of the surface. The entropy factor is also exponential in A/a2. Indeed, there is a freedom to choose orientation of neighboring plaquettes which add up to the same total area A. At some finite value of (5 there is a phase transition to percolation of the vortices. 3. SU{2)

fine tuning, seen on the lattice

As is mentioned in the Introduction, monopoles in SU(2) are defined through a projection on f/(l) fields. Similarly, vortices are defined within a projection on Z2 variables. Theoretically, it is very difficult to evaluate the effect of the projections. Phenomenologically, both lattice monopoles and vortices exhibit remarkable properties. Which we will briefly summarize here. Note that we present a simplified picture, emphasizing only the main features of the data (as we appreciate them). Details of the definitions and of results can be found in the original papers. The length of the percolating monopole cluster is usually expressed in terms of the density pperc: ^perc

=

Pperc ' '4

>

\*-®)

where V\ is the lattice volume and the the index "4" is for D = 4. The experimental observation is that pperc does not depend on the lattice spacing, for the latest results and further references see 14 . In other words, the probability for a given link on the (dual) lattice to belong to the percolating cluster is proportional to: 6{link)

~

{CL-KQCD?

,

(17)

which looks very similar to Eq (15) provided that (p — pc) itself is a power of (a • AQCD)-

184

What is amusing indeed, is that the relation (17) is perfectly gauge invariant. The same is true about the action associated with the monopoles, see Eq (2). Eqs (2) and (15) suggest that through the Abelian projection one detects gauge invariant objects e . Also, the data exhibit fine tuning. Namely, the mass (2) tends to infinity with a —> 0, at least as far as the data on presently available lattices are concerned. Numerically, the smallest value of a is a « 0.06 fm. On the other hand, the length of the percolating trajectory does not depend on o, see (17). To give a feeling, how fine the tuning is let us compare values of the mass (2) and of the monopole "free path". The point is that there are intersections in the percolating cluster. One can measure the average length along the trajectory of the percolating cluster between the two nearest intersections. It turns out that this length scales 15 : Lfree

« 1.6 fm

(18)

On the other hand, M{a)mon

> 5GeV

,

(19)

where the bound corresponds to the smallest a available. Naively, such a monopole would travel a distance of order (5 G e V ) - 1 . In reality, the free path is about 40 times larger. The story repeats itself in case of the vortices. The vortices are defined in terms of projected Z2 fields. In more detail, one first projects the original SU{2) fields into the nearest 1/(1) fields configuration, see discussion in the Introduction. And then, into the nearest Z2 configuration. The vortices are defined as unification of all the negative plaquettes evaluated on the the Zi configurations. The corresponding surfaces are closed by definition, as boundary of a boundary. For the vortices, one can also measure the total area and the associated non-Abelian action. It turns out that the total area scales in the physical units, see 13 and references therein. Numerically 16 : Avert « 2 4 ( / m ) - 2 - y 4 ,

(20)

where Avort is the total area of the vortices in the lattice volume V4. One can rewrite (20) in terms of probability to find a particular plaquette belonging to the percolating vortex: 6(plaq) ~ (a-AQCD)2 e

This suggestion is made, in particular, in Ref

n

.

,

(21)

185

compare Eq. (15). On the other hand, the non-Abelian action associated with the vortices is ultraviolet divergent 16 : < Svp^

>

-

< Spiaq >

« 0.54 [ lattice units] ,

(22)

where < S^f^ > is the average value of the action for the plaquettes belonging to the vortex and < Spiaq > is the average of the plaquette action for the whole of the lattice. Moreover, the fact that the action is finite in lattice units means that it is ultraviolet divergent in the continuum limit a ->• 0. The reason is that the lattice units for the density of action are adjusted to the dominating vacuum fluctuations, that is to zero-point fluctuations. And for the zero-point fluctuations the action density is divergent in the ultraviolet as a~ 4 . Thus, the data imply, at least at their face value, a huge cancellation between the suppression due to the UV divergent action and the entropy of the vortices which is the same UV divergent. 4. Data vs. theoretical constraints 4.1. Asymptotic

freedom

and counting

degrees of

freedom

The fine tuning exhibited by the data is just crying for interpretation. The interpretation is far from being simple, however, as mentioned a few times. The problem is that one starts with full SU(2) action to generate the field configurations and ends up with results like (22) or (20) which are also perfectly gauge invariant. However, the definitions of the monopoles and vortices, are not intrinsic to the full SU(2), but rather to its subgroups, that is £/(l) and Z2. Indeed, there is no topological definitions of trajectories or surfaces in the full SU(2) which would imply lower bounds on the action of the topologically non-trivial fluctuations. Anyhow, it looks unusual that in an asymptotically free theory there exist ultraviolet divergences associated with non-perturbative fluctuations. One feels that there should exist at least constraints imposed by the asymptotic freedom and we will dwell on the issue here. To begin with, although Eqs (15) and (21) look as typical percolation relations in the supercritical phase (i.e. at p > pc), the properties of the monopole clusters cannot in fact be similar to the uncorrelated percolation discussed in Sect. 2.3. Indeed, the uncorrelated percolation is equivalent to theory of free particles. But it is clear that the particle content of gluodynamics at short distances is fixed and we are not allowed to introduce

186

new particles. Let us look closer, what the actual constraints are. 'cosmological constant' , that is density of vacuum energy: Cvac ~

2^i k, d.o.f.

o

Consider the

^ ^

where "d. o. f." stands for "degrees of freedom". Eq (23) is nothing else, of course, but the sum over zero-point fluctuations. The point is that in an asymptotically free theory Eq (23) should be a valid approximation. The sum (23) diverges in the ultraviolet as a - 4 . The coefficient in front of the divergence depends, however, on the number of degrees of freedom. Moreover, all the divergences are regularized by the lattice, so that Eq (23) is well defined and can be used to predict the average value of the plaquette action on the lattice. To appreciate the meaning of the constraint in terms of the monopole trajectories, which are our basic observables now, notice that in the percolation picture there is nothing happening to the total monopole density at p = pc. Indeed the total density of 'open' links is simply:

pTor=V^

(24)

and there is no discontinuity or non-analyticity at p = pc. It is only the density of the infinite cluster which exhibits a threshold behaviour (15). As we mentioned in Sect 2.3 the existence of the percolating cluster is crucial for the confinement and that is why one usually concentrates on pperc- However, now we come to the conclusion that from the theoretical point of view it is the total monopole density which is constrained by the asymptotic freedom. If the density of the percolating cluster is vanishing at the point of the phase transition (see (15)) where the total density (24) goes to? Clearly, to the finite clusters. Thus, we should address theory of the finite clusters. 4.2. Finite

monopole

clusters

Since the data (2) suggest a point-like particle let us start with this case. Then the simplest Feynman graph is a closed loop. Usually, text-books say that this graph is not observable since we should normalize all the amplitudes to the vacuum-to-vacuum transition. Similarly, one is usually saying that the vacuum energy is normalized to zero by definition, However, once the lattice regularization is introduced, the 'cosmological constant'

187

turns directly observable. The same is true for the vacuum-to-vacuum transition. The vacuum monopole loops are directly observable and one can try to evaluate them theoretically 7 . For free particles, the calculations are in fact straightforward. Since it is the trajectories that are directly observable one is invited to use the polymer representation of field theory, see Sect. 2.1. There are two basic properties of finite clusters which can be predicted starting from the assumption that the monopoles at short distances can be treated as free particles 7 . First, the spectrum of the finite clusters as function of their length is given by f : AT/r.

N

&

const

= TmTi

const

. .

= -73- >

( 25 )

where we substituted the number of dimensions of the space D = 4. Second, radius of a cluster is predicted to be:

R{L) ~ VT^

.

(26)

The both predictions (25) and (26) are in perfect agreement with the data. First, properties (25) and (26) were claimed in Ref n and later confirmed on larger statistics and for smaller values of a in Ref 18 . Now one can say 18 that the properties of monopole clusters at short distances are the same as for free particles g .

4.3. Conspiracy

of the monopoles

and

vortices

The fact that the predictions (25) and (26) agree with the data, at first sight, is in contradiction with what we said earlier on counting degrees of freedom at short distances. However, Eqs (25), (26) are not yet brought into the form which would allow a direct comparison with expectations based on asymptotic freedom. Consider therefore the vacuum expectation value (11). Clearly, the asymptotic freedom requires that ( 0| |
,

(27)

while any ultraviolet divergence in this vacuum expectation value would imply that monopoles are to be introduced on equal footing with the gluons and this is not allowed. f

T h e result can be actually read off from the equations derived, e.g., in Ref. 1 7 . T h e Coulomb-like interaction in D = 4 leaves actually (25) and (26) unchanged 7 .

8

188

Constraint (27) implies in turn that the density of the finite clusters is to satisfy inequality: Pfin < const H

const

,

a

(28)

where the definition of pfin is similar to (16). It is most amusing that the constraint (28) is satisfied by the data! The first indication to the 1/a behaviour of the total density of the monopoles was obtained in Ref 14 and is now confirmed in Ref 18 . It is easy now to figure out the geometrical meaning of the relation (28). Clearly, monopoles are to be associated with a D = 2 subspace of the whole D — 4 space. Taken as a prediction, this sounds bizarre. Unfortunately, no prediction of this kind was done (and our analysis is post hoc). The fact that the vortices are populated by monopoles is known empirically since some time 19 for one value p. Most recently, it has been confirmed for all the values of a available now in Ref 16 . Thus, there exists a long-range correlation between finite clusters. While at short distances monopole clusters have the same properties as free particles living in the D = 4 space in the infrared they are correlated with a D = 2 subspace. As a result, the constraint (27) gets satisfied by the data. 4.4. Non-perturbative

completion

of the UV

renormalon

Another type of constraints is associated with analysis of divergences of perturbative expansions, for review and references see 20 . In perturbation theory, one starts with definition of an observable as a (formal) sum over powers of the coupling as(Q2) where Q2 is a generic large mass scale inherent to the problem, such that as(Q2) -C 1. Generically an observable (O) is represented perturbatively as oo

( O ) = (parton model) • (l + ^

ana" ) .

(29)

71=1

In fact the expansion (29) is only formal since the coefficients an grow factorially at large n:

Kl ~ c?-n\ ,

(30)

where Cj are constants and actually there are a few sources of the growth (30) and, respectively, there are various a. Moreover, if there is sign

189

oscillation, an ~ ( — 1)™ then the sum Borel summable while if an ~ (+1)" there is no way at all to define the sum. Asymptotic al expansions (29) can represent a physical quantity only up to some uncertainties which in case considered (that is, logarithmic dependence of the coupling on Q2) are power corrections (AQCD/QYWhich are to be added to (29). Consider two particular examples, that is, vacuum expectation value of as{Ga ) 2 , where G£„ is the non-Abelian field strength tensor, and heavy quark potential V(r). Then, if one accounts only for non-summable perturbative divergences: ( a s ( G ^ ) 2 ) « c £ A 4 ^ ( l + £ a n a ? ( A c y ) + C^KQCDI^uvf

) ,

n

(31) and limV(r) = y ( l + X > n a ? ( r )

+

c3A3QCDr3

) ,

(32)

n

where the redefined coefficients an, an do not contain the leading factorially growing contributions (without sign oscillations). It is worth emphasizing that although the power corrections are so to say detected through pure perturbative graphs the actual vacuum fluctuations which dominate the power corrections in (31), (32) can well be genuinely non-perturbative. In particular, the infrared renormalons in the cases considered could be matched by instantons. Instantons, in this sense, could be called a non-perturbative completion of the infrared renormalons. As a result of this completion the coefficients in front of the power corrections could be enhanced. There exists huge literature on the power corrections and, in particular, it has been speculated that the infrared renormalons do not exhaust all the corrections indicated phenomenologically, for review see. e.g., 21 . Moreover, careful analysis suggests that the data are adequately described if one keeps the parton model contribution and a Q~2 corrections. Such an uncertainty of expansions (29) corresponds to the ultraviolet renormalon h . In particular, analysis of the vacuum expectation (31) can be found in 22 . In case of the potential (32) the ultraviolet renormalon would h

Let us recall the reader that the ultraviolet renormalon in an asymptotically free theory generates an ~ cn • n\ • ( — l)n. It is the leading n! contribution since the constant c is the largest but it can be summed up a la Borel.

190 correspond to a linear in r piece at short distances which is quite important phenomenologically 23 \ Now, this excursion on the ultraviolet renormalon appears here by no accident. Indeed, the observations (20), (22) imply that the vortices contribute to the vacuum expectation value (31) of order (a s (G , £„) 2 ) t , or (ex ~ const •

A-UVAQCD

,

(33)

and this is exactly type of contribution whish is expected from ultraviolet renormalon. The beauty of this result is that it is formulated in an explicitly gauge invariant way. Thus, we are justified to conclude that vortices and monopoles realize a non-pereturbative completion of the ultraviolet renormalon in the same sense as instantons complete the infrared renormalon K Detailed discussion of implictaions of this observation is beyond the scope of this mini-review. 5. Conclusions We have argued that the fine tuning exhibited by the lattice data on SU(2) gluodynamics is a novel phenomenon which shares some features with the well known examples of the lattice U(l) and Zi gauge theories. The novelty is mainly due to the fact that there seems to exist an object which has never been encountered so far. This is D = 2 subspaces of the Euclidean space (vortices) populated with point-like monopoles. At short distances the monopoles behave themselves as free particles. Viewed as a whole, however, they exhibit very strong correlation with surfaces whose total area scales in the physical units. Moreover, the vortices and monopoles can be considered as a nonperturbative completion of the ultraviolet renormalon. References 1. V.G. Bornyakov, et. ai, Phys. Lett. B537 291 (2002). 2. F.V. Gubarev and V.I. Zakharov, " Interpreting lattice monopoles in the continuum terms", hep-Lat/0211033. 'Let us also mention two other examples where introduction of Q~2 corrections seems necessary. These are instanton density 2 4 and a particular current correlator 2 5 . J Actually, it was already argued that the monopoles contribution to V(r) (see (32)) corresponds to the ultraviolet renormalon since they produce a piece in potential (32 which is linear in r 2 3 . However, to have this interpretation granted one has to demonstrate that there are non-perturbative fluctuation of zero (in the limit a —> 0) size. Observation (2) does demonstrates that the monopoles are point-like indeed.

191

3. M.N. Chemodub, F.V. Gubarev, M.I. Polikarpov and A.I. Veselov, Progr. Theor. Phys. Suppl. 131, 309 (1998), (hep-lat/9802036) ; A. Di Giacomo, Progr. Theor. Phys. Suppl. 131, 161 (1998); H. Ichie and H. Suganuma, "Dual Higgs theory for color confinement in quantum chromodynamics", hep-lat/9906005 ; T. Suzuki, Progr. Theor. Phys. Suppl. 131, 633 (1998). 4. T.A. DeGrand and D. Toussaint, Phys. Rev. D22, 2478 (1980). 5. J. Ambjorn, "Quantization of geometry", hep-th/9411179. 6. V.I. Zakharov, "Hidden mass hierarchy in QCD", hep-ph/0204040. 7. M.N. Chemodub, V.I. Zakharov, "Towards understanding structure of monopole clusters", hep-th/0211267. 8. A.M. Polyakov, Phys. Lett. B59, 82 (1975); T. Banks, R. Myerson and J. Kogut, Nucl. Phys. B129, 493 (1977); H. Shiba and T. Suzuki, Phys. Lett. B343, 315 (1995). 9. M.N. Chemodub, F.V. Gubarev, M.I. Polikarpov and V.I. Zakharov, Phys. Atom. Nucl. 64, 561 (2001) (hep-th/0007135). 10. S. Fortunato, "Percolation and deconfinement in SU(2) gauge theory, heplat/0012006. 11. A. Hart and M. Teper, Phys. Rev. D58, 014504 (1998). 12. R. Savit, Rev. Mod. Phys. 52, 453 (1980). 13. J. Greensite, "The confinement problem in lattice gauge theory", heplat/0301023 ; K. Langfeld, et. al., "Vortex induced confinement and the IR properties of Green functions", hep-lat/0209040. 14. V. Bornyakov and M. Muller-Preussker, Nucl. Phys. Proc. Suppl. 106, 646 (2002) (hep-lat/0110209). 15. P.Yu. Boyko, M.I. Polikarpov and V.I. Zakharov, "Geometry of percolating monopole cluster", hep-lat/ 0209075. 16. F.V. Gubarev, A.V. Kovalenko, M.I. Polikarpov and S.N. Syritsyn, V.I. Zakharov, " Fine tuned vortices in lattice SU(2) gluodynamics", heplat/0212003. 17. A.M. Polyakov, "Gauge Fields and Strings", ch. 9, (1987). 18. V.G. Bornyakov, P.Yu. Boyko, M.I. Polikarpov and V.I. Zakharov, in preparation. 19. J. Ambjorn, J. Giedt, J. Greensite, JEEP 0002, 033, (2000). 20. V.I. Zakharov, Nucl. Phys. B385, 452 (1992); M. Beneke, Phys. Rept. 317, 1 (1999). 21. K.G. Chetyrkin, S. Narison and V. I. Zakharov, Nucl. Phys. B550, 353 (1999); M.N. Chemodub. F.V. Gubarev, M.I. Polikarpov and V. I. Zakharov , Phys. Lett. B475, 303 (2000) ; E.V. Shuryak, "Scales and Phases of Nonperturbative QCD", hep-ph/9911244. 22. F. Di Renzo, E. Onofri and G. Marchesini, Nucl. Phys. B457, 202 (1995). 23. R. Akhoury and V.I. Zakharov, Phys. Lett. B438, 165 (1998); S. Bali, Phys. Lett. B460, 170 (1999); V.I. Shevchenko, "A remark on the short distance potential in gluodynamics",

192 hep-ph/0301280; F.V. Gubarev, M.I. Polikarpov and V.I. Zakharov, Mod. Phys. Lett. A14, 2039 (1999); Yu. A. Simonov, Phys. Rept. 320, 265 (1999); A.M. Badalian and D.S. Kuzmenko, A short distance quark- anti-quark potential", hep-ph/0302072. 24. E.V. Shuryak, Probing the boundary of the nonperturbative QCD by small size instantons", hep-ph/9909458. 25. S. Narison and V.I. Zakharov, Phys. Lett. B522, 266 (2001).

S T R U C T U R E OF T H E QCD C O N F I N I N G STRING*

M. I. POLIKARPOV f ITEP, B. Cheremusahkinskaya 25, Moscow 109028, Russia E-mail: [email protected] V. G. BORNYAKOV Institute for High Energy Physics, Protvino 142284, Russia E-mail: [email protected] G. SCHIERHOLZ NIC/DESY Zeuthen, Platanenallee 6, D-15738 Zeuthen, Germany E-mail: [email protected] T. SUZUKI Institute for Theoretical Physics, Kanazawa University, Kanazawa 920-1192, Japan E-mail: [email protected]

The structure of the confining string in lattice QCD with two dynamical quarks is described in the Abelian projection. The gluon field inside baryon made from three infinitely heavy quarks is also discussed.

* All results presented in this review are obtained by DIK (DESY-ITEP-Kanazawa) collaboration. We are very grateful to the members of this collaboration: M. N. Chernodub, H. Ichie, S. Kitahara, Y. Koma, Y. Mori, Y. Nakamura, D. Pleiter, D. Sigaev, A. A. Slavnov, T. Streuer, H. Stuben, P. V. Uvarov and A. I. Veselov. + Work partially supported by grants RFBR 02-02-17308, RFBR 01-02-17456, DFGRFBR 436 RUS 113/739/0, INTAS-00-00111 and CRDF award RPI-2364-MO-02.

193

194

1. Introduction This brief review is based on the results of papers [1, 2]. Confinement of color can be naturally explained in the dual superconductor model of QCD vacuum [3, 4, 5] and we describe the confining string (flux tube) in terms of the Abelian degrees of freedom in the Maximally Abelian gauge [6]. In this gauge the dynamical degrees of freedom are monopole currents and Abelian gauge fields. The use of Abelian projection allows to reduce substantially the statistical noise. We discuss the results of the numerical simulation of lattice QCD with two dynamical quarks. To detect the effects of light dynamical quarks on the confinement mechanism we compare results obtained in the full QCD with that obtained in the quenched QCD. The lattice spacing in full and in quenched QCD is RS 0.1 fm. We use the dynamical fermion configurations produced by the QCDSF and UKQCD collaborations [7] with the improved fermionic Wilson action : S = Sw ~ ^gcSw^2ip(s)<J^Fin/(s)il)(s),

(1)

s

where Sw is the standard Wilson action including fermionic and gauge field terms and the improvement coefficient csw is determined nonperturbatively [8]. 2. Gluon Fields inside Meson In this Section we study the gluon fields in the Abelian projection in the meson made from infinitely heavy (static) quarks. 2.1. Abelian

Operators

in the Field of the Confining

String

In the Abelian projection all operators are expressed through the diagonal SU(3) link matrices diag(u/, (s),u^ (s),uji (s)), where uji (s) = exp{i8^'(s)} are Abelian variables which are constructed through the nonabelian link matrices ?7M(s) in the following way: 6»W(a) = arg(tf<»>(S)) - | * M ( « ) e [ - | T T , \K] ,

(2)

3

*„(*) = £ > r g ( [ / W ( S ) ) mod 2TT 6 [-n,n).

(3)

To calculate the expectation value of an Abelian operator O(s) — d i a g ( 0 ( 1 ) ( s ) , 0 ( 2 ) ( s ) , 0 ( 3 ) ( s ) ) in the background of the Abelian flux tube

195

we calculate the following correlators [9]:

for C-parity even operators O (the action density and the monopole density). For C-parity odd operators O (the color electric field and the monopole current) we use the following correlators: { {S))W

°

=

(TrWc)

(5)

•

These definitions can be considered as the generalization of correlators used in the U(l) and the Abelian projected SU(2) theories [10, 11, 9, 12]. They can also be considered as Abelian projection of the connected correlators of the nonabelian field strength operators with the nonabelian Wilson loops (see e.g. review [13]). The Abelian Wilson loop,

WC = Y[diag(u^(s),u^(s),u^(s)), sec

(6)

is defined on the rectangular R x T contour with smeared spatial links. We measure the profile of the Abelian flux tube with the static sources placed on the x axis at points x = —R/2 and x = R/2 and operator O(s), s = {x,y,z,t} placed at t = y . The distance from the point s to the line connecting static sources at t = y is denoted by r±. To calculate the action density, PA(S), the following operator is used: 0{s) = ! £

diag (cos(6$(*)), cos(0#( S )), cos(«>$( S ))) ,

(7)

where 6^1 are the plaquette angles: O^l = 6^ (s + i>) — 6^ '(s) — 6l (s + A) + ®v (s) • The relation between the action densities in Euclidean and in Minkowski space is discussed in refs. [10, 9]. The operator corresponding to the color electric field Ej(s) is defined similarly to SU(2) case [9]: Oj(s) = diag (t0g>( s ), i0
.

(8)

Note that only the imaginary part of the Wilson loop contributes to the numerator of (5) for this observable.

196

The monopole currents are denned in a standard way [14] through plaquette angles Qp :

fc(')rs,M) = ^ - E 0 p = { ° ' ± 1 ' ± 2 } '

(9)

where the summation is over all oriented faces P of the 3D cube C, which is dual to the site *s of the dual lattice and orthogonal to the direction p,. The plaquette angles QP are obtained from plaquette angles d^l by proper shift to satisfy condition [15]: X)/=i ©p = 0- Thus magnetic currents satisfy condition X)/=i ^l\*s,fj.) = 0, which means that there are only two independent monopole currents in compact U(l)xU(l) gauge theory. The expectation value of the operator OjCs) = 2vi diag{k^Ca,j),

k^3\*s,j))

k^(*s,j),

(10)

corresponding to the monopole current is defined via (5). The local monopole density p(*s) is calculated using (4) with the operator 0

(* s ) = j £ d i a s ( > ( 1 ) ( * s ' ^ l ' \ki2)(*s,»)\, |* (3) r*,/i)|) •

2.2. Abelian

flux

(11)

tube

Below we present numerical results obtained on 163 • 32 lattice for the Wilson loop Wab(-R) T) with R = 10, which corresponds to the test quarks separation of the order of 1 fm, and T — 6. The space-like links are smeared. We use configurations generated at /? = 5.2, K = 0.1355 for full and at /3 — 6.0 for quenched QCD. In Fig. 1 and Fig. 2 we show the action density, PA{S)- In these figures and below we express dimensional quantities through the "force parameter" r^ « (394 Mew) - 1 . It occurs that the action density in full QCD is higher than in the quenched case while their shapes are quite similar. We estimate the width S of the Abelian flux tube fitting the numerical data by the function*: (r±-6)2

PA (r±, x — 0) = const e

s2

.

(12)

We obtain equal width, S = 0.29(1) fm, for quenched and for full QCD. a

T h e fitting parameter e is the O(a) displacement which is due to the spatial asymmetry of our definition of the action density operator.

197

Figure 1. The action density PA(S)TQ quenched (below) QCD.

of the Abelian flux tube in full (above) and in

In Fig. 3 and in Fig. 4 we show the color electric field. One can see only small differences between distributions obtained in full and in quenched QCD. Fig. 3 shows that the electric field is purely longitudinal in a region between the sources as we expect for the flux tube. The fit of Ex at x = 0, z = 0 for y/ro > 0.5 by the function Ex = const e - ^

(13)

gives the penetration length A = 0.15(1) fm in full and 0.17(1) fm in the quenched QCD. In Fig. 5 the local monopole density p(s) (11) near the flux tube is shown. We see that the monopole density, outside the flux tube, is more

198

Figure 2. Distribution of the action density PA(S)VQ at the center of the flux tube in full and in quenched QCD.

v s*Z~

\ <

H"~

Figure 3. Distribution of the color electric field in full (top figure) and quenched (bottom figure) QCD.

199

- -

/

I

' - '

t

Figure 4. Distribution of the color electric field at the center of the flux tube in full and quenched QCD.

than two times larger in full QCD than in quenched case. Inside the flux tube the monopole density is strongly suppressed. This fact is in accordance with the vanishing of the expectation value of the Higgs field inside the flux tube in the dual superconductor model of the vacuum. In this model we also expect that the monopole current forms a solenoidal (i.e. azimuthal) supercurrent which squeezes the electric field into flux tube and satisfies the dual Ampere law k = VxE.

(14)

In Fig. 6 the transversal components of the monopole current in the plane perpendicular to the Abelian flux tube at x = 0 are shown in the full and in the quenched QCD. In Fig. 7 we compare the l.h.s and r.h.s. of Eq. (14) for R = 6 (we use R = 6 instead of R = 10 to get reasonably small statistical errors). It is seen that the Ampere law is reasonably well satisfied in quenched and in full QCD. The dual Ampere's law has been verified earlier in pure SU(2) gauge theory [10, 9]. 3. Gluon Fields inside Baryon In this Section we study the gluon fields in the Abelian projection in the baryon made from infinitely heavy (static) quarks. The gauge field

200

Figure 5. The local monopole density P(*S)TQ quenched (below) QCD.

near the flux tube in full (above) and

configurations were generated on 24 3 • 48 lattice at /3 = 5.29, K = 0.1355 (rn„/mp ~ 0.7). 3.1. Abelian

Operators

in the Field of

Baryon

The lattice study of the three quark system is important for the understanding of the baryon structure. The question whether there exists the genuine three body force (Y shape flux tube), or the long range force is

201

t ,

' (

. » . , . . * . . » , \ - , ' ~ . /. ' > ' ^ \ . , , /

"* 1

. v

x

, .

,

_

_

i

.

.

_

x

.

/

-N

,

.,

v

"-»

'

v

-

-

\ _

> \ -

»

^

v

' ,

,

.

'

'

/

,

.

,

-

'

x

•v

.

rN

»

,

, ,

,

'

-

,

•

..

„ ^

-

,

,

-

v

-

y

'

v

-

-

-

-

v

>

'

v

^ _

-

»

.

v

' • — > , . , -

-

'

1

\ ^ * /

» >

\

1

' -

.

,

- V

. . . / . ,

„ -

.

.

~

.

1

/ -

> '

) ' * ^ ^ ' - ' ' - -

' ' ' '

« ' ' / • , /

' - ' " /

' '

Figure 6. The solenoidal monopole current k r j at the center of the flux tube in the (yz) plane at x = 0 in full (above) and quenched (below) QCD.

the sum of the two-body forces (A shape flux tube) is widely discussed [16, 17, 18, 19, 20, 21]. Some lattice calculations support the A-shape [18, 19], whereas the others find evidence for the F-shape [21, 20] potential. This discrepancy existed for a long time due to the fact that the difference between the two ansatze for the baryonic potential is rather small, while the evaluation of the potential is difficult. Thus the direct measurement of the gluon field flux inside baryon is important [2]. The propagation of the baryon from point A to point B is described by

202

2

1

•

1

1.5 ~ .

1 s

X

1

o VxEr03

•

©

'

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x

©w

kr

3

80

*•

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0.5 ~

@© © ^x®© M e M

00

Figure 7.

;: . O

9

0.5

T

x© ©

*

«.*

•-

,

1.5

Check of the dual Ampere's law in full (above) and quenched (below) QCD.

the following operator (baryon Wilson loop [19, 21]): W3o =

3!

j U l-eijkeiljC'kx'u^Ui C2 U'u^' C3

(15)

where Uc = Yliec ^ ' ' s t n e P r °duct of the link matrices Ui along path C; Ck

203

rR

Blue quark Figure 8.

i f Green quark Three quark Wilson loop.

are trajectories of the (infinitely heavy) quarks connecting points A and B. In practical calculations it is convenient to choose staple-like trajectories Ck, see Fig. 8. If the points A and B are separated by the time interval T then the potential energy of such baryon is: V(x\, X2, £3) = — limT-j-oo Y In Wzq. In the Abelian projection the link matrices are substituted by the diagonal (3)/ .(')/(s) = e o(0 («) and the .(!)'(s),Uf,( 2 ) ,( s ) , .% matrices u* b (s) = diag{u/j (s)}, Up Abelian baryon Wilson loop is given by: (16) ! s where u,(') ^ =- FL M}ecm u^.(')/ ( ) a r e products of Abelian link variables along the paths connecting points A and B. W^ is invariant under

U(l)xU(l) gauge transformations u\i (s) —> eta^Uf,'(s),

ei0^ui2\s),

u{l!\s)-^e- • i a ( « ) - » / ? ( s ) , . ( 3 )

M}/ (s) -»

(S).

The local quantities describing the baryonic Abelian flux are defined as expectation values in the presence of the Wilson loop [2]. For C-parity even quantities like the action density and the monopole density the expectation value is defined as:

_ (0Wgb)

((D)

(17)

204

Figure 9.

Abelian action density for the baryon made from static quarks.

For C-parity odd quantities such as electric field or monopole current, which carry the color index, the expectation value is defined as:

{0{s))3q

p^-

,

(18)

where the summation over indexes i,j,k is assumed. For noise reduction we used smearing of spatial links for the baryon Abelian Wilson loop.

3.2. Numerical

Results

In Fig. 9 we show the baryon Abelian action density for the full QCD. Three peaks corresponding to the static quarks and the F-shape flux tube, which is responsible for the confining potential are clearly seen. The Abelian action density have a clear peak in the center of the 3 quark system. The electric field and the monopole current in the considered 3g system is shown in Fig. 10. It is clearly seen that the solenoidal monopole current appears for each electric flux. These results are in a agreement with the predictions of the dual Ginzburg-Landau model of QCD vacuum [22, 23, 24].

205 monopole c u r r e n t . : : : : : ::•.:•.::•.:

y±m

--\ \ > • \\

monopole current

V

'it'

x electric field

x=21

x 2=29

monopole c u r r e n t

y

Figure 10. One component of the Abelian electric field and monopole current in the 3q system in full QCD. References 1. V. Bornyakov, H. Ichie, S. Kitahara, Y. Koma, Y. Mori, Y. Nakamura, M. Polikarpov, G. Schierholz, T. Streuer, H. Stuben, T. Suzuki, Nucl.Phys. Proc.Suppl. 106, 634 (2002); V. Bornyakov, Y. Nakamura, M. Chernodub, Y. Koma, Y. Mori, M. Polikarpov, G. Schierholz, A. Slavnov, H. Stuben, T. Suzuki, P. Uvarov and A. Veselov, preprint DESY-02-135, Sep 2002, e-Print Archive: heplat/0209157; V. Bornyakov et al, DIK-collaboration, preprint ITEP-LAT-2002-23, KANAZAWA-02-32, Dec 2002, e-Print Archive: hep-lat/0212023; V. Bornyakov, M. Chernodub, Y. Koma, Y. Mori, Y. Nakamura, M. Polikarpov, G. Schierholz, D. Sigaev, A. Slavnov, H. Stuben, T. Suzuki, P. Uvarov and A. Veselov, preprint ITEP-LAT-2002-31, KANAZAWA-02-40, Jan 2003, e-Print Archive: hep-lat/0301002; Y. Mori, V. Bornyakov, M. Chernodub, Y. Koma, Y. Nakamura, T. Suzuki, M. Polikarpov, D. Sigaev, P. Uvarov, A. Veselov, A. Slavnov, G. Schierholz and H. Stuben, preprint ITEP-LAT-2002-29, KANAZAWA-02-41, Jan 2003, e-Print Archive: hep-lat/0301003. 2. H. Ichie, V. Bornyakov, T. Streuer and G. Schierholz, to appear in Nucl. Phys. B (Proc. Suppl.) (2003), hep-lat/0212024; H. Ichie, V. Bornyakov, T. Streuer and G. Schierholz, to appear in Nucl. Phys. B (2003), hep-lat/0212036. 3. Y. Nambu, Phys. Rev. D 1 0 , 4262 (1974). 4. G. 't Hooft, "High Energy Physics", ed. A. Zichichi (Editorice Compositori,

206 Bologna, 1975). 5. S. Mandelstam, Phys. Rep. C23, 245 (1976). 6. A.S. Kronfeld, M.L. Laursen, G. Schierholz and U.-J. Wiese, Phys. Lett. B198, 516 (1987). 7. S. Booth, M. Gockeler, R. Horsley, A.C. Irving, B. Joo, S. Pickles, D. Pleiter, P.E.L. Rakow, G. Schierholz, Z. Sroczynski and H. Stiiben [QCDSF-UKQCD collaboration], Phys. Lett. B519, 229 (2001). 8. K. Jansen and R. Sommer, Nucl. Phys. B530, 185 (1998). 9. G.S. Bali, Ch. Schlichter and K. Schilling, Phys. Rev. D 5 1 , 5165 (1995); Prog. Theor. Phys. Suppl. 122, 67 (1996). 10. R.W. Haymaker, V. Singh, Y.-C. Peng and J. Wosiek, Phys. Rev. D 5 3 , 389 (1996). 11. M. Zach, M. Faber, W. Kainz and P. Skala, Phys. Lett. B358, 325 (1995). 12. S. Cheluvaraja, R. W. Haymaker and T. Matsuki, e-Print Archive: heplat/021001. 13. H. G. Dosch, V. I. Shevchenko and Y. A. Simonov, Phys. Rept. 372, 319 (2002). 14. T. DeGrand and D. Toussaint, Phys. Rev. D22, 2478 (1980). 15. M. I. Polikarpov and K. Yee, Phys. Lett. B316, 333 (1993). 16. R. Sommer and J. Wosiek, Phys. Lett. B149, 497 (1984). 17. H.B. Thacker, E. Eichten and J.C. Sexton, Seillac Sympos. 0234 (1987). 18. G.S. Bali, Phys. Rep. 343, 1 (2001). 19. C. Alexandrou, Ph. de Forcrand, and A. Tsapalis, Phys. Rev. D65, 054503 (2002). 20. C. Alexandrou et a/., to appear in Nucl. Phys. (Proc. Suppl.) B (2003), ePrint Archive: hep-lat/0209062. 21. T.T. Takahashi, H. Suganuma, Y. Nemoto, and H. Matsufuru, Phys. Rev. D65, 114509 (2002); T.T. Takahashi, H. Matsufuru, Y. Nemoto, H. Suganuma, Phys. Rev. Lett. 86, 18 (2001). 22. S. Kamizawa, Y. Matsubara, H. Shiba and T. Suzuki, Nucl. Phys. B389, 563 (1993). 23. Y. Koma , E.-M. Ilgenfritz, T. Suzuki, H. Toki, Phys. Rev. D 6 4 , 014015 (2001). 24. M. N. Chernodub and D. A. Komarov, JETP Lett. 68, 117 (1998).

O N THEORY OF L A N D A U - P O M E R A N C H U K - M I G D A L EFFECT*

V. N. B A I E R A N D V. M. K A T K O V Budker Institute for Nuclear Physics, Novosibirsk, 630090, Russia E-mail: [email protected]; [email protected]

The cross section of bremsstrahlung from high-energy electron suppressed due to the multiple scattering of an emitting electron in a dense media (the LPM effect) is discussed. Development of the LPM effect theory is outlined. For the target of finite thickness one has to include boundary photon radiation as well as the interference effects. The importance of multiphoton effects is emphasized. Comparison of theory with SLAC data is presented.

1. Introduction The process of photon radiation takes place not in one point but in some domain of space-time, in which a photon (an emitted wave in the classical language) is originating. The longitudinal dimension of this domain is called the formation (coherence) length If Z / o H = 2ee' /urn2,

(1)

where e(e') is the energy of the initial (final) electron, to is the photon energy, e' = e — OJ, m is the electron mass, we employ units such that h = c — 1. So for ultrarelativistic particles the formation length extends substantially. For example for e = 25 GeV and emission of w = 100 MeV photon, // = lOyum, i.e.~ 105 interatomic distances. Landau and Pomeranchuk were the first to show that if the formation length of the bremsstrahlung becomes comparable to the distance over which the multiple scattering becomes important, the bremsstrahlung will be suppressed l. They considered radiation of soft photons. Migdal 2 "This work is partially supported by grant 03-02-16154 of the Russian Fund of Fundamental Research

207

208

developed a quantitative theory of this phenomenon. Now the common name is the Landau- Pomeranchuk -Migdal (LPM) effect. Let us estimate a disturbance of the emission process due to a multiple scattering. As it is known, the mean angle of the multiple scattering at some length If is 0 . = \ / ^ f = £s/eyJlf/2Lrad,

es = m^/An/a

= 21.2 MeV,

(2)

where a = e2 = 1/137, Lrad is the radiation length. Since we are interesting in influence of the multiple scattering on the radiation process we put here the formation length (1). One can expect that when tfs > 1/7 this influence will be substantial. From this inequality we have z'jeLP

> u/e,

eLp = miLrad/e'2s,

(3)

here ELP is the characteristic energy scale, for which the multiple scattering will influence the radiation process for the whole spectrum. It was introduced in 1 and denoted by EQ = eip. For tungsten we have ELP — 2.72 TeV, and similar values for the all heavy elements. For light elements the energy ELP is much larger. When e LP = E2/ELP, e.g. for particles with energy e = 25 GeV one has U>LP — 230 MeV. Of course, we estimate here the criterion only. For description of the effect one has to calculate the probability of the bremsstrahlung taking into account multiple scattering. 2. The LPM effect in a infinite medium We consider first the case where the formation length is much shorter than the thickness of a target 1(1 /o ^ /)• The basic formulas for probability of radiation under influence of multiple scattering were derived in 3 . The general expression for the radiation probability obtained in framework of the quasiclassical operator method (Eq.(4.2) in 4 )was averaged over all possible particle trajectories. This operation was performed with the aid of the distribution function, averaged over atomic positions in the scattering medium and satisfying the kinetic equation. The spectral distribution of the probability of radiation per unit time was reduced in 5 to the form /•OO

dW/du = (2a/7 2 )Re / Jo

dte'11 [RiMW)

+ #2P¥>(0,i)],

(4)

209

where Ri = cu2/ee', the equation i ^

= H^,

+ l n £! +

R2 = e/e' + e'/e, and the functions y?M (
H = p2-iV(g),

2C-l),

P

Q^^azyee'

= -iVQ,

V{Q) =

-QQ2(\n1H\

C = Q57?216

(5)

4 / m^u with the initial conditions <po{g,0) = 6(g),
—

+ In g2/4 + 2C) ,

(6)

l/2)> the function / = / ( Z a ) is 00

no = Re [va+io - v(i)] = e E vw* 2 +£ 2 )),

(7)

71=1

where ip(0 is the logarithmic derivative of the gamma function. In above formulas g is two-dimensional space of the impact parameters measured in the Compton wavelengths Ac, which is conjugate to space of the transverse momentum transfers measured in the electron mass m. An operator form of a solution of Eq. (5) is
= < g\exp(-iHt)\0 0) = < g\ exp(-iHt)p\0

>, >,

H = p2

-iV(g), (8)

where we introduce the following Dirac state vectors: \g > is the state vector of coordinate g, and < g\0 > = 5(g)- Substituting (8) into (4) and taking integral over t we obtain for the spectral distribution of the probability of radiation ^

= ^ImT,

T =< 0|iii (G-1 - Go1) + R2p (G~l - G^1) p|0 >, (9)

210

where G = p 2 + 1 - iV,

Go = P 2 + 1.

(10)

Here and below we consider an expression < 0|...|0 > as a limit: lim x -> 0, lim x' -» 0 of < x|...|x' >. Now we estimate the effective impact parameters gc which give the main contribution into the radiation probability. Since the characteristic values of gc can be found straightforwardly by calculation of (9), we the estimate characteristic angles $ c connected with gc by an equality gc = l / ( 7 # c ) . The mean square scattering angle of a particle on the formation length of a photon If (1) has the form Q2

4nZ2a2

, ,

C

*' = — ^ 5 — n « ' / l n ^ 2

40 = ^

l

( n

^ 2

>


where £ = 1 + 7 2 $ 2 , we neglect here the polarization of a medium. When fl2. >C I / 7 2 the contribution in the probability of radiation gives a region where £ ~ l(t9c = 1/7), in this case gc — 1. When i?s 3> I / 7 the characteristic angle of radiation is determined by self-consistency arguments: dzs ~ v% ~ —z = ——zln-z-T?, 72 Cc72 7 2 ^i

-5-ln-—-r = l, Cc2 72tf?

AQgl In 2 n9 = 1. 7 ^i^? (12)

It should be noted that when the characteristic impact parameter gc becomes smaller than the radius of nucleus Rn, the potential V(g) acquires an oscillator form (see Appendix B, Eq.(B.3) in 5 ) V(g) = Qg2(\na2JR2n-

0.041)

(13)

Allowing for the estimates (12) we present the potential V(g) (5) in the following form V(Q) =

VC(Q)

+ V{Q),

VC{Q)

= qg2, q = QLC,

Lc = L(gc) = In

2,

7 "iBc ilSi

(l) =ln-i|=1„|,

„(<,) = ^ (

2 C + ln ^).

(14)

The inclusion of the Coulomb corrections (f(Za) and -1) into ln$?, diminishes effectively the correction v(g) to the potential Vc{g). In accordance with such division of the potential we present the propagators in expression (9) as G^-GQ1

= G~l

- G~l

+ G~l

- G^1

(15)

211

where Gc = p 2 + 1 - iVc,

G = p 2 + 1 - iVc - iv

This representation of the propagator G _ 1 permits one to expand it over the "perturbation" v. Indeed, with an increase of q the relative value of the v 1 perturbation is diminished (— ~ —) since the effective impact parameters diminishes and, correspondingly, the value of logarithm Lc in (14) increases. The maximal value of Lc is determined by the size of a nucleus Rn 2

L

m o x

2

=ln^~21n^=2Li,

Lx = 2(ln(183Z- 1 / 3 ) - f(Za)),

(16)

where as2 = a s exp(—/ + 1/2). So, one can to redefine the parameters as and $i to include the Coulomb corrections. The value L\ is the important parameter of the radiation theory. The matrix elements of the operator G~l can be calculated explicitly. The exponential parametrization of the propagator is /•OO

G'1 = i /

dte-u exp(-iHct),

Hc = p 2 - iqg2

(17)

The matrix elements of the operator exp(—iHct) has the form (details see in5) < g1\exp(-iHct)\g2

>= Kc(g1,g2,t), iv

4wi sinh vt exp

{ei

+

Kc(g1,g2,t) Q2)Cothvt--^-tg1g2

(18)

where v = I^JTq (see (14)). Substituting formulae (17) and (18) in the expression for the spectral distribution of the probability of radiation (9) we have

^7 = 2^Im

^

/•CO

$(i/) = v \ dte~a [i?i ( 1 / sinh^ - 1/z) - ivR2 ( 1 / sinh2 z - 1/z2)} Jo = Rx (lnp - rl> (p + 1/2)) + R2 {ijj (p) - lnp + l/2p)

(19)

where z = i/t, p = i/(2v), let us remind that ip(x) is the logarithmic derivative of the gamma function (see Eq.(7)). If in Eq.(19) one omits the Coulomb correction, then the probability (19) coincides formally with the probability calculated by Migdal (see Eq.(49) in 2 ).

212

We now expand the expression G G - 1 - G-1 = G-\iv)G:1

1

— Gc 1 over powers of v

+ Gj1(iv)G71(iv)G71

+ ...

(20)

In accordance with (15) and (20) we present the probability of radiation in the form dW/du = dWc/du + dWi /du + dW2/du + ...

(21)

At Q > 1 the expansion (20) is a series over powers of 1/L. It is important that variation of the parameter gc by a factor order of 1 has an influence on the dropped terms in (20) only. The probability of radiation dWc/du> is denned by Eq.(19). The term dW\/du) in (21) corresponds to the term linear in v in (20). The explicit formula for the first correction to the probability of radiation 5 is dW-\ a 1 ~ = - . 2T Im F(v); - I m F[y) = I>i(*b)i?i + -D2(u0)R2; aw 47r7^Lc s 00 00 J sz rc f dze~ — - " r7r I f°° dze Di{vo) = / , , o d(z)sinsz+-g(z)cossz\, D2(v0) = / Jo smh z L 4 J y0 sinh z ( ) ~ 2^(z) (sin sz + cos sz) + —g(z) (cos sz — sin sz) f,

d z

d(z) = (lni/ 0 i?(l-i> 0 ) - l n s i n h z - C ) ^ ) - 2 c o s h 2 G ( z ) , s = l/\/2z/ 0 ,(22) where p(z) = 2coshz — sinhz,

G(z) = / (1 — ycothy)dy Jo

= z - z2/2 - TT 2 /12 - ^ In (1 - e- 2 z ) + Li2 (e" 2 2 ) / 2 ,

(23)

here Li2 (x) is the Euler dilogarithm; and v2 = H 2 = Aq = 4QL{gc) = 8irnaZ2a2£e'L{Qc)/m4LO,

(24)

As it was said above (see (12), (16)), gc = 1 at \v2\ = v2=AQLl<\. (25) The logarithmic functions Lc = L(gc) and 1^ are defined in (14) and (16). If the parameter \i/\ > 1, the value of gc is defined from the equation (12), where i?i —> d2, up to gc — Rn/Xc. So, we have two representation of \v\ depending on gc: at gc = 1 it is \v\ = v\ and at £>c < 1 it is \v\ = VQ. The

213

mentioned parameters can be presented in the following form S=ivl

vi = H 2 c v\ ( l + ^ - r f ( " l - 1 ) ) , \

ee=m

(8TrZ2a2naX3cLiyl,

Ly LC~LJI

vl = - —

j +

l

-~#{vi

£e

,

X

- l ) ] , i = - . (26)

It should be noted that in the logarithmic approximation the parameter gc entering into the parameter v is defined up to the factor ~ 1. However, we calculated the next term of the decomposition over v(g) (an accuracy up to the " next to leading logarithm") and this permitted to obtain the result which is independent of the parameter gc (in terms <x 1/L). Our definition of the parameter gc minimizes corrections to (19) practically for all values of the parameter gc. The approximate solution of Eq.(12) given in this formula has quite good numerical accuracy:it is ~ 2 % at V\ = 100 and ~ 4.5 % at V\ = 1000.The LPM effect manifests itself when vi(xc)

= l,

xc = uc/e = e/(ee + e).

(27)

So, the characteristic energy ee is the energy starting from which the multiple scattering distorts the whole spectrum of radiation including its hard part. If the radiation length Lraa is taken within the logarithmic approximation the value ee coincides with ELP Eq.(3). The formulas derived in 5 , 6 and written down above are valid for any energy. In Fig.l the spectral radiation intensity in gold (ee = 2.5 TeV) is shown for different energies of the initial electron. In the case when e -C ee (e = 25 GeV and e = 250 GeV) the LPM suppression is seen in the soft part of the spectrum only for x < xc — e/ee <S 1 while in the region e > ee (e = 2.5 TeV and e = 25 TeV) where xc ~ 1 the LPM effect is significant for any x. For relatively low energies e = 25 GeV and e = 8 GeV used in famous SLAC experiment 7 , 8 we have analyzed the soft part of spectrum, including all the accompanying effects: the boundary photon emission, the multiphoton radiation and influence of the polarization of the medium (see below). The perfect agreement of the theory and data was achieved in the whole interval of measured photon energies (200 keV< UJ <500 MeV), see below and the corresponding figures in 5 , 9 , 10 . It should be pointed out that both the correction term with F{v) and the Coulomb corrections have to be taken into account for this agreement. When a scattering is weak (i^i
214

rrm]

0.0001

1 i i IIIIIJ

1 i i Mini

1 i i iinii—i

0.0010 0.0100 Photon e n e r g y x

0.1000

i iinn

1.0000

Figure 1. The spectral intensity of radiation wdW/dui = xdW/dx, x = ui/e in gold in terms of 3L r a £ j taken with the Coulomb corrections. Curve BH is the Bethe-Maximon spectral intensity (see Eq.(29)); curve 1 is the logarithmic approximation udWc/duj Eq.(19), curve cl is the first correction to the spectral intensity uidWi/duj Eq.(22) and curve T l is the sum of the previous contributions for the electron energy e = 25 GeV; curves 2, c2, T2 are the same the electron energy e = 250 GeV; curves 3, c3, T3 are the same for the electron energy e = 2.5 TeV; curves 4, c4, T4 are the same for the electron energy e = 25 TeV.

215

a region where z -C 1. Then - I m F M = Im i/2 (i? 2 - i J i ) / 9 , 2

$(1/) ~ i/ (i?i + 2i?2) /6,

LC-*LX,

(|i/| « 1)

(28)

Combining the results obtained in (28) we obtain the spectral distribution of the probability of radiation in the case when scattering is weak {\v\ -C 1) dW _ dW^ dco du>

dW^ _ 4Z2a3na duj

rur,2

(ln(l83Z-1/3)

+ 2 f 1 + p - U l n (183Z- 1 / 3 ) + 1 - / ( Z a ) ) ] ,

- \ - f(Za))

(29)

where Li is defined in (16). This expression coincide with the known BetheMaximon formula for the probability of bremsstrahlung from high-energy electrons in the case of complete screening (if one neglects the contribution of atomic electrons) written down within power accuracy (omitted terms are of the order of powers of 1/7) with the Coulomb corrections, see e.g. Eq.(18.30) i n 1 1 . At i/0 > 1 the function Im F{v) Eq.(22) has the form - I m F(v) = irRi/A + v0{\n2-C

+ TT/4)R 2 /V2.

(30)

Under the same conditions (VQ 3> 1) the function Im 4>(i/) (19) is Im^{v)

= nRi/A + uoR2/V2^u0R2/\/2.

(31)

So, in the region where the LPM effect is strong the probability (19) can written as dWc aR2 (Z2a2ee'naT, V -7— = —yHQC) diO

Ez

\

TTU!

/ 2

•

(32)

J

This means that in this limit the emission probability is proportional to the square root of the density. This fact was pointed out by Migdal 2 (see Eq.(52)). dW\ Thus, at i/o > 1 the relative contribution of the first correction —— is du denned by r = dW1/dWc = {\n2-C where L(gc) = In

2 as2

+ 7r/4)/2L(gc)

~0A51/L(gc),

(33)

. In this expression the value r with the accuracy

*cQc

up to terms ~ \jL\ doesn't depend on the energy:L c ~ L\ + \n(e/ee)/2. Hence we can find the correction to the total probability at e S> ee. The

216

maximal value of the correction is attained at e ~ 10ee, it is ~ 6% for heavy elements. 3. Target of finite thickness 3.1. Boundary

effects for a thick

target

For the homogeneous target of finite thickness I the radiation process in a medium depends on interrelation between I and formation length l/o (1). In the case when I ^> lf0 we have the thick target where radiation on the boundary should be incorporated. In the case when I <S Ifo we have the thin target where the mechanism of radiation is changed essentially and in the case when I ~ Ifo we have intermediate thickness. The spectral distribution of the probability of radiation of boundary photons can be written in the form 5 ^ M=(-±^—), (34) = ^ R e ( 0 | r i M + r2pMp|0), au LO \H + K, pz +1J where K = 1 + K02, K02 = K o £ '/ £ > Ko — up/u), LOP = wo7, <^o = 47rcme/m, ne is the electron density, LOQ is the plasma frequency, H is defined in Eq.(5), ri = R\e'je = w 2 /e 2 , r2 — i?2e'/e = 1 + e 2/e2, the parameter K0 describes the polarization of medium. In the case when both the LPM effect and effect of polarization of a medium are weak one can decompose combination in M in (34)

HT^-TTT^TTI^-^^TI-

(35)

In the case when i/i < K„ > 1 one can omit the potential V in H (34), so that the effect of polarization of a medium is essential, then dwb 4a rxM + r2pMp\0)d2p dw LU(2

W<°'

— \ri ( l + TTLU I

V

- l n / s ) +r2 K

K—

1

1+

) In K - 2

(36)

K — 1

This result is the quantum generalization of the transition radiation probability. 3.2. A thin

target

This is a situation when the formation length of radiation is much larger than the thickness I of a target 9

l«J/=WC>

C = l + 72
T = f/i/0
(37)

217

where Z/o are defined in Eq.(l). In the case KT
[nKtie)

+ r2K2(g)] (1 - exp(-V(g)T)),

(38)

where V(g) is defined in (5), (6), Kn is the modified Bessel function.

3.3. Multiphoton

effects in energy loss

spectra 7 8

It should be noted that in the experiments , the summary energy of all photons radiated by a single electron is measured. This means that besides mentioned above effects there is an additional " calorimetric" effect due to the multiple photon radiation. This effect is especially important in relatively thick used targets. Since the energy losses spectrum of an electron is actually measured, which is not coincide in this case with the spectrum of photons radiated in a single interaction, one have to consider the distribution function of electrons over energy after passage of a target 10

We consider the spectral distribution of the energy losses. After summing over n the probability of the successive radiation of n soft photons with energies ux,u>i,.. .u>k by a particle with energy e (w* <£ e, k = l , 2 . . . n ) under condition Y^=i ^k = w o n the length I in the energy intervals duidu>2 • • • dtjn we obtain 10 1

™ r°

— Re / n J0

/^

f

f°°dw

N

/ • "Mi J i J

exp (ix) exp < — / - — 1 — exp I — ix— [ J0 dujx I V w )\

dux > dx J (39)

The formula (39) was derived by Landau 12 (see also 1) as solution of the kinetic equation under assumption that energy losses are much smaller than particle's energy (the paper 12 was devoted to the ionization losses). The energy losses are defined by the hard part of the radiation spectrum. In the soft part of the energy losses spectrum (39) the probability of radiation of one hard photon only is taken into account accurately. So,it is applicable for the thin targets only and has an accuracy l/LradWe will analyze first the interval of photon energies where the BetheMaximon formula is valid. Substituting this formula for dw/dto (within the

218

logarithmic accuracy) into Eq.(39) we have d£

,

exp(-Bu)

„

41

, e

- = PfBMlfBM = ^ m , ^ ^ - , ^ i

a

5

--

„ l

+

c1 (40)

where T(z) is the Euler gamma function. If we consider radiation of the one soft photon, we have from (39) de/dw = (3. Thus, the formula (40) gives additional "reduction factor" JBM which characterizes the distortion of the soft Bethe-Maximon spectrum due to multiple photons radiation. The emission of accompanying photons with energy much lesser or of the order of w changes the spectral distribution on quantity order of (3. However, if one photon with energy ur > UJ is emitted, at least, then photon with energy ijj is not registered at all in the corresponding channel of the calorimeter. Since mean number of photons with energy larger than to is determined by the expression oo

re

Y J nwn = ww =

(dw/dto) dio,

(41)

where wn is the probability of radiation of n soft photons. So, when radiation is described by the Bethe-Maximon formula the value wu increases as a logarithm with u decrease (wu ~ /?lne/w) and for large ratio e/to the value ww is much larger than /3. Thus, amplification of the effect is connected with a large interval of the integration (u -f- e) at evaluation of the radiation probability. If we want to improve accuracy of the formula (39) and for the case of thick targets / > Lra(i one has to consider radiation of an arbitrary number of hard photons. This problem is solved in Appendix of 10 for the case when hard part of the radiation spectrum is described by the BetheMaximon formula. In this case the formula (39) acquires the additional factor. As a result we extend this formula on the case thick targets. The Bethe-Maximon formula becomes inapplicable for the photon energies w < u>c, where LPM effect starts to manifest itself (see Eq.(27)). Calculating the integral in (39) we find for the distribution of the spectral energy losses

£= 3 "\&*" /i™=(i+IW3eip<-"0'

m

where JLPM is the reduction factor in the photon energy range where the LPM effect is essential, wc = 0 (ln£/w c + C2),

C2 ^ 1.959.

(43)

219

In this expression the terms ~ 1/L (see Sec.2) are not taken into account. In the region u p (see Eq.(34)) where the contribution of boundary photons dominates the reduction factor for transition radiation ftr was found in 10 . 3.4. Discussion

of theory and

experiment

Here we compare the experimental data 7 , 8 with theory predictions. According to Eq.(27) the LPM effect becomes significant for to < uc = e 2 /e e . The mechanism of radiation depends strongly on the thickness of the target. The thickness of used target in terms of the formation length is

/3(u) = T±- = T(v0 + T

=^2'

wp = o;o7,

K)~Tc

0L+ U)c

Tc=T(u,c)~

w V U>c

I

p U)U)C

— -4-,

(44)

where we put that I/Q — \ — (see Eq.(26)). Below we assume that wc 3> u>p v to which is true under the experimental conditions. So we have that Tc = 2Trl/aLrad > 20

at

l/Lrad

> 2 %.

(45)

For Tc 3> 1 (/3(wc) ~S> 1) the minimal value of the ratio of the thickness of a target to the formation length follows from Eq.(44):/?m ~ 2T c (w p /w c ) 2 / 3 and is attained for the heavy elements (Au, W, U) at the initial energy £ = 25 GeV. In this case one has u>c ~ 250 MeV, up ~ 4 MeV, pm > 2.5. As an example we calculated the spectrum of the energy losses in the tungsten target with thickness / = 0.088 mm (=2.5 %Lrad) for e=25 GeV. The result is shown in Fig.2. We calculated the main (Migdal type) term Eq.(19), the first correction term Eq.(22) taking into account an influence of the polarization of a medium, as well as the Coulomb corrections entering into parameter VQ Eq.(24) and value L(gc) Eq.(14). We calculated also the contribution of boundary photons (see Eq.(4.12) in 5 ) . Here in the soft part of the spectrum OJ < ud (u)d — 2 MeV ) the transition radiation term Eq.(36) dominates, while in the harder part of the boundary photon spectrum u > ud the terms depending on both the multiple scattering and the polarization of a medium give the contribution. All the mentioned contribution presented separately in Fig.2. Under conditions of the experiment the multiphoton reduction of the spectral curve is very essential. It is taken into account in the curve "T". Experimental data are

220

Photon energy u> (MeV)

Figure 2. The energy losses udW/duj in tungsten with thickness ( = 0.023 mm in units 2a/-K. Curve 1 is the main (Migdal) contribution, curve 2 is the correction term, curve 3 is the sum of previous contributions, curve 4 is the contribution of boundary photons, curve 5 is the sum of the previous contributions. Curve T is the final theory prediction with regard to the reduction factor (the multiphoton effects).

221

taken from 7 . It is seen that there is a perfect agreement of the curve T with data. 4. Conclusion The LPM effect is essential at very high energies. It will be important in electromagnetic calorimeters in TeV range. Another possible application is air showers from the highest energy cosmic rays. Let us find the lower bound of photon energy starting from which the LPM effect will affect substantially the shower development. The interaction of photon with energy lower than this bound moving vertically towards earth surface will be described by standard (Bethe-Maximon) formulas. The probability wp(h) to find the primary photon on the altitude h is dwp/dh = wpexp(-h/h0)/lp,

wp(h) = exp f -h0e~h/h°/lp)

,

(46)

where lp ~ 9lr/7, lr is the radiation length on the ground level (lr ~ 0.3 km), ho — 8.7 km. The parameter i/p (see Eq.(26)) characterizing the maximal strength of the LPM effect at pair creation by a photon (e+ = £- = w/2) has a form v\ = (to/ue) exp(-h/h0)wp(h)

= (w/w e )exp(-/i 0 e" / l / h c /l p - h/h0),

(47)

where ue is the characteristic critical energy for air on the ground level (we = 4e e ~ 1018 eV). The last expression has a maximum at hm = h0 ln(/i 0 /Jp), so that vp(hm) = Lo/ujm, wm = ujeh0e/lp ~ 6- 1019eV. Bearing in mind that the LPM effect becomes essential for photon energy order of magnitude higher than u>m (e.g. the probability of pair creation is half of Bethe-Maximon value at u> = 12we) we have that the LPM effect becomes essential for shower formation for photon energy u ~ 5 • 1020 eV. It should be noted that GZK cutoff for proton is 5 • 1019 eV. So the the photons with the mentioned energy are extremely rare. References 1. L. D. Landau and I. Ya. Pomeranchuk, Dokl.Akad.Nauk SSSR 92, 535, 735 (1953). See in English in The Collected Papers of L. D. Landau, Pergamon Press, 1965. 2. A. B. Migdal, Phys. Rev. 103, 1811 (1956). 3. V.N.Baier, V.M.Katkov and V.M.Strakhovenko, Sov. Phys. JETP 67, 70 (1988). 4. V.N.Baier, V.M.Katkov and V.M.Strakhovenko, Electromagnetic Processes at High Energies in Oriented Single Crystals, World Scientific Publishing Co, Singapore, 1998.

222 5. V. N. Baier and V. M. Katkov, Phys.Rev. D57, 3146 (1998). 6. V. N. Baier and V. M. Katkov, Phys.Rev. D62, 036008 (2000). 7. P. L. Anthony, R. Becker-Szendy, P. E. Bosted et al, Phys.Rev.D56, 1373 (1997). 8. S.Klein, Rev. Mod. Phys. 7 1 , 501 (1999). 9. V. N. Baier and V. M. Katkov, Quantum Aspects of Beam Physics, ed.P. Chen, World Scientific PC, Singapore, 1998, p.525. 10. V. N. Baier and V. M. Katkov, Phys.Rev. D59, 056003 (1999). 11. V. N. Baier, V. M. Katkov and V. S Fadin, Radiation from Relativistic Electrons (in Russian) Atomizdat, Moscow, 1973. 12. L. D. Landau J.Phys. USSR 8 (1944) 201.

ABOUT THE

LANDAU-POMERANCHUK-MIGDAL EFFECT *

N. F. SHUL'GA Institute for Theoretical Physics, National Science Center "Kharkov Institute of Physics and Technology", 1 Akademicheskaja Street, Kharkov 61108, Ukraine Belgorod State University, Belgorod, Russia E-mail: [email protected] S. P. FOMIN Institute for Theoretical Physics, National Science Center "Kharkov Institute of Physics and Technology", 1 Akademicheskaja Street, Kharkov 61108, Ukraine E-mail: [email protected]

The effect of suppression of ultra relativistic electron radiation due to the multiple scattering in matter was predicted fifty years ago by Landau and Pomeranchuk. The brief review of pioneer works and also recent progress of theoretical and experimental investigations in this field is presented.

1. I n t r o d u c t i o n At the beginning of the 1950s Ter-Mikaelian [1], as well as Landau and Pomenranchuk [2] paid attention to the fact t h a t the process of radiation of a relativistic electron develops in a large spatial region along the direction of particle motion. T h e longitudinal size of this region is called the coherence length of radiation process lc. This length grows fast with increasing of particle energy e and with decreasing of emitted photon energy u>. At enough high energy e and low u> the coherence length can exceed the mean value of the distance between atoms in the m a t t e r . If within the coherence length of radiation process an electron collides with many atoms, then the electron and atoms interaction differs from electron interaction "This work partially supported by grant 03-02-16263 of the Russian Foundation of Basic Research 223

224

with separated atoms. In this case both increase and decrease of electron radiation in matter is possible. The increase of radiation takes place when relativistic electrons pass through a crystal at a small angle to one of the crystal axes or planes. This effect was pointed up in the works of Feretti [3], Ter-Mikaelian [1] and Uberall [4]. Landau and Pomeranchuk showed that multiple scattering of a relativistic electron within the coherence length of radiation process can lead to the decrease of bremsstrahlung in the range of small particles [2]. For many years both these effects have been studies from various positions, and that can probably be explained by different methods of their description. Thus, electron radiation in an aligned crystal was studied on the basis of the first Born approximation of quantum perturbation theory. To describe the effect of multiple scattering on bremsstrahlung of electrons in an amorphous medium Landau and Pomeranchuk suggested using the formulae of the classic radiation theory with its further averaging at a random process of a multiple scattering of a particle in matter. However, the results obtained on the basis of the latter method were only of a qualitative nature. The quantitative theory of the multiple scattering effect on an electron radiation in an amorphous medium was offered later by Migdal on the base of application of the kinetic equation method to the given task [5] and the density matrix method [6]. Later, on the basis of these methods there were also taken in account a number of other factors essential for the process of electron scattering in an amorphous matter (see reviews and monographs [7-11]). Considering the important input of Migdal to the theory of this effect it is now called the Landau-Pomeranchuk-Migdal effect (the LPM-effect). In the late 1970s there appeared new tendencies in the study of the process of high energy electron radiation in crystal and amorphous matters. They are based on the development of the method of description of relativistic electron radiation in a matter, which was suggested by Landau and Pomeranchuk in Ref. [12,13]. In particular, it appeared that at electron radiation in a crystal the validity conditions of the Born approximation are broken rapidly with the particle energy increase and the decrease of the angle of their incidence to the crystal in relation to one of the crystal axes or planes. It was shown that under this condition the electron in a crystal can be studied on the basis of the eikonal and quasiclassical approximation of quantum electrodynamics, as well as within the classical radiation theory. It gave a possibility to study the process of electron radiation in crystals and in amorphous media from the same perspective. Besides, it was shown

225

that on the basis of the method suggested by Landau and Pomeranchuk it is possible to get not only qualitative but also quantitative results of the multiple scattering effect on bremsstrahlung of ultrarelativistic electrons in an amorphous medium. It became possible due to elimination of some inaccuracies of the Ref. [2] and to application of the functional integration method to the problem under consideration [14]. The present work offers a brief overlook of some of the results obtained while developing the Landau and Pomeranchuk method of description of high energy electron radiation in an amorphous matter.

2. The Landau-Pomeranchuk method The process of radiation of a relativistic electron develops in a large spatial region along the direction of particle motion, which is called the coherence length of radiation process. This length is determined by the expression [1.2] le = 2ee'/m2uj

(1)

where m is the electron mass, e and e' = e — w are the initial and final electron energies and UJ is the photon energy. (We use the system of units for which the light velocity and the Plank constant are equal to unity.) If in the limits of this region an electron interacts only with one atom of a medium then interference of the waves radiated by an electron at different atoms is not important for radiation. In this case the radiation spectral density is determined by the Bethe-Heitler formula [1,2] dEBH du

=

4L g' ( 3LR e V

3^2\ 4ee'/ '

l

;

where L is the target thickness, LR = m2/(4Z2e6n * ln{mR)) is the radiation length, Z\e\ is the charge of atomic nuclear, n is the density of atoms in a medium and R is the screening radius of atomic potential. The coherence length grows fast with particle energy increasing and with decreasing of emitted photon energy. Landau and Pomeranchuk paid attention to the fact that if in the limits of coherence length of radiation process an electron interacts with a large number of atoms, the radiation of low energy photons can be considered on the basis of the classical theory of radiation [2]. In classical electrodynamics the spectral density of radiation is

226

determined by the formula (see [7,17]) dJE du

e2u2 4TT2

CO

n x II'

1=


(3)

-OO

where v(t) and r(t) are the velocity and the position vector of the electron at a moment of time t and do is the element of solid angle near the unit vector n, which determines the direction of radiation. The characteristic angles of scattering and radiation of a relativistic electron in a matter are small, therefore in (3) we can perform an expansion in terms of these angles. For this purpose Landau and Pomerachuk first carried out the integration over solid angle in (3). The result of such integration is determined by the formula sin(cc|r(T + r ) - r ( T ) l ) |r(T + T ) - r ( T ) |

^ • ^ / ^ / ^ . - [ v d W r + rJ-ll

(4)

Using now the smallness of the particle scattering angle 0(r) on the interval of time (T, T + r) , we can represent the quantity v(T + T) in the form v(T + r)

1 v(T)(l--0»)+0(r),

(5)

where V{T)0(T) = 0 and |0 T | < 1. Using this equation, we find [7,13] dE_ du

7T7^

x sin < oj

dt9(t)

(6)

This formula is the main for the Landau-Pomeranchuk method of description of relativistic electron radiation in a medium. However, the Eq. (6) is slightly different from the corresponding results of the Ref. [2]. In Ref. [2] the multiplier is r " ^ 2 0/J" dtd{t) - T " 1 (/J" dt0{t))2\ instead of (1 + | 7 2 0 2 ( r ) ) . The difference appears because of the fact that in Ref. [2] there were left out the terms that have the same order of magnitude as those that were retained (see the discussion of this problem in Refs. [13,18]). The importance which the Eq. (6) presents is in its generality and in the possibility of describing radiation in various media and in external fields by means of this formula from a unified point of view. In the letter case

227

6(T) is a defined function of r.If the radiation occurs in a medium, then (6) must be averaged over the random process which is defined by a character of electron scattering in this media. 3. The Landau-Pomeranchuk-Migdal effect If 7 2 0j 1 Landau and Pomeranchuk offered to substitute the mean value of sine by the sine from the mean value of its argument [2]. Using this procedure we obtain the following formula for the averaged radiation spectrum dELP

e2 / 3 ^

where q = Q\jlcComparing the Eq. (7) with Bethe-Heitler formula (2) we see that dE

LP dEBH dto du> Thus, Landau and Pomeranchuk showed that character of high energy electron radiation in an amorphous medium has changed considerably at 7 2 # 2 RJ 1, e.g. in the region of electron energies and photon frequencies for which the mean square angles of multiple scattering in the limits of coherence length is compared with the square of the typical radiation angles of relativistic electrons ss 7~ 2 . However the formula (7) has only qualitative character. The quantitative theory of the multiple scattering effect on an electron radiation in an amorphous medium was offered by Migdal in Ref. [5]. This theory was based on the application of the kinetic equation method to the given task. Migdal obtained the following formula for a spectrum of electron radiation in an amorphous medium a t w < £

dE

(dE\

^

..

228

where (dE/dio)o is the spectrum of radiation without taking into account the multiple scattering influence on radiation (this value coincides with the corresponding result of Bethe and Heitler (2) with a logarithmic accuracy) and $ M ( S ) is the function, obtained by Migdal, which describes the influence of multiple scattering on radiation $M(S)=24S

2

|/

dicoth*e-2sisin2si-^j.

(10)

The parameter s is determined by the expression s = —T=W

, ULMP = 16nZ2e4nm~4e2

ln(mR)

(11)

2V2 V ULPM

The value ULMP determines the range of gamma quanta energies, starting with which the multiple scattering influences radiation spectrum essentially. The Migdal function is close to unity at s > 1, i.e. at u > UILPM- The spectrum of radiation in this case coincides with the corresponding result of Bethe and Heitler (2). Ifs«l, $M(s)«6s.

(12)

The intensity of radiation in this case is much less, than the corresponding result of Bethe and Heitler. The formula (9) with the asymptotic (12) differs from the qualitative result (7) only by numerical coefficient. The theory of the LPM effect has afterwards got its development in a number of works. In particular, on the basis of the method of density matrix the recoil effect at radiation, the influence of polarization of a medium and boundaries of the target on radiation and a number of other effects at high energy electrons and photons interaction with an amorphous medium were taken into account (see reviews [8,9] and references therein). All these researches are based on the application either the method of kinetic equation or the density matrix method to the given task. 4. The functional integration method A new interest to the LPM effect appeared in the 1970s. It was stimulated by an analysis of the coherent radiation process of relativistic electrons in crystals [12,13]. There were several reasons for this. One of them consisted in the following. The analysis of applicability conditions of the Born theory of coherent radiation of electrons in a crystal has shown that the condition of

229 applicability of the Born approximation in the given task rapidly break with increase of particle energy and with the decrease of the angle of particle falling on a crystal related to one of the crystal axes [12]. Thus, to describe the electron radiation in a crystal it was required to advance methods permitting to go beyond the frameworks of the Born perturbation theory. In Refs. [12,13] studying the process of coherent radiation of relativistic electrons in a crystal outside the domain of applicability of the Born perturbation theory, there were investigated the possibilities to develop the method of description of relativistic electron radiation in substance, offered by Landau and Pomeranchuk [2]. This method was based on the classical theory of radiation by fast electrons moving in an external field on the trajectory r(£). Both the field of a crystal lattice and the field of a set of atoms of an amorphous medium can be considered as an external field. Thus brought out the possibility to investigate the radiation processes in a crystal and in an amorphous medium from a unified point of view (on the bases of identical methods) and the possibility to find common traits and differences between the radiation processes in these cases. In an amorphous medium the trajectory of a particle is random, therefore the radiation spectral density should be averaged by random trajectories of an electron in substance. It was noted in [13], that the radiation density (6), represents a functional, which has the Gaussian form. The random process, to which the multiple scattering of an electron in an amorphous medium is connected, can be considered as the Gaussian process. It meant, that for realization of the averaging procedure the method of functional integration could be utilized. The implementation of this method in the task of description of the LPM effect was realized in [14,19,20]. In particular, on the basis of the method of functional integration there was reproduced the main result of the Migdal theory (9). It opened new possibilities to obtain the quantitative results to describe an electron interaction with the substance. Thus, on the basis of the functional integration method the spectral-angular distribution of bremsstrahlung of a high energy electron in an amorphous medium was researched [21]; the influence of polarization of a medium on radiation and the recoil effect at radiation were taken into account [7,22]; the influence of multiple scattering of relativistic electrons in a crystal on the processes of coherent [7,13,20]and parametric X-ray radiation [23,24] was investigated.

230

5. Radiation in a thin layer of substance At rather high energies of electrons and small energies of radiated photons the following condition can be carried out lc » L.

(13)

where the coherent length of radiation is greater than the target thickens L. The radiation process in this case is of special interest because the condition (13) is contrary to the condition lc ^> L that was used for the LPM-effect. The radiation process in this case was studied on the basis of kinetic equation method [25], the classical theory of radiation [26,27] and the theorem of factorization for radiation cross-section [28]. The latter two methods have a special interest because on the basis of these methods it becomes possible to investigate radiation in an amorphous medium and in a crystal from the same position. In this case the radiation spectral density is determined by the expression dE du

-^/«/(.,L)[^LLln({+^Ti)-l

(14)

where £ = 7#/2, 6 is the angle of particle scattering by the target and f(6, L) is the function of particles distribution on angles 8. The formula (14) shows that at realization of the condition (13) the spectral density of radiation is determined only by the scattering angle of a particle and does not depend on the details of its trajectory in a target. Therefore, the formula (14) can be used when studying radiation of particles, both in crystalline, and amorphous targets. The difference between the processes of radiation in these cases is determined only by distribution function of scattered particles on angles. The formula (14) has simple asymptotes at small and large values of a mean square angle of multiple scattering of particles by the target 92 :

2 dJ5

-f^¥,

72^«i,

, ,

dui

|3 _ — m( 7 2 0 2 ),

_ 7202»1-

(15)

7T

In the amorphous target the value 02 is proportional to the thickness L. Thus, if j26f <§C 1 , the formula (15) transforms into the corresponding result of Bethe and Heitler (2). If *y26f ^> 1, then according to (15), the linear dependence of dE/dw from L is replaced by a weaker logarithmic one. It means, that at realization of the condition 7 2 # 2 3> 1 the effect of

231

suppression of radiation as contrasted to the corresponding result of Bethe and Heitler takes place. Modification of character of electron radiation in a thin layer of substance happens under the same conditions, as in case of the LPM effect. However, the radiation spectral densities (9) and (14), essentially differ (different are the dependencies from L, e and u). It is related to the fact that the formula (14) is valid at realization of the condition lc 3> L, whereas the formula (9), describing the LPM effect is valid at lc -C L. In an amorphous target the function f(9,L) is the Bethe -Moliere distribution function [29] which takes into account both single and multiple scattering of particle in a medium. Thus the formula (14) takes into account both single and multiple electron scattering in a target. At L -> 0, when we can neglect the influence of the multiple scattering on radiation, the formula (14) transforms into the corresponding result of the Bethe-Heitler theory. (We need to note that the Bethe-Heitler result is obtained with logarithmic accuracy in the Migdal theory.) The analysis of the formula (14) has shown in [27] that the Bethe-Heitler result is also true at the condition 7 2 # 2 1, as it was shown in [27], first several terms of the distribution of the formula (14) on the parameter a~2, where a = j292, have the following form dE

2e2 [„

,

_w / ,

2\

2

C

1

,1C,

where C is the Euler constant, B - lnJ5 = ln(x 2 /Xi) + 1 ~ 2C> xl = AimZ2eiLI{pv)2 and xi = 1/P-RWe want to emphasize that consideration of several terms of distribution in (16) is very important for comparing the theory predictions with the experimental data [30] (see Ref. [27]). The similar effects are also possible at electron radiation in a crystal. However, in a crystal there can be carried out the conditions, at which the mean square value of multiple scattering angles is considerably larger, than in an amorphous target. Thus, if 7 2 # 2 1, accordingly to (14), there appears an effect of suppression of coherent radiation. Thus, modification of the character of electron radiation in a crystal can take place much faster, than in the amorphous target.

232

6. Conclusion We have shown only a small part of the results obtained while developing the method of description of high energy electron radiation, suggested in the work of Landau and Pomeranchuk. It was demonstrated that on the basis of this method it is possible to obtain not only qualitative, but also quantitative results. Besides that, this method also helps to study electron radiation in various media and in external fields from one and the same perspective, which is important for detecting of common laws and different traits of radiation processes in these cases. In a brief paper we couldn't concentrate on a number of other results obtained at the development of the Landau and Pomeranchuk method, such as the LPM-effect in a spectral-angular distribution of radiation of relativistic electrons in a thin layer of amorphous matter [31]; development of quasiclassical theory of high energy electron radiation in matter that takes into account the recoil effect at radiation [7, 22,28]; a comparative analysis of the LPM-effect and synchrotron radiation [13], etc. Either didn't we discuss here the question of comparison of the theory predictions and the existing data on the study of the LPM-effect in cosmic rays [32,33] and on accelerators [30,34]. In this connection we want to note that a detailed experimental study of the LPM-effect was conducted on the SLAC accelerator only in the mid 1990s [30, 35, 36]. Those experimental data showed good correlation with the predictions of the Migdal theory for the thick targets and considerable differences with the predictions of this theory for the relatively thin targets. This stimulated the development of new approaches to the description of high energy electron radiation in matter, which allow getting quantitative results (see Refs [11, 18, 27, 28, 37-41]). References 1. M. L. Ter-Mikaelian, Zh. Eksp. Teor. Fiz., 25, 296 (1953). 2. L. D. Landau and I. Ya. Pomeranchuk, Dokl. Akad. Nauk SSSR 92, 735 (1953). 3. B. Ferretti, Nuovo Cimento, 7,118 (1950). 4. H. Uberall, Phys.Rev., 103, 1055 (1956). 5. A. B. Migdal, Dokl. Akad. Nauk SSSR, 96, 49 (1954). 6. A. B. Migdal, Phys. Rev., 103, 1811 (1956). 7. A. I. Akhiezer and N. F. Shul'ga, High-Energy Electrodynamics in matter. Amsterdam: Gordon and Breach, 1996. 8. M. L. Ter-Mikaelian, High-Energy electrodynamics processes in condensed matter. New-York: Wiley Interscience, 1972. 9. M. I. Riazanov, Usp.Phys.Nauk, 114, 393 (1974).

233 10. A. I. Akhiezer and N. F. Shul'ga, Sov. Phys. Usp., 30, 197 (1987). 11. V. N. Baier and V. M. Katkov, Phys. Rev., D60, 076001 (1999). 12. A. I. Akhiezer, V. F. Boldyshev and N. F. Shul'ga, Sov. J. Particles and Nuclei, 10, 51 (1979). 13. A. I. Akhiezer and N. F. Shul'ga, Sov. Phys. Usp., 25, 451 (1982). 14. N. V. Laskin, A. S. Mazmanishvili and N. F. Shul'ga, Dokl. Akad. Nauk USSR, 277, 850 (1984). 15. H. Bethe and W. Heitler, Proc. Roy. Soc, A146,83 (1934). 16. A. I. Akhiezer and V. B. Berestetskii, Quantum Electrodynamics. New-YorkLondon: Wiley-Interscience, 1965. 17. J. D. Jackson, Classical Electrodynamics. New-York: Wiley, 1975. 18. R. Blankenbacler and S. D. Drell, Phys. Rev., D53, 6265 (1996). 19. N. V. Laskin, A. S. Mazmanishvili and N. F. Shul'ga, Phys. Lett, A112, 240 (1985). 20. N. V. Laskin, A. S. Mazmanishvili, N. N. Nasonov and N. F. Shul'ga, JETP, 62, 438 (1985). 21. N. V. Laskin, A. I. Zhukov, JETP, 98, 571 (1990). 22. A. I. Akhiezer, N. V. Laskin and N. F. Shul'ga, Dokl. Akad. Nauk SSSR, 295, 1363 (1987). 23. N. F. Shul'ga and M. Tabrizi, JETP Lett, 76, 279 (2002). 24. N. F. Shul'ga and M. Tabrizi, Phys Lett, A308, 467 (2003). 25. F. F. Ternovskii, Sov. Phys. JETP, 12, 123 (1961). 26. N. F. Shul'ga, S. P. Fomin, JETP Lett, 27, 1399 (1978). 27. N. F. Shul'ga and S. P. Fomin, JETP, 86, 32 (1998). 28. N. F. Shul'ga and S. P. Fomin, Nucl. Instr. Meth., B145, 180 (1998). 29. H. Bethe, Phys.Rev., 75, 1949 (1953). 30. R. L. Anthony et al., Phys. Rev. Lett, 89, 1256 (1995). 31. S. P. Fomin, N. F. Shul'ga and S. N. Shul'ga, Physics of Atomic Nuclei, 66, 394 (2003). 32. M. Miesowicz, O. Stanisz, W. Wolther, Nuovo Cimento, 5, 513 (1957). 33. A. Varfolomeev et al., Sov. Phys. JETP, 38, 33 (1960). 34. A. Varfolomeev et al., Sov. Phys. JETP, 69, 429 (1975). 35. R. L. Anthony et al., Phys. Rev., D56, 1373 (1997). 36. S. Klein, Rev. of Mod. Phys., 71, 150 (1999). 37. N. F. Shul'ga, S. P. Fomin, JETP Lett, 63, 873 (1996). 38. R. Blankenbacler, Phys. Rev., D55, 190 (1997). 39. V. G. Zhakharov, JETP Lett, 64, 781 (1996). 40. V. G. Zhakharov, Yadernaya Fizika, 61, 924 (1998). 41. R. Baier, Yu. L. Dokshitser, A. H. Miller, S. Peigne and D. Schiff, Nucl. Phys., B478, 577 (1996).

O N C O N T R I B U T I O N S OF F U N D A M E N T A L PARTICLES TO T H E V A C U U M ENERGY*

G. E. V O L O V I K Low Temperature Laboratory, Helsinki University of Technology P.O.Box 2200, FIN-02015 HUT, Finland and L.D. Landau Institute for Theoretical Physics Kosygin Str. 2, 117940 Moscow, Russia

Recently different regularization schemes for calculations of the vacuum energy stored in the zero-point motion of fundamental fields were discussed. We show that the contribution of the fermionic and bosonic fields to the energy of the vacuum depends on the physical realization of the vacuum state. The energy density of the homogeneous equilibrium vacuum is zero irrespective of the fermionic and bosonic content of the effective theories in the infra-red corner. The contribution of the lowenergy fermions and bosons becomes important when the coexistence of different vacua is considered, such as the bubble of the true vacuum inside the false one. We consider the case when these vacua differ only by the masses of the low-energy fermionic fields, M t r u e > Mf a i s e , while their ultraviolet structure is identical. In this geometry the energy density of the false vacuum outside the bubble is zero, Pfalse = —-Pfalse = Oi which corresponds to zero cosmological constant. The energy density of the true vacuum inside the bubble is ptrue = — -Ptrue oc — A 2 (M t 2 r u e — M f 2 a[se ), where A is the ultraviolet cut-off.

1. Introduction Recently the problem of the contribution of different fermionic and bosonic fields to the vacuum energy was revived in relation to the cosmological constant problem (see e.g. 1>2 ). The general form of the vacuum energy density under discussion was p = a4A4 + a 2 A 2 M 2 +

UQM*

A2 In — ,

(1)

' T h i s work was supported by ESF COSLAB Programme and by the Russian Foundations for Fundamental Research.

234

235

where M is the mass of the corresponding field, and A is the ultraviolet energy cut-off. Different regularization schemes were suggested in order to obtain the dimensionless parameters a^. Here we consider this problem from the point of view of the effective relativistic quantum field theory emergent in the low-energy corner of the quantum vacuum, whose ultraviolet structure is known. It appears that the parameters a^ depend on the details of the trans-Planckian physics. But in addition we find that even if the two vacua have completely identical structure throughout all the scales, the parameters a; depend on the arrangement and geometry of the vaccum state. In particular, if the vacuum is completely homogeneous and static, all the parameters vanish,
(2)

This corresponds to zero cosmological constant in any homogeneous vacuum, if the vacuum is stationary, corresponds to the extremum of the energy functional, and is isolated from the environment 3 . The energy density of the true vacuum inside the bubble, Ptrue = - P t r u e ~ ~ A 2 ( M t 2 r u e - M

2

^) ^

< 0 .

(3)

236

vacuum energy density p outside and inside the bubble of true vacuum, and in the interface

false vacuum outside contains fermionic field with mass M„ p=-P=0

interface of thickness with energy denⅈp ~ +A 2 M2 false

and surface tension O ~ p£, false Figure 1.

Critical bubble of true vacuum inside the false one.

contains the quadratic in M t r u e term with negative sign, but it also depends quadratically on the fermionic mass Mfaise in the neighboring vacuum. On the other hand, if the true vacuum occupies the whole space and becomes equilibrium, its vacuum energy density is zero again, ptrue = —-Ftrue = 0. All this can be obtained without invoking the microscopic (ultraviolet) structure of the quantum vacuum, using only the general arguments of the vacuum stability. However, here we shall use the microscopic model from which the relativistic quantum field theory with massive fermions emerges in the low-energy corner. 2. Vacuum energy in effective quantum field theory 2.1. Effective

quantum field theory with Dirac

fermions

Massive relativistic fermions are realized as the low-energy Bogoliubov quasiparticles in the class of the spin-triplet superfluids/superconductors

237

(see section 7.4.9 of 3 ) . The Bogoliubov-Nambu Hamiltonian for fermionic quasiparticles (analog of elementary particles) is the 4 x 4 matrix

Fp=f3

(f^-/U)+fV^'

(4)

< = — (x?xl + y»yl) + —z^z1

, £*- = (i . (5) PF PF 2m Here // is the chemical potential of the particles (atoms) forming the liquid (analog of the quantum vacuum); m is their mass; fa and erM are 2 x 2 Pauli matrices describing the Bogoliubov-Nambu spin and the ordinary spin of particles correspondingly; A and M are amplitudes of the order parameter One should not confuse particles which form the vacuum (atoms of the liquid) and quasiparticles - excitations above the vacuum - which form the analog of matter in quantum liquids and correspond to elementary particles. Quasiparticles do not scatter on the atoms of the liquid if the liquid is in its ground state, and thus for quasiparticles the ground state of the liquid is seen as an empty space - the vacuum - though this space is densly filled by atoms. The energy spectrum of quasiparticles in this model is

The case M = 0 corresponds to the so-called planar phase, while M = A describes the isotropic B-phase of superfluid 3 He. We assume that the interaction of atoms in the liquid (or electrons in superconductors), which leads to the formation of the order parameter, is such that the locally stable vacuum states of the liquid have M <§; A. Then in the low-energy limit the Bogoliubov-Nambu Hamiltonian for quasiparticles transforms to the Dirac Hamiltonian Hp « C||pzf3 -I- f1 (c±pxax + c±pyay + Maz) , PF A Pz =Pz-PF , C|| = , CX = , ' m PF with the relativistic spectrum E2(p)^gikpipk p = p-pFz

, g**=gvv

+ M2 ,

=
(7) , .

(8)

(9) (10)

Here cy is the 'speed of light' propagating along the z-axis, while in transverse direction the 'light' propagates with the speed c±. Though in this

238

model the effective speed of light depends on the direction of propagation, this anisotropy can be removed by rescaling along the z-axis. The speed of light is not fundamental, since it is determined by the material parameters of the microscopic system. However, it is fundamental from the point of view of all inner observers who consist of the low-energy quasiparicles. For them the speed of light does not depend on direction, and they believe in the laws of special relativity since these laws can be confirmed by all experiments (including the Michelson-Morley measurements of the speed of light) which use clock and rods made of the low-energy quasiparticles. The high-energy observer will not agree with that: for us the original model (4) has no Lorentz invariance, and the speed of light is anisotropic. After rescaling along the z-axis the quasiparticles become the complete analog of the Standard Model fermions with mass M: the left-handed and right-handed chiral quasiparticles of the planar state with M = 0 are hybridized to form the Dirac particles with the mass M. Note, that together with the effective Dirac fermions, also the effective gravity and effective U{1) and SU(2) gauge fields (with 'photons' and 'gauge bosons') emerge in the low-energy corner of this model, with all the accompaning phenomena, such as chiral anomaly, running couplings, etc. 3 . The reason for such close analogy is the momentum space topology of the quantum vacuum, which is common for the ground state of the considered liquid and for the quantum vacuum of the Standard Model. The parameter A plays the role of the Planck energy scale in the effective theory.

2.2.

Vacuum

energy

This model can serve for the consideration of the contribution of the Dirac fermions to the energy density of the vacuum in equilibrium. In the many body system (superfluid liquid which contain many particles) the relevant vacuum energy whose gradient expansion gives rise to the effective quantum field theory for quasiparticles at low energy is {% — /iAOeq v a c , 4 where H is the Hamiltonian of the system, Af is the particle number operator for atoms forming the liquid, and \i is their chemical potential. The energy density of the superfluid/superconducting ground state (analog of the quantum vacuum) with the order parameter e^ is given by (see e.g. Eq.(5.38) in 5 )

P = \ { U - ^ o v a c = \$t-/iAOvace.

=0-

JSr<4 ,

(ii)

239 where V is the volume of the system. For the order parameter (5) which gives rise to the relativistic fermions at low energy one has

/, = P ( 0 )

-IS(2A4

+ A2M2)

'

(12)

where i/^g = c^c^2 according to Eq.(lO); and p(0) = p(e^ = 0) is the energy density in the normal (non-superfluid) state of the liquid, in which e^ = 0. The equation (12) demonstrates that the amplitude A in the order parameter expression (5) is the proper ultraviolet cut-off for the estimation of the vacuum energy related to the effective relativistic fermions, while the contribution p(0), which does not depend on A and M, comes from the more fundamental physics of the normal state of the liquid at energies well above the energy scales A and M. In this model the term M 4 ln(A 2 /M 2 ) is absent, though it naturally appears in all the regularization schemes discussed in l'2. The dimensionless factors in front of A4 and A 2 M 2 were obtained using the microscopic model in Eqs. (5) and (6). In principle, the low-energy fermions in Eq.(9) can be obtained from different microscopic theories, and one finds that the dimensionless factors in front of A4 and A 2 M 2 depend on the details of the Planck physics. In other words, they depend on the regularization imposed by the ultraviolet physics, which cannot be found within the effective low-energy theory. It is rather natural to think that the equation (12) reflects the general structure (1) of the vacuum energy in terms of the massive fields discussed in l'2. Moreover, from the point of view of the low-energy observers who are made of quasiparticles and live in the liquid and for whom the effective theory is fundamental, the cosmological constant in their world must have the natural value of order A4. However, this is not so. The vacuum energy contains the contribution p(0) from the vacuum degrees of freedom with energies well above A and M. Together with the sub-Planckian modes these high-energy modes determine the behavior of the whole vacuum, and in particular the stability of the static vacuum configurations. If something happens in the low-energy corner, so that the vacuum configuration changes, the high-energy degrees of freedom respond to restore the stability of the whole vacuum in the new configuration. As a result p(0) is tuned to the low-energy degrees of freedom and thus becomes dependent on A and M after the adjustment of the trans-Planckian degrees to the sub-Planckian ones. This dependence is different for different realizations of the equilibrium vacuum state. Moreover, the adjustment

240

can cancel completely or almost completely the low-energy contributions. Below we consider how this occurs in the geometry of Fig. 1. 2.3. Energy density

of the false

vacuum

Let us assume now that the situation is similar to that of the Ising ferromagnet in applied small magnetic field which slightly discriminates between spin-up and spin-down vacua. This means that there are two vacuum states, false and true, corresponding to two nearly degenerate local minima whose fermionic masses are almost the same: 0 < Mt2rue - M f i lse « Mt2rue « A2 .

(13)

Then we can apply this to the vacuum states in the geometry of Fig. 1, where the false vacuum is separated by the domain wall (the interface) from the spherical domain containing the true vacuum. Let us start with the external domain occupied by the false vacuum. This domain is open and it occupies the whole space except for the finite volume. That is why one can apply to this domain the results known for the infinite homogeneous vacuum. First, we note that any equilibrium macroscopic system consisting of the identical elements (atoms in the case of quantum liquids) obeys the Gibbs-Duhem relation E-nN-TS=

-PV

,

(14)

where P is the pressure and S the entropy (see Chapter 10.9 in Ref. 6 ) . Applying this to the equilibrium ground state of the system at T = 0 (the quantum vacuum), one has for the energy density of the vacuum:

This equation of state, p = —P, is valid for any homogeneous vacuum irrespective of whether the system is relativistic or not. If the effective relativistic quantum field theory emerges in the low-energy corner of the system, this p becomes the cosmological cosntant in the low-energy world. Second, if the system (the Universe) is isolated from the environment, the external pressure -Pextemai = 0. Thus for the false vacuum in Fig. 1 one has Pfaise = -Pextemai = 0, which gives the vanishing energy density of the false vacuum in this geometry: Pfaise = --Pfaise = 0 .

(16)

Thus the cosmological constant in the vacuum outside the bubble is zero. This property does not depend on the low-energy physics, and is

241

determined by the physics of the whole vacuum including the degrees of freedom at energies well above the scales A and M. These high-energy degrees of freedom respond to any change in the low-energy corner exactly compensating the contribution from the low-energy degrees of freedom to ensure the zero value of the cosmological constant in the equilibrium homogeneous vacuum.

2.4. Energy and pressure

of the true

vacuum

Now let us turn to the inner domain occupied by the true vacuum. If the domain is big enough, R 3> A - 1 , the vacuum can be considered as homogeneous, and thus the equation of state p = — P in (15) is applicable. However, this vacuum is not isolated from the environment, and as a result its vacuum energy density is non-zero. This energy density ptrue can be found by comparison with the energy density of the false vacuum using Eq.(12). Since /9faise = 0 one has Ptrue = -Ptrue

= Ptrue - Pfalse = - J ^ j V ^ A

2

(Mt2rue - M f ^ g e ) < 0 . (17)

In this geometry, the true vacuum has the negative cosmological constant proportional to the difference of M2 in two vacua. The quartic term oc A4 does not appear in the considered model; it appears if the two vacua have different physics at energies below A. Note that if the true vacuum is outside and is in equilibrium, the cosmological constant in this vacuum will be zero again. Let us now estimate the radius R of the static bubble - the saddlepoint critical bubble - which is obtained when the vacuum pressure inside the bubble is compensated by the effect of the surface tension. Since the external pressure is zero, the pressure Ptrue within the domain is determined by the surface tension and the curvature of the domain wall: Ptrue = -M,rue

—

Pfalse = ~JT •

(18)

Comparing equations (17) and (18) one obtains the radius of the static bubble critical bubble. The order of magnitude of the surface tension is a ~ p(M = 0)f, where £ is the thickness of the domain wall; and p(M — 0) is the energy density of the vacuum with massless fermions. This energy can be obtained by comparing it with the energy density of either of the

242

domains, and it appears to be positive: p(M = 0) = p(M = 0) - p false =

I

^ 3 A 2 M£ l s e > 0 .

(19)

This gives the following estimation for the radius of the critical bubble: M2 *~*M» l al M2 Jw

true

m

»*•

(20)

false

Here £ ~ ficy/A in our model, and £ ~ /lc/A in its relativistic counterpart. 3. Conclusion In general, the energy density of the quantum vacuum is determined by all the vacuum degrees of freedom. They act coherently as the elements of the same medium, which results in the zero value of the cosmological constant if the vacuum is equilibrium, homogeneous and is isolated from the environment. Thus the contribution of the zero-point motion of the low-energy fermionic and bosonic fields to the vacuum energy cannot be singled out from the ultrviolet contributions and thus it cannot be obtained from the low-energy theory by any regularization scheme. The low-energy physics is capable to describe only the contribution of different ifrared perturbations of the vacuum to the vacuum energy density, such as dilute matter, space curvature, expansion of the Universe, rotation, Casimir effect, etc. 3 ' 7 . This gives rise to induced vacuum energy which is proportional to the energies of the infrared perturbations, including the energy density of matter, which is in agreement with observations 8 . If the Universe evolves in time, cosmological 'constant' becomes an evolving physical quantity, which responds to the combined action of the evolving perturbations of the vacuum state. The example which we considered here is somewhat similar to the Casimir effect, where the difference in energy density between two neghbouring vacua, in the restricted and unrestricted domains, are calculated using the low-energy physics. In our case, the two vacua across the interface (domain wall) also have identical ultraviolet (microscopic) structure and also have very close low-energy theories, whose fermionic and bosonic contents are the same, but the masses of the fields are slightly different. That is why, for the estimation of the difference one can try to explore the low-energy physics. One can use, for example, the regularized equation for the contribution of the massive quantum field to the energymomentum tensor of the vacuum - the equation (9) in Ref. l with minus

243

sign when applied to the fermionic field:

V = -g^M2 J -0^S(P2 ~ M2)6(po) .

(21)

Then for the difference in energy and pressure between the true and false vacua one obtains: Ptrue

Pfalse —~ -Mfalse

-'true

(22) = - (Mt2rue - Mllse) j ^ ^ 1 ~ - V=^A 2 (Mt2rue - Mllse) , (23) where yf-g = c - 3 . This gives for the energy and pressure difference the correct dependence on masses, the correct sign and even the correct GibbsDuhem relation Ap = — AP. However, the numerical factor is still missing because of the quadratic divergence, and this factor depends on details of the microscopic physics. This demonstrates that the regularization schemes are not applicable in a strict sense even when the difference in the vacuum energies is calculated. As for the energy density itself (the cosmological constant), it is determined (as in the case of the Einstein and Godel Universes 7 ) by the equilibrium properties of the whole system and depends on the geometry. In particular, for the equilibrium homogeneous vacuum one obtains p = —P = 0, irrespective of whether this vacuum is true or false, and whether it contains massive or massless fermions. On the other hand, the vacuum inside the bubble is not isolated from the environment, as a result the energy density of this vaccum is non-zero and is given by the difference of the quadratic terms in Eq.(3). Except for the numerical factor, this result can be reproduced without consideration of the microscopic theory. In addition to the general arguments of the vacuum stability, including the Gibbs-Duhem relation, one can use the effective low-energy theory for the order-of-magnitude estimation of the energy and pressure difference of two neighboring vacua. For our particular problem, we can use the regularization presented in Eq.(21). References 1. G. Ossola and A. Sirlin, Considerations concerning the contributions of fundamental particles to the vacuum energy density, hep-ph/0305050. 2. E. Kh. Akhmedov, Vacuum energy and relativistic invariance, hepth/0204048. 3. G.E. Volovik, The Universe in a Helium Droplet, Clarendon Press, Oxford (2003).

244 4. A.A. Abrikosov, L.P. Gorkov and I.E. Dzyaloshinskii, Quantum Field Theoretical Methods in Statistical Physics, Pergamon, Oxford (1965). 5. V.P. Mineev and K.V. Samokhin, Introduction to Unconventional Superconductivity, Gordon and Breach Science Publishers (1999). 6. B. Yavorsky and A. Detlaf, Handbook of Physics, Mir Publishers, Moscow (1975). 7. G. E. Volovik, Phenomenology of effective gravity, gr-qc/0304061. 8. A. G. Riess, et al., Astron. J. 116, 1009 (1998); S. Perlmutter, et al, Astrophys. J. 517, 565 (1999).

PART II. CONTRIBUTED PAPERS Interactions at High Energies and Quantum Chromodynamics

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T H E OBSERVATION OF MULTI-QUARK S T R A N G E METASTABLE A N D STABLE STATES

P.Z. ASLANYAN, V.N. EMELYANENKO, G.G.RIKHVITZKIY Joint Institute for Nuclear Research, LHE Dubna, Moscow region, p.o. 141980, Joliot Curie 6. E-mail :[email protected]. ru

In the metastable state sector the Ap, Airir and Apir resonances found in n-t-12Ccollisions at 7 GeV/c. Preliminary results show that the predicted peaks has been confirmed in p12C collision at 10 GeV/c. In the stable state sector candidates for S=-2 light and heavy dibaryons observed in p+ 1 2 C collision at 10 GeV/c

1. Introduction Quark models with confinement predict the existence of exotic objects as, for instance, glueballs, hybrid states of quarks and gluons as well as multiquark hadrons. 1 - 6 It is important to stress that the existence of multiquarks is a necessary consequence of modern ideas of non-perturbative QCD vacuum and is a matter of principle to comprehend the characteristics of matter both at microscopic and cosmological levels. New particles or states of matter containing 2-or more strange quarks have inspired a lot of experiments at BNL(AGS), CERN, FNAL, SEBAF, KEK and JINR. Stable and metastable strange dibaryons were searched a long time ago at LHE JINR, too. A thorough analysis of the merits and demerits of the available detection techniques and data analysis procedures has convinced us that the propane bubble chamber technique is most adequate to the physics problem at the present reconnaissance stage. A reliable identification of dibaryons needs a multivertex kinematic analysis which is in turn is feasible only using 47r-detectors and high precision measurements of the sought objects. The investigations of multiquark states have been carried out in two directions.

247

248

1.1. Multiquark resonance formation via the compression mechanism reduces to the phase transition of normal density super strange hadronic matter revealing itself as multiquark resonances [1-3].

Table 1. The effective mass spectra in collisions of 4.0 GeV/c 7r~ and neutrons of a 7.0 GeV/c average momentum with 1 2 C nuclei, have led to the discovery of the peaks presented below. Resonanse system Ap Ap Ap Ap Ap Ap ApTT*

M MeV/c 2 2095.0±2.0 2181.0±2.0 2223.6±1.8 2263.0±3.0 2356.0±4.0 2129.2±0.3 2495.2±8.7

r MeV/c 2 7.0±2.0 3.2±0.5 22.Oil.9 15.6±2.3 98.6±2.5 0.7±0.16 204.5±5.6

Significance Nsd 5.70±1.20 4.36±1.21 6.24±1.23 8.55±1.35 13.81±1.39 11.37±1.37 12.86±1.68

a

Bag model jv M MeV|

55.0±16.0 60.0±15.0 40.0±12.0 85.3±20.0 65.0±17.0 90.0±20.0 70.5±15.0

2110 2169 2230 2241 2253 2128 2500

11+ 0+ 2+ 1+ 1+ 1", 2"

AA AAp A7T + 7T + A7T + 7r + A7T+7T+

2365.3±9.6 «3568.3 1704.9±0.9 2071.6±4.0 2604.8±4.8

47.2±15.1 <60 18.0±0.5 172.9±12.4 85.9±21.5

4.2±1.40 5.3±1.6 10.3±1.5 5.2±1.4

24.2±7.0 16.1±5.2 19.0±0.6 88.0±27.0 31.9±9.0

2365 3570 1710 2120 2615

-.5/2+ 1/2" 1/2" 3/2-

The problem of experimental examination of multibaryon strange resonances was started at LHE from 1962 up to now. The effective mass spectra of 17 strange multiquark systems were studied for neutron exposure n 1 2 C —> AX at average momentum < pn >=7.0 GeV/c, and our group succeeded in finding resonance-like peaks [7-10] (Table 1) only in five of them Ap, lYpn, AA, AAp, ATT+TT+. Most significant evidence is included in the Review of Particle Properties. Up to now, the same group is going on to collect statistics(more 3 time) for strange V° particles on the photographs of the JINR 2m propane bubble chamber exposed to a 10 GeV/c proton beam because it is the necessary to improve a statistical significance of the identified resonance peaks and to search for new strange multiquark resonance states. Fig. l(a,b,c) show the preliminary experimental effective-mass spectrum of the A7r, Ap and Apn produced by pC interaction.

249

120

^

p + C^>(Ap)X

a)

™o 100

1 80 iri

\-w

60

| o °

40

n \ n1

20 0

' i i i i i i i i i i i n i i f

2.25

2.5

2.75

w (GeV/c 2 )

MAp (GeV/c 2 )

Figure 1 . a) The An effective mass spectrum, showing peaks approximately at 2095, 2 1 8 1 , 2263 MeV/c 2 . b) The A7T7T effective mass spectrum , showing peaks approximately at 1704, 2100 MeV/c 2 . c) The Apn effective mass spectrum , showing peaks approximately at 2600 MeV/c 2 .

MApn (GeV/c 2

Figure 1.

1.2. A simple consideration of symmetry and the properties of color-magnetic interactions argues in flavor of increasing binding in three flavor mattersystems containing u,d and s quarks. These objects known an H particles(uuddss). This was first proposed by Jaffe in 1977 using the MIT bag model [3-6]. The possibility that H dibaryon matter may exist in the core of a neutron star was also pointed out. The estimated binding energy of the H particle is model - dependent and ranges from positive (unbound) to negative strong bound states( -650MeV). The heavy isotriplet stable dibaryon H (1=1, J=0+,Y=0,B=2, S=-2) of a 2370 MeV/c2 mass is predicted by the soliton Skirme-like model [1,3].

250 Table 2.

Mass and weak decay channels for the registration of dibaryons.

Channel of decay

H" -

£~p

H° -> H°(2146)7 Hu -> S - p , E - - + n 7 r tfu - > £ - p , £ - - » n 7 r i/+-*p7ruA°,A°-»p7rif+ -»p7ruAu,A° - » P T T -

H + n - » £ + A ° n,A°—p7r—+P7T

H+ - » p 7 r 0 A ° , A ° - ^ p 7 r -

Mass H (MeV/c2) Dibaryon 2172 ± 15 2146 ± 1 2203 ± 6 2218 ± 12 2385 ± 31 2376 ± 10 2580 ± 108 2410 ± 90 2448 ± 47 2488 ± 48

C.L. of fit

References

% 99 30 51 69 34 87 86 6 73 72

Z.Phys.C 39, 151(1988). JINR RC, N l(69)-95-61,1995. Phys.Lett B235(1990),208. Phys.Lett B316(1993),593. Phys.Lett B316(1993),593 Nucl.Phys.75B(1999),63. JINR Com. (2002) El-2001-265

Figure 2. a)Two body weak decay of heavy, stable, newtral dibaryon H° —> p + £ ° , £ ° —> n~ + n. b) Three- body weak decay of heavy, stable, positively charged dibaryon H+ —> -fA + 7r°, A —> p + 7r~(the hypothesis H+n —> £ + + A + n, A —» P + 7T-, £ + ^ p + 7T°).

251 T h e search for weak decay channels for stable S=-2,3 dibaryon states is being continued to date. A few events, detected on the photographs of t h e propane bubble chamber exposed to a 10 G e V / c proton beam, were interpreted as H dibaryons [11-16]. There are two groups of events interpreted as S = - 2 stable dibaryons (Table 2 ): a) T h e first group is formed of three neutral, S=-2 stable dibaryons, the masses of which are below A A threshold; b) T h e second group is formed of neutral and positively charged S = - 2 heavy stable dibaryons(Figs. 2). T h e masses of all the three dibaryons coincide within the errors are over the AA, EN, AE threshold.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

Ts. Sakai et al.,Osaka U.,nucl-th/9912063,Dec. 1999. . P.J.G. Mulders, Aerts A.Th.M., de J.J. Swart, Phys.Rev., 1980, D.21, p.2553. J. Schaffner-Bielich et al., Phys. Rev. C, v.55, n. 6,p. 3038-3046, 1997. R.L. Jaffe, Phys. Rev. Lett. 38(1977)195. T.Goldman et al., Phys.Rev.Lett.,1987, 59, p.627. A. Faessler, U. Straub, Annalen der Physik, 1990, 47, p.439. B.A. Shahbazian et al., Nucl. Physics, A374(1982),p. 73c-93.c. B.A. Shahbazian ,JINR Commun., El-82-446,1982, International Conference on Hypernuclear and Kaon Physics, Heidelberg, 1982. B.A. Shakhbazian et al., (Dubna, JINR). JINR-D1-81-107, 1981. B.A. Shahbazian et al., JINR Rapid Communications, No. 2[48]-91. B.A. Shahbazian et al., JINR Rapid Communications, No. l[69]-95. B.A. Shahbazian et al.,Phys. Lett. B316(1993)593. P.Zh.Aslanian et al., JINR Rapid Communications, N 1(87)-98,1998. P.Zh.Aslanyan et a l , Nucl. Phys.B(Proc.Suppl) 75B (1999)63-65. P.Z. Aslanian et al.,ISBN 981-02-4733-8, 2001,Int. Conference, Bologna'2000, 29 May - 3 June, Italy. P.Zh.Aslanian et al., JINR Communications, El-2001-265,2002.

MANIFESTATIONS OF T H E A B E L I A N Z' B O S O N W I T H I N THE ANALYSIS OF THE LEP2 DATA

V. I. DEMCHIK, A. V. GULOV, V. V. SKALOZUB AND A. YU. TISHCHENKO Dniepropetrovsk National Dnepropetrovsk, 49050

University, Ukraine

The preliminary LEP data on the e+e~ —> l+l~ scattering are analysed to establish a model-independent search for the signals of virtual states of the Abelian Z' boson. The recently introduced observables give a possibility to pick up uniquely the Abelian Z' signals in these processes. The mean values of the observables are in accordance with the Z' existence at the lcr confidence level.

I. The recently stopped LEP2 experiments have accumulated a huge amount of data on four-fermion processes at the center-of-mass energies y/s ~ 130 - 207 GeV 1'2. Besides the precision tests of the Standard Model (SM) of elementary particles these data allow the estimation of the energy scale of a new physics beyond the SM. Various model-dependent and model-independent approaches to detect manifestations of physics beyond the SM have been proposed in the literature. It seems to us that it is reasonable to develop the modelindependent searches for the manifestations of heavy particles with specific quantum numbers. Such an approach is intended to detect the signal of some heavy particle by means of the experimental data without specifying a model beyond the SM. In this way, it is also possible to derive modelindependent constraints on the mass and the couplings of the considered heavy particle. To develop this approach one has to take into account some model-independent relations between the couplings of the heavy particle as well as some features of the kinematics of the considered scattering processes. In the present talk we focus on the problem of model-independent searches for signals of the heavy Abelian Z' boson 3 by means of the analysis of the LEP2 data on the lepton processes e + e~ -> fi+' (i~ ,T+T~ . This particle is a necessary element of different models extending the SM. The 252

253

low limits on its mass estimated for a variety of popular GUT models are found to be in the wide energy interval 600-2000 GeV *'2. In what follows we assume that the Z' boson is heavy enough to be decoupled at the LEP2 energies. In the previous papers 4 we argued that the low-energy Z' couplings to the SM particles satisfy some model-independent relations, which are the consequences of renormalizability of a theory beyond the SM remaining in other respects unspecified. These relations, called the renormalization group (RG) relations, predict two possible types of the low-energy Z' interactions with the SM fields, namely, the chiral and the Abelian Z' bosons. Each Z' type is described by a few couplings to the SM fields. Therefore, it is possible to introduce observables which uniquely pick up the Z' virtual state 4 . The signal of the Abelian Z'-boson in the four-lepton scattering process e + e~ —> l+l~ can be detected with a sign-definite observable, which is ruled by the center-of-mass energy and an additional kinematic parameter. The incorporation of the next-to-leading terms in m^,2 allows to consider the Z' effects beyond the approach of four-fermion contact interactions. As a consequence, the four-fermion contact couplings and the Z' mass can be fitted separately. Thus, the outlined analysis has to answer whether or not one could detect the model-independent signal of the Abelian Z' boson by treating the LEP2 data. As it will be shown, the LEP2 data on the scattering into fi and r pairs lead to the Abelian Z' signal at about la confidence level. I I . The Abelian Z' boson can be introduced in a model-independent (phenomenological) way by defining its effective low-energy couplings to the SM left-handed fermion doublets, jx, the right-handed fermion singlets, / # , and the scalar doublet 3 . Such a definition usually means that the covariant derivatives in the Lagrangian contain the additional terms: D% = £>£"•* - ^ F ( 0 ) B M / 2 , Df, = D ^ - igY{!)Bj2, where .BM denotes the massive Z' field before the spontaneous breaking of the electroweak symmetry, g is the charge corresponding to the Z' gauge group, D™^ and D™) = d i a g ^ i , Y ^ ) , Y(fL) = d\&g(YLju,YLjd) and numbers Y(fn) — YRJ mean the unknown Z' generators characterizing the model beyond the SM. Such a parameterization involves the Z' interactions of renormalizable type, since the non-renormalizable interactions are generated at higher energies due to radiation corrections and suppressed by

254

m^,1. The corresponding Lagrangian generally leads to the Z-Z' mixing of order mz/mz, which is proportional to Y ^ and originated from the diagonalization of the neutral vector boson states. The mixing contributes to the scattering amplitudes and cannot be neglected at the LEP2 energies. The Z' couplings to a fermion / are parameterized by two numbers YLJ and YRj. Alternatively, the couplings to the axial-vector and vector fermion currents, alz, = (YRJ - YL,I)/2

and vlz, = {YLj + YRJ)/2,

can

be used. Their values are determined by the unknown model beyond the SM. Assuming an arbitrary underlying theory one usually supposes that the parameters aj and Vf are independent numbers. However, if a theory beyond the SM is renormalizable these parameters satisfy some relations. For the Abelian Z1 boson this is reflected in the correlations between a/ and Vf 4 : Vf -a,f =vf* -af*,

af = T3jY
Y^i =Y^t2 = Y<j),

(1)

where T? is the third component of the fermion weak isospin, and / * means the isopartner of / (namely, I* = ui,u* = I,...). In what follows we will use the short notation a = ai = —Y^/2. Note also that the Z-Z' mixing is expressed in terms of the axial-vector coupling a. An important benefit of the relations (1) is the possibility to reduce the number of independent parameters of new physics. Due to a fewer number of independent Z' couplings the amplitudes and cross-sections of different scattering processes are also related. As a result, one is able to pick up the characteristic signal of the Abelian Z'-boson in these processes and to fit successfully the corresponding Z' couplings. III. We investigate the processes e+e~ —> l+l~ (I = fx,r) with the nonpolarized initial- and final-state fermions. To take into consideration the correlations (1) we introduce the observable <Ji(z) defined as the difference of cross sections integrated in some ranges of the scattering angle 6 4 : = / -dcosQ — l -dcos9, (2) v Jz dcosO j _ x dcos6 ' ' where z stands for the cosine of the boundary angle. In what follows the index I = H,T denotes the final-state lepton. The idea of introducing the zdependent observable (2) is to choose the value of the kinematic parameter z in such a way that to pick up the characteristic features of the Abelian Z' signals. The lower-order diagrams for the process describe the neutral vector boson exchange in the s-channel (e + e~ —> V* —> l+l~, V = A, Z, Z'). <TI(Z) v ;

255

As for the one-loop corrections, two classes of diagrams are taken into account. The first one includes the pure SM graphs (the mass operators, the vertex corrections, and the boxes). The second set of the one-loop diagrams improves the Born-level Z'-exchange amplitude by "dressing" the Z' propagator and and the Z'-fermion vertices. We assume that Z' states are not excited inside loops. Such an approximation means that the Z'boson is completely decoupled. Then, the differential cross-section consists of the squared tree-level amplitude and the term from the interference of the tree-level and the one-loop amplitudes. To obtain an infrared-finite result, we also take into account the processes with the soft-photon emission in the initial and final states. In the lower order in m^,2 the Z' contributions to the observable are expressed in terms of four-fermion contact couplings, only. If one takes into consideration the higher-order corrections in m^,2, it becomes possible to estimate separately the Z'-induced contact couplings and the Z' mass. In the present analysis we keep the terms of order 0(m~^, ) to fit both of these parameters. Neglecting the terms of order 0(m^f) in the observable, we have: i

7

AaL(Z) = at(z)-afM(z)

=£ £

7

+

i

j

[i'y( a ,z) +£{,-(*,z)c] a^

j=l

i=l

k

HmZXl^Un( S ' 2 ) aia J a * a "i = l j=l

k=l

(3)

n=\

where the dimensionless quantities m2z,'

4irm%,

I Q Tfl

( a i , a 2 , o 3 , o 4 , a 5 , a 6 , a 7 ) = J-

§~{a,ve,Vp,vT,vd,vs,vb)

(4)

are introduced. The coefficients A, B, C are determined by the SM couplings and masses. They are evaluated with FEYNARTS, FORMCALC and LOOPTOOLS software within the MS renormalization scheme. The factors A describe the leading-order contribution, whereas others correspond to the higher order corrections in m~^, . There is an interval of values of the boundary angle, at which the factors A[i, Bln, and C[in at the sign-definite parameters e, e(t and e2 contribute more than 95% of the observable value. It gives a possibility to construct

256

the sign-definite observable Aai(z*) < 0 by specifying the proper value of z*. We define some quantitative criterion to estimate the contributions from the sign-definite factors at a given value of the boundary angle z 5. Maximizing the criterion, we found that the cosine of the boundary angle z*(s) decreases with the growth of the center-of-mass energy from 0.45 at y/s = 130GeV to 0.37 at 207GeV. The obtained observable is negative with the accuracy 4-5%:

Aal(z*)=[A[1(s,z*)+CB[1(8,z*)\e

+ C,nn{8,z*y<0.

(5)

The introduced observable Aai(z*) selects the model-independent signal of the Abelian Z' boson in the processes e + e~ -> l+l~. It allows to use the data on scattering into fifi and TT pairs in order to estimate the Abelian Z' coupling to the axial-vector lepton currents. Although the observable can be computed from the differential crosssections directly, it is also possible to recalculate it from the total crosssections and the forward-backward asymmetries. The recalculation is based on the fact that the differential cross-section can be approximated with a good accuracy by the two-parametric polynomial in the cosine of the scattering angle z. The approximated cross-section reproduces the exact one in the limit of the massless initial- and final-state leptons and if one neglects the contributions of the box diagrams. The unknown coefficients of the polynomial are expressed in terms of the total cross-section and the forward-backward asymmetry leading to the following relation for the observable: B 2 2 Aa,(z*) = A F ( l - , * ) - £ _ ( 3 + .* ) AaJ

+ {l-z*2)al'bMAA!B.

FB

(6)

As computations show, the theoretical error of the approximation can be estimated as a quantity of order 0.003pb. At the same time, the corresponding statistical uncertainties on the observable are larger than 0.06pb. Thus, the introduced approximation is quite good. It can be successfully used to obtain more accurate experimental values of the observable, because the published data on the total cross-sections and the forward-backward asymmetries are still more precise than the data on the differential cross-sections. IV. To search for the model-independent signals of the Abelian Z'boson we will analyze the introduced observable Aai(z*) on the base of the

257

LEP2 data set. In the lower order in mz, the one flavor-independent parameter e, Aa?(z*)

=A[1(8,z')e

the observable (5) depends on

+ C[111(8,z*y,

(7)

which can be fitted from the experimental values of A 0) is by construction a signal of the Abelian Z'. To derive the central value of e and the corresponding la confidencelevel interval we will apply the usual fit method based on the likelihood function £(e) oc exp(—x 2 (e)/2) 5 . We also introduce the probability of the Abelian Z' signal, P, as the integral of the likelihood function over the positive values of e. To compare our results with those of Refs. l'2 we introduce the contact interaction scale A2 = 4mz€~x. This normalization of contact couplings is admitted in Refs. l<2. We use again the likelihood method to determine a one-sided lower limit on the scale A at the 95% confidence level. It is derived by the integration of the likelihood function over the physically allowed region e > 0. The exact definition is A = 2mz{e*)-1/2,

/ C{e')de' = 0.95 / C(e')de'. (8) Jo Jo Actually, the fitted value of the contact coupling e originates mainly from the leading-order term in the inverse Z' mass in Eq. (5). The analysis of the higher-order terms allows to estimate the constraints on the Z1 mass. Substituting e in the observable (5) by its fitted central value, e, one obtains the expression

Aat(z*) = [iL(*.0 + C£ii(*,0]*+CW«>**)e2>

(9)

which depends on the parameter ( = mz/mz,. Then, the central value on this parameter and the corresponding la confidence level interval are derived in the same way as those for e. To fit the parameters e and £ we start with the LEP2 data on the total cross-sections and the forward-backward asymmetries 1'2. Those data are converted into the experimental values of the observable Aai(z*) with the corresponding errors Sai(z*) by means of Eq. (6). We perform the fits assuming several data sets, including the (ifx, TT, and the complete /i/u and TT data, respectively. The results are presented in Table 1. As is seen, the more precise y^i data demonstrate the signal of about la level. It corresponds to the Abelian Z'-boson with the mass of order 1.2-1.5TeV if

258

one assumes the value of a = g2/4ir to be in the interval 0.01-0.02. No signal is found by the analysis of the TT cross-sections. The combined fit of the fifx and TT data leads to the signal below the la confidence level. Data set MM TT

H/j, and

TT

TT

/x/i and

TT

e

A, TeV Winter 2002 15.7 0.0000482i|;™31 16.0 o.ocxxx)i6±8:gS8gSSS 18.1 0.0000313±°;S™ Summer 2002 16.4 0.0000366i™^ 17.4 -0.0000266±S:H« 9 19.7 0.0000133inr o ^ T

P

c

0.83 0.51 0.78

0.007 ±0.215 -0.052 ± 8.463 0.006 ± 0.264

0.77 0.34 0.63

0.009 ± 0.278 -0.001 ±0.501 0.017 ±0.609

Being governed by the next-to-leading contributions in m^,2, the fitted values of £ are characterized by significant errors. The fifi data set gives the central value which corresponds to mz1 — 1.1 TeV. We also perform a separate fit of the parameters based on the direct calculation of the observable from the differential cross-sections. The complete set of the available data is used. Three of the LEP2 Collaborations demonstrate positive values of e. The combined value e = 0.00012 ± 0.0003 is also positive. As it follows from the present analysis, the Abelian Z' boson has to be light enough to be discovered at the LHC. On the other hand, the LEP2 data on the processes e+e~ -» /u + ^ _ , T+T~ do not provide the necessary statistics for the detection of the model-independent signal of the Abelian Z' boson at more than la confidence level. So, it is of interest to find the observables for other scattering processes in order to increase the data set. References 1. ALEPH Collaboration, DELPHI Collaboration, L3 Collaboration, OPAL Collaboration, the LEP Electroweak Working Group, the SLD Heavy Flavour, Electroweak Working Group, hep-ex/0112021. 2. LEP Electroweak Working Group, LEP2FF/02-03. 3. A. Leike, Phys. Rep., 317, 143 (1999). 4. A. Gulov and V. Skalozub, Eur. Phys. J. C 17, 685 (2000). 5. V. Demchik, A. Gulov, V. Skalozub, and A. Tishchenko, hep-ph/0302211.

O N G E N E R A T I O N OF M A G N E T I C FIELDS AT HIGH T E M P E R A T U R E IN A S U P E R S Y M M E T R I C THEORY

V. D E M C H I K A N D V. S K A L O Z U B The spontaneous generation of magnetic and chromomagnetic fields at high temperature in the minimal supersymmetric standard model (MSSM) is investigated. The consistent effective potential including the one-loop and the daisy diagrams of all bosons and fermions is calculated and the magnetization of the vacuum is observed. The mixing of the generated fields due to the quark and s-quark loop diagrams and the role of superpartners are studied in detail. The magnetized vacuum state is found to be stable due to the magnetic masses of gauge fields included in the daisy diagrams. Applications of the results obtained are discussed. A comparison with the standard model (SM) case is done.

1. Introduction Possible existence of strong magnetic fields in the early universe is one of the most interesting problems in high energy physics. Different mechanisms of the fields producing at different stages of the universe evolution were proposed 1 . The spontaneous vacuum magnetization at high temperature is one of the mechanisms mentioned. It was already investigated in the SM2 where the possibility of this phenomenon has been shown. The stability of the magnetized vacuum state was also studied 3 . The magnetization takes place for the non-abelian gauge fields due to a vacuum dynamics. In 3 the fermion contributions were not taken into consideration. However, at high temperature they affect the vacuum considerably. Quark posses both the electric and the color charges and therefore the quark loops change the strengths of the simultaneously generated magnetic and chromomagnetic fields2. In a supersymmetric theory new peculiarities should be accounted for. First is an influence of superpartners. They having a low spin have to decrease the generated magnetic field strengths. Second, s-quarks also possess the electric and the color charges, so the interdependence of magnetic and chromomagnetic fields is expected to be stronger as well as the fields due to their vacuum loops. Because of it some specific configurations 259

260

of the fields must be produced at high T. In the present report the spontaneous vacuum magnetization is investigated in the MSSM. All boson and fermion fields are taken into consideration. In the MSSM there are two kinds of non-abelian gauge fields - the SU(2) weak isospin gauge fields responsible for weak interactions and the SU(3) gluons mediating the strong interactions. Magnetic and chromomagnetic fields are related to these symmetry groups, respectively. To elaborate this problem we calculate the effective potential (EP) including the one-loop and the daisy diagram contributions in the constant abelian chromomagnetic and magnetic fields at high temperatures. The values of the generated fields strengths are found as the minimum position of the EP in the field strength plane. The EP of the background abelian magnetic fields is a gauge fixing independent one, while the daisy diagrams account for the most essential long-range corrections at high temperature. Therefore, such a type of EP includes the leading and the next-to-leading terms in the coupling constants. Moreover, as it was shown in 3 ' 4 , the daisy diagrams of the charged gluons and the W-bosons with their magnetic masses taken into consideration make the vacuum with nonzero magnetic fields stable at high temperatures. The obtained results are in a good agreement with the non-perturbative calculations carried out in 5 . This approximation will be used in what follows. 2. Basic Formulae In the SU(2) sector there is only one magnetic field, the third projection of the gauge field. In the SU(3)C sector there are two possible chromomagnetic fields connected with the third and the eighth generators of the group. For simplicity, we shall consider the field associated with the third generator of the SU(S)C. To obtain the EP one has to rewrite the thermodynamics potential as a sum in quantum states calculated near the nontrivial classical external field solutions Aext and Aext (see, for instance, 3 ) . The result can be written in the form V = VW(H,H3,T)

+ VM(H,H3,T)

+ ... + Vdaisy(H,H3,T)

+ ..., (1)

where V^ is the one-loop EP; the other terms present the higher loop corrections. Among these terms there are some Vdaisy responsible for the dominant contributions of long distances at high temperature - so-called daisy or ring diagrams (see 6 ) . It is nonzero in case when massless states appear

261

in a system. The ring diagrams have to be calculated when the vacuum magnetization at finite temperature is investigated. In fact, one first must assume that the fields are nonzero, calculate EP and after that check whether its minimum is located H, H3 ^ 0. On the other hand, if one investigates problems in the applied external fields, the charged fields become massive and have to be omitted. Omitting the detailed calculations we notice that the exact one-loop EP is transformed into the EP which contains the daisy diagrams as well as the one-loop diagrams if one adds to the exponent a term containing the temperature dependent mass of a particle 3 ' 6 . It is convenient for what follows to introduce the dimensionless quantities: x = H/H0 (H0 = Af^/e), y = H 3 / H ° (H° = M^/gs), B = Mw/T, v = V/HQ. The total EP consists of several terms «' = y + y + «{ + V'q + V'w + Vg +
(2)

These terms can be written down as follows (in dimensionless variables): • SM sector - leptons

.j_f(_irr*

2

2

xs coth(a;s) — 1]:

(3)

quarks 1

^

OO

< = -T-2 E E ( ~ 1 ) n

/ = ln=l

J

/>00

n

nl

2

%e-{n''+-£-)[qfX8Coth{qfX8)yacath{y8)-l]-;

/ J

<>

S

^

- W-bosons (see 6 ) .-(""ff'+^l .

sinh(a;s)

+ 4sinh(a;s) ;

(5)

- gluons (see 3 )

9

2

^ t[J°

1 + 2 sinh(ys) ; sinh(ys)

s2

(6)

• MSSM sector - s-leptons dS

_(rrP

, I a2"21

XS

sinh(xs)

-1

(7)

262

- s-quarks a2.2.

33

7T - ^ 7 % - to'^Jo *S n=l 2


x

sinh.(qfxs) • sinh(ys)

Q u

(8)

charginos 1

v

°°

a2

r°° dl

'ch = - 747r -2 £ ( - ! ) " / „=i Jo

2

Is e " ( m - s + ^ ) • [ x . c o t h M - 1]; s

(9)

gluinos

I^E^ 47r

1

-Te-«'+fi^>-[yacoth(y*)-l].

)"/

n=l

^O

(10)

S

Here, g/ = ( f r l r l ) f >"|, | ) a r e the charges of quarks; m;, m / , mw, mg, m si, msq, mch and msg are the temperature masses of leptons, quarks, W-bosons, gluons, s-leptons, s-quarks, charginos and gluinos, respectively. Since we investigate the high-temperature effects connected with the presence of external fields, we used leading in temperature terms of the Debye masses of the particles, only 3 ' 6 . The temperature masses of particles are taken as follows2 ,2

2 -(s.+

MS3\2

2 _(e

+

M*V

2 -.(S.

+

MULY

(11)

where the masses from SSB terms are taken as the low experimental limits of their masses: Msi = 40GeV, Msq = 176GeV, Mch = 62GeV, Msg = 154GeV. As it was established in numeric computation the spontaneous generation of fields depends on SSB masses fairly weak. Even in the case of zero SSB masses there is the generation of magnetic and chromomagnetic fields. In the limit of infinite SSB masses the picture conforms to the SM case. The temperature masses of gluons and VF-bosons are m 2 ^ = asw and as are the interaction couplings. In one-loop order, the neutral gluon contribution is trivial H3— independent constant which can be omitted. However, these fields are long-range states and they do give H3—dependent EP through the correlation corrections depending on the temperature and field. We include

263

the longitudinal neutral modes only because their Debye masses TL°(y,P) are nonzero. The corresponding EP is 3 (H°(y, f3)f2

vnng = ^ ^ ( y , / ? ) - ^

, (n°(y,/?)) 32TT 2

(12)

l g

° {{3(n°(y,(3)y/i)+4

7

7 is Euler's constant, U°(y, @) = U.Q0(k = 0, y, 0) is the zero-zero component of the neutral gluon field polarization operator calculated in the external field at finite temperature and taken at zero momentum u[y,P)-302

^

4?r2.

(id)

Equations (2)-(10) and (12) will be used in numeric calculations. 3. Combined generation of the fields To calculate the strengths of combined generated magnetic and chromomagnetic fields we use the perturbative computation method in 2 . First of all we find the strengths of the fields x and y when the quark and the s-quark contributions (vq) are divided in two parts, v' (x,/3) — v'q \y^o and v' (y,P) = v'q |ar_+o> where v' (x,fi) is the quark and s-quark contribution in the case of single magnetic field, and v'q{y,(}) is the one in the presence of chromomagnetic field, only. So, v'(x,y) = vi(x) + v2(y) + v3(x,y),

(14)

where x = x + Sx, y = y + Sy, and 6x and Sy are the field corrections connected with the interfusion effect of the fields in the quark and s-quark sector. Since the mixing of fields due to quark and s-quark loop is weak (this is justified in numeric calculations) one can assume that Sx
v2(y) = v2(y) +

d

-^Sy,

(lg)

v3(x,y).

After simple transformations we can find Sx and Sy, and hence may obtain x,y: r

_

f dv3{x,0)

\

dx

_ dv3(x,y)\

dx

tf

d2vi(x)\

r

) / \ 9x'2 J ' &

_ /dv 3 (0,y) _ dv3(x,y)\

y

dy

dy

If

J/y

d2v2(y)\

dy2

J'

(16)

264

These results on the field strengths determined by means of numeric investigation of the total EP are summarized in Figs. 1, 2. The solid line is the magnetic H and chromomagnetic field (3 strengths in the MSSM and the dashed line are that of in the SM. From the above analysis it follows that in the considered temperature interval the presence in the system of both fields leads to increasing of each of them in contrast with the SM case. In the latter the strengths of the combined fields are decreased as compared to the separate generation. With temperature decreasing this effect becomes less pronounced and disappears at 0 ~ 1. . . . . .

J . . 1

o.i

\ 0.08

\

\ \ \ \ \

0.06 0.04

0.02

0

0.2

0.4

0.6

0.8

P Figure 1.

The dependences of H on the inverse temperature ,

4. Discussion Let us discuss the results obtained. As we elaborated within the EP including the one-loop and the daisy diagrams, in the MSSM at high temperatures both the magnetic and chromomagnetic fields have to be generated. This vacuum is stable, as it follows from the absence of imaginary terms in the EP minima. If quark and s-quark loops are discarded, both of the fields can generated separately. All these states are stable 2 . As we have seen, the strengths of generated fields reduce due to the s-particle sector of the MSSM. This vacuum state is more stable as compare to the SM case. The result on the stabilization of the charged gauge field spectra is very important. It

265

0.2

K

1 I

0.15

-

0.1

-

0.05

-

\

\ \ •

—

.

— 0

0.2

0.4

0.6

_ 0.8

1

P Figure 2.

The dependences of H 3 on the inverse temperature /3.

has relevance not only to the problem of the consistent description of the generation of magnetic fields but also to the related problem on symmetry behavior in external magnetic fields investigated in the MSSM recently in 4 . As it is seen from Figs. 1,2, the strengths of the generated fields are increasing when the temperature is rising. It is also found, the dynamics of curves obtained in the SM2 are in a good agreement with our numeric calculations. The ground state possessing the magnetic and the chromomagnetic fields makes it reasonable to expect the existence of these fields in the electroweak transition epoch for both the SM and the MSSM. The state is stable in the whole considered temperature interval. The imaginary part in the EP exists for the external fields much stronger than the strengths of the spontaneously generated ones. The mixing of magnetic and chromomagnetic fields arising from the quark and the s-quark sectors of the EP is weak. In the MSSM, the change of the field minima in the inclusion of the field mixing does not exceed 4% (2% in the SM). During the universe cooling the strengths of the generated fields are decreasing. This is in agreement with what is expected in cosmology. One of the consequences of the results obtained is the presence of a strong chromomagnetic field in the early universe, in particular, at the electroweak phase transition. The influence of this field on the phase transitions may bring new insight to these phenomena. As our estimate showed, the chromomagnetic field is as strong as the magnetic one. So the

266

role of strong interactions in the early universe in the presence of the field needs more detailed investigations as compare to what is usually assumed 1 . We would like to notice that in the literature devoted to investigations of the quark-gluon plasma in the deconfinement phase carried out by nonperturbative methods the vacuum magnetization at high temperature has not been accounted for. From the point of view of the present analysis these investigations are incomplete. The generation of the chromomagnetic field at high temperature has to be taken into consideration. References 1. 2. 3. 4. 5.

D.Grasso, H.R.Rubinstein, Phys. Rept. 348, 163 (2001), astro-ph/0009061. V.Demchik, V.Skalozub, EPJ C 25, 291 (2002); EPJ C 27, 601 (2003). V.Skalozub, M.Bordag, Nucl. Phys. B 576, 430 (2000). M.Giovannini, M.Shaposhnikov, Phys. Rev. D 57, 2186 (1998). K.Kajantie, M.Laine, J.Peisa, K.Rummunkainen, M.E.Shaposhnikov, Nucl. Phys. B 544, 357 (1999); M.Laine (1999), hep-ph/9902282. 6. V.Skalozub, V.Demchik (1999), hep-th/9912071.

P H O T O N COLLIDERS A N D M A I N P O I N T S OF THEIR PHYSICAL P R O G R A M *

ILYA F . G I N Z B U R G Sobolev Institite of Mathematics SB RAS, prosp.ac. Koptyug, 4, Novosibirsk, 630090, Russia E-mail: [email protected]

I describe what the Photon Collider is and what the main fields are where it either has advantages in comparison with other machines or is a very important supplement.

1. Photon Colliders The next generation of high energy e + e~ colliders will be Linear Colliders - LC (see e.g. 1). The fact that each e^ bunch is used here only once led to the proposal to convert e+e~ LC into Photon Colliders with roughly the same energy and luminosity, 2 . This Photon Collider mode of LC is included in all modern projects of LC (TESLA, CLIC, JLC,... - see e.g. 3 ) To obtain photon beam for Photon Collider, the focused laser flash meets the electron bunch of LC in the conversion point C at small distance b before interaction point IP. In the conversion point a laser photon scatters 7o (laser) 7 (e)

Figure 1.

Scheme of conversion e —» 7 for Photon Collider

backward off high-energy electron taking a large portion of its energy. The ""This work is supported by grants RFBR. 02-02-17884, INTAS 00-00679 and grant 015.02.01.16 Russian Universities.

267

268

scattered photons travel along the direction of the initial electron with a small additional angular spread ~ l / 7 e = mec2/E, they are focused in the IP. Here they collide with an opposite electron (e"f collider) or photon (77collider). The laser flash with energy of a few Joule and length of a few mm becomes opaque for electrons. In this set-up the e —> 7 conversion coefficient becomes close to 1. The energy spectrum of the photon beam obtained is concentrated near its upper bound. It becomes sharper with a suitable choice of the longitudinal polarization of the initial electrons Ae and circular polarization of the laser photons Pc, see Fig. 2, where Ee is the electron energy.

1 d£p - ocdy

x.4.8

......

2KP*

o-pr b 0 c| 1

-

O',

:

'V ••p

> 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 y • u/E„

Figure 2.

yw/E.

High energy photon spectra and polarizations

The luminosity spectra differ from the simple convolution of above spectra for many reasons, (i) When photons with energy u> < ujmax propagate from the conversion point C to the interaction point IP, they get distributed over a wider area, reducing 77 luminosity in its soft part 2. (ii) The low energy part of spectra is increased due to multiple rescatterings of the electron on the other laser photons. (Hi) The nonlinear QED effects also modify spectra 4 , 5 . Therefore, the luminosity spectra will be measured during operations. Let us list the main features of a typical Photon Collider 3 . 1. Characteristic photon energy to « 0.8-Ee2. For high energy peak, uj\^ > 0-7umax (separated well from the low energy part of spectrum): • Luminosity £ 7 7 « Cee/3, Cei « £ e e/4 => 200 -f-150 fb _ 1 /year. • Mean energy spread (Aw) « 0.07cJmax (by factor 2 -=- 3 worse than in the e + e~ mode considering beamstrahlung and ISR).

269

• Mean photon helicity (A7) « 0.95, with easily variable sign. One can also obtain the linear polarization. The Photon Collider will be a good supplement to the e+e~~ LC with similar methods of observation. Besides, the Photon Collider can be treated as a hadron machine with point-like colliding "hadrons" - photons, and similar details should be added to detector. In the study of small distance phenomena the relative value of background from long distance interactions is much lower here than at the proton machines. 2.

Direct hunting for N e w Physics

I start with discussion of the problems in which the Photon Colliders can give results either unattainable or very difficult for other constructed machines. 2.1.

Discovery

of new

particles

The discovery of a new particle will be a clean signal of some definite form of a new theory. We denote the discovery bound for mass of this particle by Mb. • The ej mode provides the final states which cannot be produced

e*

K

w

s w u z s e Y

name excited e excited v new W

M <MhK, reaction 1.8E e7 —* e* e7 —> Wv*e 1.8E-MW cy —> vW 1.8E

wino cy —> Wx 1.8E- Mx 1.8E-Me e7 —> Ze zino l.8E-Mx selectron e-1 -> ex X - LSP, lightest superparticle

with similar intensity by other ways; it gives the best opportunity for discovering a lot of new particles. Some kinematical discovery limits Mb are presented in the Table above. The cross sections (and real discovery limits) of some of these e7 processes depend on unknown coupling constants. • The 7 7 mode. <> The cross section of the pair production 77 —> P+P~ (P = S - scalar,

270

10 x=W2/4M!

Figure 3.

17(77 —* P+P

20

)/[ 7 r a 2 M^p]> nonpolarized photons

P — F - fermion, P = W - gauge boson) not far from the threshold is given by QED with a reasonable accuracy. In Fig. 3 they are compared P+P cross sections (at y/s > 100 with the corresponding e++ ecGeV the Z contribution enhances the above cross section by a factor of 1.1-=-1.3 depending on the weak isospin). The 77 cross sections are obviously higher than the corresponding e+e" cross sections. With the expected LC luminosity, any new charged particle will be discovered in e+e~ mode. After that, the key problem will be to understand its nature. In this respect, the 77 production provides essential supplement to the e + e~ collisions: 1. The 77 cross sections decrease slow with energy growth. Therefore, one can study these processes relatively far from the threshold where the decay products are almost non-overlapping. 2. Near the threshold fp oc (1 + A1A2 ± l\(.2 cos2) with + sign for P = S (scalar) and - sign for P = F (fermion). This simple polarization dependence provides an opportunity to determine the spin of the produced particle independently from its charge. (This problem appears, e.g., for the discovery of SUSY particles since spin of invisible neutral is unknown). • The QED process 77 —> P+P~ with P = F or W generally cannot be observed in pure form due to instability of P. The intermediate P are polarized, and this polarization depends on production angle and on the initial photon helicity state. The subsequent decay of this P transforms this polarization to the momentum distribution of decay products. In particular, for the processes like 77 —> /i+\i~~ + neutrals (obtained from muon decay modes of 77 —> WW, 77 —> T+T~, etc.) muons should exhibit charge asymmetry related to the polarization of the initial photons 18 . It

r

271

is expected that the observation of this charge asymmetry will be a good tool for the discovery of some New Physics effects. 0 The leptoquark (£q) can be discovered in reactions like 77 —> £+t+(£t) with Mb « 1.5£ 7 . <0> The scalar or tensor resonances R that arise due to the strong interaction in the Higgs sector can be discovered in process 77 —* R with Mb « 2 £ 7 . 0 The gluino g can be produced in process 77 —> gg (via quark loop) 6 . The maximal value of this cross section is ~ (a2a^/M~) ln(M|/M^) at 2M-g < y/s^; < 2Mg. For example, at M , = 0.5 TeV, M-g = 0.25 TeV for y/s 7 7 « 1 TeV this cross section is about 1 pb. 0 If the stop mass Mi ss 100 -=- 200 GeV, the narrow stoponium with M « 200 — 400 GeV should also exist. Such states cannot be observed at hadron collider but they can be clearly seen at 77 collider with cross section averaged over photon spectrum < o~ > « 10 — 50 fb and a clear signature 7 . 2.2. 7 7 —y 7 7 . The search of effects of extra dimensions point-like Dirac monopole

or

In both cases mentioned the 77 —> 77 process is considered far below new mass scale M, and - due to the standard gauge invariance and dimensional reasoning - its cross section can be written as

a(77

^ 77) =3^(dy 4

(i)

with simple polarization dependence and angular distribution (S and D waves, roughly — isotropic). This wide angle elastic 77 scattering has very clear signature and small QED background. The observation of strong elastic 77 scattering rising quickly with energy will be the signal of one of these mechanisms. The study of polarization and angular dependence at photon collider and some similar processes can discriminate between the mechanisms. • Effects of extra dimensions 8 are considered in the scenario where gravity propagates in the (4 + Tridimensional bulk of the space-time, while gauge and matter fields are confined to the (3+l)-dimensional world volume of a brane configuration. The extra n dimensions are compactified with scale R, which produces the Kaluza-Klein excitations with masses nn/R. The corresponding scale in our world is assumed to be M ~ few TeV. The particles of our world interact (as AA —> BB) via the set of Kaluza-Klein excitations having spin 2 or 0 as e.g. T^T^/M4, where T^v is the stress-

272

energy tensor. The coefficients are accumulated in the definition of M (with A**l). The 77 initial state has numerical advantage as compared to e + e~ one due to the higher spin. The 77 final state has the best signature and the lowest SM background. The interference with SM effect enhances this anomaly for 77 —> WW process (simultaneously with enhancement of background). • The point—like Dirac monopole existence would explain mysterious quantization of the electric charge. There is no place for it in modern theories of our world but there are no convincing reasons against its existence. At s
273

these two scenarios. The SM - violated scenario is discussed now widely. We mainly skip this case in our discussion but the discussion in the preceding section is also valid for this case. The realization of the SM - like scenario looks very probable. 4.

If the SM. — like scenario is realized,

the Colliders study will have the following goals: • The insight into the £WSB mechanism. • Discovery of signals of New Physics via deviations from SM predictions. • Description of the observed phenomena in SM , especially QCD. Photon Colliders provide unique keys to these problems.

4 . 1 . The study of

£WSB.

The Higgs mechanism of £WSB can be realized in different ways: 1. Standard SM case - single Higgs boson, Mh < 400 -r 700 GeV. 2. More complicated Higgs sector, e.g. - Two Doublet Higgs Model. (2) 3. Single Higgs doublet with strong self-interaction. These variants should be discriminated with the aid of future experiments. Here (and in the study of anomalous interactions) the measurement of /177 (hZ'y) couplings is very promising since (i) In the SM these couplings appear only at the loop level. Therefore, the S/B for new signals is better than for the processes allowed at tree level. (ii) All fundamental charged particles contribute to these effective couplings. The entire structure of theory influences these vertices. (Hi) The expected accuracy in the measurement of the two-photon width is ~ 2% at Mh < 150 GeV and the luminosity integral 30 fb _ 1 (which is by 5 times lower than the anticipated annual luminosity) u . • In the cases (2.1,2) the Higgs boson will be discovered at the Tevatron or LHC, its spin and couplings to W, Z and fermions will be precisely measured at e + e~ LC. The SM - like scenario can be realized both in the SM (case (2.1)) and in other models (case (2.2)). 13 Distinguishing Styl'H'DM . The simplest extension of the Higgs sector is the 2HVM with the Model II for the Yukawa coupling (the same is realized in MSSM ). It contains 2 Higgs doublet fields <j>\ and fo with v.e.v.'s wcos/3 and vsin/3. The physical sector contains charged scalars H^ and three neutral scalars hi with no definite CV parity; in the CP conserving case these hi are scalars h and H (with Mh < MH) and pseudoscalar A.

274

Generally, 2HVM permits to have relatively strong QP and large FCNC (flavor changing neutral currents). To make these effects naturally weak - in accordance with observations - the terms in Higgs potential, giving (fa, fa) mixing should be relatively small, and the properties of the observed Higgs boson are close to those of h or H. In this case masses MH, MA and MH± are < 3 TeV due to perturbativity constraint 14 . The SM. - like scenario means that the coupling constants squared (not coupling constants themselves) are close to the SM values. In the 2'KDM it can be realized in many ways even in the CV conserving case (Table 1). The third column here marks which Higgs boson is observed, h Table 1. Allowed realizations of SM -like scenario in the 2HVM

(II)

observed type

notation

Higgs

constraint

tan/3

XV

boson <j> = Ah+

h

w+1

^ i

A,p±

AH+

H

RS+1

^ i

XV « Xu ~ Xd

Ah-

h

« -1

< I

AH-

H

RJ - 1

» l

Bj>±d •

Bh+d

h

RJ+1

XV ~ Xu ~ ~Xd

BH±d

H

«±1

B
Bh±u

h

«±1

BH+U

H

XV ~ Xd ~ ~Xu

Xi = -§KT =

^

eu =

^

^ ^ o -

ey =

1

ed =

-

^

- ^ L

-

^

«+l ±

( 1 ~ £ i)

with

Qi

i = V{= Z, W) or i = u{= t, c) or i = d,£(= b, T);

ev > 0, eued < 0 .

or H. The other scalar, H or h, and pseudoscalar A are almost decoupled from the gauge bosons and cannot be seen at e + e _ LC in the standard processes. If mass of any of these Higgs bosons is below 350 GeV and tan/3 77 process (for h - using the low energy part of photon spectrum) (and in e+e~ —> ttH at 2E > 2Mt + MH). (The widely discussed decoupling limit corresponds to solution Ah+ supplemented with a claim of unnaturally strong (4>i, fa) mixing (which allows heavy H, A and H^ with MH ~ MA ~ M^± without perturbativity limitation).) In any case, one can distinguish models via measurement of two-photon width of observed SM - like Higgs boson 13 . For MH± = 800 GeV the ratios of the two-photon Higgs width and the cross sections for e-y —> eh

process (for latter reaction - at y/s = 1.5 TeV for the left hand polar-

2HDMQQ/SM - Solution A

120

140

160

180 200 220 240 M„,„ [GeV]

Figure 4. Solutions A and Brj,±a. < 0-77 —> fi2HDM > / < 0-77 panel; aL^ej —* eh)2HDM/o\£,(e7 —* eh)SM - right panel.

> - left

ized photons 15 ) to their SM values are shown in Fig. 4 for the natural set of parameters of 27iT>M • The bands around the central line represent possible difference of Higgs-fermion and Higgs-W couplings (squared) from their SM values according to the anticipated uncertainty of future measurements 1 . The deviations from SM in Fig. 4 for solutions A and Bd are about ~ 10% (much more than anticipated 2% accuracy). These deviations are given by contribution of heavy charged Higgs bosons. For solutions Bu changing of relative sign of contributions of i-loop and W-loop increases the observable cross section more than twice in comparison with SM. Therefore, the measurement of the two-photon width at Photon Collider can distinguish these cases reliably. • In the case (2.3) a standard Higgs particle does not exist. The strong interaction in Higgs sector will manifest itself as the strong interaction of longitudinal components of gauge bosons WL and ZL with formation of new resonance states. Based on the experience with process 77 —> 7r+7r~ the following picture is expected: The strong interaction modifies weakly the cross section near the threshold in comparison with its SM value (but the phase of the amplitude reproduces that of strong interacting WL,WL scattering). It makes the observation of the strong interaction in the Higgs sector difficult below these resonances, typically, at \fs < 1.5 — 2 TeV (see e.g. 1 6 ). The charge asymmetry in the process ej —> eW+W~ will be sensitive to the phase of 77 —> W^WL amplitude even at relatively low energy of TESLA (0.8-1 TeV) 17 ' 18 , considerably below the threshold of possible res-

276

onance production. Indeed, the essential contribution to this asymmetry is given by interference of two-photon production subprocess of WW pair (in C-even state) and bremsstrahlung (one-photon) production subprocess (in C-odd state) like for the process e+e~ —> e+e~ n+n~. The interference with the axial part of Z exchange contributes additionally to this asymmetry.

4.2. Anomalous

interactions

of gauge and Higgs

bosons

If the energy is too low to discover new heavy particles, the New Physics reveals itself as some anomalies in the interactions of known particles. The goal of studies at colliders at this stage is to find these anomalous interactions and discriminate between them as well as possible. The correlation between coefficients of different anomalies will be the key for understanding what the nature of New Physics is. In this task the Photon Colliders have an exceptional potential. Anomalous interactions of gauge bosons The practically unique process under interest in the e + e~mode is e+e~ —> WW. The suitable electron polarization kills the neutrino exchange contribution, the residuary cross section (7 and Z exchanges) has maximum of about 2 pb within LEP operation interval, and decreases with energy after that. The e + e~ —> e+e~ WW, etc. cross sections are small at Vs < 1 TeV. At Photon collider main processes are e7 —> Wv, 77 —> W+W~ . Their SM cross sections are about 80 pb and energy independent at y/s > 200 GeV 20 , which gives (1 -=- 5) • 107 W's per year. Due to the high value of these basic cross sections, many processes of the 3-rd and 4-th order have large enough cross sections, Fig. 5: 7 7 -> ZWW, e-f -»• uWZ, 7 7 -»• WWWW, ... e 7 _• eWW, Large variety of these processes allows one to discover and separate well anomalies in specific processes and (or) distributions. The algorithms should be developed for the simulation of real processes of such type with real final states generated by W or Z decays to leptons or quarks (5,6,7,8 particles in the final state) and various backgrounds. The cross section e^ —> vW oc (1 — 2Ae), it is switched on or off with variation of electron helicity Ae. It gives very precise test of the absence of right handed currents in the interaction of W with the matter. Even first and incomplete modern simulations of processes ej —> vW

277 77 a n d

ye p r o c e s s e s

(tr tol , t r e e level)

CompHEPv.2.4

s*

' /"

: . _ :

eWW

zww

eZ

i>ZW •

',

/

-

,

°«fcoa<0a)

0

Figure 5.

200

400

600

600

1000

1200

1400

1600

1000

2000

Some processes with gauge boson production, unpolarized photons

and 77 —> W+W~ show that the sensitivity of this reaction to the anomalous magnetic moment and quadruple momentum of W is comparable to that attainable at e+e~ LC 21 - 22 . • The process ej —> eWW permits to study anomaly jZWW when one considers events with transverse momentum of scattered electron p± > 30 GeV. Anomalous interactions of Higgs boson with light ( 7 7 -^> h, ej —* eh) The observable anomalous interactions of Higgs boson with light (CV conserving and violating) can be summarized in an effective interaction (0i = e*<) v

Z F^ F F^F"" hv 1 6 V " : ; +2^z"T, +iOP^-^ A| A^

F^ vPy

Z„VF»V

+2i6PZ^-2

).

(3)

V

PZ

Here F^" and Z M " are the standard field strengths for the electromagnetic and Z field, F M " = s>i'/al3Fap/2 and & are the phases of couplings, generally different from 0 or ir, v = (GFV^)~1/2 = 246 GeV - v.e.v. of Higgs field 19 . The CV conserving anomalies give the deviation of the measured cross sections from SM. prediction. The CV violating anomalies give the transverse asymmetry - with the angle between directions of linear polarization

278

of collided photons and the longitudinal one, r

_

= 2

a

+ +

- a „

<J++ + a__

w

This longitudinal asymmetry is shown in Fig. 6. We see that the effects

Figure 6.

can be seen at reasonable values of anomaly scales Aj and phases £j. 5.

Hadron Physics. QCD

All problems studied at HERA and LEP will be studied there but in much wider interval of parameters and with much better accuracy. Among them I underline those which look most interesting now. • Nature of the total cross sections growth. The widespread concepts assume Regge factorization and universal energy behavior for different processes. With Photon colliders, for the first time in particle physics, one can have the set of mass shell cross sections of very high energy processes, appropriate for the testing of factorization or degree of its violation. That are app, measured at Tevatron and LHC, cr7P, measured at HERA, THERA, ajj, measurable at Photon collider. To measure large enough CT-yy , the preliminary stage of operations with low luminosity can be used to observe phenomena at small scattering angles. In this task the study of quasi-elastic process 77 —» pp is possible down to very low values of the momentum transfer squared t, which cannot be reached in pp experiments. In fact, to measure 77 —> pp cross section at \/\t\ « 20 MeV, it is necessary

279

to measure pions scattered on the angles ~ m p /2w. These angles 20 times larger than the corresponding angles for pp scattering. • The structure function of photon is a unique object of QCD calculable completely without phenomenology and auxiliary parameters at high enough photon virtuality Q2 and moderate x 23 . In modern data phenomenological hadronic component of photon dominates and accuracy of data is low. The ej Collider improves accuracy and strongly enhanced the obtainable region of Q2 that should suppress hadronic contribution in the comparison of the point-like one.

6.

Miscellaneous

• The two-loop radiative corrections to 77 —* W+W~ and e-y —> vW should be considered. They are measurable and sensitive to the following problems: (i) construction of S-matrix of theory with unstable particles and to (ii) gluon corrections like Pomeron exchange between quark components of W's. • In addition to the usually discussed problems related to the t quarks, the specific one is the study of axial anomaly in the process e-y —> ett, which exists even in the SAi . At small transverse momenta of scattered electrons p±, the cross section of subprocess ZLJ —> tt with longitudinally polarized Z does not disappear as it happens with photons but diverges as

M2lp\

2

\

References 1. R.D. Heuer et al. TESLA Technical Design Report, p. Ill DESY 2001-011, TESLA Report 2001-23, TESLA FEL 2001-05 (2001) 192p. hep-ph/0106315 2. I.F.Ginzburg, G.L.Kotkin, V.G.Serbo, V.I.Telnov. JETP Lett. 34 (1981) 514-518; Nucl. Instrum. Methods 205 (1983) 47; I.F. Ginzburg, G.L. Kotkin, S.L. Panfil, V.G. Serbo, V.I. Telnov, Nucl Instrum. Methods A 219 (1984) 5, see earlier references here. 3. B.Badelek et al. TESLA Technical Design Report, p. VI, chap.l DESY 2001-011, TESLA Report 2001-23, TESLA FEL 2001-05 (2001) hepex/0108012, p.1-98 4. A.I. Nikishov, V.I. Ritus, Sov. Phys. JETP 19 (1964) 529; N.B. Narozny, A.I. Nikishov, V.I. Ritus, Sov. Phys. JETP 20 (1964) 622; V.I. Ritus, A.I. Nikishov. Trudy FIAN 111 (1979). 5. I.F. Ginzburg, G.L. Kotkin, S.L Polityko. Yad. Fiz. 37 (1983) 368; 40 (1984)

280

6. 7. 8.

9. 10. 11. 12.

13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24.

1495; 56 (1993) 1487; M.Galynskii, E. Kuraev, M. Levchuk, V.I. Telnov. NIMR A 406 (1998) S.P. Li, H.C. Liu, D. Silverman. Phys. Rev. D 31 (1987) 1736 D.S. Gorbunov, V.A. Ilyin, hep-ph/0004092; D.S. Gorbunov, V.A. Ilyin, V.I. Telnov, hep-ph/0012175 E.g. H. Davoudiasi J. Mod. Phys. A 15 (2000) 2613; K. Cheung, Phys. Rev. D 61 (2000) 0105015; T. Rizzo, Phys. Rev. D 60 (1999) 115010, hepph/0008037 I.F. Ginzburg, S.L. Panfil, Yad. Fiz. 36 (1982) 850; I.F. Ginzburg, A. Schiller, Phys. Rev. D 60 (1999) 075016 B. Abbott et al., (DO Collab.), Phys. Rev. Lett. 81, 524(1998). G. Jikia, S. Soldner-Rembold, Nucl. Phys. B (Proc. Suppl.) 82 (2000) 373; M. Melles, W.J. Stirling, V.A. Khoze,Phys. Rev. D61(2000) 054015 E.Gabrielli, V.A.Ilyin, B.Mele, Phys. Rev. D 56 (1997) 5945; Phys. Rev. D 60 (1999) 113005; A.T. Banin, I.F. Ginzburg, LP. Ivanov, Phys. Rev. D 59 (1999) 115001; I.F. Ginzburg, LP. Ivanov, Eur. Phys. Journ. C 22 (2001) 411-421; hep-ph/0004069. I.F. Ginzburg, M. Krawczyk, P.Osland, hep-ph/9909455; hep-ph/0101229; hep-ph/0101331. I.F. Ginzburg, M. Krawczyk, P.Osland, in preparation, to be published in Proc. LCWS2002, Jeju, Korea, august 2002 I.F. Ginzburg, M.V. Vychugin, hep-ph/0201117 D. Dominici, hep-ph/0210438; M Battaglia, S. De Curtis, D. Dominici, hepph/0210351 I.F. Ginzburg, Proc. 9th Int. Workshop on Photon - Photon Collisions, San Diego (1992) 474-501, World Sc. Singapore. I.F.Ginzburg, hep-ph/0211099; D. A. Anipko, M. Cannoni,I. F. Ginzburg, O. Panella, A. V. Pak, Proc. PHOTON2003, in print I.F. Ginzburg, LP. Ivanov, hep-ph/0004069 Eur. Phys. Journ. C 22 (2001) 411-421; hep-ph/0004069 I.F. Ginzburg, G.L. Kotkin, S.L. Panfil, V.G. Serbo, Nucl. Phys. B228 (1983) 285 - 300. D. Anipko, I.Ginzburg, A. Pak, PHOTON2001, P r o c , hep-ph/0201072; Nucl. Instrum. Methods A 502/2-3 (2003) 752-754 I. Bozovic-Jelisacovic, K. Monig, J. Sekaric, hep-ph/0210308 E. Witten, Nucl. Phys. B120 (1977) 189. I.F. Ginzburg, V.A. Ilyin, in preparation

HIGH E N E R G Y QCD A N D S T R I N G / G A U G E CORRESPONDENCE*

A. G O R S K Y I n s t i t u t e of Theoretical a n d E x p e r i m e n t a l Physics, B.Cheremushkinskaya 25, Moscow, 117259, Russia

We briefly review the recent progress concerning the application of the gauge/string duality to the high energy QCD.

1. Introduction The explicit realization of the generic string/gauge correspondence program (see,for instance, *) remains the challenging problem during the last decades. It escaped the complete solution apart from the simplified twodimensional example 2 . Some time ago it was attacked in the SUGRA approximation at the string side3 where the extended N=4 SUSY in gauge theory allowed to identify the dual classical SUGRA solution. Moreover, it was argued 4 that the classical solution in the a model corresponding to the rotating long string amounts to the anomalous dimensions of the operators extended along the light-cone. In the recent papers 5 ' 6 which this talk is mainly based on it was suggested that a kind of stringy picture can be developed in another regimes, namely in the Regge and light-cone kinematics for a non supersymmetric gauge theory. This hypothetical string theory in principle should be responsible for the Veneziano type amplitudes in QCD. Since the theory is not supersymmetric one can not expect any conformal symmetry in the low energy sector. However, there is a remnant of the conformal symmetry in the high energy sector. Actually, it was exploited long time ago in calculations of the anomalous dimensions in QCD 7 . The symmetry which substitutes SO(2,4) conformal group in d=4 is SL(2,C) in the Regge *I am grateful to A. Belitsky, I. Kogan and G. Korchemsky for the collaboration. The work was supported in part by grants CRDF-RP2-2247, INTAS-00-334 and RFBR 0101-00549.

281

282

limit and collinear conformal group SL(2,R) on the light-cone. According to the string/gauge correspondence these groups of the global symmetry should be related to the isometry of the gravity background. Hence the natural gravity background to be expected should involve Euclidean AdSz factor for the Regge case and AdS2 on the light-cone. It appears that the place where the stringy picture can be captured is the RG evolution. It is the RG flow where the relevant symmetry can be traced out. The key property of RG flows to be used is the hidden integrability which can be identified in both regimes under consideration. To define the relevant integrable system we have to extract the proper degrees of freedom, set of integrals of motion and the corresponding "times". The degrees of freedom in the Regge case have been identified as the coordinates of the reggeized gluons8 while in the light-cone kinematics these are fields involved in the bare Lagrangians like quarks and gluons 9 ' 10 . The time to in the integrable system has the meaning of some evolution parameter; namely, logs in Regge case and logfi in the light-cone case. The evolution equations can be presented in the Hamiltonian form where the anomalous dimensions for the light-cone operators or intersepts of the multireggeon states in the Regge case play the role of the eigenvalues of integrable spin chain Hamiltonians. On the other hand, there is another class of (supersymmetric) YangMills theories in which integrability emerges. In the M=2 SUSY YM theory which can be solved in the low-energy limit exactly 11 the relevant integrable system was identified as the complexified periodic Toda chain. The low-energy effective action and the BPS spectrum in the theory can be described in terms of the classical Toda system 12 . In the relevant case of the superconformal Nf = 2NC theory, the solution leads to the periodic XXX chain in the magnetic field. Because these magnets are classical, the spins are not quantized and their values are determined by the matter masses. In the special case, when the matter is massless, one arrives at the classical XXX magnet of spin zero. At the first glance there is nothing in common neither between highenergy (Regge) limit of QCD and low-energy limit of SUSY YM, nor between the corresponding integrable models. However it appears that there is a deep relation between the Regge limit of QCD and some particular limit of superconformal Af=2 SUSY YM at Nf = 2NC. This relation is based on the fact that in the both cases we are dealing with the same class of integrable models - the 5L(2,C) homogenous spin zero Heisenberg magnets. There is the key difference between two theories; the integrable

283

model in the SUSY YM is a classical Heisenberg magnet, whereas the Regge limit of QCD is described by a quantum Heisenberg magnet. To bring two pictures together, it is natural to develop quasiclassical description in QCD case and quantize the classical picture in SUSY YM. In 5 we developed a brane picture for the Regge limit of the multi-colour QCD. We argue that the dynamics of the multi-reggeon compound states in multi-colour QCD is described in this picture by a single membrane wrapped around the spectral curve of the Heisenberg magnet. The fact that we are dealing with the quantum magnet implies that the moduli of the complex structure of the spectral curve can take only quantized values. Let us note that in the Regge limit there is a natural splitting of the four-momenta of reggeized gluons into two transverse and two longitudinal components, which is unambiguously determined by the kinematics of the scattering process. One can make a Fourier transform with respect to the transverse momenta and define a mixed representation: two-dimensional longitudinal momentum space and two-dimensional impact parameter space for which it is natural to use complex coordinates z and z = z*. The Riemann surface which the brane is wrapped around lives in this "mixed" coordinate-momenta space. Different projections of the wrapped brane represent hadrons and reggeons. The genus of this surface is fixed by the number of reggeons. Let us note that multi-reggeon states appear from the summation of only planar diagrams in QCD and from the point of view of topological expansion they all correspond to the cylinder-like diagram in a colour space. It contribution to the scattering amplitude is given by the sum of N—reggeon exchanges in the t—channel with JV = 2,3.... It is amusing that in result we get a picture where we sum over Riemann surfaces of arbitrary genus g = N — 2.

2. Riemann surfaces and universality class of the Regge regime In the Regge limit the two particle scattering can be described in terms of the reggeon degrees of freedom. The calculation of the N—reggeon diagrams can be performed using the Bartels-Kwiecinski-Praszalowicz approach 13 . The diagrams have a generalized ladder form and for fixed number of reggeons, N, their contribution satisfies Bethe-Salpeter like equations for the scattering of N particles. The solutions to these equations define the colour-singlet compound states |\?AT) built from N reggeons. These states propagate in the t—channel between two scattered onia states and give rise

284

to the following (Regge like) high energy asymptotics of the onium-onium cross-section

rtotw- £ (^r

x

r^s/'

P{AN)(Q2)P{BN\Q2)

(i)

JV=2,s,... y/otsNclnl/x with the exponents SN denned below. Here, each term in the sum is associated with the contribution of the N—reggeon compound states. At N = 2, the corresponding state defines the BFKL pomeron 14 with e2 = 4 1 n 2 . The N—reggeon states are defined in QCD as solutions to the Schrodinger equation Qj

UN$(zi,z2,...,zN)

= —NceNyi{zi,z
(2)

with the effective QCD Hamiltonian T-LN acting on two-dimensional transverse coordinates of reggeons, Zk {k = 1,...,N) and their colour SU(NC) charges ^•N

=

~2^

2_^

H

ij tftf •

(3)

l
Here, the sum goes over all pairs of reggeons. To get some insight into the properties of the ./V—reggeon states, it proves convenient to interpret the Feynman diagrams as describing a quantum-mechanical evolution of the system of N particles in the t—channel between two onia states |J4.) and \B) fftot(s) = E

(asNc)N(A\

eMV*)-** \B),

(4)

N>2

with the rapidity In a; = In Q2/s serving as Euclidean evolution time. To find the high energy asymptotics of (4), one has to solve the Schrodinger equation (2) for arbitrary number of reggeized gluons N, expand the operator {1/X)'HN over the complete set of the eigenstates of T-LN and, then, resum their contribution for arbitrary N. The pair-wise interaction between reggeons is reduced in the multicolour limit to a nearest-neighbor interaction and, in addition, the colour operator is replaced by a c—valued factor tffj -» —Nc/2. The two-particle reggeon Hamiltonian i?i,i+i acts on the two-dimensional (transverse) reggeon coordinates z = (x,y) and it coincides with the BFKL kernel. It becomes convenient to introduce the complex valued (anti)holomorphic coordinates, z = x + iy and z = x — iy, and parameterize the position of the

285

kth reggeon as Zk = (zk,Zk). One can rewrite the N—reggeon Hamiltonian as (HN + HN) + 0{N~2).

HN = ^

(5)

Here the hamiltonians HN and HN act on the (anti)holomorphic coordinates and describe the nearest-neighbor interaction between N reggeons N

N

HN = 2_^ H(zm,zm+i),

HN = ^

m=l

H(zm,zm+i),

m=\

with the periodic boundary conditions Zk+i — z\ and Zk+i = z\- The interaction Hamilonian between two reggeons with the coordinates {z\,z{) and (z2,Z2) in the impact parameter space, is given by the BFKL kernel H(Zl,z2) where ip(x) = d\nT(x)/dx the equation

= -r/>{J12) - V>(1 - J12) + 2V(1),

(6)

and the operator J12 is defined as a solution to

Jl2(Jl2-l)=-(zi-Z2)2d1d2

(7)

with dm = d/dzm. The expression for H(z\,z2) is obtained from (6) by substituting Zk -^ ZkIn the quasiclassical limit the solutions to the equations of motion of the spin chains can be described in terms of the Riemann surfaces15 and we will argue that just these surfaces fix the universality class of the Regge regime. Moreover it appears that it falls into the same universality class as superconformal N=2 SQCD at the strong coupling orbifold point. Comparing the spectral curve for the superconformal M=2 SUSY YM with Nf = 2NC with the spectral curve for ./V-reggeon compound states in multi-colour QCD one observes that they coincide if we make the following identification • The number of the reggeons N = Nc; • The integrals of motion of multi-reggeon system are identified as the above mentioned functions qk(u) on the moduli space of the superconformal theory; • The coupling constant of the gauge theory should be at the so called orbifold point TC\ = | + -h= Under these three conditions the both theories fall into the same universality class.

286

3. Quantum spectrum and S-duality The S—duality is a powerful symmetry in the SUSY YM theory which allows us to connect the weak and strong coupling regimes. The effective coupling in this theory coincides with the modular parameter of the spectral curve of the underlying classical integrable model. As a consequence, the S—duality transformations in the gauge theory are translated into the modular transformations of the spectral curve describing complexified integrable system. So far the 5—duality was well understood only for classical integrable models. In the case of multi-colour QCD in the Regge limit the situation is more complicated since the duality has to be formulated for a quantum integrable model. The integrals of motion take quantized set of values and the coordinates on the moduli space are not continuous anymore. Therefore the question to be answered is whether it is possible to formulate some duality transformations at the quantum level. To study this question let us propose the WKB quantization conditions which are consistent with the duality properties of the complexified dynamical system whose solution to the classical equations of motion are described by the Riemann surface. We recall that the standard WKB quantization conditions involve the real slices of the spectral curve i

pdx = 2-Kh{m + l/2)

(8)

where n, are integers and the cycles Ai correspond to classically allowed trajectories on the phase space of the system. In our case the coordinate x is complex and arbitrary point on the Riemann surface is classically allowed. As a result the general classical motion involves both A— and B—cycles on the Riemann surface. This leads to the following generalized WKB quantization conditions for actions and dual actions Rei

pdx

= „Hn>,

Ref

Pdx

=^

,

(9)

Note that in the context of the SUSY YM this condition would correspond to the nontrivial constraints on the periods and on the mass spectrum of the BPS particles . It is clear that the WKB conditions (9) imply the duality Ai -H- Bi and n; «-» nn. Let us consider the quantization conditions (9) in the simple case of the Odderon N = 3 system. The spectral curve is a torus y2 = (2x3 + q2x + q3f - 4a;6

(10)

287

where q2 is given by conformal spin while 93 is the complex integral of motion to be quantized. The quantization conditions (9) read Rea(qz) = irn,

Rea£>(<73) = nm

(11)

where n and m are integer. These equations can be solved for large values of q\/q2 3> 1, for which the expressions for the periods a(qs) and ao{qz) are simplified considerably. The explicit evaluation of the integrals in this limit yields

Substituting these expressions into (11) one finds

where t\ = n and t2= n — 2m. The WKB expressions (13) are in a good agreement with the exact expressions for quantized q% obtained from the numerical solutions of the Baxter equations in 16 . In the general multi-reggeon case we have to consider the quantization conditions (9) on the Riemann surface of the genus (N — 2) which has the same number of the A— and B—cycles. In result the spectrum of the integrals of motion 93, ..., gjv is parameterized by two (N — 2)—component vectors n and rh. In the SUSY YM case these vectors define the electric and magnetic charges of the BPS states. In the Regge case the physical interpretation of n and m is much less evident. Let us first compare the electric quantum numbers in the two cases. In Regge case it corresponds to rotation in the coordinate space around the ends of the Reggeons. This picture fits perfectly with the interpretation of the electric charge in SUSY YM case. Indeed VEVs of the complex scalar take values on the complex plane which is the counterpart of the impact parameter plane and the rotation of the phase of the complex scalar is indeed the "electric rotation". 4. Stringy picture and the calculation of the anomalous dimensions We have demonstrated that integrability properties of the Schrodinger equation for the compound state of Reggeized gluons give rise to the stringy/brane picture for the Regge limit in multi-colour QCD. There is another limit in which QCD exhibit remarkable properties of integrability. It has to do with the scaling dependence of the structure functions of deep

288

inelastic scattering and hadronic light-cone wave functions in QCD. In the both cases, the problem can be studied using the Operator Product Expansion and it can be reformulated as a problem of calculating the anomalous dimensions of the composite operators of a definite twist. The operators of the lowest twist have the following general form

O/&(0) = (yD^^myD^^myD^-^-^^iO),

(14)

where k = (ki,k2) denotes the set of integer indices fcj, y^ is a light-cone vector such that yi = 0. $& denotes elementary fields in the underlying gauge theory and D^ = d^ — iA^ is a covariant derivative. The operators of a definite twist mix under renormalization with each other. In order to find their scaling dependence one has to diagonalize the corresponding matrix of the anomalous dimension and construct linear combination of such operators, the so-called conformal operators

Oc^(0) = Y/Ck,g-ON,k(0).

(15)

k

A unique feature of these operators is that they have an autonomous RG evolution A2 ^

O ^ f ( 0 ) = -lN,q

• 0£"/(O).

(16)

Here A2 is a UV cut-off and 7jv,g is the corresponding anomalous dimension depending on some set of quantum numbers q to be specified below. It turns out that the problem of calculating the spectrum of the anomalous dimensions 7;v,g to one-loop accuracy becomes equivalent to solving the Schrodinger equation for the SL(2,R) Heisenberg spin magnet. The number of sites in the magnet is equal to the number of fields entering into the operators under consideration. To explain this correspondence it becomes convenient to introduce nonlocal light-cone operators F(zuz2)

= $i(ziy)$2(z2y),

F(zi,z2,z3)

= $i{ziy)$2(z2y)$3(z3y)

• (17)

Here y^ is a light-like vector (y^ = 0) defining certain direction on the light-cone and the scalar variables z* serve as a coordinates of the fields along this direction. The fields $i{ziy) are transformed under the gauge transformations and it is tacitly assumed that the gauge invariance of the nonlocal operators F(zi) is restored by including the Wilson lines between

289 the fields in the appropriate (fundamental or adjoint) representation. The conformal operators appear in the OPE expansion of the nonlocal operators (17) for small z\ — 23 and z2 — Z3. The field operators entering the definition of F(zi) are located on the light-cone. This leads to the appearance of the additional light-cone singularities. They modify the renormalization properties of the nonlocal light-cone operators (17) and lead to nontrivial evolution equations which as we will show below become related to integrable chain models. We notice that there exists the following relation between the conformal three-particle operators (15) and the nonlocal operators (17) 0S?°f(O) =

*N,q(dzl,dZ2,dZ3)F(zuz2,z3)

(18) Zi=0

where ^N,q{x\,x2,X3)

is a homogeneous polynomial in Xi of degree N

*N,q{xux2,xz)

= ^2Ck,q

•x*x**xZ-h-k>

(19)

fc

with the expansion coefficients Ck,q defined in (15). The problem of defining the conformal operators is reduced to finding the polynomial coefficient functions ^N,q{xi) and the corresponding anomalous dimensions 7jv,gUsing the renormalization properties of the nonlocal light-cone operators (17) one can show, that to the one-loop accuracy the QCD evolution equation for the conformal operators (17) can be rewritten in the form of a Schrodinger equation n • $N,g{xi)

= 7Ar,a¥jv,,(a:i),

(20)

where the Hamiltonian "H acts on the x\—variables which are conjugated to the derivatives dZi and, therefore, have the meaning of light-cone projection (y • Pi) of the momenta pi carried by particles described by fields $(ZJ2/). For example when $1 and $2 are quark fields of the same chirality Nc

Fap(Zl,Z2) = Y.tilfloMUliMztV)

(21)

i=i

with qj = (1 +

7 5 )<7J/2,

the two-particle Hamiltonian is given by

Hi2 = —CF [Hqq(J12) + 1/4] ,

Hqq(J12)

= V(Ji 2 ) - V(2).

(22)

where CF = (N% — 1)/(2NC). The corresponding eigenvalues define the anomalous dimensions of the twist-2 mesonic operators built from two

290 quarks with the same helicity N

7^

= -CFMN

+ 2) - V(2) + 1/4] =

7T

^CF 7T

i

E fc+l

+

i 4

(23)

jfe=i

At large N this expression has well-known asymptotic behaviour xN' ~ asCF/nlnN. It is conformal symmetry which dictates that the two-particle Hamiltonian is a function of the Casimir operator of the SL(2, M) group, but it does not fix this function. The fact that this function turns out to be the Euler i/>-function leads to a hidden integrability of the evolution equations for anomalous dimensions of baryonic operators. Namely, for baryonic operator built from three quark fields of the same chirality Fa0j(zi,z2,z3)

= ]T

e

ijk(^Qj)a(ziy)Uq})p(z2y)Uql)7(z3y)

(24)

i,j,k=l

the evolution kernel is given by9 ^(3) = £ i {(l + 1/NC) [Hqq(J12) + Hqq(J23) + Hqq(J31)} + 3CF/2}

(25)

Z7T

with Hqq given by (22). Similar to the Regge case, one can identify (25) as the Hamiltonian of a quantum XXX Heisenberg magnet of SL(2, E) spin jq = 1. The number of sites is equal to the number of quarks. Based on this identification we shall argue now that the calculation of the anomalous dimensions can be formulated entirely in terms of Riemann surfaces which in turn leads to a stringy/brane picture. It is important to stress here the key difference between Regge and light-cone limits of QCD. In the first case the impact parameter space provides the complex plane for the Reggeon coordinates and we are dealing with a (2 +1)—dimensional dynamical system. In the second case the QCD evolution occurs along the light-cone direction and is described by a (1 4- 1)—dimensional dynamical system. As a consequence, in these two cases we have two different integrable magnets: the SL(2,€) magnet for the Regge limit and the SL(2, K) magnet for the light-cone limit. The evolution parameters ("time" in the dynamical models) are also different: the rapidity In s for the Regge case and the RG scale In y, for the anomalous dimensions of the conformal operators. Our approach to calculation of the anomalous dimensions via Riemann surfaces looks as follows. For concreteness, we shall concentrate on the evolution kernel (25). Similarly to the Regge case, one starts with the

291

finite-gap solution to the classical equation of motion of the underlying SL(2,R) spin chain and identifies the corresponding Riemann surface w

= 2x3-(N

+ 2)(N + 3)x + q,

uj = xzep

(26) w where q is the integral of motion and N is the total SL(2,M) spin of the magnet, or equivalently the number of derivatives entering the definition of the conformal operator (15). Note that the Riemann surface corresponding to the three-quark operator has genus 5 = 1 , while g = 0 for the twist 2 operators. Summarizing, in our approach we have the following correspondence operator <^=>- Riemann surface twist of the operator <^=> genus of the Riemann surface calculation of the <^=> quantization of the anomalous dimension

Riemann surface (27)

5. Cusp anomaly and gauge/string duality There is another effective way to get anomalous dimensions of the lightcone operators based on the stringy representation of the cusp anomaly introduced long time ago 18 . It counts the anomalous dimensions of the nonlocal operators corresponding to the Wilson lines with cusps. It appears that cusp anomaly serves as the generating function for the anomalous dimensions of twist two operators with large Lorentzian spins. It provides the anomalous dimensions both at weak and strong coupling regimes. The explicit procedure involves the following Wilson line in adjoint representation To get the anomalous dimensions one derives the Wilson line in the cusp geometry as the function of the cusp angle and make a substitution (py) = iS for the spin S operators 17 . To develop the stringy representation one has to treat weak and strong coupling regimes separately 6 . The weak coupling regime can be treated in a stringy way by noting that cusp anomaly at one loop can be identified with the disk amplitude in two dimensional SL(2,R) Yang-Mills theory. Namely, W{6) = r c u s p (0; a.) = ^ ^

(0coth
(28)

On the other hand d=2 Yang-Mills theory admits the string representation 2 hence weak coupling cusp anomaly enjoys it too. It is important to

292 emphasize that the string responsible for the weak coupling regime is not of Nambu-Goto type and most naturally can be considered as the topological tensionless string. At strong coupling one applies the standard AdS*, geometry for NambuGoto string to get the anomalous dimensions. In the closed string picture the energy of the string with spin S amounts to the anomalous dimensions of spin S operators. To get the same anomalous dimension from the open string one has to consider the contour with the cusp on the boundary of AdS space. The minimal surface with the cusp boundary provide the anomalous dimension for spin S operators at strong coupling19 7S oc (g2N)l>2logS

(29)

The generalization of this picture for the anomalous dimensions of the multipartonic operators at strong coupling has been found in 6 . To this aim more complicated collection of Wilson lines at large N has to be considered (trWnKi,6]...trWn[&,&]> = ...

(30)

The minimal surface with the corresponding boundary conditions immediately provides the anomalous dimensions of such operators. Calculation of the total area of the minimal surface yields *

k

^ 0 j l n / z ~ 2fc0r cusp (a s )ln/x 3=1

3=1

(31) for 0i ~ . . . ~ 6k ~ 0 ^> 1. As before, the coefficient in front of the In ^ at k = 2 and k = N can be identified as the anomalous dimensions of the ./V-particle conformal operators, 7™'" and 7™ax, respectively, for 8 ~ J. It was shown that this answer agrees with the answer for the same dimensions followed from the rotating closed string in AdS background if the number of the string ends at the boundary coincides with the number of partons in the corresponding operators. References 1. A.Polyakov, hep-th/9711002, hep-th/9809057, hep-th/0110196 2. D. J. Gross and W. I. Taylor, Nucl. Phys. B 400, 181 (1993) [arXiv:hepth/9301068]. 3. J. Maldacena, Adv. Theor. Math. Phys. 2, 231 (1998) [Int. J. Theor. Phys. 38, 1113 (1998)] [arXiv:hep-th/9711200]. S. S. Gubser, I. R. Klebanov and A. M. Polyakov, Phys. Lett. B 428 105

293

4. 5. 6. 7.

8.

9.

10. 11.

12. 13. 14.

15. 16.

17. 18. 19.

(1998) [arXiv:hep-th/9802109]. E. Witten, Adv. Theor. Math. Phys. 2 253 (1998) [arXiv:hep-th/9802150]. S. S. Gubser, I. R. Klebanov and A. M. Polyakov, Nucl. Phys. B 636 99 (2002) [arXiv:hep-th/0204051]. A. Gorsky, I. I. Kogan and G. Korchemsky, JEEP 0205, 053 (2002) [arXiv:hep-th/0204183]. A. V. Belitsky, A. S. Gorsky and G. P. Korchemsky, arXiv:hep-th/0304028. S.J. Brodsky et al., Phys. Lett. B 9 1 , 239 (1980); Phys. Rev. D 3 3 (1986) 1881; Yu.M. Makeenko, Sov. J. Nucl. Phys. 33, 440 (1981); Th. Ohrndorf, Nucl. Phys. B198, 26 (1982); L. N. Lipatov, JETP Lett. 59, 596 (1994) [Pisma Zh. Eksp. Teor. Fiz. 59, 571 (1994); arXiv:hep-th/9311037]. L. D. Faddeev and G. P. Korchemsky, Phys. Lett. B 342, 311 (1995) [arXiv:hep-th/9404173]. V.M. Braun, S.E. Derkachov, A.N. Manashov, Phys. Rev. Lett. 81 2020 (1998) [arXiv:hep-ph/9805225]. V. M. Braun, S. E. Derkachov, G. P. Korchemsky and A. N. Manashov, Nucl. Phys. B 553, 355 (1999) [arXiv:hep-ph/9902375]. A.V. Belitsky, Phys. Lett. B 453,59 (1999) [arXiv:hep-ph/9902361]. S. E. Derkachov, G. P. Korchemsky and A. N. Manashov, Nucl. Phys. B 566, 203 (2000) [arXiv:hep-ph/9909539]. N. Seiberg and E. Witten, Nucl. Phys. B 426, 19 (1994) N. Seiberg and E. Witten, Nucl. Phys. B 431, 484 (1994) [arXivihepth/9408099]. A. Gorsky, I. Krichever, A. Marshakov, A. Mironov and A. Morozov, Phys. Lett. B 355, 466 (1995) [arXiv:hep-th/9505035], J. Bartels, Nucl. Phys., B175, 365 (1980). J. Kwiecinski and M. Praszalowicz, Phys. Lett., B94, 413 (1980). E.A. Kuraev, L.N. Lipatov and V.S. Fadin, Phys. Lett, B60, 50 (1975); Sov. Phys. JETP 44, 443 (1976); ibid. 45, 199 (1977); Ya.Ya. Balitsky and L.N. Lipatov, Sov. J. Nucl. Phys., 28, 822 (1978). G. P. Korchemsky, Nucl. Phys. B 462, 333, (1996) [arXiv:hep-th/9508025] G. P. Korchemsky, J. Kotanski and A. N. Manashov, Phys. Rev. Lett, 88, 122002 (2002) [arXiv:hep-ph/0111185]; S. E. Derkachov, G. P. Korchemsky, J. Kotanski and A. N. Manashov, hepth/0204124. G.P. Korchemsky, G. Marchesini, Nucl. Phys., B406, 225, (1993). A.M. Polyakov, Nucl. Phys., B164, 171 (1980). M. Kruczenski, J. High Ener. Phys. 0212, 024 (2002). Yu.M. Makeenko, J. High Ener. Phys., 0301, 007 (2003).

ANISOTROPIC COLORED SUPERFLUIDS*

j . HOSEK Dept.

Theoretical

Physics,

Nuclear Physics Institute, Czech Republic E-mail: [email protected]

250 68 Rez

(Prague),

We review characteristic aspects of selected physically distinct macroscopic quantum phases breaking spontaneously the rotation symmetry which under plausible assumptions can exist in cold QCD matter with both three and two colors at various baryon densities.

1. Introduction Walk along the astrophysics axis (i.e. the baryon number density n 1 / 3 axis, or the quark chemical potential /i axis) in the QCD matter phase diagram provides an interesting and informative view of different macroscopic quantum phases of strongly interacting matter with essentially naked eye 1 . For a three-color QCD (Nc — 3) this in practice amounts to observing the behavior of the neutron stars, for Nc = 2it amounts to observing the QCD matter in numerical lattice experiments 2 . 2. Three-color QCD matter A. At densities close to the nuclear matter density the neutrons inside the neutron stars undergo standard BCS neutron-neutron pairing due to an attractive S-wave effective interaction at the Fermi surface. As the density increases the average 3P attraction implying a spin-1 BCS-type pairing becomes dominant 3 . In this respect the system resembles the pairing in 3 He. Indeed, in a Fermi liquid of one species of non-relativistic fermions ip the BCS-type condensate due to an attractive P-wave effective interaction *I thank the organizers for invitation, and for giving me the floor. I am also grateful to Michael Buballa, Micaela Oertel, Jifi adam, and Jean-Baptiste Juin for pleasant collaboration, the work was supported by the grant GA CR 202-2/0847. 294

295

has the general form (O\tpa(x)(ia2(Ta)a0VAip0(x)\O)

~ AaA

(1)

In 3He the Hamiltonian is approximately invariant with respect to separate SU(2) spin, 50(3) orbital rotations, and an U(l) phase transformation generated by the operator of the fermion number. Many physically different phases are then distinguished by different forms of the matrix A. They characterize different patterns of spontaneous breakdown of the global 517(2) x 50(3) x U(l) symmetry. In dense neutron matter the Hamiltonian contains a strong tensor force and, consequently, it is invariant only with respect to 50(3) x U(l) transformations generated by the total angular momentum J, and the fermion number. It selects the pairing (1) into the J = 2, and forces A to be a traceless symmetric 3 x 3 matrix. To the best of our knowledge general analysis of the properties of different ordered phases characterized by generically different AaA does not exist. Knowledge of the low-lying spectra of both the quasi-particles, and of the collective excitations depending upon the details of the effective fermion-fermion interaction, is important. It determines the behavior of the condensed system as viewed from the outside. B. In a deconfined moderately dense low-T Nc = 3 quark matter anisotropic color superfiuid phases are feasible 4 ' 5 ' 6 ' 7 . For the relativistic quarks of two species distinguished by the Pauli matrices r two forms of the quark-quark condensate without derivatives are possible: (1) a polar condensate (0\iP(x)Ca03 r 2 \ip(x)\0) ~ A

(2)

(2) a ferromagnetic condensate (0\TP(x)C(a01 + ia02) T2 rP{x)\0) ~ Af

(3)

The dispersion law of the relativistic fermions undergoing Cooper pairing (2) was found explicitly 6 ' 7 , exhibiting clearly the expected spontaneous breakdown of the rotation invariance: E±(p)

= y/el + n* + \A\*±28

,

(4)

where s = ufJ^e2, + |A| 2 p£ , and p]_ = p2 + p2. Here fi is the quark chemical potential, and ep is the quark energy. Physical consequences of the form (4) discussed in6 can be summarized as follows: (i) Due to the peculiar form of the gap equation, exhibiting spontaneous breakdown

296

of rotational symmetry, the corresponding gap parameter A is extremely sensitive to details of the effective interaction and to the chemical potential, (ii) The critical temperature of the anisotropic component is approximately given by the relation Tc ~ A(T = 0)/3. (iii) In the chiral limit the gas of anisotropic Bogolyubov-Valatin quasiquarks becomes effectively gapless and two-dimensional. Consequently, its specific heat depends quadratically on temperature, (iv) All collective Nambu-Goldstone excitations of the anisotropic phase have a linear dispersion law and the whole system remains a superfluid. The alternative dispersion laws corresponding to (3) are complicated; they are available only in the form of an approximation 7 , and the analysis of physical consequences of this form is yet to be done. 3. Two-color QCD matter A. According to the confinement dogmas the ground state baryon in Nc = 2 QCD with one massive flavor should have spin one, and we are allowed to contemplate a many-baryon interacting spin-1 matter at various baryon densities. Assumption of massiveness of one quark flavor is important. We want to avoid all complications related with spontaneous breakdown of a non-Abelian chiral symmetry, subsequent degeneracy of baryons with pions, and mixture with spin-0 baryons. To the best of our knowledge nothing is known about the baryon -baryon effective interaction in such a world regardless of density. This is good. The emerging picture would be a clean prediction testable one day in lattice experiments. In particular, there should be a short-range baryon-baryon repulsion which guarantees that the smallest possible colorless quark-quark hadrons keep their identity unless dissolved by a brute-force compression. At low densities the description of the spin-1 baryon matter in terms of the second-quantized Schroedinger field ipa,a = 1,2,3

H = V&(e - M* + \gi{^M2

+ ^2(VM)(##)

(5)

together with the Bogolyubov approximation 8 ' 9 : (ipa(x)

> * Q +
I V&(a:)-»*i+¥£(*) should be appropriate. Here $ is a complex number known as the condensate wave function: $ = (0\ip(x)\0), where |0) is the ground state of the condensate,