Hadron multiplicities. (English) Zbl 0969.81652

Dremin, I. M.; Gary, J. W.

Phys. Rep. 349, No. 4, 301-393 (2001).

Summary: We review results on hadron multiplicities in high-energy particle collisions. Both theory and experiment are discussed. The general procedures used to describe particle multiplicity in quantum chromodynamics (QCD) are summarized. The QCD equations for the generating functions of the multiplicity distributions are presented both for fixed and running coupling strengths. The mean multiplicities of gluon and quark jets, their ratio, higher moments, and the slopes of multiplicities as a function of energy scale, are among the main global features of multiplicity for which QCD results exist. Recent data from high energy e\(^+\)e\(^-\) experiments, including results for separated quark and gluon jets, allow rather direct tests of these results. The theoretical predictions are generally quite successful when confronted with data. Jet and subjet multiplicities are described. Multiplicity in limited regions of phase space is discussed in the context of intermittency and fractality. The problem of singularities in the generating functions is formulated. Some special features of average multiplicities in heavy quark jets are described.

MSC:

81V05

Strong interaction, including quantum chromodynamics

Keywords:

high-energy particle collisions; quantum chromodynamics; jet multiplicities

References:

[1]	Dremin, I.M., Phys.-usp., 37, 715, (1994)
[2]	A Giovannini, L Van Hove, Z. Phys. C 30 (1986) 391; Acta Phys. Pol. B 19 (1988) 495, 917, 931.
[3]	I.V. Andreev, Chromodynamics and Hard Processes at High Energies, Moscow, Nauka, 1981 (in Russian).
[4]	B.L. Ioffe, L.N. Lipatov, V.A. Khoze, Deep Inelastic Processes, Moscow, Energoatomizdat, 1983 (in Russian).
[5]	Yndurain, F.J., Quantum chromodynamics, (1983), Springer New York · Zbl 1059.81623
[6]	M.B. Voloshin, K.A. Ter-Martirosyan, Theory of Gauge Interactions of Elementary Particles, Moscow, Energoatomizdat, 1984 (in Russian).
[7]	Yu.L. Dokshitzer, V.A. Khoze, A.H. Mueller, S.I. Troyan, Basics of Perturbative QCD, Editions Frontieres, Gif-sur-Yvette, 1991.
[8]	Azimov, Ya.I.; Dokshitzer, Yu.L.; Khoze, V.A.; Troyan, S.I., Z. phys., C 27, 65, (1985)
[9]	Amati, D.; Veneziano, G., Phys. lett., B 83, 87, (1979)
[10]	Bassetto, A.; Ciafaloni, M.; Marchesini, G., Phys. rep., C 100, 201, (1983)
[11]	Bassetto, A.; Ciafaloni, M.; Marchesini, G., Nucl. phys., B 163, 477, (1980)
[12]	Koba, Z.; Nielsen, H.B.; Olesen, P., Nucl. phys., B 40, 317, (1972)
[13]	Dokshitzer, Yu.L., Phys. lett., B 305, 295, (1993)
[14]	Dremin, I.M., Phys. lett., B 313, 209, (1993)
[15]	OPAL Collaboration, K. Ackerstaff et al., Eur. Phys. J. C 1 (1998) 479.
[16]	DeWolf, E.A.; Dremin, I.M.; Kittel, W., Phys. rep., 270, 1, (1996)
[17]	Khoze, V.A.; Ochs, W., Int. J. mod. phys., A 12, 2949, (1997)
[18]	Kaufmann, S.K.; Gyulassi, M., J. phys., A 11, 1715, (1978)
[19]	S. Hegyi, Phys. Lett. B 309 (1993) 443; B 318 (1993) 642; B 463 (1999) 126.
[20]	E.M. Friedländer, I. Stern, LBL-31354, 1991.
[21]	K. Oldham, The Fractional Calculus, Academic Press, Orlando, 1974, p. 60. · Zbl 0292.26011
[22]	B. Ross, Fractional Calculus and its Applications, Lecture Notes in Mathematics, Vol. 457, Springer, Berlin, 1975, p. 1.
[23]	Dremin, I.M., JETP lett., 59, 585, (1994)
[24]	A.M. Polyakov, Sov. Phys. JETP 32 (1971) 296; 33 (1971) 850; 34 (1972) 1177.
[25]	S. Hegyi, Proceedings of the Eighth International Conference on Correlations and Fluctuations, Matrahaza, Hungary, World Scientific, Singapore, 1998, to be published.
[26]	M. Ploszajczak, Proceedings of the Eighth International Conference on Correlations and Fluctuations, Matrahaza, Hungary, World Scientific, Singapore, 1998 to be published.
[27]	Botet, R.; Ploszajczak, M., Z. phys., C 76, 257, (1997)
[28]	Hegyi, S., Phys. lett., B 467, 126, (1999)
[29]	A.I. Golokhvastov, Sov. J. Nucl. Phys. 27 (1978) 430; 30 (1979) 253.
[30]	Wehrl, A., Rev. mod. phys., 50, 221, (1978)
[31]	Šimák, V.; Šumbera, M.; Zborovský, I., (), 205
[32]	Šimák, V.; Šumbera, M.; Zborovský, I., Phys. lett., B 206, 159, (1988)
[33]	Pachr, M.; Šimák, V.; Šumbera, M.; Zborovský, I., Mod. phys. lett., A 7, 2333, (1992)
[34]	F. Takagi, in: C.S. Liu et al. (Eds.), Proceedings of the XXI International Symposium on Multiparticle Dynamics, Wuhan, China, 1991, World Scientific, Singapore, 1992, p. 346.
[35]	Dremin, I.M., Mod. phys. lett., A 8, 2747, (1993)
[36]	Dremin, I.M.; Hwa, R.C., Phys. rev., D 49, 5805, (1994)
[37]	Giovannini, A., Nuovo cim., A 15, 543, (1973)
[38]	S. Hegyi, Phys. Lett. B 387 (1996) 642; B 417 (1998) 186.
[39]	Biyajima, M., Phys. lett., B 137, 225, (1984)
[40]	Chliapnikov, P.V.; Tchikilev, O.G.; Uvarov, V.A., Phys. lett., B 352, 461, (1995)
[41]	V.D. Rusov, T.N. Zelenzova, S.J. Kosenko, OPU-98-1-ENIN, 1998.
[42]	Carius, S.; Ingelmann, G., Phys. lett., B 252, 647, (1990)
[43]	R. Szwed, G. Wrochna, Z. Phys. C 47 (1990) 447; R. Szwed, G. Wrochna, A.K. Wroblewski, Mod. Phys. Lett. A 5 (1990) 981.
[44]	Dremin, I.M.; Dunaevskii, A.M., Phys. rep., 18, 159, (1975)
[45]	Andersson, B.; Gustafson, G.; Sjöstrand, T., Phys. lett., B 94, 211, (1980)
[46]	Andersson, B.; Gustafson, G.; Ingelman, G.; Sjöstrand, T., Phys. rep., 97, 33, (1983)
[47]	Capella, A.; Sukhatme, U.; Tan, C.I.; Tran Thanh Van, J., Phys. lett., B 81, 68, (1979)
[48]	A. Capella, J. Tran Thanh Van, Phys. Lett. B 93 (1980) 146; Z. Phys. C 10 (1981) 249.
[49]	Kaidalov, A.B., Phys. lett., B 116, 459, (1982)
[50]	A.B. Kaidalov, K.A. Ter-Martirosyan, Phys. Lett. B 117 (1982) 247; Sov. J. Nucl. Phys. 39 (1984) 979; 40 (1984) 135.
[51]	Ugoccioni, R.; Giovannini, A.; Lupia, S., (), 297
[52]	B.B. Levtchenko, in: B.B. Levchenko (Ed.), Proceedings of the VIII Workshop on High Energy Physics and Quantum Field Theory and III Workshop on Physics at VLEPP, Zvenigorod, Russia, 1993, Moscow State Univ., Moscow, 1994, p. 68.
[53]	Ugoccioni, R.; Giovannini, A., Nucl. phys. proc. suppl., 71, 201, (1999)
[54]	Bianchi, F.; Giovannini, A.; Lupia, S.; Ugoccioni, R., Z. phys., C 58, 71, (1993)
[55]	Dremin, I.M.; Chernavsky, D.S., Zh. exper. teor. fiz., 38, 229, (1960)
[56]	L. McLerran, R. Venugopalan, Phys. Rev. D 49 (1994) 2233, 3352; D 50 (1994) 2225.
[57]	Yu.V. Kovchegov, E.M. Levin, L. McLerran, BNL-NT-99/7, TAUP-2613-99.
[58]	Fermi, E., Progr. theor. phys., 5, 570, (1950)
[59]	Pomeranchuk, I.Ya., Doklady AN SSSR, 78, 78, (1951)
[60]	Landau, L.D., Izvestia AN SSSR, 17, 51, (1953)
[61]	E.A. Kuraev, J. Manjavidze, A.N. Sisakian, hep-ph/9909287; hep-ph/0003093.
[62]	UA5 Collaboration, G.J. Alner et al., Phys. Lett. B 160 (1985) 193, 199.
[63]	A. Zichichi, M. Basile, G. Bonvicini et al., Proceedings Erice 1982, p. 701.
[64]	Basile, M., Lett. nuovo cim., 41, 293, 298, (1984)
[65]	ALEPH Collaboration, D. Buskulic et al., Z. Phys. C 69 (1995) 15.
[66]	OPAL Collaboration, G. Abbiendi et al., Eur. Phys. J. C 11 (1999) 239.
[67]	Sarkisyan, E.K.G., Phys. lett., B 477, 1, (2000)
[68]	Biyajima, M.; Kawabe, T.; Suzuki, N., Phys. lett., B 189, 466, (1987)
[69]	T. Sjöstrand, Comp. Phys. Comm. 82 (1994) 74; CERN-TH.7112/93 (revised August 1995).
[70]	Marchesini, G.; Webber, B.R., Comp. phys. comm., 67, 465, (1992)
[71]	Lönnblad, L., Comp. phys. comm., 71, 15, (1992)
[72]	B. Bambah et al., in: G. Altarelli, R. Kleiss, C. Verzegnassi (Eds.), Z Physics at LEP, CERN 89-08, Geneva, Vol. 3, 1989, p. 143.
[73]	Goulianos, K., Phys. lett., B 193, 151, (1987)
[74]	Wang, X.N.; Gyulassy, M.; Plümer, M., Phys. rev., D 51, 3436, (1995)
[75]	B.I. Ermolaev, V.S. Fadin, JETP Lett. 33 (1981) 269; A.H. Mueller, Phys. Lett. B 104 (1981) 161.
[76]	Konishi, K.; Ukawa, A.; Veneziano, G., Nucl. phys., B 157, 45, (1979)
[77]	Khoze, V.A.; Lupia, S.; Ochs, W., Eur. phys. J., C 5, 77, (1998)
[78]	S. Lupia, W. Ochs, Phys. Lett. B 418 (1998) 214; Nucl. Phys. Proc. Suppl. 64 (1998) 74.
[79]	Bardeen, W.A.; Buras, A.J.; Duke, D.W.; Muta, T., Phys. rev., D 18, 3998, (1978)
[80]	PDG, R.M. Barnett et al., Phys. Rev. D 54 (1996) 1.
[81]	Dokshitzer, Yu.L.; Khoze, V.A.; Troyan, S.I., Phys. rev., D 53, 89, (1996)
[82]	V.N. Gribov, L.N. Lipatov, Sov. J. Nucl. Phys. 15 (1972) 438; L.N. Lipatov, Sov. J. Nucl. Phys. 20 (1975) 95; G. Altarelli, G. Parisi, Nucl. Phys. B 126 (1977) 298; Yu.L. Dokshitzer, Sov. Phys. JETP 46 (1977) 641.
[83]	Dokshitzer, Yu.L.; Olsson, M., Nucl. phys., B 396, 137, (1993)
[84]	Cuypers, F.; Tesima, K., Z. phys., C 54, 87, (1992)
[85]	Dremin, I.M.; Lam, C.S.; Nechitailo, V.A., Phys. rev., D 61, 074020, (2000)
[86]	Yu.L. Dokshitzer, V.A. Khoze, S.I. Troyan, in: A.H. Mueller (Ed.), Perturbative QCD, World Scientific, Singapore, 1989. · Zbl 0961.81527
[87]	Malaza, E.D.; Webber, B.R., Nucl. phys., B 267, 702, (1986)
[88]	Gaffney, J.B.; Mueller, A.H., Nucl. phys., B 250, 109, (1985)
[89]	Cuypers, F.; Tesima, K., Z. phys., C 52, 69, (1991)
[90]	Catani, S.; Dokshitzer, Yu.L.; Fiorani, F.; Webber, B.R., Nucl. phys., B 383, 419, (1992)
[91]	Olsson, M.; Gustafson, G., Nucl. phys., B 406, 293, (1993)
[92]	Catani, S.; Dokshitzer, Yu.L.; Fiorani, F.; Webber, B.R., Nucl. phys., B 377, 445, (1992)
[93]	Dremin, I.M.; Nechitailo, V.A., Mod. phys. lett., A 9, 1471, (1994)
[94]	Dokshitzer, Yu.L.; Dremin, I.M., Nucl. phys., B 402, 139, (1993)
[95]	Dremin, I.M., Phys. lett., B 341, 95, (1994)
[96]	Dremin, I.M.; Nechitailo, V.A., JETP lett., 58, 881, (1993)
[97]	Ugoccioni, R.; Giovannini, A.; Lupia, S., Phys. lett., B 342, 387, (1995)
[98]	Capella, A.; Dremin, I.M.; Gary, J.W.; Nechitailo, V.A.; Tran Thanh Van, J., Phys. rev., D 61, 074009, (2000)
[99]	Dremin, I.M.; Levtchenko, B.B.; Nechitailo, V.A., Phys. atom. nucl., 57, 1029, (1994)
[100]	Dremin, I.M.; Gary, J.W., Phys. lett., B 459, 341, (1999)
[101]	Shizuya, K.; H. Tye, S.-H., Phys. rev. lett., 41, 787, (1978)
[102]	Einhorn, M.B.; Weeks, B.G., Nucl. phys., B 146, 445, (1978)
[103]	Azimov, Ya.I.; Dokshitzer, Yu.L.; Khoze, V.A., Sov. phys. uspekhi, 23, 732, (1980)
[104]	Lupia, S., Phys. lett., B 439, 150, (1998)
[105]	Brodsky, S.J.; Gunion, J.F., Phys. rev. lett., 37, 402, (1976)
[106]	Mueller, A.H., Nucl. phys., B 241, 141, (1984)
[107]	Eden, P.; Gustafson, G., Jhep, 09, 015, (1998)
[108]	Webber, B.R., Phys. lett., B 143, 501, (1984)
[109]	Dokshitzer, Yu.L.; Khoze, V.A.; Troyan, S.I., Int. J. mod. phys., A 7, 1875, (1992)
[110]	Marciano, W.J., Phys. rev., D 29, 580, (1984)
[111]	Zakharov, V.I., Progr. theor. phys. suppl., 131, 107, (1998)
[112]	OPAL Collaboration, G. Alexander et al., Phys. Lett. B 388 (1996) 659.
[113]	OPAL Collaboration, G. Abbiendi et al., Eur. Phys. J. C 11 (1999) 217.
[114]	CLEO Collaboration, M.S. Alam et al., Phys. Rev. D 56 (1997) 17.
[115]	DELPHI Collaboration, P. Abreu et al., Z. Phys. C 56 (1992) 63.
[116]	Giovannini, A.; Lupia, S.; Ugoccioni, R., Phys. lett., B 374, 231, (1996)
[117]	Dremin, I.M.; Hwa, R.C., Phys. lett., B 324, 477, (1994)
[118]	Dokshitzer, Yu.L.; Khoze, V.A.; Mueller, A.H.; Troyan, S.I., Rev. mod. phys., 60, 373, (1988)
[119]	A. Giovannini, R. Ugoccioni, Proceedings of the XXVIII International Symposium on Multiparticle Dynamics, Delphi, Greece, (World Scientific, Singapore, 1998, to be published).
[120]	Giovannini, A.; Lupia, S.; Ugoccioni, R., Phys. lett., B 388, 639, (1996)
[121]	J.W. Gary, unpublished.
[122]	Dremin, I.M.; Nechitailo, V.A.; Biyajima, M., J. atom. nucl., 59, 2264, (1996)
[123]	Dremin, I.M., JETP lett., 31, 185, (1980)
[124]	Dremin, I.M.; Leonidov, A.V., Sov. J. nucl. phys., 35, 247, (1982)
[125]	Leonidov, A.V.; Ostrovskii, D.M., Phys. atom. nucl., 60, 119, (1997)
[126]	Leonidov, A.V.; Ostrovskii, D.M., Lebedev phys. inst. rep., 1-2, 36, (1997)
[127]	Ellis, J.; Geiger, K., Phys. rev., D 52, 1500, (1995)
[128]	Ellis, J.; Geiger, K., Nucl. phys., A 590, 609c, (1995)
[129]	Dremin, I.M., JETP lett., 60, 764, (1994)
[130]	Blankenbecler, R., Phys. rev., D 8, 1611, (1973)
[131]	Yang, C.N.; Lee, T.D., Phys. rev., 87, 404, (1952)
[132]	Lee, T.D.; Yang, C.N., Phys. rev., 87, 410, (1952)
[133]	E.A. De Wolf, in: A. Giovannini, S. Lupia, R. Ugoccioni (Eds.), Proceedings of the XXIV International Symposium on Multiparticle Dynamics, Vietri sul Mare, Italy, 1994, World Scientific, Singapore, 1995, p. 15.
[134]	Capella, A.; Dremin, I.M.; Nechitailo, V.A.; Tran Thanh Van, J., Z. phys., C 75, 89, (1996)
[135]	Brooks, T.C.; Kowalski, K.L.; Taylor, C.C., Phys. rev., D 56, 5857, (1997)
[136]	N. Brown, W.J. Stirling, Phys. Lett. B 252 (1990) 657; Z. Phys. C 53 (1992) 629.
[137]	Catani, S.; Dokshitzer, Yu.L.; Olsson, M.; Turnock, G.; Webber, B.R., Phys. lett., B 269, 432, (1991)
[138]	JADE Collaboration, W. Bartel et al., Z. Phys. C 33 (1986) 23; S. Bethke et al., Phys. Lett. B 213 (1988) 235.
[139]	Dokshitzer, Yu.L.; Khoze, V.A.; Troyan, S.I., Sov. J. nucl. phys., 47, 881, (1988)
[140]	Gary, J.W., Phys. rev., D 49, 4503, (1994)
[141]	V.A. Khoze, in: R. Orava, E. Eerola, M. Nordberg (Eds.), Proceedings of the Workshop on Physics with Linear Colliders, Saariselkä, Finland, 1991, World Scientific, Singapore, 1992, p. 547.
[142]	JADE Collaboration, W. Bartel et al., Z. Phys. C 21 (1983) 37.
[143]	UA2 Collaboration, P. Bagnaia et al., Phys. Lett. B 144 (1984) 291.
[144]	TASSO Collaboration, M. Althoff et al., Z. Phys. C 22 (1984) 307.
[145]	HRS Collaboration, M. Derrick et al., Phys. Lett. B 165 (1985) 449.
[146]	OPAL Collaboration, P.D. Acton et al., Z. Phys. C 58 (1993) 387.
[147]	DELPHI Collaboration, P. Abreu et al., Z. Phys. C 70 (1996) 179.
[148]	ALEPH Collaboration, D. Buskulic et al., Phys. Lett. B 384 (1996) 353.
[149]	OPAL Collaboration, R. Akers et al., Z. Phys. C 68 (1995) 179.
[150]	CLEO Collaboration, M.S. Alam et al., Phys. Rev. D 46 (1992) 4822.
[151]	ARGUS Collaboration, H. Albrecht et al., Z. Phys. C 54 (1992) 13.
[152]	JADE Collaboration, W. Bartel et al., Z. Phys. C 20 (1983) 187.
[153]	TASSO Collaboration, W. Braunschweig et al., Z. Phys. C 45 (1989) 193.
[154]	HRS Collaboration, M. Derrick et al., Phys. Rev. D 34 (1986) 3304.
[155]	AMY Collaboration, H.W. Zheng et al., Phys. Rev. D 42 (1990) 737.
[156]	ALEPH Collaboration, R. Barate et al., Phys. Rep. 294 (1998) 1; DELPHI Collaboration, P. Abreu et al., Eur. Phys. J. C 6 (1999) 19; L3 Collaboration, B. Adeva et al., Z. Phys. C 55 (1992) 39; OPAL Collaboration, K. Ackerstaff et al., Eur. Phys. J. C 7 (1999) 369.
[157]	ALEPH Collaboration Note 98-025, CONF 98-014.
[158]	DELPHI Collaboration, P. Abreu et al., Phys. Lett. B 372 (1996) 172; L3 Collaboration, M. Acciarri et al., Phys. Lett. B 371 (1996) 137; OPAL Collaboration, G. Alexander et al., Z. Phys. C 72 (1996) 191.
[159]	DELPHI Collaboration, P. Abreu et al., Phys. Lett. B 416 (1998) 233; L3 Collaboration, M. Acciarri et al., Phys. Lett. B 404 (1997) 390.
[160]	OPAL Collaboration, K. Ackerstaff et al., Z. Phys. C 75 (1997) 193.
[161]	OPAL Collaboration, G. Abbiendi et al., Eur. Phys. J. C 16 (2000) 185.
[162]	L3 Collaboration, M. Acciarri et al., Phys. Lett. B 444 (1998) 569.
[163]	ALEPH Collaboration Note 99-023, CONF 99-018.
[164]	L3 Collaboration Note 2505.
[165]	ALEPH Collaboration Note 2000-017, CONF 2000-014.
[166]	OPAL Collaboration, Physics Note PN 420.
[167]	G.D. Lafferty, P.I. Reeves, M.R. Whalley, J. Phys. G, Nucl. Part. Phys. 21 (1995) A1-A151; The Durham RAL Databases, http://durpdg.dur.ac.uk/HEPDATA.
[168]	DELPHI Collaboration, P. Abreu et al., Phys. Lett. B 449 (1999) 383.
[169]	Safonov, A.N., Nucl. phys. proc. suppl., 86, 55, (2000)
[170]	OPAL Collaboration, G. Abbiendi et al., CERN-EP-2000-070.
[171]	DELPHI Collaboration, CERN-OPEN-2000-134.
[172]	Dremin, I.M., JETP lett., 68, 559, (1998)
[173]	Lupia, S.; Ochs, W.; Wosiek, J., Nucl. phys. B, 540, 405, (1999)
[174]	SLD Collaboration, K. Abe et al., Phys. Lett. B 371 (1996) 149.
[175]	L3 Collaboration, L3 Note 2407; W. Metzger, Proceedings of the XXIX International Symposium on Multiparticle Dynamics, Providence, USA, 1999, World Scientific, Singapore, to be published.
[176]	Khoze, V.A.; Lupia, S.; Ochs, W., Phys. lett. B, 394, 179, (1997)
[177]	Konishi, K.; Ukawa, A.; Veneziano, G., Phys. lett. B, 78, 243, (1978)
[178]	Dremin, I.M.; Arena, V.; Boca, G.; Gianini, G., Phys. lett. B, 336, 119, (1994)
[179]	Dremin, I.M.; Nechitailo, V.A.; Biyajima, M.; Suzuki, N., Phys. lett. B, 403, 149, (1997)
[180]	Ellis, R.K.; Ross, D.A.; Terrano, A.E., Nucl. phys. B, 178, 421, (1981)
[181]	Sterman, G.; Weinberg, S., Phys. rev. lett., 39, 1436, (1977)
[182]	Bethke, S.; Kunszt, Z.; Soper, D.E.; Stirling, W.J., Nucl. phys. B, 370, 310, (1992)
[183]	L3 Collaboration, O. Adriani et al., Phys. Rep. 236 (1993) 1.
[184]	OPAL Collaboration, R. Akers et al., Z. Phys. C 63 (1994) 363.
[185]	ALEPH Collaboration, D. Buskulic et al., Phys. Lett. B 346 (1995) 389; ALEPH Collaboration, D. Barate et al., CERN-EP/98-016.
[186]	Seymour, M.H., Phys. lett. B, 378, 279, (1996)
[187]	OPAL Collaboration, M.Z. Akrawy et al., Phys. Lett. B 263 (1991) 311; OPAL Collaboration, G. Alexander et al., Phys. Lett. B 265 (1991) 462.
[188]	Gary, J.W., Phys. rev. D, 61, 114007, (2000)
[189]	Edén, P.; Gustafson, G.; Khoze, V., Eur. phys. J. C, 11, 345, (1999)
[190]	Edén, P.; Gustafson, G., Jhep, 9809, 015, (1998)
[191]	Bialas, A.; Peschanski, R., Nucl. phys. B, 273, 703, (1986)
[192]	Dremin, I.M., JETP lett., 45, 643, (1987)
[193]	R.P. Feynman, in: Casalbuoni et al. (Eds.), Proceedings of the III Workshop on Current Problems in High Energy Particle Theory, Florence, Italy, John Hopkins University Press, Baltimore, MD, 1979.
[194]	Giovannini, A., Nucl. phys. B, 161, 429, (1979)
[195]	G. Veneziano, in: Casalbuoni et al. (Eds.), Proceedings of the III Workshop on Current Problems in High Energy Particle Theory, Florence, Italy, John Hopkins University Press, Baltimore, MD, 1979.
[196]	Bialas, A.; Peschanski, R., Nucl. phys. B, 308, 857, (1988)
[197]	Ochs, W.; Wosiek, J., Phys. lett. B, 214, 617, (1988)
[198]	Dremin, I.M., Mod. phys. lett. A, 4, 1027, (1989)
[199]	Chiu, C.B.; Hwa, R., Phys. lett. B, 236, 446, (1990)
[200]	G. Gustafson, A. Nilsson, Z. Phys. C 52 (1991) 533; Nucl. Phys. B 355 (1991) 106.
[201]	W. Ochs, J. Wosiek, Phys. Lett. B 289 (1992) 159; B 304 (1993) 144.
[202]	Ochs, W.; Wosiek, J., Z. phys. C, 68, 269, (1995)
[203]	Brax, Ph.; Meunier, J.L.; Peschanski, R., Z. phys. C, 62, 649, (1994)
[204]	Dokshitzer, Yu.L.; Marchesini, G.; Oriani, G., Nucl. phys. B, 387, 675, (1992)
[205]	Meunier, J.L.; Peschanski, R., Z. phys. C, 72, 647, (1996)
[206]	Andreev, I.V.; Biyajima, M.; Dremin, I.M.; Suzuki, N., Int. J. mod. phys. A, 10, 3951, (1995)
[207]	L3 Collaboration, M. Acciarri et al., Phys. Lett. B 428 (1998) 186.
[208]	DELPHI Collaboration, P. Abreu et al., Phys. Lett. B 457 (1999) 368.
[209]	L3 Collaboration, M. Acciarri et al., Phys. Lett. B 429 (1998) 375.
[210]	Chekanov, S.V.; Kuvshinov, V.I., Acta phys. polonica B, 25, 1189, (1994)
[211]	Chekanov, S.V.; Kittel, W.; Kuvshinov, V.I., Z. phys. C, 74, 517, (1997)
[212]	Polikarpov, M.I., Phys. lett. B, 236, 61, (1990)
[213]	Dremin, I.M.; Levtchenko, B.B., Phys. lett. B, 292, 155, (1992)
[214]	Dremin, I.M., JETP lett., 34, 594, (1981)
[215]	I.M. Dremin, JETP Lett. 30 (1979) 140; Sov. J. Nucl. Phys. 33 (1981) 726.
[216]	Apanasenko, A.V.; Dobrotin, N.A.; Dremin, I.M.; Kotelnikov, K.A., JETP lett., 30, 145, (1979)
[217]	Schumm, B.A.; Dokshitzer, Yu.L.; Khoze, V.A.; Koetke, D.S., Phys. rev. lett., 69, 3025, (1992)
[218]	Dokshitzer, Yu.L.; Khoze, V.A.; Troyan, S.I., J. phys. G, 17, 1481; 1602, (1991)
[219]	Azimov, Ya.I.; Dokshitzer, Yu.L.; Khoze, V.A., Sov. J. nucl. phys., 36, 878, (1982)
[220]	Kisselev, A.V.; Petrov, V.A.; Yuschenko, O.P., Z. phys. C, 41, 521, (1988)
[221]	Petrov, V.A.; Kisselev, A.V., Z. phys. C, 66, 453, (1995)
[222]	DELPHI Collaboration, P. Abreu et al., Phys. Lett. B 479 (2000) 118.
[223]	Dremin, I.M.; Yakovlev, V.I., (), 849
[224]	C.M. Ko et al., nucl-th/0010004.
[225]	Andersson, B.; Gustafson, G.; Samuelsson, J., Nucl. phys. B, 463, 217, (1996)
[226]	E. Norrbin, T. Sjöstrand, hep-ph/0010012.