Next-to-leading order evolution of twist-two conformal operators: The Abelian case

Nuclear Physics B

Volumn 527, Issue 1-2, 1998, Pages 207-234

  Belitsky, A V ^a,b   Muller D ^a,c  

^a UNIVERSITY OF REGENSBURG   (Germany)

^b JOINT INSTITUTE FOR NUCLEAR RESEARCH   (Russian Federation)

^c UNIVERSITY OF LEIPZIG   (Germany)

Author keywords

Anomalous dimensions; Conformal anomalies; Conformal Ward identities; Evolution equations

Indexed keywords

EID: 0032584867     PISSN: 05503213     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0550-3213(98)00310-1     Document Type: Article

References (79)

1. [1] H.A. Kastrup, Ann. Physik 7 (1962) 388.
2. [2] G. Mack and A. Salam, Ann. Phys. 53 (1969) 174.
3. [3] S. Ferrara, R. Gatto and A.F. Grillo, Conformal algebra in space-time and operator product expansion, Springer tracts in modern physics, Vol. 67 (1973), and references cited therein.
4. [4] V.N. Gribov and L.N. Lipatov, Sov. J. Nucl. Phys. 15 (1972) 438; L.N. Lipatov, Sov. J. Nucl. Phys. 20 (1974) 94; A.P. Bukhvostov, L.N. Lipatov and N.P. Popov, Sov. J. Nucl. Phys. 20 (1974) 287.
5. [4] V.N. Gribov and L.N. Lipatov, Sov. J. Nucl. Phys. 15 (1972) 438; L.N. Lipatov, Sov. J. Nucl. Phys. 20 (1974) 94; A.P. Bukhvostov, L.N. Lipatov and N.P. Popov, Sov. J. Nucl. Phys. 20 (1974) 287.
6. [4] V.N. Gribov and L.N. Lipatov, Sov. J. Nucl. Phys. 15 (1972) 438; L.N. Lipatov, Sov. J. Nucl. Phys. 20 (1974) 94; A.P. Bukhvostov, L.N. Lipatov and N.P. Popov, Sov. J. Nucl. Phys. 20 (1974) 287.
7. [5] G. Altarelli and G. Parisi, Nucl. Phys. B 126 (1977) 298.
8. [6] Yu.L. Dokshitzer, Sov. Phys. JETP 46 (1977) 641.
9. [7] A.V. Efremov and A.V. Radyushkin, JINR preprint E2-11983, Dubna (October 1978); Theor. Math. Phys. 42 (1980) 97; Phys. Lett. B 94 (1980) 245.
10. [7] A.V. Efremov and A.V. Radyushkin, JINR preprint E2-11983, Dubna (October 1978); Theor. Math. Phys. 42 (1980) 97; Phys. Lett. B 94 (1980) 245.
11. [7] A.V. Efremov and A.V. Radyushkin, JINR preprint E2-11983, Dubna (October 1978); Theor. Math. Phys. 42 (1980) 97; Phys. Lett. B 94 (1980) 245.
12. [8] S.J. Brodsky and G.P. Lepage, Phys. Lett. B 87 (1979) 359; Phys. Rev. D 22 (1980) 2157.
13. [8] S.J. Brodsky and G.P. Lepage, Phys. Lett. B 87 (1979) 359; Phys. Rev. D 22 (1980) 2157.
14. [9] S.J. Brodsky and G.P. Lepage, Exclusive processes in Quantum Chromodynamics, in Perturbative Quantum Chromodynamics, ed. A.H. Mueller (World Scientific, Singapore, 1989).
15. [10] S.J. Brodsky, Y. Frishman, G.P. Lepage and C. Sachrajda, Phys. Lett. B 91 (1980) 239.
16. [11] M.K. Chase, Nucl. Phys. B 174 (1980) 109.
17. [12] Th. Ohrndorf, Nucl. Phys. B 186 (1981) 153; M.A. Shifman and M.I. Vysotsky, Nucl. Phys. B 186 (1981) 475.
18. [12] Th. Ohrndorf, Nucl. Phys. B 186 (1981) 153; M.A. Shifman and M.I. Vysotsky, Nucl. Phys. B 186 (1981) 475.
19. [13] Yu.M. Makeenko, Sov. J. Nucl. Phys. 33 (1981) 440.
20. [14] Th. Ohrndorf, Nucl. Phys. B 198 (1982) 26.
21. [15] S.J. Brodsky, P. Damgaard, Y. Frishman and G.P. Lepage, Phys. Rev. D 33 (1986) 1881.
22. [16] F.-M. Dittes and A.V. Radyushkin, Phys. Lett. B 134 (1984) 359.
23. [17] M.H. Sarmadi, Phys. Lett. B 143 (1984) 471.
24. [18] S.V. Mikhailov and A.V. Radyushkin, Evolution kernel for the pion wave function: two-loop QCD calculation in Feynman gauge, JINR preprint P2-83-271 (1983); Nucl. Phys. B 254 (1985) 89.
25. [18] S.V. Mikhailov and A.V. Radyushkin, Evolution kernel for the pion wave function: two-loop QCD calculation in Feynman gauge, JINR preprint P2-83-271 (1983); Nucl. Phys. B 254 (1985) 89.
26. [19] G.R. Katz, Phys. Rev. D 31 (1985) 652.
27. [20] S.V. Mikhailov and A.V. Radyushkin, Nucl. Phys. B 273 (1986) 161.
28. [21] F.-M. Dittes, D. Müller, D. Robaschik and B. Geyer, Phys. Lett. B 209 (1988) 325.
29. [22] D. Müller, D. Robaschik, B. Geyer, F.-M. Dittes and J. Hořejši, Fortschr. Phys. 42 (1994) 101.
30. [23] C.G. Callan, S. Coleman and R. Jackiw, Ann. Phys. 59 (1970) 42; S. Coleman and R. Jackiw, Ann. Phys. 67 (1971) 552.
31. [23] C.G. Callan, S. Coleman and R. Jackiw, Ann. Phys. 59 (1970) 42; S. Coleman and R. Jackiw, Ann. Phys. 67 (1971) 552.
32. [24] K. Symanzik, Comm. Math. Phys. 23 (1971) 49; B. Schroer, Nuovo Cim. Lett. 2 (1971) 627; G. Parisi, Phys. Lett. B 39 (1972) 643.
33. [24] K. Symanzik, Comm. Math. Phys. 23 (1971) 49; B. Schroer, Nuovo Cim. Lett. 2 (1971) 627; G. Parisi, Phys. Lett. B 39 (1972) 643.
34. [24] K. Symanzik, Comm. Math. Phys. 23 (1971) 49; B. Schroer, Nuovo Cim. Lett. 2 (1971) 627; G. Parisi, Phys. Lett. B 39 (1972) 643.
35. [25] D. Müller, Restricted conformal invariance in QCD and its predictive power for virtual two-photon processes, hep-ph/9704406.
36. [26] A.V. Belitsky and D. Müller, Phys. Lett. B 417 (1998) 129.
37. [27] A.V. Belitsky and A. Schäfer, Nucl. Phys. B 527 (1998) 235, next article in this issue.
38. [28] N.K. Nielsen, Nucl. Phys. B 65 (1973) 413.
39. [29] S. Sarkar, Nucl. Phys. B 83 (1974) 108.
40. [30] N.K. Nielsen, Nucl. Phys. B 97 (1975) 527.
41. [31] D. Müller, Z. Phys. C 49 (1991) 293.
42. [32] D. Müller, Phys. Rev. D 49 (1994) 2525.
43. [33] C. Itzykson and J. Zuber, Quantum Field Theory (McGraw-Hill, New York, 1980).
44. [34] R. Jackiw, Phys. Rev. Lett. 41 (1978) 1635; R. Jackiw, Topological investigations of quantized gauge theories, in Proceedings of Les Houches Summer School on Theoretical Physics: Relativity, Groups and Topology II, ed. B.S. DeWitt and R. Stora (North-Holland, Amsterdam, 1984) p. 221.
45. [34] R. Jackiw, Phys. Rev. Lett. 41 (1978) 1635; R. Jackiw, Topological investigations of quantized gauge theories, in Proceedings of Les Houches Summer School on Theoretical Physics: Relativity, Groups and Topology II, ed. B.S. DeWitt and R. Stora (North-Holland, Amsterdam, 1984) p. 221.
46. [35] J.A. Dixon and J.C. Taylor, Nucl. Phys. B 78 (1974) 552; H. Kluberg-Stern and J.B. Zuber, Phys. Rev. D 12 (1975) 476, 482, 3159; S.D. Joglekar and B.W. Lee, Ann. Phys. 97 (1975) 160.
47. [35] J.A. Dixon and J.C. Taylor, Nucl. Phys. B 78 (1974) 552; H. Kluberg-Stern and J.B. Zuber, Phys. Rev. D 12 (1975) 476, 482, 3159; S.D. Joglekar and B.W. Lee, Ann. Phys. 97 (1975) 160.
48. [35] J.A. Dixon and J.C. Taylor, Nucl. Phys. B 78 (1974) 552; H. Kluberg-Stern and J.B. Zuber, Phys. Rev. D 12 (1975) 476, 482, 3159; S.D. Joglekar and B.W. Lee, Ann. Phys. 97 (1975) 160.
49. [36] S.D. Joglekar, Ann. Phys. 108 (1977) 233; 109 (1977) 210.
50. [36] S.D. Joglekar, Ann. Phys. 108 (1977) 233; 109 (1977) 210.
51. [37] J.C. Collins, Renormalization (Cambridge Univ. Press, Cambridge, 1984).
52. [38] D.Z. Freedman, I.J. Muzinich and E.J. Weinberg, Ann. Phys. 87 (1974) 95; D.Z. Freedman and E.J. Weinberg, Ann. Phys. 87 (1974) 354.
53. [38] D.Z. Freedman, I.J. Muzinich and E.J. Weinberg, Ann. Phys. 87 (1974) 95; D.Z. Freedman and E.J. Weinberg, Ann. Phys. 87 (1974) 354.
54. [39] S.D. Joglekar, Ann. Phys. 100 (1976) 395.
55. [40] S.L. Adler, J.C. Collins and A. Duncan, Phys. Rev. D 15 (1977) 1712; J.C. Collins, A. Duncan and S.D. Joglekar, Phys. Rev. D 16 (1977) 438.
56. [40] S.L. Adler, J.C. Collins and A. Duncan, Phys. Rev. D 15 (1977) 1712; J.C. Collins, A. Duncan and S.D. Joglekar, Phys. Rev. D 16 (1977) 438.
57. [41] N.K. Nielsen, Nucl. Phys. B 120 (1977) 212.
58. [42] P. Minkowski, On the anomalous divergence of the dilaton current in gauge theories, Bern U. preprint 76-0813 (1976).
59. [43] C.G. Callan, Phys. Rev. D 2 (1970) 1451; K. Symanzik, Comm. Math. Phys. 18 (1970) 227.
60. [43] C.G. Callan, Phys. Rev. D 2 (1970) 1451; K. Symanzik, Comm. Math. Phys. 18 (1970) 227.
61. [44] E.G. Floratos, D.A. Ross and C.T. Sachrajda, Nucl. Phys. B 152 (1979) 493; A. Gonzales-Arroyo and C. Lopez, Nucl. Phys. B 166 (1980) 429; E.G. Floratos, C. Kounnas and R. Lacaze, Phys. Lett. B 98 (1981) 285; W. Furmanski and R. Petronzio, Phys. Lett. B 97 (1980) 437; R.K. Ellis and W. Vogelsang, The evolution of parton distributions beyond leading order: the siglet case, hep-ph/9602356.
62. [44] E.G. Floratos, D.A. Ross and C.T. Sachrajda, Nucl. Phys. B 152 (1979) 493; A. Gonzales-Arroyo and C. Lopez, Nucl. Phys. B 166 (1980) 429; E.G. Floratos, C. Kounnas and R. Lacaze, Phys. Lett. B 98 (1981) 285; W. Furmanski and R. Petronzio, Phys. Lett. B 97 (1980) 437; R.K. Ellis and W. Vogelsang, The evolution of parton distributions beyond leading order: the siglet case, hep-ph/9602356.
63. [44] E.G. Floratos, D.A. Ross and C.T. Sachrajda, Nucl. Phys. B 152 (1979) 493; A. Gonzales-Arroyo and C. Lopez, Nucl. Phys. B 166 (1980) 429; E.G. Floratos, C. Kounnas and R. Lacaze, Phys. Lett. B 98 (1981) 285; W. Furmanski and R. Petronzio, Phys. Lett. B 97 (1980) 437; R.K. Ellis and W. Vogelsang, The evolution of parton distributions beyond leading order: the siglet case, hep-ph/9602356.
64. [44] E.G. Floratos, D.A. Ross and C.T. Sachrajda, Nucl. Phys. B 152 (1979) 493; A. Gonzales-Arroyo and C. Lopez, Nucl. Phys. B 166 (1980) 429; E.G. Floratos, C. Kounnas and R. Lacaze, Phys. Lett. B 98 (1981) 285; W. Furmanski and R. Petronzio, Phys. Lett. B 97 (1980) 437; R.K. Ellis and W. Vogelsang, The evolution of parton distributions beyond leading order: the siglet case, hep-ph/9602356.
65. [44] E.G. Floratos, D.A. Ross and C.T. Sachrajda, Nucl. Phys. B 152 (1979) 493; A. Gonzales-Arroyo and C. Lopez, Nucl. Phys. B 166 (1980) 429; E.G. Floratos, C. Kounnas and R. Lacaze, Phys. Lett. B 98 (1981) 285; W. Furmanski and R. Petronzio, Phys. Lett. B 97 (1980) 437; R.K. Ellis and W. Vogelsang, The evolution of parton distributions beyond leading order: the siglet case, hep-ph/9602356.
66. [45] R. Mertig and W.L. van Neerven, Z. Phys. C 70 (1996) 637; W. Vogelsang, Nucl. Phys. B 475 (1996) 47.
67. [45] R. Mertig and W.L. van Neerven, Z. Phys. C 70 (1996) 637; W. Vogelsang, Nucl. Phys. B 475 (1996) 47.
68. [46] A.V. Radyushkin, Phys. Lett. B 380 (1996) 417; B 385 (1996) 333; Phys. Rev. D 56 (1997) 5524; I.I. Balitsky and A.V. Radyushkin, Phys. Lett. B 413 (1997) 114.
69. [46] A.V. Radyushkin, Phys. Lett. B 380 (1996) 417; B 385 (1996) 333; Phys. Rev. D 56 (1997) 5524; I.I. Balitsky and A.V. Radyushkin, Phys. Lett. B 413 (1997) 114.
70. [46] A.V. Radyushkin, Phys. Lett. B 380 (1996) 417; B 385 (1996) 333; Phys. Rev. D 56 (1997) 5524; I.I. Balitsky and A.V. Radyushkin, Phys. Lett. B 413 (1997) 114.
71. [46] A.V. Radyushkin, Phys. Lett. B 380 (1996) 417; B 385 (1996) 333; Phys. Rev. D 56 (1997) 5524; I.I. Balitsky and A.V. Radyushkin, Phys. Lett. B 413 (1997) 114.
72. [47] J. Blümlein, B. Geyer and D. Robaschik, Twist-2 light-ray operators: anomalous dimensions and evolution equations, hep-ph/9711405, revised version.
73. [48] X. Ji, Phys. Rev. D 55 (1997) 7114.
74. [49] A.P. Bukhvostov, G.V. Frolov, L.N. Lipatov and E.A. Kuraev, Nucl. Phys. B 258 (1985) 601.
75. [50] A.V. Belitsky and D. Müller, Nucl. Phys. B 503 (1997) 279; A.V. Belitsky, Leading order analysis of the twist-3 space-like and time-like cut vertices in QCD, in Proc. XXXI PNPI Winter School (St. Petersburg, 1997) p. 369; hep-ph/9703432.
76. [50] A.V. Belitsky and D. Müller, Nucl. Phys. B 503 (1997) 279; A.V. Belitsky, Leading order analysis of the twist-3 space-like and time-like cut vertices in QCD, in Proc. XXXI PNPI Winter School (St. Petersburg, 1997) p. 369; hep-ph/9703432.
77. [50] A.V. Belitsky and D. Müller, Nucl. Phys. B 503 (1997) 279; A.V. Belitsky, Leading order analysis of the twist-3 space-like and time-like cut vertices in QCD, in Proc. XXXI PNPI Winter School (St. Petersburg, 1997) p. 369; hep-ph/9703432.
78. [51] H. Bateman and A. Erdélyi, Higher Transcendental Functions, Vol. 1 (McGraw-Hill, New York, 1953).
79. [52] X. Ji, Phys. Rev. Lett. 78 (1997) 610.