B → s + γ: A QCD-consistent analysis of the photon energy distribution

International Journal of Modern Physics A

Volumn 11, Issue 3, 1996, Pages 571-611

  David Dikeman, R ^a   Shifman, M ^a   Uraltsev, N G ^a,b,c  

^a University of Minnesota   (United States)

^b UNIVERSITY OF NOTRE DAME   (United States)

^c Petersburg Nuclear Physics Institute   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0010183695     PISSN: 0217751X     EISSN: None     Source Type: Journal    
DOI: 10.1142/s0217751x96000274     Document Type: Article

References (119)

1. M. S. Alam et al. (CLEO), "First measurement of the rate for the inclusive radiative penguin decay b → sγ," Cornell preprint CLNS 94/1314; E. H. Thorndike (CLEO), Proc. XXVII Int. Conf. on High Energy Physics (Glasgow, 1994), eds. P. J. Bussey and I. G. Knowles (Institute of Physics Publishing, Bristol, 1995), Vol. 2, p. 1327.
2. M. S. Alam et al. (CLEO), "First measurement of the rate for the inclusive radiative penguin decay b → sγ," Cornell preprint CLNS 94/1314; E. H. Thorndike (CLEO), Proc. XXVII Int. Conf. on High Energy Physics (Glasgow, 1994), eds. P. J. Bussey and I. G. Knowles (Institute of Physics Publishing, Bristol, 1995), Vol. 2, p. 1327.
3. For a brief review see e.g.: J. Hewett, "Probing new physics in B guins," preprint SLAC-PUB-95-6782 (hep-ph/9505247).
4. K. Wilson, Phys. Rev. 179, 1499 (1969); D3, 1818 (1971).
5. K. Wilson, Phys. Rev. 179, 1499 (1969); D3, 1818 (1971).
6. M. Shifman, "Theory of preasymptotic effects in weak inclusive decays," in Proc. Workshop on Continuous Advances in QCD, ed. A. Smilga (World Scientific, Singapore, 1994), p. 249 (hep-ph/9405246).
7. M. Voloshin, M. Shifman, Yad. Fiz. 41, 187 (1985) [Sov. J. Nucl. Phys. 41, 120 (1985)]; Zh. Eksp. Teor. Fiz. 91, 1180 (1986) [Sov, Phys. JETP 64, 698 (1986)].
8. M. Voloshin, M. Shifman, Yad. Fiz. 41, 187 (1985) [Sov. J. Nucl. Phys. 41, 120 (1985)]; Zh. Eksp. Teor. Fiz. 91, 1180 (1986) [Sov, Phys. JETP 64, 698 (1986)].
9. M. Voloshin, M. Shifman, Yad. Fiz. 41, 187 (1985) [Sov. J. Nucl. Phys. 41, 120 (1985)]; Zh. Eksp. Teor. Fiz. 91, 1180 (1986) [Sov, Phys. JETP 64, 698 (1986)].
10. M. Voloshin, M. Shifman, Yad. Fiz. 41, 187 (1985) [Sov. J. Nucl. Phys. 41, 120 (1985)]; Zh. Eksp. Teor. Fiz. 91, 1180 (1986) [Sov, Phys. JETP 64, 698 (1986)].
11. J. Chay, H. Georgi and B. Grinstein, Phys. Lett. B247, 399 (1990).
12. I. Bigi, N. Uraltsev and A. Vamshtein, Phys. Lett. B293, 430 (1992); (E) B297, 477 (1993).
13. I. Bigi, N. Uraltsev and A. Vamshtein, Phys. Lett. B293, 430 (1992); (E) B297, 477 (1993).
14. I. Bigi, B. Blok, M. Shifman, N. Uraltsev and A. Vainshtein, The Fermilab Meeting Proc. 1992 DPF Meeting of APS eds. C. H. Albright et al. (World Scientific, Singapore, 1993), Vol. 1, p. 610 (hep-ph/9212227).
15. I. Bigi, M. Shifman, N. G. Uraltsev and A. Vainshtein, Phys. Rev. Lett. 71, 496 (1993).
16. R. Jaffe and L. Randall, Nucl. Phys. B412, 79 (1994).
17. M. Neubert, Phys.Rev. D49, 3392 (1994).
18. I. Bigi, M. Shifman, N. G. Uraltsev and A. Vainshtein, Int. J. Mod. Phys. A9, 2467 (1994).
19. A. Falk, E. Jenkins, A. Manohar and M. Wise, Phys. Rev. D49, 4553 (1994).
20. M. Neubert, Phys. Rev. D49, 4623 (1994).
21. A. Ali and E. Pietarinen, Nucl. Phys. B154, 512 (1979).
22. G. Altarelli, N. Cabibbo, G. Corbo, L. Maiani and G. Martinelli, Nucl. Phys. B208, 365 (1982).
23. Qualitative discussion of the effects due to the heavy quark motion inside heavy hadrons in the problem of the heavy quark fragmentation can be traced back to the early days of QCD; see e.g. M. Suzuki, Phys. Lett. B71, 139 (1977); V. A. Khoze, Ya. Azimov and L. Frankfurt, invited talk at XVII Int. Conf. on High Energy Physics (Tbilisi, 1976); J. D. Bjorken, Phys. Rev. D17, 171 (1978). A parton-like approach to consideration of the distribution functions in the inclusive semileptonic b → u transitions was suggested in Ref. 18.
24. Qualitative discussion of the effects due to the heavy quark motion inside heavy hadrons in the problem of the heavy quark fragmentation can be traced back to the early days of QCD; see e.g. M. Suzuki, Phys. Lett. B71, 139 (1977); V. A. Khoze, Ya. Azimov and L. Frankfurt, invited talk at XVII Int. Conf. on High Energy Physics (Tbilisi, 1976); J. D. Bjorken, Phys. Rev. D17, 171 (1978). A parton-like approach to consideration of the distribution functions in the inclusive semileptonic b → u transitions was suggested in Ref. 18.
25. Qualitative discussion of the effects due to the heavy quark motion inside heavy hadrons in the problem of the heavy quark fragmentation can be traced back to the early days of QCD; see e.g. M. Suzuki, Phys. Lett. B71, 139 (1977); V. A. Khoze, Ya. Azimov and L. Frankfurt, invited talk at XVII Int. Conf. on High Energy Physics (Tbilisi, 1976); J. D. Bjorken, Phys. Rev. D17, 171 (1978). A parton-like approach to consideration of the distribution functions in the inclusive semileptonic b → u transitions was suggested in Ref. 18.
26. A. Bareiss and E. Paschos, Nucl. Phys. B327, 353 (1989); C. Jin, W. Palmer and E. Paschos, Phys. Lett. B329, 364 (1994).
27. A. Bareiss and E. Paschos, Nucl. Phys. B327, 353 (1989); C. Jin, W. Palmer and E. Paschos, Phys. Lett. B329, 364 (1994).
28. I. Bigi, M. Shifman, N. G. Uraltsev and A. Vainshtein, Phys. Lett. B328, 431 (1994).
29. C. Csaki and L. Randall, Phys. Lett. B324, 451 (1994); G. Baillie, Phys. Lett. B324, 446 (1994).
30. C. Csaki and L. Randall, Phys. Lett. B324, 451 (1994); G. Baillie, Phys. Lett. B324, 446 (1994).
31. V. Novikov, M. Shifman, A. Vainshtein and V. Zakharov, Nucl. Phys. B249, 445 (1985).
32. 1,2 in the nomenclature of this paper). The result has been generalized to the three-generation model in: T. In ami and C. S. Lim, Prog. Theor. Phys. 65, 297 (1981); N. G. Deshpande and G. Eilam, Phys. Rev. D26, 2463 (1982); B. A. Campbell and P. J. O'Donnel, Phys. Rev. D25, 1989 (1982); N. G. Deshpande and M. Nazerimonfared, Nucl. Phys. B123, 390 (1983). From these papers one can extract the magnetic penguin for b → sγ in the Standard Model.
33. 1,2 in the nomenclature of this paper). The result has been generalized to the three-generation model in: T. In ami and C. S. Lim, Prog. Theor. Phys. 65, 297 (1981); N. G. Deshpande and G. Eilam, Phys. Rev. D26, 2463 (1982); B. A. Campbell and P. J. O'Donnel, Phys. Rev. D25, 1989 (1982); N. G. Deshpande and M. Nazerimonfared, Nucl. Phys. B123, 390 (1983). From these papers one can extract the magnetic penguin for b → sγ in the Standard Model.
34. 1,2 in the nomenclature of this paper). The result has been generalized to the three-generation model in: T. In ami and C. S. Lim, Prog. Theor. Phys. 65, 297 (1981); N. G. Deshpande and G. Eilam, Phys. Rev. D26, 2463 (1982); B. A. Campbell and P. J. O'Donnel, Phys. Rev. D25, 1989 (1982); N. G. Deshpande and M. Nazerimonfared, Nucl. Phys. B123, 390 (1983). From these papers one can extract the magnetic penguin for b → sγ in the Standard Model.
35. 1,2 in the nomenclature of this paper). The result has been generalized to the three-generation model in: T. In ami and C. S. Lim, Prog. Theor. Phys. 65, 297 (1981); N. G. Deshpande and G. Eilam, Phys. Rev. D26, 2463 (1982); B. A. Campbell and P. J. O'Donnel, Phys. Rev. D25, 1989 (1982); N. G. Deshpande and M. Nazerimonfared, Nucl. Phys. B123, 390 (1983). From these papers one can extract the magnetic penguin for b → sγ in the Standard Model.
36. 1,2 in the nomenclature of this paper). The result has been generalized to the three-generation model in: T. In ami and C. S. Lim, Prog. Theor. Phys. 65, 297 (1981); N. G. Deshpande and G. Eilam, Phys. Rev. D26, 2463 (1982); B. A. Campbell and P. J. O'Donnel, Phys. Rev. D25, 1989 (1982); N. G. Deshpande and M. Nazerimonfared, Nucl. Phys. B123, 390 (1983). From these papers one can extract the magnetic penguin for b → sγ in the Standard Model.
37. 1,2 in the nomenclature of this paper). The result has been generalized to the three-generation model in: T. In ami and C. S. Lim, Prog. Theor. Phys. 65, 297 (1981); N. G. Deshpande and G. Eilam, Phys. Rev. D26, 2463 (1982); B. A. Campbell and P. J. O'Donnel, Phys. Rev. D25, 1989 (1982); N. G. Deshpande and M. Nazerimonfared, Nucl. Phys. B123, 390 (1983). From these papers one can extract the magnetic penguin for b → sγ in the Standard Model.
38. 1,2 in the nomenclature of this paper). The result has been generalized to the three-generation model in: T. In ami and C. S. Lim, Prog. Theor. Phys. 65, 297 (1981); N. G. Deshpande and G. Eilam, Phys. Rev. D26, 2463 (1982); B. A. Campbell and P. J. O'Donnel, Phys. Rev. D25, 1989 (1982); N. G. Deshpande and M. Nazerimonfared, Nucl. Phys. B123, 390 (1983). From these papers one can extract the magnetic penguin for b → sγ in the Standard Model.
39. -1, where μ is "the end of the perturbative domain," μ ∼ several units × ΛQCD.
40. The first calculations of the hard gluon corrections were done in: M. Shifman, A. Vainshtein and V. Zakharov, Phys. Rev. D18, 2583 (1978). Since then the analysis has been repeatedly discussed in the literature; see e.g.: S. Bertolini, F. Borzumati and A. Masiero, Phys. Rev. Lett. 59, 180 (1987); N. Deshpande, P. Lo, J. Trampetic, G. Eilam and P. Singer, Phys. Rev. Lett. 59, 183 (1987); B. Grinshtein, R. Springer and M. Wise, Phys. Lett. B202, 138 (1988); R. Grigjanis, P. J. O'Donnel, M. Sutherland and H. Navelet, Phys. Lett. B213, 335 (1988); (E) Phys. Lett. B286, 413 (1992); G. Cella, G. Curci, G. Ricciardi and A. Vicere, Phys. Lett. B248, 181 (1990); B325, 227 (1994); M. Misiak, Phys. Lett. B269, 161 (1991); M. Ciuchini, E. Franco, G. Martinelli, L. Reina and L. Silvestrini, Phys. Lett B316, 127 (1993); B334, 137 (1994); M. Misiak, Phys. Lett. B321, 113 (1994); A. Ali and C. Greub, Phys. Lett. B259, 182 (1991); Z. Phys. C60, 433 (1993); A. J. Buras, M. Jamin, M. Lauten-bacher and P. Weisz, Nucl. Phys. B370, 69 (1992); B400, 37, 75 (1993);
41. The first calculations of the hard gluon corrections were done in: M. Shifman, A. Vainshtein and V. Zakharov, Phys. Rev. D18, 2583 (1978). Since then the analysis has been repeatedly discussed in the literature; see e.g.: S. Bertolini, F. Borzumati and A. Masiero, Phys. Rev. Lett. 59, 180 (1987); N. Deshpande, P. Lo, J. Trampetic, G. Eilam and P. Singer, Phys. Rev. Lett. 59, 183 (1987); B. Grinshtein, R. Springer and M. Wise, Phys. Lett. B202, 138 (1988); R. Grigjanis, P. J. O'Donnel, M. Sutherland and H. Navelet, Phys. Lett. B213, 335 (1988); (E) Phys. Lett. B286, 413 (1992); G. Cella, G. Curci, G. Ricciardi and A. Vicere, Phys. Lett. B248, 181 (1990); B325, 227 (1994); M. Misiak, Phys. Lett. B269, 161 (1991); M. Ciuchini, E. Franco, G. Martinelli, L. Reina and L. Silvestrini, Phys. Lett B316, 127 (1993); B334, 137 (1994); M. Misiak, Phys. Lett. B321, 113 (1994); A. Ali and C. Greub, Phys. Lett. B259, 182 (1991); Z. Phys. C60, 433 (1993); A. J. Buras, M. Jamin, M. Lauten-bacher and P. Weisz, Nucl. Phys. B370, 69 (1992); B400, 37, 75 (1993);
42. The first calculations of the hard gluon corrections were done in: M. Shifman, A. Vainshtein and V. Zakharov, Phys. Rev. D18, 2583 (1978). Since then the analysis has been repeatedly discussed in the literature; see e.g.: S. Bertolini, F. Borzumati and A. Masiero, Phys. Rev. Lett. 59, 180 (1987); N. Deshpande, P. Lo, J. Trampetic, G. Eilam and P. Singer, Phys. Rev. Lett. 59, 183 (1987); B. Grinshtein, R. Springer and M. Wise, Phys. Lett. B202, 138 (1988); R. Grigjanis, P. J. O'Donnel, M. Sutherland and H. Navelet, Phys. Lett. B213, 335 (1988); (E) Phys. Lett. B286, 413 (1992); G. Cella, G. Curci, G. Ricciardi and A. Vicere, Phys. Lett. B248, 181 (1990); B325, 227 (1994); M. Misiak, Phys. Lett. B269, 161 (1991); M. Ciuchini, E. Franco, G. Martinelli, L. Reina and L. Silvestrini, Phys. Lett B316, 127 (1993); B334, 137 (1994); M. Misiak, Phys. Lett. B321, 113 (1994); A. Ali and C. Greub, Phys. Lett. B259, 182 (1991); Z. Phys. C60, 433 (1993); A. J. Buras, M. Jamin, M. Lauten-bacher and P. Weisz, Nucl. Phys. B370, 69 (1992); B400, 37, 75 (1993);
43. The first calculations of the hard gluon corrections were done in: M. Shifman, A. Vainshtein and V. Zakharov, Phys. Rev. D18, 2583 (1978). Since then the analysis has been repeatedly discussed in the literature; see e.g.: S. Bertolini, F. Borzumati and A. Masiero, Phys. Rev. Lett. 59, 180 (1987); N. Deshpande, P. Lo, J. Trampetic, G. Eilam and P. Singer, Phys. Rev. Lett. 59, 183 (1987); B. Grinshtein, R. Springer and M. Wise, Phys. Lett. B202, 138 (1988); R. Grigjanis, P. J. O'Donnel, M. Sutherland and H. Navelet, Phys. Lett. B213, 335 (1988); (E) Phys. Lett. B286, 413 (1992); G. Cella, G. Curci, G. Ricciardi and A. Vicere, Phys. Lett. B248, 181 (1990); B325, 227 (1994); M. Misiak, Phys. Lett. B269, 161 (1991); M. Ciuchini, E. Franco, G. Martinelli, L. Reina and L. Silvestrini, Phys. Lett B316, 127 (1993); B334, 137 (1994); M. Misiak, Phys. Lett. B321, 113 (1994); A. Ali and C. Greub, Phys. Lett. B259, 182 (1991); Z. Phys. C60, 433 (1993); A. J. Buras, M. Jamin, M. Lauten-bacher and P. Weisz, Nucl. Phys. B370, 69 (1992); B400, 37, 75 (1993);
44. The first calculations of the hard gluon corrections were done in: M. Shifman, A. Vainshtein and V. Zakharov, Phys. Rev. D18, 2583 (1978). Since then the analysis has been repeatedly discussed in the literature; see e.g.: S. Bertolini, F. Borzumati and A. Masiero, Phys. Rev. Lett. 59, 180 (1987); N. Deshpande, P. Lo, J. Trampetic, G. Eilam and P. Singer, Phys. Rev. Lett. 59, 183 (1987); B. Grinshtein, R. Springer and M. Wise, Phys. Lett. B202, 138 (1988); R. Grigjanis, P. J. O'Donnel, M. Sutherland and H. Navelet, Phys. Lett. B213, 335 (1988); (E) Phys. Lett. B286, 413 (1992); G. Cella, G. Curci, G. Ricciardi and A. Vicere, Phys. Lett. B248, 181 (1990); B325, 227 (1994); M. Misiak, Phys. Lett. B269, 161 (1991); M. Ciuchini, E. Franco, G. Martinelli, L. Reina and L. Silvestrini, Phys. Lett B316, 127 (1993); B334, 137 (1994); M. Misiak, Phys. Lett. B321, 113 (1994); A. Ali and C. Greub, Phys. Lett. B259, 182 (1991); Z. Phys. C60, 433 (1993); A. J. Buras, M. Jamin, M. Lauten-bacher and P. Weisz, Nucl. Phys. B370, 69 (1992); B400, 37, 75 (1993);
45. The first calculations of the hard gluon corrections were done in: M. Shifman, A. Vainshtein and V. Zakharov, Phys. Rev. D18, 2583 (1978). Since then the analysis has been repeatedly discussed in the literature; see e.g.: S. Bertolini, F. Borzumati and A. Masiero, Phys. Rev. Lett. 59, 180 (1987); N. Deshpande, P. Lo, J. Trampetic, G. Eilam and P. Singer, Phys. Rev. Lett. 59, 183 (1987); B. Grinshtein, R. Springer and M. Wise, Phys. Lett. B202, 138 (1988); R. Grigjanis, P. J. O'Donnel, M. Sutherland and H. Navelet, Phys. Lett. B213, 335 (1988); (E) Phys. Lett. B286, 413 (1992); G. Cella, G. Curci, G. Ricciardi and A. Vicere, Phys. Lett. B248, 181 (1990); B325, 227 (1994); M. Misiak, Phys. Lett. B269, 161 (1991); M. Ciuchini, E. Franco, G. Martinelli, L. Reina and L. Silvestrini, Phys. Lett B316, 127 (1993); B334, 137 (1994); M. Misiak, Phys. Lett. B321, 113 (1994); A. Ali and C. Greub, Phys. Lett. B259, 182 (1991); Z. Phys. C60, 433 (1993); A. J. Buras, M. Jamin, M. Lauten-bacher and P. Weisz, Nucl. Phys. B370, 69 (1992); B400, 37, 75 (1993);
46. The first calculations of the hard gluon corrections were done in: M. Shifman, A. Vainshtein and V. Zakharov, Phys. Rev. D18, 2583 (1978). Since then the analysis has been repeatedly discussed in the literature; see e.g.: S. Bertolini, F. Borzumati and A. Masiero, Phys. Rev. Lett. 59, 180 (1987); N. Deshpande, P. Lo, J. Trampetic, G. Eilam and P. Singer, Phys. Rev. Lett. 59, 183 (1987); B. Grinshtein, R. Springer and M. Wise, Phys. Lett. B202, 138 (1988); R. Grigjanis, P. J. O'Donnel, M. Sutherland and H. Navelet, Phys. Lett. B213, 335 (1988); (E) Phys. Lett. B286, 413 (1992); G. Cella, G. Curci, G. Ricciardi and A. Vicere, Phys. Lett. B248, 181 (1990); B325, 227 (1994); M. Misiak, Phys. Lett. B269, 161 (1991); M. Ciuchini, E. Franco, G. Martinelli, L. Reina and L. Silvestrini, Phys. Lett B316, 127 (1993); B334, 137 (1994); M. Misiak, Phys. Lett. B321, 113 (1994); A. Ali and C. Greub, Phys. Lett. B259, 182 (1991); Z. Phys. C60, 433 (1993); A. J. Buras, M. Jamin, M. Lauten-bacher and P. Weisz, Nucl. Phys. B370, 69 (1992); B400, 37, 75 (1993);
47. The first calculations of the hard gluon corrections were done in: M. Shifman, A. Vainshtein and V. Zakharov, Phys. Rev. D18, 2583 (1978). Since then the analysis has been repeatedly discussed in the literature; see e.g.: S. Bertolini, F. Borzumati and A. Masiero, Phys. Rev. Lett. 59, 180 (1987); N. Deshpande, P. Lo, J. Trampetic, G. Eilam and P. Singer, Phys. Rev. Lett. 59, 183 (1987); B. Grinshtein, R. Springer and M. Wise, Phys. Lett. B202, 138 (1988); R. Grigjanis, P. J. O'Donnel, M. Sutherland and H. Navelet, Phys. Lett. B213, 335 (1988); (E) Phys. Lett. B286, 413 (1992); G. Cella, G. Curci, G. Ricciardi and A. Vicere, Phys. Lett. B248, 181 (1990); B325, 227 (1994); M. Misiak, Phys. Lett. B269, 161 (1991); M. Ciuchini, E. Franco, G. Martinelli, L. Reina and L. Silvestrini, Phys. Lett B316, 127 (1993); B334, 137 (1994); M. Misiak, Phys. Lett. B321, 113 (1994); A. Ali and C. Greub, Phys. Lett. B259, 182 (1991); Z. Phys. C60, 433 (1993); A. J. Buras, M. Jamin, M. Lauten-bacher and P. Weisz, Nucl. Phys. B370, 69 (1992); B400, 37, 75 (1993);
48. The first calculations of the hard gluon corrections were done in: M. Shifman, A. Vainshtein and V. Zakharov, Phys. Rev. D18, 2583 (1978). Since then the analysis has been repeatedly discussed in the literature; see e.g.: S. Bertolini, F. Borzumati and A. Masiero, Phys. Rev. Lett. 59, 180 (1987); N. Deshpande, P. Lo, J. Trampetic, G. Eilam and P. Singer, Phys. Rev. Lett. 59, 183 (1987); B. Grinshtein, R. Springer and M. Wise, Phys. Lett. B202, 138 (1988); R. Grigjanis, P. J. O'Donnel, M. Sutherland and H. Navelet, Phys. Lett. B213, 335 (1988); (E) Phys. Lett. B286, 413 (1992); G. Cella, G. Curci, G. Ricciardi and A. Vicere, Phys. Lett. B248, 181 (1990); B325, 227 (1994); M. Misiak, Phys. Lett. B269, 161 (1991); M. Ciuchini, E. Franco, G. Martinelli, L. Reina and L. Silvestrini, Phys. Lett B316, 127 (1993); B334, 137 (1994); M. Misiak, Phys. Lett. B321, 113 (1994); A. Ali and C. Greub, Phys. Lett. B259, 182 (1991); Z. Phys. C60, 433 (1993); A. J. Buras, M. Jamin, M. Lauten-bacher and P. Weisz, Nucl. Phys. B370, 69 (1992); B400, 37, 75 (1993);
49. The first calculations of the hard gluon corrections were done in: M. Shifman, A. Vainshtein and V. Zakharov, Phys. Rev. D18, 2583 (1978). Since then the analysis has been repeatedly discussed in the literature; see e.g.: S. Bertolini, F. Borzumati and A. Masiero, Phys. Rev. Lett. 59, 180 (1987); N. Deshpande, P. Lo, J. Trampetic, G. Eilam and P. Singer, Phys. Rev. Lett. 59, 183 (1987); B. Grinshtein, R. Springer and M. Wise, Phys. Lett. B202, 138 (1988); R. Grigjanis, P. J. O'Donnel, M. Sutherland and H. Navelet, Phys. Lett. B213, 335 (1988); (E) Phys. Lett. B286, 413 (1992); G. Cella, G. Curci, G. Ricciardi and A. Vicere, Phys. Lett. B248, 181 (1990); B325, 227 (1994); M. Misiak, Phys. Lett. B269, 161 (1991); M. Ciuchini, E. Franco, G. Martinelli, L. Reina and L. Silvestrini, Phys. Lett B316, 127 (1993); B334, 137 (1994); M. Misiak, Phys. Lett. B321, 113 (1994); A. Ali and C. Greub, Phys. Lett. B259, 182 (1991); Z. Phys. C60, 433 (1993); A. J. Buras, M. Jamin, M. Lauten-bacher and P. Weisz, Nucl. Phys. B370, 69 (1992); B400, 37, 75 (1993);
50. The first calculations of the hard gluon corrections were done in: M. Shifman, A. Vainshtein and V. Zakharov, Phys. Rev. D18, 2583 (1978). Since then the analysis has been repeatedly discussed in the literature; see e.g.: S. Bertolini, F. Borzumati and A. Masiero, Phys. Rev. Lett. 59, 180 (1987); N. Deshpande, P. Lo, J. Trampetic, G. Eilam and P. Singer, Phys. Rev. Lett. 59, 183 (1987); B. Grinshtein, R. Springer and M. Wise, Phys. Lett. B202, 138 (1988); R. Grigjanis, P. J. O'Donnel, M. Sutherland and H. Navelet, Phys. Lett. B213, 335 (1988); (E) Phys. Lett. B286, 413 (1992); G. Cella, G. Curci, G. Ricciardi and A. Vicere, Phys. Lett. B248, 181 (1990); B325, 227 (1994); M. Misiak, Phys. Lett. B269, 161 (1991); M. Ciuchini, E. Franco, G. Martinelli, L. Reina and L. Silvestrini, Phys. Lett B316, 127 (1993); B334, 137 (1994); M. Misiak, Phys. Lett. B321, 113 (1994); A. Ali and C. Greub, Phys. Lett. B259, 182 (1991); Z. Phys. C60, 433 (1993); A. J. Buras, M. Jamin, M. Lauten-bacher and P. Weisz, Nucl. Phys. B370, 69 (1992); B400, 37, 75 (1993);
51. The first calculations of the hard gluon corrections were done in: M. Shifman, A. Vainshtein and V. Zakharov, Phys. Rev. D18, 2583 (1978). Since then the analysis has been repeatedly discussed in the literature; see e.g.: S. Bertolini, F. Borzumati and A. Masiero, Phys. Rev. Lett. 59, 180 (1987); N. Deshpande, P. Lo, J. Trampetic, G. Eilam and P. Singer, Phys. Rev. Lett. 59, 183 (1987); B. Grinshtein, R. Springer and M. Wise, Phys. Lett. B202, 138 (1988); R. Grigjanis, P. J. O'Donnel, M. Sutherland and H. Navelet, Phys. Lett. B213, 335 (1988); (E) Phys. Lett. B286, 413 (1992); G. Cella, G. Curci, G. Ricciardi and A. Vicere, Phys. Lett. B248, 181 (1990); B325, 227 (1994); M. Misiak, Phys. Lett. B269, 161 (1991); M. Ciuchini, E. Franco, G. Martinelli, L. Reina and L. Silvestrini, Phys. Lett B316, 127 (1993); B334, 137 (1994); M. Misiak, Phys. Lett. B321, 113 (1994); A. Ali and C. Greub, Phys. Lett. B259, 182 (1991); Z. Phys. C60, 433 (1993); A. J. Buras, M. Jamin, M. Lauten-bacher and P. Weisz, Nucl. Phys. B370, 69 (1992); B400, 37, 75 (1993);
52. The first calculations of the hard gluon corrections were done in: M. Shifman, A. Vainshtein and V. Zakharov, Phys. Rev. D18, 2583 (1978). Since then the analysis has been repeatedly discussed in the literature; see e.g.: S. Bertolini, F. Borzumati and A. Masiero, Phys. Rev. Lett. 59, 180 (1987); N. Deshpande, P. Lo, J. Trampetic, G. Eilam and P. Singer, Phys. Rev. Lett. 59, 183 (1987); B. Grinshtein, R. Springer and M. Wise, Phys. Lett. B202, 138 (1988); R. Grigjanis, P. J. O'Donnel, M. Sutherland and H. Navelet, Phys. Lett. B213, 335 (1988); (E) Phys. Lett. B286, 413 (1992); G. Cella, G. Curci, G. Ricciardi and A. Vicere, Phys. Lett. B248, 181 (1990); B325, 227 (1994); M. Misiak, Phys. Lett. B269, 161 (1991); M. Ciuchini, E. Franco, G. Martinelli, L. Reina and L. Silvestrini, Phys. Lett B316, 127 (1993); B334, 137 (1994); M. Misiak, Phys. Lett. B321, 113 (1994); A. Ali and C. Greub, Phys. Lett. B259, 182 (1991); Z. Phys. C60, 433 (1993); A. J. Buras, M. Jamin, M. Lauten-bacher and P. Weisz, Nucl. Phys. B370, 69 (1992); B400, 37, 75 (1993);
53. The first calculations of the hard gluon corrections were done in: M. Shifman, A. Vainshtein and V. Zakharov, Phys. Rev. D18, 2583 (1978). Since then the analysis has been repeatedly discussed in the literature; see e.g.: S. Bertolini, F. Borzumati and A. Masiero, Phys. Rev. Lett. 59, 180 (1987); N. Deshpande, P. Lo, J. Trampetic, G. Eilam and P. Singer, Phys. Rev. Lett. 59, 183 (1987); B. Grinshtein, R. Springer and M. Wise, Phys. Lett. B202, 138 (1988); R. Grigjanis, P. J. O'Donnel, M. Sutherland and H. Navelet, Phys. Lett. B213, 335 (1988); (E) Phys. Lett. B286, 413 (1992); G. Cella, G. Curci, G. Ricciardi and A. Vicere, Phys. Lett. B248, 181 (1990); B325, 227 (1994); M. Misiak, Phys. Lett. B269, 161 (1991); M. Ciuchini, E. Franco, G. Martinelli, L. Reina and L. Silvestrini, Phys. Lett B316, 127 (1993); B334, 137 (1994); M. Misiak, Phys. Lett. B321, 113 (1994); A. Ali and C. Greub, Phys. Lett. B259, 182 (1991); Z. Phys. C60, 433 (1993); A. J. Buras, M. Jamin, M. Lauten-bacher and P. Weisz, Nucl. Phys. B370, 69 (1992); B400, 37, 75 (1993);
54. The first calculations of the hard gluon corrections were done in: M. Shifman, A. Vainshtein and V. Zakharov, Phys. Rev. D18, 2583 (1978). Since then the analysis has been repeatedly discussed in the literature; see e.g.: S. Bertolini, F. Borzumati and A. Masiero, Phys. Rev. Lett. 59, 180 (1987); N. Deshpande, P. Lo, J. Trampetic, G. Eilam and P. Singer, Phys. Rev. Lett. 59, 183 (1987); B. Grinshtein, R. Springer and M. Wise, Phys. Lett. B202, 138 (1988); R. Grigjanis, P. J. O'Donnel, M. Sutherland and H. Navelet, Phys. Lett. B213, 335 (1988); (E) Phys. Lett. B286, 413 (1992); G. Cella, G. Curci, G. Ricciardi and A. Vicere, Phys. Lett. B248, 181 (1990); B325, 227 (1994); M. Misiak, Phys. Lett. B269, 161 (1991); M. Ciuchini, E. Franco, G. Martinelli, L. Reina and L. Silvestrini, Phys. Lett B316, 127 (1993); B334, 137 (1994); M. Misiak, Phys. Lett. B321, 113 (1994); A. Ali and C. Greub, Phys. Lett. B259, 182 (1991); Z. Phys. C60, 433 (1993); A. J. Buras, M. Jamin, M. Lauten-bacher and P. Weisz, Nucl. Phys. B370, 69 (1992); B400, 37, 75 (1993);
55. The first calculations of the hard gluon corrections were done in: M. Shifman, A. Vainshtein and V. Zakharov, Phys. Rev. D18, 2583 (1978). Since then the analysis has been repeatedly discussed in the literature; see e.g.: S. Bertolini, F. Borzumati and A. Masiero, Phys. Rev. Lett. 59, 180 (1987); N. Deshpande, P. Lo, J. Trampetic, G. Eilam and P. Singer, Phys. Rev. Lett. 59, 183 (1987); B. Grinshtein, R. Springer and M. Wise, Phys. Lett. B202, 138 (1988); R. Grigjanis, P. J. O'Donnel, M. Sutherland and H. Navelet, Phys. Lett. B213, 335 (1988); (E) Phys. Lett. B286, 413 (1992); G. Cella, G. Curci, G. Ricciardi and A. Vicere, Phys. Lett. B248, 181 (1990); B325, 227 (1994); M. Misiak, Phys. Lett. B269, 161 (1991); M. Ciuchini, E. Franco, G. Martinelli, L. Reina and L. Silvestrini, Phys. Lett B316, 127 (1993); B334, 137 (1994); M. Misiak, Phys. Lett. B321, 113 (1994); A. Ali and C. Greub, Phys. Lett. B259, 182 (1991); Z. Phys. C60, 433 (1993); A. J. Buras, M. Jamin, M. Lauten-bacher and P. Weisz, Nucl. Phys. B370, 69 (1992); B400, 37, 75 (1993);
56. G. Buchalla and A. Buras, Nucl. Phys. B400, 225 (1993); M. Ciuchini, E. Franco, L. Reina and L. Silvestrini, Nucl. Phys. B415; 403 (1994); A. Buras, M. Misiak, M. Munz and S. Pokorski, Nucl. Phys. B424, 374 (1994).
57. G. Buchalla and A. Buras, Nucl. Phys. B400, 225 (1993); M. Ciuchini, E. Franco, L. Reina and L. Silvestrini, Nucl. Phys. B415; 403 (1994); A. Buras, M. Misiak, M. Munz and S. Pokorski, Nucl. Phys. B424, 374 (1994).
58. G. Buchalla and A. Buras, Nucl. Phys. B400, 225 (1993); M. Ciuchini, E. Franco, L. Reina and L. Silvestrini, Nucl. Phys. B415; 403 (1994); A. Buras, M. Misiak, M. Munz and S. Pokorski, Nucl. Phys. B424, 374 (1994).
59. b/2, of a width that is a few units ×ΛQCD, we get the end point domain.
60. A. Ali and C. Greub, Z. Phys. C49, 431 (1991); Phys. Lett. B287, 191 (1992).
61. A. Ali and C. Greub, Z. Phys. C49, 431 (1991); Phys. Lett. B287, 191 (1992).
62. This is a very important feature. In general, the primordial distribution function is required to fall off faster than any power of 1/x at large negative x. This requirement is due to the fact that the moments of the primordial distribution function are related to the matrix elements of certain well-defined operators. For this reason, ansätze with a slower falloff, like that of Ref. 28, have no interpretation in QCD as a primordial distribution.
63. C. H. Jin and E. A. Paschos, preprint DO-TH 95/07 (hep-ph/9504375).
64. T. Mannel, Phys. Rev. D50, 428 (1994).
65. P. Ball and V. Braun, Phys. Rev. D49, 2472 (1994).
66. E. Bagan, P. Ball, V. Braun and P. Gosdzinsky, Phys. Lett B342, 362 (1995); see also the last paper of Refs. 47.
67. M. Voloshin, preprint TPI-MINN-94/18-T (unpublished).
68. M. Shifman, N. G. Uraltsev and A. Vainshtein, Phys. Rev., D51, 2217 (1995).
69. I. Bigi, M. Shifman, N. Uraltsev and A. Vainshtein, Phys. Rev. D52, 196 (1995).
70. I. Bigi, A. Grozin, M. Shifman, N. G. Uraltsev, A. Vainshtein, Phys. Lett B339, 160 (1994).
71. S. Narison, Phys. Lett B308, 365 (1993).
72. I. Bigi and N. G. Uraltsev, Phys. Lett. B321, 412 (1994).
73. I. Bigi, M. Shifman, N. G. Uraltsev and A. Vainshtein, Phys. Rev. D50, 2234 (1994).
74. M. Beneke and V. Braun, Nucl. Phys. B426, 301 (1994).
75. M. Voloshin, Phys. Rev. D46, 3062 (1992).
76. E. Eichten and B. Hill, Phys. Lett B234, 511 (1990); H. Georgi, Phys. Lett. B240, 447 (1990). For recent reviews see e.g.: N. Isgur and M. B. Wise, in B Decays, ed. S. Stone, (World Scientific, Singapore, 1992); H. Georgi, in Proc. Theoretical Advanced Study Institute 1991, eds. R. K. Ellis, C. T. Hill and J. D. Lykken (World Scientific, Singapore, 1992).
77. E. Eichten and B. Hill, Phys. Lett B234, 511 (1990); H. Georgi, Phys. Lett. B240, 447 (1990). For recent reviews see e.g.: N. Isgur and M. B. Wise, in B Decays, ed. S. Stone, (World Scientific, Singapore, 1992); H. Georgi, in Proc. Theoretical Advanced Study Institute 1991, eds. R. K. Ellis, C. T. Hill and J. D. Lykken (World Scientific, Singapore, 1992).
78. E. Eichten and B. Hill, Phys. Lett B234, 511 (1990); H. Georgi, Phys. Lett. B240, 447 (1990). For recent reviews see e.g.: N. Isgur and M. B. Wise, in B Decays, ed. S. Stone, (World Scientific, Singapore, 1992); H. Georgi, in Proc. Theoretical Advanced Study Institute 1991, eds. R. K. Ellis, C. T. Hill and J. D. Lykken (World Scientific, Singapore, 1992).
79. E. Eichten and B. Hill, Phys. Lett B234, 511 (1990); H. Georgi, Phys. Lett. B240, 447 (1990). For recent reviews see e.g.: N. Isgur and M. B. Wise, in B Decays, ed. S. Stone, (World Scientific, Singapore, 1992); H. Georgi, in Proc. Theoretical Advanced Study Institute 1991, eds. R. K. Ellis, C. T. Hill and J. D. Lykken (World Scientific, Singapore, 1992).
80. In fact, S is the square of a form factor; we still will use this somewhat sloppy notation, because the form factor itself will never appear and so there can be no confusion.
81. For a review see e.g.: Yu. L. Dokshitzer, V. A. Khoze, A. H. Mueller and S. I. Troyan, Basics of Perturbative QCD, ed. J. Tran Thanh Van (Editions Frontières, Gif-sür-Yvette, 1991), and references therein.
82. s there. Then Eq. (22) coincides with Eq. (17) with the additional constraint ε > μ.
83. S. J. Brodsky, G. P. Lepage and P. B. Mackenzie, Phys. Rev. D28, 228 (1983); G. P. Lepage and P. B. Mackenzie, Phys. Rev. D48, 2250 (1993).
84. S. J. Brodsky, G. P. Lepage and P. B. Mackenzie, Phys. Rev. D28, 228 (1983); G. P. Lepage and P. B. Mackenzie, Phys. Rev. D48, 2250 (1993).
85. B. Smith and M. Voloshin, Phys. Lett. B340, 176 (1994).
86. M. Beneke and V. Braun, Phys. Lett. B348, 513 (1995); P. Ball, M. Beneke and V. Braun, preprint CERN-TH/95-26 (hep-ph/9502300); preprint CERN-TH/95-65 (hep-ph/9503492).
87. M. Beneke and V. Braun, Phys. Lett. B348, 513 (1995); P. Ball, M. Beneke and V. Braun, preprint CERN-TH/95-26 (hep-ph/9502300); preprint CERN-TH/95-65 (hep-ph/9503492).
88. M. Neubert, Phys. Rev. D51, 5924 (1995).
89. Simplified elements of this procedure have been in use among the Sudakov theory practitioners for years; e.g.: G. Altarelli, Phys. Rep. 81, 1 (1982) and the further review of the literature therein; Ref. 43.
90. Yu. L. Dokshitzer, D. I. Dyakonov and S. I. Troyan, Phys. Rep. 58, 270 (1980); D. Amati, A. Bassetto, M. Ciafaloni, G. Marchesini and G. Veneziano, Nucl. Phys. B173, 429 (1980).
91. Yu. L. Dokshitzer, D. I. Dyakonov and S. I. Troyan, Phys. Rep. 58, 270 (1980); D. Amati, A. Bassetto, M. Ciafaloni, G. Marchesini and G. Veneziano, Nucl. Phys. B173, 429 (1980).
92. A. Smilga, Nucl. Phys. B161, 449 (1979); G. Korchemsky, Phys. Lett. B217, 330 (1989).
93. A. Smilga, Nucl. Phys. B161, 449 (1979); G. Korchemsky, Phys. Lett. B217, 330 (1989).
94. Yu. Dokshitser and B. Webber, Phys. Lett. B352, 451 (1995).
95. N. G. Uraltsev, preprint TPI-MINN-95/5-T (hep-ph/9503404).
96. We hope the reader will not be confused by the use of one and the same letter, ε, for the critical exponent and the photon energy. The former is marked by the subscript 0.
97. We are grateful to Yu. Dokshitser for pointing out to us this purely perturbative aspect of application of double logarithmic expressions at asymptotically large energies.
98. S. Catani, G. Turnock, B. R. Webber and L. Trentadue, Phys. Lett. B263, 491 (1991) 491; S. Catani, G. Turnock and B. R. Webber, ibid. B272, 368 (1991).
99. S. Catani, G. Turnock, B. R. Webber and L. Trentadue, Phys. Lett. B263, 491 (1991) 491; S. Catani, G. Turnock and B. R. Webber, ibid. B272, 368 (1991).
100. G. Korchemsky and G. Sterman, Phys. Lett. B340, 96 (1994).
101. M. Beneke, V. Braun and V. I. Zakharov, Phys. Rev. Lett. 73, 3058 (1994).
102. 2) as a sum over polarizations, for massive gluons one must sum over three transverse polarizations rather than two in the massless case.
103. 26 which is infrared stable.
104. A similar consideration for the polarization tensor of light quark currents was made in: M. Neubert, CERN-TH.7524/94 (hep-ph/9502264). However, the technique used there cannot be applied to other problems, in particular to heavy flavor transitions. Let us parenthetically note that we disagree with certain aspects of this work.
105. M. Shifman, N. G. Uraltsev and A. Vainshtein, in preparation.
106. I. Bigi and N. G. Uraltsev, Nucl. Phys. B423, 33 (1994).
107. For a review see e.g.: B. L. Ioffe, V. A. Khoze and L. N. Lipatov, Hard Proceses (North-Holland, 1984).
108. V. N. Gribov and L. N. Lipatov, Yad. Fiz. 15, 781, 1218 (1972); [Sov. J. Nucl. Phys. 15, 438, 675 (1972)]; Yu. Dokshitser, Zh. Eksp. Teor. Fiz. 73, 1216 (1977) [Sov. Phys. JETP, 46, 641 (1977)]; G. Altarelli and G. Parisi, Nucl. Phys. B126, 298 (1977).
109. V. N. Gribov and L. N. Lipatov, Yad. Fiz. 15, 781, 1218 (1972); [Sov. J. Nucl. Phys. 15, 438, 675 (1972)]; Yu. Dokshitser, Zh. Eksp. Teor. Fiz. 73, 1216 (1977) [Sov. Phys. JETP, 46, 641 (1977)]; G. Altarelli and G. Parisi, Nucl. Phys. B126, 298 (1977).
110. V. N. Gribov and L. N. Lipatov, Yad. Fiz. 15, 781, 1218 (1972); [Sov. J. Nucl. Phys. 15, 438, 675 (1972)]; Yu. Dokshitser, Zh. Eksp. Teor. Fiz. 73, 1216 (1977) [Sov. Phys. JETP, 46, 641 (1977)]; G. Altarelli and G. Parisi, Nucl. Phys. B126, 298 (1977).
111. V. N. Gribov and L. N. Lipatov, Yad. Fiz. 15, 781, 1218 (1972); [Sov. J. Nucl. Phys. 15, 438, 675 (1972)]; Yu. Dokshitser, Zh. Eksp. Teor. Fiz. 73, 1216 (1977) [Sov. Phys. JETP, 46, 641 (1977)]; G. Altarelli and G. Parisi, Nucl. Phys. B126, 298 (1977).
112. V. N. Gribov and L. N. Lipatov, Yad. Fiz. 15, 781, 1218 (1972); [Sov. J. Nucl. Phys. 15, 438, 675 (1972)]; Yu. Dokshitser, Zh. Eksp. Teor. Fiz. 73, 1216 (1977) [Sov. Phys. JETP, 46, 641 (1977)]; G. Altarelli and G. Parisi, Nucl. Phys. B126, 298 (1977).
113. M. Voloshin, Int. J. Mod. Phys. A10, 2865 (1995).
114. M. Voloshin, in preparation.
115. To make contact with other publications we observe that this value of the running 6 quark mass corresponds to the one-loop pole mass 4.81 GeV if it is extrapolated from the point μ = 1 GeV.
116. When comparing experimental data, one also has to keep in mind some broadening of the spectrum due to slow movement of decaying B produced in Υ(4S).
117. For a review see e.g.: V. Chernyak and I. Zhitnitsky, Phys. Rep. 112, 173 (1984).
118. L. Koyrakh, N. G. Uraltsev and A. Vainshtein, to be published.
119. H.-L. Yu, privite communication to M. S. and H. Li and H.-L. Yu, to be published.