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Drell–Yan production at small q T , transverse parton distributions and the collinear anomaly

  • Thomas Becher
  • Matthias NeubertEmail author
Regular Article - Theoretical Physics

Abstract

Using methods from effective field theory, an exact all-order expression for the Drell–Yan cross section at small transverse momentum is derived directly in q T space, in which all large logarithms are resummed. The anomalous dimensions and matching coefficients necessary for resummation at NNLL order are given explicitly. The precise relation between our result and the Collins–Soper–Sterman formula is discussed, and as a by-product the previously unknown three-loop coefficient A (3) is obtained. The naive factorization of the cross section at small transverse momentum is broken by a collinear anomaly, which prevents a process-independent definition of x T -dependent parton distribution functions. A factorization theorem is derived for the product of two such functions, in which the dependence on the hard momentum transfer is separated out. The remainder factors into a product of two functions of longitudinal momentum variables and \(x_{T}^{2}\), whose renormalization-group evolution is derived and solved in closed form. The matching of these functions at small x T onto standard parton distributions is calculated at \(\mathcal{O}(\alpha_{s})\), while their anomalous dimensions are known to three loops.

Keywords

Transverse Momentum Anomalous Dimension Wilson Line Soft Function Large Logarithm 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Y.L. Dokshitzer, D.I. Dyakonov, S.I. Troyan, Phys. Rep. 58, 269 (1980) ADSCrossRefGoogle Scholar
  2. 2.
    G. Parisi, R. Petronzio, Nucl. Phys. B 154, 427 (1979) ADSCrossRefGoogle Scholar
  3. 3.
    G. Curci, M. Greco, Y. Srivastava, Nucl. Phys. B 159, 451 (1979) ADSCrossRefGoogle Scholar
  4. 4.
    J.C. Collins, D.E. Soper, G.F. Sterman, Nucl. Phys. B 250, 199 (1985) ADSCrossRefGoogle Scholar
  5. 5.
    C.T.H. Davies, W.J. Stirling, Nucl. Phys. B 244, 337 (1984) ADSCrossRefGoogle Scholar
  6. 6.
    C.T.H. Davies, B.R. Webber, W.J. Stirling, Nucl. Phys. B 256, 413 (1985) ADSCrossRefGoogle Scholar
  7. 7.
    D. de Florian, M. Grazzini, Nucl. Phys. B 616, 247 (2001). arXiv:hep-ph/0108273 ADSCrossRefGoogle Scholar
  8. 8.
    T. Aaltonen et al. (CDF Collaboration), Phys. Rev. Lett. 99, 151801 (2007). arXiv:0707.0085 [hep-ex] ADSCrossRefGoogle Scholar
  9. 9.
    T. Aaltonen et al. (CDF Collaboration), Phys. Rev. D 77, 112001 (2008). arXiv:0708.3642 [hep-ex] ADSCrossRefGoogle Scholar
  10. 10.
    V.M. Abazov et al. (D0 Collaboration), Phys. Rev. Lett. 103, 141801 (2009). arXiv:0908.0766 [hep-ex] ADSCrossRefGoogle Scholar
  11. 11.
    C.W. Bauer, S. Fleming, D. Pirjol, I.W. Stewart, Phys. Rev. D 63, 114020 (2001). hep-ph/0011336 ADSCrossRefGoogle Scholar
  12. 12.
    C.W. Bauer, D. Pirjol, I.W. Stewart, Phys. Rev. D 65, 054022 (2002). hep-ph/0109045 ADSCrossRefGoogle Scholar
  13. 13.
    M. Beneke, A.P. Chapovsky, M. Diehl, T. Feldmann, Nucl. Phys. B 643, 431 (2002). hep-ph/0206152 ADSzbMATHCrossRefGoogle Scholar
  14. 14.
    J.C. Collins, Acta Phys. Pol. B 34, 3103 (2003). arXiv:hep-ph/0304122 ADSGoogle Scholar
  15. 15.
    J.y. Chiu, F. Golf, R. Kelley, A.V. Manohar, Phys. Rev. Lett. 100, 021802 (2008). arXiv:0709.2377 [hep-ph] ADSCrossRefGoogle Scholar
  16. 16.
    J.y. Chiu, F. Golf, R. Kelley, A.V. Manohar, Phys. Rev. D 77, 053004 (2008). arXiv:0712.0396 [hep-ph] ADSCrossRefGoogle Scholar
  17. 17.
    I.W. Stewart, F.J. Tackmann, W.J. Waalewijn, Phys. Rev. D 81, 094035 (2010). arXiv:0910.0467 [hep-ph] ADSCrossRefGoogle Scholar
  18. 18.
    I.W. Stewart, F.J. Tackmann, W.J. Waalewijn, J. High Energy Phys. 1009, 005 (2010). arXiv:1002.2213 [hep-ph] ADSCrossRefGoogle Scholar
  19. 19.
    V.A. Smirnov, Phys. Lett. B 309, 397 (1993) ADSCrossRefGoogle Scholar
  20. 20.
    V.A. Smirnov, Springer Tracts Mod. Phys. 177, 1 (2002) ADSCrossRefGoogle Scholar
  21. 21.
    G. Bozzi, S. Catani, D. de Florian, M. Grazzini, Phys. Lett. B 564, 65 (2003). arXiv:hep-ph/0302104 ADSCrossRefGoogle Scholar
  22. 22.
    C. Balazs, E.L. Berger, P.M. Nadolsky, C.P. Yuan, Phys. Rev. D 76, 013009 (2007). arXiv:0704.0001 [hep-ph] ADSCrossRefGoogle Scholar
  23. 23.
    Y. Gao, C.S. Li, J.J. Liu, Phys. Rev. D 72, 114020 (2005). arXiv:hep-ph/0501229 ADSCrossRefGoogle Scholar
  24. 24.
    A. Idilbi, X.d. Ji, F. Yuan Phys. Lett. B 625, 253 (2005). arXiv:hep-ph/0507196 ADSCrossRefGoogle Scholar
  25. 25.
    S. Mantry, F. Petriello, Phys. Rev. D 81, 093007 (2010). arXiv:0911.4135 [hep-ph] ADSCrossRefGoogle Scholar
  26. 26.
    T. Becher, M. Neubert, Phys. Rev. Lett. 97, 082001 (2006). arXiv:hep-ph/0605050 ADSCrossRefGoogle Scholar
  27. 27.
    G. Sterman, Nucl. Phys. B 281, 310 (1987) ADSCrossRefGoogle Scholar
  28. 28.
    S. Catani, L. Trentadue, Nucl. Phys. B 327, 323 (1989) ADSCrossRefGoogle Scholar
  29. 29.
    T. Becher, M. Neubert, G. Xu, J. High Energy Phys. 0807, 030 (2008). arXiv:0710.0680 [hep-ph] ADSCrossRefGoogle Scholar
  30. 30.
    A.V. Manohar, Phys. Rev. D 68, 114019 (2003). arXiv:hep-ph/0309176 ADSCrossRefGoogle Scholar
  31. 31.
    T. Becher, M. Neubert, B.D. Pecjak, J. High Energy Phys. 0701, 076 (2007). arXiv:hep-ph/0607228 ADSCrossRefGoogle Scholar
  32. 32.
    M. Beneke, T. Feldmann, Phys. Lett. B 553, 267 (2003). hep-ph/0211358 ADSzbMATHCrossRefGoogle Scholar
  33. 33.
    A.V. Efremov, A.V. Radyushkin, Theor. Math. Phys. 44, 774 (1981), Teor. Mat. Fiz. 44, 327 (1980) CrossRefGoogle Scholar
  34. 34.
    J.C. Collins, D.E. Soper, Nucl. Phys. B 193, 381 (1981), Nucl. Phys. B 213, 545 (1983), Erratum ADSCrossRefGoogle Scholar
  35. 35.
    J.C. Collins, D.E. Soper, Nucl. Phys. B 194, 445 (1982) ADSCrossRefGoogle Scholar
  36. 36.
    T. Becher, M. Neubert, Phys. Rev. Lett. 102, 162001 (2009). arXiv:0901.0722 [hep-ph] ADSCrossRefGoogle Scholar
  37. 37.
    T. Becher, M. Neubert, J. High Energy Phys. 0906, 081 (2009). arXiv:0903.1126 [hep-ph] MathSciNetADSCrossRefGoogle Scholar
  38. 38.
    T. Becher, R.J. Hill, M. Neubert, Phys. Rev. D 69, 054017 (2004). arXiv:hep-ph/0308122 ADSCrossRefGoogle Scholar
  39. 39.
    M. Beneke, T. Feldmann, Nucl. Phys. B 685, 249 (2004). arXiv:hep-ph/0311335 ADSCrossRefGoogle Scholar
  40. 40.
    B.O. Lange, M. Neubert, Nucl. Phys. B 690, 249 (2004). arXiv:hep-ph/0311345. Nucl. Phys. B 723, 201 (2005), Erratum ADSzbMATHCrossRefGoogle Scholar
  41. 41.
    J.G.M. Gatheral, Phys. Lett. B 133, 90 (1983) MathSciNetADSCrossRefGoogle Scholar
  42. 42.
    J. Frenkel, J.C. Taylor, Nucl. Phys. B 246, 231 (1984) ADSCrossRefGoogle Scholar
  43. 43.
    I.O. Cherednikov, N.G. Stefanis, arXiv:0911.1031 [hep-ph]
  44. 44.
    T. Becher, M. Neubert, D. Wilhelm, in preparation Google Scholar
  45. 45.
    J.C. Collins, Adv. Ser. Dir. High Energy Phys. 5, 573 (1989). arXiv:hep-ph/0312336 Google Scholar
  46. 46.
    J.C. Collins, F. Hautmann, Phys. Lett. B 472, 129 (2000). arXiv:hep-ph/9908467 ADSCrossRefGoogle Scholar
  47. 47.
    J.C. Collins, F. Hautmann, J. High Energy Phys. 0103, 016 (2001). arXiv:hep-ph/0009286 ADSCrossRefGoogle Scholar
  48. 48.
    X.d. Ji, J.p. Ma, F. Yuan, Phys. Rev. D 71, 034005 (2005) arXiv:hep-ph/0404183 ADSCrossRefGoogle Scholar
  49. 49.
    J. Collins, PoS LC2008, 028 (2008). arXiv:0808.2665 [hep-ph]
  50. 50.
    J.C. Collins, A. Metz, Phys. Rev. Lett. 93, 252001 (2004). arXiv:hep-ph/0408249 ADSCrossRefGoogle Scholar
  51. 51.
    V. Ahrens, T. Becher, M. Neubert, L.L. Yang, Phys. Rev. D 79, 033013 (2009). arXiv:0808.3008 [hep-ph] ADSCrossRefGoogle Scholar
  52. 52.
    V. Ahrens, T. Becher, M. Neubert, L.L. Yang, Eur. Phys. J. C 62, 333 (2009). arXiv:0809.4283 [hep-ph] ADSCrossRefGoogle Scholar
  53. 53.
    L. Magnea, G.F. Sterman, Phys. Rev. D 42, 4222 (1990) ADSCrossRefGoogle Scholar
  54. 54.
    M. Neubert, Eur. Phys. J. C 40, 165 (2005). arXiv:hep-ph/0408179 ADSCrossRefGoogle Scholar
  55. 55.
    S. Frixione, P. Nason, G. Ridolfi, Nucl. Phys. B 542, 311 (1999). arXiv:hep-ph/9809367 ADSCrossRefGoogle Scholar
  56. 56.
    M. Beneke, Phys. Rep. 317, 1 (1999). arXiv:hep-ph/9807443 ADSCrossRefGoogle Scholar
  57. 57.
    J.C. Collins, D.E. Soper, Nucl. Phys. B 197, 446 (1982) ADSCrossRefGoogle Scholar
  58. 58.
    M. Beneke, V.M. Braun, Nucl. Phys. B 454, 253 (1995). arXiv:hep-ph/9506452 ADSCrossRefGoogle Scholar
  59. 59.
    R.K. Ellis, S. Veseli, Nucl. Phys. B 511, 649 (1998). arXiv:hep-ph/9706526 ADSCrossRefGoogle Scholar
  60. 60.
    S. Moch, J.A.M. Vermaseren, A. Vogt, Nucl. Phys. B 688, 101 (2004). arXiv:hep-ph/0403192 MathSciNetADSzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag / Società Italiana di Fisica 2011

Authors and Affiliations

  1. 1.Institut für Theoretische PhysikUniversität BernBernSwitzerland
  2. 2.Institut für Physik (THEP)Johannes Gutenberg-UniversitätMainzGermany
  3. 3.Institut für Theoretische PhysikRuprecht-Karls-Universität HeidelbergHeidelbergGermany

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