Advertisement

Decoupling of the DGLAP evolution equations at next-to-next-to-leading order (NNLO) at low-x

  • G. R. BorounEmail author
  • B. Rezaei
Regular Article - Theoretical Physics

Abstract

We present a set of formulas to extract two second-order independent differential equations for the gluon and singlet distribution functions. Our results extend from the LO up to NNLO DGLAP evolution equations with respect to the hard-Pomeron behavior at low-x. In this approach, both singlet quarks and gluons have the same high-energy behavior at low-x. We solve the independent DGLAP evolution equations for the functions \(F_{2}^{s}(x,Q^{2})\) and G(x,Q 2) as a function of their initial parameterization at the starting scale \(Q_{0}^{2}\). The results not only give striking support to the hard-Pomeron description of the low-x behavior, but give a rather clean test of perturbative QCD showing an increase of the gluon distribution and singlet structure functions as x decreases. We compared our numerical results with the published BDM (Block et al. Phys. Rev. D 77:094003 (2008)) gluon and singlet distributions, starting from their initial values at \(Q_{0}^{2}=1\ \mathrm{GeV}^{2}\).

Keywords

Structure Function Gluon Distribution Splitting Function Quark Distribution Proton Structure Function 
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.

Notes

Acknowledgements

G.R. Boroun would like to thank the anonymous referee of the paper for his/her careful reading of the manuscript and for the productive discussions.

References

  1. 1.
    Yu.L. Dokshitzer, Sov. Phys. JETP 6, 641 (1977) ADSGoogle Scholar
  2. 2.
    G. Altarelli, G. Parisi, Nucl. Phys. B 126, 298 (1997) ADSCrossRefGoogle Scholar
  3. 3.
    V.N. Gribov, L.N. Lipatov, Sov. J. Nucl. Phys. 28, 822 (1978) Google Scholar
  4. 4.
    L.F. Abbott, W.B. Atwood, A.M. Barnett, Phys. Rev. D 22, 582 (1980) ADSCrossRefGoogle Scholar
  5. 5.
    A.M. Cooper-Sarkar, R.C.E. Devenish, A. DeRoeck, Int. J. Mod. Phys. A 13, 3385 (1998) ADSCrossRefGoogle Scholar
  6. 6.
    M. Cluck, E. Reya, A. Vogt, Z. Phys. C 48, 471 (1990) CrossRefGoogle Scholar
  7. 7.
    A.D. Martin, W.J. Stirling, R.S. Thorne, Phys. Lett. B 636, 259 (2006) ADSCrossRefGoogle Scholar
  8. 8.
    A.D. Martin, W.J. Stirling, R.S. Thorne, G. Watt, Eur. Phys. J. C 63, 189 (2009) ADSCrossRefGoogle Scholar
  9. 9.
    A.D. Martin, W.J. Stirling, R.S. Thorne, G. Watt, arXiv:1301.6754 [hep-ph]
  10. 10.
    M. Gluck, P. Jimenez-Delgado, E. Reya, Eur. Phys. J. C 53, 355 (2008) ADSCrossRefGoogle Scholar
  11. 11.
    M. Gluck, P. Jimenez-Delgado, E. Reya, Phys. Rev. D 79, 074023 (2009) ADSCrossRefGoogle Scholar
  12. 12.
    K. Prytz, Phys. Lett. B 311, 286 (1993) ADSCrossRefGoogle Scholar
  13. 13.
    K. Prytz, Phys. Lett. B 332, 393 (1994) ADSCrossRefGoogle Scholar
  14. 14.
    M.B.G. Ducati, V.P.B. Goncalves, Phys. Lett. B 390, 401 (1997) ADSCrossRefGoogle Scholar
  15. 15.
    A.V. Kotikov, G. Parente, Phys. Lett. B 379, 195 (1996) ADSCrossRefGoogle Scholar
  16. 16.
    P. Desgrolard, A. Lengyel, E. Martynov, J. High Energy Phys. 02, 029 (2002) ADSCrossRefGoogle Scholar
  17. 17.
    A. Donnachie, P.V. Landshoff, Phys. Lett. B 533, 277 (2002) ADSCrossRefGoogle Scholar
  18. 18.
    J.R. Cudell, A. Donnachie, P.V. Landshoff, Phys. Lett. B 448, 281 (1999) ADSCrossRefGoogle Scholar
  19. 19.
    J. Kwiecinski, arXiv:hep-ph/9607221 (1996)
  20. 20.
    P.D. Collins, An Introduction to Regge Theory an High-Energy Physics (Cambridge University Press, Cambridge, 1997) Google Scholar
  21. 21.
    M. Bertini et al., Riv. Nuovo Cimento 19, 1 (1996) MathSciNetGoogle Scholar
  22. 22.
    A. Donnachie, P.V. Landshoff, Z. Phys. C 61, 139 (1994) ADSCrossRefGoogle Scholar
  23. 23.
    A. Donnachie, P.V. Landshoff, Phys. Lett. B 518, 63 (2001) ADSCrossRefGoogle Scholar
  24. 24.
    A. Donnachie, P.V. Landshoff, Phys. Lett. B 550, 160 (2002) ADSCrossRefGoogle Scholar
  25. 25.
    R.D. Ball, P.V. Landshoff, arXiv:hep-ph/9912445
  26. 26.
    G. Altarelli, G. Parisi, Nucl. Phys. B 126, 298 (1977) ADSCrossRefGoogle Scholar
  27. 27.
    W. Furmanski, R. Petronzio, Phys. Lett. B 97, 437 (1980) ADSCrossRefGoogle Scholar
  28. 28.
    R.K. Ellis, W.J. Stirling, B.R. Webber, QCD and Collider Physics (Cambridge University Press, Cambridge, 1996) CrossRefGoogle Scholar
  29. 29.
    W.L. van Neerven, A. Vogt, Nucl. Phys. B 588, 345 (2000) ADSCrossRefGoogle Scholar
  30. 30.
    W.L. van Neerven, A. Vogt, Nucl. Phys. B 568, 263 (2000) ADSCrossRefGoogle Scholar
  31. 31.
    S. Moch, J. Vermaseren, A. Vogt, Nucl. Phys. B 688, 101 (2004) MathSciNetADSzbMATHCrossRefGoogle Scholar
  32. 32.
    S. Moch, J. Vermaseren, A. Vogt, Nucl. Phys. B 691, 129 (2004) MathSciNetADSzbMATHCrossRefGoogle Scholar
  33. 33.
    A. Retey, J. Vermaseren, Nucl. Phys. B 604, 281 (2001) ADSCrossRefGoogle Scholar
  34. 34.
    M.M. Block, L. Durand, D.W. Mckay, Phys. Rev. D 77, 094003 (2008) ADSCrossRefGoogle Scholar
  35. 35.
    E.L. Berger, M.M. Block, C.-I. Tan, Phys. Rev. Lett. 98, 242001 (2007) ADSCrossRefGoogle Scholar
  36. 36.
    V. Chekanov et al. (ZEUS Collaboration), Eur. Phys. J. C 21, 443 (2001) ADSCrossRefGoogle Scholar
  37. 37.
    M. Devee, B. Baishya, J.K. Sarma, Eur. Phys. J. C 72, 2036 (2012) ADSCrossRefGoogle Scholar
  38. 38.
    S. Weinzierl, arXiv:hep-ph/0203112
  39. 39.
    M.M. Block, L. Durand, P. Ha, D.W. Mckay, arXiv:1004.1440 [hep-ph] (2010)
  40. 40.
    M.M. Block, L. Durand, P. Ha, D.W. Mckay, arXiv:1005.2556 [hep-ph] (2010)
  41. 41.
    G.R. Boroun, J. Exp. Theor. Phys. 106, 700 (2008) ADSCrossRefGoogle Scholar
  42. 42.
    B. Rezaei, G.R. Boroun, J. Exp. Theor. Phys. 112, 381 (2011) ADSCrossRefGoogle Scholar
  43. 43.
    G.R. Boroun, B. Rezaei, Eur. Phys. J. C 72, 2221 (2012) ADSCrossRefGoogle Scholar
  44. 44.
    W.L. van Neerven, E.B. Zijlstra, Phys. Lett. B 272, 127 (1991) ADSCrossRefGoogle Scholar
  45. 45.
    E.B. Zijlstra, W.L. van Neerven, Phys. Lett. B 273, 476 (1991) ADSCrossRefGoogle Scholar
  46. 46.
    E.B. Zijlstra, W.L. van Neerven, Phys. Lett. B 297, 377 (1992) ADSCrossRefGoogle Scholar
  47. 47.
    E.B. Zijlstra, W.L. van Neerven, Phys. Lett. B 383, 525 (1992) Google Scholar
  48. 48.
    R. Hamberg, W.L. van Neerven, T. Matsuura, Nucl. Phys. B 359, 343 (1991) ADSCrossRefGoogle Scholar
  49. 49.
    R. Hamberg, W.L. van Neerven, T. Matsuura, Nucl. Phys. B 644, 403 (2002), Erratum ADSCrossRefGoogle Scholar
  50. 50.
    R.V. Harlander, W.B. Kilgore, Phys. Rev. Lett. 88, 201801 (2002). hep-ph/0201206 ADSCrossRefGoogle Scholar
  51. 51.
    B.G. Shaikhatdenov, A.V. Kotikov, V.G. Krivokhizhin, G. Parente, Phys. Rev. D 81, 034008 (2010) ADSCrossRefGoogle Scholar
  52. 52.
    A. Donnachie, P.V. Landshoff, Phys. Lett. B 296, 257 (1992) ADSGoogle Scholar
  53. 53.
    P. Desgrolard, M. Giffon, E. Martynov, E. Predazzi, Eur. Phys. J. C 18, 555 (2001) ADSCrossRefGoogle Scholar
  54. 54.
    P. Desgrolard, M. Giffon, E. Martynov, Eur. Phys. J. C 7, 655 (1999) ADSCrossRefGoogle Scholar
  55. 55.
    A.D. Martin, M.G. Ryskin, G. Watt, arXiv:hep-ph/0406225
  56. 56.
    F.D. Aaron et al. (H1 Collaboration), Eur. Phys. J. C 71, 1579 (2011) ADSCrossRefGoogle Scholar
  57. 57.
    C. Adloff et al. (H1 Collaboration), Eur. Phys. J. C 21, 33 (2001) ADSCrossRefGoogle Scholar
  58. 58.
    R.G. Roberts, The Structure of the Proton (Cambridge University Press, Cambridge, 1990) CrossRefGoogle Scholar
  59. 59.
    B. Lampe, E. Reya, Phys. Rep. 332, 1 (2000) ADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg and Società Italiana di Fisica 2013

Authors and Affiliations

  1. 1.Physics DepartmentRazi UniversityKermanshahIran

Personalised recommendations