Skip to main content
SpringerLink
Log in

Generalized double-logarithmic large-x resummation in inclusive deep-inelastic scattering

  • Published:
Journal of High Energy Physics Aims and scope Submit manuscript

Abstract

We present all-order results for the highest three large-x logarithms of the splitting functions P qg and P gq and of the coefficient functions C ϕ,q, C 2,q and C L,g for structure functions in Higgs-and gauge-boson exchange DIS in massless perturbative QCD. The corresponding coefficients have been derived by studying the unfactorized partonic structure functions in dimensional regularization independently in terms of their iterative structure and in terms of the constraints imposed by the functional forms of the real-and virtual-emission contributions together with their Kinoshita-Lee-Nauenberg cancellations required by the mass-factorization theorem. The numerical resummation corrections are small for the splitting functions, but partly very large for the coefficient functions. The highest two (three for C L,g) logarithms can be resummed in a closed form in terms of new special functions recently introduced in the context of the resummation of the leading logarithms.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Particle Data Group collaboration, K. Nakamura et al., Review of particle physics, J. Phys. G 37 (2010) 075021 [SPIRES].

    ADS  Google Scholar 

  2. S.A. Larin, F.V. Tkachev and J.A.M. Vermaseren, The α s 3 correction to the Bjorken sum rule, Phys. Rev. Lett. 66 (1991) 862 [SPIRES].

    Article  ADS  Google Scholar 

  3. S.A. Larin and J.A.M. Vermaseren, The α s 3 corrections to the Bjorken sum rule for polarized electroproduction and to the Gross-Llewellyn Smith sum rule, Phys. Lett. B 259 (1991) 345 [SPIRES].

    ADS  Google Scholar 

  4. S.A. Larin, F.V. Tkachov and J.A.M. Vermaseren, The O(α s 3) QCD correction to the lowest moment of the longitudinal structure function in deep inelastic electron-nucleon scattering, Phys. Lett. B 272 (1991) 121 [SPIRES].

    ADS  Google Scholar 

  5. S.A. Larin, T. van Ritbergen and J.A.M. Vermaseren, The Next next-to-leading QCD approximation for nonsinglet moments of deep inelastic structure functions, Nucl. Phys. B 427 (1994) 41 [SPIRES].

    Article  ADS  Google Scholar 

  6. S.A. Larin, P. Nogueira, T. van Ritbergen and J.A.M. Vermaseren, The 3-loop QCD calculation of the moments of deep inelastic structure functions, Nucl. Phys. B 492 (1997) 338 [hep-ph/9605317] [SPIRES].

    ADS  Google Scholar 

  7. A. Retey and J.A.M. Vermaseren, Some higher moments of deep inelastic structure functions at next-to-next-to leading order of perturbative QCD, Nucl. Phys. B 604 (2001) 281 [hep-ph/0007294] [SPIRES].

    Article  ADS  Google Scholar 

  8. S. Moch, J.A.M. Vermaseren and A. Vogt, The three-loop splitting functions in QCD: The non-singlet case, Nucl. Phys. B 688 (2004) 101 [hep-ph/0403192] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  9. A. Vogt, S. Moch and J.A.M. Vermaseren, The three-loop splitting functions in QCD: The singlet case, Nucl. Phys. B 691 (2004) 129 [hep-ph/0404111] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  10. S. Moch, J.A.M. Vermaseren and A. Vogt, The Longitudinal Structure Function at the Third Order, Phys. Lett. B 606 (2005) 123 [hep-ph/0411112] [SPIRES].

    ADS  Google Scholar 

  11. J.A.M. Vermaseren, A. Vogt and S. Moch, The third-order QCD corrections to deep-inelastic scattering by photon exchange, Nucl. Phys. B 724 (2005) 3 [hep-ph/0504242] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  12. S. Moch, J.A.M. Vermaseren and A. Vogt, Third-order QCD corrections to the charged-current structure function F 3, Nucl. Phys. B 813 (2009) 220 [arXiv:0812.4168] [SPIRES].

    Article  Google Scholar 

  13. G.P. Korchemsky, Asymptotics of the Altarelli-Parisi-Lipatov Evolution Kernels of Parton Distributions, Mod. Phys. Lett. A 4 (1989) 1257 [SPIRES].

    ADS  Google Scholar 

  14. Y.L. Dokshitzer, G. Marchesini and G.P. Salam, Revisiting parton evolution and the large-x limit, Phys. Lett. B 634 (2006) 504 [hep-ph/0511302] [SPIRES].

    ADS  Google Scholar 

  15. A. Vogt, Leading logarithmic large-x resummation of off-diagonal splitting functions and coefficient functions, Phys. Lett. B 691 (2010) 77 [arXiv:1005.1606] [SPIRES].

    ADS  Google Scholar 

  16. A. Daleo, A. Gehrmann-De Ridder, T. Gehrmann and G. Luisoni, Antenna subtraction at NNLO with hadronic initial states: initial-final configurations, JHEP 01 (2010) 118 [arXiv:0912.0374] [SPIRES].

    Article  ADS  Google Scholar 

  17. G. Soar, S. Moch, J.A.M. Vermaseren and A. Vogt, On Higgs-exchange DIS, physical evolution kernels and fourth-order splitting functions at large x, Nucl. Phys. B 832 (2010) 152 [arXiv:0912.0369] [SPIRES].

    Article  Google Scholar 

  18. G.F. Sterman, Summation of Large Corrections to Short Distance Hadronic Cross-Sections, Nucl. Phys. B 281 (1987) 310 [SPIRES].

    Article  ADS  Google Scholar 

  19. L. Magnea, All Order Summation And Two Loop Results For The Drell-Yan Cross-Section, Nucl. Phys. B 349 (1991) 703 [SPIRES].

    Article  ADS  Google Scholar 

  20. S. Catani and L. Trentadue, Resummation of the QCD Perturbative Series for Hard Processes, Nucl. Phys. B 327 (1989) 323 [SPIRES].

    Article  ADS  Google Scholar 

  21. S. Catani and L. Trentadue, Comment on QCD exponentiation at large x, Nucl. Phys. B 353 (1991) 183 [SPIRES].

    Article  ADS  Google Scholar 

  22. H. Contopanagos, E. Laenen and G.F. Sterman, Sudakov factorization and resummation, Nucl. Phys. B 484 (1997) 303 [hep-ph/9604313] [SPIRES].

    Article  ADS  Google Scholar 

  23. S. Catani, M.L. Mangano, P. Nason and L. Trentadue, The Resummation of Soft Gluon in Hadronic Collisions, Nucl. Phys. B 478 (1996) 273 [hep-ph/9604351] [SPIRES].

    Article  ADS  Google Scholar 

  24. A. Vogt, Next-to-next-to-leading logarithmic threshold resummation for deep-inelastic scattering and the Drell-Yan process, Phys. Lett. B 497 (2001) 228 [hep-ph/0010146] [SPIRES].

    ADS  Google Scholar 

  25. S. Moch, J.A.M. Vermaseren and A. Vogt, Non-singlet structure functions at three loops: Fermionic contributions, Nucl. Phys. B 646 (2002) 181 [hep-ph/0209100] [SPIRES].

    Article  ADS  Google Scholar 

  26. S. Moch, J.A.M. Vermaseren and A. Vogt, Higher-order corrections in threshold resummation, Nucl. Phys. B 726 (2005) 317 [hep-ph/0506288] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  27. P.A. Baikov and K.G. Chetyrkin, New four loop results in QCD, Nucl. Phys. Proc. Suppl. 160 (2006) 76 [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  28. A. Vogt, Parton Distributions: Progress and Challenges, proceedings of DIS 07, Munich Germany, April 2007 [arXiv:0707.4106] [SPIRES].

  29. M. Krämer, E. Laenen and M. Spira, Soft gluon radiation in Higgs boson production at the LHC, Nucl. Phys. B 511 (1998) 523 [hep-ph/9611272] [SPIRES].

    Article  ADS  Google Scholar 

  30. R. Akhoury, M.G. Sotiropoulos and G.F. Sterman, An operator expansion for the elastic limit, Phys. Rev. Lett. 81 (1998) 3819 [hep-ph/9807330] [SPIRES].

    Article  ADS  Google Scholar 

  31. R. Akhoury and M.G. Sotiropoulos, The large-x factorization of the longitudinal structure function, hep-ph/0304131 [SPIRES].

  32. E. Laenen, L. Magnea and G. Stavenga, On next-to-eikonal corrections to threshold resummation for the Drell-Yan and DIS cross sections, Phys. Lett. B 669 (2008) 173 [arXiv:0807.4412] [SPIRES].

    ADS  Google Scholar 

  33. E. Laenen, G. Stavenga and C.D. White, Path integral approach to eikonal and next-to-eikonal exponentiation, JHEP 03 (2009) 054 [arXiv:0811.2067] [SPIRES].

    Article  ADS  Google Scholar 

  34. E. Gardi, E. Laenen, G. Stavenga and C.D. White, Webs in multiparton scattering using the replica trick, JHEP 11 (2010) 155 [arXiv:1008.0098] [SPIRES].

    Article  ADS  Google Scholar 

  35. E. Laenen, L. Magnea, G. Stavenga and C.D. White, Next-to-eikonal corrections to soft gluon radiation: a diagrammatic approach, JHEP 01 (2011) 141 [arXiv:1010.1860] [SPIRES].

    Article  ADS  Google Scholar 

  36. G. Grunberg, Threshold resummation to any order in (1 − x), arXiv:0710.5693 [SPIRES].

  37. G. Grunberg and V. Ravindran, On threshold resummation beyond leading 1 − x order, JHEP 10 (2009) 055 [arXiv:0902.2702] [SPIRES].

    Article  ADS  Google Scholar 

  38. G. Grunberg, Large-x structure of physical evolution kernels in Deep Inelastic Scattering, Phys. Lett. B 687 (2010) 405 [arXiv:0911.4471] [SPIRES].

    ADS  Google Scholar 

  39. S. Moch and A. Vogt, Threshold Resummation of the Structure Function F L , JHEP 04 (2009) 081 [arXiv:0902.2342] [SPIRES].

    Article  ADS  Google Scholar 

  40. S. Moch and A. Vogt, On non-singlet physical evolution kernels and large-x coefficient functions in perturbative QCD, JHEP 11 (2009) 099 [arXiv:0909.2124] [SPIRES].

    Article  ADS  Google Scholar 

  41. W.L. van Neerven and A. Vogt, Non-singlet structure functions beyond the next-to-next-to leading order, Nucl. Phys. B 603 (2001) 42 [hep-ph/0103123] [SPIRES].

    Article  ADS  Google Scholar 

  42. A. Vogt, G. Soar, S. Moch and J.A.M Vermaseren, On higher-order flavour-singlet splitting and coefficient functions at large x, Nucl. Phys. Proc. Suppl. 205–206 (2010) 250 [arXiv:1008.0952 [SPIRES].

    Article  Google Scholar 

  43. W. Furmanski and R. Petronzio, Lepton-Hadron Processes Beyond Leading Order in Quantum Chromodynamics, Zeit. Phys. C 11 (1982) 293 [SPIRES].

    ADS  Google Scholar 

  44. S. Catani, Physical anomalous dimensions at small x, Z. Phys. C 75 (1997) 665 [hep-ph/9609263] [SPIRES].

    Google Scholar 

  45. J. Blumlein, V. Ravindran and W.L. van Neerven, On the Drell-Levy-Yan relation to O(α s 2 ), Nucl. Phys. B 586 (2000) 349 [hep-ph/0004172] [SPIRES].

    Article  ADS  Google Scholar 

  46. J.A.M. Vermaseren, New features of FORM, math-ph/0010025 [SPIRES].

  47. M. Tentyukov and J.A.M. Vermaseren, The multithreaded version of FORM, Comput. Phys. Commun. 181 (2010) 1419 [hep-ph/0702279] [SPIRES].

    Article  ADS  Google Scholar 

  48. W.L. van Neerven and E.B. Zijlstra, Order α s 2 contributions to the deep inelastic Wilson coefficient, Phys. Lett. B 272 (1991) 127 [SPIRES].

    ADS  Google Scholar 

  49. E.B. Zijlstra and W.L. van Neerven, Contribution of the second order gluonic Wilson coefficient to the deep inelastic structure function, Phys. Lett. B 273 (1991) 476 [SPIRES].

    ADS  Google Scholar 

  50. E.B. Zijlstra and W.L. van Neerven, Order α s 2 QCD corrections to the deep inelastic proton structure functions F2 and F(L), Nucl. Phys. B 383 (1992) 525 [SPIRES].

    Article  ADS  Google Scholar 

  51. S. Moch and J.A.M. Vermaseren, Deep inelastic structure functions at two loops, Nucl. Phys. B 573 (2000) 853 [hep-ph/9912355] [SPIRES].

    Article  ADS  Google Scholar 

  52. T. Kinoshita, Mass singularities of Feynman amplitudes, J. Math. Phys. 3 (1962) 650 [SPIRES].

    Article  ADS  MATH  Google Scholar 

  53. T.D. Lee and M. Nauenberg, Degenerate Systems and Mass Singularities, Phys. Rev. B 133 (1964) 1549 [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  54. H. Kluberg-Stern and J.B. Zuber, Ward Identities and Some Clues to the Renormalization of Gauge Invariant Operators, Phys. Rev. D 12 (1975) 467 [SPIRES].

    ADS  Google Scholar 

  55. J.C. Collins, A. Duncan and S.D. Joglekar, Trace and Dilatation Anomalies in Gauge Theories, Phys. Rev. D 16 (1977) 438 [SPIRES].

    ADS  Google Scholar 

  56. S. Moch, J.A.M. Vermaseren and A. Vogt, The quark form factor at higher orders, JHEP 08 (2005) 049 [hep-ph/0507039] [SPIRES].

    Article  ADS  Google Scholar 

  57. S. Moch, J.A.M. Vermaseren and A. Vogt, Three-loop results for quark and gluon form factors, Phys. Lett. B 625 (2005) 245 [hep-ph/0508055] [SPIRES].

    ADS  Google Scholar 

  58. S. Moch and A. Vogt, Higher-order threshold resummation for semi-inclusive e + e − annihilation, Phys. Lett. B 680 (2009) 239 [arXiv:0908.2746] [SPIRES].

    ADS  Google Scholar 

  59. T. Matsuura and W.L. van Neerven, Second order logarithmic corrections to the Drell-Yan cross-section , Z. Phys. C 38 (1988) 623 [SPIRES].

    ADS  Google Scholar 

  60. T. Matsuura, S.C. van der Marck and W.L. van Neerven, The Calculation of the Second Order Soft and Virtual Contributions to the Drell-Yan Cross-Section, Nucl. Phys. B 319 (1989) 570 [SPIRES].

    Article  ADS  Google Scholar 

  61. P.A. Baikov, K.G. Chetyrkin, A.V. Smirnov, V.A. Smirnov and M. Steinhauser, Quark and gluon form factors to three loops, Phys. Rev. Lett. 102 (2009) 212002 [arXiv:0902.3519] [SPIRES].

    Article  ADS  Google Scholar 

  62. R.N. Lee, A.V. Smirnov and V.A. Smirnov, Analytic Results for Massless Three-Loop Form Factors, JHEP 04 (2010) 020 [arXiv:1001.2887] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  63. T. Gehrmann, E.W.N. Glover, T. Huber, N. Ikizlerli and C. Studerus, Calculation of the quark and gluon form factors to three loops in QCD, JHEP 06 (2010) 094 [arXiv:1004.3653] [SPIRES].

    Article  ADS  Google Scholar 

  64. T. Gehrmann, E.W.N. Glover, T. Huber, N. Ikizlerli and C. Studerus, The quark and gluon form factors to three loops in QCD through to O(eps 2), JHEP 11 (2010) 102 [arXiv:1010.4478] [SPIRES].

    Article  ADS  Google Scholar 

  65. N.H. Christ, B. Hasslacher and A.H. Mueller, Light cone behavior of perturbation theory, Phys. Rev. D 6 (1972) 3543 [SPIRES].

    ADS  Google Scholar 

  66. A.J. Buras, Asymptotic Freedom in Deep Inelastic Processes in the Leading Order and Beyond, Rev. Mod. Phys. 52 (1980) 199 [SPIRES].

    Article  ADS  Google Scholar 

  67. E. Reya, Perturbative Quantum Chromodynamics, Phys. Rept. 69 (1981) 195 [SPIRES].

    Article  ADS  Google Scholar 

  68. M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions, Dover, New York U.S.A. (1965).

    Google Scholar 

  69. D. Braodhurst, private communication.

  70. L. Mankiewicz, M. Maul and E. Stein, Perturbative part of the non-singlet structure function F2 in the large-N (F) limit, Phys. Lett. B 404 (1997) 345 [hep-ph/9703356] [SPIRES].

    ADS  Google Scholar 

  71. P.A. Baikov, K.G. Chetyrkin and J.H. Kühn, Adler Function, Bjorken Sum Rule and the Crewther Relation to Order α s 4 in a General Gauge Theory, Phys. Rev. Lett. 104 (2010) 132004 [arXiv:1001.3606] [SPIRES].

    Article  ADS  Google Scholar 

  72. W.L. van Neerven and A. Vogt, NNLO evolution of deep-inelastic structure functions: The singlet case, Nucl. Phys. B 588 (2000) 345 [hep-ph/0006154] [SPIRES].

    Article  ADS  Google Scholar 

  73. W.L. van Neerven and A. Vogt, Improved approximations for the three-loop splitting functions in QCD, Phys. Lett. B 490 (2000) 111 [hep-ph/0007362] [SPIRES].

    ADS  Google Scholar 

  74. P.J. Rijken and W.L. van Neerven, O(α s 2) Contributions to the longitudinal fragmentation function in e + e − annihilation, Phys. Lett. B 386 (1996) 422 [hep-ph/9604436] [SPIRES].

    ADS  Google Scholar 

  75. P.J. Rijken and W.L. van Neerven, Higher order QCD corrections to the transverse and longitudinal fragmentation functions in electron positron annihilation, Nucl. Phys. B 487 (1997) 233 [hep-ph/9609377] [SPIRES].

    Article  ADS  Google Scholar 

  76. A. Mitov, S. Moch and A. Vogt, Next-to-next-to-leading order evolution of non-singlet fragmentation functions, Phys. Lett. B 638 (2006) 61 [hep-ph/0604053] [SPIRES].

    ADS  Google Scholar 

  77. S. Moch and A. Vogt, On Third-Order Timelike Splitting Functions and Top-Mediated Higgs Decay into Hadrons, Phys. Lett. B 659 (2008) 290 [arXiv:0709.3899] [SPIRES].

    ADS  Google Scholar 

  78. A. Almasy, S. Moch and A. Vogt, in preparation.

  79. A.V. Manohar, Deep inelastic scattering as x → 1 using soft-collinear effective theory, Phys. Rev. D 68 (2003) 114019 [hep-ph/0309176] [SPIRES].

    ADS  Google Scholar 

  80. J. Chay and C. Kim, Deep inelastic scattering near the endpoint in soft-collinear effective theory, Phys. Rev. D 75 (2007) 016003 [hep-ph/0511066] [SPIRES].

    ADS  Google Scholar 

  81. A. Idilbi, X.-d. Ji and F. Yuan, Resummation of Threshold Logarithms in Effective Field Theory For DIS, Drell-Yan and Higgs Production, Nucl. Phys. B 753 (2006) 42 [hep-ph/0605068] [SPIRES].

    Article  ADS  Google Scholar 

  82. T. Becher, M. Neubert and B.D. Pecjak, Factorization and momentum-space resummation in deep-inelastic scattering, JHEP 01 (2007) 076 [hep-ph/0607228] [SPIRES].

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, University of Liverpool, Liverpool, L69 3BX, United Kingdom

    A. A. Almasy, G. Soar & A. Vogt

Authors
  1. A. A. Almasy
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. G. Soar
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. Vogt
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. Vogt.

Additional information

ArXiv ePrint: 1012.3352

Rights and permissions

Reprints and permissions

About this article

Cite this article

Almasy, A.A., Soar, G. & Vogt, A. Generalized double-logarithmic large-x resummation in inclusive deep-inelastic scattering. J. High Energ. Phys. 2011, 30 (2011). https://doi.org/10.1007/JHEP03(2011)030

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/JHEP03(2011)030

Keywords

Search

Navigation