Advertisement

Non-linear evolution in QCD at high-energy beyond leading order

  • B. Ducloué
  • E. Iancu
  • A. H. Mueller
  • G. Soyez
  • D. N. TriantafyllopoulosEmail author
Open Access
Regular Article - Theoretical Physics
  • 35 Downloads

Abstract

The standard formulation of the high-energy evolution in perturbative QCD, based on the Balitsky-Kovchegov equation, is known to suffer from severe instabilities associated with radiative corrections enhanced by double transverse logarithms, which occur in all orders starting with the next-to-leading one. Over the last years, several methods have been devised to resum such corrections by enforcing the time-ordering of the successive gluon emissions. We observe that the instability problem is not fully cured by these methods: various prescriptions for performing the resummation lead to very different physical results and thus lack of predictive power. We argue that this problem can be avoided by using the rapidity of the dense target (which corresponds to Bjorken x) instead of that of the dilute projectile as an evolution time. This automatically ensures the proper time-ordering and also allows for a direct physical interpretation of the results. We explicitly perform this change of variables at NLO. We observe the emergence of a new class of double logarithmic corrections, potentially leading to instabilities, which are however less severe, since disfavoured by the typical BK evolution for “dilute-dense” scattering. We propose several prescriptions for resumming these new double-logarithms to all orders and find only little scheme dependence: different prescriptions lead to results which are consistent to each other to the accuracy of interest. We restore full NLO accuracy by completing one of the resummed equations (non-local in rapidity) with the remaining NLO corrections.

Keywords

Perturbative QCD Resummation 

Notes

Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

References

  1. [1]
    I. Balitsky, Operator expansion for high-energy scattering, Nucl. Phys. B 463 (1996) 99 [hep-ph/9509348] [INSPIRE].
  2. [2]
    J. Jalilian-Marian, A. Kovner, A. Leonidov and H. Weigert, The BFKL equation from the Wilson renormalization group, Nucl. Phys. B 504 (1997) 415 [hep-ph/9701284] [INSPIRE].
  3. [3]
    J. Jalilian-Marian, A. Kovner, A. Leonidov and H. Weigert, The Wilson renormalization group for low x physics: Towards the high density regime, Phys. Rev. D 59 (1998) 014014 [hep-ph/9706377] [INSPIRE].
  4. [4]
    A. Kovner, J.G. Milhano and H. Weigert, Relating different approaches to nonlinear QCD evolution at finite gluon density, Phys. Rev. D 62 (2000) 114005 [hep-ph/0004014] [INSPIRE].
  5. [5]
    E. Iancu, A. Leonidov and L.D. McLerran, Nonlinear gluon evolution in the color glass condensate. 1., Nucl. Phys. A 692 (2001) 583 [hep-ph/0011241] [INSPIRE].
  6. [6]
    E. Iancu, A. Leonidov and L.D. McLerran, The Renormalization group equation for the color glass condensate, Phys. Lett. B 510 (2001) 133 [hep-ph/0102009] [INSPIRE].
  7. [7]
    E. Ferreiro, E. Iancu, A. Leonidov and L. McLerran, Nonlinear gluon evolution in the color glass condensate: II, Nucl. Phys. A 703 (2002) 489 [hep-ph/0109115] [INSPIRE].
  8. [8]
    Y.V. Kovchegov, Small x F 2 structure function of a nucleus including multiple Pomeron exchanges, Phys. Rev. D 60 (1999) 034008 [hep-ph/9901281] [INSPIRE].
  9. [9]
    I. Balitsky and G.A. Chirilli, Next-to-leading order evolution of color dipoles, Phys. Rev. D 77 (2008) 014019 [arXiv:0710.4330] [INSPIRE].
  10. [10]
    I. Balitsky and G.A. Chirilli, Rapidity evolution of Wilson lines at the next-to-leading order, Phys. Rev. D 88 (2013) 111501 [arXiv:1309.7644] [INSPIRE].
  11. [11]
    A. Kovner, M. Lublinsky and Y. Mulian, Jalilian-Marian, Iancu, McLerran, Weigert, Leonidov, Kovner evolution at next to leading order, Phys. Rev. D 89 (2014) 061704 [arXiv:1310.0378] [INSPIRE].
  12. [12]
    A. Kovner, M. Lublinsky and Y. Mulian, NLO JIMWLK evolution unabridged, JHEP 08 (2014) 114 [arXiv:1405.0418] [INSPIRE].ADSCrossRefGoogle Scholar
  13. [13]
    M. Lublinsky and Y. Mulian, High Energy QCD at NLO: from light-cone wave function to JIMWLK evolution, JHEP 05 (2017) 097 [arXiv:1610.03453] [INSPIRE].ADSCrossRefGoogle Scholar
  14. [14]
    T. Lappi and H. Mäntysaari, Direct numerical solution of the coordinate space Balitsky-Kovchegov equation at next to leading order, Phys. Rev. D 91 (2015) 074016 [arXiv:1502.02400] [INSPIRE].
  15. [15]
    L.D. McLerran and R. Venugopalan, Computing quark and gluon distribution functions for very large nuclei, Phys. Rev. D 49 (1994) 2233 [hep-ph/9309289] [INSPIRE].
  16. [16]
    L.D. McLerran and R. Venugopalan, Gluon distribution functions for very large nuclei at small transverse momentum, Phys. Rev. D 49 (1994) 3352 [hep-ph/9311205] [INSPIRE].
  17. [17]
    D.N. Triantafyllopoulos, The Energy dependence of the saturation momentum from RG improved BFKL evolution, Nucl. Phys. B 648 (2003) 293 [hep-ph/0209121] [INSPIRE].
  18. [18]
    E. Avsar, A.M. Stasto, D.N. Triantafyllopoulos and D. Zaslavsky, Next-to-leading and resummed BFKL evolution with saturation boundary, JHEP 10 (2011) 138 [arXiv:1107.1252] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  19. [19]
    V.S. Fadin, M.I. Kotsky and R. Fiore, Gluon Reggeization in QCD in the next-to-leading order, Phys. Lett. B 359 (1995) 181 [INSPIRE].
  20. [20]
    V.S. Fadin, M.I. Kotsky and L.N. Lipatov, One-loop correction to the BFKL kernel from two gluon production, Phys. Lett. B 415 (1997) 97 [INSPIRE].
  21. [21]
    G. Camici and M. Ciafaloni, NonAbelian \( q\overline{q} \) contributions to small x anomalous dimensions, Phys. Lett. B 386 (1996) 341 [hep-ph/9606427] [INSPIRE].
  22. [22]
    G. Camici and M. Ciafaloni, Irreducible part of the next-to-leading BFKL kernel, Phys. Lett. B 412 (1997) 396 [Erratum ibid. B 417 (1998) 390] [hep-ph/9707390] [INSPIRE].
  23. [23]
    V.S. Fadin and L.N. Lipatov, BFKL Pomeron in the next-to-leading approximation, Phys. Lett. B 429 (1998) 127 [hep-ph/9802290] [INSPIRE].
  24. [24]
    M. Ciafaloni and G. Camici, Energy scale(s) and next-to-leading BFKL equation, Phys. Lett. B 430 (1998) 349 [hep-ph/9803389] [INSPIRE].
  25. [25]
    L.N. Lipatov, Reggeization of the Vector Meson and the Vacuum Singularity in Nonabelian Gauge Theories, Sov. J. Nucl. Phys. 23 (1976) 338 [INSPIRE].Google Scholar
  26. [26]
    E.A. Kuraev, L.N. Lipatov and V.S. Fadin, The Pomeranchuk Singularity in Nonabelian Gauge Theories, Sov. Phys. JETP 45 (1977) 199 [INSPIRE].ADSMathSciNetGoogle Scholar
  27. [27]
    I.I. Balitsky and L.N. Lipatov, The Pomeranchuk Singularity in Quantum Chromodynamics, Sov. J. Nucl. Phys. 28 (1978) 822 [INSPIRE].Google Scholar
  28. [28]
    J. Kwiecinski, A.D. Martin and A.M. Stasto, A Unified BFKL and GLAP description of F 2 data, Phys. Rev. D 56 (1997) 3991 [hep-ph/9703445] [INSPIRE].
  29. [29]
    G.P. Salam, A Resummation of large subleading corrections at small x, JHEP 07 (1998) 019 [hep-ph/9806482] [INSPIRE].
  30. [30]
    M. Ciafaloni and D. Colferai, The BFKL equation at next-to-leading level and beyond, Phys. Lett. B 452 (1999) 372 [hep-ph/9812366] [INSPIRE].
  31. [31]
    M. Ciafaloni, D. Colferai and G.P. Salam, Renormalization group improved small x equation, Phys. Rev. D 60 (1999) 114036 [hep-ph/9905566] [INSPIRE].
  32. [32]
    M. Ciafaloni, D. Colferai, G.P. Salam and A.M. Stasto, Renormalization group improved small x Greens function, Phys. Rev. D 68 (2003) 114003 [hep-ph/0307188] [INSPIRE].
  33. [33]
    A. Sabio Vera, AnAll-polesapproximation to collinear resummations in the Regge limit of perturbative QCD, Nucl. Phys. B 722 (2005) 65 [hep-ph/0505128] [INSPIRE].
  34. [34]
    G. Beuf, Improving the kinematics for low-x QCD evolution equations in coordinate space, Phys. Rev. D 89 (2014) 074039 [arXiv:1401.0313] [INSPIRE].
  35. [35]
    E. Iancu, J.D. Madrigal, A.H. Mueller, G. Soyez and D.N. Triantafyllopoulos, Resumming double logarithms in the QCD evolution of color dipoles, Phys. Lett. B 744 (2015) 293 [arXiv:1502.05642] [INSPIRE].
  36. [36]
    E. Iancu, J.D. Madrigal, A.H. Mueller, G. Soyez and D.N. Triantafyllopoulos, Collinearly-improved BK evolution meets the HERA data, Phys. Lett. B 750 (2015) 643 [arXiv:1507.03651] [INSPIRE].
  37. [37]
    J.L. Albacete, Resummation of double collinear logs in BK evolution versus HERA data, Nucl. Phys. A 957 (2017) 71 [arXiv:1507.07120] [INSPIRE].
  38. [38]
    T. Lappi and H. Mäntysaari, Next-to-leading order Balitsky-Kovchegov equation with resummation, Phys. Rev. D 93 (2016) 094004 [arXiv:1601.06598] [INSPIRE].
  39. [39]
    A.M. Stasto, K.J. Golec-Biernat and J. Kwiecinski, Geometric scaling for the total γ p cross-section in the low x region, Phys. Rev. Lett. 86 (2001) 596 [hep-ph/0007192] [INSPIRE].
  40. [40]
    E. Iancu, K. Itakura and L. McLerran, Geometric scaling above the saturation scale, Nucl. Phys. A 708 (2002) 327 [hep-ph/0203137] [INSPIRE].
  41. [41]
    A.H. Mueller and D.N. Triantafyllopoulos, The Energy dependence of the saturation momentum, Nucl. Phys. B 640 (2002) 331 [hep-ph/0205167] [INSPIRE].
  42. [42]
    S. Munier and R.B. Peschanski, Geometric scaling as traveling waves, Phys. Rev. Lett. 91 (2003) 232001 [hep-ph/0309177] [INSPIRE].
  43. [43]
    S. Munier and R.B. Peschanski, Traveling wave fronts and the transition to saturation, Phys. Rev. D 69 (2004) 034008 [hep-ph/0310357] [INSPIRE].
  44. [44]
    G.P. Salam, An Introduction to leading and next-to-leading BFKL, Acta Phys. Polon. B 30 (1999) 3679 [hep-ph/9910492] [INSPIRE].
  45. [45]
    L.V. Gribov, E.M. Levin and M.G. Ryskin, Semihard Processes in QCD, Phys. Rept. 100 (1983) 1 [INSPIRE].ADSCrossRefGoogle Scholar
  46. [46]
    E. Iancu, A.H. Mueller and S. Munier, Universal behavior of QCD amplitudes at high energy from general tools of statistical physics, Phys. Lett. B 606 (2005) 342 [hep-ph/0410018] [INSPIRE].
  47. [47]
    E. Levin and K. Tuchin, Solution to the evolution equation for high parton density QCD, Nucl. Phys. B 573 (2000) 833 [hep-ph/9908317] [INSPIRE].
  48. [48]
    M. Alvioli, G. Soyez and D.N. Triantafyllopoulos, Testing the Gaussian Approximation to the JIMWLK Equation, Phys. Rev. D 87 (2013) 014016 [arXiv:1212.1656] [INSPIRE].
  49. [49]
    A.H. Mueller, Gluon distributions and color charge correlations in a saturated light cone wave function, Nucl. Phys. B 643 (2002) 501 [hep-ph/0206216] [INSPIRE].
  50. [50]
    E. Iancu and D.N. Triantafyllopoulos, JIMWLK evolution in the Gaussian approximation, JHEP 04 (2012) 025 [arXiv:1112.1104] [INSPIRE].ADSCrossRefGoogle Scholar
  51. [51]
    A.H. Mueller and G.P. Salam, Large multiplicity fluctuations and saturation effects in onium collisions, Nucl. Phys. B 475 (1996) 293 [hep-ph/9605302] [INSPIRE].
  52. [52]
    E. Iancu and A.H. Mueller, Rare fluctuations and the high-energy limit of the S matrix in QCD, Nucl. Phys. A 730 (2004) 494 [hep-ph/0309276] [INSPIRE].
  53. [53]
    I. Balitsky, Quark contribution to the small-x evolution of color dipole, Phys. Rev. D 75 (2007) 014001 [hep-ph/0609105] [INSPIRE].
  54. [54]
    Y.V. Kovchegov and H. Weigert, Triumvirate of Running Couplings in Small-x Evolution, Nucl. Phys. A 784 (2007) 188 [hep-ph/0609090] [INSPIRE].
  55. [55]
    B. Andersson, G. Gustafson and J. Samuelsson, The Linked dipole chain model for DIS, Nucl. Phys. B 467 (1996) 443 [INSPIRE].
  56. [56]
    T. Liou, A.H. Mueller and B. Wu, Radiative p -broadening of high-energy quarks and gluons in QCD matter, Nucl. Phys. A 916 (2013) 102 [arXiv:1304.7677] [INSPIRE].
  57. [57]
    E. Iancu, The non-linear evolution of jet quenching, JHEP 10 (2014) 095 [arXiv:1403.1996] [INSPIRE].ADSCrossRefGoogle Scholar
  58. [58]
    I. Balitsky and G.A. Chirilli, NLO evolution of color dipoles in N = 4 SYM, Nucl. Phys. B 822 (2009) 45 [arXiv:0903.5326] [INSPIRE].
  59. [59]
    Y. Hatta, E. Iancu, A.H. Mueller and D.N. Triantafyllopoulos, Resumming double non-global logarithms in the evolution of a jet, JHEP 02 (2018) 075 [arXiv:1710.06722] [INSPIRE].ADSCrossRefGoogle Scholar
  60. [60]
    D.A. Ross, The Effect of higher order corrections to the BFKL equation on the perturbative Pomeron, Phys. Lett. B 431 (1998) 161 [hep-ph/9804332] [INSPIRE].

Copyright information

© The Author(s) 2019

Authors and Affiliations

  • B. Ducloué
    • 1
  • E. Iancu
    • 1
  • A. H. Mueller
    • 2
  • G. Soyez
    • 1
  • D. N. Triantafyllopoulos
    • 3
    Email author
  1. 1.Institut de physique théoriqueUniversité Paris Saclay, CNRS, CEAGif-sur-YvetteFrance
  2. 2.Department of PhysicsColumbia UniversityNew YorkU.S.A.
  3. 3.European Centre for Theoretical Studies in Nuclear Physics and Related Areas (ECT*), and Fondazione Bruno KesslerVillazzano (TN)Italy

Personalised recommendations