Advertisement

Three-gluon running coupling from lattice QCD at N f = 2 + 1 + 1: a consistency check of the OPE approach

  • Ph. Boucaud
  • M. Brinet
  • F. De Soto
  • V. Morenas
  • O. Pène
  • K. Petrov
  • J. Rodríguez-QuinteroEmail author
Open Access
Article

Abstract

We present a lattice calculation of the renormalized running coupling constant in symmetric (MOM) and asymmetric \( \left( {\widetilde{\mathrm{MOM}}} \right) \) momentum substraction schemes including u, d, s and c quarks in the sea. An Operator Product Expansion dominated by the dimension-two 〈A 2〉 condensate is used to fit the running of the coupling. We argue that the agreement in the predicted 〈A 2〉 condensate for both schemes is a strong support for the validity of the OPE approach and the effect of this non-gauge invariant condensate over the running of the strong coupling.

Keywords

Lattice QCD Sum Rules QCD 

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]
    M. Lüscher, R. Sommer, P. Weisz and U. Wolff, A Precise determination of the running coupling in the SU(3) Yang-Mills theory, Nucl. Phys. B 413 (1994) 481 [hep-lat/9309005] [INSPIRE].ADSCrossRefGoogle Scholar
  2. [2]
    G. de Divitiis, R. Frezzotti, M. Guagnelli and R. Petronzio, Nonperturbative determination of the running coupling constant in quenched SU(2), Nucl. Phys. B 433 (1995) 390 [hep-lat/9407028] [INSPIRE].ADSCrossRefGoogle Scholar
  3. [3]
    ALPHA collaboration, M. Della Morte et al., Computation of the strong coupling in QCD with two dynamical flavors, Nucl. Phys. B 713 (2005) 378 [hep-lat/0411025] [INSPIRE].ADSCrossRefGoogle Scholar
  4. [4]
    PACS-CS collaboration, S. Aoki et al., Precise determination of the strong coupling constant in N f = 2 + 1 lattice QCD with the Schrödinger functional scheme, JHEP 10 (2009) 053 [arXiv:0906.3906] [INSPIRE].Google Scholar
  5. [5]
    UKQCD collaboration, S. Booth et al., The Running coupling from SU(3) lattice gauge theory, Phys. Lett. B 294 (1992) 385 [hep-lat/9209008] [INSPIRE].ADSCrossRefGoogle Scholar
  6. [6]
    N. Brambilla, X. Garcia i Tormo, J. Soto and A. Vairo, Precision determination of \( {r_0}{\varLambda_{{\overline{M}S}}} \) from the QCD static energy, Phys. Rev. Lett. 105 (2010) 212001 [Erratum ibid. 108 (2012) 269903] [arXiv:1006.2066] [INSPIRE].ADSCrossRefGoogle Scholar
  7. [7]
    M. Gockeler et al., A Determination of the Lambda parameter from full lattice QCD, Phys. Rev. D 73 (2006) 014513 [hep-ph/0502212] [INSPIRE].ADSGoogle Scholar
  8. [8]
    HPQCD Collaboration, UKQCD collaboration, Q. Mason et al., Accurate determinations of α s from realistic lattice QCD, Phys. Rev. Lett. 95 (2005) 052002 [hep-lat/0503005] [INSPIRE].ADSCrossRefGoogle Scholar
  9. [9]
    K. Maltman, D. Leinweber, P. Moran and A. Sternbeck, The Realistic Lattice Determination of α s(M Z) Revisited, Phys. Rev. D 78 (2008) 114504 [arXiv:0807.2020] [INSPIRE].ADSGoogle Scholar
  10. [10]
    HPQCD collaboration, C. Davies et al., Update: Accurate Determinations of α s from Realistic Lattice QCD, Phys. Rev. D 78 (2008) 114507 [arXiv:0807.1687] [INSPIRE].ADSGoogle Scholar
  11. [11]
    HPQCD collaboration, I. Allison et al., High-Precision Charm-Quark Mass from Current-Current Correlators in Lattice and Continuum QCD, Phys. Rev. D 78 (2008) 054513 [arXiv:0805.2999] [INSPIRE].ADSGoogle Scholar
  12. [12]
    C. McNeile, C. Davies, E. Follana, K. Hornbostel and G. Lepage, High-Precision c and b Masses and QCD Coupling from Current-Current Correlators in Lattice and Continuum QCD, Phys. Rev. D 82 (2010) 034512 [arXiv:1004.4285] [INSPIRE].ADSGoogle Scholar
  13. [13]
    K. Jansen, M. Petschlies and C. Urbach, Charm Current-Current Correlators in Twisted Mass Lattice QCD, PoS(LATTICE 2011)234 [arXiv:1111.5252] [INSPIRE].
  14. [14]
    B. Alles et al., α s from the nonperturbatively renormalised lattice three gluon vertex, Nucl. Phys. B 502 (1997) 325 [hep-lat/9605033] [INSPIRE].ADSCrossRefGoogle Scholar
  15. [15]
    P. Boucaud, J. Leroy, J. Micheli, O. Pène and C. Roiesnel, Lattice calculation of α s in momentum scheme, JHEP 10 (1998) 017 [hep-ph/9810322] [INSPIRE].ADSCrossRefGoogle Scholar
  16. [16]
    P. Boucaud et al., Lattice calculation of 1/p 2 corrections to α s and of Lambda(QCD) in the MOM scheme, JHEP 04 (2000) 006 [hep-ph/0003020] [INSPIRE].ADSCrossRefGoogle Scholar
  17. [17]
    P. Boucaud et al., Consistent OPE description of gluon two point and three point Green function?, Phys. Lett. B 493 (2000) 315 [hep-ph/0008043] [INSPIRE].ADSCrossRefGoogle Scholar
  18. [18]
    P. Boucaud et al., Testing Landau gauge OPE on the lattice with aA 2condensate, Phys. Rev. D 63 (2001) 114003 [hep-ph/0101302] [INSPIRE].ADSGoogle Scholar
  19. [19]
    P. Boucaud et al., Preliminary calculation of α s from Green functions with dynamical quarks, JHEP 01 (2002) 046 [hep-ph/0107278] [INSPIRE].ADSCrossRefGoogle Scholar
  20. [20]
    A. Sternbeck et al., Running α s from Landau-gauge gluon and ghost correlations, PoS(Lattice 2007)256 [arXiv:0710.2965] [INSPIRE].
  21. [21]
    P. Boucaud et al., The Infrared Behaviour of the Pure Yang-Mills Green Functions, Few Body Syst. 53 (2012) 387 [arXiv:1109.1936] [INSPIRE].ADSCrossRefGoogle Scholar
  22. [22]
    F. De Soto and J. Rodriguez-Quintero, Notes on the determination of the Landau gauge OPE for the asymmetric three gluon vertex, Phys. Rev. D 64 (2001) 114003 [hep-ph/0105063] [INSPIRE].ADSGoogle Scholar
  23. [23]
    P. Boucaud et al., Artefacts andA 2power corrections: Revisiting the MOM Z ψ (p 2) and Z V, Phys. Rev. D 74 (2006) 034505 [hep-lat/0504017] [INSPIRE].ADSGoogle Scholar
  24. [24]
    P. Boucaud et al., Non-perturbative power corrections to ghost and gluon propagators, JHEP 01 (2006) 037 [hep-lat/0507005] [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    P. Boucaud et al., Ghost-gluon running coupling, power corrections and the determination of Lambda(MS-bar), Phys. Rev. D 79 (2009) 014508 [arXiv:0811.2059] [INSPIRE].ADSGoogle Scholar
  26. [26]
    ETM collaboration, B. Blossier et al., Ghost-gluon coupling, power corrections and \( {\varLambda_{{\overline{\mathrm{MS}}}}} \) from twisted-mass lattice QCD at N f = 2, Phys. Rev. D 82 (2010) 034510 [arXiv:1005.5290] [INSPIRE].ADSGoogle Scholar
  27. [27]
    B. Blossier et al., Renormalisation of quark propagators from twisted-mass lattice QCD at N f = 2, Phys. Rev. D 83 (2011) 074506 [arXiv:1011.2414] [INSPIRE].ADSGoogle Scholar
  28. [28]
    B. Blossier et al., Ghost-gluon coupling, power corrections and \( {\varLambda_{{\overline{\mathrm{M}}\mathrm{S}}}} \) from lattice QCD with a dynamical charm, Phys. Rev. D 85 (2012) 034503 [arXiv:1110.5829] [INSPIRE].ADSGoogle Scholar
  29. [29]
    B. Blossier et al., The Strong running coupling at τ and Z 0 mass scales from lattice QCD, Phys. Rev. Lett. 108 (2012) 262002 [arXiv:1201.5770] [INSPIRE].ADSCrossRefGoogle Scholar
  30. [30]
    B. Blossier et al., Testing the OPE Wilson coefficient for A 2 from lattice QCD with a dynamical charm, Phys. Rev. D 87 (2013) 074033 [arXiv:1301.7593] [INSPIRE].ADSGoogle Scholar
  31. [31]
    M. Lavelle and M. Oleszczuk, The operator product expansion of the QCD propagators, Mod. Phys. Lett. A 7 (1992) 3617 [INSPIRE].ADSCrossRefGoogle Scholar
  32. [32]
    D. Dudal, S. Sorella, N. Vandersickel and H. Verschelde, New features of the gluon and ghost propagator in the infrared region from the Gribov-Zwanziger approach, Phys. Rev. D 77 (2008) 071501 [arXiv:0711.4496] [INSPIRE].ADSGoogle Scholar
  33. [33]
    D. Dudal, J.A. Gracey, S.P. Sorella, N. Vandersickel and H. Verschelde, A refinement of the Gribov-Zwanziger approach in the Landau gauge: infrared propagators in harmony with the lattice results, Phys. Rev. D 78 (2008) 065047 [arXiv:0806.4348] [INSPIRE].ADSGoogle Scholar
  34. [34]
    D. Dudal, O. Oliveira and N. Vandersickel, Indirect lattice evidence for the Refined Gribov-Zwanziger formalism and the gluon condensateA 2in the Landau gauge, Phys. Rev. D 81 (2010) 074505 [arXiv:1002.2374] [INSPIRE].ADSGoogle Scholar
  35. [35]
    F. Gubarev and V.I. Zakharov, On the emerging phenomenology of \( \left\langle {\left( {{A_{\mu }}} \right)_{\min}^2} \right\rangle \), Phys. Lett. B 501 (2001) 28 [hep-ph/0010096] [INSPIRE].ADSCrossRefGoogle Scholar
  36. [36]
    K.-I. Kondo, Vacuum condensate of mass dimension 2 as the origin of mass gap and quark confinement, Phys. Lett. B 514 (2001) 335 [hep-th/0105299] [INSPIRE].ADSCrossRefGoogle Scholar
  37. [37]
    H. Verschelde, K. Knecht, K. Van Acoleyen and M. Vanderkelen, The Nonperturbative groundstate of QCD and the local composite operator \( A_{\mu}^2 \), Phys. Lett. B 516 (2001) 307 [hep-th/0105018] [INSPIRE].ADSCrossRefGoogle Scholar
  38. [38]
    D. Dudal, H. Verschelde and S. Sorella, The Anomalous dimension of the composite operator A 2 in the Landau gauge, Phys. Lett. B 555 (2003) 126 [hep-th/0212182] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  39. [39]
    P. Boucaud et al., Instantons andA 2condensate, Phys. Rev. D 66 (2002) 034504 [hep-ph/0203119] [INSPIRE].ADSGoogle Scholar
  40. [40]
    E. Ruiz Arriola, P.O. Bowman and W. Broniowski, Landau-gauge condensates from the quark propagator on the lattice, Phys. Rev. D 70 (2004) 097505 [hep-ph/0408309] [INSPIRE].ADSGoogle Scholar
  41. [41]
    E. Megias, E. Ruiz Arriola and L. Salcedo, Dimension two condensates and the Polyakov loop above the deconfinement phase transition, JHEP 01 (2006) 073 [hep-ph/0505215] [INSPIRE].ADSCrossRefGoogle Scholar
  42. [42]
    E. Ruiz Arriola and W. Broniowski, Dimension-two gluon condensate from large-N c Regge models, Phys. Rev. D 73 (2006) 097502 [hep-ph/0603263] [INSPIRE].ADSGoogle Scholar
  43. [43]
    E. Megias, E. Ruiz Arriola and L. Salcedo, Trace Anomaly, Thermal Power Corrections and Dimension Two condensates in the deconfined phase, Phys. Rev. D 80 (2009) 056005 [arXiv:0903.1060] [INSPIRE].ADSGoogle Scholar
  44. [44]
    D. Vercauteren and H. Verschelde, A Two-component picture of the \( \left\langle {{A^{\mu }}} \right\rangle \) condensate with instantons, Phys. Lett. B 697 (2011) 70 [arXiv:1101.5017] [INSPIRE].ADSCrossRefGoogle Scholar
  45. [45]
    J. Taylor, Ward Identities and Charge Renormalization of the Yang-Mills Field, Nucl. Phys. B 33 (1971) 436 [INSPIRE].ADSCrossRefGoogle Scholar
  46. [46]
    ETM collaboration, B. Blossier et al., High statistics determination of the strong coupling constant in Taylor scheme and its OPE Wilson coefficient from lattice QCD with a dynamical charm, Phys. Rev. D 89 (2014) 014507 [arXiv:1310.3763] [INSPIRE].ADSGoogle Scholar
  47. [47]
    D. Binosi, D. Ibanez and J. Papavassiliou, QCD effective charge from the three-gluon vertex of the background-field method, Phys. Rev. D 87 (2013) 125026 [arXiv:1304.2594] [INSPIRE].ADSGoogle Scholar
  48. [48]
    A.C. Aguilar and J. Papavassiliou, Gluon mass generation in the PT-BFM scheme, JHEP 12 (2006) 012 [hep-ph/0610040] [INSPIRE].ADSCrossRefGoogle Scholar
  49. [49]
    D. Binosi and J. Papavassiliou, Pinch Technique: Theory and Applications, Phys. Rept. 479 (2009) 1 [arXiv:0909.2536] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  50. [50]
    Particle Data Group collaboration, J. Beringer et al., Review of Particle Physics (RPP), Phys. Rev. D 86 (2012) 010001 [INSPIRE].ADSGoogle Scholar
  51. [51]
    V. Gribov, Quantization of Nonabelian Gauge Theories, Nucl. Phys. B 139 (1978) 1 [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  52. [52]
    A. Cucchieri, Gribov copies in the minimal Landau gauge: The Influence on gluon and ghost propagators, Nucl. Phys. B 508 (1997) 353 [hep-lat/9705005] [INSPIRE].ADSCrossRefGoogle Scholar
  53. [53]
    P. Silva and O. Oliveira, Gribov copies, lattice QCD and the gluon propagator, Nucl. Phys. B 690 (2004) 177 [hep-lat/0403026] [INSPIRE].ADSCrossRefGoogle Scholar
  54. [54]
    A. Sternbeck, E.-M. Ilgenfritz, M. Muller-Preussker and A. Schiller, The Gluon and ghost propagator and the influence of Gribov copies, Nucl. Phys. Proc. Suppl. 140 (2005) 653 [hep-lat/0409125] [INSPIRE].ADSCrossRefGoogle Scholar
  55. [55]
    I. Bogolubsky, E. Ilgenfritz, M. Muller-Preussker and A. Sternbeck, Lattice gluodynamics computation of Landau gauge Greens functions in the deep infrared, Phys. Lett. B 676 (2009) 69 [arXiv:0901.0736] [INSPIRE].ADSCrossRefGoogle Scholar
  56. [56]
    I. Bogolubsky, E.-M. Ilgenfritz, M. Muller-Preussker and A. Sternbeck, The Landau gauge gluon propagator in 4D SU(2) lattice gauge theory revisited: Gribov copies and scaling properties, PoS(LAT2009)237 [arXiv:0912.2249] [INSPIRE].
  57. [57]
    V. Bornyakov, V. Mitrjushkin and M. Muller-Preussker, SU(2) lattice gluon propagator: Continuum limit, finite-volume effects and infrared mass scale m(IR), Phys. Rev. D 81 (2010) 054503 [arXiv:0912.4475] [INSPIRE].ADSGoogle Scholar
  58. [58]
    B. Blossier et al., A novel method for the physical scale setting on the lattice and its application to N f = 4 simulations, Phys. Rev. D 89 (2014) 034026 [arXiv:1312.1514] [INSPIRE].ADSGoogle Scholar
  59. [59]
    R. Baron et al., Light hadrons from lattice QCD with light (u, d), strange and charm dynamical quarks, JHEP 06 (2010) 111 [arXiv:1004.5284] [INSPIRE].ADSCrossRefGoogle Scholar
  60. [60]
    ETM collaboration, R. Baron et al., Light hadrons from N f = 2 + 1 + 1 dynamical twisted mass fermions, PoS(LATTICE 2010)123 [arXiv:1101.0518] [INSPIRE].
  61. [61]
    D. Becirevic et al., Asymptotic behavior of the gluon propagator from lattice QCD, Phys. Rev. D 60 (1999) 094509 [hep-ph/9903364] [INSPIRE].ADSGoogle Scholar
  62. [62]
    D. Becirevic et al., Asymptotic scaling of the gluon propagator on the lattice, Phys. Rev. D 61 (2000) 114508 [hep-ph/9910204] [INSPIRE].ADSGoogle Scholar
  63. [63]
    F. de Soto and C. Roiesnel, On the reduction of hypercubic lattice artifacts, JHEP 09 (2007) 007 [arXiv:0705.3523] [INSPIRE].CrossRefGoogle Scholar
  64. [64]
    P. Boucaud, D. Dudal, J. Leroy, O. Pene and J. Rodriguez-Quintero, On the leading OPE corrections to the ghost-gluon vertex and the Taylor theorem, JHEP 12 (2011) 018 [arXiv:1109.3803] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  65. [65]
    N. Carrasco et al., A determination of the average up-down, strange and charm quark masses from N f = 2 + 1 + 1, arXiv:1311.2793 [INSPIRE].
  66. [66]
    M.A. Shifman, A. Vainshtein and V.I. Zakharov, QCD and Resonance Physics. Sum Rules, Nucl. Phys. B 147 (1979) 385 [INSPIRE].ADSCrossRefGoogle Scholar
  67. [67]
    M.A. Shifman, A. Vainshtein and V.I. Zakharov, QCD and Resonance Physics: Applications, Nucl. Phys. B 147 (1979) 448 [INSPIRE].ADSCrossRefGoogle Scholar
  68. [68]
    P. Boucaud et al., A Transparent expression of the A 2 condensates renormalization, Phys. Rev. D 67 (2003) 074027 [hep-ph/0208008] [INSPIRE].ADSGoogle Scholar
  69. [69]
    S.J. Brodsky, C.D. Roberts, R. Shrock and P.C. Tandy, Essence of the vacuum quark condensate, Phys. Rev. C 82 (2010) 022201 [arXiv:1005.4610] [INSPIRE].ADSGoogle Scholar
  70. [70]
    L. Chang, C.D. Roberts and P.C. Tandy, Expanding the concept of in-hadron condensates, Phys. Rev. C 85 (2012) 012201 [arXiv:1109.2903] [INSPIRE].ADSGoogle Scholar
  71. [71]
    O. Pene et al., Vacuum expectation value of A 2 from LQCD, PoS(FacesQCD)010 [arXiv:1102.1535] [INSPIRE].

Copyright information

© The Author(s) 2014

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  • Ph. Boucaud
    • 1
  • M. Brinet
    • 2
  • F. De Soto
    • 3
  • V. Morenas
    • 4
  • O. Pène
    • 1
  • K. Petrov
    • 5
  • J. Rodríguez-Quintero
    • 6
    • 7
    Email author
  1. 1.Laboratoire Physique ThéoriqueUniversité de Paris XIOrsay CedexFrance
  2. 2.Laboratoire de Physique Subatomique et de Cosmologie, CNRS/IN2P3/UJFGrenobleFrance
  3. 3.Departamento de Sistemas Físicos, Químicos y NaturalesUniversidad Pablo de OlavideSevillaSpain
  4. 4.Laboratoire de Physique Corpusculaire, Université Blaise Pascal, CNRS/IN2P3Aubière CedexFrance
  5. 5.Laboratoire de l’Accélérateur Linéaire, Centre Scientifique d’OrsayORSAY Cedex,France
  6. 6.Departamento de Física Aplicada, Facultad de Ciencias ExperimentalesUniversidad de HuelvaHuelvaSpain
  7. 7.CAFPE, Universidad de GranadaGranadaSpain

Personalised recommendations