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Wilson expansion of QCD popagators at three loops: operators of dimension two and three

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Abstract

In this paper we construct the Wilson short distance operator product expansion for the gluon, quark and ghost propagators in QCD, including operators of dimension two and three, namely, A 2, m 2, mA 2, \( \bar \psi \psi \), and m 3. We compute analytically the coefficient functions of these operators at three loops for all three propagators in the general covariant gauge. Our results, taken in the Landau gauge, should help to improve the accuracy of extracting the vacuum expectation values of these operators from lattice simulation of the QCD propagators.

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References

  1. T. van Ritbergen, J.A.M. Vermaseren and S.A. Larin, The four-loop β -function in quantum chromodynamics, Phys. Lett. B 400 (1997) 379 [hep-ph/9701390] [SPIRES].

    ADS  Google Scholar 

  2. M. Czakon, The four-loop QCD β-function and anomalous dimensions, Nucl. Phys. B 710 (2005) 485 [hep-ph/0411261] [SPIRES].

    Article  ADS  Google Scholar 

  3. J.A.M. Vermaseren, S.A. Larin and T. van Ritbergen, The 4-loop quark mass anomalous dimension and the invariant quark mass, Phys. Lett. B 405 (1997) 327 [hep-ph/9703284] [SPIRES].

    ADS  Google Scholar 

  4. K.G. Chetyrkin, Quark mass anomalous dimension to O(α4 s ), Phys. Lett. B 404 (1997) 161 [hep-ph/9703278] [SPIRES].

    ADS  Google Scholar 

  5. K.G. Chetyrkin, Four-loop renormalization of QCD: Full set of renormalization constants and anomalous dimensions, Nucl. Phys. B 710 (2005) 499 [hep-ph/0405193] [SPIRES].

    Article  ADS  Google Scholar 

  6. C.D. Roberts and A.G. Williams, Dyson-Schwinger equations and their application to hadronic physics, Prog. Part. Nucl. Phys. 33 (1994) 477 [hep-ph/9403224] [SPIRES].

    Article  ADS  Google Scholar 

  7. P. Maris and C.D. Roberts, Dyson-Schwinger equations: A tool for hadron physics, Int. J. Mod. Phys. E 12 (2003) 297 [nucl-th/0301049] [SPIRES].

    ADS  Google Scholar 

  8. R. Alkofer and L. von Smekal, The infrared behavior of QCD Green’s functions: Confinement, dynamical symmetry breaking and hadrons as relativistic bound states, Phys. Rept. 353 (2001) 281 [hep-ph/0007355] [SPIRES].

    Article  ADS  MATH  Google Scholar 

  9. K.G. Wilson, Nonlagrangian models of current algebra, Phys. Rev. 179 (1969) 1499 [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  10. H.D. Politzer, Effective quark masses in the chiral limit, Nucl. Phys. B 117 (1976) 397 [SPIRES].

    Article  ADS  Google Scholar 

  11. M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, QCD and resonance physics. Sum rules, Nucl. Phys. B 147 (1979) 385 [SPIRES].

    Article  ADS  Google Scholar 

  12. 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] [SPIRES].

    ADS  Google Scholar 

  13. V. Giménez, V. Lubicz, F. Mescia, V. Porretti and J. Reyes, Operator product expansion and quark condensate from lattice QCD in coordinate space, Eur. Phys. J. C 41 (2005) 535 [hep-lat/0503001] [SPIRES].

    Article  ADS  Google Scholar 

  14. A. Cucchieri and T. Mendes, Propagators, running coupling and condensates in lattice QCD, Braz. J. Phys. 37 (2007) 484 [hep-ph/0605224] [SPIRES].

    Article  Google Scholar 

  15. P. Boucaud et al., Ghost-gluon running coupling, power corrections and the determination of \( \Lambda_{{\bar{M}{S}}} \), Phys. Rev. D 79 (2009) 014508 [arXiv:0811.2059] [SPIRES].

    ADS  Google Scholar 

  16. M.J. Lavelle and M. Schaden, Propagators and condensates in QCD, Phys. Lett. B 208 (1988) 297 [SPIRES].

    ADS  Google Scholar 

  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] [SPIRES].

    ADS  Google Scholar 

  18. P. Boucaud et al., Testing Landau gauge OPE on the lattice with a <A 2> condensate, Phys. Rev. D 63 (2001) 114003 [hep-ph/0101302] [SPIRES].

    ADS  Google Scholar 

  19. P. Boucaud et al., Non-perturbative power corrections to ghost and gluon propagators, JHEP 01 (2006) 037 [hep-lat/0507005] [SPIRES].

    Article  ADS  Google Scholar 

  20. P. Boucaud et al., Instantons and <A 2> condensate, Phys. Rev. D 66 (2002) 034504 [hep-ph/0203119] [SPIRES].

    ADS  Google Scholar 

  21. P. Boucaud et al., Evidences for instantons effects in Landau lattice Green functions, Phys. Rev. D 70 (2004) 114503 [hep-ph/0312332] [SPIRES].

    ADS  Google Scholar 

  22. P. Boucaud et al., Artefacts and <A 2> power corrections: Revisiting the MOM Z ψ (p 2) and Z V , Phys. Rev. D 74 (2006) 034505 [hep-lat/0504017] [SPIRES].

    ADS  Google Scholar 

  23. 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] [SPIRES].

    ADS  Google Scholar 

  24. M. Lavelle and M. Oleszczuk, The operator product expansion of the QCD propagators, Mod. Phys. Lett. A 7 (1992) 3617 [SPIRES].

    ADS  Google Scholar 

  25. P.O. Bowman, U.M. Heller, D.B. Leinweber, A.G. Williams and J.-b. Zhang, Infrared and ultraviolet properties of the Landau gauge quark propagator, Nucl. Phys. Proc. Suppl. 128 (2004) 23 [hep-lat/0403002] [SPIRES].

    Article  ADS  Google Scholar 

  26. P.O. Bowman, U.M. Heller, D.B. Leinweber, M.B. Parappilly and A.G. Williams, QCD propagators: Some results from the lattice, Nucl. Phys. Proc. Suppl. 161 (2006) 27 [SPIRES].

    Article  ADS  Google Scholar 

  27. G.M. Cicuta and E. Montaldi, Analytic renormalization via continuous space dimension, Nuovo Cim. Lett. 4 (1972) 329 [SPIRES].

    Article  Google Scholar 

  28. J.F. Ashmore, A method of gauge invariant regularization, Lett. Nuovo Cim. 4 (1972) 289 [SPIRES].

    Article  Google Scholar 

  29. G. ’t Hooft and M.J.G. Veltman, Regularization and renormalization of gauge fields, Nucl. Phys. B 44 (1972) 189 [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  30. G. ’t Hooft, Dimensional regularization and the renormalization group, Nucl. Phys. B 61 (1973) 455 [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  31. K.G. Chetyrkin, F.V. Tkachov and S.G. Gorishnii, Operator product expansion in the minimal subtraction scheme, Phys. Lett. B 119 (1982) 407 [SPIRES].

    ADS  Google Scholar 

  32. F.V. Tkachov, On the operator product expansion in the MS scheme, Phys. Lett. B 124 (1983) 212 [SPIRES].

    ADS  Google Scholar 

  33. F.V. Gubarev, L. Stodolsky and V.I. Zakharov, On the significance of the quantity A 2, Phys. Rev. Lett. 86 (2001) 2220 [hep-ph/0010057] [SPIRES].

    Article  ADS  Google Scholar 

  34. F.V. Gubarev and V.I. Zakharov, On the emerging phenomenology of < (A a μ)2 min >, Phys. Lett. B 501 (2001) 28 [hep-ph/0010096] [SPIRES].

    ADS  Google Scholar 

  35. K.-I. Kondo, Weak gauge-invariance of dimension two condensate in Yang- Mills theory, Phys. Lett. B 619 (2005) 377 [hep-th/0504088] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  36. K.-I. Kondo, Gauge-invariant gluon mass, infrared Abelian dominance and stability of magnetic vacuum, Phys. Rev. D 74 (2006) 125003 [hep-th/0609166] [SPIRES].

    ADS  Google Scholar 

  37. R.N. Baranov, D.V. Bykov and A.A. Slavnov, The condensate < tr(A 2μ ) > in commutative and noncommutative theories, Theor. Math. Phys. 148 (2006) 1168 [Teor. Mat. Fiz. 148 (2006) 350] [hep-th/0601142] [SPIRES] [SPIRES].

    Article  MathSciNet  MATH  Google Scholar 

  38. E.R. Arriola and W. Broniowski, Dimension-2 condensates, ζ-regularization and large-NcRegge Models, Eur. Phys. J. A 31 (2007) 739 [hep-ph/0609266] [SPIRES].

    ADS  Google Scholar 

  39. D. Dudal et al., A local non-Abelian gauge invariant action stemming from the nonlocal operator F μν (D 2)−1 F μν , Braz. J. Phys. 37 (2007) 232 [hep-th/0609184] [SPIRES].

    Google Scholar 

  40. S.P. Sorella et al., Infrared behavior of the gluon and ghost propagators in Yang-Mills theories, PoS(IC2006)026 [hep-th/0610306] [SPIRES].

  41. J.A. Gracey, One loop gluon form factor and freezing of αs in the Gribov-Zwanziger QCD Lagrangian, JHEP 05 (2006) 052 [hep-ph/0605077] [SPIRES].

    Article  ADS  Google Scholar 

  42. M.A.L. Capri et al., Quantum properties of a non-Abelian gauge invariant action with a mass parameter, Phys. Rev. D 74 (2006) 045008 [hep-th/0605288] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  43. M.N. Chernodub and E.M. Ilgenfritz, Electric-magnetic asymmetry of the A 2 condensate and the phases of Yang-Mills theory, Phys. Rev. D 78 (2008) 034036 [arXiv:0805.3714] [SPIRES].

    ADS  Google Scholar 

  44. K.G. Chetyrkin and A. Retey, Renormalization and running of quark mass and field in the regularization invariant and MS-bar schemes at three and four loops, Nucl. Phys. B 583 (2000) 3 [hep-ph/9910332] [SPIRES].

    Article  ADS  Google Scholar 

  45. O.V. Tarasov, Anomalous dimensions of quark masses in three loop approximation. (In Russian), JINR-P2-82-900 [SPIRES].

  46. S.A. Larin and J.A.M. Vermaseren, The three loop QCD β-function and anomalous dimensions, Phys. Lett. B 303 (1993) 334 [hep-ph/9302208] [SPIRES].

    ADS  Google Scholar 

  47. J.A. Gracey, Three loop \( \bar MS \) renormalization of the Curci-Ferrari model and the dimension two BRST invariant composite operator in QCD, Phys. Lett. B 552 (2003) 101 [hep-th/0211144] [SPIRES].

    ADS  Google Scholar 

  48. V.P. Spiridonov and K.G. Chetyrkin, Nonleading mass corrections and renormalization of the operators \( m\psi - \bar \psi \) and g 2μν , Sov. J. Nucl. Phys. 47 (1988) 522 [Yad. Fiz. 47 (1988) 818] [SPIRES].

    Google Scholar 

  49. K.G. Chetyrkin, C.A. Dominguez, D. Pirjol and K. Schilcher, Mass singularities in light quark correlators: The Strange quark case, Phys. Rev. D 51 (1995) 5090 [hep-ph/9409371] [SPIRES].

    ADS  Google Scholar 

  50. K.G. Chetyrkin and J.H. Kuhn, Quartic mass corrections to Rhad, Nucl. Phys. B 432 (1994) 337 [hep-ph/9406299] [SPIRES].

    Article  ADS  Google Scholar 

  51. P. Nogueira, Automatic Feynman graph generation, J. Comput. Phys. 105 (1993) 279 [SPIRES].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  52. T. Seidensticker, Automatic application of successive asymptotic expansions of Feynman diagrams, hep-ph/9905298 [SPIRES].

  53. S.G. Gorishnii, S.A. Larin, L.R. Surguladze and F.V. Tkachov, MINCER: program for multiloop calculations in quantum field theory for the schoonschip system, Comput. Phys. Commun. 55 (1989) 381 [SPIRES].

    Article  ADS  Google Scholar 

  54. S.A. Larin, F.V. Tkachov and J.A.M. Vermaseren, The FORM version of MINCER, NIKHEF-H-91-18 [SPIRES].

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

  56. S.G. Gorishnii, S.A. Larin and F.V. Tkachov, The algotithm for ope coefficient functions in the MS scheme, Phys. Lett. B 124 (1983) 217 [SPIRES].

    ADS  Google Scholar 

  57. S.G. Gorishnii and S.A. Larin, Coefficient functions of asymptotic operator expansions in minimal subtraction scheme, Nucl. Phys. B 283 (1987) 452 [SPIRES].

    Article  ADS  Google Scholar 

  58. S. Narison and V.I. Zakharov, Duality between QCD perturbative series and power corrections, Phys. Lett. B 679 (2009) 355 [arXiv:0906.4312] [SPIRES].

    ADS  Google Scholar 

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Authors and Affiliations

  1. Institut für Theoretische Teilchenphysik, Karlsruhe Institute for Technology, Wolfgang-Gaede Straße 1, D-76128, Karlsruhe, Germany

    K. G. Chetyrkin & A. Maier

  2. Institute for Nuclear Research RAS, Russian Academy of Sciences, Prospekt 60-letiya Oktyabrya 7a, Moscow, 117312, Russia

    K. G. Chetyrkin

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  1. K. G. Chetyrkin
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  2. A. Maier
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Correspondence to K. G. Chetyrkin.

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ArXiv ePrint: 0911.0594

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Chetyrkin, K.G., Maier, A. Wilson expansion of QCD popagators at three loops: operators of dimension two and three. J. High Energ. Phys. 2010, 92 (2010). https://doi.org/10.1007/JHEP01(2010)092

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