Skip to main content
SpringerLink
Log in

Theoretical description of the plaquette with exponential accuracy

  • Review
  • Published:
The European Physical Journal Special Topics Aims and scope Submit manuscript

Abstract

We review recent studies of the operator product expansion of the plaquette and of the associated determination of the gluon condensate. One first needs the perturbative expansion to orders high enough to reach the asymptotic regime where the renormalon behavior sets in. The divergent perturbative series is formally regulated using the principal value prescription for its Borel integral. Subtracting the perturbative series truncated at the minimal term, we obtain the leading non-perturbative correction of the operator product expansion, i.e., the gluon condensate, with superasymptotic accuracy. It is then explored how to increase such precision within the context of the hyperasymptotic expansion. The results fully confirm expectations from renormalons and the operator product expansion.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

Notes

  1. We define the \(\beta \)-function as \(\beta (\alpha )=d\alpha /d\ln \mu =-\beta _0/(2\pi )\alpha ^2-\beta _1/(8\pi ^2)\alpha ^3-\cdots \), i.e. \(\beta _0=11\).

  2. In the last equality, we approximate the Wilson coefficients by their perturbative expansions, neglecting the possibility of non-perturbative contributions associated to the hard scale 1/a. These would be suppressed by factors \(\sim \exp (-2\pi /\alpha )\) and, therefore, would be sub-leading relative to the gluon condensate.

  3. The labeling (D,N) in general is defined in Refs. [8, 9].

References

  1. K.G. Wilson, Phys. Rev. 179, 1499 (1969)

    Article  ADS  MathSciNet  Google Scholar 

  2. W. Zimmermann, Annals Phys. 77, 570 (1973) [Lect. Notes Phys. 558, 278 (2000)]

  3. A.I. Vainshtein, V.I. Zakharov, M.A. Shifman, JETP Lett. 27, 55 (1978) [Pi’sma Zh. Eksp. Teor. Fiz. 27, 60 (1978)]

  4. V.A. Novikov, M.A. Shifman, A.I. Vainshtein, V.I. Zakharov, Nucl. Phys. B 249, 445 (1985) [Yad. Fiz. 41, 1063 (1985)]

  5. G.S. Bali, C. Bauer, A. Pineda, Phys. Rev. D 89, 054505 (2014). arXiv:1401.7999 [hep-ph]

    Article  ADS  Google Scholar 

  6. G.S. Bali, C. Bauer, A. Pineda, Phys. Rev. Lett. 113, 092001 (2014). arXiv:1403.6477 [hep-ph]

    Article  ADS  Google Scholar 

  7. C. Ayala, X. Lobregat, A. Pineda, Phys. Rev. D 99(7), 074019 (2019). arXiv:1902.07736 [hep-th]

    Article  ADS  MathSciNet  Google Scholar 

  8. C. Ayala, X. Lobregat, A. Pineda, Phys. Rev. D 101(3), 034002 (2020). arXiv:1909.01370 [hep-ph]

    Article  ADS  Google Scholar 

  9. C. Ayala, X. Lobregat, A. Pineda, Nucl. Part. Phys. Proc. 309–311, 77–86 (2020). arXiv:1910.04090 [hep-ph]

    Article  Google Scholar 

  10. C. Ayala, X. Lobregat, A. Pineda, JHEP 12, 093 (2020). arXiv:2009.01285 [hep-ph]

    Article  ADS  Google Scholar 

  11. G.S. Bali, A. Pineda, A.I.P. Conf, AIP Conf. Proc. 1701(1), 030010 (2016). arXiv:1502.00086 [hep-ph]

    Article  Google Scholar 

  12. F. Di Renzo, G. Marchesini, P. Marenzoni, E. Onofri, Nucl. Phys. B Proc. Suppl. 34, 795 (1994)

    Article  ADS  Google Scholar 

  13. F. Di Renzo, E. Onofri, G. Marchesini, P. Marenzoni, Nucl. Phys. B 426, 675 (1994). arXiv:hep-lat/9405019

    Article  ADS  Google Scholar 

  14. F. Di Renzo, L. Scorzato, J. High Energy Phys. 0410, 073 (2004). arXiv:hep-lat/0410010

    Article  Google Scholar 

  15. M.V. Berry, C.J. Howls, Hyperasymptotics. Proc. R. Soc. Lond. A 430, 653–668 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  16. J.P. Boyd, The Devil’s invention: asymptotic, superasymptotic and hyperasymptotic series. Acta Applicandae Mathematica 56, 1 (1999)

    Article  MathSciNet  Google Scholar 

  17. K.G. Wilson, Phys. Rev. D 10, 2445 (1974)

    Article  ADS  Google Scholar 

  18. A. Di Giacomo, H. Panagopoulos, E. Vicari, Phys. Lett. B 240, 423 (1990)

    Article  ADS  Google Scholar 

  19. A. Di Giacomo, H. Panagopoulos, E. Vicari, Nucl. Phys. B 338, 294 (1990)

    Article  ADS  Google Scholar 

  20. M. Lüscher, P. Weisz, Nucl. Phys. B 452, 234 (1995). arXiv:hep-lat/9505011

    Article  ADS  Google Scholar 

  21. C. Christou, A. Feo, H. Panagopoulos, E. Vicari, Nucl. Phys. B 525, 387 (1998) [Erratum-ibid. B 608, 479 (2001)] arXiv:hep-lat/9801007

  22. A. Bode, H. Panagopoulos, Nucl. Phys. B 625, 198 (2002). arXiv:hep-lat/0110211

  23. G.S. Bali, C. Bauer, A. Pineda, Proc. Sci. LATTICE 2013, 371 (2014). arXiv:1311.0114 [hep-lat]

  24. G.S. Bali, C. Bauer, A. Pineda, C. Torrero, Phys. Rev. D 87, 094517 (2013). arXiv:1303.3279 [hep-lat]

    Article  ADS  Google Scholar 

  25. M. Guagnelli, R. Petronzio, N. Tantalo, Phys. Lett. B 548, 58 (2002). arXiv:hep-lat/0209112

    Article  ADS  Google Scholar 

  26. G. ’t Hooft, in Proceedings of the International School of Subnuclear Physics: The Whys of Subnuclear Physics, Erice 1977, ed. by A. Zichichi, Subnucl. Ser. vol 15 (Plenum, New York, 1979), p. 943

  27. M. Beneke, Phys. Rep. 317, 1–142 (1999). arXiv:hep-ph/9807443 [hep-ph]

    Article  ADS  Google Scholar 

  28. A. Di Giacomo, G.C. Rossi, Phys. Lett. 100B, 481 (1981)

    Article  ADS  Google Scholar 

  29. J. Kripfganz, Phys. Lett. 101B, 169 (1981)

    Article  ADS  Google Scholar 

  30. A. Di Giacomo, G. Paffuti, Phys. Lett. 108B, 327 (1982)

    Article  ADS  Google Scholar 

  31. E.-M. Ilgenfritz, M. Müller-Preußker, Phys. Lett. 119B, 395 (1982)

    Article  ADS  Google Scholar 

  32. B. Alles, M. Campostrini, A. Feo, H. Panagopoulos, Phys. Lett. B 324, 433 (1994). arXiv:hep-lat/9306001

    Article  ADS  Google Scholar 

  33. F. Di Renzo, E. Onofri, G. Marchesini, P. Marenzoni, Nucl. Phys. B 426, 675 (1994). arXiv:hep-lat/9405019

    Article  ADS  Google Scholar 

  34. X.-D. Ji, arXiv:hep-ph/9506413

  35. F. Di Renzo, E. Onofri, G. Marchesini, Nucl. Phys. B 457, 202 (1995). arXiv:hep-th/9502095

    Article  ADS  Google Scholar 

  36. G. Burgio, F. Di Renzo, G. Marchesini, E. Onofri, Phys. Lett. B 422, 219 (1998). arXiv:hep-ph/9706209

    Article  ADS  Google Scholar 

  37. R. Horsley, P.E.L. Rakow, G. Schierholz, Nucl. Phys. B Proc. Suppl. 106, 870 (2002). arXiv:hep-lat/0110210

    Article  ADS  Google Scholar 

  38. P. E. L. Rakow, Proc. Sci. LAT2005, 284 (2006). arXiv:hep-lat/0510046

  39. Y. Meurice, Phys. Rev. D 74, 096005 (2006). arXiv:hep-lat/0609005

    Article  ADS  Google Scholar 

  40. T. Lee, Phys. Rev. D 82, 114021 (2010). arXiv:1003.0231 [hep-ph]

    Article  ADS  Google Scholar 

  41. R. Horsley, G. Hotzel, E.-M. Ilgenfritz, R. Millo, H. Perlt, P. E. L. Rakow, Y. Nakamura, G. Schierholz and A. Schiller [QCDSF Collaboration], Phys. Rev. D 86, 054502 (2012). arXiv:1205.1659 [hep-lat]

  42. G. Boyd, J. Engels, F. Karsch, E. Laermann, C. Legeland, M. Lütgemeier, B. Petersson, Nucl. Phys. B 469, 419 (1996). arXiv:hep-lat/9602007

    Article  ADS  Google Scholar 

  43. S. Necco, R. Sommer, Nucl. Phys. B 622, 328 (2002). arXiv:hep-lat/0108008

    Article  ADS  Google Scholar 

  44. L. Del Debbio, F. Di Renzo and G. Filaci, Eur. Phys. J. C 78(11), 974 (2018). arXiv:1807.09518 [hep-lat]

  45. K.G. Chetyrkin, S. Narison, V.I. Zakharov, Nucl. Phys. B 550, 353 (1999). arXiv:hep-ph/9811275

    Article  ADS  Google Scholar 

  46. F.V. Gubarev, V.I. Zakharov, Phys. Lett. B 501, 28 (2001). arXiv:hep-ph/0010096

    Article  ADS  Google Scholar 

  47. E. Ruiz Arriola and W. Broniowski, Phys. Rev. D 73, 097502 (2006). arXiv:hep-ph/0603263

  48. O. Andreev, Phys. Rev. D 73, 107901 (2006). arXiv:hep-th/0603170

    Article  ADS  MathSciNet  Google Scholar 

  49. I. Caprini, Phys. Rev. D 102(5), 054017 (2020). arXiv:2006.16605 [hep-ph]

Download references

Acknowledgements

I thank C. Ayala, G.S. Bali, C. Bauer, X. Lobregat for collaboration in the work reviewed here. This work was supported in part by the Spanish grants FPA2017-86989-P and SEV-2016-0588 from the ministerio de Ciencia, Innovación y Universidades, and the grant 2017SGR1069 from the Generalitat de Catalunya. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No 824093. IFAE is partially funded by the CERCA program of the Generalitat de Catalunya.

Author information

Authors and Affiliations

  1. Institut de Física d’Altes Energies (IFAE), The Barcelona Institute of Science and Technology, Campus UAB, Bellaterra, 08193, Barcelona, Spain

    Antonio Pineda

  2. Grup de Física Teòrica, Dept. Física, Universitat Autònoma de Barcelona, Bellaterra, 08193, Barcelona, Spain

    Antonio Pineda

Authors
  1. Antonio Pineda
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Antonio Pineda.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Pineda, A. Theoretical description of the plaquette with exponential accuracy. Eur. Phys. J. Spec. Top. 230, 2601–2608 (2021). https://doi.org/10.1140/epjs/s11734-021-00263-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1140/epjs/s11734-021-00263-1

Search

Navigation