Analyticity domain of a quantum field theory and accelero-summation

  • Marc P. BellonEmail author
  • Pierre J. Clavier


From ’t Hooft’s argument, one expects that the analyticity domain of an asymptotically free quantum field theory is horn shaped. In the usual Borel summation, the function is obtained through a Laplace transform and thus has a much larger analyticity domain. However, if the summation process goes through the process called acceleration by Écalle, one obtains such a horn-shaped analyticity domain. We therefore argue that acceleration, which allows to go beyond standard Borel summation, must be an integral part of the toolkit for the study of exactly renormalisable quantum field theories. We sketch how this procedure is working and what are its consequences.


Renormalization Borel transform Alien calculus Accelero-summation 

Mathematics Subject Classification

81Q40 81T16 40G10 


Compliance with ethical standards

Conflict of interest

The authors declare that there is no conflict of interest.


  1. 1.
    Ecalle, J.: Les fonctions résurgentes, vol. 1. Pub. Math. d’Orsay, Orsay (1981)zbMATHGoogle Scholar
  2. 2.
    Ecalle, J.: Les fonctions résurgentes, vol. 2. Pub. Math. d’Orsay, Orsay (1981)zbMATHGoogle Scholar
  3. 3.
    Ecalle, J.: Les fonctions résurgentes, vol. 3. Pub. Math. d’Orsay, Orsay (1981)zbMATHGoogle Scholar
  4. 4.
    Dunne, G.V., Ünsal, M.: Resurgence and trans-series in quantum field theory: the \({CP(N-1)}\) model. J. High Energy Phys. 2012(11), 1–86 (2012). MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cherman, A., Dorigoni, D., Dunne, G.V., Ünsal, M.: Resurgence in quantum field theory: nonperturbative effects in the principal chiral model. Phys. Rev. Lett. 112, 021601 (2014). ADSCrossRefGoogle Scholar
  6. 6.
    Dunne, G.V., Ünsal, M.: Uniform WKB, multi-instantons, and resurgent trans-series. Phys. Rev. D 89, 105009 (2014). ADSCrossRefGoogle Scholar
  7. 7.
    Aniceto, I., Schiappa, R., Vonk, M.: The resurgence of instantons in string theory. Commun. Number Theory Phys. 6(2), 339–496 (2012). MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Couso-Santamaria, R., Edelstein, J.D., Schiappa, R., Vonk, M.: Resurgent transseries and the holomorphic anomaly: nonperturbative closed strings in local \({\mathbb{C}}{\mathbb{P}}^2\) (2014). arXiv:1407.4821
  9. 9.
    Mariño, M., Schiappa, R., Weiss, M.: Nonperturbative effects and the large-order behavior of matrix models and topological strings. Commun. Number Theory Phys. 2(2), 349–419 (2008). MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bellon, M.P., Clavier, P.J.: Alien calculus and a Schwinger–Dyson equation: two-point function with a nonperturbative mass scale. Lett. Math. Phys. 108, 391–412 (2017). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    ’t Hooft, G.: Can We Make Sense Out of “Quantum Chromodynamics”?, pp. 943–982. Springer, Boston (1979). Google Scholar
  12. 12.
    Ecalle, J.: Introduction aux fonctions analysables et preuve constructive de la conjecture de Dulac. Actualités mathématiques, Hermann (1992)zbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Laboratoire de Physique Théorique et Hautes Energies, LPTHESorbonne Université, CNRSParisFrance
  2. 2.MathematikPotsdam UniversitätGolmDeutschland

Personalised recommendations