Letters in Mathematical Physics

, Volume 105, Issue 6, pp 795–825 | Cite as

A Schwinger–Dyson Equation in the Borel Plane: Singularities of the Solution

  • Marc P. Bellon
  • Pierre J. Clavier


We map the Schwinger–Dyson equation and the renormalization group equation for the massless Wess–Zumino model in the Borel plane, where the product of functions gets mapped to a convolution product. The two-point function can be expressed as a superposition of general powers of the external momentum. The singularities of the anomalous dimension are shown to lie on the real line in the Borel plane and to be linked to the singularities of the Mellin transform of the one-loop graph. This new approach allows us to enlarge the reach of previous studies on the expansions around those singularities. The asymptotic behavior at infinity of the Borel transform of the solution is beyond the reach of analytical methods and we do a preliminary numerical study, aiming to show that it should remain bounded.


renormalization Schwinger–Dyson equation Borel transform alien calculus 

Mathematics Subject Classification

81Q40 81T16 40G10 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Mezrag, C., Moutarde, H., Rodrigues-Quintero, J., Sabatié, F.: Toward a Pion generalized parton distribution model from Dyson–Schwinger equations. arXiv:1406.7425 [hep-th] (2014)
  2. 2.
    Broadhurst D.J., Kreimer D.: Exact solutions of Dyson–Schwinger equations for iterated one-loop integrals and propagator-coupling duality. Nucl. Phys. B 600, 403–422 (2001)CrossRefADSMATHGoogle Scholar
  3. 3.
    Clavier, P.J.: Analytic results for Schwinger–Dyson equations with a mass term. Lett. Math. Phys. doi: 10.1007/s11005-015-0762-1. arXiv:1409.3351 [hep-th] (2014)
  4. 4.
    Kreimer D., Yeats K.: An etude in non-linear Dyson–Schwinger equations. Nucl. Phys. Proc. Suppl 160, 116–121 (2006)CrossRefADSMathSciNetGoogle Scholar
  5. 5.
    Bellon M.P.: An efficient method for the solution of Schwinger–Dyson equations for propagators. Lett. Math. Phys 94, 77–86 (2010)CrossRefADSMATHMathSciNetGoogle Scholar
  6. 6.
    Bellon M., Schaposnik F.A.: Higher loop corrections to a Schwinger–Dyson equation. Lett. Math. Phys 103, 881–893 (2013)CrossRefADSMATHMathSciNetGoogle Scholar
  7. 7.
    Bellon M.P., Clavier P.J.: Higher order corrections to the asymptotic perturbative solution of a Swinger–Dyson equation. Lett. Math. Phys 104, 1–22 (2014)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Ecalle, J.: Les fonctions résurgentes, vol. 1. Pub. Math. Orsay (1981)Google Scholar
  9. 9.
    Stingl M.: A systematic extended iterative solution for qcd. Z. Phys. A 353, 423–445 (1996)CrossRefADSGoogle Scholar
  10. 10.
    Stingl, M.: Field-theory amplitudes as resurgent functions. Report number: MS-TP-01-4. arXiv:hep-ph/0207349 (2002)
  11. 11.
    Cherman, A., Dorigoni, D., Dunne, G.V., Unsal, M.: Resurgence in qft: unitons, fractons and renormalons in the principal chiral model. Phys. Rev. Lett. 112, 021601 (2014)Google Scholar
  12. 12.
    Cherman, A., Dorigoni, D., Unsal, M.: Decoding perturbation theory using resurgence: Stokes phenomena, new saddle points and lefschetz thimbles. Report number: DAMTP-2014-17; UMN-TH-2239/14; FTPI-MINN-14/8. arXiv:1407.4821v2 [hep-th] (2014)
  13. 13.
    Couso-Santamaria, R., Edelstein, J., Schiappa, R., Vonk, M.: Resurgent transseries and the holomorphic anomaly: nonperturbative closed strings in local \({\mathbb{CP}^2}\) . Report number: CERN-PH-TH-2014-110. arXiv:1407.4821v2 [hep-th] (2014)
  14. 14.
    Piguet, O., Sibold, K.: Renormalized supersymmetry. Birkhauser Verlag AG (1986)Google Scholar
  15. 15.
    Yeats, K.A.: Growth estimates for Dyson–Schwinger equations. PhD thesis, Boston University (2008)Google Scholar
  16. 16.
    Bellon M., Schaposnik F.: Renormalization group functions for the Wess-Zumino model: up to 200 loops through Hopf algebras. Nucl. Phys. B 800, 517–526 (2008)CrossRefADSMATHMathSciNetGoogle Scholar
  17. 17.
    Bouillot, O.: Invariants analytiques des Difféomorphismes et MultiZêtas. PhD thesis, Université Paris-Sud 11 (2011)Google Scholar
  18. 18.
    Sauzin, D.: Introduction to 1-summability and resurgence. arXiv:1405.0356 [math.DS] (2014)
  19. 19.
    Bellon, M.P.: Approximate differential equations for renormalization group functions in models free of vertex divergencies. Nucl. Phys. B, 826[PM], 522–531 (2010)Google Scholar
  20. 20.
    Kreimer, D., Sars, M., van Suijlekom, W.D.: Quantization of gauge fields, graph polynomials and graph homology. Ann. Phys. 336. doi: 10.1016/j.aop.2013.04.019. arXiv:1208.6477 [hep-th] (2013)
  21. 21.
    Panzer, E.: On the analytic computation of massless propagators in dimensional regularization. Nucl. Phys. B 874, 567–593. doi: 10.1016/j.nuclphysb.2013.05.025. arXiv:1305.2161 [hep-th] (2013)

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Sorbonne Universités, UPMC Univ Paris 06, UMR 7589, LPTHEParisFrance
  2. 2.CNRS, UMR 7589, LPTHEParisFrance

Personalised recommendations