Abstract
We consider the effects of higher loop corrections to a Schwinger–Dyson equation for propagators. This is made possible by the efficiency of the methods we developed in preceding works, still using the supersymmetric Wess–Zumino model as a laboratory. We obtain the dominant contributions of the three and four-loop primitive divergences at high order in perturbation theory, without the need for their full evaluations. Our main conclusion is that the asymptotic behavior of the perturbative series of the renormalization function remains unchanged, and we conjecture that this will remain the case for all finite order corrections.
Similar content being viewed by others
References
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). arXiv:0801.0727
Bellon, M.P.: Approximate differential equations for renormalization group functions in models free of vertex divergencies. Nucl. Phys. B, 826[PM], 522–531 (2010). arXiv:0907.2296. doi:10.1016/j.nuclphysb.2009.11.002
Bellon, M.P.: An efficient method for the solution of Schwinger–Dyson equations for propagators. Lett. Math. Phys., 94, 77–86 (2010). arXiv:1005.0196. doi:10.1007/s11005-010-0415-3
Kreimer, D., Yeats, K.: An etude in non-linear Dyson–Schwinger equations. Nucl. Phys. Proc. Suppl., 160, 116–121 (2006). arXiv:hep-th/0605096. doi:10.1016/j.nuclphysbps.2006.09.036
Connes, A., Kreimer, D.: Renormalization in quantum field theory and the Riemann–Hilbert problem. JHEP, 09, 024 (1999) arXiv:hep-th/9909126
Connes, A., Kreimer, D.: Renormalization in quantum field theory and the Riemann– Hilbert problem. I: The Hopf algebra structure of graphs and the main theorem. Commun. Math. Phys., 210, 249–273 (2000). arXiv:hep-th/9912092. doi:10.1007/s002200050779
Connes, A., Kreimer, D.: Renormalization in quantum field theory and the Riemann– Hilbert problem. II: the beta-function, diffeomorphisms and the renormalization group. Commun. Math. Phys. 216, 215–241 (2001). arXiv:hep-th/0003188. doi:10.1007/PL00005547
Bollini C.G., Giambiagi J.J., Domínguez A.G.: Analytic regularization and the divergences of quantum field theories. Nuevo Cimento 31, 550–561 (1964)
Speer E.R.: Analytic renormalization. J. Math. Phys. 9, 1404–1410 (1969)
Speer E.R.: On the structure of analytic renormalization. Commun. Math. Phys. 23, 23–36 (1971)
Bierenbaum, I., Weinzierl, S.: The massless two-loop two-point function. Eur. Phys. J. C. 32, 67–78 (2003). arXiv:hep-ph/0308311. doi:10.1140/epjc/s2003-01389-7
Isaev A.P., Gorishnii S.G.: TMΦ 58, 343 (1984)
Isaev A.P., Gorishnii S.G.: Theor. Math. Phys. 58, 232 (1984)
Broadhurst D.J.: Exploiting the 1,440-fold symmetry of the master two-loop diagram. Z. Phys. C 32, 249–253 (1986)
Schnetz, O.: Quantum periods: a census of \({\phi^4}\)-transcendentals. Commun. Number Theory Phys. 4(1), 1–48 (2010). arXiv:0801.2856
Nakanishi N.: Graph theory and Feynman integrals. Mathematics and its Applications, vol. 11. Gordon and Breach, New York (1971)
Kreimer, D.: The core Hopf algebra. Clay Math. Proc. 11, 313–322 (2010). arXiv:0902.1223
Kreimer, D., van Suijlekom, W.D.: Recursive relations in the core Hopf algebra. Nucl. Phys. B, 820, 682–693 (2009). arXiv:0903.2849. doi:10.1016/j.nuclphysb.2009.04.025
Vermaseren, J.A.M.: New Features of Form (2000). arXiv:math-ph/0010025
Brown, F., Yeats, K.: Spanning forest polynomials and the transcendental weight of feynman graphs. Commun. Math. Phys. 301, 357–382 (2011). arXiv:0910.5429. doi:10.1007/s00220-010-1145-1
Brown, F., Schnetz, O., Yeats, K.: Properties of c 2 invariants of Feynman graphs (2012). arXiv:1203.0188
Author information
Authors and Affiliations
Corresponding author
Additional information
F. A. Schaposnik associated with CICBA.
Rights and permissions
About this article
Cite this article
Bellon, M.P., Schaposnik, F.A. Higher Loop Corrections to a Schwinger–Dyson Equation. Lett Math Phys 103, 881–893 (2013). https://doi.org/10.1007/s11005-013-0621-x
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11005-013-0621-x