Alien calculus and non perturbative effects in Quantum Field Theory

Abstract

In many domains of physics, methods for dealing with non-perturbative aspects are required. Here, I want to argue that a good approach for this is to work on the Borel transforms of the quantities of interest, the singularities of which give non-perturbative contributions. These singularities in many cases can be largely determined by using the alien calculus developed by Jean Écalle. My main example will be the two point function of a massless theory given as a solution of a renormalization group equation.

Keywords

Schwinger–Dyson equations / series resommation / alien calculus

References

[1] J. Ecalle, Les Fonctions résurgentes, Vol. 1 (Pub. Math. Orsay, 1981)

[2] D. Sauzin, Introduction to 1-summability and resurgence, arXiv: 1405.0356v1 [math.DS] (2014)

[3] M. P. Bellon and P. J. Clavier, A Schwinger–Dyson equation in the Borel plane: Singularities of the solution, Lett. Math. Phys. 105, 795(2015), arXiv: 1411.7190 [math-ph]

[4] E. R. Speer, Analytic renormalization, J. Math. Phys. 9, 1404 (1968)

[5] E. R. Speer, On the structure of analytic renormalization, Commun. Math. Phys. 23(1), 23 (1971)

[6] M. Bellon and F. A. Schaposnik, Higher loop corrections to a Schwinger–Dyson equation, Lett. Math. Phys. 103, 881 (2013), arXiv: 1205.0022 [hep-th]

[7] Convolution products have generic singularities at the positions of the singularities of each factors, but also at the sum of the positions, at least in some of the sheets of the Riemann surface on which the convolution productis defined.

[8] J. Ecalle, Introduction aux fonctions analysables et preuve constructive de la conjecture de Dulac (Hermann, 1992)

[9] M. Stingl, Field-theory amplitudes as resurgent functions, arXiv: hep-ph/0207349 (2002)

[10] M. P. Bellon and P. J. Clavier, Higher order corrections to the asymptotic perturbative solution of a Schwinger– Dyson equation, Lett. Math. Phys. 104, 1(2014), arXiv: 1311.1160v2 [hep-th]

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