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Borel-Écalle Resummation of a Two-Point Function

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Abstract

We provide an overview of the tools and techniques of resurgence theory used in the Borel-Écalle resummation method, which we then apply to the massless Wess–Zumino model. Starting from already known results on the anomalous dimension of the Wess–Zumino model, we solve its renormalisation group equation for the two-point function in a space of formal series. We show that this solution is 1-Gevrey and that its Borel transform is resurgent. The Schwinger–Dyson equation of the model is then used to prove an asymptotic exponential bound for the Borel transformed two-point function on a star-shaped domain of a suitable ramified complex plane. This proves that the two-point function of the Wess–Zumino model is Borel-Écalle summable.

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Notes

  1. In order to avoid confusion between the kinematic parameter of the two-point function and the length of the path, we will denote the former by the letter \(\Lambda \).

  2. \(f:U\subset \mathbb {C}\longrightarrow \mathbb {C}\) is real if \(f(\bar{z}) =\overline{f(z)}\) whenever both sides of the equation make sense. We require this condition since we want the resummed function to represent a physical quantity.

  3. This was not published since the perturbation around higher singularities of the Borel transform are subdominant and difficult to approach numerically.

  4. At least the first one, but since a singularities in \(\omega \in \mathbb {C}^*\) generally produces new singularities in \(\omega \mathbb {N}^*\) (as in Example 2.12), we expect that all singularities will depend on L, at least in some non-perturbative regime.

  5. This being of course an abuse of language, it is only possible if Eq. (20) is a bound not an equivalence. We are not more precise in order to not burden the text with too much technical details.

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Acknowledgements

The author thanks Marc Bellon for many exciting discussions on resurgence theory and the Wess–Zumino model. I also thank David Sauzin for having kindly answered my questions regarding his nonlinear analysis for resurgent functions and Sylvie Paycha for encouragements, discussions and suggestions. I am very grateful for Marc Bellon’s and Sylvie Paycha’s corrections an an early draft of this paper. I would also like to thank the two anonymous referees whose questions and suggestions have greatly improved the quality of this paper. This work was partly completed while at the Perimeter Institute.

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Authors and Affiliations

  1. Institute of Mathematics, University of Potsdam, 14476, Potsdam, Germany

    Pierre J. Clavier

  2. Mathematics Laboratory, Technische Universität, 10623, Berlin, Germany

    Pierre J. Clavier

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Correspondence to Pierre J. Clavier.

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Communicated by Karl-Henning Rehren.

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Clavier, P.J. Borel-Écalle Resummation of a Two-Point Function. Ann. Henri Poincaré 22, 2103–2136 (2021). https://doi.org/10.1007/s00023-021-01057-w

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