# Resurgent transseries & Dyson–Schwinger equations

## Abstract

We employ resurgent transseries as algebraic tools to investigate two self-consistent Dyson–Schwinger equations, one in Yukawa theory and one in quantum electrodynamics. After a brief but pedagogical review, we derive fixed point equations for the associated anomalous dimensions and insert a moderately generic log-free transseries ansatz to study the possible strictures imposed. While proceeding in various stages, we develop an algebraic method to keep track of the transseries’ coefficients. We explore what conditions must be violated in order to stay clear of fixed point theorems to eschew a unique solution, if so desired, as we explain. An interesting finding is that the flow of data between the different sectors of the transseries shows a pattern typical of resurgence, i.e. the phenomenon that the perturbative sector of the transseries talks to the nonperturbative ones in a one-way fashion. However, our ansatz is not exotic enough as it leads to trivial solutions with vanishing nonperturbative sectors, even when logarithmic monomials are included. We see our result as a harbinger of what future work might reveal about the transseries representations of observables in fully renormalised four-dimensional quantum field theories and adduce a tentative yet to our mind weighty argument as to why onemore »

- Authors:

- Publication Date:

- OSTI Identifier:
- 22617383

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Annals of Physics; Journal Volume: 372; Other Information: Copyright (c) 2016 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; FOUR-DIMENSIONAL CALCULATIONS; GREEN FUNCTION; MATHEMATICAL SOLUTIONS; QUANTUM ELECTRODYNAMICS; RENORMALIZATION

### Citation Formats

```
Klaczynski, Lutz, E-mail: klacz@mathematik.hu-berlin.de.
```*Resurgent transseries & Dyson–Schwinger equations*. United States: N. p., 2016.
Web. doi:10.1016/J.AOP.2016.06.003.

```
Klaczynski, Lutz, E-mail: klacz@mathematik.hu-berlin.de.
```*Resurgent transseries & Dyson–Schwinger equations*. United States. doi:10.1016/J.AOP.2016.06.003.

```
Klaczynski, Lutz, E-mail: klacz@mathematik.hu-berlin.de. Thu .
"Resurgent transseries & Dyson–Schwinger equations". United States.
doi:10.1016/J.AOP.2016.06.003.
```

```
@article{osti_22617383,
```

title = {Resurgent transseries & Dyson–Schwinger equations},

author = {Klaczynski, Lutz, E-mail: klacz@mathematik.hu-berlin.de},

abstractNote = {We employ resurgent transseries as algebraic tools to investigate two self-consistent Dyson–Schwinger equations, one in Yukawa theory and one in quantum electrodynamics. After a brief but pedagogical review, we derive fixed point equations for the associated anomalous dimensions and insert a moderately generic log-free transseries ansatz to study the possible strictures imposed. While proceeding in various stages, we develop an algebraic method to keep track of the transseries’ coefficients. We explore what conditions must be violated in order to stay clear of fixed point theorems to eschew a unique solution, if so desired, as we explain. An interesting finding is that the flow of data between the different sectors of the transseries shows a pattern typical of resurgence, i.e. the phenomenon that the perturbative sector of the transseries talks to the nonperturbative ones in a one-way fashion. However, our ansatz is not exotic enough as it leads to trivial solutions with vanishing nonperturbative sectors, even when logarithmic monomials are included. We see our result as a harbinger of what future work might reveal about the transseries representations of observables in fully renormalised four-dimensional quantum field theories and adduce a tentative yet to our mind weighty argument as to why one should not expect otherwise. This paper is considerably self-contained. Readers with little prior knowledge are let in on the basic reasons why perturbative series in quantum field theory eventually require an upgrade to transseries. Furthermore, in order to acquaint the reader with the language utilised extensively in this work, we also provide a concise mathematical introduction to grid-based transseries.},

doi = {10.1016/J.AOP.2016.06.003},

journal = {Annals of Physics},

number = ,

volume = 372,

place = {United States},

year = {Thu Sep 15 00:00:00 EDT 2016},

month = {Thu Sep 15 00:00:00 EDT 2016}

}