# Borel summation of the derivative expansion and effective actions

## Abstract

We argue that the derivative expansion of the QED effective action is a divergent but Borel summable asymptotic series, for a particular inhomogeneous background magnetic field. A duality transformation B[r arrow]iE gives a non-Borel-summable perturbative series for a time dependent background electric field, and Borel dispersion relations yield the non-perturbative imaginary part of the effective action, which determines the pair production probability. Resummations of leading Borel approximations exponentiate to give perturbative corrections to the exponents in the non-perturbative pair production rates. Comparison with a WKB analysis suggests that these divergence properties are general features of derivative expansions and effective actions. [copyright] [ital 1999] [ital The American Physical Society]

- Authors:

- (Department of Physics, University of Connecticut, Storrs, Connecticut 06269 (United States))

- Publication Date:

- OSTI Identifier:
- 6465577

- Alternate Identifier(s):
- OSTI ID: 6465577

- Resource Type:
- Journal Article

- Journal Name:
- Physical Review, D

- Additional Journal Information:
- Journal Volume: 60:6; Journal ID: ISSN 0556-2821

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; ACTION INTEGRAL; CONVERGENCE; DISPERSION RELATIONS; PERTURBATION THEORY; QUANTUM ELECTRODYNAMICS; SERIES EXPANSION; ELECTRODYNAMICS; FIELD THEORIES; INTEGRALS; QUANTUM FIELD THEORY 662110* -- General Theory of Particles & Fields-- Theory of Fields & Strings-- (1992-)

### Citation Formats

```
Dunne, G.V., and Hall, T.M.
```*Borel summation of the derivative expansion and effective actions*. United States: N. p., 1999.
Web. doi:10.1103/PhysRevD.60.065002.

```
Dunne, G.V., & Hall, T.M.
```*Borel summation of the derivative expansion and effective actions*. United States. doi:10.1103/PhysRevD.60.065002.

```
Dunne, G.V., and Hall, T.M. Wed .
"Borel summation of the derivative expansion and effective actions". United States. doi:10.1103/PhysRevD.60.065002.
```

```
@article{osti_6465577,
```

title = {Borel summation of the derivative expansion and effective actions},

author = {Dunne, G.V. and Hall, T.M.},

abstractNote = {We argue that the derivative expansion of the QED effective action is a divergent but Borel summable asymptotic series, for a particular inhomogeneous background magnetic field. A duality transformation B[r arrow]iE gives a non-Borel-summable perturbative series for a time dependent background electric field, and Borel dispersion relations yield the non-perturbative imaginary part of the effective action, which determines the pair production probability. Resummations of leading Borel approximations exponentiate to give perturbative corrections to the exponents in the non-perturbative pair production rates. Comparison with a WKB analysis suggests that these divergence properties are general features of derivative expansions and effective actions. [copyright] [ital 1999] [ital The American Physical Society]},

doi = {10.1103/PhysRevD.60.065002},

journal = {Physical Review, D},

issn = {0556-2821},

number = ,

volume = 60:6,

place = {United States},

year = {1999},

month = {9}

}