# Analytic solution of Hedin's equations in zero dimensions

## Abstract

Feynman diagrams for the many-body perturbational theory are enumerated by solving the system of Hedin's equations in zero dimension. We extend the treatment of Molinari [Phys. Rev. B 71, 113102 (2005)] and give a complete solution of the enumeration problem in terms of Whittaker functions. An important relation between the generating function of the electron propagator and anomalous dimension in quantum field theory of massless fermions and mesons in four dimensions (Yukawa theory) is found. The Hopf algebra of undecorated rooted trees yields the anomalous field dimension in terms of the solution of the same differential equation. Its relation to the mathematical problem of combinatorics of chord diagrams is discussed; asymptotic expansions of the counting numbers are obtained.

- Authors:

- Department of Physics, Kaiserslautern University of Technology, Box 3049, 67653 Kaiserslautern (Germany)

- Publication Date:

- OSTI Identifier:
- 20929702

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Mathematical Physics; Journal Volume: 48; Journal Issue: 5; Other Information: DOI: 10.1063/1.2728512; (c) 2007 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; ALGEBRA; ANALYTICAL SOLUTION; ANOMALOUS DIMENSION; DIFFERENTIAL EQUATIONS; ELECTRONS; FEYNMAN DIAGRAM; FUNCTIONS; MANY-BODY PROBLEM; MESONS; PERTURBATION THEORY; PROPAGATOR; QUANTUM FIELD THEORY

### Citation Formats

```
Pavlyukh, Y., and Huebner, W..
```*Analytic solution of Hedin's equations in zero dimensions*. United States: N. p., 2007.
Web. doi:10.1063/1.2728512.

```
Pavlyukh, Y., & Huebner, W..
```*Analytic solution of Hedin's equations in zero dimensions*. United States. doi:10.1063/1.2728512.

```
Pavlyukh, Y., and Huebner, W.. Tue .
"Analytic solution of Hedin's equations in zero dimensions". United States.
doi:10.1063/1.2728512.
```

```
@article{osti_20929702,
```

title = {Analytic solution of Hedin's equations in zero dimensions},

author = {Pavlyukh, Y. and Huebner, W.},

abstractNote = {Feynman diagrams for the many-body perturbational theory are enumerated by solving the system of Hedin's equations in zero dimension. We extend the treatment of Molinari [Phys. Rev. B 71, 113102 (2005)] and give a complete solution of the enumeration problem in terms of Whittaker functions. An important relation between the generating function of the electron propagator and anomalous dimension in quantum field theory of massless fermions and mesons in four dimensions (Yukawa theory) is found. The Hopf algebra of undecorated rooted trees yields the anomalous field dimension in terms of the solution of the same differential equation. Its relation to the mathematical problem of combinatorics of chord diagrams is discussed; asymptotic expansions of the counting numbers are obtained.},

doi = {10.1063/1.2728512},

journal = {Journal of Mathematical Physics},

number = 5,

volume = 48,

place = {United States},

year = {Tue May 15 00:00:00 EDT 2007},

month = {Tue May 15 00:00:00 EDT 2007}

}