# Analytic coupling structure of large ${N}_{f}$ (super) QED and QCD

## Abstract

We study the analytic properties of the ’t Hooft coupling expansion of the beta function at the leading nontrivial large– ${N}_{f}$ order for QED, QCD, super QED, and super QCD. For each theory, the ’t Hooft coupling expansion is convergent. We discover that an analysis of the expansion coefficients to roughly 30 orders is required to establish the radius of convergence accurately, and to characterize the (logarithmic) nature of the first singularity. We study summations of the beta-function expansion at order 1/ ${N}_{f}$ and identify the physical origin of the singularities in terms of iterated bubble diagrams. We find a common analytic structure across these theories, with important technical differences between supersymmetric and nonsupersymmetric theories. We also discuss the expected structure at higher orders in the 1/ ${N}_{f}$ expansion, which will be in the future accessible with the methods presented in this work, meaning without the need for resumming the perturbative series. Understanding the structure of the large– ${N}_{f}$ expansion is an essential step towards determining the ultraviolet fate of asymptotically nonfree gauge theories.

- Authors:

- Univ. of Southern Denmark, Odense M (Denmark)
- Univ. of Connecticut, Storrs, CT (United States)
- Univ. of Southern Denmark, Odense M (Denmark); Stanford Univ., Stanford, CA (United States)

- Publication Date:

- Research Org.:
- SLAC National Accelerator Lab., Menlo Park, CA (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), High Energy Physics (HEP) (SC-25)

- OSTI Identifier:
- 1563040

- Alternate Identifier(s):
- OSTI ID: 1546712

- Grant/Contract Number:
- AC02-76SF00515; DNRF:90; SC0010339

- Resource Type:
- Published Article

- Journal Name:
- Physical Review D

- Additional Journal Information:
- Journal Volume: 100; Journal Issue: 1; Journal ID: ISSN 2470-0010

- Publisher:
- American Physical Society (APS)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS

### Citation Formats

```
Dondi, Nicola Andrea, Dunne, Gerald V., Reichert, Manuel, and Sannino, Francesco. Analytic coupling structure of large Nf (super) QED and QCD. United States: N. p., 2019.
Web. doi:10.1103/physrevd.100.015013.
```

```
Dondi, Nicola Andrea, Dunne, Gerald V., Reichert, Manuel, & Sannino, Francesco. Analytic coupling structure of large Nf (super) QED and QCD. United States. doi:10.1103/physrevd.100.015013.
```

```
Dondi, Nicola Andrea, Dunne, Gerald V., Reichert, Manuel, and Sannino, Francesco. Thu .
"Analytic coupling structure of large Nf (super) QED and QCD". United States. doi:10.1103/physrevd.100.015013.
```

```
@article{osti_1563040,
```

title = {Analytic coupling structure of large Nf (super) QED and QCD},

author = {Dondi, Nicola Andrea and Dunne, Gerald V. and Reichert, Manuel and Sannino, Francesco},

abstractNote = {We study the analytic properties of the ’t Hooft coupling expansion of the beta function at the leading nontrivial large–Nf order for QED, QCD, super QED, and super QCD. For each theory, the ’t Hooft coupling expansion is convergent. We discover that an analysis of the expansion coefficients to roughly 30 orders is required to establish the radius of convergence accurately, and to characterize the (logarithmic) nature of the first singularity. We study summations of the beta-function expansion at order 1/Nf and identify the physical origin of the singularities in terms of iterated bubble diagrams. We find a common analytic structure across these theories, with important technical differences between supersymmetric and nonsupersymmetric theories. We also discuss the expected structure at higher orders in the 1/Nf expansion, which will be in the future accessible with the methods presented in this work, meaning without the need for resumming the perturbative series. Understanding the structure of the large–Nf expansion is an essential step towards determining the ultraviolet fate of asymptotically nonfree gauge theories.},

doi = {10.1103/physrevd.100.015013},

journal = {Physical Review D},

number = 1,

volume = 100,

place = {United States},

year = {2019},

month = {7}

}

DOI: 10.1103/physrevd.100.015013

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