# Soft factorization in QED from 2D Kac-Moody symmetry

## Abstract

The soft factorization theorem for 4D abelian gauge theory states that the S -matrix factorizes into soft and hard parts, with the universal soft part containing all soft and collinear poles. Similarly, correlation functions on the sphere in a 2D CFT with a U(1) Kac-Moody current algebra factorize into current algebra and non-current algebra factors, with the current algebra factor fully determined by its pole structure. In this paper, we show that these 4D and 2D factorizations are mathematically the same phenomena. The soft ‘t Hooft-Wilson lines and soft photons are realized as a complexified 2D current algebra on the celestial sphere at null infinity. The current algebra level is determined by the cusp anomalous dimension. The associated complex U(1) boson lives on a torus whose modular parameter is τ=$$\frac{2πi}{e^2}$$ + $$\frac{θ}{2π}$$. The correlators of this 2D current algebra fully reproduce the known soft part of the 4D S -matrix, as well as a conjectured generalization involving magnetic charges.

- Authors:

- Harvard Univ., Cambridge, MA (United States). Dept. of Physics

- Publication Date:

- Research Org.:
- Harvard Univ., Cambridge, MA (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC)

- OSTI Identifier:
- 1505226

- Grant/Contract Number:
- SC0007870

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Journal of High Energy Physics (Online)

- Additional Journal Information:
- Journal Volume: 2018; Journal Issue: 2; Journal ID: ISSN 1029-8479

- Publisher:
- Springer Berlin

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; Conformal Field Theory; Gauge Symmetry; Scattering Amplitudes

### Citation Formats

```
Nande, Anjalika, Pate, Monica, and Strominger, Andrew.
```*Soft factorization in QED from 2D Kac-Moody symmetry*. United States: N. p., 2018.
Web. doi:10.1007/jhep02(2018)079.

```
Nande, Anjalika, Pate, Monica, & Strominger, Andrew.
```*Soft factorization in QED from 2D Kac-Moody symmetry*. United States. doi:10.1007/jhep02(2018)079.

```
Nande, Anjalika, Pate, Monica, and Strominger, Andrew. Wed .
"Soft factorization in QED from 2D Kac-Moody symmetry". United States. doi:10.1007/jhep02(2018)079. https://www.osti.gov/servlets/purl/1505226.
```

```
@article{osti_1505226,
```

title = {Soft factorization in QED from 2D Kac-Moody symmetry},

author = {Nande, Anjalika and Pate, Monica and Strominger, Andrew},

abstractNote = {The soft factorization theorem for 4D abelian gauge theory states that the S -matrix factorizes into soft and hard parts, with the universal soft part containing all soft and collinear poles. Similarly, correlation functions on the sphere in a 2D CFT with a U(1) Kac-Moody current algebra factorize into current algebra and non-current algebra factors, with the current algebra factor fully determined by its pole structure. In this paper, we show that these 4D and 2D factorizations are mathematically the same phenomena. The soft ‘t Hooft-Wilson lines and soft photons are realized as a complexified 2D current algebra on the celestial sphere at null infinity. The current algebra level is determined by the cusp anomalous dimension. The associated complex U(1) boson lives on a torus whose modular parameter is τ=$\frac{2πi}{e^2}$ + $\frac{θ}{2π}$. The correlators of this 2D current algebra fully reproduce the known soft part of the 4D S -matrix, as well as a conjectured generalization involving magnetic charges.},

doi = {10.1007/jhep02(2018)079},

journal = {Journal of High Energy Physics (Online)},

issn = {1029-8479},

number = 2,

volume = 2018,

place = {United States},

year = {2018},

month = {2}

}